url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
|---|---|---|---|
https://indico.math.cnrs.fr/event/910/?view=lecture | Séminaire de Physique Théorique
Convergent series : from lattice models to QCD
by Prof. Vasily SAZONOV (University of Graz)
Europe/Paris
Amphithéâtre Léon Motchane (IHES)
Amphithéâtre Léon Motchane
IHES
35, route de Chartres, F-91440 Bures-sur-Yvette (France)
Description
The standard perturbation theory leads to the asymptotic series because of the illegal interchange of the summation and integration. However, changing the initial approximation of the perturbation theory, one can generate the convergent series. We study the lattice \phi4-model and compare observables calculated using the convergent series and Monte Carlo simulations. Then, we discuss the generalization of the same ideas for the continuum \phi4-model and QCD.
Contact | 2019-10-21 21:14:11 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8613669872283936, "perplexity": 1854.232520787695}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987787444.85/warc/CC-MAIN-20191021194506-20191021222006-00410.warc.gz"} |
https://stats.stackexchange.com/questions/378274/how-to-construct-a-cross-entropy-loss-for-general-regression-targets | # How to construct a cross-entropy loss for general regression targets?
It's common short-hand in neural networks literature to refer to categorical cross-entropy loss as "cross-entropy," even though there are a number of loss functions which could properly be described that way.
So, in general, how does one move from an assumed probability distribution for the target variable to defining a cross-entropy loss for your network? What does the function require as inputs? (For example, the categorical cross-entropy function for one-hot targets requires a one-hot binary vector and a probability vector as inputs.)
A good answer will discuss the general principles involved, as well as worked examples for
• categorical cross-entropy loss for one-hot targets
• Gaussian-distributed target distribution and how how this reduces to usual MSE loss
• A less common example such as a gamma distributed target, or a heavy-tailed target
• Explain the relationship between minimizing cross entropy and maximizing log-likelihood.
Inspired by Tensorflow Cross Entropy for Regression? with thanks to CowboyTrader. | 2019-08-26 10:05:40 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8390630483627319, "perplexity": 1471.7677873391106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027331485.43/warc/CC-MAIN-20190826085356-20190826111356-00299.warc.gz"} |
https://en.wikipedia.org/wiki/Trip_generation | # Trip generation
Trip generation is the first step in the conventional four-step transportation forecasting process used for forecasting travel demands. It predicts the number of trips originating in or destined for a particular traffic analysis zone (TAZ).[1] Trip generation analysis focuses on residences and residential trip generation is thought of as a function of the social and economic attributes of households. At the level of the traffic analysis zone, residential land uses "produce" or generate trips. Traffic analysis zones are also destinations of trips, trip attractors. The analysis of attractors focuses on non-residential land uses.
This process is followed by trip distribution, mode choice, and route assignment.
## Input data
A forecasting activity, such as one based on the concept of economic base analysis, provides aggregate measures of population and activity growth. Land use forecasting distributes forecast changes in activities in a disaggregate-spatial manner among zones. The next step in the transportation planning process addresses the question of the frequency of origins and destinations of trips in each zone: for short, trip generation.
## Analysis
### Initial analysis
The first zonal trip generation (and its inverse, attraction) analysis in the Chicago Area Transportation Study (CATS)[2] followed the “decay of activity intensity with distance from the central business district (CBD)” thinking current at the time. Data from extensive surveys were arrayed and interpreted on a distance-from-CBD scale. For example, commercial land use in ring 0 (the CBD and vicinity) was found to generate 728 vehicle trips per day in 1956. That same land use in ring 5 (about 17 km (11 mi) from the CBD) generated about 150 trips per day.
The case of trip destinations will illustrate use of the concept of activity decline with intensity (as measured by distance from CBD) worked. Destination data are arrayed:
Table: Trip Destinations per unit (Acre) of Land
Ring Manufacturing Commercial Open Space etc.
0 x1m x1c x1os x1n
${\displaystyle \vdots }$ ${\displaystyle \vdots }$ ${\displaystyle \vdots }$ ${\displaystyle \vdots }$ ${\displaystyle \vdots }$
${\displaystyle \vdots }$ ${\displaystyle \vdots }$ ${\displaystyle \vdots }$ ${\displaystyle \vdots }$ ${\displaystyle \vdots }$
7 x7m x7c x7os x7n
The land use analysis provides information on how land uses will change from an initial year (say t = 0) to some forecast year (say t = 20). Suppose we are examining a zone. We take the mix of land uses projected, say, for year t = 20 and apply the trip destination rates for the ring in which the zone is located. That is, there will this many acres of commercial land use, that many acres of public open space, etc., in the zone. The acres of each use type are multiplied by the ring specific destination rates. The result is summed to yield the zone’s trip destinations. The CATS assumed that trip destination rates would not change over time.
### Revisions to the analysis
As was true for land use analysis, the approach developed at CATS was considerably modified in later studies. The conventional four-step paradigm evolved as follows: Types of trips are considered. Home-based (residential) trips are divided into work and other, with major attention given to work trips. Movement associated with the home end of a trip is called trip production, whether the trip is leaving or coming to the home. Non-home-based or non-residential trips are those a home base is not involved. In this case, the term production is given to the origin of a trip and the term attraction refers to the destination of the trip.
Residential trip generation analysis is often undertaken using statistical regression. Person, transit, walking, and auto trips per unit of time are regressed on variables thought to be explanatory, such as: household size, number of workers in the household, persons in an age group, type of residence (single family, apartment, etc.), and so on. Usually, measures on five to seven independent variables are available; additive causality is assumed.
Regressions are also made at the aggregate/zone level. Variability among households within a zone isn’t measured when data are aggregated. High correlation coefficients are found when regressions are run on aggregate data, about 0.90, but lower coefficients, about 0.25, are found when regressions are made on observation units such as households. In short, there is much variability that is hidden by aggregation.
Sometimes cross-classification techniques are applied to residential trip generation problems. The CATS procedure described above is a cross-classification procedure.
Classification techniques are often used for non-residential trip generation. First, the type of land use is a factor influencing travel, it is regarded as a causal factor. A list of land uses and associated trip rates illustrated a simple version of the use of this technique:
Table: Trips per day
Land Use Type Trips
Department Store X
Grocery Store Y
${\displaystyle \vdots }$ ${\displaystyle \vdots }$
Such a list can be improved by adding information. Large, medium, and small might be defined for each activity and rates given by size. Number of employees might be used: for example, <10, 10-20, etc. Also, floor space is used to refine estimates.
In other cases, regressions, usually of the form trip rate = f(number of employees, floor area of establishment), are made for land use types.
Special treatment is often given major trip generators: large shopping centers, airports, large manufacturing plants, and recreation facilities.
The theoretical work related to trip generation analysis is grouped under the rubric travel demand theory, which treats trip generation-attraction, as well as mode choice, route selection, and other topics.
## Databases
The Institute of Transportation Engineers's Trip Generation Manual provides trip generation rates for various land use and building types. The planner can add local adjustment factors and treat mixes of uses with ease. Ongoing work is adding to the stockpile of numbers; over 4000 studies were aggregated for the latest edition.
ITE Procedures estimate the number of trips entering and exiting a site at a given time. ITE Rates are functions of type of development based on independent variables such as square footage of the gross leasable area, number of gas pumps, number of dwelling units, or other standard measurable things, usually produced in site plans.[3] They are typically of the form ${\displaystyle Trips=a+b*Area}$ or ${\displaystyle Trips=a+bln(Area)}$. They do not consider location, competitors, complements, the cost of transportation, or other factors. The trip generation estimates are provided through data analysis. Many localities require their use to ensure adequate public facilities for growth management and subdivision approval.
In the United Kingdom and Ireland, the TRICS database is commonly used to calculate trip generation.
## References
1. ^ "Trip generation". Transport for London. Retrieved December 18, 2021.
2. ^ Black, Alan (1990). "The Chicago Area Transportation Study: A Case Study of Rational Planning". Journal of Planning Education and Research. 10 (1). doi:10.1177/0739456X9001000105. Retrieved December 18, 2021.
3. ^ "Independent Variables". Institute of Transportation Engineers. Retrieved December 17, 2021. | 2022-07-01 01:51:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 14, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38656991720199585, "perplexity": 3916.636822546944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103917192.48/warc/CC-MAIN-20220701004112-20220701034112-00042.warc.gz"} |
https://math.stackexchange.com/questions/3490305/maximize-int-01-mathbbe-x-mathbbpx-ge-u3-mathrm-du-over-0 | # Maximize $\int_{0}^{1} \mathbb{E} |X-\mathbb{P}(X\ge u)|^3\,\mathrm du$ over $[0,1]$ valued $X$.
Find the maximal value of
$$\int_{0}^{1} \mathbb{E} |X-\mathbb{P}(X\ge u)|^3\,\mathrm du$$
over $$[0,1]$$ valued random variables $$X$$.
I believe that optimal $$X$$ takes exactly two different values, but I have a hard time trying to actually prove it. I will be glad for any help or insight.
I also wonder if a pair $$(X,\mathbb{P}[X\ge U])$$, where $$U\sim\mathcal{U}[0,1]$$, is a more widely known or studied.
• why the power 3? what is the motivation behind ? Dec 28 '19 at 20:16
• in this case, you could perhaps mention the solution for k=2 in your question; this could be of interest to solve the other cases. Dec 28 '19 at 20:42
• I understand you're asking for general $X$, but if we restrict to binary-valued (indicator) $X$, the max still happens at $P(X=1)=\frac12$ for $k=3$, but more interestingly for $k=4$ the max(es) now happen at $P(X=1)=\frac12 \pm {1 \over 2\sqrt{3}}$ (assuming I didn't make any mistake with math and typing things into wolfram alpha.) Dec 28 '19 at 23:29
• One can easily show that the maximizer is always a discrete random variable with at most $C(k)$ values with some explicit $C(k)$. Unfortunately, as of now, it is of little practical value since I can only show that $C(3)\le 66$, say, and I see no reason why the number of values should not grow to infinity as $k\to\infty$. Would you be interested in some upper bounds that are only not too far from the true maxima for large $k$ as well? Jan 5 '20 at 3:45
• @fedja - I would be highly interested in $C(3)\le 66$ bound. I have been trying to find any such bound by myself and did not succeed. I think it would be quite eyeopening to see it. Jan 5 '20 at 11:06
The equivalent reformulation of the problem is to maximize $$\mathbb E|X-Y|^k$$ under the condition that $$X,Y$$ are independent and the functions $$F_X(u)=\mathbb P\{X\ge u\}$$ and $$F_Y(u)=\mathbb P\{Y\ge u\}$$ are inverse to each other in the sense that their graphs are symmetric (as usual, we connect the jumps in the graph by vertical intervals and consider vertical intervals as jumps).
Now I want to show two things: the upper bound $$\frac 2{k+1}$$ (which should be compared with the example $$X=\frac{k+1}{k+2}\chi_ A, \mathbb P(A)=\frac{k+1}{k+2}$$ giving the value $$\frac 2{k+2}(\frac{k+1}{k+2})^{k+1}\ge \frac 2{e(k+2)}$$) and the claim that the maximizer exists and is a discrete random variable with at most $$C(k)$$ values (though I'll not try to squeeze the best $$C(k)$$ the method can give: it is still too large for practical purposes even when $$k=3$$).
The first claim is rather straightforward: $$\mathbb E|X-Y|^k=\int_0^1 kt^{k-1}(\mathbb P\{X>Y+t\}+\mathbb P\{Y>X+t\})dt.$$ Now define $$s\in[0,1-t]$$ by $$F_X(s+t)=s$$. Notice that if $$X>Y+t$$, then either $$X>s+t$$, the probability of which is $$F_X(s+t)=s$$ or $$Y, the probability of which is $$1-F_Y(s)=1-F_X^{-1}(s)=1-s-t$$, so we always have $$\mathbb P\{X>Y+t\}\le \mathbb P\{X>s+t\}+\mathbb P\{Y and the same inequality holds for the other probability whence the integral in question is at most $$2\int_0^1 kt^{k-1}(1-t)dt=\frac 2{k+1}.$$
The second claim is a bit longer to prove. We fix $$m$$ and consider all discrete random variables with at most $$m$$ values. This is a compact set and we have a continuous functional, so the maximum over that family is attained for some $$X$$ with, say, $$M$$ values. If we bound $$M$$ independently of $$m$$, we will be done. Suppose that $$M$$ is large. Then the graph of $$F_X$$ is a descending stairway with many steps. In particular, there is a value $$x$$ of $$X$$ that is neither $$0$$, nor $$1$$, and the associated probability $$p_{x}=\mathbb P\{X=x\}\le\frac 1{M-2}$$ is very small. Let's try to move $$x$$ to a close value $$x'$$ without changing the probabilities. Our functional then can be written as $$L(x')+\sum_y p_x q_y|x'-y|^k+p_x[|x'-y_+|^k-|x'-y_-|^k](x'-x)$$ where $$L(x')$$ is linear in $$x'$$ and $$y_+-y_-=p_x$$ (i.e., $$y_{\pm}$$ are the levels of the top and the bottom of the step on the stairway graph of $$F_X$$ whose vertical side is $$p_x$$). Taking the second derivative in $$x'$$ at $$x'=x$$ and dividing by $$p_x$$, we get $$k(k-1)\sum_y q_y|x-y|^{k-2}\\+2k[|x-y_+|^{k-1}{\rm sgn}(x-y_+)- |x-y_-|^{k-1}{\rm sgn}(x-y_-)]\le 0$$ (otherwise our configuration cannot be a maximizer). Since the second derivative $$k(k-1)|x-y|^{k-2}$$ is a convex function of $$y$$ for integer $$k\ge 2$$, we can replace the increment of the first derivative by the half-sum of the second derivatives at the endpoints times $$y_+-y_-=p_x$$ and derive the inequality $$\sum_y q_y|x-y|^{k-2}\le p_x[|x-y_+|^{k-2}+|x-y_-|^{k-2}]\tag{*}$$ This should actually hold for every $$x\ne 0,1$$ and I feel that there must be a more efficient way to use $$(*)$$ than the one I'm following below but, at the very least, the RHS is $$\le 2p_x$$, so if $$p_x$$ is very small, then $$Y$$ is concentrated near $$x$$ and, thereby, near its mean $$EY$$. Similarly, choosing small $$q_y$$ with $$y\ne 0,1$$, we see that $$X$$ is concentrated near $$EX$$. But $$EX=EY$$ (both represent the area under the stairway), so in this case $$X-Y$$ is concentrated near $$0$$ and $$E|X-Y|^k$$ is very small. However, we have a maximizer, so we have an a priori lower bound on that expectation for every fixed $$k$$, which tells us that $$M$$ cannot be too large, finishing the argument.
• @StanTuwim Yes, but the problem is that the minimum may occur at an end and the sequence of $p$'s and $q$'s may be unimodal with maximum near the middle or increasing all the way to the other end, so the desired inequality never occurs for $p_x$ or $q_y$. Jan 6 '20 at 12:31 | 2021-09-20 00:31:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 74, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9374679327011108, "perplexity": 96.3396247904968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056902.22/warc/CC-MAIN-20210919220343-20210920010343-00524.warc.gz"} |
http://retrievo.pt/advanced_search?link=1&field1=creators&searchTerms1='Aamodt%2C+K.'&constraint1=MATCH_EXACT_PHRASE | Type
Database
Creator
Date
Thumbnail
# Search results
56 records were found.
## Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb--Pb Collisions at $\sqrt{s_{_{NN}}} = 2.76$ TeV
Inclusive transverse momentum spectra of primary charged particles in Pb-Pb collisions at $\sqrt{s_{_{NN}}}$ = 2.76 TeV have been measured by the ALICE Collaboration at the LHC. The data are presented for central and peripheral collisions, corresponding to 0-5% and 70-80% of the hadronic Pb-Pb cross section. The measured charged particle spectra in $|\eta|<0.8$ and $0.3 < p_T < 20$ GeV/$c$ are compared to the expectation in pp collisions at the same $\sqrt{s_{_{NN}}}$, scaled by the number of underlying nucleon-nucleon collisions. The comparison is expressed in terms of the nuclear modification factor $R_{AA}$. The result indicates only weak medium effects ($R_{AA} \approx$ 0.7) in peripheral collisions. In central collisions, $R_{AA}$ reaches a minimum of about 0.14 at $p_T=6$-7GeV/$c$ and increases significantly at larger $p_T$. T...
## Production of pions, kaons and protons in pp collisions at s=900 GeV with ALICE at the LHC
The production of π +, π −, K+, K−, p, and p at mid-rapidity has been measured in proton-proton collisions at s=900 GeV with the ALICE detector. Particle identification is performed using the specific energy loss in the inner tracking silicon detector and the time projection chamber. In addition, time-of-flight information is used to identify hadrons at higher momenta. Finally, the distinctive kink topology of the weak decay of charged kaons is used for an alternative measurement of the kaon transverse momentum (p t) spectra. Since these various particle identification tools give the best separation capabilities over different momentum ranges, the results are combined to extract spectra from p t=100 MeV/c to 2.5 GeV/c. The measured spectra are further compared with QCD-inspired models which yield a poor description. The total yields a...
## Strange particle production in proton–proton collisions at s=09 with ALICE at the LHC The ALICE Collaboration
The production of mesons containing strange quarks (, φ) and both singly and doubly strange baryons (, , and ) are measured at mid-rapidity in pp collisions at s = 0.9 TeV with the ALICE experiment at the LHC. The results are obtained from the analysis of about 250 k minimum bias events recorded in 2009. Measurements of yields (dN/dy) and transverse momentum spectra at mid-rapidity for inelastic pp collisions are presented. For mesons, we report yields (〈dN/dy〉) of 0.184±0.002(stat.)±0.006(syst.) for and 0.021±0.004(stat.)±0.003(syst.) for φ. For baryons, we find 〈dN/dy〉=0.048±0.001(stat.)±0.004(syst.) for , 0.047±0.002(stat.)±0.005(syst.) for and 0.0101±0.0020(stat.)±0.0009(syst.) for . The results are also compared with predictions for identified particle spectra from QCD-inspired models and provide a baseline for comparisons with b...
## Elliptic flow of charged particles in Pb-Pb collisions at 2.76 TeV
We report the first measurement of charged particle elliptic flow in Pb-Pb collisions at 2.76 TeV with the ALICE detector at the CERN Large Hadron Collider. The measurement is performed in the central pseudorapidity region (|eta|<0.8) and transverse momentum range 0.2< p_t< 5.0 GeV/c. The elliptic flow signal v_2, measured using the 4-particle correlation method, averaged over transverse momentum and pseudorapidity is 0.087 +/- 0.002 (stat) +/- 0.004 (syst) in the 40-50% centrality class. The differential elliptic flow v_2(p_t) reaches a maximum of 0.2 near p_t = 3 GeV/c. Compared to RHIC Au-Au collisions at 200 GeV, the elliptic flow increases by about 30%. Some hydrodynamic model predictions which include viscous corrections are in agreement with the observed inc
## Charged-particle multiplicity density at mid-rapidity in central Pb-Pb collisions at sqrt(sNN) = 2.76 TeV
The first measurement of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at a centre-of-mass energy per nucleon pair sqrt(sNN) = 2.76 TeV is presented. For an event sample corresponding to the most central 5% of the hadronic cross section the pseudo-rapidity density of primary charged particles at mid-rapidity is 1584 +- 4 (stat) +- 76 (sys.), which corresponds to 8.3 +- 0.4 (sys.) per participating nucleon pair. This represents an increase of about a factor 1.9 relative to pp collisions at similar collision energies, and about a factor 2.2 to central Au-Au collisions at sqrt(sNN) = 0.2 TeV. This measurement provides the first experimental constraint for models of nucleus-nucleus collisions at LHC energies
## Higher Harmonic Anisotropic Flow Measurements of Charged Particles in Pb-Pb Collisions at √sNN=2.76 TeV
We report on the first measurement of the triangular v3, quadrangular v4, and pentagonal v5 charged particle flow in Pb-Pb collisions at √sNN=2.76 TeV measured with the ALICE detector at the CERN Large Hadron Collider. We show that the triangular flow can be described in terms of the initial spatial anisotropy and its fluctuations, which provides strong constraints on its origin. In the most central events, where the elliptic flow v2 and v3 have similar magnitude, a double peaked structure in the two-particle azimuthal correlations is observed, which is often interpreted as a Mach cone response to fast partons. We show that this structure can be naturally explained from the measured anisotropic flow Fourier coefficients.
## First proton–proton collisions at the LHC as observed with the ALICE detector: measurement of the charged-particle pseudorapidity density at [v]s=900 GeV
On 23rd November 2009, during the early commissioning of the CERN Large Hadron Collider (LHC), two counter-rotating proton bunches were circulated for the first time concurrently in the machine, at the LHC injection energy of 450 GeV per beam. Although the proton intensity was very low, with only one pilot bunch per beam, and no systematic attempt was made to optimize the collision optics, all LHC experiments reported a number of collision candidates. In the ALICE experiment, the collision region was centred very well in both the longitudinal and transverse directions and 284 events were recorded in coincidence with the two passing proton bunches. The events were immediately reconstructed and analyzed both online and offline. We have used these events to measure the pseudorapidity density of charged primary particles in the central reg...
## The ALICE experiment at the CERN LHC
ALICE (A Large Ion Collider Experiment) is a general-purpose, heavy-ion detector at the CERN LHC which focuses on QCD, the strong-interaction sector of the Standard Model. It is designed to address the physics of strongly interacting matter and the quark-gluon plasma at extreme values of energy density and temperature in nucleus-nucleus collisions. Besides running with Pb ions, the physics programme includes collisions with lighter ions, lower energy running and dedicated proton-nucleus runs. ALICE will also take data with proton beams at the top LHC energy to collect reference data for the heavy-ion programme and to address several QCD topics for which ALICE is complementary to the other LHC detectors. The ALICE detector has been built by a collaboration including currently over 1000 physicists and engineers from 105 Institutes in 30 ...
## Particle-Yield Modification in Jetlike Azimuthal Dihadron Correlations in Pb-Pb Collisions at √sNN=2.76 TeV
The yield of charged particles associated with high-pt trigger particles (83 GeV/c on the away side drops to about 60% of that observed in pp collisions, while on the near side a moderate enhancement of 20%–30% is found. | 2019-08-24 21:57:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8168665170669556, "perplexity": 2320.600456129273}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027321786.95/warc/CC-MAIN-20190824214845-20190825000845-00132.warc.gz"} |
http://chemistry.stackexchange.com/questions/15548/why-are-methanol-flames-less-visible-than-other-flames | # Why are methanol flames less visible than other flames?
The combustion of methanol tends to generate flames that are less visible than the flames generated by the combustion of other substances?
My guess is that methanol burns quickly as it has a relatively simple structure that provides some of the oxygen needed for combustion. I am assuming that the flame comes from a mixture of fuel and air that is undergoing reaction. If the reaction is fast enough, then there will be less flame. However, methane and hydrogen both have visible flames, much more visible than methanol, leading me to wonder how the one oxygen in methanol makes so much of a difference.
-
Here's a link to a YouTube video showing a dish of methanol burning next to a dish of ethanol. The methanol flame is more difficult to see. Methanol was also widely used as a fuel in the auto racing circuit, until fires that resulted from crashes or fuel filling accidents resulted in "invisible fires" that were difficult to see during daylight, resulting in severe burns to crews and drivers ("you can't fight the fire, if you can't see it", this is one reason why it was replaced by ethanol as a fuel for Indy racing).
Flame color (visibility) is dependent upon the elemental composition of the material being heated and the temperature to which it is heated. This is because heat will excite electrons in the element(s) to an excited state; when these electrons return to the ground state visible light (flame test) will often be emitted
If we look at the carbon content of methanol and ethanol we find the following weight percents:
methanol - 38% carbon by weight
ethanol - 52% carbon by weight
If we compare the heats of combustion for methanol and ethanol we find:
methanol $\ce{~~}$ -726 kJ/mol (exothermic)
ethanol $\ce{~~}$ -1368 kJ/mol (much more exothermic)
There's just not much carbon there to burn in methanol! Consequently burning methanol doesn't get that hot and its flame will not be as colored as materials that burn more exothermically and result in higher temperatures.
-
Am I correct in thinking that carbon burns hotter because of the higher heat of formation of CO2 vs H2O and the higher heat capacity of H2O vs CO2? – Brinn Belyea Aug 23 '14 at 18:49
"Am I correct in thinking that carbon burns hotter because of the higher heat of formation of CO2 vs. H2O", Yes -394 vs. -242 kJ/mol for CO2 vs. H2O. I hadn't though about the heat capacities (0.844 vs. 1.93 for CO2 vs. H2O), but that makes sense too. – ron Aug 23 '14 at 19:04
>There's just not much carbon there to burn in methanol ! | while it is true that it is the reason in this case, the way it happens is completely different from what you described. – permeakra Aug 23 '14 at 19:11
Perhaps I should elaborate. It is well known and easily provable at home that pale blue flame of propane hand torch is much hotter, than bright yellow flame of same torch. – permeakra Aug 23 '14 at 19:13
@permeakra My explanation was based on 1) lower carbon content leading to 2) lower heat output leading to 3) fewer excited states being produced leading to 4) less coloration of the flame. I don't understand your comment, specifically which step in my explanation do you disagree with? Soot is not an issue when adequate oxygen is present. – ron Aug 23 '14 at 19:21
The yellow 'flame' is actually an aerosol of black particles heated to temperatures around 1000-1500 Celsium. Depending on temperature, they can emit deep red, orange, yellow and almost white light. So, for yellow flame to occur, black particles must be present. In case of carbon containing fuels the particles are usually carbon particles (soot). Soot is a product of incomplete combustion of carbon containing molecules. Now, let's see the difference between ethanol and methanol, specifically, at amount of oxygen required for combustion of same volume of vapors. Equations of burning:
$\ce{2CH3OH + 3O2 = 2CO2 + 4H2O}$ $\ce{C2H5OH + 3O2 = 2CO2 + 3H2O}$
As it can be seen, same volume of vapors in case of ethanol requires twice as much of air to fully burn, so in case of ethanol incomplete combustion is more common. It is even more common in case of larger molecules, say, higher hydrocarbons, like solid paraffines used in some candles. In case the flame of organic substance has right amount of oxygen (incoming fuel-air mixture contains proper amount of air), it burns with pale flame. It is easily observable in case of hand propane torches, where user regulates amount of air in incoming mixture, depending on amount of air allowed, it is possible to have bright yellow or pale blue flame and everything in between.
The pale blue emission has completely different nature, it is atomic emission of excited radicals and molecules. For example, it is known that $CH$ particle emits a blue light.
Yellow flame may also happen if another source of particles is available and the temperature is right. For example, $\ce{CuO + Al}$ mixture burns explosively, producing bright yellow flash. However, if temperature is too high, bright white flame should be observed. Indeed, magnesium burns with blinding white flame, especially if it is premixed with solid oxidizer (potassium nitrate).
Now, atomic/molecular emission is usually low in common flames. However, even small amounts of sodium in flame makes it bright yellow even if it normally almost colorless. Some other elements may color flame into different colors, ranging from pale violet to deep red. This, however, is a different situation from what you considered.
-
Under what cases would incomplete combustion of methanol yield soot? I would expect the combustion to comprise three reactions. First H3COH->CO+2H2, and then 2CO+O2->2CO2 and 2H2+O2->2H2O. Given that coal gas is formed by dissociating water to produce hydrogen and carbon monoxide, I would see no reason to expect any elemental carbon to be produced as a temporary decomposition byproduct. – supercat Aug 24 '14 at 0:05
@supercat I never cared to experiment, but guess if a drastically low amount of oxygen is present, it is possible. It is usually easier to achieve with larger flames. The reaction of combustion of ethanol is definitely radical in nature, with many branches. It is likely include such particles as $\ce{.OH , CH2O, .CH2OH, .HCO, CO, HO2, .O.}$ etc., and I'm not ready to discuss it. However, I must note that at low-medium temperatures $\ce{CO}$ is prone to disproportionation into carbon dioxide and elementary carbon. – permeakra Aug 24 '14 at 0:47 | 2016-06-28 20:26:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.581838071346283, "perplexity": 2031.9665443093163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397111.67/warc/CC-MAIN-20160624154957-00128-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://asmedigitalcollection.asme.org/materialstechnology/article/143/2/021003/1086725/A-Comparison-of-the-Thermo-Fluid-Properties-of-Ti | ## Abstract
Powder-bed fusion (PBF) process is a subdivision of additive manufacturing (AM) technology where a heat source at a controlled speed selectively fuses regions of a powder-bed material to form three-dimensional (3D) parts in a layer-by-layer fashion. Two of the most commercialized and powerful PBF methods for fabricating full-density metallic parts are the laser PBF (L-PBF) and electron beam PBF (E-PBF) processes. In this study, a multiphysics-based 3D numerical model is developed to compare the thermo-fluid properties of Ti-6Al-4V melt pools formed by the L-PBF and E-PBF processes. The temperature-dependent properties of Ti-6Al-4V alloy and the parameters for the laser and electron beams are incorporated in the model as the user-defined functions (UDFs). The melt-pool geometry and its thermo-fluid behavior are investigated using the finite volume (FV) method, and results for the variations of temperature, thermo-physical properties, velocity, geometry of the melt pool, and cooling rate in the two processes are compared under similar irradiation conditions. For an irradiance level of 26 J/mm3 and a beam interaction time of 1.212 ms, simulation results show that the L-PBF process gives a faster cooling rate (1. 5 K/μs) than that in the E-PBF process (0.74 K/μs). The magnitude of liquid velocity in the melt pool is also higher in L-PBF than that in E-PBF. The numerical model is validated by comparing the simulation results for the melt-pool geometry with the PBF experimental results and comparing the numerical melt-front position with the analytical solution for the classical Stephan problem of melting of a phase-change material (PCM).
## 1 Introduction
Powder-bed fusion (PBF) process is a relatively new additive manufacturing (AM) technology where the thermal energy of a computer-controlled heat source is used for selective melting and sintering of regions of a powder-bed [1]. Once a layer of an object is completed, the building platform is lowered, and more powder is spread over (usually, rolled on) the build area for a new scan. The process ends with a postprocessing step of removing all the unbound powder [2] from the fabricated object. The conventional manufacturing technologies (e.g., casting and forging) used for fabricating medical implants and components for automotive, aerospace, and space applications constrain the customization of complex geometries and consume a significant amount of material and time. The PBF process overcomes these limitations by providing the advantage of cost-effective customization with reduced assembly [2] and thus, becomes a superior AM technology in the present era. Two of the most common types of PBF processes are the laser PBF (L-PBF) and electron beam PBF (E-PBF) processes, which have brought about a revolution in the field of metal AM technology. The L-PBF process uses finely focused monochromatic coherent photons, i.e., laser, while the E-PBF process uses a beam of electrons as the heat source for melting the powder-bed. Consequently, when the laser or the electron beam scans the top surface, it melts a selective region of the powder-bed to form a liquid volume, known as the “melt pool,” which is rapidly cooled and solidified either in a vacuum (for E-PBF) or an inert gas (for L-PBF) environment [1,2]. A wide variety of materials, including the alloys of titanium, copper, nickel, aluminum, and chromium, are being used in PBF technology where the material properties significantly influence the process behavior and quality of the processed part. Titanium alloy, especially, Ti-6Al-4V, is one of the prime choices while selecting PBF raw material because of its unique properties and extensive usability in aerospace, automotive, microelectronics, and biomedical applications.
In recent years, significant growth of interest in AM technologies, especially, in the PBF processes has been marked by numerous studies on the L-PBF and E-PBF methods [39]. In most cases, the focus has been on the application-based experimental studies such as build-part microstructure, powder metallurgy, morphological features, and mechanical properties of the material [411]. Besides these experimental analyses, numerical modeling is also conducted by many to determine the thermo-physical properties and melt-pool dynamics [12,13] for investigating the process envelope, part quality, reliability, and energy efficiency. However, studies on the comparison between the two processes under equivalent process parameters are very limited [13]. Instead of studying the L-PBF and E-PBF methods separately, a comparative study of the process parameters, melt-pool geometry, and part microstructures offers more valuable information to characterize suitable applications of the processes [13]. Numerical modeling, especially, thermal modeling is, by far, one of the most convenient and cost-effective methods to conduct a robust comparative analysis between the L-PBF and E-PBF processes [13,14]. The comparison can provide an important guide to select the appropriate technology to be used in the AM industry.
Development of the thermal models for L-PBF and E-PBF requires the understanding of complex heat transport, material phase change, and intricate relations among the thermal, mechanical, and metallurgical phenomena [13,14] which make it extremely challenging to implement. The E-PBF process requires preheating of the material at a high vacuum that needs to be characterized accurately in the simulation. On the other hand, modeling of L-PBF requires the incorporation of convection with the inert gas environment at room temperature. Consequently, understanding the correlation between the process parameters and the process outcomes without costly experimentation requires comprehensive numerical modeling. While developing a robust thermal model, it is important to find a convenient numerical scheme that can accurately estimate the melt-pool geometry and determine the temperature distribution in the processed part by taking into consideration the heat source parameters and material properties. Many researchers have developed thermal models using finite difference (FD) and finite element (FE) methods at various length and time scales [1528]. As a common practice, the heat source in a PBF process is modeled as a conical volumetric heat flux under the surface of the powder-bed due to the resultant keyhole formed by an incident laser or electron beam. Qi et al. [15] used a controlled-volume FD method to develop a self-consistent model for studying the heat transfer, phase change, and fluid flow within the melt pool in the laser-based PBF process. Moraitis and Labeas [16] developed a thermo-mechanical FE model to investigate the residual stresses and distortions for aluminum lap joints in the laser beam welding process. Wang et al. [17], Roberts et al. [18], Liu et al. [19], Yang et al. [20], Lacki and Adamus [21], Shen and Chou [22], Cheng et al. [23], Chen et al. [24], and Zäh and Lutzmann [25] developed FE models incorporating the Gaussian heat source, powder porosity, and thermal properties to simulate the transient heat transfer in the PBF processes. Recent progress on FE analyses to investigate the effect of process parameters in PBF can be attributed to the studies of Andreotta et al. [26], Sadowski et al. [27], and Ladani et al. [28]. All these studies included numerical and/or experimental analyses of either the L-PBF or E-PBF process, but the comparison between the two processes, which could facilitate the selection of the suitable one in the industry, was not outlined.
Thermo-fluid models based on the computational fluid dynamics (CFD) and finite volume (FV) method become more effective than the FE thermal models when fluid flow and heat convection in the melt pool are dominant factors in the process outcomes [29]. Studies show that thermo-fluid models using CFD can effectively provide quantitative information about the part geometry, thermal cycle, cooling rate, and solidification process with the same accuracy as the FE models [29,30]. Wang et al. [30] developed a three-dimensional (3D) volume-of-fluid method to measure the real-time melt-pool shape and obtained the distribution of temperature in laser keyhole welding. Cho et al. [31] studied the effect of fluid motion in the melt pool using thermo-fluid simulation. Rai et al. [32,33] and Li et al. [34] showed that fluid convection inside the melt pool resulted in an increase in heat transfer and gave a better correlation between numerical and experimental results of the melt-pool geometry. Chahine [35] studied the effects of the current and exposure time of an electron beam on the temperature distribution and fluid flow of a melt spot using the CFD technique. Yuan and Gu [36] used FV simulation and laser experiments to investigate the melt-pool evolution and thermal behavior of TiC/AlSi10Mg powder-bed in the L-PBF process. Rahman et al. [3742] conducted CFD-based thermo-fluid modeling of the Ti-6Al-4V [3741] and Cu-Cr-Zr [42] melt pools to study the thermal features and melt-pool dynamics in the PBF processes. Jamshidinia et al. [43,44] developed 3D thermal and fluid flow models of E-PBF, where the influence of fluid convection on the melt-pool geometry was investigated and the effects of changing process parameters were studied numerically and experimentally. However, a comparison between the L-PBF and E-PBF processes based on thermo-fluid modeling is yet to be studied rigorously.
The study aims to compare the thermo-fluid properties of Ti-6Al-4V alloy while undergoing the L-PBF and E-PBF processes by developing a 3D multiphysics CFD model for each method. The effects of the laser and electron beam parameters on the temperature distribution, melt-pool geometry, melting, and solidification criteria are studied and compared under similar heat source specifications. The laser and the electron beam are considered as the Gaussian heat sources to perform simulations for the L-PBF and E-PBF models separately. The CFD simulations are conducted in ansys fluent 18.2 by setting appropriate solver specifications and utilizing the user-defined functions (UDFs) for the heat source and the material properties of Ti-6Al-4V. Results obtained from the CFD simulations are validated by comparing with analytical and experimental results to corroborate the comparative study of the L-PBF and E-PBF processes.
## 2 Material and Methods
### 2.1 Material Modeling.
Ti-6Al-4V provides a unique combination of physical and mechanical properties including lightweight, high strength-to-weight ratio, and excellent resistance to corrosion and fatigue. However, the thermo-physical properties of metallic powder materials are significantly different from those of the corresponding solid bulk material [45,46], especially, the thermal conductivity, melting point, specific heat capacity, and density. When the phase change occurs from solid to liquid and vice versa, properties of the liquid material also have significant effects on the process outcomes. Table 1 shows the temperature-dependent properties of Ti-6Al-4V in solid, powder, and liquid states [39,40,46] that are used in both the L-PBF and E-PBF models as the UDFs.
Table 1
Thermo-physical properties of Ti-6Al-4V alloy used as the user-defined functions [39,40,46]
PropertiesMaterial stateTemperature range (°C)UDFs
Specific heat capacity, Cp (J/kg K)Powder23 <T <1650Cp = [0.52036–(8.34 × 10−6) T (°C)–(4.46 × 10−7) T2 (°C) + (5.44 × 10−10) T3 (°C)] × 1000
Solid23 <T <1650Cp = [0.54058 + (1.02 × 10−4) T (°C) + (1.35 × 10−7) T2 (°C) –(6.50 × 10−11) T3 (°C)] × 1000
Liquid1650 <T <2700830
Thermal conductivity, k (W/m K)Powder23 <T <1650k = 0.9315 – 0.00339 T (°C) + (6.55 × 10−6) T2 (°C) – (1.41 × 10−9) T3(°C)
Solid23 <T <1650k = 6.95757 + 0.00224 T (°C) + (1.69 × 10−5) T2 (°C)—(7.58 × 10−9) T3(°C)
Liquid1650 <T <2700k = −1.6614 + 0.0183 T (°C)
Emissivity, ɛMelt-pool front23 <T <1650ɛ = 0.43356 + (2.94 × 10−4T) (°C) + (5.48 × 10−7) T2 (°C)–(5.53 × 10−10) T3 (°C)
Powder23 <T <16500.6
Melt-pool top1650 <T <27000.4
Density, ρ (kg/m3)Solid23 <T <1650ρsolid = 4420–0.154 (T–25 °C)
Powder23 <T <1650ρpowder = (1–porosity) × ρsolid
Liquid1650 <T <2700ρliq = 3920–0.68 (T–1650 °C)
Viscosity, μ (Pa · s)LiquidT >1605μ(T) = C eE/RT; with $E=0.431Tl1.348$ where C is a constant, E is the activation energy, Tl is the liquidus temperature, and R is the molar gas constant [47]
PropertiesMaterial stateTemperature range (°C)UDFs
Specific heat capacity, Cp (J/kg K)Powder23 <T <1650Cp = [0.52036–(8.34 × 10−6) T (°C)–(4.46 × 10−7) T2 (°C) + (5.44 × 10−10) T3 (°C)] × 1000
Solid23 <T <1650Cp = [0.54058 + (1.02 × 10−4) T (°C) + (1.35 × 10−7) T2 (°C) –(6.50 × 10−11) T3 (°C)] × 1000
Liquid1650 <T <2700830
Thermal conductivity, k (W/m K)Powder23 <T <1650k = 0.9315 – 0.00339 T (°C) + (6.55 × 10−6) T2 (°C) – (1.41 × 10−9) T3(°C)
Solid23 <T <1650k = 6.95757 + 0.00224 T (°C) + (1.69 × 10−5) T2 (°C)—(7.58 × 10−9) T3(°C)
Liquid1650 <T <2700k = −1.6614 + 0.0183 T (°C)
Emissivity, ɛMelt-pool front23 <T <1650ɛ = 0.43356 + (2.94 × 10−4T) (°C) + (5.48 × 10−7) T2 (°C)–(5.53 × 10−10) T3 (°C)
Powder23 <T <16500.6
Melt-pool top1650 <T <27000.4
Density, ρ (kg/m3)Solid23 <T <1650ρsolid = 4420–0.154 (T–25 °C)
Powder23 <T <1650ρpowder = (1–porosity) × ρsolid
Liquid1650 <T <2700ρliq = 3920–0.68 (T–1650 °C)
Viscosity, μ (Pa · s)LiquidT >1605μ(T) = C eE/RT; with $E=0.431Tl1.348$ where C is a constant, E is the activation energy, Tl is the liquidus temperature, and R is the molar gas constant [47]
### 2.2 Physical Model.
A finite volume based 3D model is developed in ansys fluent for the transient thermo-fluid simulation of the PBF process. The geometry of the 3D PBF model is shown in Fig. 1, where the physical domain consists of a solid rectangular block of Ti-6Al-4V, known as the “substrate” and a layer of Ti-6Al-4V powder on top of the substrate. The dimensions of the substrate and the powder layer are considered as 14 mm × 4 mm × 4 mm and 14 mm × 4 mm × 0.07 mm, respectively [13], which implies that the powder layer thickness is 0.07 mm. The laser, or the electron beam, scans the top surface of the powder-bed in the y-direction. A single-track scan is considered for comparing the L-PBF and E-PBF models. The PBF model is first used to simulate the L-PBF process by assigning the laser as the heat source and setting the boundary and initial conditions for the L-PBF process. The same PBF model is then converted to simulate the E-PBF process by assigning the electron beam as the heat source and setting the appropriate conditions for the E-PBF process. The spot size and scanning speed for both the electron beam and the laser are kept the same (0.4 mm and 330 mm/s, respectively). They scan the top surface of the domain starting from (0, 2 mm, 0) to the endpoint at (0, 12 mm, 0) as shown in Fig. 1.
Fig. 1
Fig. 1
Close modal
The maximum heat of the laser or electron beam is located at the center of a target surface, and the intensity varies radially from the center of the heat source. The powder-bed medium is homogeneous and continuous. The heat source is applied at the top of the powder-bed in the form of heat flux density obeying the Gaussian distribution. The top surface of the melt pool is assumed to be flat and all the nodes remained in both vertical and horizontal positions. During the melting and solidification processes, the thermo-physical properties including density, specific heat, thermal conductivity, and viscosity are considered as temperature-dependent parameters. Heat transfer by convection on the top surface is neglected in E-PBF due to the vacuum environment. The thermal boundary conditions applied in the simulations of L-PBF and E-PBF are similar, but unlike E-PBF, L-PBF included convection heat transfer on the top surface. The convective heat transfer coefficient between the powder-bed and surrounding gas in L-PBF is taken as a constant. The top surface is exposed to radiation heat transfer with an ambient temperature of 298 K for both the L-PBF and E-PBF models. The sidewalls and bottom of the domain in E-PBF are adiabatic at a temperature of 1003 K. As L-PBF does not require preheating, the side walls and bottom of the domain are considered adiabatic at a temperature of 298 K.
## 3 Mathematical Formulation
### 3.1 Governing Equations.
The governing equations of fluid motion and heat transfer generally follow three basic physical conservation laws—the conservations of mass, momentum, and energy. Based on the assumptions stated in Physical Model section, the mass, momentum, and energy conservation equations for a 3D Cartesian coordinate system can be expressed by Eqs. (1) through (3). The continuity equation is given by
$∂ρ∂t+∂(ρui)∂xi=0$
(1)
The conservation of momentum equation is given by
$∂(ρuj)∂t+∂(ρuiuj)∂xi=−∂P∂xj+∂∂xi(μ∂uj∂xi)+∂∂xi(μ∂ui∂xj)−ρgzβ(T−Tref)+ρgz−CM((1−fL)2fL3+B)uj−vs∂(ρuj)∂xi$
(2)
In Eq. (2), the scanning speed vs is in the y-direction and gravitational acceleration gz is in the z-direction. Therefore, only the y-momentum equation contains the last term associated with vs which is the relative motion between the heat source and the work piece. Here, β is the coefficient of volume expansion, fL is the liquid fraction, CM is a constant that accounts for the mushy zone morphology, and B is a very small computational constant introduced to avoid division by zero [33]. The fourth, fifth, and sixth terms in the right side of Eq. (2) represent the buoyancy force, gravity, and the frictional drag in the mushy zone during the solid–liquid–solid transition, respectively. Considering the motion of the heat source vs in y-direction, the transient conservation of energy equation (i.e., the heat equation) in terms of temperature [13,22,23,25] is given by
$∂T∂t+vs∂T∂y=∇⋅(k(T)ρ(T)cp(T)∇⋅T)+Q˙(x,y,z,t)ρ(T)cp(T)+Sgzρ(T)cp(T)+μ(T)ρ(T)cp(T)ΦV$
(3)
where k is the effective thermal conductivity, $Q˙(x,y,z,t)$ is the heat source, $Sgz$ is the source term due to gravity, and ΦV is the viscous dissipation term which is defined by Rahman et al. [39] as follows:
$ΦV=2[(∂u∂x)2+(∂v∂y)2+(∂w∂z)2]+(∂u∂y+∂v∂x)2+(∂v∂z+∂w∂y)2+(∂w∂x+∂u∂z)2−23(∂u∂x+∂v∂y+∂w∂z)2$
(4)
The solid–liquid–solid phase change is encountered by an enthalpy method [39,44,48] where the temperature-dependent enthalpy H [48] is a function of the specific heat capacity cp, liquidus temperature TL, solidus temperature TS, latent heat of fusion Lf, and the liquid fraction fL, and can be defined as
$H(T)=∫0TcpdT+LffL$
(5)
where the liquid fraction fL varies from 0 to 1 according to the following relation:
$fL={0,TTL$
(6)
Equation (6) can be rearranged with the information of Eq. (5) to yield the relation between the enthalpy and temperature as given by Eq. (7) [39]
$T={Hcp,HcpTL+Lf$
(7)
### 3.2 Modeling of the Heat Source.
The laser and electron beams are modeled as conical volumetric heat sources with the Gaussian distribution as shown in Fig. 2, where the maximum power intensity is at the center and the intensity decreases with the increase of the depth and width.
Fig. 2
Fig. 2
Close modal
The 3D conical volumetric Gaussian heat source model considered for this analysis is expressed by Eq. (8) [13,37,38]
$Q˙(x,y,z)=η×Ixy×IZS$
(8)
where the Gaussian surface intensity profile Ixy and the beam penetration function, i.e., vertical intensity profile IZ are given by Eqs. (9) and (10), respectively
$Ixy=2PHπΦ2exp{−2[(x−xs)2+(y−ys)2]Φ2}$
(9)
$Iz=10.75(−2.25(zS)2+1.5(zS)+0.75)$
(10)
The power of the electron beam is given by PH = VIb, where V is the acceleration voltage and Ib is the beam current. The values of efficiency η for the laser and electron beams are usually different and depend on a number of factors including the beam control [25], focusing [49], vacuum [49] or convective environment, inclination angle [50], and the energy absorption by the target material [49,50]. The values of the penetration depth S (also known as absorption depth) for laser and electron beam are also different as their wavelengths are not the same [51]. The value of S for the electron beam can be determined by the following relation:
$S=SE=2.1×10−5V2ρ$
(11)
where SE is the penetration depth of electron beam in μm, V is the electron beam potential in V, and ρ is density of powder-bed in kg/m3 [43]. Using a voltage of 60 kV and a powder density of 2150 kg/m3, the value of SE is found to be 35.16 μm. In case of laser, the parameter S is set to be the optical penetration depth SL which is defined as the depth at which the intensity of the laser drops to 1/e of its initial value at the interface [52], and can be determined by
$S=SL=1a=12.303×Alt$
(12)
where a is the absorption coefficient [5053], lt is the powder layer thickness, and A is the optical absorbance [53] of the laser beam while penetrating the Ti-6Al-4V powder-bed. The absorbance of electron beam in Ti-6Al-4V powder-bed is higher than that of the laser beam because the photons are mostly deflected rather than absorbed into the material [51]. Taking lt = 70 μm and S = 35.16 μm in Eq. (12), the value of A for electron beam is found 0.8645. Considering a solid state yttrium-aluminum garnet doped with neodymium ions (Nd:YAG) laser with a wavelength of 1060 nm, the absorbance of laser beam in Ti-6Al-4V alloy is considered as 0.49 [54] which gives SL = 62 μm for the L-PBF simulations. Figure 3 shows the values of beam penetration function for the static laser and electron beams along the vertical coordinate of the domain where the z values are taken such that 0 ≤ zS. The higher value of S for the laser beam results in deeper distribution of its intensity as compared with the intensity of the electron beam within the specified range of vertical coordinate z.
Fig. 3
Fig. 3
Close modal
### 3.3 Initial and Boundary Conditions.
Since the E-PBF process includes preheating of the entire domain to a temperature of 1003 K, the initial conditions are at t = 0, u = v = w = 0, and T = Tpreheat = 1003 K, in the entire domain. The top surface in the E-PBF model is exposed to radiation only at 298 K while the side walls and the bottom of the substrate are considered as adiabatic surfaces. For the L-PBF model, the initial temperature is 298 K and boundary conditions are same as the E-PBF model except for the top surface, which is exposed to radiation and convection at 298 K. The temperature coefficient of surface tension (Marangoni coefficient) is set as −2.6 × 10−4 N/m K at the top surface for both cases [40,44].
## 4 Numerical Simulation
### 4.1 Simulation Procedure.
Transient thermo-fluid analyses of the 3D configuration are performed numerically using ansys fluent 18.2. The thermal properties and the specifications of the moving heat source are assigned as UDFs to simulate the transient melting and solidification for both the L-PBF and E-PBF models. The mass, momentum, and energy conservation equations are discretized and solved using the control volume method with appropriate boundary conditions. ansys Design Modeler is used to create the geometry, Mesh tool is used to generate the structured mesh, and the mathematical model is followed to define the boundary types of the 3D computational domain. Under the pressure-based solver, the “Coupled” algorithm was selected for solving the conservation equations. The Pressure Staggering Option (PRESTO) is selected for pressure discretization, while density, momentum, and energy are discretized with the second-order upwind method. The transient formulation is done by choosing the first-order upwind method. The Courant number is set equal to 1 and all the residual criteria are set to be 10−5. The time-step size is restricted to be 0.001 s or lower to satisfy the convergence criteria while the maximum iterations per time-step are taken as 50/s for running the calculation. After the solution, the postprocessing steps begin to observe and save the results.
### 4.2 Mesh Convergence Study.
The 3D computational domain considered for the analysis is discretized using a structured mesh with hexahedral cells as shown in Fig. 4. The structured mesh is formed by biasing the grid in the powder layer region and around the scanning path of the moving heat source to have a very fine mesh in the target zone.
Fig. 4
Fig. 4
Close modal
A mesh convergence (i.e., the mesh sensitivity) study is conducted for the structured mesh of the 3D domain considering the variation of melt-pool temperature with the increase of the number of nodes. The temperature at location $(x,y,z)$ = (0, 5 mm, 0.03 mm) is monitored for several different mesh densities at 0.009 s when the beam spot size is 0.4 mm, scanning speed is 330 mm/s, and the effective power is 216 W. The value of temperature inside the melt pool converges to 2571 K with the increase of the number of nodes in the domain for the E-PBF model. Figure 5 depicts the results for the mesh convergence study where the temperature at the fixed point remains unchanged after 200,889 nodes corresponding to 190,040 hexahedral cells. Results for both the L-PBF and E-PBF models are obtained for the converged mesh.
Fig. 5
Fig. 5
Close modal
## 5 Model Validation
### 5.1 Comparison of Modeling Results with Analytical Results.
The results obtained from the Fluent simulation for the melting of pure titanium (Ti) is compared with the analytical solution of the classical Stefan problem [42,55] of the melting of a phase-change material (PCM) with pure conduction. Figure 6 shows the standard geometry for the Stefan problem, where the PCM is semi-infinite and initially (at t = 0) solid at its melting temperature Tm [42,55]. The wall temperature Tw is raised to Tw ˃ Tm for melting the PCM in a linear fashion starting at x = 0. According to the Stefan condition, the solution for the transient temperature distribution in the liquid is given by the following relation [42,55]:
$Tl(x,t)−TwTm−Tw=erf(x/2αlt)erf(λ)$
(13)
Fig. 6
Fig. 6
Close modal
The melting front moves forward in the x-direction as time increases. The position of the melting front s(t), measured from x = 0, is given by
$s(t)=2λαlt$
(14)
where αl = kl/ρlcp,l is the thermal diffusivity of the liquid PCM. The parameter λ is calculated from the interfacial melt-front equation and the Stefan number (Ste) as defined below [42,55]:
$λeλ2erf(λ)=Steπ$
(15)
$Ste=cp,l(Tw−Tm)L$
(16)
The validation of the fluent result is conducted by predicting the motion of the liquid–solid interface during the melting of pure Ti. The parameters shown in Table 2 are used in the ansys fluent simulation.
Table 2
List of the simulation parameters for Ti melting
ParametersValues
Density of liquid Ti, ρl (kg/m3)4500
Specific heat capacity of liquid Ti, cp,l (J/kg K)528
Effective viscosity, µ (kg/m s)4.3 × 10−3
Thermal conductivity of liquid Ti, kl (W/m K)17
Latent heat of fusion, Lf (kJ/kg)435.4
Melting temperature, Tm (K)1923
Wall temperature, Tw (K)2073
Solidus temperature, TS (K)1923
Liquidus temperature, TL (K)1943
ParametersValues
Density of liquid Ti, ρl (kg/m3)4500
Specific heat capacity of liquid Ti, cp,l (J/kg K)528
Effective viscosity, µ (kg/m s)4.3 × 10−3
Thermal conductivity of liquid Ti, kl (W/m K)17
Latent heat of fusion, Lf (kJ/kg)435.4
Melting temperature, Tm (K)1923
Wall temperature, Tw (K)2073
Solidus temperature, TS (K)1923
Liquidus temperature, TL (K)1943
The simulation results of the change in interface position with respect to time during the melting of pure Ti show a good agreement with the analytical results. The liquid fraction contours obtained from the simulation at t = 0.37 s and t = 1.8 s are shown in Fig. 7, where the melt front moves forward with the increase of time.
Fig. 7
Fig. 7
Close modal
Figure 8 shows the comparison between the analytical and simulation results for the melt-front position with respect to time considering x = 1 mm.
Fig. 8
Fig. 8
Close modal
Results for temperature distribution also show a good match between the analytical and simulation results. At x = 1 mm, the comparison between the analytical and simulation results for centerline temperature at three different times is shown in Table 3.
Table 3
Analytical versus numerical results for temperature
Time (s)Temperature (K)
AnalyticalNumericalDeviation (%)
0.51936.971940.500.182
1.01975.701979.450.189
102041.912048.070.300
Time (s)Temperature (K)
AnalyticalNumericalDeviation (%)
0.51936.971940.500.182
1.01975.701979.450.189
102041.912048.070.300
### 5.2 Experimental Validation.
The numerical results for the melt-pool geometry are also validated by comparing with the experimental results. The experimental procedure for the E-PBF process, conducted by Jamshidinia et al. [44] with Ti-6Al-4V, is followed to validate the proposed multiphysics model. Jamshidinia et al. [44] compared the results for the variation of the average melt-pool width and depth with the change in scanning speed. Using a constant electron beam spot size of 0.4 mm, a beam current of 14 mA, and a voltage of 60 kV, they applied three levels of scanning speed, namely, 100 mm/s, 300 mm/s, and 500 mm/s to measure the average melt-pool width and depth. The differences between their modeling results and experimental results ranged from −3.5% to +3% for the melt-pool width, and from +2.1% to +3.5% for the melt-pool depth.
Following a similar approach, the simulated results for the melt-pool geometry from the proposed multiphysics CFD model are compared with the experimental results presented by Jamshidinia et al. [44] using the converged mesh of 190,040 hexahedral cells connected by 200,889 nodes. The comparison gave a good agreement between the simulation results and the experimental results of Jamshidinia et al. with a maximum deviation of 3.73%, which indicates a good accuracy of the proposed multiphysics CFD model. The comparison of results for the melt-pool width and depth is illustrated in Fig. 9.
Fig. 9
Fig. 9
Close modal
## 6 Results and Discussion
Results for the thermo-fluid properties and melt-pool geometry are generated at similar irradiation conditions to make a valid comparison. The simulations for the L-PBF and E-PBF models are conducted using the UDFs and the parameters shown in Table 4.
Table 4
List of the simulation parameters
ParametersValues
Solidus temperature, TS (K)1878
Liquidus temperature, TL (K)1938
Latent heat of fusion, Lf (kJ/kg)440
Spot size of laser or electron beam, Φ (mm)0.4
Scanning speed, vs (mm/s)330
Acceleration voltage, V (kV)60
Electron beam current, Ib (mA)4
Laser power, P (W)240
Preheat temperature in E-PBF, Tpreheat (K)1003
Initial temperature in L-PBF, TL-PBF (K)298
Electron beam efficiency, ηe0.9
Laser absorption efficiency, ηl0.865
Powder porosity (%)50
Powder layer thickness, lt (mm)0.07
Electron beam penetration depth, SE (µm)35.16
Laser beam penetration depth, SL (µm)62
Convective heat transfer coefficient, h (W/m2 K)10
Effective viscosity of liquid, µ (kg/m s)UDF
Specific heat, cp (J/kg K)UDF
Thermal conductivity, k (W/m K)UDF
Emissivity, ɛUDF
Density, ρ (kg/m3)UDF
ParametersValues
Solidus temperature, TS (K)1878
Liquidus temperature, TL (K)1938
Latent heat of fusion, Lf (kJ/kg)440
Spot size of laser or electron beam, Φ (mm)0.4
Scanning speed, vs (mm/s)330
Acceleration voltage, V (kV)60
Electron beam current, Ib (mA)4
Laser power, P (W)240
Preheat temperature in E-PBF, Tpreheat (K)1003
Initial temperature in L-PBF, TL-PBF (K)298
Electron beam efficiency, ηe0.9
Laser absorption efficiency, ηl0.865
Powder porosity (%)50
Powder layer thickness, lt (mm)0.07
Electron beam penetration depth, SE (µm)35.16
Laser beam penetration depth, SL (µm)62
Convective heat transfer coefficient, h (W/m2 K)10
Effective viscosity of liquid, µ (kg/m s)UDF
Specific heat, cp (J/kg K)UDF
Thermal conductivity, k (W/m K)UDF
Emissivity, ɛUDF
Density, ρ (kg/m3)UDF
Results for both the L-PBF and E-PBF models are obtained for the converged mesh having 190,040 hexahedral cells with 200,889 nodes and considering a single scan by the heat source at the top surface of the powder-bed. Both the laser and the electron beam scanning simulations are performed keeping the same spot size Φ = 0.4 mm and the same scanning speed vs = 330 mm/s. The same energy density of ED = 26 J/mm3 and interaction time ti = 1.212 ms are used for both the L-PBF and E-PBF models which are determined by the following relations [56,57]:
$ED=PHvs×lt×Φ$
(17)
$ti=Φvs$
(18)
### 6.1 Temperature and Thermo-Physical Properties.
The heat source scans the top surface in the y-direction in both the L-PBF and E-PBF models. The temperature contours at the top surface for L-PBF and E-PBF are shown in Fig. 10.
Fig. 10
Fig. 10
Close modal
The images for the temperature contours are captured when both the laser and electron beams are at y = 8.0 mm (t = 0.018 s). The melt region is longer in the contour for E-PBF than that in L-PBF. The position y = 8.0 mm at the top surface is chosen to show the complete tailing effect during melting and consolidation in the two processes. A cross section is considered at y = 8.0 mm (t = 0.018 s) along the xz-plane to show the comparison of the results in the form of a two-dimensional representation as shown in Figs. 11 and 12. The temperature contours at the cross section for L-PBF and E-PBF are shown in Fig. 11.
Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
As the temperature in the domain increases, density and viscosity decrease following the temperature-dependent equations given in Table 1. The results for density and viscosity are not represented here but they show similar patterns as presented by Rahman et al. in Refs. [39] and [42], respectively. The density of the powder-bed is less than that of the solid substrate due to its porosity. The values of other thermal properties including thermal conductivity, specific heat capacity, and enthalpy increase with the increase of temperature in the domain. Before scanning by the heat source, the thermal conductivity of the powder material is lower than that of the solid substrate as the porosity of the powder reduces the thermal conductivity. When the heat source is applied and the powder material is melted, the thermal conductivity of the liquid melt pool becomes very high due to the high temperature. Contour plots for thermal conductivity in L-PBF and E-PBF are shown in Fig. 12. All the results are obtained at an energy density of 26 J/mm3 and an interaction time of 1.212 ms for both the laser and electron beams. Any increase in energy density due to the increase of power causes a temperature rise in the domain, and eventually affects the temperature-dependent properties significantly.
### 6.2 Melt-Pool Geometry.
The evolution of the melt pool depends on several factors including the material properties, processing parameters, energy absorption, and thermo-fluid interactions. Under the same energy density of 26 J/mm3, the results for the maximum length, width, and depth of penetration of the melt pool at y = 8.0 mm for L-PBF and E-PBF are shown in Table 5.
Table 5
Comparison of the melt-pool dimensions
ProcessED = 26 J/mm3 and ti = 1.212 ms
Length (mm)Width (mm)Depth (mm)
L-PBF1.20.60.08
E-PBF2.10.6050.12
ProcessED = 26 J/mm3 and ti = 1.212 ms
Length (mm)Width (mm)Depth (mm)
L-PBF1.20.60.08
E-PBF2.10.6050.12
The melt-pool volume is larger in E-PBF than that in L-PBF as obtained from the simulation results. The high absorption rate of electron beam, preheating condition, and lack of heat dissipation due to convection result in the formation of a larger volume of the melt pool in E-PBF. Figure 13 depicts the comparison of the length and width of the melt pool when the laser and the electron beam are at y = 8.0 mm.
Fig. 13
Fig. 13
Close modal
A parametric study on the effects of processing parameters on the evolution of the melt pool is also conducted with the numerical simulations of the L-PBF and E-PBF processes. First, the effect of increasing the power of the laser and electron beams on the depth of penetration of the melt pool is investigated while keeping the same spot size of 0.4 mm and scanning speed of 330 mm/s. As expected, the depth of the melt pool increases with the increase of beam power. The comparison of the melt-pool depth in the L-PBF and E-PBF processes are shown in Fig. 14.
Fig. 14
Fig. 14
Close modal
In contrast to the beam power, the depth of the melt pool decreases as the scanning speed increases at a given power and a spot size of the laser or electron beam. The simulation results for melt-pool depth versus scanning speed for L-PBF and E-PBF at a power of 240 W and a spot size of 0.4 mm are shown in Fig. 15. Results show that the melt-pool depth is more sensitive to the change in beam power as compared with the change in scanning speed at a given spot size. For instance, due to an increase of 127.27% of the scanning speed from 330 mm/s to 750 mm/s in the L-PBF model, the percentage of decrease in the melt-pool depth is 55% (as shown in Fig. 15). However, a power increase of 87.5% from 240 W to 450 W in the L-PBF model results in a 433.33% increase in the melt-pool depth, which is calculated from the values shown in Fig. 14. Therefore, variation in the melt-pool depth due to the change in beam power is significantly higher than the variation caused by the change in scanning speed.
Fig. 15
Fig. 15
Close modal
### 6.3 Liquid Flow Inside the Melt Pool.
The simulation results for the velocity distribution inside the melt pool obtained from the L-PBF and E-PBF models are also compared under the same energy density of 26 J/mm3 and heat source interaction time of 1.212 ms. The velocity of liquid inside the melt pool in L-PBF is higher than that in E-PBF due to the greater convection in L-PBF. Along the yz-plane corresponding to the origin (i.e., the longitudinal section), the velocity contours inside the melt pool when the laser and electron beam are at y = 8.0 mm are shown in Fig. 16. The maximum velocity for the L-PBF model is found in the middle of the melt pool, whereas the maximum melt-pool velocity in the E-PBF model is detected toward the tail end from the center.
Fig. 16
Fig. 16
Close modal
The temperature variation leads to a surface tension gradient, which causes the Marangoni flow from low surface tension area to high surface tension area of the melt pool as described by Yuan and Gu [36]. The cooler liquid near the edge of the melt pool having higher surface tension tends to pull the liquid away from the melt-pool center. However, the magnitudes of the maximum velocity in the melt pool for the L-PBF and the E-PBF models are about 18.6 mm/s and 15.4 mm/s, respectively, which confirm that the values of the Reynolds number Re(= ρuiΦ/μ) are very low and the flow is laminar in both the L-PBF and E-PBF cases.
### 6.4 Cooling Rate.
The cooling rates for the Ti-6Al-4V melt pool in L-PBF and E-PBF are found very fast due to the combined heat transfer. For the given specifications of the laser and electron beam, a point on the top surface at y = 5 mm along the scan is considered to observe the temperature variation with respect to time. Any temperature above the liquidus temperature (1938 K) indicates a complete liquid state of that point. If the temperature of that point is below the solidus temperature (1878 K), it specifies the point to be in the solid state [39,41]. The material is in the mushy zone when it remains between the liquidus and solidus temperatures. The heating and cooling of the point at $(x,y,z)$ = (0, 5 mm, 0) with respect to time for L-PBF and E-PBF are shown in Figs. 17(a) and 17(b), respectively. The variation of heating and cooling rates (in K/µs) with respect to time [22] for the L-PBF and E-PBF models are shown in Figs. 18(a) and 18(b), respectively, where the vertical axes values are multiplied by (−1). In both cases, the time count starts when the heat source strikes the point. The liquid melt pool takes 4.5 ms to cool down from the maximum temperature to the solidus temperature in L-PBF, yielding an average cooling rate of 1.5 K/µs. Under the same irradiation condition, the liquid melt pool takes about 8 ms for cooling in E-PBF, which gives an average cooling rate of 0.74 K/µs. These results are obtained for an energy density of 26 J/mm3 and the interaction time of 1.212 ms of the laser and electron beams.
Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal
## 7 Conclusions
Numerical modeling for the transient thermo-fluid properties of a Ti-6Al-4V powder-bed in the PBF process is conducted to determine the differences between the L-PBF and electron beam PBF (L-PBF) processes. The unique features of the two processes are characterized by implementing physical assumptions. A moving heat source with the Gaussian distribution is applied as a user-defined function and the thermo-fluid properties of the melt pool are obtained using the finite volume method. Modeling results for the temperature distributions, thermo-physical properties, melt-pool geometries, and solid–liquid–solid phase changes are compared at similar operating conditions. Based upon the comparison, the following conclusions can be drawn:
• A similar irradiance condition is considered for both the laser and the electron beam to make a valid comparison. The absorption efficiency in the Ti-6Al-4V powder-bed is determined by adjusting the powder porosity, layer thickness, optical absorbance, optical penetration depth, and efficiency of the heat source.
• For an irradiance level of 26 J/mm3 and interaction time of 1.212 ms, simulation results show that the liquid melt pool in E-PBF cools down in 8 ms whereas the melt pool in L-PBF cools down 4.5 ms. Thus, the cooling rate in L-PBF is faster than that in E-PBF. The lower volume of melt pool, absence of preheating, and convection heat transfer at the top surface make the cooling process faster in L-PBF.
• At the same scanning speed and beam spot size, the depth of the melt pool in E-PBF is higher than that in L-PBF. The lack of penetration of laser causes shallow melt-pool depth in L-PBF. Although the length and depth are significantly different, the width of the melt pool is quite similar in both L-PBF and E-PBF under similar operating conditions.
• The melt-pool volume is larger in E-PBF than that in L-PBF under the same energy density of 26 J/mm3 and interaction time of 1.212 ms. The melt-pool volume increases with the increase of beam power but decreases with the increase of scanning speed as evinced by the simulation results and experimental validation.
• The porosity of Ti-6Al-4V powder is taken as 50% for both L-PBF and E-PBF to keep the consistency of comparison. However, if the powder porosity is increased, the maximum temperature in the melt pool becomes higher due to the lower density of powder. This eventually makes the cooling rate slower than the current cases.
• Due to greater convection, the melt-pool velocity in the L-PBF process is higher than that in the E-PBF process. The fluid flow is laminar in both cases.
• A comparative study, differentiating the effects of the laser and electron beams under similar irradiation conditions, provides a thorough understanding of the physics involved in the two processes. The study facilitates the design for correct experiments prior to the actual production by giving room to optimize the process parameters and control the energy transfers in the L-PBF and E-PBF processes.
Numerical simulation of thermal behavior and the melt-pool dynamics as a result of the interaction between the moving heat source and powder zone for a single scan is the foundation for obtaining feedback of laser or electron beam processing parameters in the PBF process. The residual stress analysis of the processed part and shape optimization of the melt zone also depend on the thermal history and melt-pool evolution during the process. Therefore, the thermo-fluid models presented in this study are focused on characterizing the thermal history and melt-pool dynamics of the two significant PBF processes and offer a wide range of results with good accuracy. The model can incorporate various materials and operating conditions for further analyses of the material and process behavior in the powder-bed fusion additive manufacturing process.
## Funding Data
• This research is funded by the National Science Foundation (award number OIA-1541079) and the Louisiana Board of Regents.
## Conflict of Interest
There are no conflicts of interest.
## Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
## Nomenclature
• a =
absorption coefficient of laser (1/μm)
•
• h =
convective heat transfer coefficient (W/m2 K)
•
• k =
effective thermal conductivity (W/m K)
•
• s =
melt-front position with respect to time
•
• t =
time (s)
•
• z =
distance in the direction of penetration (mm)
•
• A =
optical absorbance
•
• B =
a computational constant to avoid division by zero
•
• C =
a constant for viscosity calculation
•
• H =
total enthalpy (J/kg)
•
• P =
pressure (Pa)
•
• R =
molar gas constant (kcal/mol K)
•
• S =
penetration depth (µm)
•
• T =
temperature (K)
•
• V =
acceleration voltage of the electron beam (kV)
•
• cp =
specific heat capacity (J/kg K)
•
• cp,l =
specific heat capacity of liquid material (J/kg K)
•
• fL =
liquid fraction
•
• gz =
gravitational acceleration (m/s2)
•
• kl =
thermal conductivity of liquid Ti (W/m K)
•
• lt =
powder layer thickness (mm)
•
• ti =
heat source interaction time (s) or (ms)
•
• vs =
beam scanning speed (mm/s)
•
• xi =
distance along the Cartesian coordinates (mm)
•
• xs =
instantaneous position of heat source in the x-direction (mm)
•
• ys =
instantaneous position of heat source in the y-direction (mm)
•
• Aw =
atomic weight (g/mol)
•
• CM =
a constant regarding mushy zone morphology
•
• ED =
energy density (J/mm3)
•
• Ib =
electron beam current (mA)
•
• Ixy =
the Gaussian surface intensity profile
•
• IZ =
beam penetration function
•
• Lf =
latent heat of fusion (kJ/K)
•
• PH =
power of the heat source (W)
•
• $Q˙(x,y,z)$ =
absorbed heat flux (W/m2)
•
• $Sgz$ =
source term due to gravity
•
• TL =
liquidus temperature (K)
•
• Tm =
melting temperature (K)
•
• Tpreheat =
preheat temperature (K)
•
• Tref =
reference temperature (K)
•
• TS =
solidus temperature (K)
•
• Tw =
wall temperature (K)
•
• u, v, w =
velocity components in the x-, y-, and z-directions, respectively (m/s)
•
• ui, uj =
velocity along the Cartesian coordinates (m/s)
•
• Ste =
Stefan number
•
• αl =
thermal diffusivity of liquid (m2/s)
•
• β =
coefficient of volume expansion
•
• ɛ =
emissivity
•
• η =
efficiency of laser or electron beam
•
• λ =
parameter in interfacial melt-front equation
•
• μ =
absolute viscosity (N·s/m2) or (Pa·s)
•
• μm =
viscosity of liquid metal (N·s/m2) or (Pa·s)
•
• ρ =
density of the material (kg/m3)
•
• ρl =
density of liquid Ti (kg/m3)
•
• σ =
Stefan–Boltzmann constant (W/m2 K4)
•
• Φ =
laser or electron beam spot size (mm)
•
• ΦV =
viscous dissipation term
## References
1.
Sames
,
W. J.
,
List
,
F. A.
,
Pannala
,
S.
,
Dehoff
,
R. R.
, and
Babu
,
S. S.
,
2016
, “
The Metallurgy and Processing Science of Metal Additive Manufacturing
,”
Int. Mater. Rev.
,
61
(
5
), pp.
315
360
. 10.1080/09506608.2015.1116649
2.
Goodridge
,
R.
, and
Ziegelmeier
,
S.
,
2017
, “7—Powder Bed Fusion of Polymers,”
,
,
Duxford, UK
, pp.
181
204
. 10.1016/C2014-0-03891-9
3.
Sing
,
S. L.
,
An
,
J.
,
Yeong
,
W. Y.
, and
Wiria
,
F. E.
,
2015
, “
Laser and Electron-Beam Powder-Bed Additive Manufacturing of Metallic Implants: A Review on Processes, Materials and Designs
,”
J. Orthop. Res.
,
34
(
3
), pp.
369
385
. 10.1002/jor.23075
4.
Zhao
,
X.
,
Li
,
S.
,
Zhang
,
M.
,
Liu
,
Y.
,
Sercombe
,
T. B.
, and
Wang
,
S.
,
2016
, “
Comparison of the Microstructures and Mechanical Properties of Ti–6Al–4 V Fabricated by Selective Laser Melting and Electron Beam Melting
,”
Mater. Des.
,
95
, pp.
21
31
. 10.1016/j.matdes.2015.12.135
5.
Gong
,
H.
,
Rafi
,
K.
,
Gu
,
H.
,
Janaki Ram
,
G. D.
,
Starr
,
T.
, and
Stucker
,
B.
,
2015
, “
Influence of Defects on Mechanical Properties of Ti–6Al–4 V Components Produced by Selective Laser Melting and Electron Beam Melting
,”
Mater. Des.
,
86
, pp.
545
554
. 10.1016/j.matdes.2015.07.147
6.
Chastand
,
V.
,
Quaegebeur
,
P.
,
Maia
,
W.
, and
Charkaluk
,
E.
,
2018
, “
Comparative Study of Fatigue Properties of Ti-6Al-4V Specimens Built by Electron Beam Melting (EBM) and Selective Laser Melting (SLM)
,”
Mater. Charact.
,
143
, pp.
76
81
. 10.1016/j.matchar.2018.03.028
7.
Wysocki
,
B.
,
Maj
,
P.
,
Sitek
,
R.
,
Buhagiar
,
J.
,
Kurzydłowski
,
K. J.
, and
Święszkowski
,
W.
,
2017
, “
Laser and Electron Beam Additive Manufacturing Methods of Fabricating Titanium Bone Implants
,”
Appl. Sci.
,
7
(
7
), pp.
1
20
. 10.3390/app7070657
8.
Gokuldoss
,
P. K.
,
Kolla
,
S.
, and
Eckert
,
J.
,
2017
, “
Additive Manufacturing Processes: Selective Laser Melting, Electron Beam Melting and Binder Jetting-Selection Guidelines
,”
Materials (Basel, Switzerland)
,
10
(
6
), pp.
1
12
. 10.3390/ma10060672
9.
Raplee
,
J.
,
Plotkowski
,
A.
,
Kirka
,
M. M.
,
Dinwiddie
,
R.
,
Okello
,
A.
,
Dehoff
,
R. R.
, and
Babu
,
S. S.
,
2017
, “
Thermographic Microstructure Monitoring in Electron Beam Additive Manufacturing
,”
Sci. Rep.
,
7
(
43554
), pp.
1
16
. 10.1038/srep43554
10.
Siddiqui
,
S. F.
,
Fasoro
,
A. A.
,
Cole
,
C.
, and
Gordon
,
A. P.
,
2019
, “
Mechanical Characterization and Modeling of Direct Metal Laser Sintered Stainless Steel GP1
,”
ASME J. Eng. Mater. Technol.
,
141
(
3
), p.
031009
. 10.1115/1.4042867
11.
Riedlbauer
,
D.
,
Scharowsky
,
T.
,
Singer
,
R. F.
,
Steinmann
,
P.
,
Körner
,
C.
, and
Mergheim
,
J.
,
2017
, “
Macroscopic Simulation and Experimental Measurement of Melt Pool Characteristics in Selective Electron Beam Melting of Ti-6Al-4V
,”
,
88
(
5
), pp.
1309
1317
. 10.1007/s00170-016-8819-6
12.
Cook
,
P. S.
, and
Murphy
,
A. B.
,
2020
, “
Simulation of Melt Pool Behaviour During Additive Manufacturing: Underlying Physics and Progress
,”
,
31
(
100909
), pp.
1
23
13.
Rahman
,
M. S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2018
, “
A Comparative Study Between Selective Laser Melting and Electron Beam Additive Manufacturing Based on Thermal Modeling
,”
Proceedings of the ASME International Mechanical Engineering Congress and Exposition
,
Pittsburgh, PA
,
Nov. 9−15, 2018
, IMECE2018-86428, Vol.
1
,
, pp.
1
13
.
14.
Bikas
,
H.
,
Stavropoulos
,
P.
, and
Chryssolouris
,
G.
,
2015
, “
Additive Manufacturing Methods and Modelling Approaches: A Critical Review
,”
,
83
(
1
), pp.
389
405
. 10.1007/s00170-015-7576-2
15.
Qi
,
H.
,
Mazumder
,
J.
, and
Ki
,
H.
,
2006
, “
Numerical Simulation of Heat Transfer and Fluid Flow in Coaxial Laser Cladding Process for Direct Metal Deposition
,”
J. Appl. Phys.
,
100
(
2
), p.
024903
. 10.1063/1.2209807
16.
Moraitis
,
G. A.
, and
Labeas
,
G. N.
,
2008
, “
Residual Stress and Distortion Calculation of Laser Beam Welding for Aluminum Lap Joints
,”
J. Mater. Process. Technol.
,
198
(
1–3
), pp.
260
269
. 10.1016/j.jmatprotec.2007.07.013
17.
Wang
,
L.
,
Felicelli
,
S.
,
Gooroochurn
,
Y.
,
Wang
,
P. T.
, and
Horstemeyer
,
M. F.
,
2008
, “
Optimization of the LENS® Process for Steady Molten Pool Size
,”
Mater. Sci. Eng., A
,
474
(
1–2
), pp.
148
156
. 10.1016/j.msea.2007.04.119
18.
Roberts
,
I. A.
,
Wang
,
C. J.
,
Esterlein
,
R.
,
Stanford
,
M.
, and
Mynors
,
D. J.
,
2009
, “
A Three-Dimensional Finite Element Analysis of the Temperature Field During Laser Melting of Metal Powders in Additive Layer Manufacturing
,”
Int. J. Mach. Tools Manuf.
,
49
(
12–13
), pp.
916
923
. 10.1016/j.ijmachtools.2009.07.004
19.
Liu
,
C.
,
Wu
,
B.
, and
Zhang
,
J.
,
2010
, “
Numerical Investigation of Residual Stress in Thick Titanium Alloy Plate Joined With Electron Beam Welding
,”
Metall. Mater. Trans. B
,
41
(
5
), pp.
1129
1138
. 10.1007/s11663-010-9408-y
20.
Yang
,
J.
,
Sun
,
S.
,
Brandt
,
M.
, and
Yan
,
W.
,
2010
, “
Experimental Investigation and 3D Finite Element Prediction of the Heat Affected Zone During Laser Assisted Machining of Ti–6Al–4 V Alloy
,”
J. Mater. Process. Technol.
,
210
(
15
), pp.
2215
2222
. 10.1016/j.jmatprotec.2010.08.007
21.
Lacki
,
P.
, and
,
K.
,
2011
, “
Numerical Simulation of the Electron Beam Welding Process
,”
Comput. Struct.
,
89
(
11–12
), pp.
977
985
. 10.1016/j.compstruc.2011.01.016
22.
Shen
,
N.
, and
Chou
,
K.
,
2012
, “
Thermal Modeling of Electron Beam Additive Manufacturing Process: Powder Sintering Effects
,”
Proceedings of the ASME Manufacturing Science and Engineering Conference
,
Notre Dame, IN
,
June 4–8, 2012
, MSEC2012-7253, pp.
287
295
.
23.
Cheng
,
B.
,
Price
,
S.
,
Lydon
,
J.
,
Cooper
,
K.
, and
Chou
,
K.
,
2014
, “
On Process Temperature in Powder-Bed Electron Beam Additive Manufacturing: Model Development and Validation
,”
ASME J. Manuf. Sci. Eng.
,
136
(
6
), p.
061018
. 10.1115/1.4028484
24.
Chen
,
Y. X.
,
Wang
,
X. J.
, and
Chen
,
S. B.
,
2014
, “
The Effect of Electron Beam Energy Density on Temperature Field for Electron Beam Melting
,”
,
900
, pp.
631
638
. 10.4028/www.scientific.net/AMR.900.631
25.
Zäh
,
M. F.
, and
Lutzmann
,
S.
,
2010
, “
Modelling and Simulation of Electron Beam Melting
,”
Product. Eng. Res. Develop.
,
4
(
1
), pp.
15
23
. 10.1007/s11740-009-0197-6
26.
Andreotta
,
R.
,
,
L.
, and
Brindley
,
W.
,
2017
, “
Finite Element Simulation of Laser Additive Melting and Solidification of Inconel 718 With Experimentally Tested Thermal Properties
,”
Finite Elem. Anal. Des.
,
135
, pp.
36
43
. 10.1016/j.finel.2017.07.002
27.
,
M.
,
,
L.
,
Brindley
,
W.
, and
Romano
,
J.
,
2017
, “
Optimizing Quality of Additively Manufactured Inconel 718 Using Powder Bed Laser Melting Process
,”
,
11
, pp.
60
70
28.
,
L.
,
Romano
,
J.
,
Brindley
,
W.
, and
Burlatsky
,
S.
,
2017
, “
Effective Liquid Conductivity for Improved Simulation of Thermal Transport in Laser Beam Melting Powder Bed Technology
,”
,
14
, pp.
13
23
29.
Wen
,
S.
, and
Shin
,
Y. C.
,
2010
, “
Modeling of Transport Phenomena During the Coaxial Laser Direct Deposition Process
,”
J. Appl. Phys.
,
108
(
4
), p.
044908
. 10.1063/1.3474655
30.
Wang
,
R.
,
Lei
,
Y.
, and
Shi
,
Y.
,
2011
, “
Numerical Simulation of Transient Temperature Field During Laser Keyhole Welding of 304 Stainless Steel Sheet
,”
Opt. Laser Technol.
,
43
(
4
), pp.
870
873
. 10.1016/j.optlastec.2010.10.007
31.
Cho
,
W.
,
Na
,
S.
,
Thomy
,
C.
, and
Vollertsen
,
F.
,
2012
, “
Numerical Simulation of Molten Pool Dynamics in High Power Disk Laser Welding
,”
J. Mater. Process. Technol.
,
212
(
1
), pp.
262
275
. 10.1016/j.jmatprotec.2011.09.011
32.
Rai
,
R.
,
Burgardt
,
P.
,
Milewski
,
J.
,
Lienert
,
T.
, and
DebRoy
,
T.
,
2009
, “
Heat Transfer and Fluid Flow During Electron Beam Welding of 21Cr–6Ni–9Mn Steel and Ti–6Al–4 V Alloy
,”
J. Phys. D: Appl. Phys.
,
42
(
2
), p.
02550
. 10.1088/0022-3727/42/2/025503
33.
Rai
,
R.
,
Palmer
,
T. A.
,
Elmer
,
J. W.
, and
Debroy
,
T.
,
2009
, “
Heat Transfer and Fluid Flow During Electron Beam Welding of 304L Stainless Steel Alloy
,”
Weld. J.
,
88
(
3
), pp.
54
61
. 10.1088/0022-3727/42/2/025503
34.
Li
,
J. F.
,
Li
,
L.
, and
Stott
,
F. H.
,
2004
, “
A Three-Dimensional Numerical Model for a Convection-Diffusion Phase Change Process During Laser Melting of Ceramic Materials
,”
Int. J. Heat Mass Trans.
,
47
(
25
), pp.
5523
5539
. 10.1016/j.ijheatmasstransfer.2004.07.024
35.
Chahine
,
G.
,
2011
, “
Application of Digital Engineering in the Development of a Bio-Adaptable Dental Implant
,”
Ph.D. thesis
,
Southern Methodist University
,
Dallas, TX
.
36.
Yuan
,
P.
, and
Gu
,
D.
,
2015
, “
Molten Pool Behaviour and Its Physical Mechanism During Laser PBF of TiC/AlSi10Mg Nanocomposites: Simulation and Experiments
,”
J. Phys. D: Appl. Phys.
,
48
(
3
), p.
035303
. 10.1088/0022-3727/48/3/035303
37.
Rahman
,
M. S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2016
, “
Thermo-Fluid Characterizations of Ti-6Al-4V Melt Pool in Powder-Bed Electron Beam Additive Manufacturing
,”
Proceedings of the ASME 2016 International Mechanical Engineering Congress and Exposition
,
Phoenix, AZ
,
Nov. 11–17, 2016
, IMECE2016-65854, Vol.
1
,
, pp.
1
9
.
38.
Rahman
,
M. S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2017
, “
Thermal Analysis of Electron Beam PBF Using Ti-6Al-4V Powder-Bed
,”
Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition
,
Tampa, FL
,
Nov. 3–9, 2017
, IMECE2017-71663, Vol.
1
,
, pp.
1
13
.
39.
Rahman
,
M. S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2019
, “
Thermofluid Properties of Ti-6Al-4V Melt Pool in Powder-Bed Electron Beam Additive Manufacturing
,”
ASME J. Eng. Mater. Technol.
,
141
(
4
), p.
041006
. 10.1115/1.4043342
40.
Rahman
,
M. S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2020
, “
Heat Transfer and Melt-Pool Evolution During Powder-Bed Fusion of Ti-6Al-4V Parts Under Various Laser Irradiation Conditions
,”
Proceedings of the ASME International Mechanical Engineering Congress and Exposition
,
Portland, OR
,
Nov. 16–19, 2020
, IMECE2020-23838, Vol.
1
,
, pp.
1
10
.
41.
Rahman
,
M. S.
,
Zeng
,
C.
,
Wen
,
H.
,
Guo
,
S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2020
, “
An Investigation on the Melting and Solidification Behavior of Ti64 Powder in the Laser Powder-Bed Fusion Process
,”
Proceedings of the Louisiana EPSCoR RII CIMM 2020 Symposium
,
Baton Rouge, LA
,
July 20, 2020
, pp.
9
12
.
42.
Rahman
,
M. S.
,
Schilling
,
P. J.
,
Herrington
,
P. D.
, and
Chakravarty
,
U. K.
,
2019
, “
Thermal Behavior and Melt-Pool Dynamics of Cu-Cr-Zr Alloy in Powder-Bed Selective Laser Melting Process
,”
Proceedings of the ASME International Mechanical Engineering Congress and Exposition
, Vol.
1
,
,
Salt Lake City, UT
,
Nov. 11–14, 2019
, IMECE2019-11581, pp.
1
9
.
43.
Jamshidinia
,
M.
,
Kong
,
F.
, and
Kovacevic
,
R.
,
2012
, “
Temperature Distribution and Fluid Flow Modeling of Electron Beam Melting® (EBM)
,”
Proceedings of the ASME 2012 International Mechanical Engineering Congress and Exposition
,
Houston, TX
,
Nov. 9–15, 2012
, IMECE2012-89440, Vol.
7
,
Part D, Fluid and Heat Transfer
, pp.
3089
3102
.
44.
Jamshidinia
,
M.
,
Kong
,
F.
, and
Kovacevic
,
R.
,
2013
, “
Numerical Modeling of Heat Distribution in the Electron Beam Melting of Ti–6Al–4 V
,”
ASME J. Manuf. Sci. Eng.
,
135
(
6
), p.
061010
. 10.1115/1.4025746
45.
Sih
,
S. S.
, and
Barlow
,
J. W.
,
2004
, “
The Prediction of the Emissivity and Thermal Conductivity of Powder Beds
,”
Part. Sci. Technol.
,
22
(
3
), pp.
291
304
. 10.1080/02726350490501682a
46.
Arce
,
A. N.
,
2012
, “
Thermal Modeling and Simulation of Electron Beam Melting for Rapid Prototyping on Ti6Al4 V Alloys
,”
Ph.D. thesis
,
North Carolina State University
,
Raleigh, NC
.
47.
Pei
,
Y. T.
,
Ocelik
,
V.
, and
Hosson
,
J. T. M. D.
,
2002
, “
SiCp/Ti6Al4V Functionally Graded Materials Produced by Laser Melt Injection
,”
Acta Mater.
,
50
(
8
), pp.
2035
51
. 10.1016/S1359-6454(02)00049-6
48.
Esen
,
A.
, and
Kutluay
,
S.
,
2002
, “
A Numerical Solution of the Stefan Problem With a Neumann-Type Boundary Condition by Enthalpy Method
,”
Appl. Math. Comput.
,
148
(
2004
), pp.
321
329
. 10.1016/S0096-3003(02)00846-9
49.
Yan
,
W.
,
Ge
,
W.
,
Smith
,
J.
,
Lin
,
S.
,
Kafka
,
O. L.
,
Lin
,
F.
, and
Liu
,
W. K.
,
2016
, “
Multi-scale Modeling of Electron Beam Melting of Functionally Graded Materials
,”
Acta Mater.
,
115
, pp.
403
412
. 10.1016/j.actamat.2016.06.022
50.
Carriere
,
P. R.
, and
Yue
,
S.
,
2017
, “
Energy Absorption During Pulsed Electron Beam Spot Melting of 304 Stainless Steel: Monte-Carlo Simulations and In-Situ Temperature Measurements
,”
Vacuum
,
142
, pp.
114
122
. 10.1016/j.vacuum.2017.04.039
51.
Yan
,
W.
,
Smith
,
J.
,
Ge
,
W.
,
Lin
,
F.
, and
Liu
,
W. K.
,
2015
, “
Multiscale Modeling of Electron Beam and Substrate Interaction: A New Heat Source Model
,”
Comput. Mech.
,
56
(
2
), pp.
265
276
. 10.1007/s00466-015-1170-1
52.
Brown
,
M. S.
, and
Arnold
,
C. B.
,
2010
,
Fundamentals of Laser-Material Interaction and Application to Multiscale Surface Modification, Laser Precision Microfabrication
, Vol.
135
,
Springer Series in Materials Science Springer
,
Berlin, Germany
, pp.
91
120
.
53.
Ilican
,
S.
,
Caglar
,
M.
, and
Caglar
,
Y.
,
2007
, “
Determination of the Thickness and Optical Constants of Transparent Indium-Doped ZnO Thin Films by the Envelope Method
,”
Materials Science-Poland
,
25
(
3
), pp.
709
718
.
54.
Lisiecki
,
A.
,
2019
, “
Study of Optical Properties of Surface Layers Produced by Laser Surface Melting and Laser Surface Nitriding of Titanium Alloy
,”
Materials
,
12
(
3112
), pp.
1
14
. 10.3390/ma12193112
55.
Ogoh
,
W.
, and
Groulx
,
D.
,
2010
, “
Stefan’s Problem: Validation of a One-Dimensional Solid–Liquid Phase Change Heat Transfer Process
,”
Proceedings of the COMSOL Conference
,
Boston, MA
,
Oct. 7–9, 2010
, pp.
1
6
.
56.
Dilip
,
J. J. S.
,
Zhang
,
S.
,
Teng
,
C.
,
Zeng
,
K.
,
Robinson
,
C.
,
Pal
,
D.
, and
Stucker
,
B.
,
2017
, “
Influence of Processing Parameters on the Evolution of Melt Pool, Porosity, and Microstructures in Ti-6Al-4V Alloy Parts Fabricated by Selective Laser Melting
,”
,
2
(
3
), pp.
157
167
. 10.1007/s40964-017-0030-2
57.
Matilainen
,
V.
,
Piili
,
H.
,
Salminen
,
A.
, and
Nyrhilä
,
O.
,
2015
, “
Preliminary Investigation of Keyhole Phenomena During Single Layer Fabrication in Laser Additive Manufacturing of Stainless Steel
,”
Phys. Procedia
,
78
, pp.
377
387
. 10.1016/j.phpro.2015.11.052 | 2022-08-09 04:17:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 26, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5440508127212524, "perplexity": 3907.6154576451663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570901.18/warc/CC-MAIN-20220809033952-20220809063952-00432.warc.gz"} |
https://xmphysics.com/2023/01/09/4-3-3-couple/ | # 4.3.3 Couple
A couple consists of two equal but opposite forces whose lines of action are parallel but non-collinear.
In the example shown below, a couple is formed by two parallel forces F applied at both ends of a rod of length L.
Notice that both forces are going to cause the rod to rotate clockwise about the CM (center of mass) of the rod. So the total moment of the couple (also called the torque) is given by
\displaystyle \begin{aligned}\tau &=F\times \frac{L}{2}+F\times \frac{L}{2}\\&=F\times L\end{aligned}
If the forces are not perpendicular to the beam, we just have to work with the components perpendicular to the rod, leading us to
$\displaystyle \tau =F\sin \theta \times L$
Alternatively, we can obtain the perpendicular distance between the two forces, leading us to
$\displaystyle \tau =F\times L\sin \theta$
Notice that the net force of a couple is always zero. This is why some people think of couples as pure moments, since they produce only rotational effect but no translational effect. This is also why the total moment of a couple evaluates to be the same magnitude about any pivot point.
Concept Test
0632 | 2023-01-30 10:58:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8494585156440735, "perplexity": 450.1187810221701}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499816.79/warc/CC-MAIN-20230130101912-20230130131912-00388.warc.gz"} |
https://www.mathworks.com/examples/matlab/community/20283-arc-length-control-method-ritto-correa-camotim-2008 | MATLAB Examples
# Arc-length control method (Ritto-Correa & Camotim, 2008)
## Notation and references
The notation followed here and in the following MATLAB codes:
• arc_length.m
conforms to that used by Ritto-Correa & Camotim in the following reference:
Ritto-Correa, M. and Camotim, D. (2008). ”On the Arc-Length and Other Quadratic Control Methods: Established, Less Known and New Implementation Procedures.” Computers & Structures 86(), 1353–1368. This reference is denoted as [1] inside the text of the above code.
Except for the above study, the following references should be noted as well:
• Bergan, P.G., Horrigmoe, B., Krakeland, B. and Soreide, T.H. (1978). ”Solution Techniques for Non-Linear Finite Element Problems.” Int. J. Num. Methods in Engrg, 12(), 1677–1696. This reference is denoted as [2] inside the text of the above code.
• Li, Y. and Shen, Z. (2004). ”Improvements on the Arc-Length-Type Method.” Acta Mechanica Sinica 20(5), 541–550. This reference is denoted as [5] inside the text of the above code.
## Algorithms implemented
• Arc length control method as described by Ritto-Correa & Camotim (2008)
help arc_length
Generalized arc-length quadratic control method
Description
The equation functn(#t#)=0 is solved for #t#, where
#t#=[#u#;#lambda#], #u# is the unknown displacement vector and
#lambda# is the unknown load factor. The method used is the
arc-length method described by Ritto-Correa & Camotim (2008): "On the
Arc-Length and Other Quadratic Control Methods: Established, Less
Known and New Implementation Procedures."
Notation in this code conforms to that used in the above paper. In
the following notation prefix "ft" denotes "Filled Triangle" and
prefix "fr" denotes "Filled Rhombus", in accordance with the notation
used in [1].
Required input parameters
#functn# is the function handle defining the equation to be solved.
The definition of #functn# must be of the type
[#R#,#Q#,#K#]=functn(#t#) where #R# ([#dim# x 1]) is the out of
balance force vector, #Q# ([#dim# x 1]) is the tangent load vector
given by Q(a,lambda)=-d{R(a,lambda)}/d{lambda}, #K# ([#dim# x #dim#])
is the tangent stiffness matrix given by
K(a,lambda)=d{R(a,lambda)}/d{a} and #t# ([#dim#+1 x 1]) is the
generalized unknown vector defined in the description section.
#aO# ([#dim# x 1]) is the starting point of the solution.
Optional input arguments
#psiPid# (string) determines the type of the predictor that will be
used. It can take the values 'sph' (default) for the spherical
predictor, 'cyl' for the cylindrical predictor and 'ell' for the
ellipsoidal predictor (as described in [5]).
#psiCid# (string) determines the type of the corrector that will be
used. It can take the values 'sph' (default) for the spherical
corrector, 'cyl' for the cylindrical corrector and 'ell' for the
ellipsoidal corrector (as described in [5]).
#ninc# (scalar) is the maximum number of increments. Default value is
20.
#lambdaO# (scalar) is the initial value of load factor. Default value
is 1.
#Lbar# (scalar) is the arc radius. Default value is 1.
#maxit# (scalar) is the maximum number of iterations permitted for
each increment. Default value is 20.
#tol# (scalar) is the tolerance of the convergence criterion. It is
compared to norm(#R#). Default value is 1e-4.
#alpha# (scalar) is the constant controlling the distance of the
centre of the constraint surface from the last known equilibrium
point. Default value is 0.
#beta# (scalar) is the constant which controls the shape of the
ellipsoidal constraint surface. Default value is 1.
#Lbarmin# (scalar) is the minimum acceptable value of #Lbar#. Default
value is 0.
#Lbarmax# (scalar) is the maximum acceptable value of #Lbar#. Default
value is 1.
#Deltasmin# (scalar) is the minimum value of partial correction
permitted to avoid complex roots. Default value is 0.1.
#cutstep# (scalar) is the step length reducing factor. Default value
is 0.9.
Output parameters
#t_out# ([(#dim#+1) x #ninc#]) are the roots of the equation being
solved concatenated appropriately with the corresponding load factors
into generalized vectors as described in [1]
#SP_out# ([1 x #ninc#]) is the stiffness parameter of each increment.
#iter_out# ([1 x #ninc#]) is the number of iterations of each
increment.
Parents (calling functions)
None.
Children (called functions)
None.
__________________________________________________________________________
Copyright (c) 09-Mar-2014
George Papazafeiropoulos
First Lieutenant, Infrastructure Engineer, Hellenic Air Force
Civil Engineer, M.Sc., Ph.D. candidate, NTUA
Email: gpapazafeiropoulos@yahoo.gr
Website: http://users.ntua.gr/gpapazaf/
## Equations solved
The following equations are solved for and
## Function definitions
Two functions are utilized for the arc-length procedure:
The first function (, defined in the file function2.m ), needed to solve equation (1) is a cubic polynomial with the following properties:
• Function value:
• Function jacobian (derivative):
• Passes through the origin:
The second function (, defined in the file function1.m ), needed to solve equation (2) is a nonlinear smooth function with the following properties:
• Function value:
• Function jacobian:
## Function coding
• For function :
function [R,Q,K]=function2(t)
a=t(1:end-1);
lambda=t(end);
f1=a^3-57/8*a^2+51/4*a;
Rint=f1;
Rext=lambda*5;
% Out of balance force column vector (1-by-1)
R=Rint-Rext;
% Tangent force column vector (1-by-1)
Q=5;
% Jacobian matrix (1-by-1)
K=3*a^2-57/4*a+51/4;
end
• For function :
function [R,Q,K]=function1(t)
a=t(1:end-1);
lambda=t(end);
f1=a(1)^2+a(2)^2-49;
f2=a(1)*a(2)-24;
Rint=[f1;f2];
Rext=lambda*[1;1];
% Out of balance force column vector (2-by-1)
R=Rint-Rext;
% Tangent force column vector (2-by-1)
Q=[1;1];
% Jacobian matrix (2-by-2)
K=[2*a(1), 2*a(2);
a(2), a(1)];
end
## Initial definitions
In the subsequent code the following initial definitions are made (in the order presented below):
1. Define function
2. Define function
3. Set starting point () for solution of equation (1)
4. Set starting point () for solution of equation (2)
5. Set number of increments desired
6. Set initial value of load factor () for the solution of equation (1)
7. Set initial value of load factor () for the solution of equation (2)
8. Set arc radius for solution of equation (1)
9. Set arc radius for solution of equation (2) with the spherical-spherical arc-length method
10. Set arc radius for solution of equation (2) with the ellipsoidal-ellipsoidal arc-length method
11. Set maximum number of iterations permitted per increment
12. Set tolerance for convergence
13. Set constant controlling the distance of the centre of the constraint surface from the last known equilibrium point () for solution of equation (1) with the elliptical-elliptical arc-length method and solution of equation (2) with the spherical-spherical arc-length method
14. Set constant controlling the distance of the centre of the constraint surface from the last known equilibrium point () for solution of equation (2) with the ellipsoidal-ellipsoidal arc-length method
15. Set constant controlling the shape of the ellipsoidal constraint surface ()
16. Set minimum value for arc radius
17. Set maximum value for arc radius
18. Set minimum value of partial correction
19. Set step length reducing factor
functn1=@function2; %1
functn2=@function1; %2
aO1=0; %3
aO2=[4;6]; %4
ninc=10; %5
lambdaO1=0; %6
lambdaO2=1; %7
Lbar1=0.5; %8
Lbar2=1; %9
Lbar3=1.5; %10
maxit=20; %11
tol=5e-5; %12
alpha1=-0.5; %13
alpha2=0; %14
beta=1; %15
Lbarmin=0; %16
Lbarmax=1; %17
Deltasmin=0.1; %18
cutstep=0.9; %19
## Applications
1. Default application of the arc length control method as described by Ritto-Correa & Camotim (2008) to solve equation (1)
2. Non-default application of the arc length control method as described by Ritto-Correa & Camotim (2008) to solve equation (1)
3. Default application of the arc length control method as described by Ritto-Correa & Camotim (2008) to solve equation (2)
4. Non-default application of the arc length control method as described by Ritto-Correa & Camotim (2008) to solve equation (2) and plot of the results
5. Non-default application of the arc length control method as described by Ritto-Correa & Camotim (2008) to solve equation (2) and plot of the results
[t_out1,SP_out1,iter_out1] = arc_length(functn1,aO1); %1
Result1=[t_out1',iter_out1',SP_out1'] %1
[t_out2,SP_out2,iter_out2] = arc_length(functn1,aO1,...
'ell','ell',ninc,lambdaO1,Lbar1,maxit,tol,alpha1,beta,Lbarmin,Lbarmax,Deltasmin,cutstep); %2
Result2=[t_out2',iter_out2',SP_out2'] %2
[t_out3,SP_out3,iter_out3] = arc_length(functn2,aO2); %3
Result3=[t_out3',iter_out3',SP_out3'] %3
[t_out4,SP_out4,iter_out4] = arc_length(functn2,aO2,...
'sph','sph',ninc,lambdaO2,Lbar2,maxit,tol,alpha1,beta,Lbarmin,Lbarmax,Deltasmin,cutstep); %4
Result4=[t_out4',iter_out4',SP_out4'] %4
[t_out5,SP_out5,iter_out5] = arc_length(functn2,aO2,...
'ell','ell',ninc,lambdaO2,Lbar3,maxit,tol,alpha2,beta,Lbarmin,Lbarmax,Deltasmin,cutstep); %5
Result5=[t_out5',iter_out5',SP_out5'/10000] %5
Result1 =
0.9513 1.3084 3.0000
1.9196 1.0587 2.0000
2.6999 0.4334 2.0000
3.6214 0.0449 2.0000
4.4013 0.6709 3.0000
4.8194 1.5793 3.0000
5.1172 2.5339 2.0000
5.3553 3.5051 2.0000
5.5567 4.4846 2.0000
5.7330 5.4690 2.0000
5.8909 6.4564 2.0000
6.0345 7.4461 2.0000
6.1668 8.4373 2.0000
6.2897 9.4297 2.0000
6.4047 10.4231 2.0000
6.5131 11.4172 2.0000
6.6157 12.4119 1.0000
6.7132 13.4071 1.0000
6.8063 14.4028 1.0000
6.8953 15.3988 1.0000
Result2 =
0.2016 0.4577 2.0000 1.0000
0.4576 0.8877 2.0000 0.7973
0.7994 1.2300 2.0000 0.5938
1.2777 1.3490 2.0000 -0.2142
1.7486 1.1712 2.0000 -0.0923
2.1188 0.9080 2.0000 -0.2464
2.4295 0.6522 2.0000 -0.3158
2.7498 0.3955 4.0000 -0.3322
3.1104 0.1636 4.0000 -0.3145
3.5549 0.0417 2.0000 31.3806
Result3 =
4.2358 5.7599 0.5887 21.0000
4.0785 5.5435 0.4982 21.0000
4.0987 5.7049 0.4095 21.0000
4.2125 5.6412 0.3742 21.0000
4.1551 5.6876 0.3089 21.0000
4.0996 5.7327 0.2737 21.0000
4.0575 5.7649 0.2366 21.0000
4.0294 5.7845 0.1971 21.0000
4.0074 5.7995 0.1641 21.0000
3.9930 5.8083 0.1342 21.0000
3.9815 5.8151 0.1098 21.0000
3.9724 5.8205 0.0900 21.0000
3.9635 5.8261 0.0751 21.0000
3.9563 5.8306 0.0629 21.0000
3.9506 5.8342 0.0531 21.0000
3.9460 5.8371 0.0450 21.0000
3.9422 5.8394 0.0385 21.0000
3.9392 5.8412 0.0332 21.0000
3.9368 5.8427 0.0289 21.0000
3.9348 5.8439 0.0255 21.0000
Result4 =
4.9428 5.0553 0.9874 4.0000
5.4209 4.4310 0.0200 5.0000
5.5481 4.1575 -0.9339 2.0000
5.6292 3.9253 -1.9033 2.0000
5.6849 3.7152 -2.8794 2.0000
5.7234 3.5190 -3.8592 2.0000
5.7493 3.3324 -4.8413 2.0000
5.7651 3.1526 -5.8249 1.0000
5.7725 2.9780 -6.8095 1.0000
5.7728 2.8072 -7.7948 1.0000
Result5 =
5.0596 4.9382 0.9853 4.0000 0.0001
5.4265 4.4206 -0.0117 4.0000 0.4604
5.5751 4.0872 -1.2138 4.0000 1.0259
5.6722 3.7685 -2.6240 4.0000 1.0982
## Copyright
Copyright (c) 09-Mar-2014 by George Papazafeiropoulos | 2017-11-22 02:04:16 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8464968204498291, "perplexity": 6219.5432449455775}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806447.28/warc/CC-MAIN-20171122012409-20171122032409-00178.warc.gz"} |
https://simple.wikipedia.org/wiki/Formal_language | # Formal language
In mathematics, computer science and linguistics, a formal language is one that has a particular set of symbols, and whose expressions are made according to a particular set of rules. The symbol ${\displaystyle {\mathcal {L}}}$ is often used as a variable for formal languages in logic.[1]
Unlike natural languages, the symbols and formulas in formal languages are syntactically and semantically related to one another in a precise way.[2] As a result, formal languages are completely (or almost completely) void of ambiguity.[3]
## Examples
Some examples of formal languages include:
• The set of all words over ${\displaystyle {a,b}\,}$
• The set ${\displaystyle \left\{a^{n}\right\}}$, where ${\displaystyle n\,}$ is a natural number and ${\displaystyle a^{n}\,}$ means ${\displaystyle a\,}$ repeated ${\displaystyle n}$ times
• Finite languages, such as ${\displaystyle \{\{a,b\},\{a,aa,bba\}\}\,}$
• The set of syntactically correct programs in a given programming language
• The set of inputs upon which a certain Turing machine halts
## Specification
A formal language can be specified in a great variety of ways, such as:
## References
1. "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-10-09.
2. "Definition of formal language | Dictionary.com". www.dictionary.com. Retrieved 2020-10-09.
3. "1.11. Formal and Natural Languages — How to Think like a Computer Scientist: Interactive Edition". runestone.academy. Retrieved 2020-10-09.
• Helena Rasiowa and Roman Sikorski (1970). The Mathematics of Metamathematics (3rd ed. ed.). PWN. |edition= has extra text (help), chapter 6 Algebra of formalized languages. | 2021-12-04 21:25:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8740932941436768, "perplexity": 1623.8206098645262}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363006.60/warc/CC-MAIN-20211204185021-20211204215021-00183.warc.gz"} |
https://direct.mit.edu/tacl/article/doi/10.1162/tacl_a_00516/113853/Template-based-Abstractive-Microblog-Opinion | We introduce the task of microblog opinion summarization (MOS) and share a dataset of 3100 gold-standard opinion summaries to facilitate research in this domain. The dataset contains summaries of tweets spanning a 2-year period and covers more topics than any other public Twitter summarization dataset. Summaries are abstractive in nature and have been created by journalists skilled in summarizing news articles following a template separating factual information (main story) from author opinions. Our method differs from previous work on generating gold-standard summaries from social media, which usually involves selecting representative posts and thus favors extractive summarization models. To showcase the dataset’s utility and challenges, we benchmark a range of abstractive and extractive state-of-the-art summarization models and achieve good performance, with the former outperforming the latter. We also show that fine-tuning is necessary to improve performance and investigate the benefits of using different sample sizes.
Social media has gained prominence as a means for the public to exchange opinions on a broad range of topics. Furthermore, its social and temporal properties make it a rich resource for policy makers and organizations to track public opinion on a diverse range of issues (Procter et al., 2013; Chou et al., 2018; Kalimeri et al., 2019). However, understanding opinions about different issues and entities discussed in large volumes of posts in platforms such as Twitter is a difficult task. Existing work on Twitter employs extractive summarization (Inouye and Kalita, 2011; Zubiaga et al., 2012; Wang et al., 2017a; Jang and Allan, 2018) to filter through information by ranking and selecting tweets according to various criteria. However, this approach unavoidably ends up including incomplete or redundant information (Wang and Ling, 2016).
To tackle this challenge we introduce Microblog opinion summarization (MOS), which we define as a multi-document summarization task aimed at capturing diverse reactions and stances (opinions) of social media users on a topic. While here we apply our methods to Twitter data readily available to us, we note that this summarization strategy is also useful for other microblogging platforms. An example of a tweet cluster and its opinion summary is shown in Table 1. As shown, our proposed summary structure for MOS separates the factual information (story) from reactions to the story (opinions); the latter is further divided according to the prevalence of different opinions. We believe that making combined use of stance identification, sentiment analysis and abstractive summarization is a challenging but valuable direction in aggregating opinions expressed in microblogs.
Table 1:
Abridged cluster of tweets and its corresponding summary. Cluster content is color-coded to represent information overlap with each summary component: for Main Story, for Majority Opinion, and for Minority Opinion.
The availability of high quality news article datasets has meant that recent advances in text summarization have focused mostly on this type of data (Nallapati et al., 2016; Grusky et al., 2018; Fabbri et al., 2019; Gholipour Ghalandari et al., 2020). Contrary to news article summarization, our task focuses on summarizing an event as well as ensuing public opinions on social media. Review opinion summarization (Ganesan et al., 2010; Angelidis and Lapata, 2018) is related to MOS and faces the same challenge of filtering through large volumes of user-generated content. While recent work (Chu and Liu, 2019; Bražinskas et al., 2020) aims to produce review-like summaries that capture the consensus, MOS summaries inevitably include a spectrum of stances and reactions. In this paper we make the following contributions:
1. We introduce the task of microblog opinion summarization (MOS) and provide detailed guidelines.
2. We construct a corpus1 of tweet clusters and corresponding multi-document summaries produced by expert summarizers following our detailed guidelines.
3. We evaluate the performance of existing state-of-the-art models and baselines from three summarization domains (news articles, Twitter posts, product reviews) and four model types (abstractive vs. extractive, single document vs. multiple documents) on our corpus, showing the superiority of neural abstractive models. We also investigate the benefits of fine-tuning with various sample sizes.
##### Opinion Summarization
has focused predominantly on customer reviews with datasets spanning reviews on Tripadvisor (Ganesan et al., 2010), Rotten Tomatoes (Wang and Ling, 2016), Amazon (He and McAuley, 2016; Angelidis and Lapata, 2018) and Yelp (Yelp Dataset Challenge; Yelp).
Early work by Ganesan et al. (2010) prioritized redundancy control and concise summaries. More recent approaches (Angelidis and Lapata, 2018; Amplayo and Lapata, 2020; Angelidis et al., 2021; Isonuma et al., 2021) employ aspect driven models to create relevant topical summaries. While product reviews have a relatively fixed structure, MOS operates on microblog clusters where posts are more loosely related, which poses an additional challenge. Moreover, while the former generally only encodes the consensus opinion (Bražinskas et al., 2020; Chu and Liu, 2019), our approach includes both majority and minority opinions.
##### Multi-document summarization
has gained traction in non-opinion settings and for news events in particular. DUC (Dang, 2005) and TAC conferences pioneered this task by introducing datasets of 139 clusters of articles paired with multiple human-authored summaries. Recent work has seen the emergence of larger scale datasets such as WikiSum (Liu et al., 2018), Multi-News (Fabbri et al., 2019), and WCEP (Gholipour Ghalandari et al., 2020) to combat data sparsity. Extractive (Wang et al., 2020b, c; Liang et al., 2021) and abstractive (Jin et al., 2020) methods have followed from these multi-document news datasets.
is recognised by Cao et al. (2016) to be a promising direction for tracking reaction to major events. As tweets are inherently succinct and often opinionated (Mohammad et al., 2016), this task is at the intersection of multi-document and opinion summarization. The construction of datasets (Nguyen et al., 2018; Wang and Zhang, 2017) usually requires a clustering step to group tweets together under specific temporal and topical constraints, which we include within our own pipeline. Work by Jang and Allan (2018) and Corney et al. (2014) makes use of the subjective nature of tweets by identifying two stances for each topic to be summarized; we generalize this idea and do not impose a restriction on the number of possible opinions on a topic. The lack of an abstractive gold standard means that the majority of existing Twitter models are extractive (Alsaedi et al., 2021; Inouye and Kalita, 2011; Jang and Allan, 2018; Corney et al., 2014). Here we provide such an abstractive gold standard and show the potential of neural abstractive models for microblog opinion summarization.
### 3.1 Data Sources
Our MOS corpus consists of summaries of microblog posts originating from two data sources, both involving topics that have generated strong public opinion: COVID-19 (Chen et al., 2020) and UK Elections (Bilal et al., 2021).
• COVID-19: Chen et al. (2020) collected tweets by tracking COVID-19 related keywords (e.g., coronavirus, pandemic, stayathome) and accounts (e.g., @CDCemergency, @HHSGov, @DrTedros). We use data collected between January 2020 and January 2021, which at the time was the most complete version of this dataset.
• UK Elections: The Election dataset consists of all geo-located UK tweets posted between May 2014 and May 2016. The tweets were filtered using a list of 438 election- related keywords and 71 political party aliases curated by a team of journalists.
We follow the methodology in Bilal et al. (2021) to obtain opinionated, coherent clusters of between 20 and 50 tweets: The clustering step employs the GSDMM-LDA algorithm (Wang et al., 2017b), followed by thematic coherence evaluation (Bilal et al., 2021). The latter is done by aggregating exhaustive metrics BLEURT (Sellam et al., 2020), BERTScore (Zhang et al., 2020), and TF-IDF to construct a random forest classifier to identify coherent clusters. Our final corpus is created by randomly sampling 3100 clusters,2 1550 each from the COVID-19 and Election datasets.
### 3.2 Summary Creation
The summary creation process was carried out in 3 stages on the Figure Eight platform by 3 journalists experienced in sub-editing. Following Iskender et al. (2021), a short pilot study was followed by a meeting with the summarizers to ensure the task and guidelines were well understood. Prior to this, the design of the summarization interface was iterated to ensure functionality and usability (See pp1Appendix A for interface snapshots).
In the first stage, the summarizers were asked to read a cluster of tweets and state whether the opinions within it could be easily summarized by assigning one of three cluster types:
1. Coherent Opinionated: there are clear opinions about a common main story expressed in the cluster that can be easily summarized.
2. Coherent Non-opinionated: there are very few or no clear opinions in the cluster, but a main story is clearly evident and can be summarized.
3. Incoherent: no main story can be detected. This happens when the cluster contains diverse stories to which no majority of tweets refers, hence it cannot be summarized.
Following Bilal et al. (2021) on thematic coherence, we assume a cluster is coherent if and only if its contents can be summarized. Thus, both Coherent Opinionated and Coherent Non- opinionated can be summarized, but are distinct with respect to the level of subjectivity in the tweets, while Incoherent clusters cannot be summarized.
In the second stage, information nuggets are defined in a cluster as important pieces of information to aid in its summarization. The summarizers were asked to highlight information nuggets when available and categorise their aspect in terms of: WHAT, WHO, WHERE, REACTION, and OTHER. Thus, each information nugget is a pair consisting of the text and its aspect category (see pp1Appendix A for an example). Inspired by the pyramid evaluation framework (Nenkova and Passonneau, 2004) and extractive-abstractive two-stage models in the summarization literature (Lebanoff et al., 2018; Rudra et al., 2019; Liu et al., 2018), information nuggets have a dual purpose: (1) helping summarizers create the final summary and (2) constituting an extractive reference for summary informativeness evaluation (See 5.2.1).
In the third and final stage of the process, the summarizers were asked to write a short template-based summary for coherent clusters. Our chosen summary structure diverges from current summarization approaches that reconstruct the “most popular opinion” (Bražinskas et al., 2020; Angelidis et al., 2021). Instead, we aim to showcase a spectrum of diverse opinions regarding the same event. Thus, the summary template comprises three components: Main Story, Majority Opinion, Minority Opinion(s). The component Main Story serves to succinctly present the focus of the cluster (often an event), while the other components describe opinions about the main story. Here, we seek to distinguish the most popular opinion (Majority opinion) from ones expressed by a minority (Minority opinions). This structure is consistent with the work of Gerani et al. (2014) in template-based summarization for product reviews, which quantifies the popularity of user opinions in the final summary.
For “Coherent Opinionated clusters”, summarizers were asked to identify the majority opinion within the cluster and, if it exists, to summarize it, along with any minority opinions. If a majority opinion could not be detected, then the minority opinions were summarized. The final summary of “Coherent Opinionated clusters” is the concatenation of the three components: Main story + Majority Opinion (if any) + Minority Opinion(s) (if any). In 43% of opinionated clusters in our MOS corpus a majority opinion and at least one minority opinion were identified. Additionally, in 12% of opinionated clusters, 2 or more main opinions were identified (See Appendix C, Table 13), but without a majority opinion as there is a clear divide between user reactions. For clusters with few or no clear opinions (Coherent Non- opinionated), the final summary is represented by the Main Story component. Statistics regarding the annotation results are shown in Table 2.
Table 2:
Annotation statistics of our MOS corpus.
TotalCOVID-19Election
Size (#clusters) 3100 1550 1550
Coherent Opinionated 42% 41% 43%
Coherent Non-opinionated 30% 24% 37%
Incoherent 28% 35% 20%
TotalCOVID-19Election
Size (#clusters) 3100 1550 1550
Coherent Opinionated 42% 41% 43%
Coherent Non-opinionated 30% 24% 37%
Incoherent 28% 35% 20%
#### Agreement Analysis
Our tweet summarization corpus consists of 3100 clusters. Of these, a random sample of 100 clusters was shared among all three summarizers to compute agreement scores. Each then worked on 1000 clusters.
We obtain a Cohen’s Kappa score of κ = 0.46 for the first stage of the summary creation process, which involves categorising clusters as either Coherent Opinionated, Coherent Non-opinionated or Incoherent. Previous work (Feinstein and Cicchetti, 1990) highlights a paradox regarding Cohen’s kappa in that high levels of agreement do not translate to high kappa scores in cases of highly imbalanced datasets. In our data, at least 2 of the 3 summarizers agreed on the type of cluster in 97% of instances.
In addition, we evaluate whether the concept of ‘coherence/summarizability’ is uniformly assessed, that is, we check whether summarizers agree on what clusters can be summarized (Coherent clusters) and which clusters are too incoherent. We find that 83 out of 100 clusters were evaluated as coherent by the majority, of which 65 were evaluated as uniformly coherent by all.
ROUGE-1,2,L and BLEURT (Sellam et al., 2020) are used as proxy metrics to check the agreement in terms of summary similarity produced between the summarizers. We compare the consensus between the complete summaries as well as individual components such as the main story of the cluster, its majority opinion and any minority opinions in Table 3. The highest agreement is achieved for the Main Story, followed by Majority Opinion and Minority Opinions. These scores can be interpreted as upper thresholds for the lexical and semantic overlap later in Section 6.
Table 3:
Agreements between summarizers wrt to final summary, main story, majority opinion and minority opinions using ROUGE-1,2,L and BLEURT.
$R-1f1$$R-2f1$$R-Lf1$BLEURT
Summary 37.46 17.91 30.16 −.215
Main Story 35.15 12.98 34.59 −.324
Majority Opinion 27.53 6.15 25.95 −.497
Minority Opinion(s) 22.90 5.10 24.39 −.703
$R-1f1$$R-2f1$$R-Lf1$BLEURT
Summary 37.46 17.91 30.16 −.215
Main Story 35.15 12.98 34.59 −.324
Majority Opinion 27.53 6.15 25.95 −.497
Minority Opinion(s) 22.90 5.10 24.39 −.703
### 3.3 Comparison with Other Twitter Datasets
We next compare our corpus against the most recent and popular Twitter datasets for summarization in Table 4. To the best of our knowledge there are currently no abstractive summarization Twitter datasets for either event or opinion summarization. While we primarily focussed on the collection of opinionated clusters, some of the clusters we had automatically identified as opinionated were not deemed to be so by our annotators. Including the non-opinionated clusters helps expand the depth and range of Twitter datasets for summarization.
Table 4:
DatasetTime span#keywords#clustersAvg. Cluster SizeSummaryAvg. Summary Length
(#posts)(#tokens)
COVID-19 1 year 41 1003 31 Abstractive 42
Election 2 years 112 1236 30 Abstractive 36
Inouye and Kalita (20115 days 50 200 25 Extractive 17
SMERP (Ghosh et al., 20173 days N/A 359 Extractive 303
TSix (Nguyen et al., 201826 days 30 925 36 Extractive 109
DatasetTime span#keywords#clustersAvg. Cluster SizeSummaryAvg. Summary Length
(#posts)(#tokens)
COVID-19 1 year 41 1003 31 Abstractive 42
Election 2 years 112 1236 30 Abstractive 36
Inouye and Kalita (20115 days 50 200 25 Extractive 17
SMERP (Ghosh et al., 20173 days N/A 359 Extractive 303
TSix (Nguyen et al., 201826 days 30 925 36 Extractive 109
Compared to the summarization of product reviews and news articles, which has gained recognition in recent years because of the availability of large-scale datasets and supervised neural architectures, Twitter summarization remains a mostly uncharted domain with very few datasets curated. Inouye and Kalita (2011)3 collected the tweets for the top ten trending topics on Twitter for 5 days and manually clustered these. The SMERP dataset (Ghosh et al., 2017) focuses on topics on post-disaster relief operations for the 2016 earthquakes in central Italy. Finally, TSix (Nguyen et al., 2018) is the dataset most similar to our work as it covers, but on a smaller scale, several popular topics that are deemed relevant to news providers.
Other Twitter summarization datasets include: (Zubiaga et al., 2012; Corney et al., 2014) on summarization of football matches, (Olariu, 2014) on real-time summarization for Twitter streams. These datasets are either publicly unavailable or unsuitable for our summarization task.4
##### Summary Type.
These datasets exclusively contain extractive summaries, where several tweets are chosen as representative per cluster. This results in summaries which are often verbose, redundant and information-deficient. As shown in other domains (Grusky et al., 2018; Narayan et al., 2018), this may lead to bias towards extractive summarization techniques and hinder progress for abstractive models. Our corpus on COVID-19 and Election data aims to bridge this gap and introduces an abstractive gold standard generated by journalists experienced in sub-editing.
##### Size.
The average number of posts in our clusters is 30, which is similar to the TSix dataset and in line with the empirical findings by Inouye and Kalita (2011), who recommend 25 tweets/cluster. Having clusters with a much larger number of tweets makes it harder to apply our guidelines for human summarization. To the best of our knowledge, our combined corpus (COVID-19 and Election) is currently the biggest human-generated corpus for microblog summarization.
##### Time-span.
Both COVID-19 and Election partitions were collected across year-long time spans. This is in contrast to other datasets, which have been constructed in brief time windows, ranging from 3 days to a month. This emphasizes the longitudinal aspect of the dataset, which also allows topic diversity as 153 keywords and accounts were tracked through time.
As we introduce a novel summarization task (MOS), the baselines featured in our experiments are selected from domains tangential to microblog opinion summarization, such as news articles, Twitter posts, and product reviews (See Section 2). In addition, the selected models represent diverse summarization strategies: abstractive or extractive, supervised or unsupervised, multi-document (MDS) or single-document summarization (SDS). Note that most SDS models enforce a length limit (1024 characters) over the input, which makes it impossible to summarize the whole cluster of tweets. We address this issue by only considering the most relevant tweets ordered by topic relevance. The latter is computed using the Kullback- Leibler divergence with respect to the topical word distribution of the cluster in the GSDMM-LDA clustering algorithm (Wang et al., 2017b).
The summaries were generated such that their length matches the average length of the gold standard. Some model parameters (such as Lexrank) only allow sentence-level truncation, in which case the length matches the average number of sentences in the gold standard. For models that allow a word limit to the text to be generated (BART, Pegasus, T5), a minimum and maximum number of tokens was imposed such that the generated summary would be within [90%, 110%] of the gold standard length.
### 4.1 Heuristic Baselines
##### Extractive Oracle:
This baseline uses the gold summaries to extract the highest scoring sentences from a cluster of tweets. We follow Zhong et al. (2020) and rank each sentence by its average ROUGE-{1,2,L} recall scores. We then consider the highest ranking 5 sentences to form combinations of k5 sentences, which are re-evaluated against the gold summaries. k is chosen to equal the average number of sentences in the gold standard. The highest scoring summary with respect to the average ROUGE-{1,2,L} recall scores is assigned as the oracle.
##### Random:
k sentences are extracted at random from a tweet cluster. We report the mean result over 5 iterations with different random seeds.
### 4.2 Extractive Baselines
##### LexRank
(Erkan and Radev, 2004) constructs a weighed connectivity graph based on cosine similarities between sentence TF-IDF representations.
#### Hybrid TF-IDF
(Inouye and Kalita, 2011) is an unsupervised model designed for Twitter, where a post is summarized as the weighted mean of its TF-IDF word vectors.
#### BERTSumExt
(Liu and Lapata, 2019) is an SDS model comprising a BERT (Devlin et al., 2019)-based encoder stacked with Transformer layers to capture document-level features for sentence extraction. We use the model trained on CNN/Daily Mail (Hermann et al., 2015).
#### HeterDocSumGraph
(Wang et al., 2020b) introduces the heterogenous graph neural network, which is constructed and iteratively updated using both sentence nodes and nodes representing other semantic units, such as words. We use the MDS model trained on Multi-News (Fabbri et al., 2019).
#### Quantized Transformer
(Angelidis et al., 2021) combines Transformers (Vaswani et al., 2017) and Vector-Quantized Variational Autoencoders for the summarization of popular opinions in reviews. We trained QT on the MOS corpus.
#### Opinosis
(Ganesan et al., 2010) is an unsupervised MDS model. Its graph-based algorithm identifies valid paths in a word graph and returns the highest scoring path with respect to redundancy.
#### PG-MMR
(Lebanoff et al., 2018) adapts the single document setting for multi-documents by introducing ‘mega-documents’ resulting from concatenating clusters of texts. The model combines an abstractive SDS pointer-generator network with an MMR-based extractive component.
#### PEGASUS
(Zhang et al., 2020) introduces gap-sentences as a pre-training objective for summarization. It is then fine-tuned for 12 downstream summarization domains. We chose the model pre-trained on Reddit TIFU (Kim et al., 2019).
#### T5
(Raffel et al., 2020) adopts a unified approach for transfer learning on language- understanding tasks. For summarization, the model is pre-trained on the Colossal Clean Crawled Corpus (Raffel et al., 2020) and then fine-tuned on CNN/Daily Mail.
#### BART
(Lewis et al., 2020) is pre-trained on several evaluation tasks, including summarization. With a bidirectional encoder and GPT2, BART is considered a generalization of BERT. We use the BART model pre-trained on CNN/Daily Mail.
#### SummPip
(Zhao et al., 2020) is an MDS unsupervised model that constructs a sentence graph following Approximate Discourse Graph and deep embedding methods. After spectral clustering of the sentence graph, summary sentences are generated through a compression step of each cluster of sentences.
#### Copycat
(Bražinskas et al., 2020) is a Variational Autoencoder model trained in an unsupervised setting to capture the consensus opinion in product reviews for Yelp and Amazon. We train it on the MOS corpus.
Similar to other summarization work (Fabbri et al., 2019; Grusky et al., 2018), we perform both automatic and human evaluation of models. Automatic evaluation is conducted on a set of 200 clusters: Each partition of the test (COVID-19 Opinionated, COVID-19 Non-opinionated, Election Opinionated, Election Non-opinionated) contains 50 clusters uniformly sampled from the total corpus. For the human evaluation, only the 100 opinionated clusters are evaluated.
### 5.1 Automatic Evaluation
Word overlap is evaluated according to the harmonic mean F1 of ROUGE-1, 2, L6 (Lin, 2004) as reported elsewhere (Narayan et al., 2018; Gholipour Ghalandari et al., 2020; Zhang et al., 2020). Work by Tay et al. (2019) acknowledges the intractability of ROUGE in opinion text summarization as sentiment-rich language uses a vast vocabulary that does not rely on word matching. This issue is mitigated by Kryscinski et al. (2021) and Bhandari et al. (2020), who use semantic similarity as an additional assessment of candidate summaries. Similarly, we use text generation metrics BLEURT (Sellam et al., 2020) and BERTScore7 (Zhang et al., 2020) to assess semantic similarity.
### 5.2 Human Evaluation
Human evaluation is conducted to assess the quality of summaries with respect to three objectives: 1) linguistic quality, 2) informativeness, and 3) ability to identify opinions. We conducted two human evaluation experiments: the first (5.2.1) assesses the gold standard and non-fine-tuned model summaries on a rating scale, and the second (5.2.2) addresses the advantages and disadvantages of fine-tuned model summaries via Best-Worst Scaling. Four and three experts were employed for the two experiments, respectively.
#### 5.2.1 Evaluation of Gold Standard and Models
The first experiment focused on assessing the gold standard and best models from each summarization type: Gold, LexRank (best extractive), SummPip (best unsupervised abstractive), and BART (best supervised).
##### Linguistic quality
measures 4 syntactic dimensions, which were inspired by previous work on summary evaluation. Similar to DUC (Dang, 2005), each summary was evaluated with respect to each criterion below on a 5-point scale.
• Fluency (Grusky et al., 2018): Sentences in the summary “should have no formatting problems, capitalization errors or obviously ungrammatical sentences (e.g., fragments, missing components) that make the text difficult to read.”
• Sentential Coherence (Grusky et al., 2018): A sententially coherent summary should be well-structured and well-organized. The summary should not just be a heap of related information, but should build from sentence to sentence to a coherent body of information about a topic.
• Non-redundancy (Dang, 2005): A non- redundant summary should contain no duplication, that is, there should be no overlap of information between its sentences.
• Referential Clarity (Dang, 2005): It should be easy to identify who or what the pronouns and noun phrases in the summary are referring to. If a person or other entity is mentioned, it should be clear what their role is in the story.
##### Informativeness
is defined as the amount of factual information displayed by a summary. To measure this, we use a Question-Answer algorithm (Patil, 2020) as a proxy. Pairs of questions and corresponding answers are generated from the information nuggets of each cluster. Because we want to assess whether the summary contains factual information, only information nuggets belonging to the ‘WHAT’, ‘WHO’, ‘WHERE’ are selected as input. We chose not to include the entire cluster as input for the QA algorithm, as this might lead the informativeness evaluation to prioritize irrelevant details in the summary. Each cluster in the test set is assigned a question- answer pair and each system is then scored based on the percentage of times its generated summaries contain the answer to the question. Similar to factual consistency (Wang et al., 2020a), informativeness penalizes incorrect answers (hallucinations), as well as the lack of a correct answer in a summary.
As Opinion is a central component for our task, we want to assess the extent to which summaries contain opinions. Assessors report whether summaries identify any majority or minority opinions.8 A summary contains a majority opinion if most of its sentences express this opinion or if it contains specific terminology (‘The majority is/ Most users think...’, etc.), which is usually learned during the fine-tuning process. Similarly, a summary contains a minority opinion if at least one of its sentences expresses this opinion or it contains specific terminology (‘A minority/ A few users’, etc.). The final scores for each system are the percentage of times the summaries contain majority or minority opinions, respectively.
#### 5.2.2 Best-Worst Evaluation of Fine-tuned Models
The second human evaluation assesses the effects of fine-tuning on the best supervised model, BART. The experiments use non-fine-tuned BART (BART), BART fine-tuned on 10% of the corpus (BART FT10%) and BART fine-tuned on 70% of the corpus (BART FT70%).
As all the above are versions of the same neural model, Best-Worst scaling is chosen to detect subtle improvements, which cannot otherwise be quantified as reliably by traditional ranking scales (Kiritchenko and Mohammad, 2017). An evaluator is shown a tuple of 3 summaries (BART, BART FT70%, BART FT30%) and asked to choose the best/worst with respect to each criteria. To avoid any bias, the summary order is randomized for each document following van der Lee et al. (2019). The final score is calculated as the percentage of times a model is scored as the best, minus the percentage of times it was selected as the worst (Orme, 2009). In this setting, a score of 1 represents the unanimously best, while −1 is unanimously the worst.
The same criteria as before are used for linguistic quality and one new criterion is added to assess Opinion. We define Meaning Preservation as the extent to which opinions identified in the candidate summaries match the ones identified in the gold standard. We draw a parallel between the Faithfulness measure (Maynez et al., 2020), which assesses the level of hallucinated information present in summaries, and Meaning Preservation, which assesses the extent of hallucinated opinions.
### 6.1 Automatic Evaluation
Results for the automatic evaluation are shown in Table 5.
Table 5:
Performance on the test set of baseline models evaluated with automatic metrics: ROUGE-n (R-n) and BLEURT. The best model (Extractive, Abstractive, Fine-tuned) and are highlighted.
##### Fine-tuned Models
Unsurprisingly, the best performing models are ones that have been fine- tuned on our corpus: BART (FT70%) and BART (FT10%). Fine-tuning has been shown to yield competitive results for many domains (Kryscinski et al., 2021; Fabbri et al., 2021), including ours. In addition, one can see that only the fine-tuned abstractive models are capable of outperforming the Extractive Oracle, which is set as the upper threshold for extractive methods. Note that on average, the Oracle outperforms the Random summarizer by a 59% margin, which only fine-tuned models are able to improve on, with 112% for BART (FT10%) and 114% for BART (FT70%). We hypothesize that our gold summaries’ template format poses difficulties for off-the-shelf models and fine-tuning even on a limited portion of the corpus produces summaries that follow the correct structure (See Table 9 and Appendix C for examples). We include comparisons between the performance of BART (FT10%) and BART (FT70%) on the individual components of the summary in Table 6.9
Table 6:
Performance of fine-tuned models per each summary component (Main Story, Majority Opinion, Minority Opinion(s)) on the test set evaluated with automatic metrics: ROUGE-n (R-n) and BLEURT.
ModelsCOVID-19 Opinionated (CO)Election Opinionated (EO)
$R-1f1$$R-2f1$$R-Lf1$BLEURT$R-1f1$$R-2f1$$R-Lf1$BLEURT
Main Story
BART (FT 10%) 11.43 2.49 9.95 .082 9.82 1.72 8.31 −.185
BART (FT 70%) 11.18 2.29 9.57 −.137 9.55 1.70 8.19 .104
Majority Opinion
BART (FT 10%) 20.25 4.28 16.86 .487 17.88 3.11 14.57 −.442
BART (FT 70%) 19.74 4.06 16.18 −.505 19.13 3.74 15.60 .392
Minority Opinion(s)
BART (FT 10%) 19.05 4.66 15.87 .544 15.26 3.97 13.34 −.791
BART (FT 70%) 18.70 4.81 15.83 −.643 15.98 4.63 14.01 .604
ModelsCOVID-19 Opinionated (CO)Election Opinionated (EO)
$R-1f1$$R-2f1$$R-Lf1$BLEURT$R-1f1$$R-2f1$$R-Lf1$BLEURT
Main Story
BART (FT 10%) 11.43 2.49 9.95 .082 9.82 1.72 8.31 −.185
BART (FT 70%) 11.18 2.29 9.57 −.137 9.55 1.70 8.19 .104
Majority Opinion
BART (FT 10%) 20.25 4.28 16.86 .487 17.88 3.11 14.57 −.442
BART (FT 70%) 19.74 4.06 16.18 −.505 19.13 3.74 15.60 .392
Minority Opinion(s)
BART (FT 10%) 19.05 4.66 15.87 .544 15.26 3.97 13.34 −.791
BART (FT 70%) 18.70 4.81 15.83 −.643 15.98 4.63 14.01 .604
##### Non-Fine-tuned Models
Of these, SummPip performs the best across most metrics and datasets with an increase of 37% in performance over the random model, followed by LexRank with an increase of 29%. Both models are designed for the multi-document setting and benefit from the common strategy of mapping each sentence in a tweet from the cluster into a node of a sentence graph. However, not all graph mappings prove to be useful: Summaries produced by Opinosis and HeterDocSumGraph, which employ a word-to-node mapping, do not correlate well with the gold standard. The difference between word and sentence-level approaches can be partially attributed to the high amount of spelling variation in tweets, making the former less reliable than the latter.
##### ROUGE vs BLEURT
The performance on ROUGE and BLEURT is tightly linked to the data differences between COVID-19 and Election partitions of the corpus. Most models achieve higher ROUGE scores and lower BLEURTscores on the COVID-19 than on the Election dataset. An inspection of the data differences reveals that COVID-19 tweets are much longer than Election ones (169 vs 107 characters), as the latter had been collected before the increase in length limit from 140 to 280 characters in Twitter posts. This is in line with findings by Sun et al. (2019), who revealed that high ROUGE scores are mostly the result of longer summaries rather than better quality summaries.
### 6.2 Human Evaluation
##### Evaluation of Gold Standard and Models
Table 7 shows the comparison between the gold standard and the best performing models against a set of criteria (See 5.2.1). As expected, the human-authored summaries (Gold) achieve the highest scores with respect to all linguistic quality and structure-based criteria. However, the gold standard fails to capture informativeness as well as its automatic counterparts, which are, on average, longer and thus may include more information. Since BART is previously pre-trained on CNN/DM dataset of news articles, its output summaries are more fluent, sententially coherent and contain less duplication than the unsupervised models Lexrank and SummPip. We hypothesize that SummPip achieves high referential clarity and majority scores as a trade-off for its very low non-redundancy (high redundancy).
Table 7:
Evaluation of Gold Standard and Models: Results.
ModelFluencySentential CoherenceNon-redundancyReferential ClarityInformativenessMajorityMinority
Gold 4.52 4.63 4.85 4.31 57% 86% 64%
Lexrank 3.03 2.43 3.10 2.55 58% 15% 62
BART 3.24 2.76 3.46 3.01 67% 8% 60%
SummPip 2.73 2.70 2.53 3.37 693236%
ModelFluencySentential CoherenceNon-redundancyReferential ClarityInformativenessMajorityMinority
Gold 4.52 4.63 4.85 4.31 57% 86% 64%
Lexrank 3.03 2.43 3.10 2.55 58% 15% 62
BART 3.24 2.76 3.46 3.01 67% 8% 60%
SummPip 2.73 2.70 2.53 3.37 693236%
##### Best-Worst Evaluation of Fine-tuned Models
The results for our second human evaluation are shown in Table 8 using the guidelines presented in 5.2.2. The model fine-tuned on more data BART (FT70%) achieves the highest fluency and sentential coherence scores. As seen in Table 9, the summary produced by BART (FT70%) contains complete and fluent sentences, unlike its counterparts. Most importantly, fine-tuning yields better alignment with the gold standard with respect to meaning preservation, as the fine-tuned models BART (FT70%) and BART (FT10%) learn how to correctly identify and summarize the main story and the relevant opinions in a cluster of tweets. In the specific example, non-fine-tuned BART introduces a lot of irrelevant information (‘industrial air pollution’,‘google, apple rolling out covid’) to the main story and offers no insight into the opinions found in the cluster of tweets, whereas both fine-tuned models correctly introduce the Main Story and both partially identify the Majority Opinion (‘great idea’ for anti-maskers ‘to dig graves’). However, we note that the fine-tuning process does not lead to increased performance with respect to all criteria; non-redundancy is compromised and referential clarity stops improving after a certain amount of training data. As observed in the example, BART (FT70%) contains duplicated content ‘think this is a great idea. What a great idea!’. Wilcoxon signed rank tests with p < 0.05 and p < 0.10 are used for significance testing between all pairs of models. We note that most pairwise differences are significant at p < 0.05, while BART (FT70%) and BART NFT differences are significant at p < 0.10 for non-redundancy. The only two exceptions are referential clarity and non-redundancy between BART (FT70%) and BART (FT10%) where both fine-tuned models perform similarly.
Table 8:
Best-Worst Evaluation of Fine-tuned models: Results.
ModelFluencySentential CoherenceNon-redundancyReferential ClarityMeaning Preservation
BART −0.76 −0.65 0.15 −0.42 −0.54
BART FT 10% 0.30 0.22 −0.11 0.25 0.14
BART FT 70% 0.44 0.43 −0.04 0.17 0.40
ModelFluencySentential CoherenceNon-redundancyReferential ClarityMeaning Preservation
BART −0.76 −0.65 0.15 −0.42 −0.54
BART FT 10% 0.30 0.22 −0.11 0.25 0.14
BART FT 70% 0.44 0.43 −0.04 0.17 0.40
Table 9:
BART Summary Examples for the same cluster of tweets.
Human Summary: Anti-maskers are forced to dig graves for Covid19 victims in Indonesia. The majority of Twitter users think it is a good idea that will cause a reality check and make people rethink their ideas.
BART FT 70%: Anti-maskers in Indonesia are forced to dig graves for Covid-19 victims as punishment for not wearing mask. The majority think this is a great idea. What a great idea! It’s a good way to get people to reconsider their misinformed opinions. We should do this in the US.
BART FT 10%: Anti-maskers forced to dig graves for Covid-19 victims in Indonesia as punishment for refusing to wear a mask. The majority believe that this is a good idea, and that it will hopefully bring about an end to the need for masks. A minority do not believe it is necessary to wear a
BART: Covid-19 can be more deadly for people living in areas with industrial air pollution. Anyone refusing to wear a mask is made to dig graves for covid-19 victims as punishment in east java. as domestic violence spikes, many victims and their children have nowhere to live. google, apple rolling out covid-
Human Summary: Anti-maskers are forced to dig graves for Covid19 victims in Indonesia. The majority of Twitter users think it is a good idea that will cause a reality check and make people rethink their ideas.
BART FT 70%: Anti-maskers in Indonesia are forced to dig graves for Covid-19 victims as punishment for not wearing mask. The majority think this is a great idea. What a great idea! It’s a good way to get people to reconsider their misinformed opinions. We should do this in the US.
BART FT 10%: Anti-maskers forced to dig graves for Covid-19 victims in Indonesia as punishment for refusing to wear a mask. The majority believe that this is a good idea, and that it will hopefully bring about an end to the need for masks. A minority do not believe it is necessary to wear a
BART: Covid-19 can be more deadly for people living in areas with industrial air pollution. Anyone refusing to wear a mask is made to dig graves for covid-19 victims as punishment in east java. as domestic violence spikes, many victims and their children have nowhere to live. google, apple rolling out covid-
Error analysis is carried out on 30 fine-tuned BART summaries from a set of 15 randomly sampled clusters. The results are found in Table 10.
Table 10:
Error Analysis: Frequency of errors and representative summary examples for each error type.
Error typeFreq.Example
Intrinsic 4/30 Example 1
Hallucination Generated Summary: United States surpasses six million coronavirus cases and deaths and remains at the top of the global list of countries with the most cases and deaths The majority are pleased to see the US still leads the world in terms of cases and deaths, with 180,000 people succumbing to Covid-19.
Extrinsic 4/30 Example 2
Hallucination Generated Summary: Sex offender Rolf Harris is involved in a prison brawl after absconding from open jail. The majority think Rolf Harris deserves to be spat at and called a “nonce” and a “terrorist” for absconding from open prison. A minority are putting pressure on
Information 12/30 Example 3
Loss Human Summary: Miley Cyrus invited a homeless man on stage to accept her award. Most people thought it was a lovely thing to do and it was emotional. A minority think that it was a publicity stunt.
Generated Summary: Miley Cyrus had homeless man accept Video of the Year award at the MTV Video Music Awards. The majority think it was fair play for Miley Cyrus to allow the homeless man to accept the award on her behalf. She was emotional and selfless. The boy band singer cried and thanked him for accepting the
Error typeFreq.Example
Intrinsic 4/30 Example 1
Hallucination Generated Summary: United States surpasses six million coronavirus cases and deaths and remains at the top of the global list of countries with the most cases and deaths The majority are pleased to see the US still leads the world in terms of cases and deaths, with 180,000 people succumbing to Covid-19.
Extrinsic 4/30 Example 2
Hallucination Generated Summary: Sex offender Rolf Harris is involved in a prison brawl after absconding from open jail. The majority think Rolf Harris deserves to be spat at and called a “nonce” and a “terrorist” for absconding from open prison. A minority are putting pressure on
Information 12/30 Example 3
Loss Human Summary: Miley Cyrus invited a homeless man on stage to accept her award. Most people thought it was a lovely thing to do and it was emotional. A minority think that it was a publicity stunt.
Generated Summary: Miley Cyrus had homeless man accept Video of the Year award at the MTV Video Music Awards. The majority think it was fair play for Miley Cyrus to allow the homeless man to accept the award on her behalf. She was emotional and selfless. The boy band singer cried and thanked him for accepting the
##### Hallucination
Fine-tuning on the MOS corpus introduces hallucinated content in 8 out of 30 manually evaluated summaries. Generated summaries contain opinions that prove to be either false or unfounded after careful inspection of the cluster of tweets. We follow the work of Maynez et al. (2020) in classifying hallucinations as either intrinsic (incorrect synthesis of information in the source) or extrinsic (external information not found in the source). Example 1 in Table 10 is an instance of an intrinsic hallucination: The majority opinion is wrongly described as ‘pleased’, despite containing the correct facts regarding US coronavirus cases. Next, Example 2 shows that Rolf Harris ‘is called a terrorist’, which is confirmed to be an extrinsic hallucination as none of the tweets in the source cluster contain this information.
##### Information Loss
Information loss is the most frequent error type. As outlined in Kryscinski et al. (2021), the majority of current summarization models face length limitations (usually 1024 characters) which are detrimental for long-input documents and tasks. Since our task involves the detection of all opinions within the cluster, this weakness may lead to incomplete and less informative summaries, as illustrated in Example 3 from Table 10. The candidate summary does not contain the minority opinion identified by the experts in the gold standard. An inspection of the cluster of tweets reveals that most posts expressing this opinion are indeed not found in the first 1024-character allowed limit of the cluster input.
We have introduced the task of Twitter opinion summarization and constructed the first abstractive corpus for this domain, based on template- based human summaries. Our experiments show that existing extractive models fall short on linguistic quality and informativeness while abstractive models perform better but fail to identify all relevant opinions required by the task. Fine- tuning on our corpus boosts performance as the models learn the summary structure.
In the future, we plan to take advantage of the template-based structure of our summaries to refine fine-tuning strategies. One possibility is to exploit style-specific vocabulary during the generation step of model fine-tuning to improve on capturing opinions and other aspects of interest.
This work was supported by a UKRI/EPSRC Turing AI Fellowship to Maria Liakata (grant no. EP/V030302/1) and The Alan Turing Institute (grant no. EP/N510129/1) through project funding and its Enrichment PhD Scheme. We are grateful to our reviewers and action editor for reading our paper carefully and critically and thank them for their insightful comments and suggestions. We would also like to thank our annotators for their invaluable expertise in constructing the corpus and completing the evaluation tasks.
Ethics approval to collect and to publish extracts from social media datasets was sought and received from Warwick University Humanities & Social Sciences Research Ethics Committee. When the corpus will be released to the research community, only tweet IDs will be made available along with associated cluster membership and summaries. Compensation rates were agreed with the annotators before the annotation process was launched. Remuneration was fairly paid on an hourly rate at the end of task.
### Stage 1: Reading and choosing cluster type
The majority of the tweets in the cluster revolve around the subject of Trident nuclear submarines. The cluster contains many opinions which can be summarized easily, hence this cluster is Coherent Opinionated. Choose ‘Yes’ and proceed to the next step.
Figure 1:
Fragment of a cluster of tweets for keyword ‘nuclear’.
Figure 1:
Fragment of a cluster of tweets for keyword ‘nuclear’.
Close modal
Figure 2:
Choose type of cluster ‘Coherent Opinionated’.
Figure 2:
Choose type of cluster ‘Coherent Opinionated’.
Close modal
Figure 3:
Example of information nuggets: ‘a cornerstone of peace and security’ describes the nuclear submarine (WHAT information nugget), while ‘defence secretary Michael Fallon’ describes a person (WHO information nugget).
Figure 3:
Example of information nuggets: ‘a cornerstone of peace and security’ describes the nuclear submarine (WHAT information nugget), while ‘defence secretary Michael Fallon’ describes a person (WHO information nugget).
Close modal
### Stage 2: Highlighting information nuggets
Highlight important information and select the relevant aspect each information nugget belongs to.
### Stage 3: Template-based Summary Writing
Most user reactions dismiss the Trident plan and view it as an exaggerated security measure. This forms the Majority Opinion. A few users express fear for UK’s potential future in a nuclear war. This forms a Minority Opinion.
Figure 4:
Choose whether there exists a majority opinion in the cluster.
Figure 4:
Choose whether there exists a majority opinion in the cluster.
Close modal
Write cluster summary following the structure: Main Story + Majority Opinion (+ Minority Opinions).
### Model Implementation Details
T5, Pegasus, and BART were implemented using the HuggingFace Transformer package (Wolf et al., 2020) with max sequence length of 1024 characters.
Figure 5:
Summary template of a Coherent Opinionated cluster with a majority opinion.
Figure 5:
Summary template of a Coherent Opinionated cluster with a majority opinion.
Close modal
Table 11:
Performance on test set of baseline models evaluated with BERTScore.
ModelsCOVID-19COVID-19ElectionElection
OpinionatedNon-opinionatedOpinionatedNon-opinionated
Heuristics
Random Sentences 0.842 0.838 0.846 0.861
Extractive Oracle 0.858 0.867 0.871 0.904
Extractive Models
LexRank 0.851 0.849 0.856 0.868
Hybrid TF-IDF 0.851 0.853 0.856 0.879
BERTSumExt 0.848 0.851 0.859 0.874
HeterDocSumGraph 0.839 0.840 0.847 0.853
Quantized Transformer 0.840 0.827 0.850 0.856
Abstractive Models
Opinosis 0.845 0.853 0.846 0.860
PG-MMR 0.853 0.857 0.851 0.863
Pegasus 0.850 0.856 0.852 0.869
T5 0.850 0.851 0.853 0.872
BART 0.852 0.854 0.856 0.868
SummPip 0.852 0.858 0.854 0.878
Copycat 0.848 0.852 0.848 0.872
Fine-tuned Models
BART (FT 10%) 0.873 0.870 0.875 0.893
BART (FT 70%) 0.873 0.870 0.878 0.892
ModelsCOVID-19COVID-19ElectionElection
OpinionatedNon-opinionatedOpinionatedNon-opinionated
Heuristics
Random Sentences 0.842 0.838 0.846 0.861
Extractive Oracle 0.858 0.867 0.871 0.904
Extractive Models
LexRank 0.851 0.849 0.856 0.868
Hybrid TF-IDF 0.851 0.853 0.856 0.879
BERTSumExt 0.848 0.851 0.859 0.874
HeterDocSumGraph 0.839 0.840 0.847 0.853
Quantized Transformer 0.840 0.827 0.850 0.856
Abstractive Models
Opinosis 0.845 0.853 0.846 0.860
PG-MMR 0.853 0.857 0.851 0.863
Pegasus 0.850 0.856 0.852 0.869
T5 0.850 0.851 0.853 0.872
BART 0.852 0.854 0.856 0.868
SummPip 0.852 0.858 0.854 0.878
Copycat 0.848 0.852 0.848 0.872
Fine-tuned Models
BART (FT 10%) 0.873 0.870 0.875 0.893
BART (FT 70%) 0.873 0.870 0.878 0.892
Fine-tuning parameters for BART are: 8 batch size, 5 training epochs, 4 beams, enabled early stopping, 2 length penalty, and no trigram repetition for the summary generation. The rest of the parameters are set as default following the configuration of BartForConditionalGeneration: activation function gelu, vocabulary size 50265, 0.1 dropout, early stopping, 16 attention heads, 12 layers with feed forward layer dimension set as 4096 in both decoder and encoder. Quantized Transformer and Copycat models are trained for 5 epochs.
### Cluster examples and summaries from the MOS Corpus
Table 12:
Example of excerpt from tweet cluster “CDC”, human summary and best generated summary candidates.
Tweet cluster fragment for keyword “CDC”
Gosh i hope these cases are used for the negligent homicide class action suit that’s being constructed against trump. cdc warns against drinking hand sanitizer amid reports of deaths
the cdc has also declared, b̈eing stupid is hazardous to your health.ÜRLLINK
cdc warning! do not drink hand sanitizer! what the hell! people be idiots!
cdc warns against drinking hand sanitizer amid reports of deaths seriously omg?!
if the cdc has to put out a health bulletin to inform people not to try drinking hand sanitizers, how stupid are those people?
from the “if you had any doubt” department: the cdc is alerting your fellow americans not to drink hand sanitizer. obviously more than a couple of people have had to be treated for it. I wonder were they poisoned in the womb, too many concussions, mt. dew in their milk bottle when they were babies?
oh my...the cdc actually had to warn people not to drink hand sanitizer. only under a trump presidency have people acted so stupidly.
@realdonaldtrump you should try drinking the hand sanitizer. After your ridiculous suggestion to inject disinfectants, people have decided to drink it and are dying. CDC now issued a warning not to drink it. since u don’t believe anything the scientists say go ahead and drink it. First get kids out of cages
@USER i think this actually speaks more to the stupidity of the cdc.
@USER trump is in control of the cdc. don’t believe a single word that they are saying
this is sadly what happens when you put an idiot like @realdonaldtrump in the white house...people had seizures, lost vision and dead after drinking hand sanitizer, cdc warns URLLINK
@cdcgov @usfda @USER is it really necessary to tell people not to ingest hand sanitizer????? jesus
cdc gives out warning that people are going blind and are dying from drinking hand sanitizer. #darwinism URLLINK
Human Written Summary
Main Story: CDC warns people not to drink hand sanitiser after people fall ill and die after doing so. Majority Opinion: The majority are shocked by the stupidity of people thinking it’s OK to drink hand sanitiser. Minority Opinions: A minority blame the Trump administration for putting forward the idea of people disinfecting their insides. A minority think the warning is an act of stupidity by the CDC.
Lexrank Summary: cdc just issued a warning not to drink it. the cdc just issued an official warning to never drink hand sanitizer and i just. …people are having seizures, losing vision and dying after drinking hand sanitizer, cdc warns via @yahoo. if you need a warning from the cdc not to drink hand sanitizer…please go ahead and drink hand sanitizer.
SummPip Summary: people have decided to drink it and are dying after swallowing hand sanitizer …children masks . people are having seizures, losing vision and dying after drinking hand sanitizer, cdc warns via @yahoo .
BART Summary: People are getting sick and even dying after swallowing hand sanitizer, the us centers for disease control and prevention said wednesday. four died and others have suffered impaired vision or seizures, the cdc says. The cdc has also declared, “being stupid is hazardous to your health” @realdonaldtrump you
BART FT 10% Summary: People are dying and even dying after drinking hand sanitizer, the Centers for Disease Control warns. The majority are shocked and appalled by the CDC’s warning against people drinking hand Sanitizer. A minority believe the CDC is exaggerating the danger of drinking hand Sanitizinger to distract people from the
BART FT 70% Summary: CDC warns that people are dying and losing vision and blindness after drinking hand sanitiser. The majority believe that people are extremely stupid for even thinking about doing this, and it should not be allowed to happen. A minority do not believe the CDC’s warning and think it is not necessary to take any action
Tweet cluster fragment for keyword “CDC”
Gosh i hope these cases are used for the negligent homicide class action suit that’s being constructed against trump. cdc warns against drinking hand sanitizer amid reports of deaths
the cdc has also declared, b̈eing stupid is hazardous to your health.ÜRLLINK
cdc warning! do not drink hand sanitizer! what the hell! people be idiots!
cdc warns against drinking hand sanitizer amid reports of deaths seriously omg?!
if the cdc has to put out a health bulletin to inform people not to try drinking hand sanitizers, how stupid are those people?
from the “if you had any doubt” department: the cdc is alerting your fellow americans not to drink hand sanitizer. obviously more than a couple of people have had to be treated for it. I wonder were they poisoned in the womb, too many concussions, mt. dew in their milk bottle when they were babies?
oh my...the cdc actually had to warn people not to drink hand sanitizer. only under a trump presidency have people acted so stupidly.
@realdonaldtrump you should try drinking the hand sanitizer. After your ridiculous suggestion to inject disinfectants, people have decided to drink it and are dying. CDC now issued a warning not to drink it. since u don’t believe anything the scientists say go ahead and drink it. First get kids out of cages
@USER i think this actually speaks more to the stupidity of the cdc.
@USER trump is in control of the cdc. don’t believe a single word that they are saying
this is sadly what happens when you put an idiot like @realdonaldtrump in the white house...people had seizures, lost vision and dead after drinking hand sanitizer, cdc warns URLLINK
@cdcgov @usfda @USER is it really necessary to tell people not to ingest hand sanitizer????? jesus
cdc gives out warning that people are going blind and are dying from drinking hand sanitizer. #darwinism URLLINK
Human Written Summary
Main Story: CDC warns people not to drink hand sanitiser after people fall ill and die after doing so. Majority Opinion: The majority are shocked by the stupidity of people thinking it’s OK to drink hand sanitiser. Minority Opinions: A minority blame the Trump administration for putting forward the idea of people disinfecting their insides. A minority think the warning is an act of stupidity by the CDC.
Lexrank Summary: cdc just issued a warning not to drink it. the cdc just issued an official warning to never drink hand sanitizer and i just. …people are having seizures, losing vision and dying after drinking hand sanitizer, cdc warns via @yahoo. if you need a warning from the cdc not to drink hand sanitizer…please go ahead and drink hand sanitizer.
SummPip Summary: people have decided to drink it and are dying after swallowing hand sanitizer …children masks . people are having seizures, losing vision and dying after drinking hand sanitizer, cdc warns via @yahoo .
BART Summary: People are getting sick and even dying after swallowing hand sanitizer, the us centers for disease control and prevention said wednesday. four died and others have suffered impaired vision or seizures, the cdc says. The cdc has also declared, “being stupid is hazardous to your health” @realdonaldtrump you
BART FT 10% Summary: People are dying and even dying after drinking hand sanitizer, the Centers for Disease Control warns. The majority are shocked and appalled by the CDC’s warning against people drinking hand Sanitizer. A minority believe the CDC is exaggerating the danger of drinking hand Sanitizinger to distract people from the
BART FT 70% Summary: CDC warns that people are dying and losing vision and blindness after drinking hand sanitiser. The majority believe that people are extremely stupid for even thinking about doing this, and it should not be allowed to happen. A minority do not believe the CDC’s warning and think it is not necessary to take any action
Table 13:
Example of excerpt from tweet cluster “mental health”, human summary and best generated summary candidates.
Tweet cluster for keyword “mental health”
A ’landmark moment’? Nick Clegg (Lib Dems) promise to put mental health on par with physical #health URLLINK #inclusion #care
All of a sudden, Nick Clegg is concerned about people with mental health issues. Nothing at all to do with trying to win voters and save his job.
Delighted that nick is finally doing something about mental health in our nhs
Nick Clegg promises ’dignity and respect’ in NHS mental health treatment video URLLINK — Guardian
I have been hearing very positive noises on the radio today from Lib Dems re: mental health treatment. Certainly long overdue but great to hear!
But if you are patting Nick Clegg on the back for new mental health reforms, consider this:
Mate, Clegg could have stood up to Cameron before his harmful reductive mental health policies got implemented.
Awesome that Clegg highlighted mental health to rapturous applause, but sure he did that with tuition fees once.
.nickclegg speech #libdemconf focusing on mental health was cool. Araith Nick Clegg yn canolpwyntio ar iechyd meddyliol yn wych.
Nick Clegg’s pandering towards the treatment of mental health illness is kinda sad and pathetic#hecantbuyavote
One immediate victory of Clegg’s speech; putting mental health issues on the agenda and in the media. #ldconf #bbcnews
LibDems are back to promising the unachievable because they know they’re safe away from power. Shame because mental health is in dire state.
His position in government could have been used to stop the reductive mental health reforms Cameron put in years back. Did he? no.
Human Written Summary
Main Story: Nick Clegg promises to focus on mental health provision in the NHS. Minority Opinions: Some Twitter users are pleased something is ’finally’ being done about it and that it is great, it is highlighting mental health. Others are asking why he didn’t do it when he was in power and say that Clegg is doing it for personal gain.
Lexrank Summary: Nick Clegg promises ’dignity and respect’ in NHS mental health treatment video Speaking before his speech to the…Been hearing very positive noises on the radio today from Lib Dems re: mental health treatment. One immediate success of Clegg’s speech; getting mental health issues on the agenda and in the media. nickclegg a six week wait for mental health related treatment, but didn’t hear how you’ll resource the #NHS to achieve the needed care!
SummPip Summary: happy about nick clegg could have been used to stop the reductive mental health treatment . but if you are patting nick clegg is all of a sudden concerned about people with mental health issues . nick clegg promises ’ dignity and respect ’ in nhs mental health treatment video speaking before his speech to the …been hearing very positive noises on the radio today from lib dems re: mental health treatment .
BART Summary: Lib Dems promise to put mental health on par with physical health. Nick Clegg promises ’dignity and respect’ in NHS mental health treatment video. But if you are patting Nick Clegg on the back for new mental health reforms, consider this: Feeling blessed, trying to eradicate mental health stigma and getting lifetime opportunities
BART FT 10% Summary:Lib Dem Nick Clegg makes a speech about mental health in the NHS. The majority are pleased that the Lib Dem leader is trying to tackle the stigma attached to mental health. A minority are disappointed that he is pandering to the far right and anti-gay groups. A minority believe he is setting us up for a
BART FT 70% Summary: Lib Dem leader Nick Clegg makes a speech about putting mental health on a par with physical health in the manifesto. The majority are pleased that Nick Clegg is taking a lead on mental health and saying that mental health needs to be treated with dignity and respect. A minority are dismayed by Nick Clegg
Tweet cluster for keyword “mental health”
A ’landmark moment’? Nick Clegg (Lib Dems) promise to put mental health on par with physical #health URLLINK #inclusion #care
All of a sudden, Nick Clegg is concerned about people with mental health issues. Nothing at all to do with trying to win voters and save his job.
Delighted that nick is finally doing something about mental health in our nhs
Nick Clegg promises ’dignity and respect’ in NHS mental health treatment video URLLINK — Guardian
I have been hearing very positive noises on the radio today from Lib Dems re: mental health treatment. Certainly long overdue but great to hear!
But if you are patting Nick Clegg on the back for new mental health reforms, consider this:
Mate, Clegg could have stood up to Cameron before his harmful reductive mental health policies got implemented.
Awesome that Clegg highlighted mental health to rapturous applause, but sure he did that with tuition fees once.
.nickclegg speech #libdemconf focusing on mental health was cool. Araith Nick Clegg yn canolpwyntio ar iechyd meddyliol yn wych.
Nick Clegg’s pandering towards the treatment of mental health illness is kinda sad and pathetic#hecantbuyavote
One immediate victory of Clegg’s speech; putting mental health issues on the agenda and in the media. #ldconf #bbcnews
LibDems are back to promising the unachievable because they know they’re safe away from power. Shame because mental health is in dire state.
His position in government could have been used to stop the reductive mental health reforms Cameron put in years back. Did he? no.
Human Written Summary
Main Story: Nick Clegg promises to focus on mental health provision in the NHS. Minority Opinions: Some Twitter users are pleased something is ’finally’ being done about it and that it is great, it is highlighting mental health. Others are asking why he didn’t do it when he was in power and say that Clegg is doing it for personal gain.
Lexrank Summary: Nick Clegg promises ’dignity and respect’ in NHS mental health treatment video Speaking before his speech to the…Been hearing very positive noises on the radio today from Lib Dems re: mental health treatment. One immediate success of Clegg’s speech; getting mental health issues on the agenda and in the media. nickclegg a six week wait for mental health related treatment, but didn’t hear how you’ll resource the #NHS to achieve the needed care!
SummPip Summary: happy about nick clegg could have been used to stop the reductive mental health treatment . but if you are patting nick clegg is all of a sudden concerned about people with mental health issues . nick clegg promises ’ dignity and respect ’ in nhs mental health treatment video speaking before his speech to the …been hearing very positive noises on the radio today from lib dems re: mental health treatment .
BART Summary: Lib Dems promise to put mental health on par with physical health. Nick Clegg promises ’dignity and respect’ in NHS mental health treatment video. But if you are patting Nick Clegg on the back for new mental health reforms, consider this: Feeling blessed, trying to eradicate mental health stigma and getting lifetime opportunities
BART FT 10% Summary:Lib Dem Nick Clegg makes a speech about mental health in the NHS. The majority are pleased that the Lib Dem leader is trying to tackle the stigma attached to mental health. A minority are disappointed that he is pandering to the far right and anti-gay groups. A minority believe he is setting us up for a
BART FT 70% Summary: Lib Dem leader Nick Clegg makes a speech about putting mental health on a par with physical health in the manifesto. The majority are pleased that Nick Clegg is taking a lead on mental health and saying that mental health needs to be treated with dignity and respect. A minority are dismayed by Nick Clegg
2
Limited resources available for annotation determined the size of the MOS corpus.
3
It is unclear whether the full corpus is available: Our statistics were calculated based on a sample of 100 posts for each topic, but the original paper mentions that 1500 posts for each topic were initially collected.
4
Comparing to live stream summarization where millions of posts are used as input, we focus on summarization of clusters of maximum 50 posts.
5
For opinionated clusters, we set k=3 and for non- opinionated k=1.
6
We use ROUGE-1.5.5 via the pyrouge package: https://github.com/bheinzerling/pyrouge.
7
BERTScore has a narrow score range, which makes its interpretation more difficult than for BLEURT. Because both metrics produce similar rankings, BERTScore can be found in Appendix C.
8
Note that whether the identified minority or majority opinions are correct is not evaluated here. This is done in Section 5.2.2.
9
We do not include other models in the summary component-wise evaluation because it is impossible to identify the Main Story, Majority Opinion, and Minority Opinions in non-fine-tuned models.
Nasser
Alsaedi
,
Pete
Burnap
, and
Omer
Rana
.
2021
.
Automatic summarization of real world events using Twitter
.
Proceedings of the International AAAI Conference on Web and Social Media
,
10
(
1
):
511
514
.
Reinald Kim
Amplayo
and
Mirella
Lapata
.
2020
.
Unsupervised opinion summarization with noising and denoising
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
,
Online
.
Association for Computational Linguistics
.
Stefanos
Angelidis
,
Reinald Kim
Amplayo
,
Yoshihiko
Suhara
,
Xiaolan
Wang
, and
Mirella
Lapata
.
2021
.
Extractive opinion summarization in quantized transformer spaces
.
Transactions of the Association for Computational Linguistics
,
9
:
277
293
.
Stefanos
Angelidis
and
Mirella
Lapata
.
2018
.
Summarizing opinions: Aspect extraction meets sentiment prediction and they are both weakly supervised
. In
Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing
, pages
3675
3686
,
Brussels, Belgium
.
Association for Computational Linguistics
.
Manik
Bhandari
,
Pranav Narayan
Gour
,
Atabak
Ashfaq
,
Pengfei
Liu
, and
Graham
Neubig
.
2020
.
Re-evaluating evaluation in text summarization
. In
Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)
, pages
9347
9359
,
Online
.
Association for Computational Linguistics
.
Iman Munire
Bilal
,
Bo
Wang
,
Maria
Liakata
,
Rob
Procter
, and
Tsakalidis
.
2021
.
Evaluation of thematic coherence in microblogs
. In
Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)
, pages
6800
6814
,
Online
.
Association for Computational Linguistics
.
Arthur
Bražinskas
,
Mirella
Lapata
, and
Ivan
Titov
.
2020
.
Unsupervised opinion summarization as copycat-review generation
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
5151
5169
,
Online
.
Association for Computational Linguistics
.
Ziqiang
Cao
,
Chengyao
Chen
,
Wenjie
Li
,
Sujian
Li
,
Furu
Wei
, and
Ming
Zhou
.
2016
.
Tgsum: Build tweet guided multi-document summarization dataset
. In
Proceedings of the AAAI Conference on Artificial Intelligence
, volume
30
.
Emily
Chen
,
Kristina
Lerman
, and
Emilio
Ferrara
.
2020
.
Tracking social media discourse about the covid-19 pandemic: Development of a public coronavirus twitter data set
.
JMIR Public Health Surveill
,
6
(
2
):
e19273
. ,
[PubMed]
Wen-Ying Sylvia
Chou
,
April
Oh
, and
William MP
Klein
.
2018
.
Addressing health- related misinformation on social media
.
JAMA
,
320
(
23
):
2417
2418
. ,
[PubMed]
Eric
Chu
and
Peter J.
Liu
.
2019
.
Meansum: A neural model for unsupervised multi-document abstractive summarization
. In
ICML
.
D.
Corney
,
Carlos
Martin
, and
Ayse
Göker
.
2014
.
Two sides to every story: Subjective event summarization of sports events using twitter
. In
SoMuS@ICMR
.
Hoa Trang
Dang
.
2005
.
Overview of DUC 2005
. In
Proceedings of the Document Understanding Conf. Wksp. 2005 (DUC 2005) at the Human Language Technology Conf./Conf. on Empirical Methods in Natural Language Processing (HLT/EMNLP
.
Jacob
Devlin
,
Ming-Wei
Chang
,
Kenton
Lee
, and
Kristina
Toutanova
.
2019
.
BERT: Pre-training of deep bidirectional transformers for language understanding
. In
NAACL
.
Günes
Erkan
and
Dragomir R.
.
2004
.
Lexrank: Graph-based lexical centrality as salience in text summarization
.
Journal of Artificial Intelligence Research
,
22
(
1
):
457
479
.
Alexander
Fabbri
,
Simeng
Han
,
Haoyuan
Li
,
Haoran
Li
,
Marjan
,
Shafiq
Joty
,
Dragomir
, and
Yashar
.
2021
.
Improving zero and few-shot abstractive summarization with intermediate fine-tuning and data augmentation
. In
Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies
, pages
704
717
,
Online
.
Association for Computational Linguistics
.
Alexander
Fabbri
,
Irene
Li
,
Tianwei
She
,
Suyi
Li
, and
Dragomir
.
2019
.
Multi-news: A large-scale multi-document summarization dataset and abstractive hierarchical model
. In
Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics
, pages
1074
1084
,
Florence, Italy
.
Association for Computational Linguistics
.
Alvan R.
Feinstein
and
Dominic V.
Cicchetti
.
1990
.
High agreement but low kappa: I. the problems of two paradoxes.
Journal of Clinical Epidemiology
,
43
(
6
):
543
549
.
Kavita
Ganesan
,
ChengXiang
Zhai
, and
Jiawei
Han
.
2010
.
Opinosis: A graph based approach to abstractive summarization of highly redundant opinions
. In
Proceedings of the 23rd International Conference on Computational Linguistics (Coling 2010)
, pages
340
348
,
Beijing, China
.
Coling 2010 Organizing Committee
.
Shima
Gerani
,
Yashar
,
Giuseppe
Carenini
,
Raymond T.
Ng
, and
Bita
Nejat
.
2014
.
Abstractive summarization of product reviews using discourse structure
. In
Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)
, pages
1602
1613
,
Doha, Qatar
.
Association for Computational Linguistics
.
Demian Gholipour
Ghalandari
,
Chris
Hokamp
,
Nghia
The Pham
,
John
Glover
, and
Georgiana
Ifrim
.
2020
.
A large-scale multi-document summarization dataset from the Wikipedia current events portal
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
1302
1308
,
Online
.
Association for Computational Linguistics
.
Saptarshi
Ghosh
,
Kripabandhu
Ghosh
,
Tanmoy
Chakraborty
,
Debasis
Ganguly
,
Gareth
Jones
, and
Marie-Francine
Moens
.
2017
.
First International Workshop on Exploitation of Social Media for Emergency Relief and Preparedness (SMERP)
. In
Proceedings of the 39th European Conference on IR Research – J.M. Jose et al. (Eds.): ECIR 2017, LNCS 10193
,
ECIR 2017
, pages
779
783
.
Springer International Publishing AG
.
Max
Grusky
,
Mor
Naaman
, and
Yoav
Artzi
.
2018
.
Newsroom: A dataset of 1.3 million summaries with diverse extractive strategies
. In
Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)
, pages
708
719
,
New Orleans, Louisiana
.
Association for Computational Linguistics
.
Ruining
He
and
Julian
McAuley
.
2016
.
Ups and downs: Modeling the visual evolution of fashion trends with one-class collaborative filtering
. In
Proceedings of the 25th International Conference on World Wide Web
,
WWW ’16
, pages
507
517
,
Republic and Canton of Geneva, CHE
.
International World Wide Web Conferences Steering Committee
.
Karl Moritz
Hermann
,
Tomáš
Kočiský
,
Edward
Grefenstette
,
Lasse
Espeholt
,
Will
Kay
,
Mustafa
Suleyman
, and
Phil
Blunsom
.
2015
.
Teaching machines to read and comprehend
. In
Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1
,
NIPS’15
, pages
1693
1701
,
Cambridge, MA, USA
.
MIT Press
.
David
Inouye
and
Jugal K.
Kalita
.
2011
.
Comparing Twitter summarization algorithms for multiple post summaries
. In
2011 IEEE Third International Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third International Conference on Social Computing
, pages
298
306
.
Neslihan
Iskender
,
Tim
Polzehl
, and
Sebastian
Möller
.
2021
.
Reliability of human evaluation for text summarization: Lessons learned and challenges ahead
. In
Proceedings of the Workshop on Human Evaluation of NLP Systems (HumEval)
, pages
86
96
,
Online
.
Association for Computational Linguistics
.
Masaru
Isonuma
,
Junichiro
Mori
,
Danushka
Bollegala
, and
Ichiro
Sakata
.
2021
.
Unsupervised abstractive opinion summarization by generating sentences with tree-structured topic guidance
.
Myungha
Jang
and
James
Allan
.
2018
.
Explaining controversy on social media via stance summarization
. In
The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval
,
SIGIR ’18
, pages
1221
1224
,
New York, NY, USA
.
Association for Computing Machinery
.
Hanqi
Jin
,
Tianming
Wang
, and
Xiaojun
Wan
.
2020
.
Multi-granularity interaction network for extractive and abstractive multi-document summarization
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
6244
6254
,
Online
.
Association for Computational Linguistics
.
Kyriaki
Kalimeri
,
Mariano G.
Beiró
,
Alessandra
Urbinati
,
Andrea
Bonanomi
,
Alessandro
Rosina
, and
Ciro
Cattuto
.
2019
.
Human values and attitudes towards vaccination in social media
. In
Companion Proceedings of The 2019 World Wide Web Conference
, pages
248
254
.
Byeongchang
Kim
,
Hyunwoo
Kim
, and
Gunhee
Kim
.
2019
.
Abstractive summarization of Reddit posts with multi-level memory networks
. In
NAACL-HLT
.
Svetlana
Kiritchenko
and
Saif
.
2017
.
Best-worst scaling more reliable than rating scales: A case study on sentiment intensity annotation
. In
Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)
, pages
465
470
,
.
Association for Computational Linguistics
.
Wojciech
Kryscinski
,
Nazneen Fatema
Rajani
,
Divyansh
Agarwal
,
Caiming
Xiong
, and
Dragomir R.
.
2021
.
Booksum: A collection of datasets for long-form narrative summarization
.
CoRR
,
abs/2105.08209
.
Logan
Lebanoff
,
Kaiqiang
Song
, and
Fei
Liu
.
2018
.
Adapting the neural encoder-decoder framework from single to multi-document summarization
. In
Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing
, pages
4131
4141
,
Brussels, Belgium
.
Association for Computational Linguistics
.
Mike
Lewis
,
Yinhan
Liu
,
Naman
Goyal
,
Marjan
,
Abdelrahman
Mohamed
,
Omer
Levy
,
Veselin
Stoyanov
, and
Luke
Zettlemoyer
.
2020
.
BART: Denoising sequence- to-sequence pre-training for natural language generation, translation, and comprehension
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
7871
7880
,
Online
.
Association for Computational Linguistics
.
Xinnian
Liang
,
Shuangzhi
Wu
,
Mu
Li
, and
Zhoujun
Li
.
2021
.
Improving unsupervised extractive summarization with facet-aware modeling
. In
Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021
, pages
1685
1697
,
Online
.
Association for Computational Linguistics
.
Chin-Yew
Lin
.
2004
.
ROUGE: A package for automatic evaluation of summaries
. In
Text Summarization Branches Out
, pages
74
81
,
Barcelona, Spain
.
Association for Computational Linguistics
.
Peter J.
Liu
,
Saleh
,
E.
Pot
,
Ben
Goodrich
,
Ryan
Sepassi
,
Lukasz
Kaiser
, and
Noam M.
Shazeer
.
2018
.
Generating Wikipedia by summarizing long sequences
.
International Conference on Learning Representations
,
abs/1801.10198
.
Yang
Liu
and
Mirella
Lapata
.
2019
.
Text summarization with pretrained encoders
. In
Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)
, pages
3730
3740
,
Hong Kong, China
.
Association for Computational Linguistics
.
Joshua
Maynez
,
Shashi
Narayan
,
Bernd
Bohnet
, and
Ryan
McDonald
.
2020
.
On faithfulness and factuality in abstractive summarization
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
1906
1919
,
Online
.
Association for Computational Linguistics
.
Saif
,
Svetlana
Kiritchenko
,
Parinaz
Sobhani
,
Xiaodan
Zhu
, and
Colin
Cherry
.
2016
.
SemEval-2016 task 6: Detecting stance in tweets
. In
Proceedings of the 10th International Workshop on Semantic Evaluation (SemEval-2016)
, pages
31
41
,
San Diego, California
.
Association for Computational Linguistics
.
Ramesh
Nallapati
,
Bowen
Zhou
,
Cicero dos
Santos
,
Çağlar
Gulçehre
, and
Bing
Xiang
.
2016
.
Abstractive text summarization using sequence-to-sequence RNNs and beyond
. In
Proceedings of The 20th SIGNLL Conference on Computational Natural Language Learning
, pages
280
290
.
Shashi
Narayan
,
Shay B.
Cohen
, and
Mirella
Lapata
.
2018
.
Don’t give me the details, just the summary! Topic-aware convolutional neural networks for extreme summarization
. In
Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing
, pages
1797
1807
,
Brussels, Belgium
.
Association for Computational Linguistics
.
Ani
Nenkova
and
Rebecca J.
Passonneau
.
2004
.
Evaluating content selection in summarization: The pyramid method
. In
Proceedings of the Human Language Technology Conference of the North American Chapter of the Association for Computational Linguistics: HLT-NAACL 2004
, pages
145
152
.
Minh-Tien
Nguyen
,
Dac Viet
Lai
,
Huy-Tien
Nguyen
, and
Le-Minh
Nguyen
.
2018
.
TSix: A human-involved-creation dataset for tweet summarization
. In
Proceedings of the Eleventh International Conference on Language Resources and Evaluation (LREC 2018)
,
Miyazaki, Japan
.
European Language Resources Association (ELRA)
.
Andrei
Olariu
.
2014
.
Efficient online summarization of microblogging streams
. In
Proceedings of the 14th Conference of the European Chapter of the Association for Computational Linguistics, volume 2: Short Papers
, pages
236
240
,
Gothenburg, Sweden
.
Association for Computational Linguistics
.
Suraj
Patil
.
2020
.
Question generation
. https://github.com/patil-suraj/question_generation.
Rob
Procter
,
Jeremy
Crump
,
Susanne
Karstedt
,
Alex
Voss
, and
Marta
Cantijoch
.
2013
.
Policing and Society
,
23
(
4
):
413
436
.
Colin
Raffel
,
Noam
Shazeer
,
Roberts
,
Katherine
Lee
,
Sharan
Narang
,
Michael
Matena
,
Yanqi
Zhou
,
Wei
Li
, and
Peter J.
Liu
.
2020
.
Exploring the limits of transfer learning with a unified text-to-text transformer
.
Koustav
Rudra
,
Pawan
Goyal
,
Niloy
Ganguly
,
Imran
, and
Prasenjit
Mitra
.
2019
.
Summarizing situational tweets in crisis scenarios: An extractive-abstractive approach
.
IEEE Transactions on Computational Social Systems
,
6
(
5
):
981
993
.
Thibault
Sellam
,
Dipanjan
Das
, and
Ankur
Parikh
.
2020
.
BLEURT: Learning robust metrics for text generation
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
7881
7892
.
Association for Computational Linguistics
,
Online
.
Simeng
Sun
,
Ori
Shapira
,
Ido
Dagan
, and
Ani
Nenkova
.
2019
.
How to compare summarizers without target length? Pitfalls, solutions and re-examination of the neural summarization literature
. In
Proceedings of the Workshop on Methods for Optimizing and Evaluating Neural Language Generation
, pages
21
29
,
Minneapolis, Minnesota
.
Association for Computational Linguistics
.
Wenyi
Tay
,
Joshi
,
Xiuzhen
Zhang
,
Sarvnaz
Karimi
, and
Stephen
Wan
.
2019
.
Red-faced ROUGE: Examining the suitability of ROUGE for opinion summary evaluation
. In
Proceedings of the The 17th Annual Workshop of the Australasian Language Technology Association
, pages
52
60
,
Sydney, Australia
.
Australasian Language Technology Association
.
Chris
van der Lee
,
Albert
Gatt
,
Emiel
van Miltenburg
,
Sander
Wubben
, and
Emiel
Krahmer
.
2019
.
Best practices for the human evaluation of automatically generated text
. In
Proceedings of the 12th International Conference on Natural Language Generation
, pages
355
368
,
Tokyo, Japan
.
Association for Computational Linguistics
.
Ashish
Vaswani
,
Noam
Shazeer
,
Niki
Parmar
,
Jakob
Uszkoreit
,
Llion
Jones
,
Aidan N.
Gomez
,
Łukasz
Kaiser
, and
Illia
Polosukhin
.
2017
.
Attention is all you need
. In
Proceedings of the 31st International Conference on Neural Information Processing Systems
,
NIPS’17
, pages
6000
6010
,
Red Hook, NY, USA
.
Curran Associates Inc.
Alex
Wang
,
Kyunghyun
Cho
, and
Mike
Lewis
.
2020a
.
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
5008
5020
,
Online
.
Association for Computational Linguistics
.
Bo
Wang
,
Maria
Liakata
,
Tsakalidis
,
Spiros Georgakopoulos
Kolaitis
,
Symeon
,
Lazaros
Apostolidis
,
Arkaitz
Zubiaga
,
Rob
Procter
, and
Yiannis
Kompatsiaris
.
2017a
.
Totemss: Topic-based, temporal sentiment summarization for Twitter
. In
Proceedings of the IJCNLP 2017, System Demonstrations
, pages
21
24
.
Bo
Wang
,
Maria
Liakata
,
Arkaitz
Zubiaga
, and
Rob
Procter
.
2017b
.
A hierarchical topic modelling approach for tweet clustering
. In
International Conference on Social Informatics
, pages
378
390
.
Springer International Publishing
.
Danqing
Wang
,
Pengfei
Liu
,
Yining
Zheng
,
Xipeng
Qiu
, and
Xuanjing
Huang
.
2020b
.
Heterogeneous graph neural networks for extractive document summarization
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
6209
6219
,
Online
.
Association for Computational Linguistics
.
Kexiang
Wang
,
Baobao
Chang
, and
Zhifang
Sui
.
2020c
.
A spectral method for unsupervised multi-document summarization
. In
Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)
, pages
435
445
,
Online
.
Association for Computational Linguistics
.
Lu
Wang
and
Wang
Ling
.
2016
.
Neural network-based abstract generation for opinions and arguments
. In
Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies
, pages
47
57
,
San Diego, California
.
Association for Computational Linguistics
.
Zhongqing
Wang
and
Yue
Zhang
.
2017
.
A neural model for joint event detection and summarization
. In
Proceedings of the Twenty- Sixth International Joint Conference on Artificial Intelligence, IJCAI-17
, pages
4158
4164
.
Thomas
Wolf
,
Lysandre
Debut
,
Victor
Sanh
,
Julien
Chaumond
,
Clement
Delangue
,
Anthony
Moi
,
Pierric
Cistac
,
Tim
Rault
,
Remi
Louf
,
Morgan
Funtowicz
,
Joe
Davison
,
Sam
Shleifer
,
Patrick
von Platen
,
Clara
Ma
,
Yacine
Jernite
,
Julien
Plu
,
Canwen
Xu
,
Teven Le
Scao
,
Sylvain
Gugger
,
Mariama
Drame
,
Quentin
Lhoest
, and
Alexander
Rush
.
2020
.
Transformers: State-of-the-art natural language processing
. In
Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations
, pages
38
45
,
Online
.
Association for Computational Linguistics
.
Yelp
.
Yelp dataset challenge
. https://www.yelp.com/dataset.
Jingqing
Zhang
,
Yao
Zhao
,
Saleh
, and
Peter J.
Liu
.
2020
.
Pegasus: Pre-training with extracted gap-sentences for abstractive summarization
.
ICML
,
abs/1912.08777
.
Tianyi
Zhang
,
Varsha
Kishore
,
Felix
Wu
,
Kilian Q.
Weinberger
, and
Yoav
Artzi
.
2020
.
Bertscore: Evaluating text generation with BERT
. In
International Conference on Learning Representations
.
Jinming
Zhao
,
Ming
Liu
,
Longxiang
Gao
,
Yuan
Jin
,
Lan
Du
,
He
Zhao
,
He
Zhang
, and
Gholamreza
Haffari
.
2020
.
Summpip: Unsupervised multi-document summarization with sentence graph compression
. In
Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval
,
SIGIR ’20
, pages
1949
1952
,
New York, NY, USA
.
Association for Computing Machinery
.
Ming
Zhong
,
Pengfei
Liu
,
Yiran
Chen
,
Danqing
Wang
,
Xipeng
Qiu
, and
Xuanjing
Huang
.
2020
.
Extractive summarization as text matching
. In
Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics
, pages
6197
6208
,
Online
.
Association for Computational Linguistics
.
Arkaitz
Zubiaga
,
Damiano
Spina
,
Enrique
Amigó
, and
Julio
Gonzalo
.
2012
.
Towards real-time summarization of scheduled events from twitter streams
.
## Author notes
Action Editor: Ivan Titov
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a full description of the license, please visit https://creativecommons.org/licenses/by/4.0/legalcode. | 2023-01-31 18:37:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 18, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3101774752140045, "perplexity": 4710.1208253650775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499888.62/warc/CC-MAIN-20230131154832-20230131184832-00682.warc.gz"} |
https://www.semanticscholar.org/paper/Antipodal-symmetry-of-two-loop-MHV-amplitudes-Liu/f35c57665676d35a3f3b10220ea19981d87958b7 | # Antipodal symmetry of two-loop MHV amplitudes
@article{Liu2022AntipodalSO,
title={Antipodal symmetry of two-loop MHV amplitudes},
author={Yu-Ting Liu},
journal={Journal of High Energy Physics},
year={2022},
volume={2022}
}
• Yu-Ting Liu
• Published 24 July 2022
• Mathematics
• Journal of High Energy Physics
I present a conjecture that all two-loop MHV amplitudes in planar N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}$$\end{document} = 4 super-Yang-Mills theory possess an antipodal symmetry when evaluated on parity-even kinematics. The symmetry acts as a change of basis on the symbol letters, followed…
2 Citations
• Physics, Education
• 2022
We bootstrap the symbol of the maximal-helicity-violating four-particle form factor for the chiral part of the stress-tensor supermultiplet in planar N = 4 super-Yang-Mills theory at two loops. When
Homotopy Lie algebras are a generalization of differential graded Lie algebras encod-ing both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras
## References
SHOWING 1-10 OF 26 REFERENCES
• Physics
Journal of High Energy Physics
• 2022
We bootstrap the three-point form factor of the chiral stress-tensor multiplet in planar N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}
• Mathematics
• 2014
A bstractIn this paper we study motivic amplitudes — objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free
A bstractScattering amplitudes in superconformal field theories do not enjoy this symmetry, because the definition of asymptotic states involve a notion of infinity. Concentrating on planar \$
• Mathematics
Journal of High Energy Physics
• 2020
We study tropical Grassmanians Tr(k, n) in relation to cluster algebras, and assess their applicability to n-particle amplitudes for n = 7, 8. In N\documentclass[12pt]{minimal} \usepackage{amsmath}
• Mathematics
Journal of High Energy Physics
• 2019
A bstractWe review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N=4$$\mathcal{N}=4$$ supersymmetric Yang-Mills theory. In
• Physics
• 2005
We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 - 2{epsilon} dimensions, as a Laurent expansion about {epsilon} = 0 including the
• Mathematics
Journal of High Energy Physics
• 2021
Two-loop MHV amplitudes in planar N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}
• Mathematics
Journal of High Energy Physics
• 2021
We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}
• Mathematics
Journal of High Energy Physics
• 2012
We derive a set of first-order differential equations obeyed by the S-matrix of planar maximally supersymmetric Yang-Mills theory. The equations, based on the Yangian symmetry of the theory, involve | 2023-02-09 08:54:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7585521340370178, "perplexity": 4865.695677296532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00214.warc.gz"} |
http://www.physicspages.com/2016/12/06/ | # Canonical transformations: a few more examples
References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 2.7; Exercises 02.07.06 – 02.07.07, 02.07.08(4).
Here are a few more examples of canonical variable transformations.
Example 1 First, we revisit the two-body problem, in which we simplified the problem by transforming from the coordinates ${\mathbf{r}_{1}}$ and ${\mathbf{r}_{2}}$ of the masses ${m_{1}}$ and ${m_{2}}$ to two new position vectors:
$\displaystyle \mathbf{r}$ $\displaystyle \equiv$ $\displaystyle \mathbf{r}_{1}-\mathbf{r}_{2}\ \ \ \ \ (1)$ $\displaystyle \mathbf{r}_{CM}$ $\displaystyle \equiv$ $\displaystyle \frac{m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{2}}{M} \ \ \ \ \ (2)$
Here ${M\equiv m_{1}+m_{2}}$ is the total mass, ${\mathbf{r}}$ is the relative position, and ${\mathbf{r}_{CM}}$ is the position of the centre of mass. The conjugate momenta in the original system are
$\displaystyle \mathbf{p}_{i}=m\dot{\mathbf{r}}_{i} \ \ \ \ \ (3)$
The conjugate momenta transform according to
$\displaystyle \mathbf{p}_{CM}$ $\displaystyle =$ $\displaystyle M\mathbf{r}_{CM}=\mathbf{p}_{1}+\mathbf{p}_{2}\ \ \ \ \ (4)$ $\displaystyle \mathbf{p}$ $\displaystyle =$ $\displaystyle \mu\dot{\mathbf{r}}\ \ \ \ \ (5)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{m_{2}\mathbf{p}_{1}-m_{1}\mathbf{p}_{2}}{M} \ \ \ \ \ (6)$
where ${\mu=m_{1}m_{2}/M}$ is the reduced mass.
To check that this is a canonical transformation, we need to calculate the Poisson brackets. To make things easier, note that the new coordinates depend only on the old coordinates (and not on the momenta), and conversely, the new momenta depend only on the old momenta (and not on the coordinates). Since the Poisson brackets ${\left\{ \overline{q}_{i},\overline{q}_{j}\right\} }$and ${\left\{ \overline{p}_{i},\overline{p}_{j}\right\} }$all involve taking derivatives of coordinates with respect to momenta (in the first case) or momenta with respect to coordinates (in the second case), all these brackets are zero. We need, therefore, to check only the mixed brackets between coordinates and momenta.
Because we’re dealing with 3-d vector equations, there are 3 components to each vector and to be thorough, we need to calculate all possible brackets between all pairs of components. However, if we do the ${x}$ component of each, it should be obvious that the ${y}$ and ${z}$ components behave in the same way.
First, consider
$\displaystyle \left\{ r_{x},p_{x}\right\} =\sum_{i}\left(\frac{\partial r_{x}}{\partial q_{i}}\frac{\partial p_{x}}{\partial p_{i}}-\frac{\partial r_{x}}{\partial p_{i}}\frac{\partial p_{x}}{\partial q_{i}}\right) \ \ \ \ \ (7)$
In the RHS, the term ${q_{i}}$ stands for all 6 components of the original position vectors, that is ${q_{i}=\left\{ r_{1x},r_{1y},\ldots,r_{2z}\right\} }$ and the term ${p_{i}}$ in the denominators refers to all 6 components of the original momentum vectors. The ${p_{x}}$ in the numerators refers to the ${x}$ component of ${\mathbf{p}}$ in 6. Hopefully this won’t cause too much confusion.
The second term on the RHS is zero because it involves derivatives of coordinates with respect to momenta (and vice versa). In the first term, ${r_{x}}$ depends only the ${x}$ components of ${\mathbf{r}_{1}}$ and ${\mathbf{r}_{2}}$, and ${p_{x}}$ depends only on the ${x}$ components of ${\mathbf{p}_{1}}$and ${\mathbf{p}_{2}}$, so we have
$\displaystyle \left\{ r_{x},p_{x}\right\}$ $\displaystyle =$ $\displaystyle \frac{\partial r_{x}}{\partial r_{1x}}\frac{\partial p_{x}}{\partial p_{1x}}+\frac{\partial r_{x}}{\partial r_{2x}}\frac{\partial p_{x}}{\partial p_{2x}}\ \ \ \ \ (8)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(1\right)\frac{m_{2}}{M}+\left(-1\right)\left(-\frac{m_{1}}{M}\right)\ \ \ \ \ (9)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{m_{1}+m_{2}}{M}\ \ \ \ \ (10)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 1 \ \ \ \ \ (11)$
The same result is obtained for the ${y}$ and ${z}$ components. If we look at mixing two different components, we have, for example
$\displaystyle \left\{ r_{x},p_{y}\right\} =\frac{\partial r_{x}}{\partial r_{1x}}\frac{\partial p_{y}}{\partial p_{1x}}+\frac{\partial r_{x}}{\partial r_{2x}}\frac{\partial p_{y}}{\partial p_{2x}}+\frac{\partial r_{x}}{\partial r_{1y}}\frac{\partial p_{y}}{\partial p_{1y}}+\frac{\partial r_{x}}{\partial r_{2y}}\frac{\partial p_{y}}{\partial p_{2y}}=0 \ \ \ \ \ (12)$
This is zero because each term in the sum contains a derivative of an ${x}$ component with respect to a ${y}$ component (or vice versa), all of which are zero.
For the centre of mass components, we have
$\displaystyle \left\{ r_{CMx},p_{CMx}\right\}$ $\displaystyle =$ $\displaystyle \frac{\partial r_{CMx}}{\partial r_{1x}}\frac{\partial p_{CMx}}{\partial p_{1x}}+\frac{\partial r_{CMx}}{\partial r_{2x}}\frac{\partial p_{CMx}}{\partial p_{2x}}\ \ \ \ \ (13)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{m_{1}}{M}\left(1\right)+\frac{m_{2}}{M}\left(1\right)\ \ \ \ \ (14)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 1\ \ \ \ \ (15)$ $\displaystyle \left\{ r_{CMx},p_{CMy}\right\}$ $\displaystyle =$ $\displaystyle \frac{\partial r_{CMx}}{\partial r_{1x}}\frac{\partial p_{CMy}}{\partial p_{1x}}+\frac{\partial r_{CMx}}{\partial r_{2x}}\frac{\partial p_{CMy}}{\partial p_{2x}}+\frac{\partial r_{CMx}}{\partial r_{1y}}\frac{\partial p_{CMy}}{\partial p_{1y}}+\frac{\partial r_{CMx}}{\partial r_{2y}}\frac{\partial p_{CMy}}{\partial p_{2y}}\ \ \ \ \ (16)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 0 \ \ \ \ \ (17)$
where the last bracket is zero for the same reason as ${\left\{ r_{x},p_{y}\right\} }$: we’re mixing ${x}$ and ${y}$ in the derivatives. Again, it should be obvious that the brackets for the other combinations of ${x}$, ${y}$ and ${z}$ components work out the same way.
Example 2 A bizarre transformation of variables in one dimension is given by
$\displaystyle \overline{q}$ $\displaystyle =$ $\displaystyle \ln\frac{\sin p}{q}=\ln\sin p-\ln q\ \ \ \ \ (18)$ $\displaystyle \overline{p}$ $\displaystyle =$ $\displaystyle q\cot p \ \ \ \ \ (19)$
To show this is canonical, we need calculate only ${\left\{ \overline{q},\overline{p}\right\} }$ (since the Poisson bracket of a function with itself is always zero, we have ${\left\{ \overline{q},\overline{q}\right\} =\left\{ \overline{p},\overline{p}\right\} =0}$). We need one rather obscure derivative of a trig function.
$\displaystyle \frac{d}{dp}\cot p$ $\displaystyle =$ $\displaystyle \frac{d}{dp}\left(\frac{\cos p}{\sin p}\right)\ \ \ \ \ (20)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{-\sin^{2}p-\cos^{2}p}{\sin^{2}p}\ \ \ \ \ (21)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -1-\cot^{2}p \ \ \ \ \ (22)$
We get
$\displaystyle \left\{ \overline{q},\overline{p}\right\}$ $\displaystyle =$ $\displaystyle \frac{\partial\overline{q}}{\partial q}\frac{\partial\overline{p}}{\partial p}-\frac{\partial\overline{q}}{\partial p}\frac{\partial\overline{p}}{\partial q}\ \ \ \ \ (23)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(-\frac{1}{q}\right)\left(q\left(-1-\cot^{2}p\right)\right)-\frac{\cos p}{\sin p}\cot p\ \ \ \ \ (24)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 1+\cot^{2}p-\cot^{2}p\ \ \ \ \ (25)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 1 \ \ \ \ \ (26)$
Thus the transformation is canonical.
Example 3 Finally, we return to the point transformation, which is given in general by
$\displaystyle \overline{q}_{i}$ $\displaystyle =$ $\displaystyle \overline{q}_{i}\left(q_{1},\ldots,q_{n}\right)\ \ \ \ \ (27)$ $\displaystyle \overline{p}_{i}$ $\displaystyle =$ $\displaystyle \sum_{j}\frac{\partial q_{j}}{\partial\overline{q}_{i}}p_{j} \ \ \ \ \ (28)$
In this case, the coordinate transformation to ${\overline{q}}$ is completely arbitrary, but the momentum transformation must follow the formula given. The derivatives ${\frac{\partial q_{i}}{\partial\overline{q}_{j}}}$ in the formula for ${\overline{p}_{i}}$ are taken at constant ${\overline{q}}$. As in the earlier examples, since the coordinate formulas depend only on the old coordinates, and the momentum formulas depend only on the old momenta, the Poisson brackets satisfy
$\displaystyle \left\{ \overline{q}_{i},\overline{q}_{j}\right\} =\left\{ \overline{p}_{i},\overline{p}_{j}\right\} =0 \ \ \ \ \ (29)$
For the mixed brackets, we have
$\displaystyle \left\{ \overline{q}_{i},\overline{p}_{j}\right\}$ $\displaystyle =$ $\displaystyle \sum_{k}\left(\frac{\partial\overline{q}_{i}}{\partial q_{k}}\frac{\partial\overline{p}_{j}}{\partial p_{k}}-\frac{\partial\overline{q}_{i}}{\partial p_{k}}\frac{\partial\overline{p}_{j}}{\partial q_{k}}\right)\ \ \ \ \ (30)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{k}\frac{\partial\overline{q}_{i}}{\partial q_{k}}\frac{\partial q_{k}}{\partial\overline{q}_{j}}\ \ \ \ \ (31)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\partial\overline{q}_{i}}{\partial\overline{q}_{j}}\ \ \ \ \ (32)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \delta_{ij} \ \ \ \ \ (33)$
The second term in the first line is zero (mixed derivatives again). We used 28 to calculate the derivative ${\frac{\partial\overline{p}_{j}}{\partial p_{k}}}$ and get the second line and then notice that the sum is an expansion of the chain rule for the derivative in line 3. Since ${\overline{q}_{i}}$ and ${\overline{q}_{j}}$ are independent variables, the result is that given in the last line. Thus a point transformation is a canonical transformation. | 2017-06-28 19:05:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 137, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9502826929092407, "perplexity": 104.5570710831082}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323730.30/warc/CC-MAIN-20170628185804-20170628205804-00176.warc.gz"} |
http://theironmountain.com/ypfpm2/hvaua0.php?id=annuity-meaning-with-example-84374b | The annuity for ₹1 for 5 years at 6% interest is 0.237396. A Fixed Annuity is a personal retirement account in which the earnings are based on a fixed rate set by the insurance company. A Ltd. purchased a 5 years lease on 1 April 2013 for ₹500000. The same is true for the annuity payments. Homeowners Insurance: Protect Your Investment, Travel Insurance: Protection from Your Worst Trip Nightmares, How to Pick the Best Life Insurance Policy. Compound Savings Calculator: How Much Should I Save Each Year? Learn more. An annuity is a financial contract written by an insurance company that provides for a series of guaranteed payments, either for a specific period of time or for the lifetime of one or more individuals. How to Calculate a Monthly Loan Payment in Excel (Mortgage, Car Loan, and More), Why You Shouldn't Play It Safe In Your IRA. In fact, Excel has a function built into the program that calculates monthly... Increasing numbers of people are setting up self-directed Individual Retirement Accounts (IRAs), allowing them to call their own ... Sign up for our weekly newsletter and get our most popular content delivered straight to your inbox. For example, in case of NPS 40% of the total amount accrued needs to be mandatorily utilized for annuity purchase and cannot be withdrawn as a lump sum. Mrs Danielson is taking out a business loan requiring payments of $5000 at the beginning of each month for 12 months. Should You Sell Your Own Home or Use a Realtor? Let’s take a look at both of these examples. After they win, they often have to make the choice whether … Home » Accounting Dictionary » What is an Annuity? An annuity is a financial contract written by an insurance company that provides for a series of guaranteed payments, either for a specific period of time or for the lifetime of one or more individuals. Define Annuities: Annuity means a regular payment stream of equal amounts over a stated period. Times, Sunday Times (2014) Currently men get higher annuity incomes because they are … What makes an annuity fixed is that the insurance company promises that your money will earn a predetermined, fixed return per year for as long as you live. During the term of a guarantee your annuity income continues to be paid to your dependants after your death. Cancel anytime. Fixed annuities are susceptible to inflation risk due to the fact that there is no adjustment provided for runaway inflation. This guarantees that, should the investor die during the accumulation phase of the variable annuity, the account owner's beneficiary will receive at least the amount of the investor's contributions minus withdrawals or the current market value of the account. Annuity sentence examples. Let’s break it down to identify the meaning and value of the different variables in this problem. 6 of America's Most Expensive Summer Camps For Kids. The Best Stock To Profit From America's 'New Competitive Advantage', Simple Savings Calculator: See How You Can Grow Your Money, Calculate Cost of Monthly Used Car Loan Payments, Your Guide to Mortgages and Finding the Best Rates, Your Guide to Home Equity Loans and HELOC, 90,000 Reasons Why You've Got The Wrong Mortgage, Your FICO Score: 5 Things You Didn't Know Could Hurt It, 5 Secrets to Surviving the Mortgage Process, 5 Owner Financing Options for Home Buyers, 5 Devastating Mistakes That Turn 0% Credit Cards into Nightmares, Lower Your Credit Card Interest Rate with These Magic Words, How to Find a Personal Loan with the Best Rate, Using Leverage and Debt to Juice Your Investment Strategy, Good Debt: The 5 Best Reasons to Borrow Money, How to Create Your Own Loan Amortization Calculator, A Full Breakdown of Savings Accounts to Help You Find the Best Rates, Money Market Accounts & How to Find the Best Rates, Your Guide to CD Accounts and Finding the Best Rates, How to Find the Right Checking Account for You, How to Avoid Paying Bank Fees Once and For All, 7 Ways to Hack Proof Your Online Bank Account, How To Protect Your Assets When Your Bank Fails. “An annuity is a contract with an insurance company generally purchased for future income in retirement.” – Washington Post. This is because over time money should earn interest. Present Value of an Annuity Due Example. Under the terms of an annuity, however, the company makes its payments during the lifetime of the individual. Letâs take a look at both of these examples. If in our ordinary annuity example, if the payments were instead paid at the beginning of each period, then the future value of the payments would be: $$\text{FV}_{N}=\text{A}\left[\frac{\left(1+r\right)^{N}-1}{d}\right]=2000\left[\frac{\left(1.09\right)^{9}-1}{\frac{0.09}{109}}\right]=33,120.5868$$ Future Values of Unequal Series of Payments. Under the terms of a life insurance policy, the insurer will generally make a payment upon the death of the insured. Unfortunately, most people donât win it big, but an extremely small percentage of people do. How Many Years Will It Take to Save a Million Dollars? Example: An annuity of$400 a month for 5 years. Are you familiar with the S&P/TSX Venture Composite Index? Mortgage Calculator: What Will My Monthly Principal & Interest Payment Be? What is the definition of annuity? Thus, $600,000 today will equal$1,000,000 in the future after interest is added up over the years. Prepare the Lease A/c and the Profit & … Annuities are often obtained from a structured settlement of a personal injury lawsuit. For example, if you take out an annuity with a 10-year guarantee period and die after three years, the payments would continue for seven more years. What is the definition of present value annuity?An annuity is a financial instrument that provides regular payments to the holder each period until the end of the contract. An annuity is a series of payments made at equal intervals. Deferred Annuities offer benefit payments that begin at some future date. The annual loan rate is 12%. If the fixed annuity is at 8%, for example, the $175,000 earns 8% per year no matter what, and when it comes time to start receiving your$1,167 per month, the insurance company is obligated to pay 8% on the money remaining in the account. 6. Copyright © 2020 MyAccountingCourse.com | All Rights Reserved | Copyright |. On the other hand, lease rentals, corporate stock dividends are the examples of perpetuity. Search 2,000+ accounting terms and topics. Unlike an IRA, with an annuity there are no restrictions on the amount of the annual investment. Why the EAC Matters. This option takes the time value of money into consideration. Money Market vs Savings: Which Account is Best for You? 12 months a year, 5 years, that is 60 payments ... and a LOT of calculations. After they win, they often have to make the choice whether to be paid in a lump sum or in an annuity. We'll never sell or share your email address. An annuity with a guarantee period means your retirement income will be paid out for a specific number of years from the time you take out the policy, even if you die. In an effort to thwart boredom and to teach kids important skills in handicrafts and making friends, many parents consider the summer camp as a delightful alternative to a home-bound babysitter.... Those of us familiar with selling property know real estate agents don't come cheap. In addition, variable annuities offer the potential for greater returns and the opportunity for the investor to make his/her own decisions regarding how the assets are invested. … Traditional Annuities, pension payment, mortgage payments are some example for an annuity which will give uniform and predictable returns over a limited number of years. With real estate agent commission and fees amounting to as much as 6% of the selling price (that's $18,... Let me tell you, if you are a contrarian investor and looking for a place to hunt for bargains, this is it. Use a Monthly interest rate of 1%. EAC = NPV/A t, r where A= the present value of an annuity factor t = number of periods r = interest rate. An annuity is an investment contract made between an investor and an insurance company. They are not selected or validated by us and can contain inappropriate terms or ideas. Example: Future Value of an Annuity Due. It is decided to write off depreciation on lease using the Annuity Method. In other words, itâs a system of making or receiving payments where the payment amount and time period between payments is equal. annuity définition, signification, ce qu'est annuity: 1. a fixed amount of money paid to someone every year, usually until their death, or the insurance…. Calculate the present value of the annuity due. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Equivalent annual cost (EAC) is the annual cost of owning, operating, and maintaining an asset over its entire life. Many people play the lottery in hopes to cash in on the big jackpot. An annuity is similar to a life insurance product, but there are important differences between the two. Payment: Belongs to the period preceding its date. annuity. Loans are also set up as annuities. En savoir plus. In exchange for one or more payments, known as premiums, the insurance company agrees to make regular payments to the investor, either immediately or at some date … Firms often use EAC for capital budgeting decisions. Rude or colloquial translations are usually marked in red or orange. Annuity due is described as the series of cash flows occurring at the beginning of each period. Is This The Ultimate Value Investing Model? The present value of these payments is the amount that an investor would have to invest today at a given interest rate to equate to the total amount of payments in the future discounted by the same interest rate. Described as the name suggests, is designed to provide the subscriber with payouts at regular intervals offer benefit that... Should earn interest contingent annuities t = number of periods r = rate! Keep the terms of an annuity is similar to a life insurance.... Corporate stock dividends are the examples of perpetuity Danielson is taking out full..., operating, and compounding interest arrangement from a structured settlement of a your. Be edited or not to be 6 % p.a your Own home or Use a Realtor year, 5,. Of$ 5000 at the end of each period you translate the word or expression searched various. Unfortunately, most people donât think of them as annuities because they not...: which account is Best for you payments where the payment amount and time period payments... Their pension plan each year investment and loans are set up as annuities keep... Hopes to cash in on the other hand, lease rentals, corporate stock dividends are the examples perpetuity. The earnings are based on a tax-deferred basis in the interim annuity payments cease upon death. Eac is calculated by dividing the NPV of a guarantee your annuity income continues be... Not selected or validated by us and can contain inappropriate terms or ideas between. ' ĭ-tē, ə-nyo͝o'- the definition of an annuity is Buffett 's big Four '' Sleep-At-Night strategy the... Value tables are often obtained from a structured settlement of a personal injury lawsuit, operating and., itâs a system of making or receiving payments where the payment amount and time intervals How Much I... Another important feature of the recipient dollars or tax-deductible contributions would leave you free to spend other,! Or not to be 6 % p.a company makes its payments during the term of a your! Protection, or at any other regular interval of time leave you free to spend other assets, you. At age 60, you deposit $100,000 in a lump sum or in an annuity is an insurance that... This would leave you free to spend other assets, knowing you had a guaranteed stream income!$ 400 a month for 12 months a year, 5 years never sell or your. Market vs Savings: which account is Best for you at 6 % interest added. A stated period it agrees to make a payment made at equal during! ItâS actually relatively easy to calculate terms without a financial Calculator suppose at age 60 you. Examples to be edited or not to be paid in a lump sum or in an annuity meaning with example vary! Sell or share your email address payments in annuity meaning with example obtained from a structured settlement of a life insurance policy an. Email address the e… this is because over time money should earn interest annuity payments cease upon the death the... I typically explain annuities to the uninitiated is that prevent value and future tables! Age 85 classified by the frequency of payment dates should I Save each year Million! Colloquial translations are usually marked in red or orange depreciation on lease using annuity... In this problem dollars or tax-deductible contributions to your dependants after your death the years Danielson is out. Period of time payment terms, and maintaining an asset over its entire life, we will assume an... Unless the annuity contract specifies a beneficiary, most people donât think of them annuities! Buffett 's big Four '' Sleep-At-Night strategy you deposit $100,000 in a annuity. Life insurance company generally purchased for future income payments are important differences between two. Grows tax-deferred until the investor is ready to withdraw the assets will My monthly &... Way I typically explain annuities to keep the terms of a project by the insurance generally! Annuitiesand annuities due family protection, or death benefit, that often comes along with such.! For you amounts over annuity meaning with example stated period the annuity for ₹1 for years... Compound Savings Calculator: How Much interest will I Pay My Lender is! Offerâ benefit payments that begin at some future date to cash in on the agreement, terms. And future value tables are often needed to calculate terms without a financial Calculator its entire life annuity... Until the loan principle is paid at regular intervals at a fixed annuity is a retirement strategy. Will assume that an investor funds with either pre-tax dollars or tax-deductible contributions many people the... Age, with an insurance company generally purchased for future income in retirement. ” – Post! A full$ 1,000,000 My rehab was n't fully covered by insurance and most of the individual Principal & payment... Investment contract made between an investor funds with either pre-tax dollars or tax-deductible contributions,. Ira, with an insurance company rehab was n't fully covered by insurance and most of the variables. Annuity means a regular payment stream of equal amounts over a stated period would leave you to! Years will it take to Save a Million dollars EAC = NPV/A,. Into consideration deposits to a Savings account, monthly insurance payments and time period between is. Covered by insurance and most of the recipient money into consideration present contributions for future income retirement.. A tax-deferred basis in the future after interest is presumed to be displayed money Market Savings... Are susceptible to inflation risk due to the period preceding its date annuity I draw each month option... That prevent value and future value tables are often needed to calculate terms without a Calculator.: an annuity factor t = number of periods r = interest rate of the recipient or tax-deductible.. The different variables in this problem ə-nyo͝o'- the definition of an annuity is an insurance policy the. Identify the meaning and value of the individual company makes its payments the! You familiar with the s & P/TSX Venture Composite Index a contract an... '' Sleep-At-Night strategy we 'll never sell or share your email address is Best for you regular deposits to life... Principal & interest payment be payouts at regular intervals marked in red or orange be! Dollars or tax-deductible contributions Best for you a guaranteed stream of equal payments and pension payments to. Factor t = number of periods r = interest rate the family protection, or benefit. Of cash flows occurring at the beginning of each month for 12 months amount, we will assume that annuity! The e… this is a payment upon the death of the individual not selected or by... Regular payment stream of equal amounts over a stated period and value of money into consideration...! Each year annuity of $400 a month for 12 months age 60, you deposit$ 100,000 in lump! Is 0.237396 compounding interest arrangement remember annuities are susceptible annuity meaning with example inflation risk due to the preceding! The lifetime of the different variables in this problem main categorized: ordinary annuitiesand due! Find My mortgage Repayment Schedule the opposite of life insurance product, but an extremely small percentage of do... Cash in on the amount of life-long income starting at age 60, you deposit $100,000 in a sum... Them as annuities to keep the terms simple term of a guarantee your annuity income continues be! Payment dates often obtained from a structured settlement of a life insurance: annuities and! Beneficiary, most people donât win it big, but an extremely small percentage of people.... Will it take to Save a Million dollars involves a series of payments! Prevent value and future value tables are often obtained from a structured settlement of a your! The recipient from a structured settlement of a project by the frequency of payment.. That often comes along with such contracts the definition of an annuity I each... Periods r = interest rate it is decided to write off depreciation on lease using annuity... That exchanges present contributions for future income in retirement. ” – Washington Post that paid... Cost of owning, operating, and maintaining an asset over its life... Edited or not to be 6 % interest is added up over the years to in! Depreciation on lease using the annuity contract specifies a beneficiary, most people donât think of as!  most investment and loans are set up as annuities to the fact that there no... Depending on the agreement, payment terms, and compounding interest arrangement LOT of calculations may vary in dollar,! Amounts over a stated period for 12 months a year, 5 years that. Unfortunately, most annuity payments cease upon the death of the annual investment equal intervals the costs potential! ’ t win it big, but an extremely small percentage of people do over. Most annuity payments cease upon the death of the annual investment retirement account in which the investment grows tax-deferred the. Each period payment terms, an annuity I draw each month EAC calculated! Neither option actually pays out a business loan requiring payments of$ 5000 at the e… this because! Cash flows occurring at the beginning of each month support you later in life continues be! Or tax-deductible contributions opposite of life insurance policy, the insurer will make! The annuity for a surviving spouse vary in dollar amount, we will that... Compounding interest arrangement ’ t win it big, but there are no on... Is paid at regular intervals this is a contract with an immediate annuity for ₹1 for 5 years lease 1. Are susceptible to inflation risk due to the uninitiated is that prevent value and value... Of equal payments for example, suppose at age 60, you deposit \$ 100,000 in a sum! | 2021-06-13 00:06:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1726636290550232, "perplexity": 2570.4593582819443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586465.3/warc/CC-MAIN-20210612222407-20210613012407-00405.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-algebra/chapter-8-coordinate-geometry-and-linear-systems-8-3-slope-of-a-line-problem-set-8-3-page-349/16 | ## Elementary Algebra
Published by Cengage Learning
# Chapter 8 - Coordinate Geometry and Linear Systems - 8.3 - Slope of a Line - Problem Set 8.3: 16
#### Answer
$m=\text{undefined}$
#### Work Step by Step
Using $m=\dfrac{y_1-y_2}{x_1-x2}$ the slope of the line determined by the given points, $( -4,-5 ) \text{ and } ( -4,9 ),$ is \begin{array}{l}\require{cancel} m=\dfrac{-5-9}{-4-(-4)} \\\\ m=\dfrac{-5-9}{-4+4} \\\\ m=\dfrac{-14}{0} \\\\ m=\text{undefined} .\end{array}
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-08-18 05:27:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9342223405838013, "perplexity": 3829.853346217922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221213286.48/warc/CC-MAIN-20180818040558-20180818060558-00237.warc.gz"} |
https://search.r-project.org/R/refmans/base/html/Extract.html | Extract {base} R Documentation
Extract or Replace Parts of an Object
Description
Operators acting on vectors, matrices, arrays and lists to extract or replace parts.
Usage
x[i]
x[i, j, ... , drop = TRUE]
x[[i, exact = TRUE]]
x[[i, j, ..., exact = TRUE]]
x$name getElement(object, name) x[i] <- value x[i, j, ...] <- value x[[i]] <- value x$name <- value
Arguments
x, object object from which to extract element(s) or in which to replace element(s). i, j, ... indices specifying elements to extract or replace. Indices are numeric or character vectors or empty (missing) or NULL. Numeric values are coerced to integer as by as.integer (and hence truncated towards zero). Character vectors will be matched to the names of the object (or for matrices/arrays, the dimnames): see ‘Character indices’ below for further details. For [-indexing only: i, j, ... can be logical vectors, indicating elements/slices to select. Such vectors are recycled if necessary to match the corresponding extent. i, j, ... can also be negative integers, indicating elements/slices to leave out of the selection. When indexing arrays by [ a single argument i can be a matrix with as many columns as there are dimensions of x; the result is then a vector with elements corresponding to the sets of indices in each row of i. An index value of NULL is treated as if it were integer(0). name A literal character string or a name (possibly backtick quoted). For extraction, this is normally (see under ‘Environments’) partially matched to the names of the object. drop For matrices and arrays. If TRUE the result is coerced to the lowest possible dimension (see the examples). This only works for extracting elements, not for the replacement. See drop for further details. exact Controls possible partial matching of [[ when extracting by a character vector (for most objects, but see under ‘Environments’). The default is no partial matching. Value NA allows partial matching but issues a warning when it occurs. Value FALSE allows partial matching without any warning. value typically an array-like R object of a similar class as x.
Details
These operators are generic. You can write methods to handle indexing of specific classes of objects, see InternalMethods as well as [.data.frame and [.factor. The descriptions here apply only to the default methods. Note that separate methods are required for the replacement functions [<-, [[<- and $<- for use when indexing occurs on the assignment side of an expression. The most important distinction between [, [[ and $ is that the [ can select more than one element whereas the other two select a single element.
The default methods work somewhat differently for atomic vectors, matrices/arrays and for recursive (list-like, see is.recursive) objects. $ is only valid for recursive objects (and NULL), and is only discussed in the section below on recursive objects. Subsetting (except by an empty index) will drop all attributes except names, dim and dimnames. Indexing can occur on the right-hand-side of an expression for extraction, or on the left-hand-side for replacement. When an index expression appears on the left side of an assignment (known as subassignment) then that part of x is set to the value of the right hand side of the assignment. In this case no partial matching of character indices is done, and the left-hand-side is coerced as needed to accept the values. For vectors, the answer will be of the higher of the types of x and value in the hierarchy raw < logical < integer < double < complex < character < list < expression. Attributes are preserved (although names, dim and dimnames will be adjusted suitably). Subassignment is done sequentially, so if an index is specified more than once the latest assigned value for an index will result. It is an error to apply any of these operators to an object which is not subsettable (e.g., a function). Atomic vectors The usual form of indexing is [. [[ can be used to select a single element dropping names, whereas [ keeps them, e.g., in c(abc = 123)[1]. The index object i can be numeric, logical, character or empty. Indexing by factors is allowed and is equivalent to indexing by the numeric codes (see factor) and not by the character values which are printed (for which use [as.character(i)]). An empty index selects all values: this is most often used to replace all the entries but keep the attributes. Matrices and arrays Matrices and arrays are vectors with a dimension attribute and so all the vector forms of indexing can be used with a single index. The result will be an unnamed vector unless x is one-dimensional when it will be a one-dimensional array. The most common form of indexing a k-dimensional array is to specify k indices to [. As for vector indexing, the indices can be numeric, logical, character, empty or even factor. And again, indexing by factors is equivalent to indexing by the numeric codes, see ‘Atomic vectors’ above. An empty index (a comma separated blank) indicates that all entries in that dimension are selected. The argument drop applies to this form of indexing. A third form of indexing is via a numeric matrix with the one column for each dimension: each row of the index matrix then selects a single element of the array, and the result is a vector. Negative indices are not allowed in the index matrix. NA and zero values are allowed: rows of an index matrix containing a zero are ignored, whereas rows containing an NA produce an NA in the result. Indexing via a character matrix with one column per dimensions is also supported if the array has dimension names. As with numeric matrix indexing, each row of the index matrix selects a single element of the array. Indices are matched against the appropriate dimension names. NA is allowed and will produce an NA in the result. Unmatched indices as well as the empty string ("") are not allowed and will result in an error. A vector obtained by matrix indexing will be unnamed unless x is one-dimensional when the row names (if any) will be indexed to provide names for the result. Recursive (list-like) objects Indexing by [ is similar to atomic vectors and selects a list of the specified element(s). Both [[ and $ select a single element of the list. The main difference is that $ does not allow computed indices, whereas [[ does. x$name is equivalent to x[["name", exact = FALSE]]. Also, the partial matching behavior of [[ can be controlled using the exact argument.
getElement(x, name) is a version of x[[name, exact = TRUE]] which for formally classed (S4) objects returns slot(x, name), hence providing access to even more general list-like objects.
[ and [[ are sometimes applied to other recursive objects such as calls and expressions. Pairlists (such as calls) are coerced to lists for extraction by [, but all three operators can be used for replacement.
[[ can be applied recursively to lists, so that if the single index i is a vector of length p, alist[[i]] is equivalent to alist[[i1]]...[[ip]] providing all but the final indexing results in a list.
Note that in all three kinds of replacement, a value of NULL deletes the corresponding item of the list. To set entries to NULL, you need x[i] <- list(NULL).
When $<- is applied to a NULL x, it first coerces x to list(). This is what also happens with [[<- where in R versions less than 4.y.z, a length one value resulted in a length one (atomic) vector. Environments Both $ and [[ can be applied to environments. Only character indices are allowed and no partial matching is done. The semantics of these operations are those of get(i, env = x, inherits = FALSE). If no match is found then NULL is returned. The replacement versions, $<- and [[<-, can also be used. Again, only character arguments are allowed. The semantics in this case are those of assign(i, value, env = x, inherits = FALSE). Such an assignment will either create a new binding or change the existing binding in x. NAs in indexing When extracting, a numerical, logical or character NA index picks an unknown element and so returns NA in the corresponding element of a logical, integer, numeric, complex or character result, and NULL for a list. (It returns 00 for a raw result.) When replacing (that is using indexing on the lhs of an assignment) NA does not select any element to be replaced. As there is ambiguity as to whether an element of the rhs should be used or not, this is only allowed if the rhs value is of length one (so the two interpretations would have the same outcome). (The documented behaviour of S was that an NA replacement index ‘goes nowhere’ but uses up an element of value: Becker et al p. 359. However, that has not been true of other implementations.) Argument matching Note that these operations do not match their index arguments in the standard way: argument names are ignored and positional matching only is used. So m[j = 2, i = 1] is equivalent to m[2, 1] and not to m[1, 2]. This may not be true for methods defined for them; for example it is not true for the data.frame methods described in [.data.frame which warn if i or j is named and have undocumented behaviour in that case. To avoid confusion, do not name index arguments (but drop and exact must be named). S4 methods These operators are also implicit S4 generics, but as primitives, S4 methods will be dispatched only on S4 objects x. The implicit generics for the $ and $<- operators do not have name in their signature because the grammar only allows symbols or string constants for the name argument. Character indices Character indices can in some circumstances be partially matched (see pmatch) to the names or dimnames of the object being subsetted (but never for subassignment). Unlike S (Becker et al p. 358), R never uses partial matching when extracting by [, and partial matching is not by default used by [[ (see argument exact). Thus the default behaviour is to use partial matching only when extracting from recursive objects (except environments) by $. Even in that case, warnings can be switched on by options(warnPartialMatchDollar = TRUE).
Neither empty ("") nor NA indices match any names, not even empty nor missing names. If any object has no names or appropriate dimnames, they are taken as all "" and so match nothing.
Error conditions
Attempting to apply a subsetting operation to objects for which this is not possible signals an error of class notSubsettableError. The object component of the error condition contains the non-subsettable object.
Subscript out of bounds errors are signaled as errors of class subscriptOutOfBoundsError. The object component of the error condition contains the object being subsetted. The integer subscript component is zero for vector subscripting, and for multiple subscripts indicates which subscript was out of bounds. The index component contains the erroneous index.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
names for details of matching to names, and pmatch for partial matching.
list, array, matrix.
[.data.frame and [.factor for the behaviour when applied to data.frame and factors.
Syntax for operator precedence, and the ‘R Language Definition’ manual about indexing details.
NULL for details of indexing null objects.
Examples
x <- 1:12
m <- matrix(1:6, nrow = 2, dimnames = list(c("a", "b"), LETTERS[1:3]))
li <- list(pi = pi, e = exp(1))
x[10] # the tenth element of x
x <- x[-1] # delete the 1st element of x
m[1,] # the first row of matrix m
m[1, , drop = FALSE] # is a 1-row matrix
m[,c(TRUE,FALSE,TRUE)]# logical indexing
m[cbind(c(1,2,1),3:1)]# matrix numeric index
ci <- cbind(c("a", "b", "a"), c("A", "C", "B"))
m[ci] # matrix character index
m <- m[,-1] # delete the first column of m
li[[1]] # the first element of list li
y <- list(1, 2, a = 4, 5)
y[c(3, 4)] # a list containing elements 3 and 4 of y
y$a # the element of y named a ## non-integer indices are truncated: (i <- 3.999999999) # "4" is printed (1:5)[i] # 3 ## named atomic vectors, compare "[" and "[[" : nx <- c(Abc = 123, pi = pi) nx[1] ; nx["pi"] # keeps names, whereas "[[" does not: nx[[1]] ; nx[["pi"]] ## recursive indexing into lists z <- list(a = list(b = 9, c = "hello"), d = 1:5) unlist(z) z[[c(1, 2)]] z[[c(1, 2, 1)]] # both "hello" z[[c("a", "b")]] <- "new" unlist(z) ## check$ and [[ for environments
e1 <- new.env()
e1$a <- 10 e1[["a"]] e1[["b"]] <- 20 e1$b
ls(e1)
## partial matching - possibly with warning :
stopifnot(identical(li$p, pi)) op <- options(warnPartialMatchDollar = TRUE) stopifnot( identical(li$p, pi), #-- a warning
inherits(tryCatch (li\$p, warning = identity), "warning"))
## revert the warning option:
if(is.null(op[[1]])) op[[1]] <- FALSE; options(op)
[Package base version 4.2.0 Index] | 2022-05-19 15:00:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4594196677207947, "perplexity": 3801.10573831437}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662529538.2/warc/CC-MAIN-20220519141152-20220519171152-00623.warc.gz"} |
http://openstudy.com/updates/55fdf2dde4b0b395cadc2b00 | ## anonymous one year ago Express the complex number in trigonometric form. -3 + 3 square root of three i
1. jim_thompson5910
I'm assuming the expression given is $$\Large -3+3\sqrt{3}*i$$ If so, then it's the same as last time but now $\Large a = -3$ $\Large b = 3\sqrt{3}$
2. anonymous
yeah, that is the correct expression
3. Plasmataco
quick question... what is the square root of i?
4. Plasmataco
I know... im a little slow with the noggin but still...
5. jim_thompson5910
the i isn't part of the square root
6. Plasmataco
i know, just curios.
7. anonymous
-1
8. Plasmataco
...
9. Plasmataco
i thought that was the square... exponent 2.
10. Plasmataco
-1*-1=1 not i.
11. jim_thompson5910
$\Large i = \sqrt{-1}$ $\Large i^2 = (\sqrt{-1})^2$ $\Large i^2 = -1$
12. anonymous
$\iota ^{2} = -1$
13. Plasmataco
am i just the stupid idiot here or u genuinely dont know or connection problemos?
14. Plasmataco
15. jim_thompson5910
sorry @Plasmataco I'm not following
16. Plasmataco
i know but like -1 under a radical of 4
17. jim_thompson5910
were you able to find r and theta, @lxoser ?
18. Plasmataco
...
19. jim_thompson5910
oh you mean $\Large \sqrt{i} = \sqrt[4]{-1}$ @Plasmataco ??
20. Plasmataco
$\sqrt[4]{-1}$
21. Plasmataco
yeah
22. anonymous
im gonna need help finding r and theta @jim_thompson5910
23. Plasmataco
sry for interrupting...
24. anonymous
its okay
25. jim_thompson5910
$\Large r = \sqrt{a^2 + b^2}$ $\Large r = \sqrt{3^2 + (3\sqrt{3})^2}$ ... ... ... $\Large r = ??$
26. anonymous
r = 30?
27. Plasmataco
dont think so.
28. Plasmataco
simplify.
29. Plasmataco
it should be somthing a lot lower
30. jim_thompson5910
Hint: $\Large (3\sqrt{3})^2=(3\sqrt{3})*(3\sqrt{3})$ $\Large (3\sqrt{3})^2=(3*3)*(\sqrt{3}*\sqrt{3})$ $\Large (3\sqrt{3})^2=(3*3)*\sqrt{3*3}$ $\Large (3\sqrt{3})^2=9\sqrt{9}$ $\Large (3\sqrt{3})^2=9*3$ $\Large (3\sqrt{3})^2=27$
31. anonymous
r = 6
32. jim_thompson5910
yes
33. Plasmataco
yup.
34. Plasmataco
horay! now my question.
35. anonymous
for theta, i got -60?
36. jim_thompson5910
incorrect
37. anonymous
is the result suppose to be in radians or degrees?
38. jim_thompson5910
it depends on what the instructions say
39. jim_thompson5910
does it say which mode they want?
40. anonymous
no, but these are the answer choices
41. jim_thompson5910
ok so they want radian form
42. anonymous
so would theta be - pi/3
43. jim_thompson5910
yes, now because our point is in Q2 (see graph) we add pi radians to our angle to land in the right quadrant |dw:1442707471746:dw|
44. jim_thompson5910
- pi/3 is in Q4 |dw:1442707591362:dw|
45. jim_thompson5910
so we have to add 180 degrees or pi radians to get it into the right quadrant |dw:1442707637525:dw|
46. anonymous
2pi/3?
47. anonymous
@jim_thompson5910
48. jim_thompson5910
correct
49. anonymous
so would the answer be 6(cos 2pi/3 + i sin 2pi/3) ?
50. jim_thompson5910
yes
51. anonymous
thank you so much!! :")
52. jim_thompson5910
no problem | 2016-10-27 15:40:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8038841485977173, "perplexity": 14187.056019417261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721347.98/warc/CC-MAIN-20161020183841-00257-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3062086/proof-by-induction-inductive-step-problem | # Proof by induction, inductive step problem
Prove by induction:
$$\sum_{i=0}^n 3^i =\frac {1}{2} (3^{n+1}-1)$$
Now, i know how to do the first step and i understand it but then i have a problem with the second step which is showing that its true for n+1.
My question is:
Is this notation corect:
$$\sum_{i=0}^{n+1} 3^i =\frac {1}{2} (3^{n+2}-1)$$ Which of these is corect and why? $$\sum_{i=0}^{n+1} 3^i =\sum_{i=0}^{n} 3^i+n+1$$ or $$\sum_{i=0}^{n+1} 3^i =\sum_{i=0}^{n} 3^{n+1}$$
Neither of those is correct, but the first one is closer. What you want to say is $$\sum_{i=0}^{n+1}3^{i}=\sum_{i=0}^{n}3^{i}+3^{n+1}$$ since $$3^{n+1}$$ is the extra term missing from the sum on the right hand side. To complete the proof, the induction hypothesis implies that the right hand side is $$\frac{1}{2}(3^{n+1}-1)+3^{n+1}=\frac{3}{2}\cdot3^{n+1}-\frac{1}{2}=\frac{1}{2}\cdot 3^{n+2}-\frac{1}{2}=\frac{1}{2}(3^{n+2}-1)$$
Notation looks good.
$$\sum_{i=0}^0{3^i}=3^0=1$$ $$\frac12(3^1-1)=\frac12(2)=1$$ so true for $$n=0$$
Inductive Step:
Assume the statement true for $$n=k$$, that is:
$$\sum_{i=0}^k{3^i}=\frac12(3^{k+1}-1)$$ Then show for $$n=k+1$$. We use that:
$$\sum_{i=0}^{k+1}{3^i}=\sum_{i=0}^{k}{3^i}+3^{k+1}$$ This is the correct expansion of that sum as opposed to what you suggested. We achieve this by finding the $$(k+1)$$th term of the sum and adding it to the summation to $$k$$ terms.
So you need to show that:
$$\frac12(3^{k+1}-1)+3^{k+1}=\frac12(3^{k+2}-1)$$ | 2019-12-07 09:17:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9243256449699402, "perplexity": 97.38254523951392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540497022.38/warc/CC-MAIN-20191207082632-20191207110632-00112.warc.gz"} |
https://www.mathdoubts.com/integral-rules/ | # Integration Rules
### Properties
$\displaystyle \int udv$ $\,=\,$ $uv$ $-$ $\displaystyle \int vdu$
### Formulas
There are some standard results formed by some functions in integral calculus
#### Integration of Algebraic functions
$(1)\,\,\,$ $\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}}{n+1}+c$
$(2)\,\,\,$ $\displaystyle \int{\dfrac{1}{x}\,}dx$ $\,=\,$ $\log_{e}{x}+c$
$(3)\,\,\,$ $\displaystyle \int{a^x\,}dx$ $\,=\,$ $\dfrac{a^x}{\log_{e}{a}}+c$
$(4)\,\,\,$ $\displaystyle \int{e^x\,}dx$ $\,=\,$ $e^x+c$
$(5)\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2-a^2}\,}dx$ $\,=\,$ $\dfrac{1}{2a}\log_{e}{\Bigg|\dfrac{x-a}{x+a}\Bigg|}+c$
$(6)\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2+a^2}\,}dx$ $\,=\,$ $\dfrac{1}{a}\tan^{-1}{\Big(\dfrac{x}{a}\Big)}+c$
#### Integration of Trigonometric functions
$\Large \int \normalsize \sin{x} dx = -\cos{x}+c$
$\Large \int \normalsize \cos{x} dx = \sin{x}+c$
$\Large \int \normalsize \tan{x} dx = -\log{(\cos{x})}+c$
$\Large \int \normalsize \cot{x} dx = \log{(\sin{x})}+c$
$\Large \int \normalsize \sec^2{x} dx = \tan{x}+c$
$\Large \int \normalsize \csc^2{x} dx = -\cot{x}+c$
$\Large \int \normalsize \sec{x}\tan{x} dx = \sec{x}+c$
$\Large \int \normalsize \csc{x}\cot{x} dx = -\csc{x}+c$
#### Integration of Hyperbolic functions
$\Large \int \normalsize \sinh{x} dx = \cosh{x}+c$
$\Large \int \normalsize \cosh{x} dx = \sinh{x}+c$
$\Large \int \normalsize \tanh{x} dx = \log_{e}{|\cosh{x}|}+c$
$\Large \int \normalsize \coth{x} dx = \log_{e}{|\sinh{x}|}+c$
$\Large \int \normalsize \operatorname{sech}{x} dx = 2\tan^{-1}{(e^x)}+c$
$\Large \int \normalsize \operatorname{csch}{x} dx = 2\cosh^{-1}{(e^x)}+c$
$\Large \int \normalsize \sec^2h{x} dx = \tanh{x}+c$
$\Large \int \normalsize \csc^2h{x} dx = -\cot{x}+c$
$\Large \int \normalsize \operatorname{sech}{x}\tanh{x} dx = -\operatorname{sech}{x}+c$
$\Large \int \normalsize \operatorname{csch}{x}\coth{x} dx = -\csc{x}+c$
Latest Math Problems
A best free mathematics education website for students, teachers and researchers.
###### Maths Topics
Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.
###### Maths Problems
Learn how to solve the maths problems in different methods with understandable steps.
Learn solutions
###### Subscribe us
You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites. | 2022-12-04 23:03:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3938436210155487, "perplexity": 2878.4808365588415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710980.82/warc/CC-MAIN-20221204204504-20221204234504-00819.warc.gz"} |
http://mathoverflow.net/feeds/question/73159 | Does a fixed-point free "homotopy involution" imply that a manifold bounds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:59:25Z http://mathoverflow.net/feeds/question/73159 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73159/does-a-fixed-point-free-homotopy-involution-imply-that-a-manifold-bounds Does a fixed-point free "homotopy involution" imply that a manifold bounds? Scott Van Thuong 2011-08-18T15:08:46Z 2011-08-18T19:19:57Z <p>Let $M^n$ be a closed (compact, connected, without boundary) smooth manifold. It is known that if there exists a fixed point free involution $\tau:M \rightarrow M$, then M bounds. That is, there exists a compact manifold $W^{n+1}$ such that $\partial W = M$.</p> <p>But now suppose $\tau$ is only a "homotopy involution". That is $\tau^2$ is only homotopic to the identity on $M$ rather than equal to the identity. Can we say that $M$ bounds?</p> <p>For some reason I feel this statement is not true..., but I have not been able to construct a counterexample yet. For a counterexample, maybe an aspherical, nonbounding manifold would be the best candidate?</p> <p>On a related question, what if we say that $\tau^2$ is <em>isotopic</em> to the identity on M. Then does M bound?</p> <p>Thanks, I appreciate any responses.</p> http://mathoverflow.net/questions/73159/does-a-fixed-point-free-homotopy-involution-imply-that-a-manifold-bounds/73176#73176 Answer by Bruno Martelli for Does a fixed-point free "homotopy involution" imply that a manifold bounds? Bruno Martelli 2011-08-18T19:19:57Z 2011-08-18T19:19:57Z <p>A manifold with zero Euler characteristic admits a nowhere-vanishing vector field, which generates a one-parameter group of diffeomorphisms that are (smoothly) isotopic to the identity. A sufficiently small element $\tau$ is fixed-point free since the vector field does not vanish and the manifold is compact.</p> <p>There are manifolds with zero Euler characteristic that do not bound, for instance the unoriented cobordism group in dimension 5 is not trivial, see the <a href="http://en.wikipedia.org/wiki/Cobordism" rel="nofollow">Wikipedia page</a> on cobordisms.</p> | 2013-05-26 07:59:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8811526298522949, "perplexity": 999.1396374404076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706637439/warc/CC-MAIN-20130516121717-00099-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.wyzant.com/resources/answers/users/88612407 | 10/25/21
#### How can I graph this? {Parabolas} y=2x^{2}+3x-3
I never learned how to graph standard form only vertex.
10/24/21
#### Find an equation of the line that passes through (-1,-5) And (2,6)
Hello Guys i really need help with this question can someone submit a response ASP
10/23/21
#### Hi I really need help. Plz do all the problem
g(a)=a2-3a h(a)=3a-1 Find g(h(a))2.g(t)=-tFind g(g(t))3.g(x)=3x+3 f(x)=x3-3x2 Find g(f(x))
10/22/21
## Still looking for help? Get the right answer, fast.
Get a free answer to a quick problem.
Most questions answered within 4 hours.
#### OR
Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. | 2022-05-18 02:29:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30413544178009033, "perplexity": 3786.747761443508}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662521041.0/warc/CC-MAIN-20220518021247-20220518051247-00693.warc.gz"} |
http://gmatclub.com/forum/gmat-data-sufficiency-ds-141/index-450.html?sk=ku&sd=a | Find all School-related info fast with the new School-Specific MBA Forum
It is currently 26 Jul 2016, 19:33
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# GMAT Data Sufficiency (DS)
new topic Question banks Downloads My Bookmarks Reviews Important topics Go to page Previous 1 ... 8 9 10 11 12 ... 162 Next Search for:
Topics Author Replies Views Last post
Announcements
99
150 Hardest and easiest questions for DS Tags:
Bunuel
3
14800
07 Dec 2015, 09:09
425
DS Question Directory by Topic and Difficulty Tags: Coordinate Geometry
bb
0
122890
07 Mar 2012, 08:58
Topics
Of the guests at a charity fundraiser, 180 own both a house
siddhans
1
2517
19 Jun 2011, 18:11
Is |x| < 1 ? 1. x^4 - 1 > 0 2.
sag
3
1349
28 Jul 2010, 00:24
What is the value of the expression below?
pzazz12
5
1145
20 Oct 2010, 07:08
At a certain university, if 50 percent of the people who Tags: Poor Quality
lahoosaher
10
4345
31 Oct 2014, 05:11
If n is a positive integer, is n^3 - n divisible by 4? (1) n
btm3
2
1033
19 Jun 2011, 01:58
If a is equal to one of the numbers 5/11, 7/12,9/13 what is
vishalranka
3
1238
06 Apr 2010, 10:10
Is x .y>0? (I) x^2 7x + 10= 0 (II) y^2 + 5y + 4 = 0 Tags: Inequalities
jinx83
3
1072
29 Aug 2010, 17:40
What is the ratio of the number of burgers to the number of
guygmat
1
1004
20 Jun 2011, 22:03
If z is a positive integer, does z have more than 2 positive Tags: Algebra, Source: Grockit
guygmat
4
1436
24 Jun 2011, 03:20
If f(g(3))=12, what is f(g(4))? 1) g(x) = 2x + 2 2) f(g(5))
guygmat
3
1244
21 Jun 2011, 01:17
Moved: when a certain tree was first planted it was 4 ft tall. the height of
-
-
-
-
If y 3 and 2x/y is a prime integer greater than 2, which of Tags: Algebra
utin
7
2078
09 Apr 2010, 12:01
(a^3)b+(a^2)(b^2)+a(b^3)>0? 1. ab>0 2. b<0
Creeper300
4
1994
09 Nov 2010, 08:53
is the sum of integers a and b divisible by 7? (1) a is not
386390
9
1430
25 Jun 2011, 07:41
If n is an integer, then n is divisible by how many positive
vinayrsm
6
3535
04 Jul 2011, 02:21
Is N divisible by 4? 1. N is a product of two even integers Tags: Number Properties
prab
5
981
21 Oct 2010, 22:58
What is the area of rectangular region R? Tags: Geometry
priyanka116
2
1229
29 Jul 2010, 19:23
Is r/s^2 a terminating decimal?
study
3
1227
15 Jan 2010, 02:15
A sum of $200,000 from a certain estate was divided among a harshavmrg 1 1363 25 Jun 2011, 23:20 If i and j are integers, is i + j an even integer? sirrock 8 3944 03 May 2012, 00:28 If A, B, and C are digits and AB = 0, What is the value of B Tags: Arithmetic conair83 3 1370 07 Apr 2010, 04:42 If x is a prime number, what is the value of x? gsaxena26 14 1849 25 Nov 2013, 21:36 If k is a line in xy-plane, what is the slope of k? anilnandyala 3 4928 01 May 2012, 01:08 3 Moved: On level farmland, two runners leave at the same time from - - - - The third-place finisher of the Allen County hot dog eating siddhans 1 1168 19 Jun 2011, 15:09 Is the positive integer N a perfect square? (1) The number Tags: mojorising800 2 1213 11 Jul 2011, 01:40 gmatclub math test13 vaivish1723 12 2381 15 Jul 2010, 23:32 If n = p + r, where n, p, and r are positive integers and n abhi758 5 1504 20 Dec 2015, 05:03 GMATPrep: Isosceles Triangle appy001 2 1819 31 Jul 2010, 23:45 The price of computer was reduced by 25 percent. What was rxs0005 1 1317 04 Aug 2010, 05:15 125 passengers commute everyday. Is the median commute per day less th abdullahkhan19 3 1205 28 Oct 2010, 16:36 A person buys a share for$ 50 and sells it for $52 after a Tags: Word Problems saurabhkowley18 9 3667 17 Apr 2010, 11:56 Joanna bought only$0.15 stamps and $0.29 stamps. How many kirankp 2 1100 05 Jan 2010, 11:17 What is the ratio of x to y? (1) 5x = 2y (2) 2y = 7x 8 puneetj 8 4743 08 Aug 2011, 00:40 What is the value of {M - N} / {M^2 - N^2}? 1. {M^2 - N^2} = kylexy 14 1798 06 Jan 2011, 09:39 If a^b=c, what is the units digit of c? rait_m 3 1458 25 Mar 2011, 06:38 If zy < xy < 0, is abs(x - z) + abs(x) = abs(z)? (1) z Tags: japped187 4 3976 29 Mar 2011, 08:48 What is the value of a? 1. a - b + 4 = 0 2. a = -a Since HustleHarder 4 1387 01 Sep 2010, 07:40 If x is a positive integer, is x^3 - 3x^2 + 2x divisible by JimmyWorld 16 2762 26 Sep 2011, 05:10 If w,x,y and z are integres and w+x=y, is y divisible by z? enigma123 1 1098 05 Jul 2011, 10:52 If x and y are non-zero integers and |x| + |y| = 32, what is Tags: mojorising800 2 1225 30 Jul 2011, 03:44 Is X + Y Negative? i) X is Positive. ii) Y is Negative. It's hitmoss 6 1359 19 May 2011, 15:47 What is the greatest integer in a set of five different jamifahad 5 1171 17 May 2011, 08:33 What are the unique values of b and c in the equation 4x^2 + zerotoinfinite2006 2 1158 28 Oct 2010, 13:34 $10,000 is deposited in a certain account that pays r
praveenism
2
1447
07 Jun 2010, 10:21
Is a/b < 2?
Creeper300
1
1192
07 Nov 2010, 14:34
In the triangle above, what is the length of the side BC?
LM
5
1936
16 Apr 2010, 04:58
What is the value of x^2 - y^2 ? S1 x + y = 2x S2 x - y = 0
rxs0005
3
1383
24 Apr 2011, 22:21
In the problem statement given it says n is an integer
siddhans
7
1265
09 Jul 2011, 18:42
If A, B, C are points on a plane, is AB >15? 1) BC + Tags: Geometry
joemama142000
3
1958
13 Apr 2010, 12:54
new topic Question banks Downloads My Bookmarks Reviews Important topics Go to page Previous 1 ... 8 9 10 11 12 ... 162 Next Search for:
Who is online In total there are 4 users online :: 0 registered, 0 hidden and 4 guests (based on users active over the past 15 minutes) Users browsing this forum: No registered users and 4 guests Statistics Total posts 1513222 | Total topics 183893 | Active members 454862 | Our newest member phuongle0213
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2016-07-27 02:33:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18687045574188232, "perplexity": 3680.8845652580303}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257825358.53/warc/CC-MAIN-20160723071025-00322-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://annals.math.princeton.edu/toappear | Articles to appear in forthcoming issues
(Note: The PDF files below do not reflect the final published format.)
## $p$-Adic families of Siegel modular cuspforms
by Fabrizio Andreatta, Adrian Iovita, Vincent Pilloni | From volume To appear in forthcoming issues
## Une version relative de la conjecture des périodes de Kontsevich-Zagier
by Joseph Ayoub | From volume To appear in forthcoming issues
## Potential automorphy and change of weight
by Thomas Barnet-Lamb, Toby Gee, David Geraghty, Richard Taylor | From volume To appear in forthcoming issues
## On Zaremba’s conjecture
by Jean Bourgain, Alex Kontorovich | From volume To appear in forthcoming issues
## Limit theorems for translation flows
by Alexander Bufetov | From volume To appear in forthcoming issues
## Stable logarithmic maps to Deligne–Faltings pairs I
by Qile Chen | From volume To appear in forthcoming issues
## Calabi flow, geodesic rays and uniqueness of constant scalar curvature Kähler metrics
by Xiuxiong Chen, Song Sun | From volume To appear in forthcoming issues
## Recovering the good component of the Hilbert scheme
by Torsten Ekedahl, Roy Skjelnes | From volume To appear in forthcoming issues
## The Hodge theory of Soergel bimodules
by Ben Elias, Geordie Williamson | From volume To appear in forthcoming issues
## Spherical Hecke algebras for Kac-Moody groups over local fields
by Stéphane Gaussent, Guy Rousseau | From volume To appear in forthcoming issues
## ACC for log canonical thresholds
by Christopher D. Hacon, James McKernan, Chenyang Xu | From volume To appear in forthcoming issues
## Counting points on hyperelliptic curves in average polynomial time
by David Harvey | From volume To appear in forthcoming issues
## On the diameter of permutation groups
by Harald A. Helfgott, Ákos Seress | From volume To appear in forthcoming issues
## On self-similar sets with overlaps and inverse theorems for entropy
by Michael Hochman | From volume To appear in forthcoming issues
## Dispersion for the wave equation inside strictly convex domains I: The Friedlander model case
by Oana Ivanovici, Gilles Lebeau, Fabrice Planchon | From volume To appear in forthcoming issues
## Sharp vanishing thresholds for cohomology of random flag complexes
by Matthew Kahle | From volume To appear in forthcoming issues
## The André-Oort conjecture
by Bruno Klingler, Andrei Yafaev | From volume To appear in forthcoming issues
## Euler systems for Rankin-Selberg convolutions of modular forms
by Antonio Lei, David Loeffler, Sarah Livia Zerbes | From volume To appear in forthcoming issues
## Special test configuration and K-stability of Fano varieties
by Chi Li, Chenyang Xu | From volume To appear in forthcoming issues
## Min-max theory and the Willmore conjecture
by Fernando C. Marques, André Neves | From volume To appear in forthcoming issues
## Cyclic extensions and the local lifting problem
by Andrew Obus, Stefan Wewers | From volume To appear in forthcoming issues
## Ax-Lindemann for $\mathcal{A}_g$
by Jonathan PIla, Jacob Tsimerman | From volume To appear in forthcoming issues
## The Oort Conjecture on lifting covers of curves
by Florian Pop | From volume To appear in forthcoming issues
## Kodaira dimension and zeros of holomorphic one-forms
by Mihnea Popa, Christian Schnell | From volume To appear in forthcoming issues
## A product theorem in free groups
by Alexander A. Razborov | From volume To appear in forthcoming issues
## Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture
by Emmanuel Ullmo, Andrei Yafaev | From volume To appear in forthcoming issues
## Image of the Burau representation at $d$-th roots of unity
by T. N. Venkataramana | From volume To appear in forthcoming issues
## Truncations of level 1 of elements in the loop group of a reductive group
by Eva Viehmann | From volume To appear in forthcoming issues
## A general regularlity theory for stable codimension 1 integral varifolds
by Neshan Wickramasekera | From volume To appear in forthcoming issues
## Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups
by Wei Zhang | From volume To appear in forthcoming issues, Uncategorized
## Bounded gaps between primes
by Yitang Zhang | From volume To appear in forthcoming issues
## On the coherence conjecture of Pappas and Rapoport
by Xinwen Zhu | From volume To appear in forthcoming issues | 2013-12-10 10:58:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7280909419059753, "perplexity": 8489.584073690161}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164016462/warc/CC-MAIN-20131204133336-00049-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/tags/cdf/new | # Tag Info
1
You can analyze this graphically with a ROC curve which relates as well to areas under Gaussian curves to the right of some value. You will be looking for the point where the ROC curve intersects the diagonal given by $P(TP) = Y\cdot P(FP)$ (There can be zero, one, or two solutions). You can do this easily computationally with a search algorithm. I doubt ...
2
You computed the CDF by using the proper integral of the PDF $$\int 2x^{-2} dx = \frac{-2}{x} + C$$ But what you forgot is to use the correct integration constant (or use a definite integral). Your CDF is not $$F(x) = \frac{-2}{x}$$ But instead $$F(x) = \begin{cases} 0 &\quad \text{if} \quad x \leq 2 \\ \int_2^x 2u^{-2} du = 2 - \frac{2}{x} &\quad ... 6 The mean of a variable X can be computed as$$\mu_X = \int_{0}^{\infty}1-F(x)dx - \int_{-\infty}^{0} F(x)dx $$The mean of a shifted variable X-\mu_X (which equals zero) is computed as$$0 = \int_{0}^{\infty}1-F(x+\mu_X)dx - \int_{-\infty}^{0} F(x+\mu_X)dx $$Or$$ 0 = \int_{\mu_X}^{\infty}1-F(x)dx -\int_{-\infty}^{\mu_X} F(x)dx$$Which is ... 1 We may draw insight from considering a generalization of truncation. As a point of departure, then, I propose viewing truncation as an extreme example of locally modifying the probability. (This is known as change of measure in the literature on stochastic processes.) That is, in the Wikipedia setting where a distribution \lambda is truncated to the left ... 3 As noted in comments, Wikipedia gives$$x = \Phi^{-1}( \Phi(\alpha) + U\cdot(\Phi(\beta)-\Phi(\alpha)))\sigma + \mu$$for generating a random variate x from a truncated normal distribution Translating this expression to your question, I suspect you want$$F^{-1}(x; \mu, \sigma,\alpha,\beta) = \Phi^{-1}\left( \Phi(\alpha) + x\cdot(\Phi(\beta)-\Phi(\alpha))\...
1
$$F^{-1}(p; \mu, \sigma,a,b) = \Phi^{-1}\left(\Phi\left(\frac{a-\mu}{\sigma}\right) + p\left(\Phi\left(\frac{b-\mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)\right)\right)\sigma + \mu,\quad p \in (0, 1)$$
11
The CDF is defined the probability that your random variable takes a value less than or equal to a real number: $$F\colon\mathbb{R}\to\mathbb{R}, x\mapsto F(x):=P(X\leq x).$$ And of course if $X$ only has probability mass on the natural numbers, the probability that $X$ is (e.g.) less than or equal to $3$ is the same as the probability that it is less than ...
-1
In practice, the first step would be to generate a random deviate for the first branch of the specified CDF with half of the probability. So, let ½ U (where U is a Uniform random deviate on (0,1] ) = F(x) = ½ ${x^2}$, where x lies between 0 and 1, producing the obvious answer (per the Monte Carlo inversion approach for deriving random deviates) that ${X = ... 5 Here's some intuition. Let's use a discrete example. Say after an exam the students' scores are$X = [10, 50, 60, 90]$. But you want the scores to be more even or uniform.$h(X) = [25, 50, 75, 100]$looks better. One way to achieve this is to find the percentiles of each student's score. Score$10$is$25\%$, score$50$is$50\%\$, and so on. Note that the ...
1
As @whuber comments, your CDF is not correct. My plot of your CDF (from R) is shown below. It is OK for a PDF to be discontinuous, but the CDF of a continuous random variable must be continuous. First, you might try differentiating your CDF (piecewise) to see whether the result is your PDF. curve((exp(x)-1)/exp(x), 0,2, xlim=c(0,10), ylim=0:1,ylab="...
Top 50 recent answers are included | 2020-10-25 17:19:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8321111798286438, "perplexity": 505.7869433979413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107889574.66/warc/CC-MAIN-20201025154704-20201025184704-00624.warc.gz"} |
http://connection.ebscohost.com/c/articles/9823013/effect-hydrogen-electromigration-1-f-noise-gold-films | TITLE
Effect of hydrogen on electromigration and 1/f noise in gold films
AUTHOR(S)
Rodbell, K. P.; Ficalora, P. J.; Koch, Roger
PUB. DATE
May 1987
SOURCE
Applied Physics Letters;5/18/1987, Vol. 50 Issue 20, p1415
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The 1/f noise of polycrystalline gold films (5 μm wide and 0.5 μm thick) was found to decrease in the presence of hydrogen, to a level comparable with that in a single-crystal gold film. Additionally, hydrogen was found to segregate to the metal-substrate interface. On the basis of these results and recent evidence in the literature, we propose that hydrogen interacting with interface defects is responsible for both the observed 1/f noise decrease and the previously reported electromigration enhancements.
ACCESSION #
9823013
Related Articles
• Reduction of electromigration in gold thin films in the presence of hydrogen. Rodbell, K. P.; Ficalora, P. J. // Applied Physics Letters;11/1/1985, Vol. 47 Issue 9, p1010
Electromigration lifetime experiments have been completed on Au thin-film interconnects (Au, 0.5 μm thick and 25 μm wide) in vacuum (5×10-8 Torr) and in both hydrogen (H2) and helium (He) ambients (10 Torr). Results show that electromigration is drastically suppressed in the H2...
• Ultrahigh Au/p-GaAs Schottky barriers due to plasma hydrogenation. Ashok, S.; Wang, Y. G.; Nakagawa, O. S. // Applied Physics Letters;10/8/1990, Vol. 57 Issue 15, p1560
GaAs surface modification caused by rf plasma hydrogenation has been studied by electrical characterization of subsequently fabricated Au/GaAs Schottky barriers. While the Schottky barrier height on n-GaAs is found to reduce slightly, exceptionally high barriers have been seen for p-GaAs. The...
• Comparison of the Effects of Magnetic Field on Low Noise MoAu and TiAu TES Bolometers. Hijmering, R.; Khosropanah, P.; Ridder, M.; Gao, J.; Hoevers, H.; Jackson, B.; Goldie, D.; Withington, S.; Kozorezov, A. // Journal of Low Temperature Physics;Aug2014, Vol. 176 Issue 3/4, p316
Recently we have reported on the effects of magnetic field on our low noise (NEP = 4 $$\times 10^{-19}$$ W/ $$\surd$$ Hz) [] TiAu TES bolometers that are being developed at SRON for the SAFARI FIR Imaging Spectrometer on SPICA telescope that will be operated in three different wavelength bands:...
• Two orders of magnitude increase in metal piezoresistor sensitivity through nanoscale inhomogenization. Mohanasundaram, S. M.; Pratap, Rudra; Ghosh, Arindam // Journal of Applied Physics;Oct2012, Vol. 112 Issue 8, p084332
Metal-based piezoresistive sensing devices could find a much wider applicability if their sensitivity to mechanical strain could be substantially improved. Here, we report a simple method to enhance the strain sensitivity of metal films by over two orders of magnitude and demonstrate it on...
• Growth and Characterization of GaN Nanowires for Hydrogen Sensors. Johnson, Jason L.; Yongho Choi; Ural, Ant; Lim, Wantae; Wright, J. S.; Gila, B. P.; Ren, F.; Pearton, S. J. // Journal of Electronic Materials;Apr2009, Vol. 38 Issue 4, p490
We report on the growth and characterization of high-quality GaN nanowires for hydrogen sensors. We grew the GaN nanowires by catalytic chemical vapor deposition (CVD) using gold thin films as a catalyst on a Si wafer with an insulating SiO2 layer. Structural characterization of the as-grown...
• A study on the direct electrochemistry and electrocatalysis of microperoxidase-11 immobilized on a porous network-like gold film: Sensing of hydrogen peroxide. Zhang, Qian-Li; Wang, Ai-Jun; Meng, Zi-Yan; Lu, Ya-Hui; Lin, Hong-Jun; Feng, Jiu-Ju // Microchimica Acta;Jun2013, Vol. 180 Issue 7/8, p581
We have prepared porous and network-like nanofilms of gold by galvanic replacement of a layer of copper particles acting as a template. The films were first characterized by scanning electron microscopy and X-ray diffraction, and then modified with cysteamine so to enable the covalent...
• Investigation of intermolecular interactions in perylene films on Au(111) by infrared spectroscopy. Ding, Li; Schulz, Philip; Farahzadi, Azadeh; Shportko, Kostiantyn V.; Wuttig, Matthias // Journal of Chemical Physics;2/4/2012, Vol. 136 Issue 5, p054503
Intermolecular interactions in crystalline perylene films on Au(111) have been investigated by Fourier transform infrared spectroscopy. Dimer modes of vibrations are observed in the crystalline film, in contrast to the monomer modes found for isolated perylene molecules. These dimers are formed...
• A density functional theory study of the dissociation of H2 on gold clusters: Importance of fluxionality and ensemble effects. Barrio, L.; Liu, P.; Rodríguez, J. A.; Campos-Martín, J. M.; Fierro, J. L. G. // Journal of Chemical Physics;10/28/2006, Vol. 125 Issue 16, p164715
Density functional theory was employed to calculate the adsorption/dissociation of H2 on gold surfaces, Au(111) and Au(100), and on gold particles from 0.7 (Au14) to 1.2 nm (Au29). Flat surfaces of the bulk metal were not active towards H2, but a different effect was observed in gold...
• Gold (Film). Bardwell, John D. // Library Journal;4/1/1981, Vol. 106 Issue 7, p729
Reviews the film 'Gold: The Sacred Metal,' from International Film Foundation.
Share | 2020-05-31 14:31:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4623172879219055, "perplexity": 10877.188256600955}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347413406.70/warc/CC-MAIN-20200531120339-20200531150339-00136.warc.gz"} |
http://www.journaltocs.hw.ac.uk/index.php?action=browse&subAction=subjects&publisherID=48&journalID=9473&pageb=5 | for Journals by Title or ISSN for Articles by Keywords help
Subjects -> COMMUNICATIONS (Total: 359 journals) - AUDIO/VIDEO RECORDING AND REPRODUCTION (14 journals) - COMMUNICATIONS (304 journals) - DIGITAL AND WIRELESS COMMUNICATION (23 journals) - MEETINGS AND CONGRESSES (7 journals) - RADIO, TELEVISION AND CABLE (11 journals) COMMUNICATIONS (304 journals) First | 1 2 3 4
The end of the list has been reached. Please navigate to previous pages.
Magnetic Resonance Materials in Physics, Biology and Medicine [SJR: 0.928] [H-I: 40] [1 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0968-5243 - ISSN (Online) 1352-8661 Published by Springer-Verlag [2276 journals]
• Effect of $$T_{2}^{*}$$ T 2 ∗ correction on contrast kinetic model
analysis using a reference tissue arterial input function at 7 T
• Abstract: Objectives We aimed to investigate the effect of $$T_{2}^{*}$$ correction on estimation of kinetic parameters from T 1-weighted dynamic contrast enhanced (DCE) MRI data when a reference-tissue arterial input function (AIF) is used. Materials and methods DCE-MRI data were acquired from seven mice with 4T1 mouse mammary tumors using a double gradient echo sequence at 7 T. The AIF was estimated from a region of interest in the muscle. The extended Tofts model was used to estimate pharmacokinetic parameters in the enhancing part of the tumor, with and without $$T_{2}^{*}$$ correction of the lesion and AIF. The parameters estimated with $$T_{2}^{*}$$ correction of both the AIF and lesion time-intensity curve were assumed to be the reference standard. Results For the whole population, there was significant difference (p < 0.05) in transfer constant (K trans) between $$T_{2}^{*}$$ corrected and not corrected methods, but not in interstitial volume fraction (v e). Individually, no significant differences were found in K trans and v e of four and six tumors, respectively, between the $$T_{2}^{*}$$ corrected and not corrected methods. In contrast, K trans was significantly underestimated, if the $$T_{2}^{*}$$ correction was not used, in other tumors for which the median K trans was larger than 0.4 min−1. Conclusion $$T_{2}^{*}$$ effect on tumors with high K trans may not be negligible in kinetic model analysis, even if AIF is estimated from reference tissue where the concentration of contrast agent is relatively low.
PubDate: 2015-12-01
• The repeatability of T2 relaxation time measurement of human knee
articular cartilage
• Abstract: Objectives To assess short- and long-term repeatability of T2 relaxation time measurements of the knee articular cartilage. Materials and methods The right knees of nine asymptomatic volunteers (age 30–38 years, five male, four female) were imaged at 1.5 T in three sessions 1 and 2 weeks apart. To observe short-term repeatability, the measurements were repeated three times within one of the three imaging sessions for each volunteer. T2 relaxation time was mapped using a multi-slice multi-echo spin echo sequence in axial and sagittal planes. Cartilage was manually segmented and repeatability, as measured by root-mean-square coefficient of variation (CVRMS) was evaluated both for the entire bulk cartilage of each joint surface in the slice and separately for each region of interest (ROI) at different topographical locations and separately for the superficial and deep half of each ROI. Results For bulk T2, the long-term repeatability was 3.2, 5.4, and 3.7 %, and the short-term reproducibility was 3.9, 3.9, and 3.4 % for bulk femoral, tibial, and patellar cartilage, respectively. There were no significant differences between long-term and short-term repeatability in superficial or deep cartilage when comparing CVRMS values (p = 0.338 and 0.700, respectively). For individual ROIs, the repeatability varied between 2.5 and 22.2 % depending on the topographical location. Conclusion The current results show mostly good repeatability. However, there were remarkable variations of T2 between bulk cartilage and different ROIs, bulk cartilage showing better repeatability. With careful patient positioning T2 can be accurately determined for different cartilage surfaces.
PubDate: 2015-12-01
• Optical tracking with two markers for robust prospective motion correction
for brain imaging
• Abstract: Objective Prospective motion correction (PMC) during brain imaging using camera-based tracking of a skin-attached marker may suffer from problems including loss of marker visibility due to the coil and false correction due to non-rigid-body facial motion, such as frowning or squinting. A modified PMC system is introduced to mitigate these problems and increase the robustness of motion correction. Materials and methods The method relies on simultaneously tracking two markers, each providing six degrees of freedom, that are placed on the forehead. This allows us to track head motion when one marker is obscured and detect skin movements to prevent false corrections. Experiments were performed to compare the performance of the two-marker motion correction technique to the previous single-marker approach. Results Experiments validate the theory developed for adaptive marker tracking and skin movement detection, and demonstrate improved image quality during obstruction of the line-of-sight of one marker when subjects squint or when subjects squint and move simultaneously. Conclusion The proposed methods eliminate two common failure modes of PMC and substantially improve the robustness of PMC, and they can be applied to other optical tracking systems capable of tracking multiple markers. The methods presented can be adapted to the use of more than two markers.
PubDate: 2015-12-01
• Repeatability of in vivo quantification of atherosclerotic carotid artery
plaque components by supervised multispectral classification
• Abstract: Objective To evaluate the agreement and scan–rescan repeatability of automated and manual plaque segmentation for the quantification of in vivo carotid artery plaque components from multi-contrast MRI. Materials and methods Twenty-three patients with 30–70 % stenosis underwent two 3T MR carotid vessel wall exams within a 1 month interval. T1w, T2w, PDw and TOF images were acquired around the region of maximum vessel narrowing. Manual delineation of the vessel wall and plaque components (lipid, calcification, loose matrix) by an experienced observer provided the reference standard for training and evaluation of an automated plaque classifier. Areas of different plaque components and fibrous tissue were quantified and compared between segmentation methods and scan sessions. Results In total, 304 slices from 23 patients were included in the segmentation experiment, in which 144 aligned slice pairs were available for repeatability analysis. The correlation between manual and automated segmented areas was 0.35 for lipid, 0.66 for calcification, 0.50 for loose matrix and 0.82 for fibrous tissue. For the comparison between scan sessions, the coefficient of repeatability of area measurement obtained by automated segmentation was lower than by manual delineation for lipid (9.9 vs. 17.1 mm2), loose matrix (13.8 vs. 21.2 mm2) and fibrous tissue (24.6 vs. 35.0 mm2), and was similar for calcification (20.0 vs. 17.6 mm2). Conclusion Application of an automated classifier for segmentation of carotid vessel wall plaque components from in vivo MRI results in improved scan–rescan repeatability compared to manual analysis.
PubDate: 2015-12-01
• Acoustic-noise-optimized diffusion-weighted imaging
• Abstract: Objective This work was aimed at reducing acoustic noise in diffusion-weighted MR imaging (DWI) that might reach acoustic noise levels of over 100 dB(A) in clinical practice. Materials and methods A diffusion-weighted readout-segmented echo-planar imaging (EPI) sequence was optimized for acoustic noise by utilizing small readout segment widths to obtain low gradient slew rates and amplitudes instead of faster k-space coverage. In addition, all other gradients were optimized for low slew rates. Volunteer and patient imaging experiments were conducted to demonstrate the feasibility of the method. Acoustic noise measurements were performed and analyzed for four different DWI measurement protocols at 1.5T and 3T. Results An acoustic noise reduction of up to 20 dB(A) was achieved, which corresponds to a fourfold reduction in acoustic perception. The image quality was preserved at the level of a standard single-shot (ss)-EPI sequence, with a 27–54 % increase in scan time. Conclusions The diffusion-weighted imaging technique proposed in this study allowed a substantial reduction in the level of acoustic noise compared to standard single-shot diffusion-weighted EPI. This is expected to afford considerably more patient comfort, but a larger study would be necessary to fully characterize the subjective changes in patient experience.
PubDate: 2015-12-01
• Active decoupling of RF coils using a transmit array system
• Abstract: Objective Implementation of a decoupling method for isolation of transmit and receive radio frequency (RF) coils for concurrent excitation and acquisition (CEA) MRI in samples with ultra-short T2*. Materials and methods The new phase and amplitude (PA) decoupling method is implemented in a clinical 3T-MRI system equipped with a parallel transmit array system. For RF excitation, two transmit coils are used in combination with a single receive coil. The transmit coil is geometrically decoupled from the receive coil, and the remaining B 1-induced voltages in the receive coil during CEA are minimized by the second transmit coil using a careful adjustment of the phase and amplitude settings in this coil. Isolation of the decoupling scheme and transmit noise behavior are analyzed for different loading conditions, and a CEA MRI experiment is performed in a rubber phantom with sub-millisecond T2* and in an ex vivo animal. Results Geometrical (20 dB) and PA decoupling (50 dB) provided a total isolation of 70 dB between the transmit and receive coils. Integration of a low-noise RF amplifier was necessary to minimize RF transmit noise. CEA MR images could be reconstructed from a rubber phantom and an ex vivo animal. Conclusion CEA MRI can be implemented in clinical MRI systems using active decoupling with parallel transmit array capabilities with minor hardware modifications.
PubDate: 2015-12-01
• The effect of water suppression on the hepatic lipid quantification, as
assessed by the LCModel, in a preclinical and clinical scenario
• Abstract: Objective To investigate the effect of water suppression on the hepatic lipid quantification, using the LCModel. Materials and methods MR spectra with and without water suppression were acquired in the liver of mice at 4.7 T and patients at 3 T, and processed with the LCModel. The Cramér–Rao Lower Bound (CRLB) values of the seven lipid resonances were determined to assess the impact of water suppression on hepatic lipid quantification. A paired t test was used for comparison between the CRLBs obtained with and without water suppression. Results For the preclinical data, in the high (low) fat fraction subset an overall impairment in hepatic lipid quantification, i.e. an increase of CRLBs (no significant change of CRLBs) was observed in spectra acquired with water suppression. For the clinical data, there were no substantial changes in the CRLB with water suppression. Because (1) the water suppression does not overall improve the quantification of the lipid resonances and (2) the MR spectrum without water suppression is always acquired for fat fraction calculation, the optimal data-acquisition strategy for liver MRS is to acquire only the MR spectrum without water suppression. Conclusion For quantification of hepatic lipid resonances, it is advantageous to perform MR spectroscopy without water suppression in a clinical and preclinical scenario (at moderate fields).
PubDate: 2015-11-21
• Contrast-optimized composite image derived from multigradient echo cardiac
magnetic resonance imaging improves reproducibility of myocardial contours
and T2* measurement
• Abstract: Objectives Reproducibility of myocardial contour determination in cardiac magnetic resonance imaging is important, especially when determining T2* values per myocardial segment as a prognostic factor of heart failure or thalassemia. A method creating a composite image with contrasts optimized for drawing myocardial contours is introduced and compared with the standard method on a single image. Materials and methods A total of 36 short-axis slices from bright-blood multigradient echo (MGE) T2* scans of 21 patients were acquired at eight echo times. Four observers drew free-hand myocardial contours on one manually selected T2* image (method 1) and on one image composed by blending three images acquired at TEs providing optimum contrast-to-noise ratio between the myocardium and its surrounding regions (method 2). Results Myocardial contouring by method 2 met higher interobserver reproducibility than method 1 (P < 0.001) with smaller Coefficient of variance (CoV) of T2* values in the presence of myocardial iron accumulation (9.79 vs. 15.91 %) and in both global myocardial and mid-ventricular septum regions (12.29 vs. 16.88 and 5.76 vs. 8.16 %, respectively). Conclusion The use of contrast-optimized composite images in MGE data analysis improves reproducibility of myocardial contour determination, leading to increased consistency in the calculated T2* values enhancing the diagnostic impact of this measure of iron overload.
PubDate: 2015-11-03
• A semi-automated “blanket” method for renal segmentation from
non-contrast T1-weighted MR images
• Abstract: Objective To investigate the precision and accuracy of a new semi-automated method for kidney segmentation from single-breath-hold non-contrast MRI. Materials and methods The user draws approximate kidney contours on every tenth slice, focusing on separating adjacent organs from the kidney. The program then performs a sequence of fully automatic steps: contour filling, interpolation, non-uniformity correction, sampling of representative parenchyma signal, and 3D binary morphology. Three independent observers applied the method to images of 40 kidneys ranging in volume from 94.6 to 254.5 cm3. Manually constructed reference masks were used to assess accuracy. Results The volume errors for the three readers were: 4.4 % ± 3.0 %, 2.9 % ± 2.3 %, and 3.1 % ± 2.7 %. The relative discrepancy across readers was 2.5 % ± 2.1 %. The interactive processing time on average was 1.5 min per kidney. Conclusions Pending further validation, the semi-automated method could be applied for monitoring of renal status using non-contrast MRI.
PubDate: 2015-10-29
• Acoustic noise reduction in T 1 - and proton-density-weighted turbo
spin-echo imaging
• Abstract: Objective To reduce acoustic noise levels in T 1-weighted and proton-density-weighted turbo spin-echo (TSE) sequences, which typically reach acoustic noise levels up to 100 dB(A) in clinical practice. Materials and methods Five acoustic noise reduction strategies were combined: (1) gradient ramps and shapes were changed from trapezoidal to triangular, (2) variable-encoding-time imaging was implemented to relax the phase-encoding gradient timing, (3) RF pulses were adapted to avoid the need for reversing the polarity of the slice-rewinding gradient, (4) readout bandwidth was increased to provide more time for gradient activity on other axes, (5) the number of slices per TR was reduced to limit the total gradient activity per unit time. We evaluated the influence of each measure on the acoustic noise level, and conducted in vivo measurements on a healthy volunteer. Sound recordings were taken for comparison. Results An overall acoustic noise reduction of up to 16.8 dB(A) was obtained by the proposed strategies (1–4) and the acquisition of half the number of slices per TR only. Image quality in terms of SNR and CNR was found to be preserved. Conclusions The proposed measures in this study allowed a threefold reduction in the acoustic perception of T 1-weighted and proton-density-weighted TSE sequences compared to a standard TSE-acquisition. This could be achieved without visible degradation of image quality, showing the potential to improve patient comfort and scan acceptability.
PubDate: 2015-10-22
• Parameterization of hyperpolarized 13 C-bicarbonate-dissolution dynamic
nuclear polarization
• Abstract: Objective 13C metabolic MRI using hyperpolarized 13C-bicarbonate enables preclinical detection of pH. To improve signal-to-noise ratio, experimental procedures were refined, and the influence of pH, buffer capacity, temperature, and field strength were investigated. Materials and methods Bicarbonate preparation was investigated. Bicarbonate was prepared and applied in spectroscopy at 1, 3, 14 T using pure dissolution, culture medium, and MCF-7 cell spheroids. Healthy rats were imaged by spectral–spatial spiral acquisition for spatial and temporal bicarbonate distribution, pH mapping, and signal decay analysis. Results An optimized preparation technique for maximum solubility of 6 mol/L and polarization levels of 19–21 % is presented; T1 and SNR dependency on field strength, buffer capacity, and pH was investigated. pH mapping in vivo is demonstrated. Conclusion An optimized bicarbonate preparation and experimental procedure provided improved T1 and SNR values, allowing in vitro and in vivo applications.
PubDate: 2015-10-08
• Surface coil with reduced specific absorption rate for rat MRI at 7 T
• Abstract: Objective A scaled-down slotted surface radio frequency (RF) coil was built, and the specific absorbance rate (SAR) in 100 mg of tissue (SAR100 mg) produced in a rat brain phantom was computed at 7 T. Materials and methods A slotted coil 2-cm in diameter with six circular slots was developed. Its theoretical and experimental performance was computed and compared using the signal-to-noise ratio (SNR) expression and phantom images obtained with a spin echo sequence. Electromagnetic simulations were performed using the finite integral method with saline sphere and rat brain phantoms. SAR100 mg was computed for the circular coil, by varying its radius, and was also computed for the slotted coil. Results The slotted coil quality factor gave a twofold increment over the circular coil, and noise was reduced by 17 %. The experimental SNR of the slotted coil produced a 30 % improvement for points near the coil plane. The theoretical and experimental results showed substantial agreement. Axial map histograms and profiles showed greater SAR100 mg values for the circular coil than for the slotted coil. Conclusions The slotted surface coil offers improved performance and low SAR100 mg for rat brain imaging at 7 T. This approach may be used with new RF coils to investigate SAR in humans.
PubDate: 2015-10-08
• Highly undersampled peripheral Time-of-Flight magnetic resonance
angiography: optimized data acquisition and iterative image reconstruction
• Abstract: Object The aim of this study was to investigate the acceleration of peripheral Time-of-Flight magnetic resonance angiography using Compressed Sensing and parallel magnetic resonance imaging (MRI) while preserving image quality and vascular contrast. Materials and methods An analytical sampling pattern is proposed that combines aspects of parallel MRI and Compressed Sensing. It is used in combination with a dedicated Split Bregman algorithm. This approach is compared with current state-of-the-art patterns and reconstruction algorithms. Results The acquisition time was reduced from 30 to 2.5 min in a study using ten volunteer data sets, while showing improved sharpness, better contrast and higher accuracy compared to state-of-the-art techniques. Conclusion This study showed the benefits of the proposed dedicated analytical sampling pattern and Split Bregman algorithm for optimizing the Compressed Sensing reconstruction of highly accelerated peripheral Time-of-Flight data.
PubDate: 2015-10-01
• Reproducibility of pharmacological ASL using sequences from different
vendors: implications for multicenter drug studies
• Abstract: Object The current study assesses the multicenter feasibility of pharmacological arterial spin labeling (ASL) by comparing a caffeine-induced relative cerebral blood flow decrease (%CBF↓) measured with two pseudo-continuous ASL sequences as provided by two major vendors. Materials and methods Twenty-two healthy volunteers were scanned twice with both a 3D spiral (GE) and a 2D EPI (Philips) sequence. The inter-session reproducibility was evaluated by comparisons of the mean and within-subject coefficient of variability (wsCV) of the %CBF↓, both for the total cerebral gray matter and on a voxel level. Results The %CBF↓ was larger when measured with the 3D spiral sequence (23.9 ± 5.9 %) than when measured with the 2D EPI sequence (19.2 ± 5.6 %) on a total gray matter level (p = 0.02), and on a voxel level in the posterior watershed area (p < 0.001). There was no difference between the gray matter wsCV of the 3D spiral (57.3 %) and 2D EPI sequence (66.7 %, p = 0.3), whereas on a voxel level, the wsCV was visibly different between the sequences. Conclusion The observed differences between ASL sequences of both vendors can be explained by differences in the employed readout modules. These differences may seriously hamper multicenter pharmacological ASL, which strongly encourages standardization of ASL implementations.
PubDate: 2015-10-01
• Recreational alcohol use induces changes in the concentrations of
choline-containing compounds and total creatine in the brain: a 1 H MRS
study of healthy subjects
• Abstract: Objective It has previously been reported that even social alcohol consumption affects the magnetic resonance spectroscopy (MRS) signals of choline-containing compounds (tCho). The purpose of this study was to investigate whether the consumption of alcohol affects the concentrations of the metabolites tCho, N-acetylaspartate, creatine, or myo-inositol and/or their T 2 relaxation times. Materials and methods 1H MR spectra were obtained at 3 T from a frontal white matter voxel of 25 healthy subjects with social alcohol consumption (between 0 and 25.9 g/day). Absolute brain metabolite concentrations and T 2 relaxation times of metabolites were examined via MRS measurements at different echo times. Metabolite concentrations and their T 2 relaxation times were correlated with subjects’ alcohol consumption, controlling for age. Results We observed positive correlations of absolute tCho and phosphocreatine and creatine (tCr) concentrations with alcohol consumption but no correlation between any metabolite T 2 relaxation time and alcohol consumption. Conclusions This study shows that even social alcohol consumption affects the concentrations of tCho and tCr in cerebral white matter. Future studies assessing brain tCho and tCr levels should control for the confounding factor alcohol consumption.
PubDate: 2015-10-01
• Physiological noise in human cerebellar fMRI
• Abstract: Objectives To compare physiological noise contributions in cerebellar and cerebral regions of interest in high-resolution functional magnetic resonance imaging (fMRI) data acquired at 7T, to estimate the need for physiological noise removal in cerebellar fMRI. Materials and methods Signal fluctuations in high resolution (1 mm isotropic) 7T fMRI data were attributed to one of the following categories: task-induced BOLD changes, slow drift, signal changes correlated with the cardiac and respiratory cycles, signal changes related to the cardiac rate and respiratory volume per unit of time or other. $$R_{\text{adj}}^{2}$$ values for all categories were compared across regions of interest. Results In this high-resolution data, signal fluctuations related to the phase of the cardiac cycle and cardiac rate were shown to be significant, but comparable between cerebellar and cerebral regions of interest. However, respiratory related signal fluctuations were increased in the cerebellar regions, with explained variances that were up to 80 % higher than for the primary motor cortex region. Conclusion Even at a millimetre spatial resolution, significant correlations with both cardiac and respiratory RETROICOR components were found in all healthy volunteer data. Therefore, physiological noise correction is highly likely to improve the temporal signal-to-noise ratio (SNR) for cerebellar fMRI at 7T, even at high spatial resolution.
PubDate: 2015-10-01
• Localized semi-LASER dynamic 31 P magnetic resonance spectroscopy of the
soleus during and following exercise at 7 T
• Abstract: Objectives This study demonstrates the applicability of semi-LASER localized dynamic 31P MRS to deeper lying areas of the exercising human soleus muscle (SOL). The effect of accurate localization and high temporal resolution on data specificity is investigated. Materials and methods To achieve high signal-to-noise ratio (SNR) at a temporal resolution of 6 s, a custom-built human calf coil array was used at 7T. The kinetics of phosphocreatine (PCr) and intracellular pH were quantified separately in SOL and gastrocnemius medialis (GM) muscle of nine volunteers, during rest, plantar flexion exercise, and recovery. Results The average SNR of PCr at rest was $$64\pm 15$$ in SOL ( $$83\pm 12$$ in GM). End exercise PCr depletion in SOL ( $$19\pm 9$$ %) was far lower than in GM ( $$74\pm 14$$ %). The pH in SOL increased rapidly and, in contrast to GM, remained elevated until the end of exercise. Conclusion 31P MRS in single-shots every 6 s localized in the deeper-lying SOL enabled quantification of PCr recovery times at low depletions and of fast pH changes, like the initial rise. Both high temporal resolution and accurate spatial localization improve specificity of Pi and, thus, pH quantification by avoiding multiple, and potentially indistinguishable sources for changing the Pi peak shape.
PubDate: 2015-10-01
• ESMRMB 2015, 32nd Annual Scientific Meeting, Edinburgh, UK, 1-3 October:
Author Index
• PubDate: 2015-10-01
• ESMRMB 2015, 32nd Annual Scientific Meeting, Edinburgh, UK, 1-3 October:
Abstracts, Thursday
• PubDate: 2015-10-01
• Experience with magnetic resonance imaging of human subjects with passive
implants and tattoos at 7 T: a retrospective study
• Abstract: Object Over the last decade, the number of clinical MRI studies at 7 T has increased dramatically. Since only limited information about the safety of implants/tattoos is available at 7 T, many centers either conservatively exclude all subjects with implants/tattoos or have started to perform dedicated tests for selected implants. This work presents our experience in imaging volunteers with implants/tattoos at 7 T over the last seven and a half years. Materials and methods 1796 questionnaires were analyzed retrospectively to identify subjects with implants/tattoos imaged at 7 T. For a total of 230 subjects, the type of local transmit/receive RF coil used for examination, imaging sequences, acquisition time, and the type of implants/tattoos and their location with respect to the field of view were documented. These subjects had undergone examination after careful consideration by an internal safety panel consisting of three experts in MR safety and physics. Results None of the subjects reported sensations of heat or force before, during, or after the examination. None expressed any discomfort related to implants/tattoos. Artifacts were reported in 52 % of subjects with dental implants; all artifacts were restricted to the mouth area and did not affect image quality in the brain parenchyma. Conclusion Our initial experience at 7 T indicates that a strict rejection of subjects with tattoos and/or implants is not justified. Imaging can be conditionally performed in carefully selected subjects after collection of substantial safety information and evaluation of the detailed exposure scenario (RF coil/type and position of implant). Among the assessed subjects with tattoos, no side effects from the exposure to 7 T MRI were reported.
PubDate: 2015-09-26
JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762
Fax: +00 44 (0)131 4513327 | 2015-11-30 06:10:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.444965660572052, "perplexity": 5526.307871737384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398461113.77/warc/CC-MAIN-20151124205421-00351-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsoverflow.org/759/decidability-algorithm-checking-universality-quantum-gate?show=2960 | # Decidability/algorithm for checking universality of a quantum gate set
+ 8 like - 0 dislike
303 views
Given a finite set of quantum gates $\mathcal{G} = \{G_1, \dots, G_n\}$, is it decidable (in computation theoretic sense) whether $\mathcal{G}$ is a universal gate set? On one hand, "almost all" gate sets are universal, on the other, non-universal gate sets are still not well understood (in particular, of course, it is not known whether every non-universal gate set is classically simulatable), so I imagine giving an explicit algorithm for checking universality could be nontrivial.
This post has been migrated from (A51.SE)
Can you clarify the question? Joe's answer assumes you have a fixed number of qubits and all gates act on those, but for universality, we often assume gates can act on any subset of qubits. E.g., CNOT + all one-qubit gates are not universal if the one-qubit gates can only act on the first qubit, and CNOT is only from qubit 1 to qubit 2. In the latter case, we might want to extrapolate to many qubits to get universality. In that case, I think the anwer may be unknown.
This post has been migrated from (A51.SE)
@DanielGottesman: I agree about the limitations of my answer. Indeed, I believe it is undecidable in the latter case as follows: Take a cellular automata on an infinite lattice of qubits and use it to encode the halting problem (call this update unitary $U_1$). Then take a second lattice with a universal QCA (with update unitary $U_2$). We can define a new unitary $CU_2 = |0\rangle\langle0|_H\otimes I + |1\rangle\langle1|\otimes U_2$, where the subscript $H$ denotes a qubit which is set to $|1\rangle$ iff the first cellular automata halts.
This post has been migrated from (A51.SE)
Thus the gate $CU_2 \times U_1$ is universal if and only if the first Turing machine halts, and is hence undecidable.
This post has been migrated from (A51.SE)
+ 4 like - 0 dislike
For the case of Hamiltonians, rather than gates the answer is trivially yes: you simply enumerate the independent elements of the Lie algebra. Since the Lie algebra is a vector space with the addition of the Lie bracket operator. Since the space is finite, it has a finite basis, and which can easily be checked as to whether it is closed or open under the Lie bracket operation. Simply checking the Lie bracket of all pairs of orthogonal operators can be done in time polynomial in the dimensionality of the space, and a suitable operator basis can be found by the Gram-Schmidt method.
For gates, you don't really have the same option to resort to infinitesimals straight off, and need to construct gates with irrational eigenvalues so that you can arbitrarily well approximate the required infinitesimal generators. I guess that there is a relatively simple way to do this, but it is not immediately obvious to me.
In any case, taking the log of the gates to obtain a set of operators which generate them when exponentiated and checking whether these generated the full Lie algebra would provide a simple necessary but not sufficient criteria for universality.
This post has been migrated from (A51.SE)
answered Jan 20, 2012 by (3,575 points)
Why we should check only pairs?
This post has been migrated from (A51.SE)
@AlexV: Because the Lie bracket takes operates on 2 inputs. Every time you produce a new linearly independent operator you produce an orthogonal one and repeat until you get closure.
This post has been migrated from (A51.SE)
I meant you should consider $[\ldots[H_k,H_j],H_l],\ldots]$, but not only pairs, e.g. see my own paper http://arxiv.org/abs/quant-ph/0010071
This post has been migrated from (A51.SE)
@AlexV: You don't need to. It's a vector space, so a vector is orthogonal to a given subspace if and only if it is orthogonal to any basis for that subspace.
This post has been migrated from (A51.SE)
Likely we are talking about different things - which vector space you are talking about? You do not know from very beginning the subalgebra generated by your gates - you need to construct that from given Hamiltonians to check if it whole Lie algebra.
This post has been migrated from (A51.SE)
I did not see a work of Daniel Burgarth and Alastair Kay on the particular theme, but in works of other authors most often appear the same $spin(n)$ theme, but it is necessary some time, to check the isomorphism. Anyway, in many classical problems we should consider $n \to \infty$ to lost decidability.
Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register. | 2020-04-06 05:53:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8270519971847534, "perplexity": 422.57482771899714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371618784.58/warc/CC-MAIN-20200406035448-20200406065948-00009.warc.gz"} |
https://kimsereylam.com/typescript/vscode/2021/11/19/how-to-debug-typescript-node-application-with-vscode.html | # How To Debug Typescript Node Application With Vscode
Nov 19th, 2021 - written by Kimserey with .
In today’s post we will look at how we can enable the debugger with a Typescript Node application which uses nodemon and ts-node with the goal of being able to breakpoint in VSCode our Typescript Node application.
## Setup The Debugger
When using nodemon and ts-node, the regular start script would be:
1
nodemon --exec 'ts-node src/index.ts'
Now neither nodemon nor ts-node support the node --inspect command to enable the debugger The solution to that is to require ts-node directly while passing --inspect to node:
1
nodemon --exec 'node --inspect --require ts-node/register src/index.ts'
Once we start nodemon this way we will see the following message in the terminal:
1
2
Debugger listening on ws://127.0.0.1:9229/fbfa7597-16a3-4745-a88d-23c8ba4c21b1
For help, see: https://nodejs.org/en/docs/inspector
We managed to start the debugger properly.
## Breakpoint In Vscode
Our application is now setup, the next step is to attach the VSCode debugger to it. To do that we can levarage the launch.json:
1
2
3
4
5
6
7
8
9
10
11
12
{
"version": "0.2.0",
"configurations": [
{
"name": "Attach by Process ID",
"processId": "\${command:PickProcess}",
"request": "attach",
"skipFiles": ["<node_internals>/**"],
"type": "pwa-node"
}
]
}
This launch setting can automatically be generated by selecting NodeJS: Attach by process ID template in the list provided. Then once we start the debugger, we can select our nodemon process.
Once we select the right process, we should see the following message in the terminal:
1
Debugger attached.
And that’s it, all we have left to do is put breakpoints in our application in VSCode and the debugger should kick in!
## Conclusion
In today’s post we looked at how we can enable the debugger on a Node application written in Typescript and running with nodemon. We then saw how we could attach a VSCode debugger to the nodemon process in order to enable breakpoints in VSCode. I hope you liked this post and I’ll see you on the next one!
Designed, built and maintained by Kimserey Lam. | 2022-05-20 13:26:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26337647438049316, "perplexity": 8688.65253450497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662532032.9/warc/CC-MAIN-20220520124557-20220520154557-00033.warc.gz"} |
https://math.stackexchange.com/questions/1404691/reducing-or-avoiding-the-gibbs-phenomenon | # Reducing or avoiding the Gibbs phenomenon.
What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms? This could mean pre-processing or post-processing or altering the transform.
With the Gibbs phenomenon I mean the "overshoot" close to a step discontinuity like in the image below.
Ordinary Fourier transform ( Dirichlet kernel ): As proposed in comments ( Fejér kernel ).
Own work:
Let $$f_n$$ be the $$n$$'th Dirichlet kernel (multiplying with a box function which is $$1$$ for the $$n$$ lowest frequencies and $$0$$ otherwise).
Inspired by the Fejér kernel above, realizing we can write it recursively as:
$$s_n = \frac{n}{n+1} s_{n-1} + \frac{1}{n+1} f_n = \frac{n}{n+1} s_{n-1} + \left(1-\frac{n}{n+1}\right) f_n$$ we introduce a family of weighted averages ( which obviously will sum to $$1$$ ): $$s_n(k) = \left(\frac{n}{n+1}\right)^k s_{n-1}(k) + \left(1-\left(\frac{n}{n+1}\right)^k\right) f_n$$
For $$k = 2$$: For $$k = \sqrt 2$$
• For Fourier series, how about taking the arithmetic mean of the partial sums (replace the Dirichlet kernel with the Fejér kernel)? Since the Fejér kernel is positive, you get no overshoot. – Daniel Fischer Aug 21 '15 at 11:25
• By linearity the Fejér kernel should be possible to view as multiplying the frequencies by a triangle where the Dirichlet multiplies with a box function. As the triangle is the convolution of two boxes, I grow curious to try such convolutions of higher order (or other weighted averages). Do you know if that could be useful or any references to more kernels? – mathreadler Aug 29 '15 at 16:07
• Try to use frame methods proposed in the following paper: Gibbs phenomenon using tight framelets expans sciencedirect.com/science/article/pii/S100757041730237X – Mutaz Mohammad Apr 2 '19 at 7:15 | 2021-03-06 15:15:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7944110631942749, "perplexity": 858.602431939058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178375096.65/warc/CC-MAIN-20210306131539-20210306161539-00110.warc.gz"} |
https://astronomy.stackexchange.com/questions/14148/how-does-gravity-really-work/14165#14165 | # How does gravity really work
I am only 12 years old and I'm constantly wondering and trying understand how gravity really works. On YouTube everyone always talks about objects warping space time around themselves and uses the analogy of a trampoline. I still don't understand gravity because if space were like a trampoline, then earth would be spiraling in towards the sun along with all the other planets, right? So could someone explain to me how gravity really works without the trampoline analogy?
• Duplicated on physics.SE: physics.stackexchange.com/questions/243317/… Mar 14 '16 at 9:42
• The "real world" trampoline has friction acting on things on its surface, so they gradually lose energy and spiral inwards. In space there is no friction, so the planets stay orbiting for near-forever. Mar 14 '16 at 11:32
• Mandatory XKCD reference. Mar 14 '16 at 13:54
• Mandatory Feynman reference. (Talks about magnets, but the lesson about how you think about things applies in any science.) Mar 14 '16 at 19:43
• @Zaibis Don't feel too enlightened -- I think Luaan is simply wrong. If you brake an orbiting object it will indeed lose energy and altitude. In the new lowest position (the perigee) it will have a higher velocity, but at all times the sum of new velocity and "potential energy" from the gravity field is smaller than it was in the original orbit, demonstrated by the fact that it will be too slow at apogee to sustain the original higher orbit (we slowed it down there!). Cf. my reply to a comment Luaan made under my post below. He confuses the effect of two unrelated tidal forces on the moon. Mar 15 '16 at 10:10
First of all: "How gravity really works" is a deep question, and any serious scientist would quickly concede that all we have is an incomplete working model. You certainly have heard about General Relativity; the first image on the page is your trampoline.
Our working model, General Relativity, is working because it explains a lot of observations very nicely. (Careful, here is another deep question lingering: "Explains" means that we can predict some observations from other observations with the model of gravity we have in our mind. It does not necessarily mean that we understand the "real nature" of the underlying issues.) But we are very confident that the model is working over a wide range of observations. One of the last "first-time" observations which followed the predictions and thus gave us more confidence in the model was the two black holes colliding lately. Lately? Well, billions of years ago. We just learned about it lately. Here is a link to a New York Times article with an impressive video. (I think one can still read a limited number of Times articles for free, so try it out.)
Our model of gravity is incomplete because it doesn't connect well to the model of nature we have for other things (elementary particles, quantum physics). For a while (like 70 years or so) it didn't connect at all; Einstein himself completely failed to connect the dots, which was probably not encouraging since he had received the Nobel Price for laying one of the foundations of quantum physics and was the obvious authority about gravity. If he couldn't do it, who could?
If I'm not mistaken, the physicists today are making progress, slowly. This connection between quantum physics and gravity is one of the main unresolved problems in modern physics.
Last, let me address your concern about the planets spiraling into the sun. This idea probably comes from actual balls on an actual trampoline spiraling in, I suppose. You probably know that the balls lose speed due to friction, much the same way you slow down on your bike when you stop pedaling. Some of the kinetic energy is transformed into heat.
And you know what? You are right. Given enough time, the planets would eventually fall into the sun. Low-flying satellites fall back to earth after a few years, because there are still traces of atmosphere slowing them down out there. The reason is that there is "friction" in the wider sense involved in all large-scale processes in the universe. That is actually one of the fundamental physical principles making up the world we know. It's just that the near-vacuum between the planets doesn't provide that much friction, and the planets are fairly massive bodies with an enormous mass and kinetic energy. It will take a long long time for them to lose enough energy that they'll be so close as to touch the sun. (Perhaps too long to happen at all.) In fact, over human life times the planets, moons and stuff are almost perfect examples for movement without friction. But in the astronomical time scale -- billions of years --, there certainly is friction. For example, the moon is showing us always the same side because friction slowed its rotation so that the rotation is now "locked" with its orbit.
Bottom line: The idea that gravity bends space and time "explains" all large-scale observations so far; the "trampoline" is a good model for a 2-dimensional "space", i.e. a surface, if you ignore friction.
• The Moon is also much further than it was in the past. Tidal friction decreased its orbital speed, which increases the orbital radius. The radius is increasing by about four centimeters a year, nowadays. Mar 14 '16 at 13:55
• +1 for "We're not entirely sure, but here's some of our best guesses based on observations." Mar 14 '16 at 14:18
• Please correct the statement about moon. The moon is not synchronized ("locked") by accident but due to tidal forces of gravity - the closer side of moon suffers higher gravity than the other one. This force can actuall force object to increase its rotation, if it would rotate slower than the orbiting speed. Mar 14 '16 at 14:29
• @libik I cannot see anything which would need correction (in particular, I did not say or imply "by accident" -- on the contrary, I mentioned friction as a cause) . One could mention tidal forces but I thought friction is good enough without detouring too much. You make an interesting point with a possible accelerating rotation due to tidal forces; but it's still a slowing down (to close to 0) relative to it's orbital reference frame. Mar 14 '16 at 15:27
• @Luaan There are two tidal forces in play. (1) The moon receives energy from earth's rotation by the tidal forces the rotating earth exerts on it, accelerating it in the direction of earth's rotation. This lifts it (slowly) higher in earth's gravity well, as you say correctly. (2) The moon's cyclical deformation ("kneading") caused by the moon's rotation in earth's inhomogeneous gravity field convert(ed) some of the rotational energy into heat, eventually synchronizing orbit and rotation, at which point there are almost no moon tides any more (short of those due to libration, I believe). Mar 14 '16 at 15:37
Why are objects not escaping?
Consider first an object with velocity, and no gravity in action:
Then, that blue object will become more and more distant, if it continues in the same direction.
But it does not continue in the same direction, after a while, the gravity of the big black object has changed its course:
That happens again, and again and again:
Your question is: Why does not the object spiral in? You are perhaps thinking that as it comes closer, the gravity becomes stronger, and therefore the object is forced to come even closer.
But when it falls closer, its velocity increases. As we have seen, the velocity of the objects tries to make it escape. So when it is closer, it has more velocity to counteract the increased gravity.
Edit: Just in case of a more literal interpretation of your question, the trampoline in the original analogy causes friction, and therefore spiralling, but space is a vacuum.
• I think the key as to why it doesn't fall in is that in space we don't have friction - on a trampoline energy is constantly being removed from the ball via friction, whereas in space there's nothing to slow our planet down, so it just keeps going
– Jeff
Mar 14 '16 at 22:42
• @Jeff Edited in Mar 14 '16 at 22:45
• My physics teacher in high school said "the earth falls towards the sun all the time, but continues missing it because of its speed." Mar 15 '16 at 8:32
The trampoline analogy is useful if you understand gravity within a General Relativity framework. The conceptual problem there is that actually the space-time is wrapped in 4, not in 3 dimensions, i.e. including time.
In fact, when the Earth rotates around the Sun, it loses a very tiny amount of energy in form of gravitational waves. So, the Earth is actually spiraling towards the Sun. The thing is that this gravitational wave emission is so small, that by the time we observe any considerable spiraling, the Earth and the Sun would have already ceased to exist. Much before that, the Solar System becomes unstable due to chaotic effects already contained in classical Newtonian mechanics.
Great question!
Have you heard of Newton's First Law? It says that an object in motion continues moving at the same speed and in the same direction unless acted upon by a force.
When we roll a ball along the ground, it will eventually stop. Before Newton, many people believed that everything slows down by itself. Newton's insight was that this is not true, and actually the only reason a rolling ball will slow down is because the ground and the air rub or push against the ball to slow it down.
On a trampoline, a ball will rub against the trampoline material and against the air, which slows it down. This is the only reason that the ball ends up spiralling towards the centre.
When there is nothing to slow the object down, it won't spiral towards the middle, it will just keep going round and round forever. In space there is (almost) nothing to slow an object down.
If you find this hard to believe, you can write a computer program to do all the calculations and see what happens! I have made an example simulation for you. You will see that without friction, the planet will end up where it started each time it goes around the sun. If you change the planet's initial yspeed from 20 to 40 and then click "Run" up the top you will see a more circular orbit. You can change other things and see what happens. I hope you find this useful!
• Nice simulation. (Although the planet escaped the sun after it got close. :-) ) Mar 15 '16 at 8:30
• It's an easter egg ;) Actually it's a good discussion point---it reminds us that the simulation is ONLY simulating gravity, not collisions, and also that when the planet gets really close to the sun, the time-step of the simulation causes big inaccuracies. This can be reduced by more sophisticated numerical methods like Runge-Kutta, but now I'm well beyond the scope of the question! Mar 15 '16 at 9:02
• I don't know if it's the same simulation anymore when you do this, but if you change the for-loop condition to i < 1 rather than i < 5 and you change the timeout parameter to setInterval to 10 rather than 100, the simulation gets a whole lot more pleasant to watch. It runs slightly faster, but the framerate is much higher, so the movement of the outer body isn't so jagged.
– Alex
Mar 16 '16 at 11:34
• Thanks Alex! Actually the timeout parameter should be 20 and then (assuming your CPU is fast enough) it is the same simulation. On my computer this slows the simulation down by 25%, presumably because my CPU isn't fast enough. Still, it does look smoother; here is a new simplified version: jsfiddle.net/0erknpk8/38 Mar 16 '16 at 23:51
Neutrino String Induction-Refraction is the cause of gravity. Some will say that neutrinos are insignificant, but Dirac, Hawking, and Tyson think otherwise and most discount the effect of a charged particle traveling at the speed of light. Keep in mind that no one can, or has proved that mass is a property of matter, more of an effect.
Go to www.themechanismofreality.com , this site explains exactly how gravity works. Every Physicist who examines this agrees that this is correct. From CERN to the University of Beijing's Physic Department agree that this is a 'Fantastic connection between graviton physics and string theory"! This was also confirmed , indirectly, by LIGO and the gravity wave announcement. Enjoy!
• This is a link only answer. (not encouraged), also, that paper seems a little weird towards the end. Mar 15 '16 at 16:50
• I have the nagging feeling that the lack of math, references and collaboration indicates it's not revolutionary science but at most a popular science article. It's tough; one should, of course, not gratuitously add math just to appear serious. But this kind of lone paradigm-changing breakthrough (which, I think, is claimed here, since I never heard of it before) is exceedingly rare. Mar 15 '16 at 17:50
• In order to make the theory in the article more palatable you could try to put it in context. Like, start with what the conventional theory (and its famous proponents) thinks a Neutrino is and how it interacts, and why a different assumption could explain gravity. Mar 15 '16 at 17:53 | 2021-11-30 02:57:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5074042677879333, "perplexity": 624.4115740569447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358903.73/warc/CC-MAIN-20211130015517-20211130045517-00007.warc.gz"} |
http://breedingbetterbananas.iita.org/toby-alexander-zyfho/ljkp6o6.php?ea41c5=what-is-newton%27s-third-law | # what is newton's third law
According newton’s third law of motion: To every action there is an equal and opposite reaction. Explain why it is difficult for a fireman to hold a hose which ejects large amounts of water at a high velocity? The writers, via the character, are using Newton's third law as a metaphor for emotional and psychological progress. For every action, there is an equal (in size) and opposite (in direction) reaction. three laws of mechanics describing the motion of a body. Action is equal to reaction but the two forces act on two different bodies hence, they do not cancel the effect of each other. Introduction: Isaac Newton’s Third Law of physics states that for every action there is an equal and opposite reaction. What was … (Use for #4A-C) 67 kilogram Jody swats at 0.012 kg bug that landed on her applying Newton's third law of motion - action and reaction are equal and opposite law of action and reaction, Newton's third law, third law of motion law of Newton’s third law of motion is all about understanding these two terms: 1. Consider the flying motion of birds. But that's no problem for rockets. Teachers: Use the following demonstrations to introduce Newton’s Third Law to your class. A variety of action-reaction force pairs are evident in nature. In rocketry, Newton's third law applies in the sense that the momentum of the material ejected from the back of the rocket imparts equal momentum in … Formally stated, Newton's third law is: The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. What is Newton’s third law of motion? Rahul while putting anentirely new explanation said that both the motorcar and theinsect experienced the same force and a change in theirmomentum. By using this website, you agree to our use of cookies. If you pull on a … Fly to Mars! Comment on this logic and explain why the truck does not move. (2) 21d. Newton’s third law of motion states that: The force applied is not able to overcome the force of friction hence, the truck does not move. The force is sufficient to push you up or in other words to accelerate you up by F=ma or a=F/m. According to Newton’s third law of motion: To every action there is an equal and opposite reaction. Newtons Third Law. To every action there is an equal and opposite reaction. Newton’s Third Law of Motion Newton gave three remarkable laws for understanding the motion of bodies and relate it with force. Newton's third law of motion synonyms, Newton's third law of motion pronunciation, Newton's third law of motion translation, English dictionary definition of Newton's third law of motion. There are two forces resulting from this interaction - a force on the chair and a force on your body. Force can be classified into two categories: contact force such as frictional force and non-contact force such as gravitational force. This law needs to be understood carefully because many times people confuse that the action … Simply put, this law states that for every action there is an equal and opposite reaction. Newton's third law says that if object A exerts a force on object B, then object B must necessarily exert an … So the book may not be factually wrong, but I would say it is didactically wrong for not explaining itself clearly. Or every action always reacts in the opposite direction. And as a result, the insect died. There you have it. Even … Whenever one body exerts a force on another body, the second body exerts an equal and opposite force on the first body. More than 50 million students study for free with the Quizlet app each month. 3. Work 18d. Consider the propulsion of a fish through the water. Newton’s third law of motion states that: “To every action, there is always an equal and opposite reaction” states that for each action, there is an equal and opposite reaction. Whenever a force acts, an equal force acts in the opposite direction. These forces are called action and reaction forces. This is … Students are introduced to Newton's third law of motion: For every action, there is an equal and opposite reaction. (equal!). These two laws are laws of motion. The bullet has a greater acceleration due to the fact that it has a smaller mass. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite the direction of the force on the fish (forwards). The justification given by the student is wrong because action and reaction forces act on two different bodies hence, they cannot cancel each other. Work against Electric Forces 19.Motion in a Circle 20. A student justifies this by answering that the two opposite and equal forces cancel each other. In a scientific sense, you can compare the law of karma with Newton’s third law of motion. Newtons third law.notebook 11 April 23, 2019 The rocket car is propelled along the floor according to the principle stated in Isaac Newton's third law of motion. Newton's Third-law is without a doubt the law that is seen most in the launch of the catapult. Momentum 18c. The balloon Kiran suggested that the insect suffered a greater change in momentum ascompared to the change in momentum of the motorcar (becausethe change in the velocity of the insect was much more than that of the motorcar). This is the Law of Equilibrium, the Law of inertia. Newton’s third law of motion is all about understanding the term force pairs. Newton's third law of motion Our mission is to provide a free, world-class education to anyone, anywhere. Yet, acceleration depends on both force and mass. Trick Question! 2. The third law is also known as the law of action and reaction. Donate or volunteer today! Forces always come in pairs - equal and opposite action-reaction force pairs. 1. action and reaction are equal and opposite Familiarity information: NEWTON'S THIRD LAW used as a noun is very rare. The Third Law states that “For every action, there is an equal and opposite reaction.” It was developed by Sir Issac Newton in the 17 th century. Newton's Third-law of motion states that for every action force, there is a reaction force that is equal in strength and opposite in direction. The fact is that rockets do accelerate. Now, read the statement of Newton’s 3 rd law of motion mentioned below. Which of the two forces is greater: the force on the firefly or the force on the bus? The third law of motion describes what happens to the body when it exerts a force on another body. Many explanations can be provided using Newton’s third law of motion for rocket propulsion, the backward movement of hose pipe, the movement of horse cart, backward movement of boat when we jump out of it. The skaters' forces on each other are equal in magnitude, and in opposite directions. At the same time, water also pushes the boy forward, which is the reaction force. Newton’s third law of motion focuses on action reaction pair of forces. We have previously add three different projects that verify this law. b. with more force when the rope is attached to the elephant. Thus, if one body exerts a force F on a second body, the first body also undergoes a force of the same strength but in the opposite direction. Newton's third law is also essential for understanding and developing automobiles, airplanes, rockets, boats, and many other technologies. Third law. The Newton’s 3rd law states that for every action there is an equal and opposite reaction. A force is a push or a pull that acts upon an object as a results of its interaction with another object. It is essential to any student pilot to have a strong grasp on this basic understanding of physics. Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. Newton's Third Law of Motion states that any time a force acts from one object to another, there is an equal force acting back on the original object. They practice identifying action-reaction force pairs for a variety of real-world examples, and draw and explain simplified free-body diagram vectors (arrows) of force, velocity and acceleration for them. Newton's Third Law Two boats A A A and B , B, B , connected with a string, are floating on a calm lake. When a fire hose pipe ejects large amounts of water in the forward direction at a high velocity then the forward going stream exerts a backward reaction due to which the hose pipe tends to go backwards and slips from the hand of fireman. Newton’s Second Law of Motion The third law of motion describes what happens to the body when it exerts a force on another body. Newton's third law. Newton's 3rd Law is about the Conservation of Energy or the Condition of zero Net Force. • NEWTON'S THIRD LAW (noun) The noun NEWTON'S THIRD LAW has 1 sense:. For years, space travel was believed to be impossible because there was nothing that rockets could push off of in space in order to provide the propulsion necessary to accelerate. As discussed in Lesson 2, some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces are the result of action-at-a-distance interactions (gravitational, electrical, and magnetic forces). 3. The force exerted by the first body on the second body is known as “action” and the force exerted by second body on the first body is called “reaction”. Newton’s third law of motion states “if an object gives force to another object then the object which receive the force will give force as big as it receives from the first object in the opposite direction”. According to Newton’s third law of motion: To every action there is an equal and opposite reaction. “For every action there is an equal and opposite reaction,” according to the Newtonian law. Newton's Third Law of Motion. The third law states that an applied force creates an equal and opposite force. If you pull on a … A gunpowder explosion creates hot gases that expand outward allowing the rifle to push forward on the bullet. The Newton’s 3rd law states that for every action there is an equal and opposite reaction. Students are asked to identify the action force and reaction force in several diagrams, and research how a squid uses Newton's third law of motion. newton’s third law of motion is also called action – reaction law. Akhtar said that since the motorcar wasmoving with a larger velocity, it exerted a larger force on the insect. The force acting on horse determines whether the cart will be pulled by it or not. If you understand these two terms properly, you’ll definitely understand the whole statement of newton’s third law of motion. c. ... space is void of air and so there is no air resistance in space. Newton's Third Law of Motion: Action Reaction Pairs. Action-reaction force pairs make it possible for cars to move along a roadway surface. Windshields don't have guts. 2. These two forces are called action and reaction forces and are the subject of Newton's third law of motion. In the top picture (below), Kent Budgett is pulling upon a rope that is attached to a wall. This is a better way to say it: A force is exerted by one object on another object. (And "the downward force due to gravity and the upward force that the book exerts on the Earth due to gravity are an action/reaction pair of Newton's third law.") Besides, fireflies have guts and bug guts have a tendency to be splatterable. This law needs to be understood carefully because many times people confuse that the action reaction forces will cancel the effect of each other. Quizlet is the easiest way to study, practice and master what you’re learning. The truck does not move because the force applied is less than the friction force between the wheels of truck and road. A bird flies by use of its wings. Newton's third law of motion means that, for every force applied, there is always an equal and opposite force. If the object is a massive truck parked along the roadside, it will probably not move. Unfortunately, this statement lacks some necessary detail. Newton’s Third Law of Motion Definition Newton’s Third Law of Motion says that if a body A exerts a force on body B, then the body B exerts a force of equal magnitude, in the opposite direction, on the body A. This law represents a certain symmetry in nature: forces always occur in pairs, and one body cannot exert a … Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. Newton's third law. (adsbygoogle = window.adsbygoogle || []).push({}); If action is always equal to the reaction, explain how a horse can pull a cart. When you jump you apply a force on the ground and by Newton’s 3rd law the ground apply a force of same magnitude on you. Four Reasons why Friction is useful Friction produces heat. Stated in modern language, Newton's Third Law says every action has an equal and opposite reaction. Sir Isaac Newton first presented his three laws of motion in the "Principia Mathematica Philosophiae Naturalis" in 1686. Newton’s Third Law Of Motion Force is a push or pull acting on an object resulting in its interaction with another object. Common examples of newton’s third law of motion are: A horse pulls a cart, a person walks on the ground, hammer pushes a nail, magnets attract paper clip. Action. Newton's third law: If an object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A. Rockets are able to accelerate due to the fact that they burn fuel and push the exhaust gases in a direction opposite the direction which they wish to accelerate. We use cookies to provide you with a great experience and to help our website run effectively. Each force is the same size. [After Isaac Newton .] © 2020, Arinjay Academy. This recoil is the result of action-reaction force pairs. d. ... nonsense! His third law states that for every action (force) in nature there is an equal and opposite reaction. Although the explanation of the law is simple, STEMists often find the concept hard to comprehend. Rockets do accelerate in space and have been able to do so for a long time. Kepler's 3rd Law 21a.Applying 3rd Law 21b. A force is exerted by one object on another object. The fact that the firefly splatters only means that with its smaller mass, it is less able to withstand the larger acceleration resulting from the interaction. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. Newton’s third law, which states that every action has an equal and opposite reaction, can be related to an egg drop because the forces pushing the egg up and down must follow the law. Everyone knows that every action has an equal and opposite reaction, right? Look it up now! Review Newton's third law of motion with this printable. © 1996-2020 The Physics Classroom, All rights reserved. Review Newton's third law of motion with this printable. As the wheels spin, they grip the road and push the road backwards. The size of the force on the road equals the size of the force on the wheels (or car); the direction of the force on the road (backwards) is opposite the direction of the force on the wheels (forwards). Examples of Newton’s Third Law of Motion : There are lots of projects regarding the Newtons 3rd law of motion. This makes it difficult for the fireman to hold the pipe strongly. The size of the force on the air equals the size of the force on the bird; the direction of the force on the air (downwards) is opposite the direction of the force on the bird (upwards). What is Newton's Third Law of Motion?. This inability of a rocket to provide propulsion is because ... a. Noun 1. Akhtar, Kiran and Rahul were riding in a motor car that was moving with a high velocity on an expressway when an insecthit the windshield and got stuck on the windscreen. What this means is that pushing on an object causes that object to push back against you, the exact same amount, but in the opposite direction. 30 \text{ m}. Newton's third law of motion means that, for every force applied, there is always an equal and opposite force. Newton's third law of motion describes the nature of a force as the result of a mutual and simultaneous interaction between an object and a second object in its surroundings. The acceleration of the recoiling rifle is ... a. greater than the acceleration of the bullet. For every action, there is an equal and opposite reaction. Dictionary entry overview: What does Newton's third law mean? It is displayed countless moments in the experiment. But a push on the water will only serve to accelerate the water. The force on the rifle equals the force on the bullet. The first two laws are also important to know and remember; however, the Third Law is uniquely at work every second an aircraft is in flight. According to Newton’s third law of motion, “To every action, there is always an equal and opposite reaction” When this boy pushes the water backwards it is known as action force. BROWSE SIMILAR CONCEPTS. Consider the motion of a car on the way to school. Forces and Motion. “For every action there is an equal and opposite reaction,” according to the Newtonian law. Newton's Third Law of Motion. Force is a result of an interaction. Create your own flashcards or choose from millions created by other students. Newton's first two laws of motion refer to single bodies. Newton's Laws - Lesson 4 - Newton's Third Law of Motion. Some implementation using Newton’s third law of motion are electric force and magnetic force. They practice identifying action-reaction force pairs for a variety of real-world examples, and draw and explain simplified free-body diagram vectors (arrows) of … If the force applied on the horse is greater than the force of friction between road and the cart, the horse is able to pull cart. Unfortunately, this statement lacks some necessary detail. Newton gave three remarkable laws for understanding the motion of bodies and relate it with force. According to Newton's third law of motion, whenever two objects interact, they exert equal and opposite forces on each other. A car is equipped with wheels that spin. Reaction. Trajectory - Horizontally Launched Projectiles Questions, Vectors - Motion and Forces in Two Dimensions, Circular, Satellite, and Rotational Motion. (Action and Reaction forces) How? Action and reaction are two different forces but they act on two different bodies simultaneously. A good pilot must understand how Newton’s Third Law applies to thrust and how an aircraft flies. Newton's first two laws of motion refer to single bodies. In action: when you jump, you push the ground with your feet, and the ground pushes back, propelling you into the air. and what does that mean? The firefly hit the bus and the bus hits the firefly. Force and friction:- Newton’s third law. Since forces result from mutual interactions, the air must also be pushing the bird upwards. Action-reaction force pairs make it possible for fish to swim. Kent is pulling ... a. with more force when the rope is attached to the wall. The size of the forces on the first object equals the size of the force on the second object. A fish uses its fins to push water backwards. That I’ll show you later. In each case, the force scale reads 500 Newton. What you are missing is that the action and reaction forces apply to two different objects. Akhtarand Kiran started pondering over the situation. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features The Third Law, familiar to anyone who has ever been in a collision, explains why rockets work. But it is not true since action and reaction forces act on two different bodies simultaneously. n. The principle stating that for every action there is an equal and opposite reaction. While driving down the road, a firefly strikes the windshield of a bus and makes a quite obvious mess in front of the face of the driver. In the bottom picture, Kent is pulling upon a rope that is attached to an elephant. The direction of the force on the first object is opposite to the direction of the force on the second object. Many people are familiar with the fact that a rifle recoils when fired. "For every action there is an opposite and equal reaction." Fly to Mars! In other words, every force involves the interaction of two objects. Everyone knows that every action has an equal and opposite reaction, right? When two bodies interact, they apply force on each other that are equal in magnitude and opposite in direction. If object A applies force ##\vec F## to object B at a point that is displaced by vector ##\vec r## from point O, then A has applied a torque of ##\vec r\times\vec F## to B around point O. Newton's third law tells us that object B applies a force of ##-\vec F## to … 18. Newton’s third law represents a certain symmetry in nature: Forces always occur in pairs, and one body cannot exert a force on another without experiencing a force itself. Newton's third law of motion is probably the easiest of the three laws to understand. Newton's third law. Newton's third law of motion is not a law about motion but a law about forces. Fly to Mars! According to the third law of motion when we push on an object, the object pushes back on us with am equal and opposite force. But before that, read the statement of Newton’s 3 rd law of motion. Comment on these suggestions. So Newtons's Third Law is universal, but people still have trouble identifying these third law partner forces. b. smaller than the acceleration of the bullet. Newton's second law states that the resultant force acting on a particle equals the time rate of change of momentum of the particle For a particle of fixed mass (constant m), which is the F = ma equation above in vector form. We sometimes refer to this law loosely as “action-reaction,” where the force exerted is the action and the force experienced as a consequence is the reaction. When two bodies interact, they apply force on each other that are equal in magnitude and opposite in direction. Newton’s 3rd law of motion states that action and reaction are always equal but opposite in direction. Remember: acceleration and mass are inversely proportional. is exerted by one object on another object. According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. Students are introduced to Newton's third law of motion: For every action, there is an equal and opposite reaction. This principle describes interactions between bodies, and an experiment has been conducted to study these relations. For every action, there is an equal (in size) and opposite (in direction) reaction. Forces result from interactions! Chapter 8: Newton’s Third law External forces (e.g., nT on m) are also part of an action-reaction pair, but the reaction force (e.g., nm on T, the normal force m exerts on the table) is irrelevant to the dynamics of m or M. Thus Doing so will strongly contribute to safe and efficient flying. Newton's Gravity 21. Newton's laws of motion definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. For every action, there is an equal (in size) and opposite (in direction) reaction force. Students are asked to identify the action force and reaction force in several diagrams, and research how a squid uses Newton's third law of motion. Khan Academy is a 501(c)(3) nonprofit organization. Newton's Third Law. So one of the best ways to do it, is by listing both objects, as soon as you list both objects, well to figure out where the partner force is, you can just reverse these labels. Newton's 2nd Law 18a. The Third Law 18b. Newton third law states that: if there is action then there will be opposite & equal reaction. Consistent with Newton's third law of motion, the bullet pushes backwards upon the rifle. Action-reaction force pairs make it possible for birds to fly. Pin Wheel Experiment: Pin wheel is a cool science experiment where a balloon is fixed in a straw in arrangement. Newton's Second Law of Motion defines the relationship between acceleration, force, and mass. Often, one of these forces is called “action” and the other one “reaction”. Newton's third law is probably the most familiar. These two laws are laws of motion. Newton proposed it in order to describe the laws of physics in the material universe—but it also expresses the … c. the same size as the acceleration of the bullet. Watch this physics video on forces and motion to understand it better! Examples of Newton's third law of motion are ubiquitous in everyday life. Newton's third law of motion says that when two objects push or pull against each other, the forces that they feel are equal and opposite. Newton's third law is probably the most familiar. Newton’s Third Law of Motion explains the physics behind this technical maneuver: For every force, there is an equal and opposite force. The two forces together are called an interaction pair.. One example of Newton's third law is when you are sitting still on a chair. All rights reserved. The wings of a bird push air downwards. Newton's third law states that when a body exerts a force on a second body, the second one simultaneously exerts a force equal in magnitude and opposite in direction to that of the first body. For example , when you jump, your legs apply a force to the ground, and the ground applies and equal and opposite reaction force that propels you into the air. There is indeed nothing for rockets to push off of in space - at least nothing which is external to the rocket. And non-contact force such as frictional force and non-contact force such as frictional force and non-contact force as... Been able to do so for a fireman to hold a hose which ejects large amounts water! Smaller mass the relationship between acceleration, force, and an experiment has been conducted to,! Third-Law is without a doubt the law of motion by using this website, you ll... Be understood carefully because many times people confuse that the action … forces and motion the that... Has ever been in a collision, explains why rockets work to a.. Each case, the water forces apply to two different bodies simultaneously million students for., are using Newton ’ s third law of Equilibrium, the force on the insect in... And efficient flying of motion: for every action there is an opposite equal! In its interaction with another object for emotional and psychological progress that are and! Action-Reaction force pairs make it possible for cars to move along a surface. Are electric force and friction: - Newton ’ s third law of motion mentioned below explanation... Wheel experiment: pin Wheel experiment: pin Wheel experiment: pin Wheel experiment: Wheel... Own what is newton's third law or choose from millions created by other students the truck does move. Are familiar with the quizlet app each month the balloon Examples of Newton 's third as! Academy is a better way to school is seen most in the opposite direction, of. Relationship between acceleration, force, and mass hence, the truck not. Grip the road and push the road and push the road backwards ) nonprofit.... ’ re learning everyday life: for every action there is an equal and opposite reaction.:.... By answering that the action and reaction are equal in magnitude, and.... Clear case of what is newton's third law 's third law of motion Newton gave three remarkable laws for understanding and automobiles... Whenever objects a and B interact with each other that are equal in magnitude, and.! A noun is very rare categories: contact force such as frictional force and mass this video. Law, familiar to anyone who has ever been in a simultaneously exerted Newton 's Third-law is without a the... Greater acceleration due to the Newtonian law expand outward allowing the rifle equals the force the. Friction produces heat, rockets, boats, and Rotational motion motion electric! Itself when the rope is attached to a wall often, one of these is... Use of cookies terms: 1 home » Science » Newton ’ third... Whenever one body exerts an equal and opposite reaction. 1 sense: be factually wrong, but would. Is opposite to the Newtonian law very rare many people are familiar with the fact a. The recoiling rifle is... a. with more force when the coriolis force found Newton! Reads 500 Newton grasp on this logic and explain why the truck does not.. Of bodies and relate it with force in its interaction with another object, this states! Between bodies, and mass rockets to push forward on the rifle equals the force scale 500. It or not two categories: contact force such as frictional force and friction: Newton! Trajectory - Horizontally Launched Projectiles Questions, Vectors - motion and forces in two Dimensions, Circular,,! Probably not move useful friction produces heat hit the bus three laws to understand it better is pulling... greater... So the book may not be factually wrong, but I would it., and Rotational motion propulsion is because... a along the roadside, it will not! Contact force such as gravitational force chair and a change in theirmomentum the force on the way study. Describing the motion of bodies and relate it with force provide a free world-class... ) reaction force people confuse that the two opposite and equal reaction. upon each other are always equal opposite! Be classified into two categories: contact force such as frictional force and non-contact force such as force! » Newton ’ s 3rd law of motion, whenever two objects is! Larger velocity, it will probably not move according Newton ’ s third law as a for. Parked along the roadside, it will probably not move that whenever an as... In size ) and opposite reaction. a greater acceleration due to the of. According Newton ’ s third law is simple, STEMists often find the concept hard comprehend! Action – reaction law is that the action and reaction. does Newton third! That acts upon an object resulting in its interaction with another object, this law a simultaneously exerted Newton laws... Newton proposed it in order to describe the laws of physics: contact force such as force... Push water backwards explanation said that both the motorcar and theinsect experienced the same force a! Vectors - motion and forces in two Dimensions, Circular, Satellite, many... Work against electric forces 19.Motion in a Circle 20 motion our mission is to provide is. Force ) in nature there is an equal and opposite force is very rare the. Action and reaction forces and motion to understand it better is seen most in the picture. Are called action and reaction forces will cancel the effect of each other, familiar to anyone who ever! The laws of motion you agree to our what is newton's third law of cookies has ever been in a,! Factually wrong, but I would say it is a massive truck parked the... You understand these two terms: 1 both force and non-contact force such as gravitational force stated: always. We have previously add three different projects that verify this law needs to be understood carefully because many times confuse..., all rights reserved acceleration depends on both force and mass pairs make it possible for fish swim. Expand outward allowing the rifle equals the size of the forces on each other are in... And road rockets are unable to accelerate in space balloon is fixed a. Circular, Satellite, and many other technologies these two terms properly, you ’ ll definitely understand the statement! Before that, read the statement of Newton 's laws - Lesson -! Explain why the truck does not move ) the noun Newton 's third law of motion is a... A fireman to hold the pipe strongly very rare words to accelerate in.... Reaction force massive truck parked along the roadside, it exerted a larger force on the.. That for every action there is an equal and opposite reaction. than 50 million students for... & equal reaction. and many other technologies hit the bus ' on! Circle 20 3rd law states that an applied force creates an equal and opposite.... And friction: - Newton ’ s 3rd law states that for every action has equal... Anyone who has ever been in a simultaneously exerted Newton 's third law states that an force. From this interaction results in a simultaneously exerted Newton 's second law of motion one. Off of in space and have been able to overcome the force applied, is! Ubiquitous in everyday life used as a noun is very rare each other a massive truck parked along the,. Two bodies interact, they apply force on the second object indeed nothing rockets... Many other technologies force and mass student justifies this by answering that the action … forces motion... Pilot to have a strong grasp on this logic and explain why the truck does move! Is always an equal and opposite reaction, right didactically wrong for not explaining itself.... For birds to fly the what is newton's third law wasmoving with a larger velocity, it will probably not move,., for every action, there is an equal and opposite in direction applied on horse by cart is for... The easiest of the force on another object by Newton 's third law motion! Describing the motion of a rocket to provide propulsion is because... a a strong grasp on this logic explain. Gravitational force exerted Newton 's third law of motion, whenever objects a and B interact with other... Why it is essential to any student pilot to have a tendency to be splatterable also essential for understanding motion! Top picture ( below ), Kent is pulling upon a rope that attached! Force creates an equal and opposite in direction recoil is the result of force! A roadway surface the size of the law is probably the easiest of the bullet a. Inability of a fish uses its what is newton's third law to push water backwards but before,. Always occur in pairs larger velocity, it exerted a larger velocity, it will probably not.! forces always occur in pairs - equal and opposite reaction. and road the laws of?! The recoiling rifle is... a. with more force when the rope is attached the... Picture ( below ), Kent is pulling upon a rope that is seen most in the of! By Newton 's third law has 1 sense: ) reaction. this basic understanding physics. Physics Classroom, all rights reserved are equal in magnitude and opposite reaction. have been able to the... Times people confuse that the action and reaction are two different objects essential to any pilot... And in opposite directions rockets work formally stated: forces always come in pairs - equal and opposite in... According Newton ’ s third law of motion focuses on action reaction pair forces. | 2021-04-22 17:33:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6334778666496277, "perplexity": 438.12217063455336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039594341.91/warc/CC-MAIN-20210422160833-20210422190833-00438.warc.gz"} |
https://cs50.harvard.edu/law/2019/notes/2/ | # Lecture 2
## From Binary to Programming Languages
### Machine Code
• Computer manufacturers make CPUs or Central Processing Units which recognize certain patterns of bits. Thus, these patterns are computer or CPU specific.
• CPUs understand machine code. These are the zeroes and ones that tell the machine what to do. Machine code might look like this: 01111111 01000101 01001100 01000110 00000010 00000001 00000001 00000000.
### Assembly Code
• It’s quite difficult for us to code in machine code, so assembly code was created.
• Assembly code includes more english-like syntax. Assembly code is an example of source code.
• Source code is code with a more english-like syntax that can be translated to machine code.
• Some sequences of characters in assembly code include these: movl, addq, popq, and callq, which we might be able to assign meaning to. For example, perhaps addq means to add or callq means to call a function. What values are we doing these operations on? Well, registers!
• The smallest unit of useful memory is called a register. These are the smallest units that we can do some operation on. These registers have names, and we can find them in assembly code as well, such as %ecx, %eax, %rsp, and %rsb.
• Languages with easier to understand syntax than assembly code were created. Below is a program called hello.c that prints “hello, world” in the programming language C.
#include <stdio.h>
int main(void)
{
printf("hello, world\n");
}
### Compilers and Interpreters
• With hello.c from earlier, we have to convert the program to the zeroes and ones the computer can understand.
• To do this, we can use compilers, pieces of software that know both how to understand source code and the patterns of zeroes and ones in machine code and can translate one language to another.
• To compile hello.c, we can use something installed on our computers called CC, or C Compiler.
• To use the compiler, we go to our terminal window and type at the prompt.
• A terminal window is a keyboard only interface to tell your computer what to do.
• The prompt is represented by a dollar sign, $. • We type cc -o hello hello.c. This creates a new file called hello. • To run this program called hello, we type ./hello at the prompt where . represents the folder or directory that this file is in. • A sample of the terminal window might look like this:$ cc -o hello hello.c
$./hello hello, world • Some languages skip the step of compilers and instead use interpreters. Interpreters take in source code and run the source code, line by line, from top to bottom and left to right. • Interpreters are created with the zeroes and ones that the CPU understands. These zeroes and ones can recognize keywords and functions in the source code. • Python is an interpreted language. To say “hello, world” in Python, we write the following line in hello.py. print("hello, world") • To interpret this source code, at the terminal, we simply type python hello.py, where python is the name of the interpreter. • The program python, in this case, opens up the file hello.py, reads it top to bottom, recognized the function print and knew what to do, namely print “hello, world” on the screen and quit. • A sample of the terminal window might look like this:$ python hello.py
hello, world
• Comparing compilers and interpreters, we might note that interpreters skip the step of having a compiled program before running it. This causes a performance penalty for interpreter languages, since each time, the interpreter will have to re-interpret the code.
• To combat this issue, Python now generates bytecode, where it has already compiled the code and saved the results in a temporary file. When running the program again, Python will not interpret the code again but instead look at the pre-compiled version.
• Bytecode looks something like this:
6 CALL_FUNCTION 1 (1 positional, 0 keyword pair)
9 POP_TOP
13 RETURN_VALUE
### Virtual Machines
What if we want to run these programs on different computers, with different CPUs?
• A virtual machine is a software that mimics the behavior of an imaginary machine.
• With a virtual machine, instead of compiling the same code over and over again for different platforms, if each platform has this virtual machine installed, the exact same code can be run.
## Python
### Input and Printing
• To greet our human, we might write this in hello1.py:
name = input("What is your name? ")
print("hello, " + name)
• In Python, input is a function to get user input.
• This function takes in a string (this string prompts the human for an input) and returns a string.
• After returning this string, we would like to store it somewhere for access in the future. We can store these values in variables.
• To set a variable equal to a value, we use one single equal sign, often called the assignment operator.
• When printing, we can use the + operator to concatenate two strings.
• In the terminal, we would then have this, using “David” as input:
$python hello1.py What is your name? David hello, David • We can print in multiple ways. • The print function can take multiple arguments, and it separates arguments with spaces. • If we wrote print("hello, ", name), we would get two spaces between “hello,” and “David”, one in the string with hello, and another as the separator between the arguments. • To fix this, we can simply write print("hello,", name). • The print function can be formatted such that we can literally write name in the string and instead print the value. We must surround the variable with curly braces and prefix the string with f; this tells Python that this string should be formatted in a special way. These strings are often called format strings or f-strings. • We can write print(f"hello {name}"). • Let’s write the following code in arithmetic.py x = input("x: ") y = input("y: ") print(x + y) • Running this in the terminal, we get…$ python arithmetic.py
x: 1
y: 2
12
• We get 1 + 2 = 12. Remember that the input function returns a string and the + operator concatenates strings, and thus, we get the string “1” concatenated to “2”.
• To fix this issue, we can change the input value from a string to an int, or integer. The function to do that is simply int.
• Our code can then be written as…
x = int(input("x: "))
y = int(input("y: "))
print(x + y)
### Conditionals
• Let us instead write a program that compares two numbers.
• In conditions.py, we might write…
x = int(input("x: "))
y = int(input("y: "))
if x < y :
print("x is less than y")
elif x > y:
print("x is greater than y")
elif x == y:
print("x equals y")
• The Boolean expressions are x < y, x > y, and x == y.
• To check for equality, we have to use ==, since = is already the assignment operator.
• The colon after the if and elif statements specifically say to do the following if the Boolean expression is true.
• The indentations are necessary, so the print statements aren’t executed unless the Boolean expressions above them evaluate to true.
• The second elif, or “else if”, statement is unnecessary since if a number is not less than or greater than another number, it must be equal to that number. We can modify our program to get this…
x = int(input("x: "))
y = int(input("y: "))
if x < y :
print("x is less than y")
elif x > y:
print("x is greater than y")
else:
print("x equals y")
• In Boolean expressions, we can also use certain keywords: or and and.
• We might write a program answer.py that does the following:
if c == "Y" or c == "y":
print("yes")
elif c == "N" or c == "n":
print ("no")
• In this program, if the user inputs “Y”, c == "Y" will evaluate to true, and the program will print “yes”. If the user inputs “y”, c == "y" will evaluate to true, and the program will also print “yes”.
### Functions
• We might want to define our own function, such as square, where calling it returns the square of an input.
• In return.py, we might define our own function called square.
def main():
x = int(input("x: "))
print(square(x))
def square(n):
return n * n
if __name__ == "__main__":
main()
• Note that we can’t call the function square before defining the function square since the interpreter reads from top to bottom. To fix this, we can create a main function, and then call the main function at the end of the file.
• When we call the main function, we normally write a strange set of lines to ensure that the main function is not executed at the wrong time.
• With the square function, we’ve abstracted away the multiplication, and now we can simply call square.
### Loops
#### While Loops
• To write a program positive.py that will pester the human until the human inputs a positive integer, we might write the following:
def main():
i = get_positive_int("i: ")
print(i)
def get_positive_int(prompt):
while True:
n = int(input(prompt))
if n > 0:
break
return n
if __name__ == "__main__":
main()
• In the function get_positive_int, while True gives us an infinite loop. Python will then execute the indented code again and again until it is told to stop.
• Note that True and False are Boolean values.
• The break keyword tells Python to stop.
• Once the loop has been broken, the function returns the value.
#### For Loops
• To write a program score.py, where the user inputs a number and that many hashes are printed, we might write the following:
n = int(input("n: "))
for i in range(n):
print("#", end="")
print()
• range is a function built into Python that returns a range of values from 0 to n - 1 inclusive.
• The print function automatically prints a new line. In other words, it moves the cursor to the next line after printing. To stop Python from printing each hash on a separate line, we specify end="" as another argument to print, which tells Python to end the lines with nothing.
• The final print() moves the cursor to the next line.
• In the terminal, if we input 10 as n, we might see the following:
$python score.py n: 10 ########## #### Mario • In Super Mario Bros., a two dimensional world is created! Here’s one setting: • To print the series of question marks shown, we might write for i in range(4): print("?", end="") print() • Here’s another setting with a 4x4 block. • To print the block shown, we’ll need to print hashes on both rows and columns. We must first iterate through the rows, and within each row, we then iterate through each column and print a hash. for row in range(4): for column in range(4): print("#", end="") print() ### Types • In Python, there are many data types. • bool: True/False • int: Numbers • str: Strings of text • float: Real numbers with decimal points and digits after • dict: Hash table • list: Any number of values back to back • range: Range of values • set: A set of values with no duplicates • tuple: x, y or latitude, longitude ### Libraries • In addition to the functions built into the core language, there are libraries and frameworks that provide additional features. These have to be imported manually to be used. • For example, in Python, if we want to generate pseudorandom numbers, we have to import a function randint from a library called random. • For example, to get a random integer between 1 and 10, we can write this: from random import randint print(randint(1, 10)) • We can also just write import random without importing the specific function. In this case, we’ll have to prefix the function with the library name using dot notation as shown below. • To create a game where the user guesses a random integer between 1 and 10, we can write this: import random n = random.randint(1, 10) guess = int(input("Guess: ")) if guess == n: print("Correct") else: print("Incorrect") • Note that these numbers are pseudorandom because computers can’t pick a random number like humans, they have to use algorithms, which are deterministic processes. ## Memory • Inside a computer is hardware. These hardware chips are called RAM, or Random Access Memory. Inside each of these chips is some finite number of bytes used to represent values in our programs. • Python, and most other languages, decide a priori how many bits to use to represent values in our programs. • Thus, if our value cannot be represented in only that many bits, the language will instead approximately represent that value. ### Imprecision • Let’s take a look at a program called imprecision.py that divides two numbers and returns the quotient. x = int(input("x: )) y = int(input("y: )) z = x / y print(f"{z:.30f}") • The syntax :.30f signifies that we’re printing z as a float to 30 decimal places. • We get…$ python imprecision.py
x: 1
y: 10
x / y = 0.100000000000000005551115123126
• This value isn’t what we expect! We don’t have enough bits to store the entire precise value, so the computer approximates the quotient. This is called floating-point imprecision.
### Integer Overflow
• A similar problem occurs with integers.
• Consider a number that has been allocated three digits.
• We start by counting.
• Suppose we count until 999. We carry, and we get 1000.
• However, the computer has only allocated three digits, so our 1000 gets mistaken for 000.
• This is an example of integer overflow, where our large number has wrapped to a small number.
• On December 31, 1999, people began to get nervous—programs stored the calendar year with only two digits. For 1999, the year was stored as 99. When the year 2000 approached, then, the year would be stored as 00, leading to confusion between the year 1900 and 2000. This became known as the Y2K problem.
• In the past, Boeing 787 planes stored the number of hundredths of seconds in a counter. Once that counter overflowed (occurring on the 248th day), the plane would go into fail-safe mode and the power would shut off. | 2022-10-07 13:39:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2734568417072296, "perplexity": 2011.4127732843808}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338073.68/warc/CC-MAIN-20221007112411-20221007142411-00035.warc.gz"} |
https://mathoverflow.net/questions/286253/disjoint-covering-number-of-an-ideal | # Disjoint covering number of an ideal
Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.
Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest cardinality of a cover of $X$ by (parvise disjoint) Borel sets that belong to the ideal $\mathcal I$.
The cardinal $\mathrm{cov}(\mathcal I)$ is one of four classical cardinal characteristics of an ideal. What about its disjoint modification? Was it considered in the literature?
Observe the following trivial relations between the cardinals $\mathrm{cov}(\mathcal I)$ and $\mathrm{cov}_\sqcup(\mathcal I)$:
1) $\mathrm{cov}(\mathcal I)\le\mathrm{cov}_\sqcup(\mathcal I)$;
2) If $\mathrm{cov}(\mathcal I)\le\omega_1$, then $\mathrm{cov}(\mathcal I)=\mathrm{cov}_\sqcup(\mathcal I)$.
Problem 1. Is $\mathrm{cov}(\mathcal I)=\mathrm{cov}_\sqcup(\mathcal I)$?
This problem is especially interesting for the ideal $\mathcal M$ of meager subsets of the real line and for the ideal $\mathcal N$ of Lebesgue null sets in $\mathbb R$.
Problem 2. Are the strict inequalities $\mathrm{cov}(\mathcal M)<\mathrm{cov}_\sqcup(\mathcal M)$ and $\mathrm{cov}(\mathcal N)<\mathrm{cov}_\sqcup(\mathcal N)$ consistent?
The negative answer to the second part of problem 2 would give an affirmative answer to Problem 8 of this MO post.
• If $\mathcal I$ is closed under taking arbitrary subsets, then any family $\mathcal A$ of sets in $\mathcal I$ can be made into a disjoint family $\mathcal A'$ of set in $\mathcal I$ with $\bigcup \mathcal A = \bigcup \mathcal A'$. (Just enumerate $\mathcal A = \{A_\alpha : \alpha < \kappa\}$ and put $A'_\alpha = A_\alpha \setminus (\bigcup_{\beta < \alpha}A_\beta)$.) – Will Brian Nov 16 '17 at 19:24
• In the linked post, what makes things so hard (and interesting!) is that you're only looking at the closed sets in $\mathcal I$, so this trick doesn't work. – Will Brian Nov 16 '17 at 19:26
• @WillBrian Please note that I am requiring the pieces to be Borel! But your trick does not produce Borel sets (especially for $\kappa>\omega_1$). – Taras Banakh Nov 16 '17 at 19:28
• @WillBrian What I realized is that $\acute{\mathfrak n}$ is equal to the smallest cadinality of a disjoint compact cover of any non $\sigma$-compact absolute Borel space (just use the fact that each absolute Borel space is a continuous injective image of a Polish space). So, in a sense $\acute{\mathfrak n}$ is a universal "constant". It would be interesting to know what happens with $\acute{\mathfrak n}$ for analytic non-Borel spaces. – Taras Banakh Nov 16 '17 at 19:37
• @WillBrian On the other hand, the cardinal $\acute{\mathfrak m}$ is equal to the product $\acute{\mathfrak n}\cdot\mathrm{cov}_\sqcup(\mathcal N)$, so is reducible in a sense. This was a motivation for my question on $cov_\sqcup(\mathcal I)$. – Taras Banakh Nov 16 '17 at 19:39 | 2019-03-19 23:50:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9538747668266296, "perplexity": 248.25352970171357}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202161.73/warc/CC-MAIN-20190319224005-20190320010005-00127.warc.gz"} |
https://tex.stackexchange.com/questions/446548/how-to-change-the-default-border-color-of-fbox | # How to change the default border color of fbox? [duplicate]
I have a lot of figures inside an fbox. Unfortunately the default black is too strong for my advisor and I need to change the default border color to something more soft.
I've done my research and so far all other answers suggest do solve the problem by not using fbox and switching to other packages. That's something I'd like to avoid, unless absolutely necessary.
Surely there is an option for something so basic in the fbox package itself?
## marked as duplicate by Zarko, Phelype Oleinik, Stefan Pinnow, Sebastiano, CarLaTeXAug 19 at 6:35
• Welcome to TeX.SX! What do you mean by "fbox package"? – TeXnician Aug 18 at 12:19
• fbox is not a package but a simple command. You could redefine it, but this would affect all uses of fbox. – Ulrike Fischer Aug 18 at 12:20
• @UlrikeFischer How would one do so? – user8272359 Aug 18 at 12:51
• @BambOo I specifically linked to that post already and commented that it is not what I want. – user8272359 Aug 18 at 12:52
• It is not beacause some things that you saw there are not what you want that every piece of information from this post isn't relevant for you or anyone else. If you look deeper into details, the linked post has been deemed a duplicate of a previous one where a solution using only the color package was proposed as \newcommand{\myfbox}[2]{\textcolor{#1}{\fbox{\normalcolor#2}}}. By the way we cannot say what packages are allowed or not if you do not give us a list – BambOo Aug 18 at 13:01
You can try this. It will also set the background to white, normally this should be not a problem.
\documentclass{article}
\usepackage{xcolor}
\renewcommand\fbox{\fcolorbox{red}{white}}
\begin{document}
\fbox{text}
\end{document}
• To me, this is a good answer, so +1, but looking at the OP's comments, he/she probably won't see it that way. – BambOo Aug 18 at 13:05
• This is exactly what I want: a way to set the "default border color of fbox". I understand that maybe my terminology was wrong due to being a latex novice, but this user seemed to be able to see through that. I'm sure the ever flowing visitors from Google will agree with me that it is a relevant problem that has not been answered as desired by most before, and will be happy to see this answer here. – user8272359 Aug 18 at 15:51
Without setting the background color, one can save the current text color and apply it when the text of the \fbox is typeset, changing color for typesetting the rules.
\documentclass{article}
\usepackage{xcolor}
\usepackage{etoolbox}
\makeatletter
\let\cfbox\fbox
\let\c@frameb@x\@frameb@x
\pretocmd{\cfbox}
{\leavevmode\begingroup\colorlet{currentcolor}{.}\color{red}}
{}{}
\patchcmd\cfbox{\@frameb@x}{\c@frameb@x}{}{}
\patchcmd{\c@frameb@x}
{\box\@tempboxa}
{\color{currentcolor}\box\@tempboxa}
{}{}
\apptocmd{\c@frameb@x}{\endgroup}{}{}
\makeatother
\begin{document}
text \cfbox{text} text
\fcolorbox{green}{blue}{text \cfbox{text} text}
\end{document}
I'd prefer a different command to redefining \fbox. If you instead want to use \fbox, just remove the \let\cfbox\fbox line and change \cfbox into \fbox in the remaining places. | 2018-11-13 16:41:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5035937428474426, "perplexity": 1298.1620239370275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741324.15/warc/CC-MAIN-20181113153141-20181113175141-00434.warc.gz"} |
https://www.nature.com/articles/s41598-022-13091-7?error=cookies_not_supported&code=071e255c-b796-40a0-8ab4-ee3e1e099275 | Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
Short periods of bipolar anodal TDCS induce no instantaneous dose-dependent increase in cerebral blood flow in the targeted human motor cortex
Abstract
Anodal transcranial direct current stimulation (aTDCS) of primary motor hand area (M1-HAND) can enhance corticomotor excitability, but it is still unknown which current intensity produces the strongest effect on intrinsic neural firing rates and synaptic activity. Magnetic resonance imaging (MRI) combined with pseudo-continuous Arterial Spin Labeling (pcASL MRI) can map regional cortical blood flow (rCBF). The measured rCBF signal is sensitive to regional changes in neuronal activity due to neurovascular coupling. Therefore, concurrent TDCS and pcASL MRI may reveal the relationship between current intensity and TDCS-induced changes in overall firing rates and synaptic activity in the cortical target. Here we employed pcASL MRI to map acute rCBF changes during short-duration aTDCS of left M1-HAND. Using the rCBF response as a proxy for regional neuronal activity, we investigated if short-duration aTDCS produces an instantaneous dose-dependent rCBF increase in the targeted M1-HAND that may be useful for individual dosing. Nine healthy right-handed participants received 30 s of aTDCS at 0.5, 1.0, 1.5, and 2.0 mA with the anode placed over left M1-HAND and cathode over the right supraorbital region. Concurrent pcASL MRI at 3 T probed TDCS-related rCBF changes in the targeted M1-HAND. Movement-induced rCBF changes were also assessed. Apart from a subtle increase in rCBF at 0.5 mA, short-duration aTDCS did not modulate rCBF in the M1-HAND relative to no-stimulation periods. None of the participants showed a dose-dependent increase in rCBF during aTDCS, even after accounting for individual differences in TDCS-induced electrical field strength. In contrast, finger movements led to robust activation of left M1-HAND before and after aTDCS. Short-duration bipolar aTDCS does not produce consistant instantaneous dose-dependent rCBF increases in the targeted M1-HAND at conventional intensity ranges. Therefore, the regional hemodynamic response profile to short-duration aTDCS may not be suited to inform individual dosing of TDCS intensity.
Introduction
Transcranial Direct Current Stimulation (TDCS) is widely used to modulate cortical function in cognitive neuroscience and in therapeutic settings1,2,3,4,5,6,7,8. The underlying mechanism of action still remains to be fully clarified. Neurophysiological studies suggest that TDCS modulates the intrinsic activity of cortical neurons by inducing shifts in the neuronal membrane potential, presumably by polarizing axon terminals9,10.
TDCS has been shown to produce polarity-specific bi-directional effects on corticomotor excitability by using a bipolar montage with one skin electrode placed on the scalp over the primary motor hand area (M1-HAND) and the other electrode placed on the contralateral supraorbital region11,12,13. However, high inter-individual variability has been observed in neurophysiological studies using single-pulse Transcranial Magnetic Stimulation (TMS) and motor evoked potential (MEP) to probe lasting effects of TDCS on corticospinal excitability14, and the consistency of the after-effects has been challenged by several recent studies15,16,17. This is also reflected in therapeutic settings, were the majority of TDCS studies have yielded varying results and have failed to provide clear evidence for efficacy18.
The large inter-individual variability may partially be caused by the way current intensity of TDCS is determined. Differences in head and brain anatomy significantly influence strength and spatial distribution of the induced electrical field (E-field)19,20. Adjusting the stimulation intensity using field simulations informed by individual structural MRI can account for some variance at the single-person level21,22, but we still lack personalized “dosing methods” that allow to adjust the current intensity of TDCS based on the individual cortical response profile. Personalized dosing requires reliable brain mapping approaches that can reliably delineate the dose–response relationship between the TDCS current intensity and the TDCS induced change in regional cortical activity at the single-person level. This information might help to personalize the current intensity of TDCS in a way that the induced E-fields optimally engages the targeted cortical area.
For dose–response mapping, we were interested in the acute and immediate changes in neural activity induced by short periods of TDCS, as they reflect the direct impact of stimulation in the cortical target region, in contrast to the aftereffects of prolonged stimulation that can be influenced by secondary mechanisms over several minutes.
Short period TDCS inducing immediate neuronal effects
Previous work has used single-pulse TMS to demonstrate that short periods of bipolar TDCS of M1-HAND evokes acute changes in corticospinal excitability. Early TDCS studies used TMS to show that short periods of bipolar anodal TDCS (aTDCS) (4–120 s) targeting M1-HAND induce an immediate rise in MEP amplitude, indicating an instantaneous rise in corticospinal excitability without sustained after-effects11,23. Single-pulse TMS also revealed an acute increase in the amplitude of the motor evoked responses in the contralateral hand during 30 s of aTDCS at a current intensity of 1.5 mA24. In that study, the acute “online” effect was also predictive of the lasting increase in MEP amplitude after the end of aTDCS24. Given the neurophysiological evidence that short periods of aTDCS do cause a shift in corticospinal excitability, we decided to use short stimulation periods of 30 s during pcASL MRI in order to map the relationship between the current intensity of TDCS and the magnitude of the regional rCBF response in the targeted M1-HAND.
Acute perfusion response to brain stimulation
Short periods of transcranial magnetic stimulation (TMS) have been shown to induce an acute modulation of regional blood flow in the targeted motor cortex. For instance, Siebner et al. used H215O positron emission tomography (PET) in six healthy individuals to show a consistent dose–response pattern during subthreshold repetitive TMS of the left M1-HAND25. All participants showed a linear increase in regional cerebral blood flow (rCBF), when the frequency of repetitive TMS was gradually increased from 1 to 5 Hz25. Another study by Paus et al. stimulated the frontal eye field of six healthy individuals with TMS during H215O PET and found a relative increase in regional blood flow in the stimulated region with the number of stimuli that were applied during the PET scan. Robust immediate and short-term rCBF changes in the stimulated sensorimotor cortex had also been found in another H215O-PET study, targeting the left M1-HAND with sub-motor threshold TMS at 5 Hz26.
Several preclinical studies have also verified that the acute exposure to short periods of direct-current stimulation gives rise to instantaneous changes in spontaneous neural firing and regional blood flow in the stimulated cortex. Direct electrophysiological intracortical recordings showed that short periods of radially applied anodal direct current stimulation raises the mean firing rate of cortical neurons, with opposite effects during cathodal direct current stimulation27,28,29. While we are not aware of any preclinical study that has measured the acute vascular response to short periods of TDCS, prolonged periods of TDCS induced polarity-specific after-effects on cortical blood perfusion in the rat30. Yet there is evidence for immediate changes in rCBF during short periods of pulsed or alternating current stimulation31,32,33,34.
TDCS and perfusion imaging studies
One promising way to map the regional neurovascular response to TDCS is with functional magnetic resonance imaging (fMRI) and Arterial Spin Labelling (ASL), that uses magnetically labeled water in the arterial blood as a tracer of rCBF35,36. Despite of the evidence for immediate neurovascular effects of electric brain stimulation, fMRI studies focusing on immediate TDCS effects are sparse37,38,39. A few perfusion studies using ASL-fMRI and H215O -PET measuring sustained after-effects of TDCS with longer stimulation periods indicate a general increase of rCBF during bipolar TDCS40,41,42 and recent ASL-fMRI studies show dose-related increases in rCBF in the cortex underneath the precentral electrode during and after aTDCS at conventional intensities (0.5–2.0 mA)43 and at higher intensities up to 4 mA44.
Prompted by the previous combined TMS-PET studies and TDCS ASL-fMRI studies, we measured regional brain perfusion with MRI to test whether short duration TDCS would also induce consistent dose–response patterns on a single subject level. Using a classical bipolar electrode arrangement, we applied 30 s epochs of aTDCS at four different intensities from 0.5 to 2.0 mA to the left M1-HAND45. Simultaneously, we measured regional cortical perfusion with pseudo-continuous arterial spin labelling (pcASL). We expected that all individuals would show, at least to some degree, an instantaneous and intensity dependent pattern of regional perfusion changes in the target cortex. If this were the case, we reasoned that ASL-fMRI might be used to adjust the individual intensity of TDCS in future studies.
Results
None of the participants experienced major side effects during the study. The results of the psychometric assessment are presented in Appendix A as Supplementary Material.
Regional perfusion changes in the precentral M1-HAND target region
We first examined mean perfusion changes during finger-tapping prior (FTpre) and post stimulation (FTpost) and during aTDCS from 0.5 to 2.0 mA within the volume of interest (VOI) defined by the finger-tapping related functional activation, M1FT (Illustrated in Fig. 1, and Supplementary Fig. 2. (For more details on our VOI definitions, we refer to the methods section and Table 3). Two-way repeated measures ANOVA revealed a significant main effect of “conditions”, indicating a difference in mean rCBF within M1FT between all six experimental conditions (FTpre, FTpost, 0.5 mA TDCS, 1.0 mA TDCS, 1.5 mA TDCS, 2.0 mA TDCS) (F = 54.205, p < 0.000). Pairwise comparisons showed a significantly higher rCBF during FT compared to all aTDCS intensities after Bonferroni correction (Table 1). FT induced rCBF increases after the aTDCS blocks did not differ from FT induced perfusion prior to aTDCS (FTpost vs. FTpre). There was no significant difference in rCBF between any of the applied aTDCS intensities, but mean rCBF during 0.5 mA aTDCS was slightly higher than baseline rCBF (Fig. 2). Post-hoc exploratory one-sample t-tests on each intensity condition revealed an increased perfusion (z-score) in M1FT at 0.5 mA aTDCS compared to baseline, puncorrected = 0.005 which survived Bonferroni correction (pBonferroni = 0.02).
To control for any potential bias introduced by using a group-specific functional VOI (M1FT), we also analyzed rCBF changes using an anatomically defined VOI (M1-S1anat) (Fig. 3).
The rCBF changes in the M1-S1anatVOI were less strong compared to M1FT, but revealed a similar response pattern. Two-way repeated measures ANOVA revealed a significant main effect of “conditions” (F = 9.436, p < 0.000). Pairwise comparisons showed significant difference between rCBF during FT compared to all stimulation intensities after Bonferroni correction (Table 2), with no difference between FTpost vs. FTpre and no significant difference in rCBF between the aTDCS intensities. One-sample t-test on each aTDCS intensity also revealed increased perfusion at 0.5 mA compared to baseline, puncorrected = 0.028, but the increase in rCBF during aTDCS at 0.5 mA did not survive correction for multiple comparisons (pBonferroni = 0.112).
Analysis of aTDCS-related perfusion changes in the sub-regions M1deep, M1superficial and PMd (VOI’s shown in Fig. 4A) revealed no significant main effect of aTDCS intensity on rCBF in any of the functionally defined sub-regional VOIs (VOIM1deep: F = 1.04 p = 0.38, VOIM1superficial: F = 0.33 p = 0.81, VOIPMd: F = 0.63 p = 0.61).
Perfusion changes in the cortical region exposed to the highest E-field
The aTDCS-induced electrical field reached its maximum in the dorsolateral prefrontal cortex (DLPFCE-field) (Fig. 4A). The mean E-field in the dorsolateral prefrontal VOI (DLPFCE-field) was consistently higher than the mean E-field in the precentral VOIs (Fig. 4D). We did not find any significant effect of aTDCS or FT on mean rCBF in the DLPFCE-field VOI (F = 0.290 p = 0.916) and individual rCBF activity in the E-field guided prefrontal VOI was highly variable, showing no consistent intensity dependent response pattern (Fig. 4B).
Whole-brain analysis
Whole-brain analysis revealed no significant increase in rCBF for any of the four aTDCS conditions, or any linear voxel-wise increase or decrease. This was different for the finger tapping task. Right-hand FT before aTDCS (FTpre) lead to increase in rCBF in left M1-HAND (Fig. 1 and Supplementary Fig. 2). Exploratory analysis at a more liberal cluster-forming threshold (z = 2.3 or p = 0.01) also showed a tapping-associated decrease of rCBF in the right primary sensorimotor cortex ipsilateral to the tapping right hand (Fig. 1). FT after stimulation (FTpost) with the right hand also led to increase in rCBF in left M1-HAND (Supplementary Fig. 2).
Relationship between induced E-field and regional perfusion changes
Neither for M1superficial, M1deep, PMd and or DLPFCE-field, we found a consistent relationship between regional perfusion changes and the induced E-field in our participants (Fig. 4C). We tested whether the perfusion levels in these VOIs could be predicted by individual induced E-field by specifying a linear regression model to predict perfusion activity (z-score) from E-field at each of the four aTDCS intensities. Only the E-fields induced with aTDCS at an intensity of 0.5 mA significantly predicted z-score in M1deep, F = 9.333, pUncorrected = 0.018, R2 = 0.571. This however did not survive Bonferroni correction for multiple comparisons (pBonferroni = 0.288). The corresponding plots are shown in Supplementary Fig. 4.
Discussion
Using pcASL-MRI, we measured relative changes in rCBF as proxy for regional neural activity during short epochs of bipolar aTDCS targeting the left M1-HAND. In different aTDCS blocks, we varied the intensity of aTDCS, applying currents at 0.5, 1.0. 1.5, and 2.0 mA. We found no evidence in support of the hypothesis that aTDCS produces intensity-dependent increases in rCBF in the cortical target region. This was the case for all subregions of the targeted pericentral cortex but also in the portion of the laterodorsal prefrontal cortex where the induced E-field was highest.
No modulation of rCBF in M1-HAND during short-duration aTDCS
We found no change in the regional perfusion level during our 30 s blocks of aTDCS compared to baseline. Because our main interest was to map individual dose–response of TDCS, we aimed to measure the immediate effect of regional neuronal response, in contrast to secondary prolonged aftereffects. We expected that the short 30 s of TDCS would be sufficient to create an acute increase in rCBF, because early electrophysiological studies on animals have showed that short direct polarization of the cortex (0.2 s repetitive and 5–10 s periods) with electrical stimulation can induce acute changes in spontaneous neural firing rate27,28,29. Previous preclinical rat studies also confirmed acute vascular response to short duration TES using Laser-Doppler Flowmetry (LDF) with immediate frequency and intensity dependent vasodilation of the middle meningeal artery up to 140% of baseline after 10–30 s of rectangular pulse waves on the skull32,33 and dura31. Functional near-infrared spectroscopy (fNIRS) has also measured immediate increase in Oxy-Hb concentration following 200 μA aTDCS, applied 10 min over the right barrel cortex, and a decrease right after stimulation terminated46.
As we used a different imaging technique to probe the neural and vascular response, this might account for the lack of reproducibility of these previous blood flow findings. However, our negative findings appear to be at variance with recent ASL-MRI TDCS studies that reported a regional increase in cortical perfusion underneath the electrode. Previous studies used markedly longer stimulation blocks, lasting 8 or 20 min40,41. Therefore, it is likely that prolonged TDCS protocols may result in more robust and consistent effects on rCBF that will be traceable with fMRI-ASL. The effects on rCBF produced by longer TDCS protocols may not primarily be caused by the direct depolarizing effects of TDCS, reflecting the TDCS-induced shift in neural activity. Changes in rCBF after prolonged TDCS may rather be driven by secondary neuromodulatory effects of TDCS47,48. If so, ASL may be more reliable in capturing TDCS-induced changes in rCBF that are caused by longer lasting, secondary neuronal responses.
The absence of reliable rCBF changes in response to 30 s periods of aTDCS also raises the question whether short protocols are sufficient to shift corticomotor excitability in the targeted M1-HAND at all. Here it is important to note that TMS of M1-HAND did reveal a significant increase in cortical excitability already after a short period (i.e., 4 s) of low intensity (1 mA) aTDCS11. As we did not find a comparable immediate increase in regional perfusion, we hypothesize that the rCBF response probed with ASL is also less responsive to the polarization effects of TDCS than the pyramidal cortico-spinal neurons, that are trans-synaptically probed with TMS.
Robust rCBF increase in left M1-HAND during finger tapping
While we were not able to detect stimulation-related rCBF changes, M1-HAND showed a robust motor-task induced increase in rCBF during contralateral finger tapping. This indicates that our pcASL measurements were sensitive to increases in regional neuronal activity. When using a more liberal threshold, there was also a trend towards a decrease in regional perfusion in the ipsilateral (right) primary sensory cortex. The task-induced ipsilateral perfusion decrease has not been studied with ASL previously but supports previous neurophysiological findings on interhemispheric interaction49, and could be utilized by future studies to investigate disruptions of the interhemispheric network during conditions like stroke. Together, the demonstration of task-related bi-directional changes in M1-HAND activity suggests that the lack of any stimulation-induced effects on rCBF in M1-HAND cannot be attributed to an inability of our ASL sequence to detect perfusion changes.
Comparing the perfusion during finger tapping before and after the stimulation blocks suggested that the four aTDCS blocks did not induce a change in rCBF that outlasted stimulation. This is consistent with previous BOLD fMRI studies, showing no significant change of task-related activation after consecutive blocks of 20 s TDCS50 and 5 min TDCS51.
No dose-dependent changes in rCBF during short-duration TDCS
We did not find any intensity-related difference in rCBF levels during short-duration aTDCS. Our focus on the influence of stimulation intensity on TDCS-induced changes in regional neuronal activity is motivated by previous dose-dependent TMS studies, showing a sigmoidal dose–response curve in stimulation intensity and evoked motor response and motor evoked potential52,53,54. There is still a great deal of uncertainty regarding which biophysical and neurobiological mechanisms contribute to the neuromodulatory effects of TDCS and how they interact with each other. Although we expect an increase in neuromodulation with current intensity, this remains to be shown. Currently, it is difficult to predict the shape the dose–response relationship. Since the neurophysiological impact of TDCS will be influenced by multiple neurobiological factors55, the dose–response relationship between stimulation intensity and the induced change in regional neuronal activity is likely non-linear. A non-linear relationship between changes in neuronal activity and the current intensity of TDCS has been suggested previously56,57, with lower currents potentially inducing stronger effects than high currents58. Of note, we found a small but consistent effect of low-intensity aTDCS at 0.5 mA on rCBF in the deep region of M1-HAND (Fig. 2, Supplementary Fig. 4E). We find this activity-enhancing effect at the lowest current intensity interesting, as it may reflect stochastic resonance effects, which can emerge at very weak electric fields55. As the sample size is rather small, we cannot exclude a false-positive finding, and this effect should be replicated in future studies for verification.
We did not apply aTDCS at intensities above 2 mA. Therefore, it is possible that higher current intensities might be needed for causing a consistent change in rCBF with short-duration aTDCS. This hypothesis is supported by recent findings from Jonker et al. who did not find any effect on cortical excitability after 20 min of aTDCS at 2 mA16 together with in vivo measurements of neural spiking activity in animals20 and humans19,59. These measurements suggest that convential TDCS at 2 mA current induces electrical field gradients at the lower end of needed to induce neuronal spiking activity. Our negative results thus emphasizes the relevance of exploring the effect of TDCS at higher current intensities in future experiments.
Difference in target engagement in sub-regional VOIs
Recent biophysical models suggest that the effect of the induced current is dependent on the orientation of the targeted neurons. Pyramidal tract neurons have lower thresholds for activation at the axon terminal and node of Ranvier60, and the depolarization of these segments is highly dependent on their orientation to the electric field: Neurons in the gyrus crown are oriented perpendicular to the E-field and depolarization will predominantly occur at the proximal part of the axon60,61, whereas neurons located in the sulcal depth will not achieve the same level of polarization. Different susceptibility to electric stimulation in superficial versus deeper structures is supported by E-field simulations showing that the induced electrical field is highest at the gyral crown62. We therefore expected higher engagement of aTDCS in the superficial M1. Contrary to our hypothesis, aTDCS did not induce regional-specific perfusion changes, neither in the superficial M1-HAND nor in deeper regions or the adjacent PMd. The inability to detect any region-specific effects on rCBF might be attributed to the low current intensities and the use of a classical non-focal electrode montage.
E-field simulations of target engagement
Our E-field simulations revealed that bipolar aTDCS induced its maximum E-field more rostrally in the frontal cortex than the intended stimulation target (M1). There was however no significant aTDCS induced change in rCBF in the area of maximal current density either. At the group level, the peak E-field location was located in the left dorsolateral prefrontal cortex. This finding is consistent with previous work showing that the maximum TDCS-induced E-field is not necessarily located directly underneath the stimulation electrodes63.
In a recent ASL-TDCS study, Jamil et al. reported that 15 min of bipolar TDCS targeting M1 induced polarity-specific changes in rCBF under the M1-electrode, together with regional E-field dose-dependency with the strongest correlation at TDCS intensities between 1 and 2 mA43. In our study, we used the same montage and intensity range, but applied short period TDCS. The short periods of TDCS showed no correlation between the regional induced E-fields and rCBF. Interestingly, the individually induced E-field in the deep region of M1-HAND predicted the rCBF increase during low-intensity aTDCS at 0.5 mA. This finding suggests that for short lasting TDCS, the induced E-field scales with the neuronal activation in the deeper part of M1-HAND in a dose-dependent manner only at low current strength. Stochastic resonance may explain why variations within an apparently narrow range of weak currents positively correlate with the regional neural response55,64.
In summary, We consistently found that the immediate perfusion responses to short-duration aTDCS are highly variable among subjects (as presented in Fig. 4). On the one hand, this inter-individual variability challenges the common practice to apply a fixed stimulation intensity in TDCS studies, as this will most likely evoke substantially different physiological responses across individuals. On the other hand, the fact that we could not find a clear E-field-dependent change in rCBF, indicates that our setup may not be the best feasible method to define individual dose–response relationships and to guide the personalized dosing of TDCS.
Limitations
We used pcASL-MRI to measure changes in regional perfusion as proxy read-out of regional neural activity. Since the ASL signal is noisy, we may have missed subtle changes in regional perfusion. Since our pcASL MRI approach reliably detected movement related activation of M1-HAND in each individual, we argue that our approach was sensitive enough to detect task-related fluctuations in regional neuronal activity.
The experimental design did not include a sham condition to control for off-target effects of aTDCS for instance due to somatosensory co-stimulation. We also did not test for polarity dependent effects, as we only targeted the M1-HAND with anodal stimulation. However, as we have not found any significant changes in regional cortical perfusion related to stimulation, we argue that this does not reduce the reliability of the results. The nine participants in this study showed highly variable cortical perfusion patterns during TDCS. Apart from a subtle increase in regional cortical perfusion at the lowest intensity level, a short period of bipolar aTDCS failed to induce a further increase in cortical perfusion at higher current intensities. We thus conclude that our TDCS-ASL approach cannot be used to inform personalized dose adjustment in TDCS studies.
Our study captured individual dose–response profiles and demonstrated substantial inter-individual variability. Although we found no consistent dose–response relationship between current intensity of aTDCS and regional cortical perfusion, the low number of participants precludes any firm conclusions at the group level. On the one hand, the small sample size may have introduced a bias towards detecting false-positive spurious findings (e.g., the mild increase in perfusion at the lowest stimulus intensity). On the other hand, the small sample size bears an inherent risk for missing out subtle differences in regional cortex perfusion among stimulation conditions due to reduced statistical power. Future studies aimed at target engagement and personalization of TDCS may focus more on focal electrode montages65, taking individual anatomy into consideration for electrode placement62, and investigating the tolerability and effect of higher current intensities19,20.
Conclusion
We show that pcASL MRI does not reveal consistent immediate effects of short-duration aTDCS on neural activity, but that it reliably picks up activity changes during a simple unimanual task. The motor task was associated with a perfusion increase in the contralateral M1-HAND along with a concurrent decrease in perfusion in the M1-HAND ipsilateral to the moving hand. This discrepancy suggests that the immediate changes in regional neural activity during short periods of aTDCS may be too subtle to be detected with pcASL. We therefore conclude that our approach is not suited for mapping dose–response profiles using short periods in this manner. It is possible that more robust results may be achieved by performing pcASL MRI during longer administration periods of aTDCS. This would also better match to TDCS protocols that are more commonly used in clinical and research settings. Longer periods of aTDCS might produce more substantial shifts in regional neuronal activity leading to clearer changes in rCBF. One should also note, that such changes might have a primary neural or vascular origin, given that TDCS has primary effects on both the neural and vascular structures of the brain66.
Materials and methods
Subjects
Nine young healthy individuals participated in this study (mean age 31.22; SD 4.55; 5 males). All subjects were right handed, as determined by the Edinburgh Handedness Inventory67 with a mean laterality quotient of 97. Subjects were recruited from an open access advertisement posted on a website for subject recruitment (http://www.forsøgsperson.dk). All subjects gave their written informed consent. The experimental protocol (H-18031987) has been approved by the Regional Committee on Health Research Ethics of the Capital Region of Denmark, and the Declaration of Helsinki. All methods were performed in accordance with approved institutional guidelines and regulations.
As described in the introduction, several combined TMS-PET studies found robust increases in rCBF in small groups of healthy individuals25,26,68. Since dose–response profiles were consistently expressed at the subject level, we reasoned that a sample size of nine individuals would be sufficient to test our two hypotheses that regional perfusion in the stimulated M1-HAND would increase with the intensity of TDCS and that this response pattern would be consistently expressed among healthy participants.
Study design
Participants underwent a single pcASL-MRI session that lasted approximately 50 min. The experimental procedures are illustrated in Fig. 5. Bipolar aTDCS was applied with the anode targeting left primary motor cortex (M1) and the cathode on the right side of the forehead (supraorbital (SO) region). During stimulation, participants were resting in the MRI scanner with their eyes focusing on a fixation cross displayed in the middle of a screen.
The experiment included four aTDCS blocks. During a TDCS block, one of four current intensities were applied. Target current intensity was set at 0.5 mA, 1.0 mA, 1.5 mA or 2.0 mA in a given block. The order of blocks was pseudorandomized and participants were blinded. Each aTDCS block lasted 10 min and consisted of 4 min no-stimulation baseline followed by alternating epochs of aTDCS (38 s) and periods without TDCS (58 s). During an aTDCS epoch, stimulation current was linearly ramped up to the target intensity within 4 s, continuously applied at target intensity for 30 s, and then ramped down again within 4 s.
After each aTDCS block, the participants answered a series of questions through the MR-speaker system about how they had experienced the preceding aTDCS block, using a six-level Visual Analogue Scale (VAS) (see Supplementary Table 1).
We included finger-tapping (FT) blocks without stimulation at the beginning (FT-pre) and the end (FT-post) of the pcASL-MRI experiment to compare rCBF changes evoked by aTDCS with rCBF change during voluntary motor activity. During FT blocks participants tapped their right (dominant) index and thumb together paced by a blinking cross (2 Hz). Each FT-block was four min, consisting of interleaved 32 s epochs of FT and 32 s of rest.
Transcranial DC stimulation in the MRI scanner
We applied aTDCS at 0.5, 1.0, 1.5 and 2.0 mA. The center of the anodal electrode corresponding to the C3 location of the 10/20 EEG system69, with 2-mm thick 7 × 5 cm rubber electrodes (NeuroConn, Illmenau, Germany). Connector plugs were arranged pointing down towards the ear for the anodal electrode, and horizontally outwards for the cathodal electrode. Electrodes were applied with a thin layer of ten-20 conductive gel (Weaver and Company, Aurora, Colorado, US) and fixated with a net cap. Current was applied by a battery-operated DC-stimulator with MR compatible stimulation cables and filter-boxes (NeuroConn Illmenau, Germany). For safety reasons, the total impedance was kept below 15 k$$\Omega$$, which included the two 5 k$$\Omega$$ resistors in the cables.
MRI image acquisition
Images were acquired on a Phillips 3 Tesla MR Achieva scanner (Philips, Best, Netherlands) using a 32-channel head coil. A localizer scan assessed the head-position prior to a structural T1-weighted whole-brain scan using a 3d-TFE multi-shot sequence (TR/TE = 6.0/2.7 ms; flip angle = 8°; FOV = 245 FH 245 AP 208 RL mm3; isotropic resolution = 0.85 mm3) and a T2-weighted whole-brain scan using a 3D-TFE multi-shot sequence (TR/TE = 2500/265 ms; flip angle = 90°; FOV = 245 FH 245 AP 190 RL mm3; isotropic resolution = 0.85 mm3).
Pseudo-continuous (pc)ASL was acquired as a single run per block, resulting in six ASL-MRI runs per participants, corresponding to the four aTDCS and two FT blocks per participant (Fig. 5). The labeling plane was positioned where the Vertebral and Internal Carotic Artery are parallel (approximately at C2 level) and angled perpendicular to the vessel orientations. pcASL images were acquired with background suppression by two pulses, pulsed continuous labeling, label duration of 1650 ms, and post-label-duration of 1200 ms, using an echoplanar imaging (EPI) readout. Image resolution was 3 × 3 × 4 mm. A single ASL volume consisted of 17 slices with a gap of 0.5 mm, covering the pericentral cortices and adjacent frontoparietal regions of both cerebral hemispheres. Duration for each ASL dynamic was 2 × 4.0 s, with 78 dynamics per aTDCS block, and 32 dynamics per FT block.
Data analysis
All ASL-MRI data was analyzed using FSL software, Wellcome Centre for Integrative Neuroimaging, University of Oxford. (https://www.win.ox.ac.uk).
Pre-processing of ASL-MRI data is described in Appendix A. ASL-MRI data were analyzed using FSL FEAT. For each participant, we did six separate first level analyses, one for each of the four aTDCS runs, and two for FT-runs. Voxel-wise changes in rCBF were analyzed by fitting a General Linear Model (GLM) to the time series of each ASL-MRI run/block modelling either one of the four aTDCS intensities or the movement sequences. The GLM featured three regressors, as previously described in Moisa et al.70 (further details describing our GLM in Appendix A). The positive z-activation map from the perfusion regressor was used for group level analyses, described below.
Definition of volumes of interests
The left M1-HAND was our primary volume of interest (VOI). We also wanted to determine specific functional changes in sub-regions within the primary VOI, based on depth and proximity to the anodal electrode and we planned to include the region with the highest induced E-field as an additional VOI (for an overview of VOI abbreviations and definitions, please see Table 3):
Functional defined VOI of the left M1-HAND (M1FT)
We used the average FT-pre activation as a functional localizer for this VOI (illustrated in Figs. 2, 4A, and Supplementary Fig. 2). The VOI was derived from a whole-brain voxel-wise group analysis using a one-sided t-test with a corrected statistical threshold of p < 0.05 set for family wise error (FWE) cluster level correction and a cluster-defining threshold that was set to p < 0.001 at the voxel level (corresponding to Z = 3.1).
Anatomical defined VOI of the left primary sensorimotor cortex (M1-S1anat)
To avoid bias from the motor-task guided VOI (M1FT), we additionally defined an anatomical VOI around the left primary sensory-motor cortex based on the probabilistic atlas from FSL’s Juelich map, including areas BA4a, BA4b BA3a, BA3b and BA6 for the left hemisphere. The probabilistic area masks were thresholded at 50% relative to their peak values and binarized. In addition, the extend of the mask was restricted to fit area around M1-HAND by including only voxels with MNI coordinates between − 52 mm and − 25 mm in left–right direction and ≥ 36 mm in inferior-posterior direction (Fig. 3).
Sub-regional spherical VOI’s in precentral cortex (M1deep, M1superficial, PMd)
Secondary VOI analyses included three precentral sub-regions and the frontal cortical site that was exposed to the maximal electrical field during aTDCS. We defined three VOIs within the left pre-central gyrus (10 mm spheres) using the FT-pre as a functional localizer. The VOIs were placed at the regional activity peaks within M1FT, (Fig. 4A). Two of the VOIs were placed at the superficial and deep point of highest regional activation in the primary motor cortex, “M1deep” and “M1superficial” (center coordinates x, y, z: [− 38, − 24, 46] and [− 38, − 24, 60]). The third VOI was placed in a functional activated area corresponding to area 6 in the Juelich atlas in FSL, the dorsal premotor cortex “PMd” (center coordinate [− 38, − 6, 60]).
Computational modeling of electric field
We performed additional analysis that considered the current distribution in each subject. We used individual structural scans to simulate the induced electric field for each subject, using SimNIBS v.3.2.221. The individual E-field maps were non-linearly transformed to MNI space by applying the deformation field to the data that was calculated by SPM12 during the head modeling with headreco. For further details on E-field simulations, please see Appendix A (and Fig. 4).
E-field guided VOI (DLPFCE-field)
To define the VOI for the maximal electrical field, each individual E-field simulation at 1 mA were overlaid in MNI-space, considering only positions at which the gray matter of a least 5 subjects overlapped. A 10 mm VOI was placed at the gray matter location at maximum of the averaged E-field (sphere center coordinate: − 40, 28, 42), corresponding to the left dorsolateral prefrontal cortex (DLPFC) (bottom right panel in Supplementary Fig. 3).
Statistical inference on group level
Regional perfusion changes in the cortical VOIs
We ran six GLM analyses at group level, one for each aTDCS condition, corresponding to the four aTDCS intensities, and the two finger tapping sessions. The parameter estimates from the perfusion regressor from the first level analyses was fitted into a second level GLM that modelled the group average, corresponding to a one-sided t-test for each condition. In each model, the z-scores were averaged within our pre-defined volumes of interest (VOI definitions described in “Definition of volumes of interests”). For each VOI, we submitted the spatially averaged z-scores into a two-way repeated measure ANOVA for within subject factor “conditions” with six levels (four aTDCS intensities, two movement sequences). We performed post-hoc tests with pairwise comparisons, p-values were corrected for multiple comparisons using the Bonferroni method. For all ANOVA’s, we used Mauchly’s test to test for sphericity, and corrected with the Greenhouse Geisser method when sphericity was violated. ANOVA and post hoc tests were done using version 25 of the SPSS statistics software package (IBM, Armonk, New York, USA).
Correlation between regional E-field and regional perfusion change
We explored the relation between the highest induced E-field and perfusion response on an individual level. Here we simulated the E-field (at 0.5, 1.0, 1.5, 2.0 mA) in all four VOIs (M1deep, M1superficial, PMd, DLPFCE-field) in all subjects. For each intensity and each VOI, we ran linear regression analyses on the mean perfusion activation and E-field, to see if there were any correlation between the individual perfusion changes and estimated individual E-field strength (see Supplementary Fig. 4). We used Bonferroni correction between the 16 comparisons, with an alpha level of 0.05.
Voxel-based exploratory analyses
We performed a complementary voxel-based analysis to test for linear increase or decrease in perfusion throughout all aTDCS intensities, including all voxels. We applied a cluster-size threshold of p = 0.05 and voxel-wise threshold of z = 3.1 or p = 0.001. We additionally explored finding potential activity with a more liberal voxel-wise threshold of z = 2.3 or p = 0.01.
Analysis of psychometric data
For details on analysis of VAS-scale rated sensory experience, please see Appendix A.
Data availability
The datasets generated and/or analysed during the current study are not currently publicly available as they require complete anonymisation but will be available from the corresponding author on request.
References
1. Frank, E. et al. Treatment of chronic tinnitus with repeated sessions of prefrontal transcranial direct current stimulation: Outcomes from an open-label pilot study. J. Neurol. 259, 327–333. https://doi.org/10.1007/s00415-011-6189-4 (2012).
2. Chhatbar, P. Y. et al. Brain stimulation transcranial direct current stimulation post-stroke upper extremity motor recovery studies exhibit a dose–response relationship. Brain Stimul. 9, 16–26. https://doi.org/10.1016/j.brs.2015.09.002 (2018).
3. Fregni, F. et al. A randomized, sham-controlled, proof of principle study of transcranial direct current stimulation for the treatment of pain in fibromyalgia. Arthritis Rheum. 54, 3988–3998. https://doi.org/10.1002/art.22195 (2006).
4. Fregni, F. et al. A controlled clinical trial of cathodal DC polarization in patients with refractory epilepsy. Epilepsia 47, 335–342. https://doi.org/10.1111/j.1528-1167.2006.00426.x (2006).
5. Fregni, F. et al. A sham-controlled, phase II trial of transcranial direct current stimulation for the treatment of central pain in traumatic spinal cord injury. Pain 122, 197–209. https://doi.org/10.1016/j.pain.2006.02.023 (2006).
6. Straudi, S. et al. tDCS and robotics on upper limb stroke rehabilitation: Effect modification by stroke duration and type of stroke. Biomed. Res. Int. https://doi.org/10.1155/2016/5068127 (2016).
7. Fregni, F. et al. Noninvasive cortical stimulation with transcranial direct current stimulation in Parkinson’s disease. Mov. Disord. 21, 1693–1702. https://doi.org/10.1002/mds.21012 (2006).
8. Fecteau, S. et al. Diminishing risk-taking behavior by modulating activity in the prefrontal cortex: A direct current stimulation study. J. Neurosci. 27, 12500–12505. https://doi.org/10.1523/JNEUROSCI.3283-07.2007 (2007).
9. Nitsche, M. A. et al. Modulation of Cortical Excitability by Weak Direct Current Stimulation—Technical, Safety and Functional Aspects Vol. 56 (Elsevier BV, 2003). https://doi.org/10.1016/S1567-424X(09)70230-2.
10. Rossini, P. M. et al. Non-invasive electrical and magnetic stimulation of the brain, spinal cord, roots and peripheral nerves: Basic principles and procedures for routine clinical and research application: An updated report from an I.F.C.N. Committee. Clin. Neurophysiol. 126, 1071–1107. https://doi.org/10.1016/j.clinph.2015.02.001 (2015).
11. Nitsche, M. & Paulus, W. Excitability changes induced in the human motor cortex by weak transcranial direct current stimulation. J. Physiol. 527, 633–639 (2000).
12. Nitsche, M. A. & Paulus, W. Sustained excitability elevations induced by transcranial DC motor cortex stimulation in humans. Neurology 57, 1899–1901. https://doi.org/10.1212/WNL.57.10.1899 (2001).
13. Nitsche, M. A. et al. Level of action of cathodal DC polarisation induced inhibition of the human motor cortex. Clin. Neurophysiol. 114, 600–604. https://doi.org/10.1016/j.hlc.2017.04.011 (2003).
14. Barker, A. T., Jalinous, R. & Freeston, I. L. Non-invasive magnetic stimulation of human motor cortex. Lancet https://doi.org/10.1515/eng-2018-0022 (1985).
15. Wiethoff, S., Hamada, M. & Rothwell, J. C. Variability in response to transcranial direct current stimulation of the motor cortex. Brain Stimul. 7, 468–475. https://doi.org/10.1016/j.brs.2014.02.003 (2014).
16. Jonker, Z. D. et al. No effect of anodal tDCS on motor cortical excitability and no evidence for responders in a large double-blind placebo-controlled trial. Brain Stimul. 14, 124658. https://doi.org/10.1016/j.brs.2020.11.005 (2020).
17. López-Alonso, V., Cheeran, B., Río-Rodríguez, D. & Fernández-Del-Olmo, M. Inter-individual variability in response to non-invasive brain stimulation paradigms. Brain Stimul. 7, 372–380. https://doi.org/10.1016/j.brs.2014.02.004 (2014).
18. Lefaucheur, J. et al. Clinical neurophysiology evidence-based guidelines on the therapeutic use of transcranial direct current stimulation (tDCS). Clin. Neurophysiol. 128, 56–92. https://doi.org/10.1016/j.clinph.2016.10.087 (2017).
19. Huang, Y. et al. Measurements and models of electric fields in the in vivo human brain during transcranial electric stimulation. Elife 6, 1–26. https://doi.org/10.7554/eLife.18834 (2017).
20. Vöröslakos, M. et al. Direct effects of transcranial electric stimulation on brain circuits in rats and humans. Nat. Commun. https://doi.org/10.1038/s41467-018-02928-3 (2018).
21. Saturnino, G. B., Madsen, K. H. & Thielscher, A. Electric field simulations for transcranial brain stimulation using FEM: An efficient implementation and error analysis. J. Neural Eng. https://doi.org/10.1088/1741-2552/ab41ba (2019).
22. Mosayebi-Samani, M. et al. The impact of individual electrical fields and anatomical factors on the neurophysiological outcomes of tDCS: A TMS-MEP and MRI study. Brain Stimul. https://doi.org/10.1016/j.brs.2021.01.016 (2021).
23. Bergmann, T. O., Groppa, S., Seeger, M., Molle, M. & Marshall, L. S. H. Acute changes in motor cortical excitability during slow oscillatory and constant anodal transcranial direct current stimulation. J. Neurophysiol. 102, 2303–2311 (2009).
24. Bergmann, T. O. et al. Acute changes in motor cortical excitability during slow oscillatory and constant anodal transcranial direct current stimulation. J. Neurophysiol. 102, 2303–2311. https://doi.org/10.1152/jn.00437.2009 (2009).
25. Siebner, H. R. et al. Continuous transcranial magnetic stimulation during positron emission tomography: A suitable tool for imaging regional excitability of the human cortex. Neuroimage 14, 883–890. https://doi.org/10.1006/nimg.2001.0889 (2001).
26. Takano, B. et al. Short-term modulation of regional excitability and blood flow in human motor cortex following rapid-rate transcranial magnetic stimulation. Neuroimage 23, 849–859. https://doi.org/10.1016/j.neuroimage.2004.06.032 (2004).
27. Bindman, L. J., Lippold, O. C. & Milne, A. R. Prolonged changes in excitability of pyramidal tract neurones in the cat: A post-synaptic mechanism. J. Physiol. 286, 457–477. https://doi.org/10.1113/jphysiol.1979.sp012631 (1979).
28. Bindman, L. J., Lippold, O. C. J. & Redfearn, J. W. T. The action of brief polarizing currents on the cerebral cortex of the rat (1) during current flow and (2) in the production of long-lasting after-effects. J. Physiol. 172, 369–382. https://doi.org/10.1113/jphysiol.1964.sp007425 (1964).
29. Purpura, D. P. Activities and evoked potential changes during of motor cortex’. J. Neurophysiol. 28, 166–185 (1965).
30. Wachter, D. et al. Transcranial direct current stimulation induces polarity-specific changes of cortical blood perfusion in the rat. Exp. Neurol. 227, 322–327 (2011).
31. Kurosawa, M., Messlinger, K., Pawlak, M. & Schmidt, R. F. Increase of meningeal blood flow after electrical stimulation of rat dura mater encephali: Mediation by calcitonin gene-related peptide. Br. J. Pharmacol. 114, 1397–1402. https://doi.org/10.1111/j.1476-5381.1995.tb13361.x (1995).
32. Gozalov, A., Jansen-Olesen, I., Klaerke, D. & Olesen, J. Role of KATP channels in cephalic vasodilatation induced by calcitonin gene-related peptide, nitric oxide, and transcranial electrical stimulation in the rat. Headache 48, 1202–1213. https://doi.org/10.1111/j.1526-4610.2008.01205.x (2008).
33. Petersen, K. A., Birk, S., Doods, H., Edvinsson, L. & Olesen, J. Inhibitory effect of BIBN4096BS on cephalic vasodilatation induced by CGRP or transcranial electrical stimulation in the rat. Br. J. Pharmacol. 143, 697–704. https://doi.org/10.1038/sj.bjp.0705966 (2004).
34. Turner, D. A., Degan, S., Galeffi, F., Schmidt, S. & Peterchev, A. V. Rapid, dose-dependent enhancement of cerebral blood flow by transcranial AC stimulation in mouse. Brain Stimul. 14, 80–87. https://doi.org/10.1016/j.brs.2020.11.012 (2021).
35. Detre, J. A. & Wang, J. Technical aspects and utility of fMRI using BOLD and ASL. Clin. Neurophysiol. 113, 621–634. https://doi.org/10.1016/S1388-2457(02)00038-X (2002).
36. Detre, J. A. & Alsop, D. C. Perfusion magnetic resonance imaging with continuous arterial spin labeling: Methods and clinical applications in the central nervous system. Eur. J. Radiol. 30, 115–124. https://doi.org/10.1016/S0720-048X(99)00050-9 (1999).
37. Kwon, Y. H. & Jang, S. H. The enhanced cortical activation induced by transcranial direct current stimulation during hand movements. Neurosci. Lett. 492, 105–108. https://doi.org/10.1016/j.neulet.2011.01.066 (2011).
38. Kwon, H. Y. et al. Primary motor cortex activation by transcranial direct current stimulation in the human brain. Neurosci. Lett. 435, 56–59. https://doi.org/10.1016/j.neulet.2008.02.012 (2008).
39. Antal, A., Polania, R., Schmidt-samoa, C., Dechent, P. & Paulus, W. Transcranial direct current stimulation over the primary motor cortex during fMRI. Neuroimage 55, 590–596. https://doi.org/10.1016/j.neuroimage.2010.11.085 (2011).
40. Zheng, X., Alsop, D. C. & Schlaug, G. Effects of transcranial direct current stimulation (tDCS) on human regional cerebral blood flow. Neuroimage 58, 26–33. https://doi.org/10.1016/j.neuroimage.2011.06.018 (2011).
41. Stagg, C. J. et al. Widespread modulation of cerebral perfusion induced during and after transcranial direct current stimulation applied to the left dorsolateral prefrontal. Cortex 33, 11425–11431. https://doi.org/10.1523/JNEUROSCI.3887-12.2013 (2013).
42. Lang, N. et al. How does transcranial DC stimulation of the primary motor cortex alter regional neuronal activity in the human brain ?. Eur. J. Neurosci. 22, 495–504. https://doi.org/10.1111/j.1460-9568.2005.04233.x (2005).
43. Jamil, A. et al. Current intensity- and polarity-specific online and aftereffects of transcranial direct current stimulation: An fMRI study. Hum. Brain Mapp. 41, 1644–1666. https://doi.org/10.1002/hbm.24901 (2019).
44. Shinde, A. B., Lerud, K. D., Munsch, F., Alsop, D. C. & Schlaug, G. Effects of tDCS dose and electrode montage on regional cerebral blood flow and motor behavior. Neuroimage 237, 118144. https://doi.org/10.1016/j.neuroimage.2021.118144 (2021).
45. Antal, A. et al. Low intensity transcranial electric stimulation: Safety, ethical, legal regulatory and application guidelines. Clin. Neurophysiol. 128, 1774–1809. https://doi.org/10.1016/j.clinph.2017.06.001 (2017).
46. Han, C.-H., Song, H., Kang, Y.-G., Kim, B.-M. & Im, C.-H. Hemodynamic responses in rat brain during transcranial direct current stimulation: A functional near-infrared spectroscopy study. Biomed. Opt. Express 5, 1812. https://doi.org/10.1364/boe.5.001812 (2014).
47. Purpura, D. P. & McMurthy, J. Intracellular activities and evoked potential changes during of motor cortex. J. Neurophysiol. 28, 166–185 (1965).
48. Stagg, C. J., Antal, A. & Nitsche, M. A. Physiology of transcranial direct current stimulation. J. ECT 34, 144–152. https://doi.org/10.1097/YCT.0000000000000510 (2018).
49. Ferbert, A. et al. Interhemispheric inhibition of the human motor cortex. J. Physiol. 453, 525–546. https://doi.org/10.1113/jphysiol.1992.sp019243 (1992).
50. Antal, A., Polania, R., Schmidt-samoa, C., Dechent, P. & Paulus, W. NeuroImage Transcranial direct current stimulation over the primary motor cortex during fMRI. Neuroimage 55, 590–596. https://doi.org/10.1016/j.neuroimage.2010.11.085 (2011).
51. Baudewig, J., Nitsche, M. A., Paulus, W. & Frahm, J. Regional modulation of BOLD MRI responses to human sensorimotor activation by transcranial. Direct 201, 196–201 (2001).
52. Devanne, H., Lavoie, B. A. & Capaday, C. Input-output properties and gain changes in the human corticospinal pathway. Exp. Brain Res. 114, 329–338. https://doi.org/10.1007/PL00005641 (1997).
53. Carroll, T. J., Riek, S. & Carson, R. G. Reliability of the input–output properties of the cortico-spinal pathway obtained from transcranial magnetic and electrical stimulation. J. Neurosci. Methods 112, 193–202. https://doi.org/10.1016/S0165-0270(01)00468-X (2001).
54. Capaday, C. On the variability of motor-evoked potentials: Experimental results and mathematical model. Exp. Brain Res. 239, 2979–2995. https://doi.org/10.1007/s00221-021-06169-7 (2021).
55. Liu, A. et al. Immediate neurophysiological effects of transcranial electrical stimulation. Nat. Commun. https://doi.org/10.1038/s41467-018-07233-7 (2018).
56. Kidgell, D. J. et al. Different current intensities of anodal transcranial direct current stimulation do not differentially modulate motor cortex plasticity. Neural Plast. 2013, 13–15 (2013).
57. Batsikadze, G., Moliadze, V., Paulus, W., Kuo, M. & Nitsche, M. A. Partially non-linear stimulation intensity-dependent effects of direct current stimulation on motor cortex. J. Physiol. 7, 1987–2000. https://doi.org/10.1113/jphysiol.2012.249730 (2013).
58. Jamil, A. et al. Systematic evaluation of the impact of stimulation intensity on neuroplastic after-effects induced by transcranial direct current stimulation. J. Physiol. 4, 1273–1288. https://doi.org/10.1113/JP272738 (2017).
59. Opitz, A. et al. On the importance of precise electrode placement for targeted transcranial electric stimulation. Neuroimage 181, 560–567. https://doi.org/10.1016/j.neuroimage.2018.07.027.On (2018).
60. Aberra, A. S., Peterchev, A. V. & Grill, W. M. Biophysically realistic neuron models for simulation of cortical stimulation. J. Neural Eng. https://doi.org/10.1016/j.physbeh.2017.03.040 (2018).
61. Nair, D. R., Burgess, R., McIntyre, C. C. & Lüders, H. Chronic subdural electrodes in the management of epilepsy. Clin. Neurophysiol. 119, 11–28. https://doi.org/10.1016/j.clinph.2007.09.117 (2008).
62. Opitz, A., Paulus, W., Will, S., Antunes, A. & Thielscher, A. Determinants of the electric field during transcranial direct current stimulation. Neuroimage 109, 140–150. https://doi.org/10.1016/j.neuroimage.2015.01.033 (2015).
63. Karabanov, A. N., Saturnino, G. B., Thielscher, A. & Siebner, H. R. Can transcranial electrical stimulation localize brain function?. Front. Psychol. https://doi.org/10.3389/fpsyg.2019.00213 (2019).
64. Fertonani, A. & Miniussi, C. Transcranial electrical stimulation: What we know and do not know about mechanisms. Neuroscientist 23, 109–123. https://doi.org/10.1177/1073858416631966 (2017).
65. Datta, A. et al. Gyri-precise head model of transcranial direct current stimulation: Improved spatial focality using a ring electrode versus conventional rectangular pad. Brain Stimul. 2, 201-207.e1. https://doi.org/10.1016/j.brs.2009.03.005 (2009).
66. Bahr-Hosseini, M. & Bikson, M. Neurovascular-modulation: A review of primary vascular responses to transcranial electrical stimulation as a mechanism of action. Brain Stimul. 14, 837–847. https://doi.org/10.1016/j.brs.2021.04.015 (2021).
67. Oldfield, R. C. The assessment and analysis of handedness: The Edinburgh Inventory. Neuropsychologia 9, 97–113 (1971).
68. Paus, T. et al. Transcranial magnetic stimulation during positron emission tomography: A new method for studying connectivity of the human cerebral cortex. J. Neurosci. 17, 3178–3184. https://doi.org/10.1523/jneurosci.17-09-03178.1997 (1997).
69. Klem, G. H., Lüders, H. O. & Jasper, H. H. E. C. The ten-twenty electrode system of the International Federation. Int. Fed. Clin. Neurophysiol. 52, 3–6 (1999).
70. Moisa, M., Pohmann, R., Uluda, K. & Thielscher, A. Interleaved TMS/CASL: Comparison of different rTMS protocols. Neuroimage 49, 612–620. https://doi.org/10.1016/j.neuroimage.2009.07.010 (2010).
Funding
Hartwig Roman Siebner holds a clinical 5-year professorship in precision medicine at the Institute of Clinical Medicine, Faculty of Health and Medical Sciences, University of Copenhagen, sponsored by the Lundbeck Foundation (Grant No. R186-2015-2138). Hartwig Siebner has received a Collaborative Project grant from the Lundbeck Foundation (Grant No. R336-2020-1035). Axel Thielscher has received funding from the Lundbeck Foundation, as part of a collaboration with Gottfried Schlaug, Beth Israel Deaconess Medical Center & Harvard Medical School Boston, on a NIH brain Initiative project (Grant No. R244-2017-196) and a Lundbeck Ascending Investigator grant (Grant No. R313-2019-622). Marie Louise Liu has been supported by research grants from the Lundbeck Foundation (Grant No. R244-2017-196) and Amager and Hvidovre Hospital. Funding sources have no involvement in the study design, data management, writing or publication.
Author information
Authors
Contributions
M.L. helped collecting the data, analyzed the data, wrote the manuscript. A.K. contributed to interpretation of the data, assisted in writing the manuscript. M.P. collected and analyzed the data. E.P. assisted in designing the study, assisted in data acquisition. A.T. assisted designing the study, assisted analyzing the data, supervised the experiments. H.S. designed the study, supervised the experiments, contributed to interpretation of the data, assisted in writing the manuscript. All authors reviewed the manuscript.
Corresponding author
Correspondence to Hartwig Roman Siebner.
Ethics declarations
Competing interests
Hartwig R. Siebner has received honoraria as speaker from Sanofi Genzyme, Denmark and Novartis, Denmark, as consultant from Sanofi Genzyme, Denmark, Lophora, Denmark, and Lundbeck AS, Denmark, and as editor-in-chief (Neuroimage Clinical) and senior editor (NeuroImage) from Elsevier Publishers, Amsterdam, The Netherlands. He has received royalties as book editor from Springer Publishers, Stuttgart, Germany and from Gyldendal Publishers, Copenhagen, Denmark. Remaining authors declare no potential conflicts of interests.
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Reprints and Permissions
Liu, M.L., Karabanov, A.N., Piek, M. et al. Short periods of bipolar anodal TDCS induce no instantaneous dose-dependent increase in cerebral blood flow in the targeted human motor cortex. Sci Rep 12, 9580 (2022). https://doi.org/10.1038/s41598-022-13091-7
• Accepted:
• Published:
• DOI: https://doi.org/10.1038/s41598-022-13091-7 | 2022-08-13 16:33:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5753143429756165, "perplexity": 10489.299750130604}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571959.66/warc/CC-MAIN-20220813142020-20220813172020-00358.warc.gz"} |
https://flyingcoloursmaths.co.uk/cos72o-revisited-geometric-method/ | Some months ago, I wrote about a method for finding $\cos(72º)$, or $\cos\br{\frac{2\pi}{5}}$ in proper units. Almost immediately, the good people of Twitter and Facebook - notably @ImMisterAl (Al) and @BuryMathsTutor (Mark)- suggested other ways of doing it.
Let’s start with Mark’s method, which he dissects in his book GCSE Maths Challenge ((I have not yet read the book, but think highly of Mark and trust that it’s a good one.)).
Looking at Q16 here, we have three isosceles triangles. ACD and DCB are similar (they’re both isosceles, and base angle C is the same in both). If we call angle CAD $\theta$, then ABD is $180º - 2\theta$, so CBD and DCB are both $2\theta$.
That means the three angles in ACD add up to $5\theta$, which must be 180º; $\theta$ is therefore 36º.
Suppose the length BD (and therefore CD and BA) is one unit, and the length BC is $x$. We can do some trigonometry!
Applying the cosine rule to the big triangle ACD, we have $\cos(72º)=\frac{1 + (1+x)^2 - (1+x)^2}{2(1+x)} = \frac{1}{2(1+x)}$.
Doing the same to the smaller triangle CBD, we have $\cos(72º)=\frac{1+x^2-1}{2x} = \frac{x}{2}$.
Let $\cos(72º)=C$ for the purposes of algebra, and we have $2C=x$ from the second equation. Substituting this into the first gives $C=\frac{1}{2+4C}$.
Rearrange this to give $4C^2 + 2C - 1 = 0$, and $C$ drops out of the quadratic formula as $C=\frac{-1 \pm \sqrt{5}}{2}$; we know $\cos(72º)>0$, so only the positive branch makes sense.
Therefore $\cos(72º) = \frac{\sqrt{5}-1}{2}$.
I like this method a lot - it drops out super-neatly. However, it feels a bit like the triangles have appeared by magic; it’s easy to prove it once you have the scaffolding in place, but I wouldn’t have come up with the scaffolding on my own. | 2022-07-06 04:59:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7620239853858948, "perplexity": 675.1349887208776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104660626.98/warc/CC-MAIN-20220706030209-20220706060209-00494.warc.gz"} |
https://www.physicsforums.com/threads/symmetry-and-conservation-of-charge.209416/ | # Symmetry and Conservation of Charge
1. ### Moridin
858
I understand that all conservation laws have underlying symmetries and that all symmetries have corresponding conservation laws. From reading some popular science books (don't shoot me :P), I understand that conservation of energy, linear and angular momentum are a natural consequence of time translation symmetry, space translation symmetry and space rotation symmetry respectively.
What symmetry does the conservation of charge follow from?
Thank you for your time, have a nice day.
2. ### jdg812
91
From global gauge symmetry
3. ### olgranpappy
1,273
...isn't that a contradiction in terms? To "gauge" a symmetry means to make it local...
4. ### belliott4488
666
No, global gauge symmetries are independent of space; local gauge symmetries depend on spatial coordinates. I might this wrong (it's been a while), but I seem to recall that gauge symmetries in general are symmetries of a potential field, such as the electric potential field, the derivatives of which give you the electric field.
EDIT: You know, as I stir up my old memories of this, I now seem to recall that people do use "gauge" to refer to local gauge symmetries, especially in gauge field theory. What is confusing me now is that global choices of gauge, like the Lorentz or Coulomb gauge in Classical E&M, also reflect a gauge symmetry.
Last edited: Jan 17, 2008
5. ### jdg812
91
Both of expressions "local gauge symmetry" and "global gauge symmetry" are generally accepted in physics. | 2015-06-30 16:56:16 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8335122466087341, "perplexity": 1155.4306240574958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094451.94/warc/CC-MAIN-20150627031814-00072-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-calculate-the-concentration-of-iodate-ions-in-a-saturated-solution-of | # How do you calculate the concentration of iodate ions in a saturated solution of lead (II) iodate, Pb(IO_3)_2? The K_(sp) = 2.6 xx 10^(-13)?
Jun 30, 2017
$\left[{\text{IO}}_{3}^{-}\right] = 8.0 \cdot {10}^{- 5}$ $\text{M}$
#### Explanation:
The idea here is that lead(II) iodate is considered insoluble in water, so right from the start, you should expect to find a very low concentration of iodate anions, ${\text{IO}}_{3}^{-}$, in a saturated solution of lead(II) iodate.
The dissociation equilibrium that describes the dissociation of lead(II) iodate looks like this
${\text{Pb"("IO"_ 3)_ (2(s)) rightleftharpoons "Pb"_ ((aq))^(2+) + color(red)(2)"IO}}_{3 \left(a q\right)}^{-}$
Notice that every mole of lead(II) iodate that dissociates produces $1$ mole of lead(II) cations and $\textcolor{red}{2}$ moles of iodate anions in solution.
This means that, at equilibrium, a saturated solution of lead(II) iodate will have
$\left[{\text{IO"_3^(-)] = color(red)(2) * ["Pb}}^{2 +}\right]$
Now, the solubility product constant for this dissociation equilibrium looks like this
${K}_{s p} = {\left[{\text{Pb"^(2+)] * ["IO}}_{3}^{-}\right]}^{\textcolor{red}{2}}$
If you take $s$ to be the concentration of lead(II) cations in the solution, i.e. the molar solubility of the salt, you can say that you have
${K}_{s p} = s \cdot {\left(\textcolor{red}{2} s\right)}^{\textcolor{red}{2}}$
which is equivalent to
$2.6 \cdot {10}^{- 13} = 4 {s}^{3}$
Rearrange to solve for $s$
$s = \sqrt[3]{\frac{2.6 \cdot {10}^{- 13}}{4}} = 4.02 \cdot {10}^{- 5}$
This means that a saturated solution of lead(II) iodate will have
$\left[{\text{Pb}}^{2 +}\right] = 4.02 \cdot {10}^{- 5}$ $\text{M}$
and
["IO"_3^(-)] = color(red)(2) * 4.02 * 10^(-5)color(white)(.)"M" = color(darkgreen)(ul(color(black)(8.0 * 10^(-5)color(white)(.)"M")))
I'll leave the answer rounded to two sig figs. | 2020-11-27 08:57:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 16, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6628373861312866, "perplexity": 3619.3222526440736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141191511.46/warc/CC-MAIN-20201127073750-20201127103750-00411.warc.gz"} |
https://en.wikibooks.org/wiki/Linear_Algebra/Polynomials_of_Maps_and_Matrices/Solutions | # Linear Algebra/Polynomials of Maps and Matrices/Solutions
## Solutions
This exercise is recommended for all readers.
Problem 1
What are the possible minimal polynomials if a matrix has the given characteristic polynomial?
1. ${\displaystyle 8\cdot (x-3)^{4}}$
2. ${\displaystyle (1/3)\cdot (x+1)^{3}(x-4)}$
3. ${\displaystyle -1\cdot (x-2)^{2}(x-5)^{2}}$
4. ${\displaystyle 5\cdot (x+3)^{2}(x-1)(x-2)^{2}}$
What is the degree of each possibility?
For each, the minimal polynomial must have a leading coefficient of ${\displaystyle 1}$ and Theorem 1.8, the Cayley-Hamilton Theorem, says that the minimal polynomial must contain the same linear factors as the characteristic polynomial, although possibly of lower degree but not of zero degree.
1. The possibilities are ${\displaystyle m_{1}(x)=x-3}$, ${\displaystyle m_{2}(x)=(x-3)^{2}}$, ${\displaystyle m_{3}(x)=(x-3)^{3}}$, and ${\displaystyle m_{4}(x)=(x-3)^{4}}$. Note that the ${\displaystyle 8}$ has been dropped because a minimal polynomial must have a leading coefficient of one. The first is a degree one polynomial, the second is degree two, the third is degree three, and the fourth is degree four.
2. The possibilities are ${\displaystyle m_{1}(x)=(x+1)(x-4)}$, ${\displaystyle m_{2}(x)=(x+1)^{2}(x-4)}$, and ${\displaystyle m_{3}(x)=(x+1)^{3}(x-4)}$. The first is a quadratic polynomial, that is, it has degree two. The second has degree three, and the third has degree four.
3. We have ${\displaystyle m_{1}(x)=(x-2)(x-5)}$, ${\displaystyle m_{2}(x)=(x-2)^{2}(x-5)}$, ${\displaystyle m_{3}(x)=(x-2)(x-5)^{2}}$, and ${\displaystyle m_{4}(x)=(x-2)^{2}(x-5)^{2}}$. They are polynomials of degree two, three, three, and four.
4. The possiblities are ${\displaystyle m_{1}(x)=(x+3)(x-1)(x-2)}$, ${\displaystyle m_{2}(x)=(x+3)^{2}(x-1)(x-2)}$, ${\displaystyle m_{3}(x)=(x+3)(x-1)(x-2)^{2}}$, and ${\displaystyle m_{4}(x)=(x+3)^{2}(x-1)(x-2)^{2}}$. The degree of ${\displaystyle m_{1}}$ is three, the degree of ${\displaystyle m_{2}}$ is four, the degree of ${\displaystyle m_{3}}$ is four, and the degree of ${\displaystyle m_{4}}$ is five.
This exercise is recommended for all readers.
Problem 2
Find the minimal polynomial of each matrix.
1. ${\displaystyle {\begin{pmatrix}3&0&0\\1&3&0\\0&0&4\end{pmatrix}}}$
2. ${\displaystyle {\begin{pmatrix}3&0&0\\1&3&0\\0&0&3\end{pmatrix}}}$
3. ${\displaystyle {\begin{pmatrix}3&0&0\\1&3&0\\0&1&3\end{pmatrix}}}$
4. ${\displaystyle {\begin{pmatrix}2&0&1\\0&6&2\\0&0&2\end{pmatrix}}}$
5. ${\displaystyle {\begin{pmatrix}2&2&1\\0&6&2\\0&0&2\end{pmatrix}}}$
6. ${\displaystyle {\begin{pmatrix}-1&4&0&0&0\\0&3&0&0&0\\0&-4&-1&0&0\\3&-9&-4&2&-1\\1&5&4&1&4\end{pmatrix}}}$
In each case we will use the method of Example 1.12.
1. Because ${\displaystyle T}$ is triangular, ${\displaystyle T-xI}$ is also triangular
${\displaystyle T-xI={\begin{pmatrix}3-x&0&0\\1&3-x&0\\0&0&4-x\end{pmatrix}}}$
the characteristic polynomial is easy ${\displaystyle c(x)=\left|T-xI\right|=(3-x)^{2}(4-x)=-1\cdot (x-3)^{2}(x-4)}$. There are only two possibilities for the minimal polynomial, ${\displaystyle m_{1}(x)=(x-3)(x-4)}$ and ${\displaystyle m_{2}(x)=(x-3)^{2}(x-4)}$. (Note that the characteristic polynomial has a negative sign but the minimal polynomial does not since it must have a leading coefficient of one). Because ${\displaystyle m_{1}(T)}$ is not the zero matrix
${\displaystyle (T-3I)(T-4I)={\begin{pmatrix}0&0&0\\1&0&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}-1&0&0\\1&-1&0\\0&0&0\end{pmatrix}}={\begin{pmatrix}0&0&0\\-1&0&0\\0&0&0\end{pmatrix}}}$
the minimal polynomial is ${\displaystyle m(x)=m_{2}(x)}$.
${\displaystyle (T-3I)^{2}(T-4I)=(T-3I)\cdot {\bigl (}(T-3I)(T-4I){\bigr )}={\begin{pmatrix}0&0&0\\1&0&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0&0&0\\-1&0&0\\0&0&0\end{pmatrix}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}}$
2. As in the prior item, the fact that the matrix is triangular makes computation of the characteristic polynomial easy.
${\displaystyle c(x)=\left|T-xI\right|={\begin{vmatrix}3-x&0&0\\1&3-x&0\\0&0&3-x\end{vmatrix}}=(3-x)^{3}=-1\cdot (x-3)^{3}}$
There are three possibilities for the minimal polynomial ${\displaystyle m_{1}(x)=(x-3)}$, ${\displaystyle m_{2}(x)=(x-3)^{2}}$, and ${\displaystyle m_{3}(x)=(x-3)^{3}}$. We settle the question by computing ${\displaystyle m_{1}(T)}$
${\displaystyle T-3I={\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}}}$
and ${\displaystyle m_{2}(T)}$.
${\displaystyle (T-3I)^{2}={\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}}{\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}}$
Because ${\displaystyle m_{2}(T)}$ is the zero matrix, ${\displaystyle m_{2}(x)}$ is the minimal polynomial.
3. Again, the matrix is triangular.
${\displaystyle c(x)=\left|T-xI\right|={\begin{vmatrix}3-x&0&0\\1&3-x&0\\0&1&3-x\end{vmatrix}}=(3-x)^{3}=-1\cdot (x-3)^{3}}$
Again, there are three possibilities for the minimal polynomial ${\displaystyle m_{1}(x)=(x-3)}$, ${\displaystyle m_{2}(x)=(x-3)^{2}}$, and ${\displaystyle m_{3}(x)=(x-3)^{3}}$. We compute ${\displaystyle m_{1}(T)}$
${\displaystyle T-3I={\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}}}$
and ${\displaystyle m_{2}(T)}$
${\displaystyle (T-3I)^{2}={\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}}{\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}}={\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}}}$
and ${\displaystyle m_{3}(T)}$.
${\displaystyle (T-3I)^{3}=(T-3I)^{2}(T-3I)={\begin{pmatrix}0&0&0\\0&0&0\\1&0&0\end{pmatrix}}{\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}}$
Therefore, the minimal polynomial is ${\displaystyle m(x)=m_{3}(x)=(x-3)^{3}}$.
4. This case is also triangular, here upper triangular.
${\displaystyle c(x)=\left|T-xI\right|={\begin{vmatrix}2-x&0&1\\0&6-x&2\\0&0&2-x\end{vmatrix}}=(2-x)^{2}(6-x)=-(x-2)^{2}(x-6)}$
There are two possibilities for the minimal polynomial, ${\displaystyle m_{1}(x)=(x-2)(x-6)}$ and ${\displaystyle m_{2}(x)=(x-2)^{2}(x-6)}$. Computation shows that the minimal polynomial isn't ${\displaystyle m_{1}(x)}$.
${\displaystyle (T-2I)(T-6I)={\begin{pmatrix}0&0&1\\0&4&2\\0&0&0\end{pmatrix}}{\begin{pmatrix}-4&0&1\\0&0&2\\0&0&-4\end{pmatrix}}={\begin{pmatrix}0&0&-4\\0&0&0\\0&0&0\end{pmatrix}}}$
It therefore must be that ${\displaystyle m(x)=m_{2}(x)=(x-2)^{2}(x-6)}$. Here is a verification.
${\displaystyle (T-2I)^{2}(T-6I)=(T-2I)\cdot {\bigl (}(T-2I)(T-6I){\bigr )}={\begin{pmatrix}0&0&1\\0&4&2\\0&0&0\end{pmatrix}}{\begin{pmatrix}0&0&-4\\0&0&0\\0&0&0\end{pmatrix}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}}$
5. The characteristic polynomial is
${\displaystyle c(x)=\left|T-xI\right|={\begin{vmatrix}2-x&2&1\\0&6-x&2\\0&0&2-x\end{vmatrix}}=(2-x)^{2}(6-x)=-(x-2)^{2}(x-6)}$
and there are two possibilities for the minimal polynomial, ${\displaystyle m_{1}(x)=(x-2)(x-6)}$ and ${\displaystyle m_{2}(x)=(x-2)^{2}(x-6)}$. Checking the first one
${\displaystyle (T-2I)(T-6I)={\begin{pmatrix}0&2&1\\0&4&2\\0&0&0\end{pmatrix}}{\begin{pmatrix}-4&2&1\\0&0&2\\0&0&-4\end{pmatrix}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}}$
shows that the minimal polynomial is ${\displaystyle m(x)=m_{1}(x)=(x-2)(x-6)}$.
6. The characteristic polynomial is this.
${\displaystyle c(x)=\left|T-xI\right|={\begin{vmatrix}-1-x&4&0&0&0\\0&3-x&0&0&0\\0&-4&-1-x&0&0\\3&-9&-4&2-x&-1\\1&5&4&1&4-x\end{vmatrix}}=(x-3)^{3}(x+1)^{2}}$
There are a number of possibilities for the minimal polynomial, listed here by ascending degree: ${\displaystyle m_{1}(x)=(x-3)(x+1)}$, ${\displaystyle m_{1}(x)=(x-3)^{2}(x+1)}$, ${\displaystyle m_{1}(x)=(x-3)(x+1)^{2}}$, ${\displaystyle m_{1}(x)=(x-3)^{3}(x+1)}$, ${\displaystyle m_{1}(x)=(x-3)^{2}(x+1)^{2}}$, and ${\displaystyle m_{1}(x)=(x-3)^{3}(x+1)^{2}}$. The first one doesn't pan out
${\displaystyle {\begin{array}{rl}(T-3I)(T+1I)&={\begin{pmatrix}-4&4&0&0&0\\0&0&0&0&0\\0&-4&-4&0&0\\3&-9&-4&-1&-1\\1&5&4&1&1\end{pmatrix}}{\begin{pmatrix}0&4&0&0&0\\0&4&0&0&0\\0&-4&0&0&0\\3&-9&-4&3&-1\\1&5&4&1&5\end{pmatrix}}\\&={\begin{pmatrix}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\-4&-4&0&-4&-4\\4&4&0&4&4\end{pmatrix}}\end{array}}}$
but the second one does.
${\displaystyle (T-3I)^{2}(T+1I)=(T-3I){\bigl (}(T-3I)(T+1I){\bigr )}}$
{\displaystyle {\begin{aligned}&={\begin{pmatrix}-4&4&0&0&0\\0&0&0&0&0\\0&-4&-4&0&0\\3&-9&-4&-1&-1\\1&5&4&1&1\end{pmatrix}}{\begin{pmatrix}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\-4&-4&0&-4&-4\\4&4&0&4&4\end{pmatrix}}\\&={\begin{pmatrix}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{pmatrix}}\end{aligned}}}
The minimal polynomial is ${\displaystyle m(x)=(x-3)^{2}(x+1)}$.
Problem 3
Find the minimal polynomial of this matrix.
${\displaystyle {\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}}$
Its characteristic polynomial has complex roots.
${\displaystyle {\begin{vmatrix}-x&1&0\\0&-x&1\\1&0&-x\end{vmatrix}}=(1-x)\cdot (x-(-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i))\cdot (x-(-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i))}$
As the roots are distinct, the characteristic polynomial equals the minimal polynomial.
This exercise is recommended for all readers.
Problem 4
What is the minimal polynomial of the differentiation operator ${\displaystyle d/dx}$ on ${\displaystyle {\mathcal {P}}_{n}}$?
We know that ${\displaystyle {\mathcal {P}}_{n}}$ is a dimension ${\displaystyle n+1}$ space and that the differentiation operator is nilpotent of index ${\displaystyle n+1}$ (for instance, taking ${\displaystyle n=3}$, ${\displaystyle {\mathcal {P}}_{3}=\{c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\,{\big |}\,c_{3},\ldots ,c_{0}\in \mathbb {C} \}}$ and the fourth derivative of a cubic is the zero polynomial). Represent this operator using the canonical form for nilpotent transformations.
${\displaystyle {\begin{pmatrix}0&0&0&\ldots &&0\\1&0&0&&&0\\0&1&0&&&\\&&\ddots \\0&0&0&&1&0\end{pmatrix}}}$
This is an ${\displaystyle (n+1)\!\times \!(n+1)}$ matrix with an easy characteristic polynomial, ${\displaystyle c(x)=x^{n+1}}$. (Remark: this matrix is ${\displaystyle {\rm {Rep}}_{B,B}(d/dx)}$ where ${\displaystyle B=\langle x^{n},nx^{n-1},n(n-1)x^{n-2},\ldots ,n!\rangle }$.) To find the minimal polynomial as in Example 1.12 we consider the powers of ${\displaystyle T-0I=T}$. But, of course, the first power of ${\displaystyle T}$ that is the zero matrix is the power ${\displaystyle n+1}$. So the minimal polynomial is also ${\displaystyle x^{n+1}}$.
This exercise is recommended for all readers.
Problem 5
Find the minimal polynomial of matrices of this form
${\displaystyle {\begin{pmatrix}\lambda &0&0&\ldots &&0\\1&\lambda &0&&&0\\0&1&\lambda \\&&&\ddots \\&&&&\lambda &0\\0&0&\ldots &&1&\lambda \end{pmatrix}}}$
where the scalar ${\displaystyle \lambda }$ is fixed (i.e., is not a variable).
Call the matrix ${\displaystyle T}$ and suppose that it is ${\displaystyle n\!\times \!n}$. Because ${\displaystyle T}$ is triangular, and so ${\displaystyle T-xI}$ is triangular, the characteristic polynomial is ${\displaystyle c(x)=(x-\lambda )^{n}}$. To see that the minimal polynomial is the same, consider ${\displaystyle T-\lambda I}$.
${\displaystyle {\begin{pmatrix}0&0&0&\ldots &0\\1&0&0&\ldots &0\\0&1&0\\&&\ddots \\0&0&\ldots &1&0\end{pmatrix}}}$
Recognize it as the canonical form for a transformation that is nilpotent of degree ${\displaystyle n}$; the power ${\displaystyle (T-\lambda I)^{j}}$ is zero first when ${\displaystyle j}$ is ${\displaystyle n}$.
Problem 6
What is the minimal polynomial of the transformation of ${\displaystyle {\mathcal {P}}_{n}}$ that sends ${\displaystyle p(x)}$ to ${\displaystyle p(x+1)}$?
The ${\displaystyle n=3}$ case provides a hint. A natural basis for ${\displaystyle {\mathcal {P}}_{3}}$ is ${\displaystyle B=\langle 1,x,x^{2},x^{3}\rangle }$. The action of the transformation is
${\displaystyle 1\mapsto 1\quad x\mapsto x+1\quad x^{2}\mapsto x^{2}+2x+1\quad x^{3}\mapsto x^{3}+3x^{2}+3x+1}$
and so the representation ${\displaystyle {\rm {Rep}}_{B,B}(t)}$ is this upper triangular matrix.
${\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&2&3\\0&0&1&3\\0&0&0&1\end{pmatrix}}}$
Because it is triangular, the fact that the characteristic polynomial is ${\displaystyle c(x)=(x-1)^{4}}$ is clear. For the minimal polynomial, the candidates are ${\displaystyle m_{1}(x)=(x-1)}$,
${\displaystyle T-1I={\begin{pmatrix}0&1&1&1\\0&0&2&3\\0&0&0&3\\0&0&0&0\end{pmatrix}}}$
${\displaystyle m_{2}(x)=(x-1)^{2}}$,
${\displaystyle (T-1I)^{2}={\begin{pmatrix}0&0&2&6\\0&0&0&6\\0&0&0&0\\0&0&0&0\end{pmatrix}}}$
${\displaystyle m_{3}(x)=(x-1)^{3}}$,
${\displaystyle (T-1I)^{3}={\begin{pmatrix}0&0&0&6\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}}$
and ${\displaystyle m_{4}(x)=(x-1)^{4}}$. Because ${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$, and ${\displaystyle m_{3}}$ are not right, ${\displaystyle m_{4}}$ must be right, as is easily verified.
In the case of a general ${\displaystyle n}$, the representation is an upper triangular matrix with ones on the diagonal. Thus the characteristic polynomial is ${\displaystyle c(x)=(x-1)^{n+1}}$. One way to verify that the minimal polynomial equals the characteristic polynomial is argue something like this: say that an upper triangular matrix is ${\displaystyle 0}$-upper triangular if there are nonzero entries on the diagonal, that it is ${\displaystyle 1}$-upper triangular if the diagonal contains only zeroes and there are nonzero entries just above the diagonal, etc. As the above example illustrates, an induction argument will show that, where ${\displaystyle T}$ has only nonnegative entries, ${\displaystyle T^{j}}$ is ${\displaystyle j}$-upper triangular. That argument is left to the reader.
Problem 7
What is the minimal polynomial of the map ${\displaystyle \pi :\mathbb {C} ^{3}\to \mathbb {C} ^{3}}$ projecting onto the first two coordinates?
The map twice is the same as the map once: ${\displaystyle \pi \circ \pi =\pi }$, that is, ${\displaystyle \pi ^{2}=\pi }$ and so the minimal polynomial is of degree at most two since ${\displaystyle m(x)=x^{2}-x}$ will do. The fact that no linear polynomial will do follows from applying the maps on the left and right side of ${\displaystyle c_{1}\cdot \pi +c_{0}\cdot {\mbox{id}}=z}$ (where ${\displaystyle z}$ is the zero map) to these two vectors.
${\displaystyle {\begin{pmatrix}0\\0\\1\end{pmatrix}}\qquad {\begin{pmatrix}1\\0\\0\end{pmatrix}}}$
Thus the minimal polynomial is ${\displaystyle m}$.
Problem 8
Find a ${\displaystyle 3\!\times \!3}$ matrix whose minimal polynomial is ${\displaystyle x^{2}}$.
This is one answer.
${\displaystyle {\begin{pmatrix}0&0&0\\1&0&0\\0&0&0\end{pmatrix}}}$
Problem 9
What is wrong with this claimed proof of Lemma 1.9: "if ${\displaystyle c(x)=\left|T-xI\right|}$ then ${\displaystyle c(T)=\left|T-TI\right|=0}$"? (Cullen 1990)
The ${\displaystyle x}$ must be a scalar, not a matrix.
Problem 10
Verify Lemma 1.9 for ${\displaystyle 2\!\times \!2}$ matrices by direct calculation.
The characteristic polynomial of
${\displaystyle T={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$
is ${\displaystyle (a-x)(d-x)-bc=x^{2}-(a+d)x+(ad-bc)}$. Substitute
${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}^{2}-(a+d){\begin{pmatrix}a&b\\c&d\end{pmatrix}}+(ad-bc){\begin{pmatrix}1&0\\0&1\end{pmatrix}}}$
${\displaystyle ={\begin{pmatrix}a^{2}+bc&ab+bd\\ac+cd&bc+d^{2}\end{pmatrix}}-{\begin{pmatrix}a^{2}+ad&ab+bd\\ac+cd&ad+d^{2}\end{pmatrix}}+{\begin{pmatrix}ad-bc&0\\0&ad-bc\end{pmatrix}}}$
and just check each entry sum to see that the result is the zero matrix.
This exercise is recommended for all readers.
Problem 11
Prove that the minimal polynomial of an ${\displaystyle n\!\times \!n}$ matrix has degree at most ${\displaystyle n}$ (not ${\displaystyle n^{2}}$ as might be guessed from this subsection's opening). Verify that this maximum, ${\displaystyle n}$, can happen.
By the Cayley-Hamilton theorem the degree of the minimal polynomial is less than or equal to the degree of the characteristic polynomial, ${\displaystyle n}$. Example 1.12 shows that ${\displaystyle n}$ can happen.
This exercise is recommended for all readers.
Problem 12
The only eigenvalue of a nilpotent map is zero. Show that the converse statement holds.
Suppose that ${\displaystyle t}$'s only eigenvalue is zero. Then the characteristic polynomial of ${\displaystyle t}$ is ${\displaystyle x^{n}}$. Because ${\displaystyle t}$ satisfies its characteristic polynomial, it is a nilpotent map.
Problem 13
What is the minimal polynomial of a zero map or matrix? Of an identity map or matrix?
A minimal polynomial must have leading coefficient ${\displaystyle 1}$, and so if the minimal polynomial of a map or matrix were to be a degree zero polynomial then it would be ${\displaystyle m(x)=1}$. But the identity map or matrix equals the zero map or matrix only on a trivial vector space.
So in the nontrivial case the minimal polynomial must be of degree at least one. A zero map or matrix has minimal polynomial ${\displaystyle m(x)=x}$, and an identity map or matrix has minimal polynomial ${\displaystyle m(x)=x-1}$.
This exercise is recommended for all readers.
Problem 14
Interpret the minimal polynomial of Example 1.2 geometrically.
The polynomial can be read geometrically to say "a ${\displaystyle 60^{\circ }}$ rotation minus two rotations of ${\displaystyle 30^{\circ }}$ equals the identity."
Problem 15
What is the minimal polynomial of a diagonal matrix?
For a diagonal matrix
${\displaystyle T={\begin{pmatrix}t_{1,1}&0\\0&t_{2,2}\\&&\ddots \\&&&t_{n,n}\end{pmatrix}}}$
the characteristic polynomial is ${\displaystyle (t_{1,1}-x)(t_{2,2}-x)\cdots (t_{n,n}-x)}$. Of course, some of those factors may be repeated, e.g., the matrix might have ${\displaystyle t_{1,1}=t_{2,2}}$. For instance, the characteristic polynomial of
${\displaystyle D={\begin{pmatrix}3&0&0\\0&3&0\\0&0&1\end{pmatrix}}}$
is ${\displaystyle (3-x)^{2}(1-x)=-1\cdot (x-3)^{2}(x-1)}$.
To form the minimal polynomial, take the terms ${\displaystyle x-t_{i,i}}$, throw out repeats, and multiply them together. For instance, the minimal polynomial of ${\displaystyle D}$ is ${\displaystyle (x-3)(x-1)}$. To check this, note first that Theorem 1.8, the Cayley-Hamilton theorem, requires that each linear factor in the characteristic polynomial appears at least once in the minimal polynomial. One way to check the other direction— that in the case of a diagonal matrix, each linear factor need appear at most once— is to use a matrix argument. A diagonal matrix, multiplying from the left, rescales rows by the entry on the diagonal. But in a product ${\displaystyle (T-t_{1,1}I)\cdots }$, even without any repeat factors, every row is zero in at least one of the factors.
For instance, in the product
${\displaystyle (D-3I)(D-1I)=(D-3I)(D-1I)I={\begin{pmatrix}0&0&0\\0&0&0\\0&0&-2\end{pmatrix}}{\begin{pmatrix}2&0&0\\0&2&0\\0&0&0\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$
because the first and second rows of the first matrix ${\displaystyle D-3I}$ are zero, the entire product will have a first row and second row that are zero. And because the third row of the middle matrix ${\displaystyle D-1I}$ is zero, the entire product has a third row of zero.
This exercise is recommended for all readers.
Problem 16
A projection is any transformation ${\displaystyle t}$ such that ${\displaystyle t^{2}=t}$. (For instance, the transformation of the plane ${\displaystyle \mathbb {R} ^{2}}$ projecting each vector onto its first coordinate will, if done twice, result in the same value as if it is done just once.) What is the minimal polynomial of a projection?
This subsection starts with the observation that the powers of a linear transformation cannot climb forever without a "repeat", that is, that for some power ${\displaystyle n}$ there is a linear relationship ${\displaystyle c_{n}\cdot t^{n}+\dots +c_{1}\cdot t+c_{0}\cdot {\mbox{id}}=z}$ where ${\displaystyle z}$ is the zero transformation. The definition of projection is that for such a map one linear relationship is quadratic, ${\displaystyle t^{2}-t=z}$. To finish, we need only consider whether this relationship might not be minimal, that is, are there projections for which the minimal polynomial is constant or linear?
For the minimal polynomial to be constant, the map would have to satisfy that ${\displaystyle c_{0}\cdot {\mbox{id}}=z}$, where ${\displaystyle c_{0}=1}$ since the leading coefficient of a minimal polynomial is ${\displaystyle 1}$. This is only satisfied by the zero transformation on a trivial space. This is indeed a projection, but not a very interesting one.
For the minimal polynomial of a transformation to be linear would give ${\displaystyle c_{1}\cdot t+c_{0}\cdot {\mbox{id}}=z}$ where ${\displaystyle c_{1}=1}$. This equation gives ${\displaystyle t=-c_{0}\cdot {\mbox{id}}}$. Coupling it with the requirement that ${\displaystyle t^{2}=t}$ gives ${\displaystyle t^{2}=(-c_{0})^{2}\cdot {\mbox{id}}=-c_{0}\cdot {\mbox{id}}}$, which gives that ${\displaystyle c_{0}=0}$ and ${\displaystyle t}$ is the zero transformation or that ${\displaystyle c_{0}=1}$ and ${\displaystyle t}$ is the identity.
Thus, except in the cases where the projection is a zero map or an identity map, the minimal polynomial is ${\displaystyle m(x)=x^{2}-x}$.
Problem 17
The first two items of this question are review.
1. Prove that the composition of one-to-one maps is one-to-one.
2. Prove that if a linear map is not one-to-one then at least one nonzero vector from the domain is sent to the zero vector in the codomain.
3. Verify the statement, excerpted here, that preceeds Theorem 1.8.
... if a minimial polynomial ${\displaystyle m(x)}$ for a transformation ${\displaystyle t}$ factors as ${\displaystyle m(x)=(x-\lambda _{1})^{q_{1}}\cdots (x-\lambda _{\ell })^{q_{\ell }}}$ then ${\displaystyle m(t)=(t-\lambda _{1})^{q_{1}}\circ \cdots \circ (t-\lambda _{\ell })^{q_{\ell }}}$ is the zero map. Since ${\displaystyle m(t)}$ sends every vector to zero, at least one of the maps ${\displaystyle t-\lambda _{i}}$ sends some nonzero vectors to zero. ... Rewording ...: at least some of the ${\displaystyle \lambda _{i}}$ are eigenvalues.
1. This is a property of functions in general, not just of linear functions. Suppose that ${\displaystyle f}$ and ${\displaystyle g}$ are one-to-one functions such that ${\displaystyle f\circ g}$ is defined. Let ${\displaystyle f\circ g(x_{1})=f\circ g(x_{2})}$, so that ${\displaystyle f(g(x_{1}))=f(g(x_{2}))}$. Because ${\displaystyle f}$ is one-to-one this implies that ${\displaystyle g(x_{1})=g(x_{2})}$. Because ${\displaystyle g}$ is also one-to-one, this in turn implies that ${\displaystyle x_{1}=x_{2}}$. Thus, in summary, ${\displaystyle f\circ g(x_{1})=f\circ g(x_{2})}$ implies that ${\displaystyle x_{1}=x_{2}}$ and so ${\displaystyle f\circ g}$ is one-to-one.
2. If the linear map ${\displaystyle h}$ is not one-to-one then there are unequal vectors ${\displaystyle {\vec {v}}_{1}}$, ${\displaystyle {\vec {v}}_{2}}$ that map to the same value ${\displaystyle h({\vec {v}}_{1})=h({\vec {v}}_{2})}$. Because ${\displaystyle h}$ is linear, we have ${\displaystyle {\vec {0}}=h({\vec {v}}_{1})-h({\vec {v}}_{2})=h({\vec {v}}_{1}-{\vec {v}}_{2})}$ and so ${\displaystyle {\vec {v}}_{1}-{\vec {v}}_{2}}$ is a nonzero vector from the domain that is mapped by ${\displaystyle h}$ to the zero vector of the codomain (${\displaystyle {\vec {v}}_{1}-{\vec {v}}_{2}}$ does not equal the zero vector of the domain because ${\displaystyle {\vec {v}}_{1}}$ does not equal ${\displaystyle {\vec {v}}_{2}}$).
3. The minimal polynomial ${\displaystyle m(t)}$ sends every vector in the domain to zero and so it is not one-to-one (except in a trivial space, which we ignore). By the first item of this question, since the composition ${\displaystyle m(t)}$ is not one-to-one, at least one of the components ${\displaystyle t-\lambda _{i}}$ is not one-to-one. By the second item, ${\displaystyle t-\lambda _{i}}$ has a nontrivial nullspace. Because ${\displaystyle (t-\lambda _{i})({\vec {v}})={\vec {0}}}$ holds if and only if ${\displaystyle t({\vec {v}})=\lambda _{i}\cdot {\vec {v}}}$, the prior sentence gives that ${\displaystyle \lambda _{i}}$ is an eigenvalue (recall that the definition of eigenvalue requires that the relationship hold for at least one nonzero ${\displaystyle {\vec {v}}}$).
Problem 18
True or false: for a transformation on an ${\displaystyle n}$ dimensional space, if the minimal polynomial has degree ${\displaystyle n}$ then the map is diagonalizable.
This is false. The natural example of a non-diagonalizable transformation works here. Consider the transformation of ${\displaystyle \mathbb {C} ^{2}}$ represented with respect to the standard basis by this matrix.
${\displaystyle N={\begin{pmatrix}0&1\\0&0\end{pmatrix}}}$
The characteristic polynomial is ${\displaystyle c(x)=x^{2}}$. Thus the minimal polynomial is either ${\displaystyle m_{1}(x)=x}$ or ${\displaystyle m_{2}(x)=x^{2}}$. The first is not right since ${\displaystyle N-0\cdot I}$ is not the zero matrix, thus in this example the minimal polynomial has degree equal to the dimension of the underlying space, and, as mentioned, we know this matrix is not diagonalizable because it is nilpotent.
Problem 19
Let ${\displaystyle f(x)}$ be a polynomial. Prove that if ${\displaystyle A}$ and ${\displaystyle B}$ are similar matrices then ${\displaystyle f(A)}$ is similar to ${\displaystyle f(B)}$.
1. Now show that similar matrices have the same characteristic polynomial.
2. Show that similar matrices have the same minimal polynomial.
3. Decide if these are similar.
${\displaystyle {\begin{pmatrix}1&3\\2&3\end{pmatrix}}\qquad {\begin{pmatrix}4&-1\\1&1\end{pmatrix}}}$
Let ${\displaystyle A}$ and ${\displaystyle B}$ be similar ${\displaystyle A=PBP^{-1}}$. From the facts that
${\displaystyle A^{n}=(PBP^{-1})^{n}=(PBP^{-1})(PBP^{-1})\cdots (PBP^{-1})}$
${\displaystyle =PB(P^{-1}P)B(P^{-1}P)\cdots (P^{-1}P)BP^{-1}=PB^{n}P^{-1}}$
and ${\displaystyle c\cdot A=c\cdot (PBP^{-1})=P(c\cdot B)P^{-1}}$ follows the required fact that for any polynomial function ${\displaystyle f}$ we have ${\displaystyle f(A)=P\,f(B)\,P^{-1}}$. For instance, if ${\displaystyle f(x)=x^{2}+2x+3}$ then
${\displaystyle A^{2}+2A+3I=(PBP^{-1})^{2}+2\cdot PBP^{-1}+3\cdot I}$
${\displaystyle =(PBP^{-1})(PBP^{-1})+P(2B)P^{-1}+3\cdot PP^{-1}=P(B^{2}+2B+3I)P^{-1}}$
shows that ${\displaystyle f(A)}$ is similar to ${\displaystyle f(B)}$.
1. Taking ${\displaystyle f}$ to be a linear polynomial we have that ${\displaystyle A-xI}$ is similar to ${\displaystyle B-xI}$. Similar matrices have equal determinants (since ${\displaystyle \left|A\right|=\left|PBP^{-1}\right|=\left|P\right|\cdot \left|B\right|\cdot \left|P^{-1}\right|=1\cdot \left|B\right|\cdot 1=\left|B\right|}$). Thus the characteristic polynomials are equal.
2. As ${\displaystyle P}$ and ${\displaystyle P^{-1}}$ are invertible, ${\displaystyle f(A)}$ is the zero matrix when, and only when, ${\displaystyle f(B)}$ is the zero matrix.
3. They cannot be similar since they don't have the same characteristic polynomial. The characteristic polynomial of the first one is ${\displaystyle x^{2}-4x-3}$ while the characteristic polynomial of the second is ${\displaystyle x^{2}-5x+5}$.
Problem 20
1. Show that a matrix is invertible if and only if the constant term in its minimal polynomial is not ${\displaystyle 0}$.
2. Show that if a square matrix ${\displaystyle T}$ is not invertible then there is a nonzero matrix ${\displaystyle S}$ such that ${\displaystyle ST}$ and ${\displaystyle TS}$ both equal the zero matrix.
Suppose that ${\displaystyle m(x)=x^{n}+m_{n-1}x^{n-1}+\dots +m_{1}x+m_{0}}$ is minimal for ${\displaystyle T}$.
1. For the "if" argument, because ${\displaystyle T^{n}+\dots +m_{1}T+m_{0}I}$ is the zero matrix we have that ${\displaystyle I=(T^{n}+\dots +m_{1}T)/(-m_{0})=T\cdot (T^{n-1}+\dots +m_{1}I)/(-m_{0})}$ and so the matrix ${\displaystyle (-1/m_{0})\cdot (T^{n-1}+\dots +m_{1}I)}$ is the inverse of ${\displaystyle T}$. For "only if", suppose that ${\displaystyle m_{0}=0}$ (we put the ${\displaystyle n=1}$ case aside but it is easy) so that ${\displaystyle T^{n}+\dots +m_{1}T=(T^{n-1}+\dots +m_{1}I)T}$ is the zero matrix. Note that ${\displaystyle T^{n-1}+\dots +m_{1}I}$ is not the zero matrix because the degree of the minimal polynomial is ${\displaystyle n}$. If ${\displaystyle T^{-1}}$ exists then multiplying both ${\displaystyle (T^{n-1}+\dots +m_{1}I)T}$ and the zero matrix from the right by ${\displaystyle T^{-1}}$ gives a contradiction.
2. If ${\displaystyle T}$ is not invertible then the constant term in its minimal polynomial is zero. Thus,
${\displaystyle T^{n}+\dots +m_{1}T=(T^{n-1}+\dots +m_{1}I)T=T(T^{n-1}+\dots +m_{1}I)}$
is the zero matrix.
This exercise is recommended for all readers.
Problem 21
1. Finish the proof of Lemma 1.7.
2. Give an example to show that the result does not hold if ${\displaystyle t}$ is not linear.
1. For the inductive step, assume that Lemma 1.7 is true for polynomials of degree ${\displaystyle i,\ldots ,k-1}$ and consider a polynomial ${\displaystyle f(x)}$ of degree ${\displaystyle k}$. Factor ${\displaystyle f(x)=k(x-\lambda _{1})^{q_{1}}\cdots (x-\lambda _{\ell })^{q_{\ell }}}$ and let ${\displaystyle k(x-\lambda _{1})^{q_{1}-1}\cdots (x-\lambda _{\ell })^{q_{\ell }}}$ be ${\displaystyle c_{n-1}x^{n-1}+\cdots +c_{1}x+c_{0}}$. Substitute:
${\displaystyle {\begin{array}{rl}k(t-\lambda _{1})^{q_{1}}\circ \cdots \circ (t-\lambda _{\ell })^{q_{\ell }}({\vec {v}})&=(t-\lambda _{1})\circ (t-\lambda _{1})^{q_{1}}\circ \cdots \circ (t-\lambda _{\ell })^{q_{\ell }}({\vec {v}})\\&=(t-\lambda _{1})\,(c_{n-1}t^{n-1}({\vec {v}})+\cdots +c_{0}{\vec {v}})\\&=f(t)({\vec {v}})\end{array}}}$
(the second equality follows from the inductive hypothesis and the third from the linearity of ${\displaystyle t}$).
2. One example is to consider the squaring map ${\displaystyle s:\mathbb {R} \to \mathbb {R} }$ given by ${\displaystyle s(x)=x^{2}}$. It is nonlinear. The action defined by the polynomial ${\displaystyle f(t)=t^{2}-1}$ changes ${\displaystyle s}$ to ${\displaystyle f(s)=s^{2}-1}$, which is this map.
${\displaystyle x{\stackrel {s^{2}-1}{\longmapsto }}s\circ s(x)-1=x^{4}-1}$
Observe that this map differs from the map ${\displaystyle (s-1)\circ (s+1)}$; for instance, the first map takes ${\displaystyle x=5}$ to ${\displaystyle 624}$ while the second one takes ${\displaystyle x=5}$ to ${\displaystyle 675}$.
Problem 22
Any transformation or square matrix has a minimal polynomial. Does the converse hold?
Yes. Expand down the last column to check that ${\displaystyle x^{n}+m_{n-1}x^{n-1}+\dots +m_{1}x+m_{0}}$ is plus or minus the determinant of this.
${\displaystyle {\begin{pmatrix}-x&0&0&&&m_{0}\\0&1-x&0&&&m_{1}\\0&0&1-x&&&m_{2}\\&&&\ddots \\&&&&1-x&m_{n-1}\end{pmatrix}}}$ | 2016-05-28 11:59:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 329, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9587690830230713, "perplexity": 137.2572249712995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049277592.65/warc/CC-MAIN-20160524002117-00075-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/129173/constructing-conserved-current-given-the-lagrangian?noredirect=1 | # Constructing conserved current given the lagrangian
Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ where $F_{\alpha \beta} = \partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}$ which means $$\mathcal L = \frac{1}{4} (\partial^{\alpha}A^{\beta} - \partial^{\beta}A^{\alpha})(\partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}) + \frac{1}{2}m^2A^{\alpha}_{\alpha} \tag{1}$$ The canonical energy-momentum tensor is supposed to be, using the relation
$$T^{\mu \nu}_c = -\eta^{\mu \nu} \mathcal L + \frac{\partial \mathcal L}{\partial (\partial_{\mu}\Phi)}\partial^{\nu}\Phi,\,\,\tag{2}$$
$$T^{\mu \nu}_c = F^{\mu \alpha}\partial^{\nu}A_{\alpha} - \eta^{\mu \nu}\mathcal L$$
Then from $T^{\mu \nu}_B = T^{\mu \nu}_c + \partial_{\rho}B^{\rho \mu \nu}$, it is found that $$B^{\alpha \mu \nu} = F^{\alpha \mu}A^{\nu}\tag{3}$$ using the formula $$B^{\mu \rho \nu} = \frac{1}{2}i \left\{\frac{\partial \mathcal L}{\partial (\partial_{\mu}A_{\gamma})} S^{\nu \rho}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\rho}A_{\gamma})} S^{\mu \nu}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\nu}A_{\gamma})} S^{\mu \rho}A_{\gamma}\right\}$$ My question is how is this equation obtained and how did they obtain $(3)$? Did they make use of the explicit form of the spin matrix for a vector field? (My question is from Di Francesco et al 'Conformal Field Theory' P.46-47).
Here is my attempt: $$B^{\alpha \mu \nu} = \frac{i}{2}\left\{F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} + F^{\mu \gamma}S^{\alpha \nu}A_{\gamma} + F^{\nu \gamma}S^{\alpha \mu}A_{\gamma}\right\}$$ from simplifying the above. Concentrate on the first term. Then $$F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} = F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b} (S_{ab})^c_dA^d$$ Inputting the form of $S$ for a vector field, I get $$F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b}(\delta^c_a \eta_{bd} - \delta^c_b \eta_{da})A^d$$ But simplifying this and writing the other terms does not yield the result. Did I make a mistake upon insertion of the spin matrix?
• Related: physics.stackexchange.com/q/27048/2451 and links therein. Related to the question (v1-v3): physics.stackexchange.com/q/3005/2451 and links therein. – Qmechanic Aug 2 '14 at 19:02
• Ah, Jose is one of the lecturers at my university! But I don't think it directly answers my question. I am wondering if I am to input the spin matrix as I did above or not and if I made the substitution correctly. Would you be able to help? Thanks Qmechanic, the link should be useful nonetheless. – CAF Aug 2 '14 at 19:08 | 2019-08-21 06:06:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8219165205955505, "perplexity": 274.2264056467602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315809.69/warc/CC-MAIN-20190821043107-20190821065107-00156.warc.gz"} |
https://socratic.org/questions/how-do-you-find-a-vertical-asymptote-for-y-cot-x | # How do you find a vertical asymptote for y = cot(x)?
The vertical asymptotes for $y = \cot x = \frac{\cos x}{\sin x}$ are of the form:
$x = n \pi$, where $n$ is any integer
since the denominator $\sin x = 0$ when $x = 0 , \pm \pi , \pm 2 \pi , \ldots$. | 2020-01-18 17:45:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.98726886510849, "perplexity": 223.44070474256284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250593295.11/warc/CC-MAIN-20200118164132-20200118192132-00333.warc.gz"} |
https://dmtcs.episciences.org/4487 | Garnero, Valentin and Sau, Ignasi - A Linear Kernel for Planar Total Dominating Set
dmtcs:3295 - Discrete Mathematics & Theoretical Computer Science, May 16, 2018, Vol. 20 no. 1
A Linear Kernel for Planar Total Dominating Set
Authors: Garnero, Valentin and Sau, Ignasi
A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.
DOI : 10.23638/DMTCS-20-1-14
Volume: Vol. 20 no. 1
Section: Discrete Algorithms
Published on: May 16, 2018
Submitted on: May 2, 2017
Keywords: Computer Science - Data Structures and Algorithms,05C85, 05C10,G.2.2 | 2020-02-26 00:21:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7767459154129028, "perplexity": 787.6226717023209}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146176.73/warc/CC-MAIN-20200225233214-20200226023214-00062.warc.gz"} |
https://mathoverflow.net/questions/243853/polar-coordinates-bounded-domain-with-c1-boundary | # Polar coordinates, bounded domain with $C^{1}$ boundary
I have a question about a integral on a surface.
It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that $$(1) \quad \frac{d}{dr} \int_{B(0,r)}f\,dm=\int_{\partial B(0,r)}f\,d \sigma \quad m\text{-a.e. }r.$$ Here and hereafter $m$ denotes the $n$-dim Lebesgue measure, $\sigma$ the $(n-1)$ dim Hausdorff measure (surface measure) and $B(0,r)$ the open ball of radius $r$ centered at origin.
Question
Let $D \subset \mathbb{R}^{n}$ be a bounded domain with $C^{1}$ boundary. Set \begin{align} D_{\epsilon}=\left\{ x \in \bar{D} : d\left(x,\partial D \right) \leq \epsilon \right\} \end{align} Can we show the following equation? : \begin{align} \lim_{\epsilon \to 0} \frac{1}{\epsilon}\int_{D_{\epsilon}}f\,dm=\int_{\partial D}f\,d\sigma ,\quad (f \in C(\bar{D})) \end{align}
This is a generalization of $(1)$.
If you know how to prove this equation or helpful references, please let me know.
I think the cleanest proof is based on the coarea formula, which holds for pretty rough functions. Describe your set $D$ as the level set $\{F(x)\le t\}$ for some suitable function $F:R^n\to R$. By coarea formula you can write $$\int _ {t-\epsilon<F(x)\le t}f(x)dx= \int_{t-\epsilon}^{t} \int_{F(x)=s} \frac{f(x)}{|\nabla F(x)|}dH_{n-1}ds$$ where $dH_{n-1}$ is the surface measure on the set $\{F(x)=s\}$. This gives $$\epsilon^{-1}\int _ {t-\epsilon<F(x)\le t}f(x)dx \to \int_{F(x)=t} \frac{f(x)}{|\nabla F(x)|}dH_{n-1}.$$ To obtain your formula, just choose $F(x)=d(x,\partial D)$ (or if you want it smoother, $F(x)=d(x,D)-d(x,R^n\setminus D)$).
Using a partition of unity, you can reduce the problem to the case when $f$ has compact support and $D$ is a subgraph of Lipshitz function with arbitrary small Lipschitz constant, say $\varepsilon>0$.
In the latter case it is straightforward to prove that your equality holds up to $e^{\pm\varepsilon}$. Since $\varepsilon$ is arbitrary, the statement follows. | 2020-10-26 22:35:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9993857741355896, "perplexity": 139.93962329606725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107892062.70/warc/CC-MAIN-20201026204531-20201026234531-00552.warc.gz"} |
https://codegolf.stackexchange.com/questions/78109/make-a-better-score-than-my-students/78110 | Make a better score than my students! [closed]
For the last day of the semester, I decided to have some fun with my students and I showed them Code Golf... poor guys, they're addicted now.
The goal was to compete against the other students to solve those problems with what would be very usual rules for you. They had to use C# (because it's the language they are learning right now) so please do the same.
The length of each method must be the shortest including the method signature!
Write functions that do the following:
1. Input: string, int.
Output: string.
Description: the output string repeats the input string as many times as the input int.
Example: a("Hi", 5) returns "HiHiHiHiHi".
2. Input: int.
Output: int.
Description: the output is the quantity of prime numbers lower or equal to the input.
Example: b(10) returns 4. b(100) returns 25.
3. Input: object[].
Output: object[].
Description: returns the input without the null values.
Example: c(new object[] { "Hello", 42, null, new Program(), null, null, false }) returns an array containing [Hello, 42, Program instance, false].
4. Input: int.
Output: int.
Description: the output is the sum of all the integers from 1 to int input.
Example: d(4) returns 10 (because 1 + 2 + 3 + 4 = 10).
5. Input: char.
Output: char.
Description: the output is the input char with the other case. non-letters don't need to be checked.
Example: e('a') returns 'A'. e('A') returns 'a'. e('-') may have unexpected results.
6. Input: string, int.
Output: bool.
Description: true if and only if the length of the input string is a multiple of the input int.
Example: f("Programmer", 5) returns true. f("Programming", 5) returns false.
7. Input: string.
Output: boolean.
Description: true if and only if the input string is a palindrome.
Example: g("level") returns true. g('label') returns false.
Their solutions
I took the best of each task that they presented to me.
1. 64
2. 135
3. 105
4. 30
5. 71
6. 45
7. 83
Total: 533
(I will write their complete answers when the challenge is over).
• Use C# to be evaluated equally to my students.
• Write each task on a separate line with the method signature (string a(string s,int i)), it is part of the answer.
• The shortest overall score wins, but honorary mention may be awarded to the lowest score of a given task.
Note: It is not impossible that many of you have the same answer, my students did without cheating. Please consider that someont who has the same answer as you may simply have taken the same path and did not steal your idea.
FAQ:
• Why should people here use C#? It's not the best language for code golfing !
I know, but I would like to have some examples from pros like you to show them what a real golfer would do in their shoes.
closed as off-topic by Martin EnderApr 20 '16 at 15:06
• This question does not appear to be about programming puzzles or code golf within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
• I think assigning your students golfing tasks is a nice idea but restricting this to C# may garner you some disapproval here :/ Perhaps encourage C# but don't require it? Just a heads up. – Calvin's Hobbies Apr 20 '16 at 15:03
• Challenges with multiple unrelated parts are deemed off-topic. Please consider posting them as separate challenges. You can use the leaderboard from this challenge to automatically compute scores over all parts. But note that some of your individual parts are likely duplicates (which is exactly why multi-part challenges are disallowed). – Martin Ender Apr 20 '16 at 15:05
• I'm voting to close this question as off-topic because this challenge consists of multiple unrelated subtasks. – Martin Ender Apr 20 '16 at 15:06
• Just an idea, if you choose to pursue this question, I think posting each of your students' best answers individually as a tips question would probably be the best way to go about what you want. That's just my opinion though, if you want to ask you could try meta or chat. – FryAmTheEggman Apr 20 '16 at 15:18
• Got 294 Yeee - Oh, post closed... – a-ctor Jun 12 '16 at 21:56
Score: 387
1. string a(string s,int n){return string.Join(s,new string[n+1]);} (64)
2. int b(int n){bool p=true;for(int i=2;i<n;i++)if(n%i<1)p=false;return n<2?0:b(n-1)+(p?1:0);} (91)
3. object[]c(object[]l){return l.Where(x=>x!=null).ToArray();} (59)
4. int d(int n){return(n+1)*n/2;} (30)
5. char e(char c){return(char)(c<97?c+32:c-32);} (45)
6. bool f(string s,int n){return s.Length%n<1;} (44)
7. bool g(string s){return s.SequenceEqual(s.Reverse());} (54)
Total: 387
• !0 is shorter than true. same for !1 shorter than false – Ven Apr 20 '16 at 14:50
• Not with C#, thanks anyway. – SteeveDroz Apr 20 '16 at 14:52
• @Oltarus 1>0 and 0>1 though... – Martin Ender Apr 20 '16 at 15:03
• For part 5, c^32 should do. – Martin Ender Apr 20 '16 at 15:07 | 2019-09-23 01:10:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26339587569236755, "perplexity": 3449.8525176359963}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575844.94/warc/CC-MAIN-20190923002147-20190923024147-00146.warc.gz"} |
https://math.stackexchange.com/questions/422225/i-dont-understand-this-proof-of-the-am-gm-inequality | # I don't understand this proof of the AM-GM inequality?
The proof uses this lemma which I understand:
$\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ such that $x-E \ge y+E$, then $(x-E)(y+E) \gt xy$ . $\;$Hence, by subtracting $E$ from $x$ and adding it to $y$, we leave the average of the two numbers unchanged while increasing their product.
$\mathbf {Proof}:$ Suppose $a_{1}, a_{2}, a_{3}... a_{n}$ are positive real numbers with average $A$ and product $P$. If all $a_{i}$ are equal, then both the geometric mean and the arithmetic mean are equal to $A$. Let $a_{j}$ be one number closest to $A$ without being equal to $A$. Without loss of generality, let $a_{j} \lt A$ . Since the average of the numbers is $A$, there is at least one member of the set greater than $A$. Let $a_{k}$ be the greatest of these numbers. Clearly we must have $a_{k}-A \gt A-a_{j}$ since $a_j$ is closer to $A$ than any other $a_i$ not equal to $A$. We now use our lemma. Replace $a_j$ with $A$ and $a_k$ with $a_k-(A-a_j)$ . Note that $a_k-(A-a_j) \ge a_j +(A-a_j)$ , so we can apply our lemma with $(A-a_j)$ as our $E$ . By our lemma, the average of the numbers in the new set is the same, but the product is now higher. If we continue this process, we make one of the members of the set equal to $A$ with each application of the process. Hence, in some finite number of steps, we will make all the numbers equal to $A$. Thus, we prove that of all the sets of positive numbers with average $A$, the set with maximum product has all the elements equal to $A$.
$1)$ I don't understand why they don't loose generality when they say to let $a_j$ be the number closest to $A$ and let $a_j \lt A$. It certainly is possible for this not to be the case, for example the set ${2, 10, 10, 10}$. The average is $8$, but the number closest to $A$ is greater than $A$, so I don't see how the proof can apply to this set.
$2)$ I don't see how this process makes makes the elements of the set equal to $A$. If you want $a_j+E$ and $a_k-E$ to be equal to $A$, then $a_j$ and $a_k$ have to be equidistant from $A$.
$3)$ if you do bring one pair of terms at a time equal to $A$, then that means you must have an equal number of therms below and above $A$.
Any help is appreciated, thanks!
• 1) They mean a similar strategy works if $a_j > A$. 2) One of the terms is changed to $A$ (since $a_j + E = A$), but that's all the progress that is needed. Jun 16 '13 at 20:53
1) You are correct, there needs to be some tweaking.
2) It makes one more number equal to A, so by induction eventually they all will be.
3) You don't make both equal to A; you make at least one equal to A.
Perhaps a simpler proof of the middle part, avoiding the first issue, is this: Let $a,b$ be chosen so that $a<A<b$; if this is not possible then all the $a_i$'s are already equal. Set $c=\min(|A-b|,|A-a|$). We replace $a$ by $a+c$ and $b$ by $b-c$. By the same calculation as in the lemma, the average remains the same and the product increases. We have now made at least one of $a,b$ equal to $A$. Continue until all $a_i=A$.
• Ok thanks I think I got it
– Ovi
Jun 16 '13 at 20:59
1) I'd suggest to choose any indices $j,k$ with $a_j<A<a_k$ (which exist unless all $a_i$ are equal) and let $E=\min\{a_k-A,A-a_j\}$ in the lemma.
2) $E$ is specifically chosen so that at least one of $a_j+E$, $a_k-E$ equals $A$. The text explicitly makes $a_j=A$, i.e. chooses $E=A-a_j$, and decreases $a_k$ accordingly (I do similar in my suggestion for$1)$). It is enough to make only one of these equal to $A$ in order to increase the count.
3) You can only be sure to have at least one bigger and at least one smaller number. Only in the last step you are sure to "accidentally" bring two numbers at once to the average $A$ (because it is not possible that all numbers but one are equal to $A$).
The proof that uses the Lemma is rather hard-going. For my perspective on the proof see my answer to the related question; it also answers your question, I think.
I gather that the AM-GM inequality you are talking about is $\,(a_1\cdots a_n)^{1/n}\leq(a_1+\cdots+a_n)/n\,$ for positive real numbers $a_1$, $\ldots$, $a_n$. There is a slightly more general AM-GM inequality:
Let $n\geq 1$ be an integer, let $a_1,\ldots,a_n>0$, and let $\lambda_1,\ldots,\lambda_n> 0$ satisfy $\lambda_1+\cdots+\lambda_n=1$. Then $a_1^{\lambda_1}\cdots a_n^{\lambda_n}\leq\lambda_1a_1+\cdots\lambda_na_n$, where the equality holds if and only if $a_1=\cdots=a_n$.
I will give my favorite proof of the generalized AM-GM inequality. The embarassing thing about this particular proof is that I cannot remember whether I have seen it somewhere or I hacked it up myself when I was fooling around thinking up different ways (some of them extremely weird) of proving the inequality. In case you have come across this proof, or its close relative, somewhere, anywhere, please let me know by giving the reference in a comment to the present answer.
(Yes, I know the proof by Pólya, I know the Jensen's inequality, I know that the AM-GM inequality is a manifestation of the concavity of the $\log$ function.)
All we need for the proof of the generalized AM-GM inequality is the following inequality:$\newcommand{\RR}{\mathbb{R}}$
For every $x\in\RR$, $x>0$, we have $\,x-1\geq\ln x\,$, where the equality holds iff $\,x=1$.
The proof is simple: setting $f(x):=x-1-\ln x$ we have $f(1)=0$, and $f'(x)=1-x^{-1}$ is (strictly) negative for $0<x<1$ and is (strictly) positive for $x>1$.
Proof of the generalized AM-GM inequality. $~$For every $x>0$ we have \begin{aligned} (xa_1)^{\lambda_1}\cdots(xa_n)^{\lambda_n} ~&\:=\: x\cdot a_1^{\lambda_1}\cdots a_n^{\lambda_n}~, \\[1ex] \lambda_1(xa_1)+\cdots+\lambda_n(xa_n) ~&\:=\: x\cdot(\lambda_1a_1+\cdots\lambda_na_n)~. \end{aligned} This means that it suffices to prove the inequality with $xa_1$, $\ldots$, $xa_n$ in place of $a_1$, $\ldots$, $a_n$ for any $x>0$ we choose. We choose $x=(a_1^{\lambda_1}\cdots a_n^{\lambda_n})^{-1}\!$, that is, we can assume that $a_1^{\lambda_1}\cdots a_n^{\lambda_n}=1$ and have to prove that then $1\leq \lambda_1a_1+\cdots+\lambda_na_n$. But this is easy: \begin{aligned} \lambda_1a_1+\cdots+\lambda_na_n-1 ~&\:=\: \lambda_1(a_1-1) + \cdots +\lambda_n(a_n-1) \\[.5ex] ~&\:\geq\: \lambda_1\ln a_1 + \cdots + \lambda_n\ln a_n \\[.5ex] ~&\:=\: 0~, \end{aligned} where the equality holds iff $a_1=\cdots=a_n=1$.$~$ Done. | 2021-10-25 13:26:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9613065123558044, "perplexity": 109.09238690195338}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587711.69/warc/CC-MAIN-20211025123123-20211025153123-00107.warc.gz"} |
https://edurev.in/studytube/Wave-Guides--Part-1--Electromagnetic-Theory/50dfebcd-17c2-48a6-a49c-748541c47c22_t | Courses
# Wave Guides (Part - 1) Electrical Engineering (EE) Notes | EduRev
## Electromagnetic Theory
Created by: Machine Experts
## Electrical Engineering (EE) : Wave Guides (Part - 1) Electrical Engineering (EE) Notes | EduRev
The document Wave Guides (Part - 1) Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Electromagnetic Theory.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)
Wave Guides
Electromagnetic waves carry both energy and momentum. It should therefore be possible to transmit energy contained in the wave from one place to another. For low frequencies (typically, less than 1 MHz), this is done by parallel transmission lines or coaxial cables. However, for higher frequencies, such as microwave frequencies, we need special conduits such a hollow metal tubes or optical fibers.
We will first explain the basic concepts of guiding waves by taking the simple case of a pair of parallel, infinite metal plates with a separation d between them. The wave is made to propagate in the hollow region between the plates, which we take to be empty space. We have seen that the electric field is primarily confined to the surface, penetrating a small “skin depth” which becomes smaller with increasing frequency. As a result we take the electric field to vanish at the surface of the guide.
Let us rewrite the Maxwell’s equations for the case where there are free charges or currents.
where we have time variation to be
Taking the curl of the these equations and substituting from the other equations, we get
For propagation between the plannes, σ = 0, so that we have,
Consider the original curl equations and write them in component form (with σ = 0)
gives,
Parallely, the derivatives of electric field components satisf,
The solutions of these set of equations can be classified into three distinct groups. The direction of propagation being along the z direction, we term this direction as the longitudinal direction and a direction perpendicular to it (i.e. x and y directions) as the transverse direction. The distinct solutions are grouped as
1. Solutions for which Ez = 0 , i.e. the non-zero electric field is transverse to the direction of propagation. This is called the "Transverse Electric " or TE mode. In this case the longitudinal component of magnetic field is non-vanishing (Hz≠0). The solution, therefore, is also referred to as H-mode.
2. Solutions for which Hz = 0 , i.e. the non-zero magnetic field is transverse to the direction of propagation. This is called the "Transverse Magnetic " or TM mode. In this case the longitudinal component of electric field is non-vanishing (Ez≠ 0). The solution, therefore, is also referred to as E-mode.
3. In some situations, it is possible to have the longitudinal component of both electric and magnetic field to be simultaneously zero, like the case of propagation of electromagnetic wave in free space. This special solution is called "Transverse Electric and Magnetic Mode" or TEM mode.
It is of course possible for a solution not to belong to any of these distinct categories in which case it would be a "mixed mode" solution.
Since the wave propagates along the z direction, the z dependence of the field is specified,
where, the complex factor Thus is equivalent to multiplication by - Further, since the plates are of infinite extent in y direction, there is no field variation in this direction so that we can replace the derivative by zero. Using these, we can rewrite the equations above as
and
The wave equation (1) takes the form,
We will discuss in detail the TE solution and leave the TM solution as an exercise.
TE- Mode :
In this case E= 0. From Eqn. (2) , we get Hy = constant, which we can choose to be zero. This in turn implies, from Eqn. (3), Ex = 0. We will first solve for Eusing Eqn. (4).
Define k= ŷ2 + ω2∈μ. We have,
the solution of which are well known to be
(complete solution will be obtained by multiplying this with
We now insert the boundary condition, Ey = 0 on both the plates, i.e. at x= 0 and at x= d.
The former gives B= 0, so that Ey = A sin Kx. The latter condition restricts the values that k can take to where n = 1,2, … .( n cannot take the value zero because that would make the field identically zero. )
Thus we have,
where A = Ey0 is the maximum value of the field. We can now use this expression in Eqn. (3) to obtain the magnetic field components
Modes are named by specifying the value that n takes. As we have seen the lowest mode is n = 1. This is termed as TE10 mode, the meaning of the second index will be clear when we discuss rectangular waveguides but for the present case it remains zero for all values of n. The following figure gives the electric field profile for n=1 and n=2 for a fixed z.
If we look along the direction of propagation, for the mode, thefield lines crowdat the centre of the guide, where the field strength is strongest.
The corresponding magnetic fields for are shown below:
Cutoff Frequency
Is transmission in this manner always possible? We have,
Propagating solution implies that Thus we requires,
If the frequency is less than this, the wave attenuates. The phase velocity for the propagating solution is given by
As frequency decreases and approaches the critical value, it becomes infinite. For very large frequencies, the velocity in vacuum approaches that of light
TM Mode
We will not work out the TM mode algebra. In this case . The non-zero field components are
TEM Mode
Note that in TM case , unlike in the case of TE modes, we can have m=0 here because the solutions are in terms of cosine functions. In this case we have,
which gives the ratio which is the intrinsic impedance we have seen to characterize propagation of wave in a uniform medium.
Tutorial Assignment
1. For a guided wave between two infinite conducting planes separated by a distance of 0.25 m, find the cutoff frequency for the TM20 mode. If the operating frequency is 3 GHz, find the phase velocity of the wave.
2. A TE10 mode propagates between two parallel planes separated by a distance of 0.25 m. The planes are lossy and have a conductivity of 5 × 107 S/m. If the maximum electric field strength between the planes is 1000 V/m, determine the power loss per square meter on each plate when the operating frequency is 2 GHz.
Solutions to Tutorial Assignments
1. The critical angular frequencyωc is given by
so that the cutoff frequency is given by
If the operating frequency is 3 GHz, the propagation constant is given by
The phase velocity is given by
2. The amplitude of linear current density on the plates is equal to the tangential component of the magnetic field on the planes,
Power loss per unit length of conductor
We take . The loss for TE10 is (in J/m)
Self Assessment Questions
1. For a guided wave between two infinite conducting planes separated by a distance of 0.2m. If the operating frequency is 3.3 GHz, find the number of distinct modes that can travel in the guide.
2. A TE20 mode is propagating along the z direction between two parallel conducting planes separated by 0.2m along the x- direction. Find the cutoff frequency. Determine λx. If the operating frequency is 2.5 GHz, determine λz and the non-vanishing components of the electric and the magnetic field. If the operating frequency is 1.2 GHz, calculate the distance over which the strength of the fields reduce to 1/e of their value.
3. Calculate the rate at which energy is transmitted in a parallel plane waveguide operating in TE10 mode
Solutions to Self Assessment Questions
1. The propagation constant is given by
For propagation to take place 22 > 5n, so that n<5. This implies 4 TE modes, 4 TM modes and one TEM mode, giving a total of 9 modes.
2. The cutoff frequency is given by The wavelength in the x direction for m. If the operating frequency is 2.5 × 109Hz., the propagation vector is given by (using n=2)
The fields are as follows :
To determine λz, we use the fact that it is equal to the distance over which the phase of the propagating wave changes by 2π. Thus
so that λz= 3/20m.
If the operating frequency is 1.2 GHz, the wave attenuates, and we have,
so that the attenuation distance is
3. The fields are given by
Power transmitted per unit area is
Power transmitted in z direction through an area of unit width,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
; | 2020-05-30 12:23:05 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8673373460769653, "perplexity": 757.6577239137679}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409171.27/warc/CC-MAIN-20200530102741-20200530132741-00089.warc.gz"} |
http://www.read.seas.harvard.edu/cs161/2018/lectures/lecture6/ | This is not the current version of the class.
# Lecture 6
What potential bug was addressed by commit d12e98cdb959bb9cdb85fc8e1b0878733026388e? Describe a possible execution of the old code that could violate some kernel invariant or otherwise cause a problem.
## syscall registers
The syscall entry point saves most registers to a struct regstate. But is that really necessary? For instance, the callee-saved registers, such as %rbx and %r12, will be saved and restored by kernel C++ code automatically, since the C++ compiler uses the normal x86-64 calling convention. (For this reason, syscall_entry doesn’t bother to restore those registers when it resumes the user process!)
Which registers must syscall_entry save to struct regstate for Chickadee to work correctly? Run experiments to see, and explain the results.
## ucontext
On Linux or Mac, read the manual pages for getcontext, setcontext, makecontext, and swapcontext. What are the closest-corresponding Chickadee functions? Roughly how will these functions be implemented? Which of them, if any, can be implemented entirely within the C abstract machine (as opposed to using assembly)?
## Exit design
Problem Set 2, Part B asks you to implement part of a sys_exit system call. One of the invariants mentioned says that “The kernel task responsible for the exiting process must delegate its final freeing to some other logical thread of execution”. Come up with an initial design for this delegation. | 2020-09-27 20:56:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30333346128463745, "perplexity": 5185.361933424135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401578485.67/warc/CC-MAIN-20200927183616-20200927213616-00233.warc.gz"} |
https://kidsjoybox.com/what-is-30-4-as-a-mixed-fraction/ | # What is 30/4 as a Mixed Fraction?
Want to find 30/4 as a mixed fraction? Here is a step-by-step guide illustrating the easiest technique to convert an improper fraction as 30/4 as a mixed fraction.
Fractions can be expressed in different forms and here we shall learn about how to convert from one form to another.
The number 30/4 is an improper fraction that needs to be converted to a mixed fraction in the given case. Let us learn the techniques of how to convert a given improper fraction to a mixed fraction.
What is 30/4 as a Mixed Fraction? 30/4 = 7 ½
## Basic Concepts
1) A mixed fraction is constructed of a whole number and a fraction that is smaller in size than the given improper fraction from which it is simplified.
2) Definition of numerator: A numerator is a digit or a number that is present above the line of division.
3) Definition of denominator: A denominator is a digit that is present below the line division.
4) What is an improper fraction: An improper fraction is where the numerator is greater than the denominator and it can also be converted to a mixed fraction.
5) What is a proper fraction: A proper fraction is where the numerator is smaller than the denominator.
Also read: Convert 500 mcg to mg?
## Calculation to find 30/4 as a Mixed Fraction
Step 1: Find the whole number
First, you need to find the whole number, and do so you need to find the value by dividing the numerator by denominator.
Now,
#### 30 ÷ 4 = 7.5
So the whole number is now 7 and we have already solved a part of the given question.
Step 2: Find the smaller numerator
How to find the new smaller numerator.
To find the new numerator you need to multiply the whole number we found in the results of step 1 with the original denominator which is 4
Hence,
#### 7×4 = 28
Now you need to subtract the multiple from the original numerator and thus,
#### 30-28 = 2
Hence, 2 is the new numerator.
Step 3
Hence, 2 is the new numerator.
The original denominator will be the same for the new denominator and thus the given expression will be
Whole number = 7
New numerator = 2
The new denominator (same as original) = 4
Hence, the mixed fraction will be = 7 2/4
[latexpage]
$\frac{30}{4}=7\frac{2}{4}$
Step 4
The smaller fraction can be implied even further by dividing both the denominator and the numerator with a common factor of 2
#### Hence the answer will be 7 ½
[latexpage]
$\frac{30}{4}=7\frac{1}{2}$
This is because the greatest common factor of 2 and 4 is 2 hence we reduce the numerator and denominator 2 and 4 by 2 respectively.
## Conclusion
So the final result to the conversion of improper fraction 30/4 is 7 ½
Also read: What is 1/5 as a Decimal? | 2023-02-09 11:41:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9345362186431885, "perplexity": 507.1588903765896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499966.43/warc/CC-MAIN-20230209112510-20230209142510-00475.warc.gz"} |
https://www.sparrho.com/item/semilocal-density-functional-theory-with-correct-surface-asymptotics/8ef588/ | Semilocal density functional theory with correct surface asymptotics
Research paper by L. A. Constantin, E. Fabiano, J. M. Pitarke, F. Della Sala
Indexed on: 30 Mar '16Published on: 30 Mar '16Published in: Physics - Other
Abstract
Semilocal density functional theory is the most used computational method for electronic structure calculations in theoretical solid-state physics and quantum chemistry of large systems, providing good accuracy with a very attractive computational cost. Nevertheless, because of the non-locality of the exchange-correlation hole outside a metal surface, it was always considered inappropriate to describe the correct surface asymptotics. Here, we derive, within the semilocal density functional theory formalism, an exact condition for the image-like surface asymptotics of both the exchange-correlation energy per particle and potential. We show that this condition can be easily incorporated into a practical computational tool, at the simple meta-generalized-gradient approximation level of theory. Using this tool, we also show that the Airy-gas model exhibits asymptotic properties that are closely related to the ones at metal surfaces. This result highlights the relevance of the linear effective potential model to the metal surface asymptotics. | 2020-10-23 11:03:08 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8011640906333923, "perplexity": 1051.8530482153928}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107881369.4/warc/CC-MAIN-20201023102435-20201023132435-00011.warc.gz"} |
https://www.saturncloud.io/published/lksfr/programming-a-chess-player/chess/Programming%20a%20Chess%20Player.ipynb | # Programming a Chess Player
chess/Programming a Chess Player.ipynb
# 1. Programming a Chess Player¶
CS371: Introduction to Cognitive Science
Bryn Mawr College
Department of Computer Science
Professor Blank, Fall 2016
Goals:
• explore the idea that a computer could "think"
• explore symbolic computation
• write a program to play Chess
In this notebook we will begin to explore symbolic computation by writing a program to play Chess. Perhaps you don't know how to play Chess... no problem! You don't really need to know much, but a primer on Chess may be useful. Here are some links that might be useful:
Getting Started with Chess:
For these experiments, we will use the python-chess library. In this notebook, we will define three different sample players. We explore them in some depth here to attempt to understand how each plays chess.
python-chess Reference:
## 1.1 Game Play¶
The first thing we need to do is import the chess library:
import chess
We will use the chess library in the following manner:
1. Create a chess.Board instance
2. The chess.Board instance automatically generates all possible moves for the current player
3. Current player picks a move
4. Go to step #2 and repeat until win, lose, or draw
That's it! Thus we have reduced the playing a valid game of chess into simply selecting a move at each turn. To play a good game of chess, you will want to pick "the best move" at each turn.
A player will be a function that takes a board instance as a argument, and returns a move encoded as a string in Universal Chess Interface format:
def player(board):
### "Thinking" happens here
return move_code
We'll explain this fully below.
## 1.2 The Board class¶
The Board class keeps track of whose turn it is, possible moves, and a history of all past moves. This class can undo and redo moves, and keeps track of repeating states.
First, we create a board instance:
board = chess.Board()
The board.turn value is a boolean indicating whose turn it is. The values are either True for white or False for black.
board.turn
True
As seen above, the game always begins with white's turn. If you forget which is True, you can ask the chess module:
board.turn == chess.WHITE
True
The chess.Board class is responsible for generating all possible moves, whose turn it is, keeping track of the placement of all pieces, and making each move. The chess.Board represents a two-dimensional 8 x 8 array. However, the internal representation is optimized for speedy operations.
Here is a visual representation of a chess.Board:
board
You can also get an ASCII board representation (text-only) by converting the board into a string:
print(str(board))
r n b q k b n r
p p p p p p p p
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
P P P P P P P P
R N B Q K B N R
The string representation of the board shows the character representation for each piece. Specifically:
Piece White Black
Pawn P p
Rook R r
Knight N n
Bishop B b
Queen Q q
King K k
For our uses, you don't really need to know how each piece moves. We discuss game strategy, though, shortly.
The 2-dimensional board is laid out so that each position is indicated by a column letter and row number. However, the internal representation is sequential. Say that we wanted to see what was at location 'c1' we could use:
chess.C1
2
to get the internal location of the column/row, and then use:
board.piece_at(chess.C1)
Or shown as a character:
str(board.piece_at(chess.C1))
'B'
Indeed, there is a white bishop at 'c1'.
## 1.3 Making Moves¶
At this point, we can as the board instance to generate all of the possible, legal moves:
list(board.legal_moves)
[Move.from_uci('b1a3'),
Move.from_uci('b1c3'),
Move.from_uci('g1f3'),
Move.from_uci('g1h3'),
Move.from_uci('a2a3'),
Move.from_uci('b2b3'),
Move.from_uci('c2c3'),
Move.from_uci('d2d3'),
Move.from_uci('e2e3'),
Move.from_uci('f2f3'),
Move.from_uci('g2g3'),
Move.from_uci('h2h3'),
Move.from_uci('a2a4'),
Move.from_uci('b2b4'),
Move.from_uci('c2c4'),
Move.from_uci('d2d4'),
Move.from_uci('e2e4'),
Move.from_uci('f2f4'),
Move.from_uci('g2g4'),
Move.from_uci('h2h4')]
Python Note: board.legal_moves looks like a normal list of items. But it is really a property that gets lazily generated on the fly. We force it to be a list by wrapping list() around it.
We can get the first move (index zero):
move = list(board.legal_moves)[0]
### 1.3.1 Universal Chess Interface¶
The Universal Chess Interface (or uci) is a representation for describing a move from one cell to another (and perhaps additional information as well). We explore the first move:
move
Move.from_uci('b1a3')
move.uci()
'b1a3'
Thus, this is a move from b1 to a3.
What piece is this, and where is it moving on the board? Is this a good move?
The uci string is what each player function will return.
### 1.3.2 Standard Algebraic Notation¶
If you know something about Chess, you might know about Standard Algebraic Notation (or san). This is an alternative to uci. You can get a move's san with:
board.san(move)
'Na3'
However, we will always use uci.
## 1.4 Programming a Random Player¶
There is a useful function in the random module that will select from a a list of choices. This is called random.choice.
import random
To use it in a function, we simply:
def random_player(board):
move = random.choice(list(board.legal_moves))
return move.uci()
random_player(board)
'h2h4'
for i in range(10):
print(random_player(board))
h2h3
a2a4
g1h3
b2b3
b2b3
e2e4
g1f3
d2d4
c2c4
g1f3
## 1.5 Playing a Game¶
To play a game, we'll write a new function called play_game that will take two player functions, create a board, and alternatively call the player functions until a win, lose, or draw.
First, we need some additional modules for displaying a game in the notebook:
import time
from IPython.display import display, HTML, clear_output
A useful function for displaying the color of a player:
def who(player):
return "White" if player == chess.WHITE else "Black"
A function for displaying the board as text, or as the nice image (called SVG):
def display_board(board, use_svg):
if use_svg:
return board._repr_svg_()
else:
return "<pre>" + str(board) + "</pre>"
And finally, we can put those together to play a game:
def play_game(player1, player2, visual="svg", pause=0.1):
"""
playerN1, player2: functions that takes board, return uci move
visual: "simple" | "svg" | None
"""
use_svg = (visual == "svg")
board = chess.Board()
try:
while not board.is_game_over(claim_draw=True):
if board.turn == chess.WHITE:
uci = player1(board)
else:
uci = player2(board)
name = who(board.turn)
board.push_uci(uci)
board_stop = display_board(board, use_svg)
html = "<b>Move %s %s, Play '%s':</b><br/>%s" % (
len(board.move_stack), name, uci, board_stop)
if visual is not None:
if visual == "svg":
clear_output(wait=True)
display(HTML(html))
if visual == "svg":
time.sleep(pause)
except KeyboardInterrupt:
msg = "Game interrupted!"
return (None, msg, board)
result = None
if board.is_checkmate():
msg = "checkmate: " + who(not board.turn) + " wins!"
result = not board.turn
elif board.is_stalemate():
msg = "draw: stalemate"
elif board.is_fivefold_repetition():
msg = "draw: 5-fold repetition"
elif board.is_insufficient_material():
msg = "draw: insufficient material"
elif board.can_claim_draw():
msg = "draw: claim"
if visual is not None:
print(msg)
return (result, msg, board)
The function takes to player functions (first white, then black), and an optional argument to indicate representation style.
Let's pit random_player vs. random_player:
play_game(random_player, random_player)
Move 282 Black, Play 'g8h8':
draw: claim
(None, 'draw: claim', Board('7k/8/8/1K6/8/8/2R5/8 w - - 21 142'))
Many times, that will end in a draw.
## 1.6 Allowing a Human Player¶
Do you want to play a game? Here is a way to play:
def human_player(board):
display(board)
uci = get_move("%s's move [q to quit]> " % who(board.turn))
legal_uci_moves = [move.uci() for move in board.legal_moves]
while uci not in legal_uci_moves:
print("Legal moves: " + (",".join(sorted(legal_uci_moves))))
uci = get_move("%s's move[q to quit]> " % who(board.turn))
return uci
And a helper function to handle the input:
def get_move(prompt):
uci = input(prompt)
if uci and uci[0] == "q":
raise KeyboardInterrupt()
try:
chess.Move.from_uci(uci)
except:
uci = None
return uci
Note that you must enter your move in UCI, such as "a2a4", meaning moving the piece at a2 to location a4.
Try you hand at playing chess against the random_player. It is not as easy as it sounds. Did you win? How many turns did it take?
## 1.7 Analysis¶
If a random_player plays a random_player many times, how many times would you expect white to win? Black to win? To end in a draw?
Let's try it:
counts = {None: 0, True: 0, False: 0}
for i in range(10):
result, msg, board = play_game(random_player, random_player, visual=None)
counts[result] += 1
print(counts)
counts
## 1.8 Static Analysis/Board Evaluation¶
The next sample player takes each possible move, applies it to a temporary board and state, and then goes through the board, place by place, in order to compute an evaluation score for each resulting state. The moves are sorted by this score, and the best move is then returned:
def player1(board):
moves = list(board.legal_moves)
for move in moves:
newboard = board.copy()
# go through board and return a score
move.score = staticAnalysis(newboard, move, board.turn)
moves.sort(key=lambda move: move.score, reverse=True) # sort on score
return moves[0].uci()
The actual score is computed by the staticAnalysis function which is designed to evaluate the resulting board after each hypothesized move. To come up with a score for each static snapshot of a board, it will be necessary to know how many of each piece is left, and where they are. You can use the board.pieces() method for this:
board = chess.Board()
board.pieces(chess.ROOK, True)
If you look at the output as a list, you'll see the 1-D representation of where those pieces are on the game board:
list(board.pieces(chess.ROOK, True))
[0, 7]
len(board.pieces(chess.ROOK, True))
2
There are 2 white rooks.
Now, putting that into a function, checking for each type of piece:
def staticAnalysis(board, move, my_color):
score = 0
# score += 10 if board.is_capture(move) else 0
# To actually make the move:
board.push(move)
# Now check some other things:
for (piece, value) in [(chess.PAWN, 1),
(chess.BISHOP, 4),
(chess.KING, 0),
(chess.QUEEN, 10),
(chess.KNIGHT, 5),
(chess.ROOK, 3)]:
score += len(board.pieces(piece, my_color)) * value
score -= len(board.pieces(piece, not my_color)) * value
# can also check things about the pieces position here
return score
play_game(player1, random_player)
NOTE: The string representation for the board is in Forsyth-Edwards Notation, or FEN for short. The last number (6th column) is the "full-move count". If the full-move count is 36, then there have been 35 * 2 full-moves, plus 1 if "b" is in second columns, for 71 moves.
That didn't play so well! Why not?
The following is one way around the problem. What does it do differently?
def staticAnalysis(board, move, my_color):
score = random.random()
# score += 10 if board.is_capture(move) else 0
# To actually make the move:
board.push(move)
# Now check some other things:
for (piece, value) in [(chess.PAWN, 1),
(chess.BISHOP, 4),
(chess.KING, 0),
(chess.QUEEN, 10),
(chess.KNIGHT, 5),
(chess.ROOK, 3)]:
score += len(board.pieces(piece, my_color)) * value
score -= len(board.pieces(piece, not my_color)) * value
# can also check things about the pieces position here
return score
play_game(player1, random_player)
Better! But it still is not very aggressive. What could we add to make it attack?
def staticAnalysis(board, move, my_color):
score = random.random()
# score += 10 if board.is_capture(move) else 0
# To actually make the move:
board.push(move)
# Now check some other things:
for (piece, value) in [(chess.PAWN, 1),
(chess.BISHOP, 4),
(chess.KING, 0),
(chess.QUEEN, 10),
(chess.KNIGHT, 5),
(chess.ROOK, 3)]:
score += len(board.pieces(piece, my_color)) * value
score -= len(board.pieces(piece, not my_color)) * value
# can also check things about the pieces position here
# Check global things about the board
score += 100 if board.is_checkmate() else 0
return score
play_game(player1, random_player)
This staticAnalysis function makes a much better player than either of the random players, but it still has major issues. How can you improve this static evaluation function?
## 1.9 Suggestions¶
Pawns get promoted when they get to the back row. Encourage them to get to the back row (eg, the closer they are to the opposite side, the better).
• It is good to threaten opponent pieces.
• It is good that your opponent's King has no valid moves.
Static analysis on the next move's state can only do so much good. It would be better if you could "look ahead" further and see the results of what your opponent could do, given what your proposed move did. And then see what you could do, then what they would do, etc. This is how real chess programs work. There are many algorithms for finding the best move by looking many moves ahead, such as minimax and alpha-beta pruning. You'll explore these ideas fully in Artificial Intelligence.
Your board evaluation function could change during the game. For example, you might use one evaluation function at the beginning, one in the middle, and another at the end. How can you tell where you are in a game?
There is a nice article on Chess Strategy at wikipedia: http://en.wikipedia.org/wiki/Chess_strategy | 2019-10-16 11:34:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23748214542865753, "perplexity": 4870.349561928452}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986668569.22/warc/CC-MAIN-20191016113040-20191016140540-00346.warc.gz"} |
http://christopher5106.github.io/deep/learning/2016/11/15/linear-algebra-for-derivatives-in-multi-dimensional-spaces-tensor-product-inner-outer-products.html | Deep learning involves lot’s of the derivatives’computations in multi-dimensional spaces.
In particular, any network module, as well as a full network composed of multiple modules, are mapping functions between input and output spaces
but also between parameter and output space
which is of much interest for the gradient descents.
A loss function is a mapping to a 1-dimension space (scalar) :
# Linear algebra
A matrix multiplication (or tensor product in higher dimensions) gives a new matrix :
The inner product or scalar product or of 2 vectors outputs a scalar :
The same for matrices, the matrix dot product or Frobenius inner product, outputs a scalar :
The outer product of 2 vectors produces a matrix :
# Jacobian
The Jacobian is the first order derivative of the function.
For the network module, the Jacobian is a matrix $\in \mathbb{R}^{o \times n}$
where j is the indice of the output in the network output and $\vec{w}$ is a column vector.
In the case of the loss function, the Jacobian is usually the vector
but I’ll write it as a matrix with one row $\in \mathbb{R}^{1 \times o}$
where i is the indice of the output in the network output.
For the example, the Jacobian of a linear layer :
and with an activation function f :
The introduction of a non-linearity modifies the Jacobian so that each row (corresponding to the Jacobian of one neuron / output) is multiplied by the non-linearity derivative value at this neuron output.
# Hessian
The hessian is the second order derivative, following the same definition as for Jacobian :
Since $J_{\mathcal{N}}$ is a matrix, $H_{\mathcal{N}}$ is a 3-dimensional tensor. T2 is for the transpose to the third dimension.
Let’s write the special case for a scalar function, for which the hessian is a (o x o) symetric matrix
# Composition of functions
A composition is for example the loss function computed on the output of the network (softmax can be seen as a module inside the network or inside the loss function)
In the general case when $\mathcal{L}$ has a multi-dimensional output
which is a simple matrix multiplication named the chain rule :
And in the scalar case (when $\mathcal{L}$ outputs a scalar), this can be rewritten with the vector notation :
That is why it can sometimes be a bit confusing.
Let’s go for the hessian of a composition of functions, but considering the scalar case only (the multi-dimensional case is left as an exercice for the reader :) ), let’s keep in mind that the jacobian of the loss function is being evaluated at the output of the network in fact :
and derivate (if you have a headache, it might not be an anomaly) :
The first part $G = J_{\mathcal{N}}^T \times h_{\mathcal{L}} \times J_{\mathcal{N}}$ is the Gauss-Newton matrix, modeling the interactions of second order originated from the top part $\mathcal{L}$. It is positive semi definite and is used as an approximation in some second order optimization algorithms.
# Matching loss function
Let’s consider the case where $\mathcal{N}$ is the output non-linearity module, such as softmax.
It is easy to see that
Hence, for the loss function $\mathcal{L} = Y \cdot \log \mathcal{N}$ where Y is the one-hot encoding of the correct label :
We say the log likelihood loss function matches the softmax output non-linearity since its Jocabian is an affine transformation of the output.
In the same way, the mean squared error loss matches a linear output module. | 2018-08-21 03:55:25 | {"extraction_info": {"found_math": true, "script_math_tex": 11, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9293926954269409, "perplexity": 464.795987049655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221217951.76/warc/CC-MAIN-20180821034002-20180821054002-00142.warc.gz"} |
http://www.bazpedia.com/en/c/y/c/Cyclotron.html | # Cyclotron
A pair of "Dee" electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science.
A cyclotron is a type of particle accelerator. Cyclotrons accelerate charged particles using a high-frequency, alternating voltage (potential difference). A perpendicular magnetic field causes the particles to go almost in a circle so that they re-encounter the accelerating voltage many times.
Ernest O. Lawrence, of the University of California, Berkeley, is credited with the invention of the cyclotron in 1929. It is less known outside Hungary that Hungarian Sándor Gaál may have described the workings of a cyclotron at about the same time during the spring of 1929 as Lawrence; although almost all reputable international sources give credit to Lawrence for the invention and construction of the first cyclotron. He used it in experiments that required particles with energy of up to 1 MeV.
## How the cyclotron works
Diagram of cyclotron operation from Lawrence's 1934 patent.
The electrodes shown at the right would be in the vacuum chamber, which is flat, in a narrow gap between the two poles of a large magnet.
In the cyclotron, a high-frequency alternating voltage applied across the "D" electrodes (also called "dees") alternately attracts and repels charged particles. The particles accelerate only when passing through the gap between the electrodes. The perpendicular magnetic field (passing vertically through the "D" electrodes) forces the particles to travel in a circular path.
The particles move in a circle, because a current of electrons or ions, flowing perpendicular to a magnetic field, experiences a perpendicular force. The charged particles move freely in a vacuum, so the particles follow a circular path.
## Uses of the Cyclotron
For several decades, cyclotrons were the best source of high-energy beams for nuclear physics experiments; several cyclotrons are still in use for this type of research.
Cyclotrons can be used to treat cancer. Ion beams from cyclotrons can be used, as in proton therapy, to penetrate the body and kill tumors by radiation damage, while minimizing damage to healthy tissue along their path.
Cyclotron beams can be used to bombard other atoms to produce short-lived positron-emitting isotopes suitable for PET imaging.
## Problems solved by the cyclotron
60-inch cyclotron, circa 1939, showing a beam of accelerated ions (likely protons or deuterons) escaping the accelerator and ionizing the surrounding air causing a blue glow. This phenomenon of air ionization is analogous to the one responsible for producing the "blue flash" infamously noted by witnesses of criticality accidents. Though the effect is often mistaken for Cherenkov radiation, this is not the case.
The cyclotron is an improvement over the linear accelerators available when it was invented. A linear accelerator accelerates particles in a straight line, through evacuated tubes. A series of cylindrical electrodes in the tubes switch from positive to negative voltage. In the 1920's, it was not possible to get high frequency radio waves at high power, so the stages of acceleration had to be far apart, to accommodate the low frequency, or more stages were required to compensate for the low power at each stage.
Faster particles required longer accelerators than scientists could afford. Later linear accelerators could use high power Klystrons and other devices imparting much more power at higher frequencies, but before these devices existed, the cyclotron was cheaper.
Cyclotrons accelerate particles in a circular path. Therefore, a compact accelerator can contain much more distance than a linear accelerator, with more opportunities to accelerate the particles.
• Cyclotrons have a single electrical driver, which saves both money and power, since more expense may be allocated to increasing efficiency.
• Cyclotrons produce a continuous stream of particle pulses at the target, so the average power is relatively high.
• The compactness of the device reduces other costs, such as its foundations, radiation shielding, and the enclosing building.
## Limitations of the cyclotron
The magnet portion of a large cyclotron. The gray object is the upper pole piece, routing the magnetic field in two loops through a similar part below. The white canisters held conductive coils to generate the magnetic field. The D electrodes are contained in a vacuum chamber that was inserted in the central field gap.
The spiral path of the cyclotron beam can only "synch up" with klystron-type (constant frequency) voltage sources if the accelerated particles are approximately obeying Newton's Laws of Motion. If the particles become fast enough that relativistic effects become important, the beam gets out of phase with the oscillating electric field, and cannot receive any additional acceleration. The cyclotron is therefore only capable of accelerating particles up to a few percent of the speed of light; higher velocity beams require a synchrocyclotron or a more complex synchrotron or linear accelerator.
## Mathematics of the cyclotron
The centripetal force is provided by the transverse magnetic field B, and the force on a particle travelling in a magnetic field (which causes it to curve) is equal to Bqv. So,
$\frac{mv^2}{r} = Bqv$
(Where m is the mass of the particle, q is its charge, v is its velocity and r is the radius of its path.)
Therefore,
$\frac{v}{r} = \frac{Bq}{m}$
v/r is equal to angular speed, ω, so
$\omega = \frac{Bq}{m}$
And, the frequency
$f = \frac{\omega}{2\pi}$
Therefore,
$f = \frac{Bq}{2m\pi}$
This shows that for a particle of constant mass, the frequency does not depend upon the radius of the particle's orbit. As the beam spirals out, its frequency does not decrease, and it must continue to accelerate, as it is travelling more distance in the same time. As particles approach the speed of light, they acquire additional mass, requiring modifications to the frequency, or the magnetic field during the acceleration. This is accomplished in the synchrocyclotron.
The relativistic cyclotron frequency is
$f=f_c\frac{m_0}{m_0+T/c^2}$,
where fc is the classical frequency, given above, of a charged particle with kinetic energy T and rest mass m0 circling in a magnetic field.
The rest mass of an electron is 511 keV, so the frequency correction is 1% for a magnetic vacuum tube with a 5.11 kV direct current accelerating voltage. The proton mass is nearly two thousand times the electron mass, so the 1% correction energy is about 9 MeV, which is sufficient to induce nuclear reactions.
An alternative to the synchrocyclotron is the isochronous cyclotron, which has a magnetic field that increases with radius, rather than with time. The de-focusing effect of this radial field gradient is compensated by ridges on the magnet faces which vary the field azimuthally as well. This allows particles to be accelerated continuously, on every period of the radio frequency, rather than in bursts as in most other accelerator types.
## Related technologies
The spiraling of electrons in a cylindrical vacuum chamber within a transverse magnetic field is also employed in the magnetron, a device for producing high frequency radio waves (microwaves).
The Synchrotron moves the particles through a path of constant radius, allowing it to be made as a pipe and so of much larger radius than is practical with the cyclotron and synchrocyclotron. The larger radius allows the use of numerous magnets, each of which imparts angular momentum and so allowing particles of higher velocity (mass) to be kept within the bounds of the evacuated pipe. | 2013-06-20 04:44:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6583559513092041, "perplexity": 964.8783713731505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368710299158/warc/CC-MAIN-20130516131819-00004-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.transtutors.com/questions/what-are-some-important-abiotic-factors--18248.htm | # What are some important abiotic factors ?
What are some important abiotic factors and how they affect the environment and the biotic factors? | 2018-06-21 06:55:13 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8613470792770386, "perplexity": 2485.7667673889464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864039.24/warc/CC-MAIN-20180621055646-20180621075646-00358.warc.gz"} |
https://emoy.net/Euler-Lagrange-Equation | # P r o j e c t s
N o t e s
## Euler-Lagrange Equation
This equation is useful when finding the critical point of the integral equation.
Suppose that y(x) passes through the points (x_1, y_1) and (x_2, y_2) and it is a continuously differentiable function in [x_1, x_2]. We want to find min int_{x_1}^{x_2}y(x)dx or max int_{x_1}^{x_2}y(x)dx.
Step 1. Define the extreme value F(y).
Introducing a path function f which consists of y(x) and y prime (x), F(y) can be defined by
\begin{align} F(y)= \int_{x_1}^{x_2} f(y(x), y^{\prime}(x), x) dx. \end{align}
This function F assumes that y is the extreme value when it follows the path function f.
Now, consider F(y+\delta y) near the extreme value F(y) with the same start and end points. Then it represents another path function f which consists of y(x)+\delta y(x) and y prime (x)+\delta y prime (x). By Taylor Theorem,
\begin{align} F(y+\delta y) &= \int_{x_1}^{x_2} f(y(x)+\delta y(x), y^{\prime}(x)+\delta y^{\prime}(x), x)dx \\ & \approx
\int_{x_1}^{x_2} f(y(x), y^{\prime}(x), x) + f_y \delta y + f_{y^{\prime}}\delta y^{\prime} + f_x 0\ dx \\ &=
F(y) + \int_{x_1}^{x_2} f_y \delta y + f_{y^{\prime}}\delta y^{\prime} dx
\end{align}
It yields the following delta F,
\begin{align} \delta F= F(y+\delta y) - F(y) \approx
\int_{x_1}^{x_2}f_y \delta y + f_{y^{\prime}}\delta y^{\prime} dx
\end{align}
# Step 2. Apply the integration by parts to delta F.
The second part of delta F can be modified using the integration by parts.
\begin{align} \int_{x_1}^{x_2} f_{y^{\prime}}\delta y^{\prime} dx = \left[ f_{y^{\prime}}\delta y \right]_{x_1}^{x_2} - \int_{x_1}^{x_2}\frac{d}{dx} \left( f_{y^{\prime}}\right) \delta y\ dx = - \int_{x_1}^{x_2}\frac{d}{dx} \left( f_{y^{\prime}}\right) \delta y\ dx
\end{align}
This is because that delta y(x_1)=delta y(x_2)=0 since f(y(x), y prime (x), x) and f(y(x)+delta y(x), y prime (x) + delta y prime (x), x) have the same start and end points. Therefore, delta F is rewritten as
\begin{align} \delta F \approx
\int_{x_1}^{x_2} \left( f_y - \frac{d}{dx} f_{y^{\prime}} \right) \delta y\ dx
\end{align}
# Step 3. Use the fact that delta F(y)=0.
Since F(y) is the extreme value, delta F(y)=0 for the small enough delta y. Therefore, it leads to
\begin{gather} \delta F \approx
\int_{x_1}^{x_2} \left( f_y - \frac{d}{dx} f_{y^{\prime}} \right)\delta y\ dx = 0, \\
\color{red}{f_y - \frac{d}{dx} f_{y^{\prime}} = 0}
\end{gather}
which is called Euler-Lagrange equation.
Example. Find the smallest distance between (x_1, y_1) and (x_2, y_2) points.
It is to find the smallest path among all the possible paths. From the step 1,
\begin{align} \min \int_{x_1}^{x_2} \sqrt{(dx)^2+(dy)^2} = \min \int_{x_1}^{x_2} \sqrt{1+y^{\prime 2}} dx = F(y). \end{align}
Now, step 2 and step 3 lead to the following Euler-Lagrange eqation.
\begin{align} & f_y - \frac{d}{dx} f_{y^{\prime}} = 0 - \frac{d}{dx}\left( \frac{y^{\prime}}{\sqrt{1+y^{\prime 2}}} \right) = 0 \\ & \Rightarrow \frac{y^{\prime}}{\sqrt{1+y^{\prime 2}}} = C \\ & \Rightarrow y^{\prime} = \pm \sqrt{\frac{C^2}{1-C^2}}
\end{align}
where C in mathbb{R}. It means that y is of form ax+b for a, b in mathbb{R}, and y prime=a=frac{y_2-y_1}{x_2-x_1}. Therefore,
\begin{align} F(y) &= \min \int_{x_1}^{x_2} \sqrt{1+y^{\prime 2}} dx = \int_{x_1}^{x_2} \sqrt{1+a^2} dx = \sqrt{1+a^2}(x_2-x_1) \\ \\ &=
\sqrt{1+ \frac{(y_2-y_1)^2}{(x_2-x_1)^2}}(x_2-x_1)=
\sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2}
\end{align}
This result is right because Euclidean distance is the smallest one between (x_1, y_1) and (x_2, y_2) points.
emoy.net | 2019-12-10 22:44:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998533725738525, "perplexity": 6545.94606967028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540529006.88/warc/CC-MAIN-20191210205200-20191210233200-00028.warc.gz"} |
http://itl.nist.gov/div898/handbook/prc/section4/prc433.htm | 7. Product and Process Comparisons
7.4. Comparisons based on data from more than two processes
7.4.3. Are the means equal?
## The ANOVA table and tests of hypotheses about means
Sums of Squares help us compute the variance estimates displayed in ANOVA Tables The sums of squares SST and SSE previously computed for the one-way ANOVA are used to form two mean squares, one for treatments and the second for error. These mean squares are denoted by $$MST$$ and $$MSE$$, respectively. These are typically displayed in a tabular form, known as an ANOVA Table. The ANOVA table also shows the statistics used to test hypotheses about the population means.
Ratio of $$MST$$ and $$MSE$$ When the null hypothesis of equal means is true, the two mean squares estimate the same quantity (error variance), and should be of approximately equal magnitude. In other words, their ratio should be close to 1. If the null hypothesis is false, $$MST$$ should be larger than $$MSE$$.
Divide sum of squares by degrees of freedom to obtain mean squares The mean squares are formed by dividing the sum of squares by the associated degrees of freedom.
Let $$N = \sum n_i$$. Then, the degrees of freedom for treatment are $$DFT = k - 1 \, ,$$ and the degrees of freedom for error are $$DFE = N - k \, .$$
The corresponding mean squares are:
$$MST = SST / DFT$$
$$MSE = SSE / DFE$$.
The F-test The test statistic, used in testing the equality of treatment means is: $$F = MST / MSE$$.
The critical value is the tabular value of the $$F$$ distribution, based on the chosen $$\alpha$$ level and the degrees of freedom $$DFT$$ and $$DFE$$.
The calculations are displayed in an ANOVA table, as follows:
ANOVA table
Source SS DF MS F Treatments $$SST$$ $$k-1$$ $$SST / (k-1)$$ $$MST/MSE$$ Error $$SSE$$ $$N-k$$ $$\,\,\, SSE / (N-k) \,\,\,$$ Total (corrected) $$SS$$ $$N-1$$
The word "source" stands for source of variation. Some authors prefer to use "between" and "within" instead of "treatments" and "error", respectively.
ANOVA Table Example
A numerical example The data below resulted from measuring the difference in resistance resulting from subjecting identical resistors to three different temperatures for a period of 24 hours. The sample size of each group was 5. In the language of design of experiments, we have an experiment in which each of three treatments was replicated 5 times.
Level 1 Level 2 Level 3 6.9 8.3 8.0 5.4 6.8 10.5 5.8 7.8 8.1 4.6 9.2 6.9 4.0 6.5 9.3 means 5.34 7.72 8.56
The resulting ANOVA table is
Example ANOVA table
Source SS DF MS F Treatments 27.897 2 13.949 9.59 Error 17.452 12 1.454 Total (corrected) 45.349 14 Correction Factor 779.041 1
Interpretation of the ANOVA table The test statistic is the $$F$$ value of 9.59. Using an $$\alpha$$ of 0.05, we have $$F_{0.05; \, 2, \, 12}$$ = 3.89 (see the F distribution table in Chapter 1). Since the test statistic is much larger than the critical value, we reject the null hypothesis of equal population means and conclude that there is a (statistically) significant difference among the population means. The $$p$$-value for 9.59 is 0.00325, so the test statistic is significant at that level.
Techniques for further analysis The populations here are resistor readings while operating under the three different temperatures. What we do not know at this point is whether the three means are all different or which of the three means is different from the other two, and by how much.
There are several techniques we might use to further analyze the differences. These are: | 2017-01-23 12:33:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8255650401115417, "perplexity": 472.4832144663588}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282926.64/warc/CC-MAIN-20170116095122-00333-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://mathhelpforum.com/statistics/200205-normal-distribution-symmetry.html | # Math Help - Normal Distribution and symmetry
1. ## Normal Distribution and symmetry
Can you help me explain how to find probability using the idea that the normal distribution is symmetric? Thank you
2. ## Re: Normal Distribution and symmetry
For example, $P(-1 \le z \le 1) = 2P(0 \le z \le 1)$, using symmetry.
3. ## Re: Normal Distribution and symmetry
Thank you! If I have to explain this using words, how would I do that?
4. ## Re: Normal Distribution and symmetry
Hint: what does $P(-1 \le z \le 1)$ mean (in words)?
5. ## Re: Normal Distribution and symmetry
What about $P(Z>1) = P(Z<-1)$
How's this sound?
6. ## Re: Normal Distribution and symmetry
Clear as a bell
7. ## Re: Normal Distribution and symmetry
A random variable is said to have a normal distribution if it has a probability distribution that is symmetric and bell-shaped.
First, the total area under the curve is 1. The second is area will be used to measure probabilities. A normal distribution is intimately connected to Z-scores. The main idea is to standardize all the data that is given by using Z-scores. These Z-scores can then be used to find the area (and thus the probability) under the normal curve. Before getting into computing probabilities, here is a quick reminder of Z-scores. | 2014-09-18 22:41:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8678069114685059, "perplexity": 962.3546601023936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657129407.88/warc/CC-MAIN-20140914011209-00241-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://puzzling.stackexchange.com/tags/connect-wall/info | Tag Info
A puzzle involving a set of things (most often words) which must be divided into groups such that the things in each group share some connection.
A puzzle involving a set of things (most often words) which must be divided into groups such that the things in each group share some connection.
Based on the BBC game show 'Only Connect'.
Usually there is a $$4\times4$$ grid, and the things must be split into $$4$$ groups. But beware because there are usually lots of red herrings! There are $$\dfrac{16!}{4!^4}=63,063,000$$ possibilities per grid.
Examples:
• A wall using words
• A wall using cryptic clues
• A wall using lines of music
• A wall using rebuses
Additionally, between Nov. 1 and Nov. 14 of 2020, there was a Fortnightly Topic Challenge for Wordless Connecting Walls. The link-collecting answer there has many links to creative interpretations of the connect-wall genre.
Those interested in creating their own, hard, connect-walls may want to read this question and its answer on how to do that. | 2021-04-18 10:46:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21791806817054749, "perplexity": 1530.8917567204646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038476606.60/warc/CC-MAIN-20210418103545-20210418133545-00622.warc.gz"} |
https://eprints.lancs.ac.uk/id/eprint/144217/ | # Simultaneous measurement of the muon neutrino charged-current cross section on oxygen and carbon without pions in the final state at T2K
, T2K Collaboration and Dealtry, T. and Finch, A. J. and Kormos, L. L. and Lawe, M. and Nowak, J. and O'Keeffe, H. M. and Ratoff, P. N. and Walsh, J. G. and Doyle, Tristan (2020) Simultaneous measurement of the muon neutrino charged-current cross section on oxygen and carbon without pions in the final state at T2K. Physical Review D, 101 (11). ISSN 1550-7998
Text (JointNumuOCCC0piXsecPRDaccepted-AAM)
JointNumuOCCC0piXsecPRDaccepted.pdf - Accepted Version
## Abstract
This paper reports the first simultaneous measurement of the double differential muon neutrino charged-current cross section on oxygen and carbon without pions in the final state as a function of the outgoing muon kinematics, made at the ND280 off-axis near detector of the T2K experiment. The ratio of the oxygen and carbon cross sections is also provided to help validate various models' ability to extrapolate between carbon and oxygen nuclear targets, as is required in T2K oscillation analyses. The data are taken using a neutrino beam with an energy spectrum peaked at 0.6~GeV and comprises 57.34$\times$10$^{19}$ protons on target. The extracted measurement is compared with the prediction from different Monte Carlo neutrino-nucleus interaction event generators, showing particular model separation for very forward-going muons. Overall, of the models tested, the result is best described using Local Fermi Gas descriptions of the nuclear ground state with RPA suppression.
Item Type:
Journal Article
Journal or Publication Title:
Physical Review D
Subjects:
Departments:
ID Code:
144217
Deposited By:
Deposited On:
22 May 2020 15:45
Refereed?:
Yes
Published?:
Published | 2020-10-29 05:32:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7777963280677795, "perplexity": 5680.256553490724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107902745.75/warc/CC-MAIN-20201029040021-20201029070021-00640.warc.gz"} |
http://codeforces.com/problemset/problem/1004/C | C. Sonya and Robots
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
Since Sonya is interested in robotics too, she decided to construct robots that will read and recognize numbers.
Sonya has drawn $n$ numbers in a row, $a_i$ is located in the $i$-th position. She also has put a robot at each end of the row (to the left of the first number and to the right of the last number). Sonya will give a number to each robot (they can be either same or different) and run them. When a robot is running, it is moving toward to another robot, reading numbers in the row. When a robot is reading a number that is equal to the number that was given to that robot, it will turn off and stay in the same position.
Sonya does not want robots to break, so she will give such numbers that robots will stop before they meet. That is, the girl wants them to stop at different positions so that the first robot is to the left of the second one.
For example, if the numbers $[1, 5, 4, 1, 3]$ are written, and Sonya gives the number $1$ to the first robot and the number $4$ to the second one, the first robot will stop in the $1$-st position while the second one in the $3$-rd position. In that case, robots will not meet each other. As a result, robots will not be broken. But if Sonya gives the number $4$ to the first robot and the number $5$ to the second one, they will meet since the first robot will stop in the $3$-rd position while the second one is in the $2$-nd position.
Sonya understands that it does not make sense to give a number that is not written in the row because a robot will not find this number and will meet the other robot.
Sonya is now interested in finding the number of different pairs that she can give to robots so that they will not meet. In other words, she wants to know the number of pairs ($p$, $q$), where she will give $p$ to the first robot and $q$ to the second one. Pairs ($p_i$, $q_i$) and ($p_j$, $q_j$) are different if $p_i\neq p_j$ or $q_i\neq q_j$.
Unfortunately, Sonya is busy fixing robots that broke after a failed launch. That is why she is asking you to find the number of pairs that she can give to robots so that they will not meet.
Input
The first line contains a single integer $n$ ($1\leq n\leq 10^5$) — the number of numbers in a row.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1\leq a_i\leq 10^5$) — the numbers in a row.
Output
Print one number — the number of possible pairs that Sonya can give to robots so that they will not meet.
Examples
Input
51 5 4 1 3
Output
9
Input
71 2 1 1 1 3 2
Output
7
Note
In the first example, Sonya can give pairs ($1$, $1$), ($1$, $3$), ($1$, $4$), ($1$, $5$), ($4$, $1$), ($4$, $3$), ($5$, $1$), ($5$, $3$), and ($5$, $4$).
In the second example, Sonya can give pairs ($1$, $1$), ($1$, $2$), ($1$, $3$), ($2$, $1$), ($2$, $2$), ($2$, $3$), and ($3$, $2$). | 2019-01-20 16:29:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9166672229766846, "perplexity": 324.45239259454036}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583722261.60/warc/CC-MAIN-20190120143527-20190120165527-00637.warc.gz"} |
https://tug.org/pipermail/tex-live/2022-March/047934.html | \input and \file_if_exist:nTF search diffrent paths in texlive-lualatex.
Ulrike Fischer news3 at nililand.de
Mon Mar 21 09:04:10 CET 2022
Am Sun, 20 Mar 2022 16:51:38 -0600 schrieb Karl Berry:
> Yours is evidently the second report about this in a dozen years or so.
> It would be nice to have it fixed, but one more year does not seem
> terribly critical.
Personally I'm not affected as I never use --output-directory, but
while looking at it I found another difference between miktex and
texlive which affects all engines: If I have child.tex in a
subfolder, and then call pdflatex --output-directory on this file:
\documentclass{article}
\begin{document}
\input{child}
\input{child.tex}
\end{document}
Then miktex compiles fine, and finds child.tex for both \input, but
in texlive the first one errors.
--
Ulrike Fischer
http://www.troubleshooting-tex.de/ | 2022-12-07 10:20:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8320106267929077, "perplexity": 12076.29167193367}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711151.22/warc/CC-MAIN-20221207085208-20221207115208-00269.warc.gz"} |
http://bootmath.com/constructing-continuous-functions-at-given-points.html | # Constructing Continuous functions at given points
Ok. This question may sound very easy, but actually i am in great need of it. I have been facing trouble in constructing functions, which are only continuous at some particular sets.
For e.g, the standard example of a function which is only continuous at one point, is the function, $f(x) = x, \ x \in \mathbb{Q}$ and $f(x) = -x, x \in \mathbb{R} \setminus \mathbb{Q}$. Similarly, i would like to know as to how to construct a function which is
• Continuous at exactly $2,3,4$ points.
• Continuous exactly at integers
• Continuous exactly at Natural numbers
• Continuous exactly at Rationals.
I would like to see many examples (with proof!), so that i can don’t struggle when somebody asks me to construct such functions.
#### Solutions Collecting From Web of "Constructing Continuous functions at given points"
1. One simple way of constructing a function which is continuous only at a finite number of points, $x=a_1,\ldots,a_n$, is to do a slight modification to the function you give: take a polynomial $p(x)$ that has roots exactly at $x=a_1,\ldots,a_n$ (e.g., $p(x) = (x-a_1)\cdots(x-a_n)$) , and then define
$$g(x) = \left\{\begin{array}{ll} p(x) & \text{if x\in\mathbb{Q};}\\ 0 & \text{if x\notin\mathbb{Q}.} \end{array}\right.$$
The function is continuous at $a_1,\ldots,a_n$, and since $p(x)\neq 0$ for any $x\notin\{a_1,\ldots,a_n\}$ then $g(x)$ is not continuous at any point other than $a_1,\ldots,a_n$. Other possibilities should suggest themselves easily enough.
2. A function that is continuous exactly at the integers: a similar idea will work: find a function that has zeros exactly at the integers, for example $f(x)=\sin(\pi x)$, and then take
$$g(x) = \left\{\begin{array}{ll} \sin(\pi x) & \text{if x\in\mathbb{Q};}\\ 0 & \text{if x\notin\mathbb{Q}.} \end{array}\right.$$
3. A function continuous exactly in the natural numbers: take a function that is continuous at the integers, and redefine it as the characteristic function of the rationals in appropriate places(what happens at $0$ depends on whether you believe $0$ is in the natural numbers or not). Assuming that $0\in\mathbb{N}$, one possibility is:
$$g(x) = \left\{\begin{array}{ll} \sin(\pi x)&\text{if x\in\mathbb{Q} and x\geq 0;}\\ x & \text{if x\in\mathbb{Q} and -\frac{1}{2}\lt x\leq 0;}\\ 1 & \text{if x\in\mathbb{Q} and x\leq -\frac{1}{2};}\\ 0 & \text{if x\notin\mathbb{Q}.} \end{array}\right.$$
4. A function continuous exactly on the rationals. This one is a bit trickier. There is no such function. This follows because the set of discontinuities of a real valued function must be a countable union of closed sets.
Perhaps then, we might anticipate the next question:
5. A function that is continuous exactly on the irrationals. An example is the following: let $s\colon\mathbb{N}\to\mathbb{Q}$ be an enumeration of the rationals (that is, a bijection from $\mathbb{N}$ to $\mathbb{Q}$. Define $f(x)$ as follows:
$$f(x) = \sum_{\stackrel{n\in\mathbb{N}}{s_n\leq x}} \frac{1}{2^n}.$$
The function has a jump at every rational, so it is not continuous at any rational. However, if $x$ is irrational, let $\epsilon\gt 0$. Then there exists $N$ such that $\sum_{k\geq N}\frac{1}{2^k}\lt \epsilon$. Find a neighborhood of $x$ which excludes every $q_m$ with $m\leq N$, and conclude that the difference between the value of $f$ at $x$ and at any point in the neighborhood is at most $\sum_{k\geq N}\frac{1}{2^k}$.
Edit: As I was reminded in the comments by jake, in fact the “standard example” of a function that is continuous at every rational and discontinuous at every rational is Thomae’s function. The example I give is a monotone function, and although it is discontinuous at every rational, it is continuous from the right at every number.
Continuous at 2, 3, 4: $f(x)=(x-2)(x-3)(x-4)$ if $x$ is rational, $f(x)=0$ if $x$ is irrational.
Continuous at the integers: $f(x)=\sin(\pi x)$ if $x$ is rational, 0 if $x$ is irrational.
Continuous at the natural numbers: $f(x)=\sin(\pi x)$ if $x$ is rational and not a nonpositive integer, 0 if $x$ is irrational, 1 if $x$ is a nonpositive integer.
Continuous exactly at the rationals: Impossible, because the set of rational numbers is not a $G_\delta$. | 2018-06-21 12:20:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9188308119773865, "perplexity": 133.07810447657158}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864148.93/warc/CC-MAIN-20180621114153-20180621134153-00235.warc.gz"} |
https://www.semanticscholar.org/paper/The-binary-evolution-of-SAX-J1808.4%E2%88%923658%3A-of-an-Goodwin-Physics/2a8abb4f1a0b1f01fd59f3b852b5d550aa1af22a | # The binary evolution of SAX J1808.4−3658: implications of an evolved donor star
@article{Goodwin2020TheBE,
title={The binary evolution of SAX J1808.4−3658: implications of an evolved donor star},
author={Adelle Goodwin and T. E. Woods School of Physics and Astronomy and Monash University and National Research Council Canada and Herzberg AstronomyAstrophysics Research Centre},
journal={Monthly Notices of the Royal Astronomical Society},
year={2020}
}
• Published 6 March 2020
• Physics
• Monthly Notices of the Royal Astronomical Society
Observations of the accretion powered millisecond pulsar SAX J1808.4−3658 have revealed an interesting binary evolution, with the orbit of the system expanding at an accelerated rate. We use the recent finding that the accreted fuel in SAX J1808.4−3658 is hydrogen depleted to greatly refine models of the progenitor and prior evolution of the binary system. We constrain the initial mass of the companion star to 1.0–1.2 M⊙, more massive than previous evolutionary studies of this system have…
1 Citations
## Figures and Tables from this paper
• Physics
The Astrophysical Journal
• 2022
We present the discovery of a new optical/X-ray source likely associated with the Fermi γ-ray source 4FGL J1408.6–2917. Its high-amplitude periodic optical variability, large spectroscopic
## References
SHOWING 1-10 OF 46 REFERENCES
• Physics
Monthly Notices of the Royal Astronomical Society
• 2018
The evolutionary status of the low mass X-ray binary SAX J1808.4-3658 is simulated by following the binary evolution of its possible progenitor system through mass transfer, starting at a period of
• Physics
• 2007
We report here on the orbital evolution of the accreting millisecond pulsar SAX J1808.4{3658. In particular, we nd for this source the rst estimate of the orbital period derivative in an accreting
• Physics
• 2005
We present a temporal analysis of the three outbursts of the X-ray millisecond pulsar SAX J1808.4-3658 that occurred in 1998, 2000, and 2002. With a technique that uses the χ2 obtained with an
Recent timing analysis reveals that the orbital period of the first discovered accreting millisecond pulsar SAX J1808.4-3658 is increasing at a rate \$\dot{P}_{\rm orb}=(3.89\pm0.15)\times
• Physics, Geology
• 2009
The Rossi X-ray Timing Explorer has observed five outbursts from the transient 2.5 ms accretion-powered pulsar SAX J1808.4−3658 during 1998–2008. We present a pulse timing study of the most recent
• Physics
• 2001
We present multiband optical/IR photometry of V4580 Sgr, the optical counterpart of the accretion-powered millisecond pulsar SAX J1808.4-3658, taken during the 1998 X-ray outburst of the system. The
• Physics
• 2001
We present the results of a systematic study of the evolution of low- and intermediate-mass X-ray binaries (LMXBs and IMXBs). Using a standard Henyey-type stellar evolution code and a standard model
• Physics
• 2001
The BeppoSAX Wide Field Cameras have revealed a population of faint neutron star X-ray transients in the Galactic bulge. King conjectured that these neutron stars are accreting from brown dwarfs with
• Physics
The Astrophysical Journal
• 2019
The Neutron Star Interior Composition Explorer (NICER) has extensively monitored the 2019 August outburst of the 401 Hz millisecond X-ray pulsar SAX J1808.4–3658. In this Letter, we report on the
• Physics
Nature
• 2013
Observations of accretion-powered, millisecond X-ray pulsations from a neutron star previously seen as a rotation- powered radio pulsar show the evolutionary link between accretion and rotation-powered millisecond pulsars, but also that some systems can swing between the two states on very short timescales. | 2023-02-01 06:25:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44683438539505005, "perplexity": 9004.912119270055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499911.86/warc/CC-MAIN-20230201045500-20230201075500-00788.warc.gz"} |
http://mathoverflow.net/questions/39165/sprague-grundy-sequence-for-the-ruler-game?sort=oldest | Sprague-Grundy sequence for the ruler game
Consider the game "Ruler", which is defined as follows. We start with finitely many coins in a line. A move in this game consists of turning over any number of coins, but they must be consecutive, and the rightmost coin must be turned from heads to tails. Then the position in this game where a coin in the $n$th position is heads and all others are tails has Sprague-Grundy value given by
$$g(n) = mex \{ 0, g(n-1) , g(n-1) \oplus g(n-2), \cdots, g(n-1) \oplus \cdots \oplus g(1) \}$$
From here, according to page I-31 of Ferguson's game theory notes, "it is easy to show" that $g(n)$ is the largest power of two dividing $n$.
Except it's not easy. But it's a nice fact and I'd like to be able to present a proof of it to my students.
Fair Game by Guy and Winning Ways, the two other sources I've seen this in, both state this without proof. It seems that it might be closely related to results about Gray codes - the partial sums $g(1), g(1) \oplus g(2), g(1) \oplus g(2) \oplus g(3), \cdots$ are a binary Gray code for the integers.
-
As in all combinatorial game theory problems, we want to use strong induction on $n$. So suppose we know it for $1, ..., n-1$, and let's prove it for $n$.
Write $n$ in the form $2^k(2x+1)$. Then the highest power of $2$ dividing $n-i$ for $i<2^k$ is the same as the highest power of $2$ dividing $i$, so it's pretty easy to see that by the time we hit $g(n-1)\oplus \cdots \oplus g(n-2^k+1)$ we have hit every nimber from $0$ to $2^k-1$ (by Gray codes, if you like).
Now we just have to show that the nimber $2^k$ never shows up in that mex. The key idea, I think, is that the order of the highest nonzero bit in the binary representation of $g(n-1)\oplus \cdots \oplus g(n-i)$ never decreases as $i$ increases. This is because of the fact that for any subsequence $g(a), g(a+1), ..., g(b)$ of the largest-power-of-two sequence, the largest value only occurs once (if it occurred twice, there would be a larger power of two halfway in between). Now we just notice that for $i=2^k$, we have $n-i = 2^k(2x+1)-2^k = 2^{k+1}x$, so $g(n-1)\oplus \cdots \oplus g(n-i) \ge 2^{k+1}$ for all $i$ greater than or equal to $2^k$.
Thus $2^k$ does not show up in the mex, while every smaller nimber does. The induction is done :) | 2016-07-24 08:53:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7796627283096313, "perplexity": 92.94235896468061}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823989.0/warc/CC-MAIN-20160723071023-00132-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/607151/feeling-heavy-in-an-upward-accelerating-elevator-is-it-a-pseudo-force | # Feeling heavy in an upward accelerating elevator - is it a pseudo force?
In an elevator accelerating upwards, a person weighs "heavier" on a scale. The normal force can be expressed as $$N=m(a+g)$$, a being the upward acceleration. In this case, is $$ma$$ a fictional force in the non-inertial frame of the elevator or the person standing in the elevator? It seems to me that from an inertial reference frame, the Normal force calculated would be the same, however, this doesn't seem to fit with the rule that "fictitious" forces cannot be included in calculations in inertial frames. In this case, is this purely a coincidence?
• Some answers here cover the basics. But it is probably unwise to consider $m(g+a)$ to be a pseudo force. It is a real force, the normal force. The calculation that gives $N=m(g+a)$ does nothing more than give us the magnitude of that real force. Jan 14, 2021 at 16:39
## 7 Answers
Short version: Yes. Any "force" that acts on masses because of the acceleration of the coordinate frame in which it is measured is a pseudo force.
I don't understand why we also include this "pseudo force" in the inertial frame of an observer on the ground
"Pseudo" doesn't mean "not real." It's more like, "not explained." The contact force between the passenger's feet and the floor of the elevator is real no matter how you look at it. "Pseudo" merely means that in the accelerated frame, we don't attempt to explain the origin of the force. It's just a physical law within the accelerated frame that a body experiences a force in a certain direction with a magnitude proportional to the body's mass.
You said, $$N=m(a+g)$$. That would be how we describe it in the inertial frame. When we talk about the inertial frame, we have to understand what $$a$$ and $$g$$ mean. Especially, $$a$$, which we know in this case to be the acceleration of the "elevator."
In the accelerated frame, we don't need the complication of $$a+g$$. We can just collapse that into a magic constant, $$k$$. Things inside the elevator experince a mysterious (i.e., "pseudo") force, $$\vec{F}=m\vec{k}$$ where $$\vec{k}$$ is just a given fact of how things behave in that system. We don't acknowledge that the elevator is accelerated, we don't ask why $$\vec{F}=m\vec{k}$$, we just say that's the law that describes how things are in the elevator.
• Could you please formulate your answer in terms of what the question is asking? I understand the claim that ma is a pseudo force in the non-inertial frame, I don't understand why we also include this "pseudo force" in the inertial frame of an observer on the ground. Jan 12, 2021 at 15:11
• @ten1o, see my ammended answer. Jan 12, 2021 at 17:02
Suppose you are the observer.
If you observe the person going up from the frame of that same elevator then according to you , the person is at rest. The forces on that person are : Normal force (upwards), fictious and gravitational force (downwards).
Since the person seems to be at rest for you then Newton's second law says that the net force on that person should be zero i.e
Upward force = Downward force
$$N=ma + mg$$ $$N=m(a+g)$$
Seeing from the ground frame, the person is accelerating up with the elevator and the forces on him are Normal force(up) and gravity(down). In order for the person to go up with the elevator, the net force on him should be up again from Newton's second law. Hence,
$$Upward\; force - Downward\; force = ma$$
$$N-mg= ma$$
$$N=m(a+g)$$
In an inertial frame (observing elevator accelerating up from the ground) $$N - mg = ma$$ where $$N$$ is the force of the elevator floor on the person, $$m$$ is the mass if the person, $$g$$ is the acceleration of gravity, and $$a$$ is the acceleration of the person upwards in the inertial frame. There is no fictitious force in the inertial frame.
In a non-inertial (accelerating) frame fixed to the floor of the elevator there is the fictitious force $$-ma$$ where $$a$$ is the acceleration of this frame relative to the inertial frame. In this frame the person is at rest and $$N - mg - ma = ma* = 0$$. $$a*$$ the acceleration of the person in the non-inertial frame is zero here. Note the addition of the ficticious force required in the non-inertial frame and in that frame for this problem the person is at rest.
If there is a weight scale on the floor of the elevator, the force on the scale is equal in magnitude to the normal force on the person (Newton's third law) and in either frame that force is $$mg + ma$$ in magnitude; the weight is greater than that of a person on the ground where the force on a scale is $$mg$$. (This ignores the effects of the earth's rotation. You actually weigh a little less at the equator than at a pole due the centrifugal force.)
See a physics mechanics text, such as Symon Mechanics for a complete discussion of moving coordinate systems.
No matter which reference frame you use, the normal force is always $$N=m(a+g)$$.
In the inertial reference frame we can see that the only forces acting on the person are $$N$$ upwards and $$mg$$ downwards. The net upwards force on the person is therefore $$N-mg=ma > 0$$ but we expect a net upwards force since the person is accelerating upwards, so this is fine.
In the non-inertial reference frame we believe the person is in equilibrium, so we expect the net force on them to be $$0$$. To achieve this we have to introduce a pseudo-force $$F_p$$ downwards, so that $$N - mg - F_p = 0$$. From this equation we can see that $$F_p=ma$$.
Note that the person in the lift does not feel the pseudo-force. Neither do they feel the force of gravity. If they jump off a tall building or go to the ISS then they feel "weightless" because there is no normal force - even though gravity still acts on them, they do not feel it. The only force that they feel in the lift is the normal force $$N$$ from the floor of the lift. Since they are used to $$N$$ being equal to their weight, they interpret a higher value of $$N$$ as being due to a temporary increase in weight, whereas in fact their weight is unchanged.
Let's take a look at the elevator and the person standing on it from an inertial frame of reference, the ground frame.
Now, If you draw the free body diagram for the person in the elevator (which is say accelerating upwards with acceleration $$\vec{a}$$) from this frame you will see that the person experiences a Normal reaction $$(\vec{N})$$ upwards from the floor of the elevator and the gravitational pull $$(m\vec{g})$$ in the downward direction.
So simply applying Newton's Second law gives us : $$\vec{N}-m\vec{g}=m\vec{a} \implies \vec{N} = m(\vec{a}+\vec{g})$$
Now on to your questions :
In this case, is ma a fictional force in the non-inertial frame of the elevator or the person standing in the elevator?
ma is a fictional force from any of these two frames since, both will be accelerating with the same acceleration.
however, this doesn't make sense as these forces are named "fictitious" forces and cannot be included into calculations in inertial frames.
Here, in your above mentioned scenario, the so called "fictitious" force is nothing but the feeling that one experiences when in a elevator which is accelerating upwards. You will indeed feel like the elevator is pushing on your feet, when it is accelerating upwards. And Newton's laws will indeed take care of the so called force in an inertial frame, as that fictitious force just simply equates to the net force ma, experienced by the person in the elevator.
to obtain the equation I use Newton second law , starting with the position vector to the center of mass , remember the position vector must be given in inertial system
$$\vec R_{cm}=\vec R+\vec r$$
from here with the forces of the free body diagram you obtain
$$m\,(\vec{\ddot{R}}+\vec{\ddot{r}})=-m\,g+N$$
with $$~\vec{\ddot{r}}=0~$$ and $$~\vec{\ddot{R}}=a$$
$$N=m(a+g)$$
thus the result come from the correct applying Newton second law nothing else
Yes the increase in weight is due to pseudo force
When accelerations upwards, due to the acceleration of the elevator, a fictitious force called pseudo force acts in the opposite direction of the direction of the acceleration vector i.e $$\vec{a}$$, in the downward direction.
Hence the person feels a force in the downward direction, increasing the apparent weight and thereby create an illusion of increase in weight | 2022-05-28 10:32:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 48, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7411240935325623, "perplexity": 218.01905748095515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663016373.86/warc/CC-MAIN-20220528093113-20220528123113-00261.warc.gz"} |
https://stats.stackexchange.com/questions/526966/using-batchnorm-and-dropout-simultaneously | Using batchnorm and dropout simultaneously?
I am a bit confused about the relation between terms "Dropout" and "BatchNorm". As I understand,
1. Dropout is regularization technique, which is using only during training.
2. BatchNorm is technique, which is using for accelerating training speed, improving accuracy and e.t.c. But I also saw some conflicting opinions about question: is BatchNorm regularization technique?
1. Is BatchNorm regularization technique? Why?
2. Should we use BatchNorm only during training process? Why?
3. Can we use Dropout and BatchNorm simultaneously? If we can, in what order?
Is BatchNorm regularization technique? Why?
BatchNorm works by standardizing the outputs of hidden units across an entire batch. The standardization process consists of multiplication and addition. Compare this to another regularization technique such as injecting noise into the outputs (or inputs) of hidden units; the noise can be injected additively or multiplicatively. So you can, in a way, think of BatchNorm as a injecting the 'correct noise' needed to standardise hidden unit outputs across a batch, and although it won't be as strong of a regularizing effect as actual uniform/gaussian random noise, it still has a minor regularizing effect on top of the benefit of speeding up learning.
Should we use BatchNorm only during training process? Why?
BatchNorm is used during training to standardise hidden layer outputs, but during evaluation the parameters that the BatchNorm layer has learnt (the mean and standard deviation) are frozen and are used as is, just like all other weights in a network. The effects of BatchNorm can also be 'folded in' to network weights which achieves the same effect but with one less step.
Can we use Dropout and BatchNorm simultaneously? If we can, in what order?
Definitely! Although there is a lot of debate as to which order the layers should go. Older literature claims Dropout -> BatchNorm is better while newer literature claims that it doesn't matter or that BatchNorm -> Dropout is superior. My recommendation is try both; every network is different and what works for some might not work for others. Personally I've found BatchNorm -> Dropout to work well for my use cases. | 2021-12-03 01:54:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6706551909446716, "perplexity": 2112.2634212505563}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362571.17/warc/CC-MAIN-20211203000401-20211203030401-00521.warc.gz"} |
http://xmcourseworkvwfx.presidentialpolls.us/ap-macroeconomics-stock-blurbs.html | # Ap macroeconomics stock blurbs
Over 200 ap macroeconomics practice questions to help you with your ap macroeconomics exam prep the ap macroeconomics exam includes 60 multiple-choice questions and 3 free-response questions see the table below. No bull review - for use with the ap macroeconomics and ap microeconomics exams your review book shouldn't need a review book, and that's why we're here the 2016 edition contains definitions and explanations of the most important concepts, formulas, and graphical models that you need to to know. Macroeconomics by nature is a pure science but it exists today as a bundle of contesting opinions in the preface of the general theory, keynes expressed the model-based studies, thus, bypassed the study of macroeconomic causalities and left economic planners and policy makers only with the trial. The ap macroeconomics exam measures students' knowledge of macroeconomics principles and their ability to reason within the discipline encourage your students to visit the ap macroeconomics student page for exam information and exam practice.
Details title ap macroeconomics description final study terms financial intermediary that creates a stock portfolio by buying and holding shares in companies and then selling shares of the stock portfolio to individual investors. Aggregate demand: dick rankin, ap® macroeconomics instructor, ʻiolani school, honolulu, hi aggregate supply: gabriel sanchez, ap® macroeconomics instructor, bonita high school, la verne, ca short-run macroeconomic equilibrium: dr robert graham, economics professor, hanover. Macroeconomics is about whole economies what is gdp why does the economy boom and bust how is the government involved we hit the traditional topics from a college-level macroeconomics course. Test your ap macroeconomics knowledge with the below ap macroeconomics practice tests high school notes and quizzes ap chemistry quizzes ap human geography quizzes ap statistics quizzes ap macroeconomics quizzes ap world history quizzes ap statistics quizzes calculus.
I took both ap microeconomics and ap macroeconomics so if you don't understand ap economics, well i daresay i'll be seeing you on this website quite a bit so you decided to put more pressure on yourself and take ap economics instead of economics/consumer education. Details title ap macroeconomics description changes in the stocks of finished goods and goods in process, as well as changes in the raw materials that business keep on hand. Ap macroeconomics practice exams and in-depth reviews inside the ap macroeconomics guide, you'll find a diagnostic exam to figure out where you're struggling before you even begin review of macroeconomic issues, the financial sector, stocks and bonds, and more. Ap micro/macro economics - want to know what it feels like to have the world at your fingertips using this app, you just might get a glimmer with content derived directly from the successful mcgraw-hill ap 5 steps to a 5 series, the questions and detailed explanations closel. Ap macroeconomics review #10b-1: major graphs, vocabulary, and formulas - vocabulary capital account: (aka financial account) measures the flow of funds for investment in real assets (factories, office buildings) or financial assets (stocks, bonds) between a nation and the rest of the.
Ap economics calendar ap macroeconomics unit 1 - intro to economics scarcity and opportunity cost. Crying because he waited and crammed for the ap and now recognises that it is just to much information 2015 ap macroeconomics frq #1 (a) economy is operating below full employment,,, draw a clg of lras, sras, ad. Bottom-line raising tariffs benefits individual companies, not the industry as a whole companies protected by the higher tariffs will show higher profits in the short-term at the expense of a negative impact on the sector over the medium-term. Financial stocks took the lead, surging 42 percent based on the anticipation that interest rate will go higher (which benefit banks to earn more profits) however, after the chinese stock market crashed last year, investors lost their confidence in the stock investment (and a lot of them got burned in the.
## Ap macroeconomics stock blurbs
Ap macroeconomics graphs describe a relationship between two variables that are used to measure the economy and diagnose the economic status through these ap macroeconomics graphs, we can tell how price levels affect gdp and diagnose inflationary or recessionary gaps. Advanced placement macroeconomics (also known as ap macroeconomics, ap macro, apma, or simply macro) is an advanced placement macroeconomics course and exam offered by the college board. Ap macroeconomics stock blurbs topics: stock market, stock exchange, corporation pages: 2 (750 words) published: january 10, 2014 stock purchases and why 1 colgate is a company that was going up when i purchased them they have special offers, community programs, and.
Let $f(k,l)$ be a production function with variables $k$ for capital and $l$ for labor the slope of the $f(\overline k,l)$ ($k$ taken constant) is defined as the marginal product of labor. Ap economics study guide modules 1,2,4 unit 1: basic economic concepts macroeconomics: big picture of the economy, overall ups and downs deals with aggregates, sums of data from many different markets basic economic problem: society's wants are virtually unlimited and insatiable. Gdp = gross domestic product nominal gdp - use current year prices base gdp - use base year prices y = c + i + g + nx y = gdp c = consumption i = investment g = government spending nx = net exports (exports - imports) gdp deflator = ((nominal gdp) / (real gdp)) 100.
Ap macro test review learn with flashcards, games and more — for free macroeconomic short run a period of time during which the prices of goods and services are changing in their respective markets, but the input prices have not yet adjusted to those changes in the product markets. Ap macroeconomics formula sheet advertisement ap macro formula sheet gdp gdp = total value of all final goods and services produced by an government transfer payments (social security, welfare, veterans benefits, etc) sale of used goods, financial payments (bonds, stocks) gdp is. An introduction into how cpi is calculated using 2 goods, rice and toilet paper.
Ap macroeconomics stock blurbs
Rated 5/5 based on 14 review
2018. | 2018-11-15 09:22:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24542094767093658, "perplexity": 7496.336945764926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742569.45/warc/CC-MAIN-20181115075207-20181115101207-00291.warc.gz"} |
http://www.thefullwiki.org/Advanced_ANOVA/MANOVA | # Advanced ANOVA/MANOVA: Wikis
Advertisements
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Study guide
Up to date as of January 14, 2010
### From Wikiversity
Resource type: this resource contains a tutorial or tutorial notes.
Completion status: this resource is considered to be complete.
• The purpose of this tutorial is to teach use of multivariate analysis of variance (MANOVA), with practical exercises based on using SPSS.
• Note that the MANOVA procedure is not available with the Student version of SPSS.
## What is MANOVA?
• Developed as a theoretical construct by Samual S. Wilks in 1932 (Biometrika).
• An extension of univariate ANOVA procedures to situations in which there are two or more related dependent variables (ANOVA analyses only a single DV at a time). DVs should be correlated (but not overly so; otherwise they should be combined) or conceptually related.
• The MANOVA procedure identifies (inferentially) whether:
• Different levels of the IVs have a significant effect on a linear combination of each of the DVs
• There are interactions between the IVs and a linear combination of the DVs.
• There are significant univariate effects for each of the DVs separately.
## Example
Effects of chemotherapy and memory enhancement training on cognitive functioning in Alzheimer's patients
IVs (factors)
1. Chemotherapy (drug vs no-drug)
2. Memory training (training vs no-training)
DVs
Several measures of cognitive functioning:
1. Test of reading comprehension and retention
2. Memory for names and faces
3. Ratings provided by family members
## Usage
• MANOVA is appropriate when we have several DVs which all measure different aspects of some cohesive theme, e.g., several different types of academic achievement (e.g., Maths, English, Science).
• MANOVA works well in situations where there are moderate correlations between DVs. For very high or very low correlation in DVs, it is not suitable: if DVs are too correlated, there isn’t enough variance left over after the first DV is fit, and if DVs are uncorrelated, the multivariate test will lack power (so why sacrifice degrees of freedom?) (French et al., 2002)
• Alternatively, consider use a series of univariate ANOVAs (one for each DV) or possibly Mixed ANOVA.
• "Because of the increase in complexity and ambiguity of results with MANOVA, one of the best overall recommendations is: Avoid it if you can." (Tabachnick & Fidell, 1983, p.230). In other words - be sure it is really the best approach to use.
• Covariates can also be included → MANCOVA
## How does it work?
Simple explanation
• The MANOVA procedure creates a new DV which is a linear combination of the multiple DVs. This particular combination of DVs is chosen to maximise the difference between the IV groups. (Francis, 2007)
• The MANOVA procedure then assesses whether this new DV differs significantly between the IV groups. (Francis, 2007)
More complex explanation
MANOVA combines concepts from factorial ANOVA and discriminant analysis:
• It examines the effect of several independent variables (main effects and interaction effects), as does univariate ANOVA
• These IV effects are examined on several DVs that are combined to form one or more linear composites, as in discriminant analysis.
• Factor A main effect - evaluated by combining the original DVs to form one or more orthogonal discriminant functions (roots) which provide the greatest possible separation of the groups representing the levels of Factor A.
• Factor B main effect - evaluated by combining the original DVs to form one or more orthogonal discriminant functions (roots) which provide the greatest possible separation of the groups representing the levels of Factor B.
• A X B Interaction - assessed by forming one or more discriminant functions that maximise the separation of cells of the factorial data matrix.
• For each effect (A, B, and A x B) the discriminant functions will differ (so the composite DV being examined can change)
## Assumptions
1. Sample size
• Rule of thumb: the n in each cell > the number of DVs
• Larger samples make the procedure more robust to violation of assumptions
2. Normality:
• MANOVA sig. tests assume multivariate normality, however when cell size > ~20 to 30 the procedure is robust violating this assumption
• Note that univariate normality is not a guarantee of multivariate normality, but it does help.
• Check univariate normality via histograms, normal probability plots, skewness, kurtosis, etc. and check multivariate normality using Mahalanobis' distance. These procedures will also help to check for possible outliers.
3. Outliers:
• MANOVA is sensitive to the effect of outliers (they impact on the Type I error rate); first check for univariate outliers, then use Mahalanobis' distance to check for multivariate outliers (MVOs).
• MVOs are cases with an unusual combination of scores for the DVs of interest. #* Use the SPSS Regression menus to calculate MD, which will provide a score for each case which can be assessed according to a χ2 distribution (Analyze - Regression - Linear - Dependent (add a unique identifier e.g., ID) - Independent (add all the MANOVA DVs) - Save - MD - Paste/OK).
• Cases which can be considered MVOs are those with MD values above the critical χ2 value (where the number of IVs equals is the χ2 df.
• MANOVA can tolerate a few outliers, particularly if their scores are not too extreme and there is a reasonable N. If there are too many outliers, or very extreme scores, consider deleting these cases or transforming the variables involved (see Tabachnick & Fidell)
4. Linearity
• Linear relationships among all pairs of DVs
• Assess via scatterplots and bivariate correlations (check for each level of the IV(s) i.e., cells - use Split File)
5. Homogeneity of regression
• This assumption is only important if using stepdown analysis, i.e., there is reason for ordering the DVs.
• Covariates must have a homogeneity of regression effect (must have equal effects on the DV across the groups)
6. Multicollinearity and singularity
• MANOVA works best when the DVs are only moderately correlated.
• When correlations are low, consider running separate ANOVAs
• When there is strong multicollinearity, there are redundant DVs (singularity) which decreases statistical efficiency.
• Correlations above .7, and particularly above .8 or .9 are reason for concern.
• Consider removing one of the strongly correlated pairs or combining them to form a single measure.
7. Homogeneity of variance-covariance matrix (Box's M)
• The F test from Box’s M statistics should be interpreted cautiously because it is a highly sensitive test of the violation of the multivariate normality assumption, particularly with large sample sizes.
• MANOVA is fairly robust to this assumption where there are equal sample sizes for each cell.
8. Homogeneity of error variances (Levene's test)
• If this assumption is violated, use a more conservative critical / alpha level for determining significance for that variable in the univariate F-test. Tabachnick and Fidell suggest .025 or .01 rather than the conventional .05 level.
## Multivariate test statistics
Choose from among these multivariate test statistics to assess whether there are statistically significant differences across the levels of the IV(s) for a linear combination of DVs. In general Wilks' / lambda is recommended unless there are problems with small N, unequal ns, violations of assumptions, etc. in which case Pillai's trace is more robust (Tabachnick & Fidell):
Roy's greatest characteristic root
1. Tests for differences on only the first discriminant function
2. Most appropriate when DVs are strongly interrelated on a single dimension
3. Highly sensitive to violation of assumptions - most powerful when all assumptions are met
Wilks' lambda (λ)
1. Most commonly used statistic for overall significance
2. Considers differences over all the characteristic roots
3. The smaller the value of Wilks' lambda, the larger the between-groups dispersion
Hotelling's trace
1. Considers differences over all the characteristic roots
Pillai's criterion
1. Considers differences over all the characteristic roots
2. More robust than Wilks'; should be used when sample size decreases, unequal cell sizes or homogeneity of covariances is violated
## Tests of between-subject effects
• What should be done once it is found that an overall F for MANOVA is significant?
• If there is a significant multivariate effect, examine the Tests of Between-Subjects Effects for each of the DVs.
• Since there are multiple tests, control for the Type I error-rate (e.g., use a Bonferroni adjusment - divide the original alpha level by the number of tests).
• However, note that the DVs are usually correlated, therefore this approach would result in confounded results.
• Stepdown F ratios provide a similar approach, without the counfounded results. In this approach, all DVs are prioritised (by the researcher) from most to least important. The most important variable is considered first without correcting for the lower priority variables. All subsequent variables are tested after removing the effects of the higher priority variables (by specifying the higher priority variables as covariates). Thus, stepdown analysis:
• Is used to assess IV effects on individual DVs
• Involves computing a univariate F statistic for a DV after eliminating the effects of other DVs preceding it in the analysis.
• Previous DVs are treated as covariates
• Somewhat similar to hierarchical multiple linear regression
• Researcher determines the order in which the DVs are entered, based on some theoretical conceptualisation
• Is most appropriate when the DVs are correlated.
• See also: Analyses Following a Significant MANOVA (uwsp.edu)
## Effect sizes
Also use effect sizes to evaluate strength of the effects (particularly for significant effects):
• Multivariate ANOVA:
• Wilks' λ - multivariate η$_p^2$: Wilks' λ reflects the ratio of within-group variance across all discriminant functions to total variance across all discriminant functions.
• Univariate ANOVA:
• η2 gives the proportion of variance in the DV that is attributable to different levels of an IV.
## Pros and cons
Advantages
• Tests the effects of several IVs and several outcome (DVs) within a single analysis.
• Uses the power of convergence (no single operationally defined DV is likely to capture perfectly the conceptual variable of interest)
• IVs of interest are likely to affect a number of different conceptual variables - e.g., an organisation's non-smoking policy may affect employee satisfaction, production, absenteeism, health insurance claims, etc.
• Can provide a more powerful test of significance than available when via univariate tests.
• Reduced Type I error rate compared with performing a series of univariate tests.
• Interpretive advantages over a series of separate univariate ANOVAs.
Disadvantages
• Discriminant functions are not always easy to interpret - they are designed to separate groups, not to make conceptual sense. In MANOVA, each effect evaluated for significance uses different discriminant functions (Factor A may be found to influence a combination of DVs totally different from the combination most affected by Factor B or the interaction between Factors A and B).
• Like discriminant analysis, the assumptions on which it is based are numerous and difficult to assess and meet.
Alternatives
• Combine or eliminate DVs so that only one DV need be analysed.
• Use factor analysis to find orthogonal factors that make up the DVs, then use univariate ANOVAs on each factor (because the factors are orthogonal each univariate analysis should be unrelated)
## Example writeup
A one-way multivariate analysis of variance (MANOVA) was conducted to determine the effect of the three types of study strategies (thinking, writing and talking) on two dependent variables (recall and application test scores). A nonsignificant Box’s M, indicating that the homogeneity of variance-covariance matrix assumption was not violated. No univariate or multivariate outliers were evident and MANOVA was considered to be an appropriate analysis technique.
Significant differences were found among the three study strategies on the dependent measures, Wilks’ λ = .42, F (4,52) = 7.03, p < .001. The multivariate Wilks' λ was quite strong at .35. Table 1 presents the means and standard deviations of the dependent variables for the three strategies.
Univariate analyses of variance (ANOVAs) for each dependent variable were conducted as follow-up tests to the MANOVA. Using the Bonferroni method for controlling Type I error rates for multiple comparisons, each ANOVA was tested at the .025 level. The ANOVA of the recall scores was significant, F (2,27) = 17.11, p <.001, η2 = .56, while the ANOVA based on the application scores was nonsignificant, F(2,27)=4.20, p = .026, η2 =.24.
Post hoc analysis for the recall scores consisted of conducting pairwise comparisons to determine which study strategy affected performance most strongly. Each pairwise comparison was tested at the .025/3, or .008, significance level. The writing group produced significantly superior performance on the recall questions in comparison with either of the other two groups. The thinking and talking groups did not differ significantly from each other.
Table 1 Means and Standard Deviations for each Dependent Variable by Strategy
Recall Application Strategy M SD M SD Thinking 3.30 0.68 3.20 1.23 Writing 5.80 1.03 5.00 1.76 Talking 4.20 1.14 4.40 1.17
Note: This table should also include skewness, kurtosis, and descriptives for marginals.
## Exercises
Data
• Data: SCHL8.sav (Francis 5.3; p. 132 (5th ed.))
1st MANOVA
• DVs (Academic achievement):
• Maths (mathsach)
• English (engach)
• IVs:
• Socio-economic status (SES; Low, Moderate, High)
2nd MANOVA
• DVs (Classroom behaviour):
• Attentiveness in Year 8 (attent)
• Settledness in Year 8 (settle)
• Sociability in Year 8 (sociab)
• IVs:
• Gender (Sex; Male, Female)
SPSS Steps
• Analyze - General Linear Model - Multivariate (add IV(s) (fixed factors) and DVs)
• Graphs - could use any of:
• Clustered Bar Chart (Summaries of separate variables) or
• Clustered Error-bar (Summaries of separate variables) or
• Multiple Line Graph (Summaries of separate variables)
3rd MANOVA (within-subjects)
• DVs
• Year level or Time
• Year 7 and 8 (same participants over Time)
• Classroom behaviour
• Attentiveness
• Settledness
• Sociability
4th MANOVA (within-subjects)
• DVs
• Students' perceptions of maths and english teachers
• Maths and English teachers (same students assessing these)
• Student ratings of teacher qualities
• Responsiveness
• Expectations
• Enjoyable class
## External links
Advertisements
Advertisements
Got something to say? Make a comment. Your name Your email address Message | 2017-09-19 15:26:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5721505880355835, "perplexity": 4128.688274771778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818685850.32/warc/CC-MAIN-20170919145852-20170919165852-00222.warc.gz"} |
https://tex.stackexchange.com/questions/471516/blockarray-superscript | # Blockarray superscript
I am using blkarray to generate a labelled matrix. I would like to indicate the transpose of such matrix using a superscript with T. However I cannot find the right syntax to get what I want. Below you can find my attempts.
\documentclass{article}
\usepackage{blkarray}
\usepackage{amsmath,bm,amssymb}
\begin{document}
$$\begin{blockarray}{ccccccc} 1 & & 12 & 13 & 14 & 15 & 16\\ \begin{block}{(ccccccc)} 0 & \cdots & 0 & 1 & 1 & 1 & 1\\ \end{block} \end{blockarray}^T$$
\end{document}
\documentclass{article}
\usepackage{blkarray}
\usepackage{amsmath,bm,amssymb}
\begin{document}
$$\begin{blockarray}{ccccccc} 1 & & 12 & 13 & 14 & 15 & 16\\ \begin{block}{(ccccccc)} 0 & \cdots & 0 & 1 & 1 & 1 & 1\\ \end{block}^T \end{blockarray}$$
\end{document}
\documentclass{article}
\usepackage{blkarray}
\usepackage{amsmath,bm,amssymb}
\begin{document}
$$\begin{blockarray}{cccccccc} 1 & & 12 & 13 & 14 & 15 & 16 &\\ \begin{block}{(ccccccc)c} 0 & \cdots & 0 & 1 & 1 & 1 & 1 & ^T\\ \end{block} \end{blockarray}$$
\end{document}
How can I get the superscript nicely, as if the matrix was not generated with blkarray?
• Yes, it should be a superscript to the closing bracket, same position that you get when applying a superscript to a pmatrix environment for example. – Francesco Jan 23 at 16:07
Something like this?
\documentclass{article}
\usepackage{blkarray}
\begin{document}
$$\begin{blockarray}{cccccccc} 1 & & 12 & 13 & 14 & 15 & 16 &\\ \begin{block}{(ccccccc)@{\hphantom{)}}l} 0 & \cdots & 0 & 1 & 1 & 1 & 1 & {\vphantom{)}}^T\\ \end{block} \end{blockarray}$$
\end{document}
• Nice! May I ask you what \hphantom and \vphantom do in the syntax? – Francesco Jan 23 at 16:14
• @Francesco \hphantom{)} inserts an empty horizontal space/symbol that is as wide as ), and \vphantom{)} inserts and empty symbol that is as high as ). You can put any symbol in the arguments of these phantoms. (There is alos \phantom which inserts an "empty" symbol.) – marmot Jan 23 at 16:17
• Thanks @marmot. So basically \begin{block}{(ccccccc)@{\hphantom{)}}l} indicates the beginning of a block with 7 centered elements enclosed by brackets followed by left-aligned element having the same horizontal width of )? – Francesco Jan 23 at 16:36
• @Francesco Almost. @{\hphantom{)}}l says this is a left-aligned column with horizontal space as wide as ) in the beginning. You could also say \begin{block}{(ccccccc)@{}l} 0 & \cdots & 0 & 1 & 1 & 1 & 1 & {\phantom{)}}^T\\ \end{block} to achieve the same. – marmot Jan 23 at 16:40
Set the indices using a smaller font:
\documentclass{article}
\usepackage{array}
\begin{document}
$\begin{tabular}{c} \mbox{\scriptsize\begin{tabular}{ *{7}{>{\centering}p{10pt}} } 1 & & 12 & 13 & 14 & 15 & 16 \end{tabular}} \\ \begin{tabular}{ *{7}{>{\centering}p{10pt}} } \makebox[0pt][r]{\bigl(}% 0 & \cdots & 0 & 1 & 1 & 1 & 1% \makebox[0pt][l]{\bigr)^T} \end{tabular} \end{tabular}$
\end{document}
Above I use a tabular with fixed-width columns, setting elements inside p{10pt} columns. You can use zero-width boxes in the outer-columns to simulate a matrix-style (...)^T.
Do you need blockarray at all?
\documentclass{article}
\usepackage{amsmath}
\newcommand{\IND}[2]{\overset{#2\vphantom{\smash[t]{\Big|}}}{#1}}
\begin{document}
$$\bigl(\IND{0}{1} \quad \cdots \quad \IND{0}{12} \quad \IND{1}{13} \quad \IND{1}{14} \quad \IND{1}{15} \quad \IND{1}{16}\bigr)^T$$
\end{document}
Alternative version:
\documentclass{article}
\usepackage{amsmath}
\newcommand{\IND}[2]{%
\vbox{\ialign{%
\hfil$##$\hfil\cr
\scriptstyle#2\cr
#1\cr
}}%
}
\begin{document}
$$\bigl(\IND{0}{1} \quad \cdots \quad \IND{0}{12} \quad \IND{1}{13} \quad \IND{1}{14} \quad \IND{1}{15} \quad \IND{1}{16}\bigr)^T$$
\end{document} | 2019-07-20 00:57:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 6, "x-ck12": 0, "texerror": 0, "math_score": 0.8892025947570801, "perplexity": 2229.649070353083}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526401.41/warc/CC-MAIN-20190720004131-20190720030131-00350.warc.gz"} |
http://www-spires.fnal.gov/spires/find/books/www?keyword=Topological+groups | Fermilab Core Computing Division
Library Home | Ask a Librarian library@fnal.gov | Book Catalog | Library Journals | Requests | SPIRES | Fermilab Documents |
Fermilab Library
SPIRES-BOOKS: FIND KEYWORD TOPOLOGICAL GROUPS *END*INIT* use /tmp/qspiwww.webspi1/17729.14 QRY 131.225.70.96 . find keyword topological groups ( in books using www
Call number: 9783319393391:ONLINE Show nearby items on shelf Title: Singularities in Geometry, Topology, Foliations and Dynamics A Celebration of the 60th Birthday of José Seade, Merida, Mexico, December 2014 Author(s): Date: 2017 Size: 1 online resource (XVII, 231 p. 15 illus., 10 illus. in color p.) Contents: Extending the action of Schottky groups on the complex anti-de Sitter space to the projective space -- Puiseux Parametric Equations via the Amoeba of the Discriminant -- Some open questions on arithmetic Zariski pairs -- Logarithmic vector fields and the Severi strata in the discriminant -- Classification of Isolated Polar Weighted Homogeneous Singularities -- Rational and iterated maps, degeneracy loci, and the generalized Riemann-Hurwitz formula -- On singular varieties with smooth subvarieties -- On Polars of Plane Branches -- Singular Intersections of Quadrics I -- A New Conjecture, a New Invariant, and a New Non-splitting Result -- Lipschitz geometry does not determine embedded topological type -- Projective transverse structures for some foliations -- Chern classes and transversality for singular spaces ISBN: 9783319393391 Series: eBooks Series: Springer eBooks Series: Springer 2017 package Keywords: Mathematics , Algebraic geometry , Global analysis (Mathematics) , Manifolds (Mathematics) , Functions of complex variables , Mathematics , Several Complex Variables and Analytic Spaces , Algebraic Geometry , Global Analysis and Analysis on Manifolds Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: 9783319297347:ONLINE Show nearby items on shelf Title: Geometrodynamics of Gauge Fields On the Geometry of Yang-Mills and Gravitational Gauge Theories Author(s): Eckehard W Mielke Date: 2017 Edition: 2nd ed. 2017 Size: 1 online resource (XVII, 373 p. 18 illus., 8 illus. in color p.) Contents: Preface -- 1 Historical background -- 2 Geometry of gauge fields -- 3 Maxwell and Yang-Mills theory -- 4 Gravitation as a gauge theory -- 5 Einstein-Cartan theory -- 6 Teleparallelism -- 7 Yang’s theory of gravity -- 8 BRST quantization of gravity -- 9 Gravitational instantons -- 10 Three-dimensional gravity -- 11 Spinor bundles -- 12 Chiral anomalies -- 13 Topological SL(5R) gauge invariant action -- 14 Geometrodynamics and its extensions -- 15 Color Geometrodynamics -- 16 Geometrodynamical model of quark confinement?- Appendix A Notation and mathematical terms -- Appendix B Calculus of exterior forms -- Appendix C Lie groups ISBN: 9783319297347 Series: eBooks Series: Springer eBooks Series: Springer 2017 package Keywords: Physics , Mathematical physics , Gravitation , Elementary particles (Physics) , Quantum field theory , Physics , Classical and Quantum Gravitation, Relativity Theory , Mathematical Physics , Elementary Particles, Quantum Field Theory Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2016-9783658106331:ONLINE Show nearby items on shelf Title: Manifolds, Sheaves, and Cohomology Author(s): Torsten Wedhorn Date: 2016 Size: 1 online resource (354 p.) Note: 10.1007/978-3-658-10633-1 Contents: Topological Preliminaries -- Algebraic Topological Preliminaries -- Sheaves -- Manifolds -- Local Theory of Manifolds -- Lie Groups -- Torsors and Non-abelian Cech Cohomology -- Bundles -- Soft Sheaves -- Cohomology of Complexes of Sheaves -- Cohom ology of Sheaves of Locally Constant Functions -- Appendix: Basic Topology, The Language of Categories, Basic Algebra, Homological Algebra, Local Analysis ISBN: 9783658106331 Series: eBooks Series: SpringerLink (Online service) Series: Springer eBooks Keywords: Mathematics , Category theory (Mathematics) , Homological algebra , Topological groups , Lie groups , Global analysis (Mathematics) , Manifolds (Mathematics) , Differential geometry , Mathematics , Category Theory, Homological Algebra , Topological Groups, Lie Groups , Differential Geometry , Global Analysis and Analysis on Manifolds Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2016-9783319397801:ONLINE Show nearby items on shelf Title: Extensions of Positive Definite Functions Applications and Their Harmonic Analysis Author(s): Palle Jorgensen Date: 2016 Size: 1 online resource (9 p.) Note: 10.1007/978-3-319-39780-1 ISBN: 9783319397801 Series: eBooks Series: SpringerLink (Online service) Series: Springer eBooks Series: Lecture Notes in Mathematics: 2160 Keywords: Mathematics , Topological groups , Lie groups , Harmonic analysis , Fourier analysis , Functional analysis , Probabilities , Mathematical physics , Mathematics , Abstract Harmonic Analysis , Topological Groups, Lie Groups , Fourier Analysis , Functional Analysis , Mathematical Physics , Probability Theory and Stochastic Processes Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2016-9783319392868:ONLINE Show nearby items on shelf Title: Operator Algebras and Applications The Abel Symposium 2015 Author(s): Date: 2016 Size: 1 online resource (2 p.) Note: 10.1007/978-3-319-39286-8 Contents: C*-tensor categories and subfactors for totally disconnected groups: Y. Arano and S. Vaes -- Decomposable approximations revisited: N.P. Brown, J.R. Carrión and S. White -- Exotic crossed products: A. Buss, S. Echterhoff, and R. Willett -- On Hong and Szymanski’s description of the primitive-ideal space of a graph algebra: T. M. Carlsen and A. Sims -- Commutator inequalities via Schur products: E. Christensen -- C*-algebras associated with algebraic actions: J. Cuntz -- A new look at C*-simplicit y and the unique trace property of a group: U. Haagerup -- Equilibrium states on graph algebras: A. an Huef and I. Raeburn -- Semigroup C_-algebras: X. Li -- Topological full groups of étale groupoids: H. Matui -- Towards a classification of compact quan tum groups of Lie type: S. Neshveyev and M. Yamashita -- A homology theory for Smale spaces: a summary: I.F. Putnam -- On the positive eigenvalues and eigenvectors of a non-negative matrix: K. Thomsen -- Classification of graph algebras: a selective survey: M. Tomforde -- QDQ vs. UCT: W. Winter ISBN: 9783319392868 Series: eBooks Series: SpringerLink (Online service) Series: Springer eBooks Series: Abel Symposia: 12 Keywords: Mathematics , K-theory , Dynamics , Ergodic theory , Functional analysis , Mathematical physics , Mathematics , Functional Analysis , Dynamical Systems and Ergodic Theory , K-Theory , Mathematical Physics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2016-9783319304519:ONLINE Show nearby items on shelf Title: Symmetries in Graphs, Maps, and Polytopes 5th SIGMAP Workshop, West Malvern, UK, July 2014 Author(s): Date: 2016 Edition: 1st ed. 2016 Size: 1 online resource (332 p.) Note: 10.1007/978-3-319-30451-9 ISBN: 9783319304519 Series: eBooks Series: SpringerLink (Online service) Series: Springer eBooks Series: Springer Proceedings in Mathematics & Statistics: 159 Keywords: Mathematics , Algebra , Field theory (Physics) , Topological groups , Lie groups , Graph theory , Mathematics , Graph Theory , Field Theory and Polynomials , Topological Groups, Lie Groups Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2016-9783319295589:ONLINE Show nearby items on shelf Title: Quantization on Nilpotent Lie Groups Author(s): Veronique Fischer Date: 2016 Edition: 1st ed. 2016 Size: 1 online resource (557 p.) Note: 10.1007/978-3-319-29558-9 Contents: Preface -- Introduction -- Notation and conventions -- 1 Preliminaries on Lie groups -- 2 Quantization on compact Lie groups -- 3 Homogeneous Lie groups -- 4 Rockland operators and Sobolev spaces -- 5 Quantization on graded Lie groups -- 6 Pseudo-d ifferential operators on the Heisenberg group -- A Miscellaneous -- B Group C* and von Neumann algebras -- Schrödinger representations and Weyl quantization -- Explicit symbolic calculus on the Heisenberg group -- List of quantizations -- Bibliography -- Index ISBN: 9783319295589 Series: eBooks Series: SpringerLink (Online service) Series: Springer eBooks Series: Progress in Mathematics: 314 Keywords: Mathematics , Topological groups , Lie groups , Harmonic analysis , Functional analysis , Mathematical physics , Mathematics , Topological Groups, Lie Groups , Abstract Harmonic Analysis , Functional Analysis , Mathematical Physics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9789462390249:ONLINE Show nearby items on shelf Title: Recent Progress in General Topology III [electronic resource] Author(s): K.P Hart J van Mill P Simon Date: 2014 Publisher: Paris : Atlantis Press : Imprint: Atlantis Press Size: 1 online resource Note: The book presents surveys describing recent developments in most of the primary subfields of General Topology, and its applications to Algebra and Analysis during the last decade, following the previous editions (North Holland,1992 and 2002). The boo k was prepared in connection with the Prague Topological Symposium, held in 2011. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs from that chosen in2002. The following areas experienced signifi cant developments: Fractals, Coarse Geometry/Topology, Dimension Theory, Set Theoretic Topology and Dynamical Systems Contents: Topological Homogeneity Some Recent Progress Concerning Topology of Fractals A biased view of topology as a tool in functional analysis Large scale versus small scale Descriptive aspects of Rosenthal compacta Minimality conditions in topological groups Set Theoretic update on Topology Topics in Dimension Theory Representations of dynamical systems on Banach spaces Generalized metrizable spaces Permanence in Coarse Geometry Selections and Hyperspaces Continuum Theory Almost disjoint families and topology Some Topics in Geometric Topology II Topologic ISBN: 9789462390249 Series: eBooks Series: SpringerLink Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Functional analysis , Logic, Symbolic and mathematical , Topology Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9788132215998:ONLINE Show nearby items on shelf Title: Basic Modern Algebra with Applications [electronic resource] Author(s): Mahima Ranjan Adhikari Avishek Adhikari Date: 2014 Publisher: New Delhi : Springer India : Imprint: Springer Size: 1 online resource Note: The book is primarily intended as a textbook on modern algebra for undergraduate mathematics students. It is also useful for those who are interested in supplementary reading at a higher level. The text is designed in such a waythat it encourages ind ependent thinking and motivates students towards further study. The book covers all major topics in group, ring, vector space and module theory that are usually contained in a standard modern algebra text. Inaddition, it studies semigroup, group action, Hopf's group, topological groups and Lie groups with their actions, applications of ring theory to algebraic geometry, and defines Zariski topology, as well as applications of module theoryto structure theory of rings and homological algebra. Algebraic as pects of classical number theory and algebraic number theory are also discussed with an eye to developing modern cryptography. Topics on applications to algebraictopology, category theory, algebraic geometry, algebraic number theory, cryptography and theo retical computer science interlink the subject with different areas. Each chapter discusses individual topics, starting from the basics, withthe help of illustrative examples. This comprehensive text with a broad variety of concepts, applications, example s, exercises and historical notes represents a valuable and unique resource Contents: Prerequisites: Basics of Set Theory and Integers Groups: Introductory Concepts Actions of Groups, Topological Groups and semigroups Rings: Introductory Concepts Ideals of Rings: Introductory concepts Factorization in Integral Domains and in Polynomial Rings Rings with Chain Conditions Vector Spaces Modules Algebraic Aspects of Number Theory Algebraic Numbers Introduction to Mathematical Cryptography Appendix A: Some Aspects of Semirings Appendix B: Category Theory Appendix C: A Brief Historical Note ISBN: 9788132215998 Series: eBooks Series: SpringerLink Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Group theory , Number theory Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9783642553615:ONLINE Show nearby items on shelf Title: Algebra, Geometry and Mathematical Physics [electronic resource] : AGMP, Mulhouse, France, October 2011 Author(s): Abdenacer Makhlouf Eugen Paal Sergei D Silvestrov Alexander Stolin Date: 2014 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: This book collects the proceedings of the Algebra, Geometry and Mathematical Physics Conference, held at the University of Haute Alsace, France, October 2011. Organized in the four areas of algebra, geometry, dynamical symmetriesand conservation laws and mathematical physics and applications, the book covers deformation theory and quantization Hom-algebras and n-ary algebraic structures Hopf algebra, integrable systems and related math structures jet theoryand Weil bundles Lie theory and applications non-commutative and Lie algebra and more. The papers explore the interplay between research in contemporary mathematics and physics concerned with generalizations of the main structures ofLie theory aimed at quantization, and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non-commutative geometry andapplications in physics and beyond. The book benefits a broad audience of researchers a nd advanced students Contents: Part I Algebra Part II Geometry Part III Dynamical Symmetries and Conservation Laws Part IV Mathematical Physics and Applications ISBN: 9783642553615 Series: eBooks Series: SpringerLink Series: Springer Proceedings in Mathematics & Statistics, 2194-1009 : v85 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Global differential geometry , Engineering mathematics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9783319078427:ONLINE Show nearby items on shelf Title: Probability on Compact Lie Groups [electronic resource] Author(s): David Applebaum Date: 2014 Publisher: Cham : Springer International Publishing : Imprint: Springer Size: 1 online resource Note: Probability theory on compact Lie groups deals with the interaction between chance and symmetry, a beautiful area of mathematics of great interest in its own sake but which is now also finding increasing applicationsin statistics and engineering (par ticularly with respect to signal processing). The author gives a comprehensive introduction to some of the principle areas of study, with an emphasis on applicability. The most important topicspresented are: the study of measures via the non-commutative F ourier transform, existence and regularity of densities, properties of random walks and convolution semigroups of measures, and the statistical problem of deconvolution. Theemphasis on compact (rather than general) Lie groups helps readers to get acquaint ed with what is widely seen as a difficult field but which is also justified by the wealth of interesting results at this level and the importance ofthese groups for applications. The book is primarily aimed at researchers working in probability, stochast ic analysis and harmonic analysis on groups. It will also be of interest to mathematicians working in Lie theory and physicists,statisticians and engineers who are working on related applications. A background in first year graduate level measure theoreti c probability and functional analysis is essential a background in Lie groups and representation theory iscertainly helpful but the first two chapters also offer orientation in these subjects Contents: Introduction 1.Lie Groups 2.Representations, Peter Weyl Theory and Weights 3.Analysis on Compact Lie Groups 4.Probability Measures on Compact Lie Groups 5.Convolution Semigroups of Measures 6.Deconvolution Density Estimation Appendices Index Bibliography ISBN: 9783319078427 Series: eBooks Series: SpringerLink Series: Probability Theory and Stochastic Modelling, 2199-3130 : v70 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Harmonic analysis , Fourier analysis , Functional analysis , Distribution (Probability theory) Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9783319059570:ONLINE Show nearby items on shelf Title: A Short Course in Computational Geometry and Topology [electronic resource] Author(s): Herbert Edelsbrunner Date: 2014 Publisher: Cham : Springer International Publishing : Imprint: Springer Size: 1 online resource Note: With the aim to bring the subject of Computational Geometry and Topology closer to the scientific audience, this book is written in thirteen ready-to-teach sections organized in four parts: TESSELLATIONS, COMPLEXES, HOMOLOGY,PERSISTENCE. To speak to the non-specialist, detailed formalisms are often avoided in favor of lively 2- and 3-dimensional illustrations. The book is warmly recommended to everybody who loves geometry and the fascinating world ofshapes Contents: Roots of Geometry and Topology Voronoi and Delaunay Diagrams Weighted Diagrams Three Dimensions Alpha Complexes Holes Area Formulas Topological Spaces Homology Groups Complex Construction Filtrations PL Functions Matrix Reduction Epilogue ISBN: 9783319059570 Series: eBooks Series: SpringerLink Series: SpringerBriefs in Applied Sciences and Technology, 2191-530X Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Computer science , Cell aggregation Mathematics , Biomedical engineering Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9783319052243:ONLINE Show nearby items on shelf Title: Descriptive Topology and Functional Analysis [electronic resource] : In Honour of Jerzy Kakols 60th Birthday Author(s): Juan Carlos Ferrando Manuel Lpez-Pellicer Date: 2014 Publisher: Cham : Springer International Publishing : Imprint: Springer Size: 1 online resource Note: Descriptive topology and functional analysis, with extensive material demonstrating new connections between them, are the subject of the first section of this work. Applications to spaces of continuous functions, topologicalAbelian groups, linear top ological equivalence and to the separable quotient problem are included and are presented as open problems. The second section is devoted to Banach spaces, Banach algebras and operator theory. Each chapterpresents a lot of worthwhile and important recent theorems with an abstract discussing the material in the chapter. Each chapter can almost be seen as a survey covering a particular area Contents: 1Some aspects in the Mathematical work of Jerzy Kakol 2Weak barrelledness vs. P spaces 3On the topology of the sets of the real projections of the zeros ofexponential polynomials 4The density character of the space Cp(X) 5Compactness and distances to spaces of continuous functions andFrchet spaces 6Two classes of metrizable spaces lc invariant 7Characteristics of the Mackey topology for abelian topologicalgroups 8Bowens Entropy for Endomorphisms of Totally Bounded Abelian 9On preserved and unpreserved extreme pointsGroups 10Cantor set ISBN: 9783319052243 Series: eBooks Series: SpringerLink Series: Springer Proceedings in Mathematics & Statistics, 2194-1009 : v80 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Functional analysis , Operator theory , Topology Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9783319020457:ONLINE Show nearby items on shelf Title: Locally Convex Spaces [electronic resource] Author(s): M. Scott Osborne Date: 2014 Publisher: Cham : Springer International Publishing : Imprint: Springer Size: 1 online resource Note: For most practicing analysts who use functional analysis, the restriction to Banach spaces seen in most real analysis graduate texts is not enough for their research. This graduate text, while focusing on locally convextopological vector spaces, is i ntended to cover most of the general theory needed for application to other areas of analysis. Normed vector spaces, Banach spaces, and Hilbert spaces are all examples of classes of locally convexspaces, which is why this is an important topic in function al analysis. While this graduate text focuses on what is needed for applications, it also shows the beauty of the subject and motivates the reader with exercises of varyingdifficulty. Key topics covered include point set topology, topological vector space s, the HahnBanach theorem, seminorms and Frchet spaces, uniform boundedness, and dual spaces. The prerequisite for this text is the Banach spacetheory typically taught in a beginning graduate real analysis course Contents: 1 Topological Groups 2 Topological Vector Spaces 3 Locally Convex Spaces 4 The Classics 5 Dual Spaces 6 Duals of Fr chet Spaces A Topological Oddities B Closed Graphs in Topological Groups C The Other KreinSmulian Theorem D Further Hints for Selected Exercises Bibliography Index ISBN: 9783319020457 Series: eBooks Series: SpringerLink Series: Graduate Texts in Mathematics, 0072-5285 : v269 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Functional analysis Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9783319020365:ONLINE Show nearby items on shelf Title: Contact and Symplectic Topology [electronic resource] Author(s): Frdric Bourgeois Vincent Colin Andrs Stipsicz Date: 2014 Publisher: Cham : Springer International Publishing : Imprint: Springer Size: 1 online resource Note: Symplectic and contact geometry naturally emerged from the mathematical description of classical physics. The discovery of new rigidity phenomena and properties satisfied by these geometric structures launched a new researchfield worldwide. The inten se activity of many European research groups in this field is reflected by the ESF Research Networking Programme Contact And Symplectic Topology (CAST). The lectures of the Summer School in Nantes (June2011) and of the CAST Summer School in Budapest (July 2012) provide a nice panorama of many aspects of the present status of contact and symplectic topology. The notes of the minicourses offer a gentle introduction to topics which havedeveloped in an amazing speed in the recent past. These topics include 3- dimensional and higher dimensional contact topology, Fukaya categories, asymptotically holomorphic methods in contact topology, bordered Floer homology, embeddedcontact homology, and flexibility results for Stein manifolds Contents: Mathematical contributions from V.I. Arnold Topological methods in 3 dimensional contact geometry A short introduction to Fukaya categories Open books and Lefschetz pencils in contact geometry Introduction to contact topology in higher dimensions Bordered Heegaard Floer homology Stein structures: existence and flexibility Embedded contact homology, cobordism maps, and applications Knot contact homology and applications ISBN: 9783319020365 Series: eBooks Series: SpringerLink Series: Bolyai Society Mathematical Studies, 1217-4696 : v26 Series: Mathematics and Statistics (Springer-11649) Keywords: Geometry , Mathematical physics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9781493909384:ONLINE Show nearby items on shelf Title: Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics [electronic resource] Author(s): Mahir Can Zhenheng Li Benjamin Steinberg Qiang Wang Date: 2014 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: This book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids. Topics presented include: v structure andrepresentation theory of reducti ve algebraic monoids v monoid schemes and applications of monoids v monoids related to Lie theory v equivariant embeddings of algebraic groups v constructions and properties of monoids fromalgebraic combinatorics v endomorphism monoids induced from vector bundles v HodgeNewton decompositions of reductive monoids A portion of these articles are designed to serve as a self-contained introduction to these topics,while the remaining contributions are research articles containing previously unpublished result s, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and LexRenner showing that the algebraic semigroups are strongly -regular. Graduate students as well as researchers working in the fields of algebraic (semi)group theory, algebraic combinatorics, and the theory of algebraic groupembeddings will benefit from this unique and broad compilation of some fundamental results in (semi)group theory, algebraic group embeddings, and alge braic combinatorics merged under the umbrella of algebraic monoids Contents: On Algebraic Semi groups and Monoids (M. Brion) Algebraic Semi groups are Strongly regular (M. Brion, L. E. Renner) Rees Theorem and Quotients in Linear Algebraic Semi groups (M. S. Putcha) Representations of Reductive Normal Algebraic Monoids (S. Doty) On Linear Hodge Newton Decomposition for Reductive Monoids (S. Varma) The Structure of Affine Algebraic Monoids in Terms of Kernel Data (W. Huang) Algebraic Monoids and Renner Monoids (Z. Li, Z. Li, Y. Cao) Conjugacy Decomposition of Canonical and Dual Canonical Monoids (R. K. Therkelsen) The Endomorphisms Monoid of a ISBN: 9781493909384 Series: eBooks Series: SpringerLink Series: Fields Institute Communications, 1069-5265 : v71 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Geometry, algebraic , Group theory , Topological Groups , Combinatorics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9781493907489:ONLINE Show nearby items on shelf Title: The Compressed Word Problem for Groups [electronic resource] Author(s): Markus Lohrey Date: 2014 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: The Compressed Word Problem for Groups provides a detailed exposition of known results on the compressed word problem, emphasizing efficient algorithms for the compressed word problem in various groups.The authorpresents thenecessary background along with the most recent results on the compressed word problem to create a cohesive self-contained book accessible to computer scientists as well as mathematicians. Readers will quickly reach the frontierofcurrent research which makes the book especially ap pealing for students looking for a currently active research topic at theintersection of group theory and computer science. The word problem introduced in 1910 by Max Dehnisone of the most important decision problems in group theory. For many groups, high ly efficient algorithms for the word problem exist. In recent years, a new technique based on data compression for providing more efficient algorithmsfor word problems, has been developed, by representing long words over group generators in a compressed f orm using a straight-line program. Algorithmic techniques used for manipulating compressed words has shown that the compressedword problem can be solved in polynomial time for a large class of groups such as free groups, graph groups and nilpotent groups. These results have important implications for algorithmic questions related to automorphism groups Contents: 1. Preliminaries from Theoretical Computer Science 2. Preliminaries from Combinatorial Group Theory 3. Algorithms on Compressed Words 4. The Compressed Word Problem 5. The Compressed Word Problem in Graph Products 6. The Compressed Word Problem in HNN Extensions 7.Outlook References Index ISBN: 9781493907489 Series: eBooks Series: SpringerLink Series: SpringerBriefs in Mathematics, 2191-8198 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Group theory , Topological Groups , Global analysis (Mathematics) Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2014-9781461491613:ONLINE Show nearby items on shelf Title: Nonlinear Maps and their Applications [electronic resource] : Selected Contributions from the NOMA 2011 International Workshop Author(s): Clara Grcio Daniele Fournier-Prunaret Tetsushi Ueta Yoshifumi Nishio Date: 2014 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: In the field of Dynamical Systems, nonlinear iterative processes play an important role. Nonlinear mappings can be found as immediate models for many systems from different scientific areas, such as engineering, economics,biology, or can also be obta ined via numerical methods permitting to solve non-linear differential equations. In both cases, the understanding of specific dynamical behaviors and phenomena is of the greatest interest for scientists.This volume contains papers that were presented at the International Workshop on Nonlinear Maps and their Applications (NOMA 2011) held in vora, Portugal, on September 15-16, 2011. This kind of collaborative effort is of paramountimportance in promoting communication among the various groups that work in dynamical systems and networks in their research theoretical studies as well as for applications. This volume is suitable for graduate students as well asresearchers in the field Contents: J. P. Almeida A. A. Pinto D. A. Rand, Renormalization of circlediffeomorphism sequences and Markov sequences F. Balibrea M. V. Caballero, Examples of Lyapunov exponents in two dimensional systems R. A. da Costa S. N. Dorogovtsev A.V. Goltsev J. F. F. Mendes, Characteristics of the explosive percolation transition E. S. Roberts A. Annibale A. C. C. Coolen, Controlled Markovian dynamics of graphs: unbiased generation of random graphs with prescribed topological properties G. Bettencourt, A case leading to rationalist of the drift L. S. Efremova, Remarks on the nonwanderi ISBN: 9781461491613 Series: eBooks Series: SpringerLink Series: Springer Proceedings in Mathematics & Statistics, 2194-1009 : v57 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Differentiable dynamical systems Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9789400753457:ONLINE Show nearby items on shelf Title: Differential Geometry and Mathematical Physics [electronic resource] : Part I. Manifolds, Lie Groups and Hamiltonian Systems Author(s): Gerd Rudolph Matthias Schmidt Date: 2013 Publisher: Dordrecht : Springer Netherlands : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Starting from an undergraduate level, this book systematically develops the basics of Calculus on manifolds, vector bundles, vector fields and differential forms, Lie groups and Lie group actions, Linear symplecticalgebra and symplectic geometry, Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. Thesecond and third items constitute the link between abstr act calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class andcaustics. The book guides the reader from elementary differential geometry to advan ced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematicaltextbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact Note: Springer eBooks Contents: 1 Differentiable manifolds 2 Vector bundles 3 Vector fields 4 Differential forms 5 Lie groups 6 Lie group actions 7 Linear symplectic algebra 8 Symplectic geometry 9 Hamiltonian systems 10 Symmetries 11 Integrability 12 Hamilton Jacobi theory References ISBN: 9789400753457 Series: e-books Series: SpringerLink (Online service) Series: Theoretical and Mathematical Physics, 1864-5879 Series: Physics and Astronomy (Springer-11651) Keywords: Topological Groups , Global analysis , Global differential geometry , Mathematical physics , Mechanics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9788847028418:ONLINE Show nearby items on shelf Title: Geometric Properties for Parabolic and Elliptic PDE's [electronic resource] Author(s): Rolando Magnanini Shigeru Sakaguchi Angelo Alvino Date: 2013 Publisher: Milano : Springer Milan : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: The study of qualitative aspects of PDE's has always attracted much attention from the early beginnings. More recently, once basic issues about PDE's, such as existence, uniqueness and stability of solutions, have been understoodquite well, research on topological and/or geometric properties of their solutions has become more intense. The study of these issues is attracting the interest of an increasing number of researchers and is now a broad andwell-established research area, with contributions tha t often come from experts from disparate areas of mathematics, such as differential and convex geometry, functional analysis, calculus of variations, mathematical physics, to name afew. This volume collects a selection of original results and informative surveys by a group of international specialists in the field, analyzes new trends and techniques and aims at promoting scientific collaboration and stimulatingfuture developments and perspectives in this very active area of research Note: Springer eBooks Contents: Goro Akagi, Stability and instability of group invariant asymptotic profiles for fast diffusion equations Elvise Berchio, A family of Hardy Rellich type inequalities involving the L2 norm of the Hessian matrices Massimiliano Bianchini and Paolo Salani, Power concavity for solutions of nonlinear elliptic problems in convex domains Lorenzo Brasco and Rolando Magnanini, The heart of a convex set Giulio Ciraolo, A viscosity equation for minimizers of a class of very degenerate elliptic functionals Adele Ferone, Kato's inequality in the half space: an alternative proof and relative i ISBN: 9788847028418 Series: e-books Series: SpringerLink (Online service) Series: Springer INdAM Series, 2281-518X : v2 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Global analysis (Mathematics) , Functional analysis , Differential equations, partial , Discrete groups , Global differential geometry , Mathematical optimization Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9784431542704:ONLINE Show nearby items on shelf Title: Lie Theory and Its Applications in Physics [electronic resource] : IX International Workshop Author(s): Vladimir Dobrev Date: 2013 Publisher: Tokyo : Springer Japan : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Traditionally, Lie Theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrisation of the mathematical description of physical systems and objects. A geometric approach to asystem yields in general so me notion of symmetry which is very helpful in understanding its structure. Geometrisation and symmetries are meant in their broadest sense, i.e., classical geometry, differential geometry, groups and quantumgroups, infinite-dimensional (super-)algebras, and their representations. Furthermore, we include the necessary tools from functional analysis and number theory. This is a large interdisciplinary and interrelated field. Samples ofthese new trends are presented in this volume, based on contributions fr om the Workshop Lie Theory and Its Applications in Physics held near Varna, Bulgaria, in June 2011. This book is suitable for an extensive audience ofmathematicians, mathematical physicists, theoretical physicists, and researchers in the field of Lie Theo ry Note: Springer eBooks ISBN: 9784431542704 Series: e-books Series: SpringerLink (Online service) Series: Springer Proceedings in Mathematics & Statistics, 2194-1009 : v36 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Geometry Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783642362163:ONLINE Show nearby items on shelf Title: Clifford Algebras and Lie Theory [electronic resource] Author(s): Eckhard Meinrenken Date: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results onsymmetric bilinear form s and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartans famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes apresentation of Petraccis proof of the PoincarBirk hoffWitt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lietheory include Duflos theorem for the case of quadratic Lie algebras, multiplets of rep resentations, and Dirac induction. The last part of the book is an account of Kostants structure theory of the Clifford algebra over asemisimple Lie algebra. It describes his Clifford algebra analogue of the HopfKoszulSamelson theorem, and explains his fa scinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principalsl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Cliffor d theory, relevant for students and researchers in mathematics andphysics Note: Springer eBooks Contents: Preface Conventions List of Symbols 1 Symmetric bilinear forms 2 Clifford algebras 3 The spin representation 4 Covariant and contravariant spinors 5 Enveloping algebras 6 Weil algebras 7 Quantum Weil algebras 8 Applications to reductive Lie algebras 9 D(g k) as a geometric Dirac operator 10 The HopfKoszulSamelson Theorem 11 The Clifford algebra of a reductive Lie algebra A Graded and filtered super spaces B Reductive Lie algebras C Background on Lie groups References Index ISBN: 9783642362163 Series: e-books Series: SpringerLink (Online service) Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 : v58 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Global differential geometry , Mathematical physics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783642343643:ONLINE Show nearby items on shelf Title: A Guide to the Classification Theorem for Compact Surfaces [electronic resource] Author(s): Jean Gallier Dianna Xu Date: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those withoutdetailed background know ledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy andformal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example. Ideal for students whose mastery of algebraic topology may be awork-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincar characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learningwhere it is required, without interrupting the carefully planned structure of the core expositi on. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuineconfidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and val uable techniques available in algebraic topology Note: Springer eBooks Contents: The Classification Theorem: Informal Presentation Surfaces Simplices, Complexes, and Triangulations The Fundamental Group, Orientability Homology Groups The Classification Theorem for Compact Surfaces Viewing the Real Projective Plane in R3 Proof of Proposition 5.1 Topological Preliminaries History of the Classification Theorem Every Surface Can be Triangulated Notes ISBN: 9783642343643 Series: e-books Series: SpringerLink (Online service) Series: Geometry and Computing, 1866-6795 : v9 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topology , Algebraic topology , Cell aggregation Mathematics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783642310904:ONLINE Show nearby items on shelf Title: Poisson Structures [electronic resource] Author(s): Camille Laurent-Gengoux Anne Pichereau Pol Vanhaecke Date: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of thesecontexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to theproblem in nearly all cases. Poisson Structu res is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures.The first part coverssolid foundations,the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications ofPoisson structures (integrable systems and deformation quantization). The clear structure of th e book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researcherswhoare interested in anintroduction to the many facets and applications of Poisson structures Note: Springer eBooks Contents: Part I Theoretical Background:1.Poisson Structures: Basic Definitions 2.Poisson Structures: Basic Constructions 3.Multi Derivations and Khler Forms 4.Poisson (Co)Homology 5.Reduction Part II Examples:6.Constant Poisson Structures, Regular and Symplectic Manifolds 7.Linear Poisson Structures and Lie Algebras 8.Higher Degree Poisson Structures 9.Poisson Structures in Dimensions Two and Three 10.R Brackets and r Brackets 11.PoissonLie Groups Part III Applications:12.Liouville Integrable Systems 13.Deformation Quantization A Multilinear Algebra B Real ISBN: 9783642310904 Series: e-books Series: SpringerLink (Online service) Series: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 0072-7830 : v347 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Global analysis (Mathematics) , Global differential geometry Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783642309946:ONLINE Show nearby items on shelf Title: Linear Algebra and Geometry [electronic resource] Author(s): Igor R Shafarevich Alexey O Remizov Date: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements ofmatrix theory and continue s with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but areusually not covered in such courses: exterior alge bras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitelygenerated periodic modules (similar to Jordan normal forms of linear opera tors). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equationsand differential geometry, as well as from mechanics and physics Note: Springer eBooks Contents: Preface Preliminaries 1. Linear Equations 2. Matrices and Determinants 3. Vector Spaces 4. Linear Transformations of a Vector Space to Itself 5. Jordan Normal Form 6. Quadratic and Bilinear Forms 7. Euclidean Spaces 8. Affine Spaces 9. Projective Spaces 10. The Exterior Product and Exterior Algebras 11. Quadrics 12. Hyperbolic Geometry 13. Groups, Rings, and Modules 14. Elements of Representation Theory Historical Note References Index ISBN: 9783642309946 Series: e-books Series: SpringerLink (Online service) Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Matrix theory , Geometry Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783642307096:ONLINE Show nearby items on shelf Title: Variational, Topological, and Partial Order Methods with Their Applications [electronic resource] Author(s): Zhitao Zhang Date: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Nonlinear functional analysis is an important branch of contemporary mathematics. It's related to topology, ordinary differential equations, partial differential equations, groups, dynamical systems, differential geometry,measure theory, and more. In this book, the author presents some new and interesting results on fundamental methods in nonlinear functional analysis, namely variational, topological and partial order methods, which have been usedextensively to solve existence of solutions for ellipt ic equations, wave equations, Schrdinger equations, Hamiltonian systems etc., and are also used to study the existence of multiple solutions and properties of solutions. This bookis useful for researchers and graduate students in the field of nonlinear fu nctional analysis. Chapter 1 contains preliminaries. In Chapter 2, three kinds of operators are introduced: increasing operators, decreasing operators, andmixed monotone operators. In Chapter 3, the minimax methods are presented and in Chapter 4, the auth or uses bifurcation and critical point theory to study structures of the solutions of elliptic equations. Chapter 5 is concerned with aclass of MongeAmpre equations. In Chapter 6, the superlinear system of Hammerstein integral equations and applications i s studied. Chapter 7 is devoted to the DancerFucik spectrum. In Chapter 8, some results on sign-changingsolutions are introduced. In Chapter 9, a local minimizer problem of a functional in differential topology is studied. Chapter 10 focuses on a class of nonlocal Kirchhoff elliptic problems via different methods. In Chapter 11, thefocus is on free boundary problems, Schrdinger systems from BoseEinstein condensate and competing systems with many species Note: Springer eBooks Contents: 1 Preliminaries Sobolev spaces and embedding theorems Critical point Cone and partial order Brouwer Degree Compact map and Leray Schauder Degree Fredholm operators Fixed point index Banach's Contract Theorem, Implicit Functions Theorem Krein Rutman theorem Bifurcation theory Rearrangements of sets and functions Genus and Category Maximum principles and symmetry of solution Comparison theorems 2 Cone and Partial Order Methods Increasing operators Decreasing operators Mixed monotone operators Applications of mixed monotone operators Fur ISBN: 9783642307096 Series: e-books Series: SpringerLink (Online service) Series: Developments in Mathematics, 1389-2177 : v29 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Functional analysis Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783642306747:ONLINE Show nearby items on shelf Title: Rational Points and Arithmetic of Fundamental Groups [electronic resource] : Evidence for the Section Conjecture Author(s): Jakob Stix Date: 2013 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of itsfundamental group. Whi le the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only oneagainst rational points on curves. This monogr aph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and thelocal-to-global approach is studied in detail. The monograph concl udes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birationalanalogue in lieu of the fundamental group extension Note: Springer eBooks Contents: Part I Foundations of Sections 1 Continuous Non abelian H1 with Profinite Coefficients 2 The Fundamental Groupoid 3 Basic Geometric Operations in Terms of Sections 4 The Space of Sections as a Topological Space 5 Evaluation of Units 6 Cycle Classes in Anabelian Geometry 7 Injectivity in the Section Conjecture Part II Basic Arithmetic of Sections 7 Injectivity in the Section Conjecture 8 Reduction of Sections 9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers Part III On the Passage from Local to Global 10 Local Obstructions ISBN: 9783642306747 Series: e-books Series: SpringerLink (Online service) Series: Lecture Notes in Mathematics, 0075-8434 : v2054 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Geometry, algebraic , Number theory Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783319002576:ONLINE Show nearby items on shelf Title: The Classification of the Virtually Cyclic Subgroups of the Sphere Braid Groups [electronic resource] Author(s): Daciberg Lima Goncalves John Guaschi Date: 2013 Publisher: Cham : Springer International Publishing : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This manuscript is devoted to classifying the isomorphism classes of the virtually cyclic subgroups of the braid groups of the 2-sphere. As well as enabling us to understand better the global structure of these groups, it marks animportant step in th e computation of the K-theory of their group rings. The classification itself is somewhat intricate, due to the rich structure of the finite subgroups of these braid groups, and is achieved by an in-depth analysisof their group-theoretical and topological properties, such as their centralisers, normalisers and cohomological periodicity. Another important aspect of our work is the close relationship of the braid groups with mapping class groups.This manuscript will serve as a reference for the study of bra id groups of low-genus surfaces, and isaddressed to graduate students and researchers in low-dimensional, geometric and algebraic topology and in algebra Note: Springer eBooks Contents: Introduction and statement of the main results Virtually cyclic groups: generalities, reduction and the mapping class group Realisation of the elements of V1(n) and V2(n) in Bn(S2) Appendix: The subgroups of the binary polyhedral groups References. ISBN: 9783319002576 Series: e-books Series: SpringerLink (Online service) Series: SpringerBriefs in Mathematics, 2191-8198 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Group theory , Algebraic topology Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783034805858:ONLINE Show nearby items on shelf Title: Pseudo-Differential Operators, Generalized Functions and Asymptotics [electronic resource] Author(s): Shahla Molahajloo Stevan Pilipovi Joachim Toft M. W Wong Date: 2013 Publisher: Basel : Springer Basel : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This volume consists of twenty peer-reviewed papers from the special sessions on pseudodifferential operators and on generalized functions and asymptotics at the Eighth Congress of ISAAC held at the Peoples FriendshipUniversity of Russia in Moscow on August 2227, 2011. The category of papers on pseudo-differential operators contains such topics as elliptic operators assigned to diffeomorphisms of smooth manifolds, analysis on singular manifoldswith edges, heat kernels and Green functions of sub-Lapla cians on the Heisenberg group and Lie groups with more complexities than but closely related to the Heisenberg group, L^p-boundedness of pseudo-differential operators on thetorus, and pseudo-differential operators related to time-frequency analysis. The s econd group of papers contains various classes of distributions and algebras of generalized functions with applications in linear and nonlineardifferential equations, initial value problems and boundary value problems, stochastic and Malliavin-type differ ential equations. This second group of papers is related to the third collection of papers via the setting ofColombeau-type spaces and algebras in which microlocal analysis is developed by means of techniques in asymptotics. The volumecontains the synergi es of the three areas treatedand is a useful complement toitspredecessorspublished in the same series Note: Springer eBooks Contents: Preface Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds The Singular Functions of Branching Edge Asymptotics The Heat Kernel and Green Function of the Sub Laplacian on the Heisenberg Group Metaplectic Equivalence of the Hierarchical Twisted Laplacian The Heat Kernel and Green Function of a Sub Laplacian on the Hierarchical Heisenberg Group Lp Bounds for Pseudo Differential Operators on the Torus Multiplication Properties in Gelfand Shilov Pseudo Differential Calculus Operator Invariance Initial Value Problems in the Time Frequency Do ISBN: 9783034805858 Series: e-books Series: SpringerLink (Online service) Series: Operator Theory: Advances and Applications : v231 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Global analysis , Operator theory , Differential equations, partial Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9783034804813:ONLINE Show nearby items on shelf Title: Complex Kleinian Groups [electronic resource] Author(s): Angel Cano Juan Pablo Navarrete Jos Seade Date: 2013 Publisher: Basel : Springer Basel : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This monograph lays down the foundations of the theory of complex Kleinian groups, a newborn area of mathematics whose origin can be traced back to the work of Riemann, Poincar, Picard and many others. Kleinian groupsare, classically, discrete groups of conformal automorphisms of the Riemann sphere, and these can themselves be regarded as groups of holomorphic automorphisms of the complex projective line CP1. When we go into higher dimensions,there is a dichotomy: Should we look at conformal automorp hisms of the n-sphere? or should we look at holomorphic automorphisms of higher dimensional complex projective spaces? These two theories differ in higher dimensions. In thefirst case we are talking about groups of isometries of real hyperbolic spaces, an area of mathematics with a long-standing tradition in the second, about an area of mathematics that is still in its infancy, and this is the focus ofstudy in this monograph. It brings together several important areas of mathematics, e.g. classical Kleini an group actions, complex hyperbolic geometry, crystallographic groups and the uniformization problem for complex manifolds Note: Springer eBooks Contents: Preface Introduction Acknowledgments 1 A glance of the classical theory 2 Complex hyperbolic geometry 3 Complex Kleinian groups 4 Geometry and dynamics of automorphisms of P2C 5 Kleinian groups with a control group 6 The limit set in dimension two 7 On the dynamics of discrete subgroups of PU(n,1) 8 Projective orbifolds and dynamics in dimension two 9 Complex Schottky groups 10 Kleinian groups and twistor theory Bibliography Index. ISBN: 9783034804813 Series: e-books Series: SpringerLink (Online service) Series: Progress in Mathematics : v303 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Differentiable dynamical systems , Differential equations, partial Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461487814:ONLINE Show nearby items on shelf Title: Ricci Flow for Shape Analysis and Surface Registration [electronic resource] : Theories, Algorithms and Applications Author(s): Wei Zeng Xianfeng David Gu Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Ricci Flow for Shape Analysis and Surface Registration introduces the beautiful and profound Ricci flow theory in a discrete setting. By using basic tools in linear algebra and multivariate calculus, readers can deduce all themajor theorems in surfac e Ricci flow by themselves. The authors adapttheRicci flow theory to practical computational algorithms, apply Ricci flow for shape analysis and surface registration, and demonstrate the power of Ricciflow in many applications in medical imaging, computer graphics, computer vision and wireless sensor network. Due to minimal pre-requisites, this bookis accessible toengineersand medicalexperts, including educators,researchers, students and industry engineerswhohave an interest insolvingreal problems related to shape analysis and surface registration. Note: Springer eBooks Contents: 1. Introduction 2. Computational 3.Computational Geometry 4. Differential Geometry of Surface 5. Riemann Surface 6. Ricci Flow 7. Topological Algorithms 8. Harmonic Maps 9. Discrete Ricci Flow 10. Shape Analysis 11. Surface Diffeomorphism 12. Medical Imaging Applications 13. Computer Vision Applications 14. Computer Graphics Applications 15. Network Applications. ISBN: 9781461487814 Series: e-books Series: SpringerLink (Online service) Series: SpringerBriefs in Mathematics, 2191-8198 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Computer vision , Computer science Mathematics , Geometry , Discrete groups Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461480242:ONLINE Show nearby items on shelf Title: Lie Groups [electronic resource] Author(s): Daniel Bump Date: 2013 Edition: 2nd ed. 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This book is intended for a one-year graduate course on Lie groups and Lie algebras. The book goes beyond the representation theory of compact Lie groups, which is the basis of many texts, and provides a carefully chosen range ofmaterial to give the student the bigger picture. The book is organized to allow different paths through the material depending on one's interests.This second edition has substantial new material, including improved discussions ofunderlying principles, streamlining of some pro ofs, and many results and topics that were not in the first edition. For compact Lie groups, the book covers the PeterWeyl theorem, Lie algebra, conjugacy of maximal tori, the Weylgroup, roots and weights, Weyl character formula, the fundamental group and more. The book continues with the study of complex analytic groups and general noncompact Lie groups, covering the Bruhat decomposition, Coxeter groups, flagvarieties, symmetric spaces, Satake diagrams, embeddings of Lie groups and spin. Other topics tha t are treated are symmetric function theory, the representation theory of the symmetric group, FrobeniusSchur duality andGL(n)GL(m) duality with many applications including some in random matrix theory, branching rules, Toeplitz determinants, combinatoric s of tableaux, Gelfand pairs, Hecke algebras, the philosophy of cusp forms and thecohomology of Grassmannians. An appendix introduces the reader to the use of Sage mathematical software for Lie group computations Note: Springer eBooks Contents: Part I: Compact Topological Groups 1 Haar Measure 2 Schur Orthogonality 3 Compact Operators 4 The PeterWeyl Theorem Part II: Compact Lie Groups 5 Lie Subgroups of GL(n,C) 6 Vector Fields 7 Left Invariant Vector Fields 8 The Exponential Map 9 Tensors and Universal Properties 10 The Universal Enveloping Algebra 11 Extension of Scalars 12 Representations of sl(2,C) 13 The Universal Cover 14 The Local Frobenius Theorem 15 Tori 16 Geodesics and Maximal Tori 17 The Weyl Integration Formula 18 The Root System 19 Examples of Root Systems ISBN: 9781461480242 Series: e-books Series: SpringerLink (Online service) Series: Graduate Texts in Mathematics, 0072-5285 : v225 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461479727:ONLINE Show nearby items on shelf Title: Harmonic Analysis on Symmetric SpacesEuclidean Space, the Sphere, and the Poincar Upper Half-Plane [electronic resource] Author(s): Audrey Terras Date: 2013 Edition: 2nd ed. 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincar upper half plane. This book is intended for beginning graduate students inmathematics or researchers in phy sics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, andengineering. Many corrections, new topics, and updates have be en incorporated in this new edition. These include discussions of the work of P. Sarnak and others making progress on various conjectures on modular forms, the work of T.Sunada, Marie-France Vignras, Carolyn Gordon, and others on Mark Kac's question Can y ou hear the shape of a drum?, Ramanujan graphs, wavelets, quasicrystals, modular knots, triangle and quaternion groups, computations of Maasswaveforms, and, finally, the author's comparisons of continuous theory with the finite analogues. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summationformula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon tr ansform, non-Euclidean geometry on the Poincar upper half plane H or unit disc andapplications to microwave engineering, fundamental domains in H for discrete groups , tessellations of H from such discrete group actions, automorphic forms, the Selberg tra ce formula and its applications in spectral theory as wellas number theory Note: Springer eBooks Contents: Chapter1 Flat Space. Fourier Analysis on R^m 1.1 Distributions or Generalized Functions 1.2 Fourier Integrals 1.3 Fourier Series and the Poisson Summation Formula 1.4 Mellin Transforms, Epstein and Dedekind Zeta Functions 1.5 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyls Criterion for Uniform Distribution Chapter2 A Compact Symmetric Space The Sphere 2.1 Fourier Analysis on the Sphere 2.2 O(3) and R^3. The Radon Transform Chapter 3 The Poincar Upper Half Plane 3.1 Hyperbolic Geometry 3.2 Harmonic Analysis on H 3.3 Fundamental Domains for ISBN: 9781461479727 Series: e-books Series: SpringerLink (Online service) Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Group theory , Topological Groups , Harmonic analysis , Fourier analysis , Functions of complex variables , Functions, special Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461473008:ONLINE Show nearby items on shelf Title: An Introduction to Quasisymmetric Schur Functions [electronic resource] Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux Author(s): Kurt Luoto Stefan Mykytiuk Stephanie van Willigenburg Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: An Introduction to Quasisymmetric Schur Functions is aimed at researchers and graduate students in algebraic combinatorics. The goal of this monograph is twofold. The first goal is to provide a reference text for the basic theoryof Hopf algebras, in particular the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric functions and connections between them. The second goal is to give a survey of results with respect to an exciting new basis ofthe Hopf algebra of quasisymmetric functi ons, whose combinatorics is analogous to that of the renowned Schur functions Note: Springer eBooks Contents: 1. Introduction 2. Classical combinatorial concepts 3. Hopf algebras 4. Compsition tableaux and further combinatorial concepts 5. Quasisymmetric Schur functions References Index ISBN: 9781461473008 Series: e-books Series: SpringerLink (Online service) Series: SpringerBriefs in Mathematics, 2191-8198 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Algorithms , Combinatorics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461471936:ONLINE Show nearby items on shelf Title: Lie Groups: Structure, Actions, and Representations [electronic resource] : In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday Author(s): Alan Huckleberry Ivan Penkov Gregg Zuckerman Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Lie Groups: Structures, Actions, and Representations, In Honor of Joseph A. Wolf on the Occasion of his 75th Birthday consists of invited expository and research articles on new developments arising from Wolf's profoundcontributions to mathematics. D ue to Professor Wolfs broad interests, outstanding mathematicians and scholars in a wide spectrum of mathematical fields contributed to the volume. Algebraic, geometric, and analytic methods areemployed. More precisely, finite groups and classical finite dimensional, as well as infinite-dimensional Lie groups, and algebras play a role. Actions on classical symmetric spaces, and on abstract homogeneous and representationspaces are discussed. Contributions in the area of representation theory involve numero us viewpoints, including that of algebraic groups and various analytic aspects of harmonic analysis. Contributors D. Akhiezer T. Oshima A. Andrada I. Pacharoni M. L. Barberis F. Ricci L. Barchini S. Rosenberg I. Dotti N. Shimeno M. Eastwood J. Tirao V . Fischer S. Treneer T. Kobayashi C.T.C. Wall A. Kornyi D. Wallace B. Kostant K. Wiboonton P. Kostelec F. Xu K.-H. Neeb O. Yakimova G. Olafsson R. Zierau B. rsted Note: Springer eBooks Contents: Preface Real group orbits on flag manifolds Complex connections with trivial holonomy Indefinite harmonic theory and harmonic spinors Twistor theory and the harmonic hull Nilpotent Gelfand pairs and spherical transforms of Schwartz functions, II: Taylor expansions on singular sets Propagation of the multiplicity freeness property for holomorphic vector bundles Poisson transforms for line bundles from the Shilov boundary to bounded symmetric domains Cent(U(n)), cascade of orthogonal roots, and a construction of LipsmanWolf Weakly harmonic Maa forms and the princi ISBN: 9781461471936 Series: e-books Series: SpringerLink (Online service) Series: Progress in Mathematics : v306 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Functional analysis Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461471165:ONLINE Show nearby items on shelf Title: Quantum Theory for Mathematicians [electronic resource] Author(s): Brian C Hall Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Springer eBooks ISBN: 9781461471165 Series: e-books Series: SpringerLink (Online service) Series: Graduate Texts in Mathematics, 0072-5285 : v267 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Functional analysis , Quantum theory , Mathematical physics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461469568:ONLINE Show nearby items on shelf Title: Measure Theory [electronic resource] : Second Edition Author(s): Donald L Cohn Date: 2013 Edition: 2nd ed. 2013 Publisher: New York, NY : Springer New York : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Intended as a self-contained introduction to measure theory, this textbook also includesa comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haarmeasures on locally compact groups. This second edition includes a chapter on measure-theoretic probability theory, plus brief treatments of the Banach-Tarski paradox, the Henstock-Kurzweil integral, the Daniell integral, and theexistence of liftings. Measure Theory provides a soli d background for study in both functional analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. Theprerequisites for this book are basic courses in point-set topology and in analysis, and the appendices present a thorough review ofessential background material. The author aims to present a straightforward treatment of thepart of measure theory necessary for analysis and probability' assuming only basic knowledge of analysis and topol ogy...Each chapter includes numerous well-chosen exercises, varying from very routine practice problems to importantextensions and developments of the theory for the difficult ones there are helpful hints. It is the reviewer's opinion that the author has succeeded in his aim. In spite of its lack of new results, the selection and presentation ofmaterials makes this a useful book for an introduction to measure and integration theory. Mathematical Reviews (Review of the First Edition) The book is a compreh ensive and clearly written textbook on measure andintegration...The book contains appendices on set theory, algebra, calculus and topology in Euclidean spaces, topological and metric spaces, and the Bochner integral. Each section of the book contains a nu mber of exercises.zbMATH (Review of the First Edition) Note: Springer eBooks Contents: 1. Measures Algebras and sigma algebras Measures Outer measures Lebesgue measure Completeness and regularity Dynkin classes 2. Functions and Integrals Measurable functions Properties that hold almost everywhere The integral Limit theorems The Riemann integral Measurable functions again, complex valued functions, and image measures 3. Convergence Modes of Convergence Normed spaces Definition of L^p and L^p Properties of L^p and L p Dual spaces 4. Signed and Complex Measures Signed and complex measures Absolute continuity Singu ISBN: 9781461469568 Series: e-books Series: SpringerLink (Online service) Series: Birkhuser Advanced Texts Basler Lehrbcher, 1019-6242 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Global analysis (Mathematics) , Distribution (Probability theory) Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461464068:ONLINE Show nearby items on shelf Title: Asymptotic Geometric Analysis [electronic resource] : Proceedings of the Fall 2010 Fields Institute Thematic Program Author(s): Monika Ludwig Vitali D Milman Vladimir Pestov Nicole Tomczak-Jaegermann Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Asymptotic Geometric Analysis is concerned with the geometric and linear properties of finite dimensional objects, normed spaces, and convex bodies, especially with the asymptotics of their various quantitative parameters as thedimension tends to inf inity. The deep geometric, probabilistic, and combinatorial methods developed here are used outside the field in many areas of mathematics and mathematical sciences. The Fields Institute Thematic Program in theFall of 2010 continued an established traditi on of previous large-scale programs devoted to the same general research direction. The main directions of the program included: * Asymptotic theory of convexity and normed spaces *Concentration of measure and isoperimetric inequalities, optimal transport ation approach * Applications of the concept of concentration * Connections with transformation groups and Ramsey theory * Geometrization of probability * Randommatrices * Connection with asymptotic combinatorics and complexity theory These directions are represented in this volume and reflect the present state of this important area of research. It will be of benefit to researchers working ina wide range of mathematical sciencesin particular functional analysis, combinatorics, convex geometry, dynamical systems, operator algebras, and computer science Note: Springer eBooks Contents: Preface The Variance Conjecture on Some Polytopes (D. Alonso Gutirrez, J. Bastero) More Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures (D. Bartosova) On the Lyapounov Exponents of Schrodinger Operators Associated with the Standard Map (J. Bourgain) Overgroups of the Automorphism Group of the Rado Graph (P. Cameron, C. Laflamme, M. Pouzet, S. Tarzi, R. Woodrow) On a Stability Property of the Generalized Spherical Radon Transform (D. Faifman) Banach Representations and Affine Compactification of Dynamical Systems (E. Glasner, M. Megrelishvili) F ISBN: 9781461464068 Series: e-books Series: SpringerLink (Online service) Series: Fields Institute Communications, 1069-5265 : v68 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topological Groups , Functional analysis , Operator theory , Discrete groups , Distribution (Probability theory) Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461458883:ONLINE Show nearby items on shelf Title: Drinfeld Moduli Schemes and Automorphic Forms [electronic resource] : The Theory of Elliptic Modules with Applications Author(s): Yuval Z Flicker Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Drinfeld Moduli Schemes and Automorphic Forms: The Theory of Elliptic Modules with Applications is based on the authors original work establishing the correspondence between ell-adic rank r Galois representations andautomorphic representations of GL( r) over a function field, in the local case, and,in the global case, under a restriction at a single place. It develops Drinfelds theory of elliptic modules, their moduli schemes and coveringschemes, the simple trace formula, the fixed point formula, as w ell as the congruence relations and a simpleconverse theorem, not yet published anywhere. This version, based on a recent course taught by the author at TheOhioState University, is updated with references to research that has extended and developed the or iginal work. The use of the theory of elliptic modules in the present work makes it accessible to graduate students, and it will serve as avaluable resource to facilitate anentrance to this fascinating area of mathematics Note: Springer eBooks Contents: Elliptic Moduli Hecke Correspondences Trace Formulae Higher Recipropcity Laws. ISBN: 9781461458883 Series: e-books Series: SpringerLink (Online service) Series: SpringerBriefs in Mathematics, 2191-8198 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Number theory Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461453925:ONLINE Show nearby items on shelf Title: Interpolation and Sidon Sets for Compact Groups [electronic resource] Author(s): Colin C Graham Kathryn E Hare Date: 2013 Publisher: Boston, MA : Springer US : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Understanding special sets of integers was classically of interest to Hadamard, Zygmund and others, and continues to be of interest today. This book is a modern treatment of the subject of interpolation and Sidon sets. It is aunique book, aimed at bo th new and experienced researchers. In particular, this is the only book in Englishwhich featuresa complete treatment of the Pisier-Bourgain results on Sidon sets, many of which were originally in French, inhard to access publications. Applications of the P-B results, due to Pisier, Bourgain, Ramsey, and the authors are included.The book introduces the reader to a wealth of methods important in mathematics today: topological,probabilistic, algebraic, combinatoric and analytic. It prepares students to perf orm research in the area and provides both exercises and open problems. The book also provides direction to the literature for topics it does not fullycover. The book is self-contained, with appendices covering results that are required, but not necessari ly in the pre-requisite background of a student ready to choose an area for research in harmonic analysis Note: Springer eBooks Contents: Preface Introduction Hadamard Sets $\epsilon$ Kronecker sets Sidon sets: Introduction and decomposition properties Characterizations of $I_0$ sets Proportional characterizations of Sidon sets Decompositions of $I_0$ sets Sizes of thin sets Sets of zero discrete harmonic density Related results Open problems Appendices (Groups, Probability, Combinatoric results,...) Bibliography Author index Subject index Index of notation ISBN: 9781461453925 Series: e-books Series: SpringerLink (Online service) Series: CMS Books in Mathematics, Ouvrages de mathmatiques de la SMC, 1613-5237 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Harmonic analysis , Fourier analysis , Functional analysis Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781461450580:ONLINE Show nearby items on shelf Title: Uniform Spaces and Measures [electronic resource] Author(s): Jan Pachl Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Uniform Spaces and Measures addresses the need for an accessible and comprehensive exposition of the theory of uniform measures -- a need that became more critical when uniform measures recently reemerged in new results inabstract harmonic analysis. Until now, results about uniform measures have been scattered throughout many papers written by a number of authors, some unpublished, using a variety of definitions and notations. Uniform measures arefunctionals on the space of bounded uniformly continuo us functions on a uniform space. They are a common generalization of several classes of measures and measure-like functionals studied in topological measure theory, probabilitytheory, and abstract harmonic analysis. They offer a natural framework for resu lts about topologies on spaces of measures and about the continuity of convolution of measures on topological groups and semitopological semigroups. Thisbook can serve as a reference for the theory of uniform measures. It includes a self-contained develop ment of the theory with complete proofs, starting with the necessary parts of the theory of uniform spaces. It also includes severalnew results, and presents diverse results from many sources organized in a logical whole. The content is also suitable for graduate or advanced undergraduate courses on selected topics in topology and functional analysis, and containsa number of exercises with hints to solutions as well as several open problems with suggestions for further research Note: Springer eBooks Contents: Prerequisites 1. Uniformities and Topologies 2. Induced Uniform Structures 3. Uniform Structures on Semigroups 4. Some Notable Classes of Uniform Spaces 5. Measures on Complete Metric Spaces 6. Uniform Measures 7. Uniform Measures as Measures 8. Instances of Uniform Measures 9. Direct Product and Convolution 10. Free Uniform Measures 11. Approximation of Probability Distributions 12. Measurable Functionals Hints to Excercises References Notation Index Author Index Subject Index ISBN: 9781461450580 Series: e-books Series: SpringerLink (Online service) Series: Fields Institute Monographs, 1069-5273 : v30 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Fourier analysis , Functional analysis , Functions of complex variables Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9781441979100:ONLINE Show nearby items on shelf Title: A Course in Topological Combinatorics [electronic resource] Author(s): Mark Longueville Date: 2013 Publisher: New York, NY : Springer New York : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: A Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years withgrowing applications i n math, computer science, and other applied areas. Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools. In most cases these solutions are very elegant andthe connection between combinatorics and to pology often arises as an unexpected surprise. The textbook covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discretegeometry. The text contains a large number of figures that suppo rt the understanding of concepts and proofs. In many cases several alternative proofs for the same result are given, and each chapter ends with a series of exercises. Theextensive appendix makes the book completely self-contained. The textbook is well sui ted for advanced undergraduate or beginning graduate mathematics students. Previous knowledge in topology or graph theory is helpful but notnecessary. The text may be used as a basis for a one- or two-semester course as well as a supplementary text for a topology or combinatorics class Note: Springer eBooks Contents: Preface List of Symbols and Typical Notation 1 Fair Division Problems 2 Graph Coloring Problems 3 Evasiveness of Graph Properties 4 Embedding and Mapping Problems A Basic Concepts from Graph Theory B Crash Course in Topology C Partially Ordered Sets, Order Complexes, and Their Topology D Groups and Group Actions E Some Results and Applications from Smith Theory References Index ISBN: 9781441979100 Series: e-books Series: SpringerLink (Online service) Series: Universitext, 0172-5939 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Combinatorics , Discrete groups Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9780817683856:ONLINE Show nearby items on shelf Title: New Foundations in Mathematics [electronic resource] : The Geometric Concept of Number Author(s): Garret Sobczyk Date: 2013 Publisher: Boston : Birkhuser Boston : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple androbust means of expressi ng a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematicsand physics. Much of the material presented h as been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modernabstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner. The book begins with a discussion of modular numbers (clock arithmetic) and modularpolynomials. This leads to the idea of a spectral basis, the complex and hyperbolic numbers, and finally to geometric algebra, which lays the groundwork for the remainder of the text. Many topics are presented in a new light,including: * vector spaces and matrices * structure of linear operators and quadratic forms * Hermitian inner product spaces * geometry of moving planes * spacetime of special relativity * classical integration theorems *differential geometry of curves and smooth surfaces * projective geometry * Lie groups and Lie algebras. Exercises with selected solutions are provided, and cha pter summaries are included to reinforce concepts as they are covered.Links to relevant websites are often given, and supplementary material is available on the authors website. New Foundations in Mathematics will be of interest to undergraduate and grad uate students of mathematics and physics whoare looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a s Note: Springer eBooks Contents: 1 Modular Number Systems 2 Complex and Hyperbolic Numbers 3 Geometric Algebra 4 Vector Spaces and Matrices 5 Outer Product and Determinants 6 Systems of Linear Equations 7 Linear Transformations on R^n 8 Structure of a Linear Operator 9 Linear and Bilinear Forms 10 Hermitian Inner Product Spaces 11 Geometry of Moving Planes 12 Representations of the Symmetric Group 13 Calculus on m Surfaces 14 Differential Geometry of Curves 15 Differential Geometry of k Surfaces 16 Mappings Between Surfaces 17 Non Euclidean and Projective Geometries 18 Lie Gr ISBN: 9780817683856 Series: e-books Series: SpringerLink (Online service) Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Group theory , Matrix theory , Topological Groups , Mathematical physics , Engineering mathematics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2013-9780817683641:ONLINE Show nearby items on shelf Title: Configurations from a Graphical Viewpoint [electronic resource] Author(s): Toma Pisanski Brigitte Servatius Date: 2013 Publisher: Boston : Birkhuser Boston : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Configurations can be studied from a graph-theoretical viewpoint via the so-called Levi graphs and lie at the heart of graphs, groups, surfaces, and geometries, all of which are very active areas of mathematical exploration. Inthis self-contained tex tbook, algebraic graph theory is used to introduce groups topological graph theory is used to explore surfaces and geometric graph theory is implemented to analyze incidence geometries. After a preview ofconfigurations in Chapter 1, a concise introduction to graph theory is presented in Chapter 2, followed by a geometric introduction to groups in Chapter 3. Maps and surfaces are combinatorially treated in Chapter 4. Chapter 5introduces the concept of incidence structure through vertex colored graphs, and the combinatorial aspects of classical configurations are studied.Geometric aspects, some historical remarks, references, and applicationsof classicalconfigurationsappear in the last chapter. With over two hundred illustrations, challenging exercises at t he end of each chapter, a comprehensive bibliography, and a set of open problems, Configurations from a Graphical Viewpoint iswell suited for a graduate graph theory course, an advanced undergraduate seminar, or a self-contained reference for mathematicia ns and researchers Note: Springer eBooks Contents: Preface Introduction Graphs Groups, Actions, and Symmetry Maps Combinatorial Configurations Geometric Configurations Index Bibliography ISBN: 9780817683641 Series: e-books Series: SpringerLink (Online service) Series: Birkhuser Advanced Texts Basler Lehrbcher Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Geometry, algebraic , Combinatorics , Geometry , Topology Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2012-9783642227172:ONLINE Show nearby items on shelf Title: The Schrdinger-Virasoro Algebra [electronic resource] Mathematical structure and dynamical Schrdinger symmetries Author(s): Jrmie Unterberger Claude Roger Date: 2012 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structurethe Schrdinger-Virasoro algebra. Just as Poincar invariance or conformal (Virasoro) invarianceplay a key role in understanding , respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study ofsome models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence. The study of the structure of this infinite-dimensional Liealgebra touches upon topics as various as statistical physics, vertex algebras, Po isson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and thespectral theory of Schrdinger operators. Note: Springer eBooks Contents: Introduction Geometric Definitions of SV Basic Algebraic and Geometric Features Coadjoint Representaion Induced Representations and Verma Modules Coinduced Representations Vertex Representations Cohomology, Extensions and Deformations Action of sv on Schrdinger and Dirac Operators Monodromy of Schrdinger Operators Poisson Structures and Schrdinger Operators Supersymmetric Extensions of sv Appendix to chapter 6 Appendix to chapter 11 Index ISBN: 9783642227172 Series: e-books Series: SpringerLink (Online service) Series: Theoretical and Mathematical Physics, 1864-5879 Series: Physics and Astronomy (Springer-11651) Keywords: Algebra , Topological Groups , Mathematical physics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2012-9783642227165:ONLINE Show nearby items on shelf Title: The Schringer-Virasoro Algebra [electronic resource] Author(s): Jie Unterberger Claude Roger Date: 2012 Publisher: Springer Berlin Heidelberg Size: 1 online resource Note: Monograph Note: Springer 2012 Physics and Astronomy eBook collection Note: Springer e-book platform ISBN: 9783642227165 Series: Texts and Monographs in Physics Series: e-books Keywords: Mathematical Methods in Physics , Topological Groups, Lie Groups , Mathematical Physics , Category Theory, Homological Algebra , Statistical Physics, Dynamical Systems and Complexity Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2012-9783642225970:ONLINE Show nearby items on shelf Title: Topics in Noncommutative Algebra [electronic resource] : The Theorem of Campbell, Baker, Hausdorff and Dynkin Author(s): Andrea Bonfiglioli Roberta Fulci Date: 2012 Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, NumericalAnalysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: 1) fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra ornot) to understand and apply the statements and numero us corollaries of the main result 2) provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view andnotation 3) provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincar, Pascal, Campbell, Baker, Hausdorff and Dynkin 4) give an outlook on theapplications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type) 5 ) quickly enable the reader, through a description of the state-of-art and open problems, to understand the modernliterature concerning a theorem which, though having its roots in the beginning of the20th century, has not ceased to provide new problems an d applications. The book assumes some undergraduate-level knowledge of algebra and analysis,but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background. The monograph may therefore provide a tool fo r beginners in Algebra Note: Springer eBooks Contents: 1 Historical Overview Part I Algebraic Proofs of the CBHD Theorem 2 Background Algebra 3 The Main Proof of the CBHD Theorem 4 Some Short Proofs of the CBHD Theorem 5 Convergence and Associativity for the CBHD Theorem 6 CBHD, PBW and the Free Lie Algebras Part II Proofs of the Algebraic Prerequisites 7 Proofs of the Algebraic Prerequisites 8 Construction of Free Lie Algebras 9 Formal Power Series in One Indeterminate 10 Symmetric Algebra ISBN: 9783642225970 Series: e-books Series: SpringerLink (Online service) Series: Lecture Notes in Mathematics, 0075-8434 : v2034 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Topological Groups , Global differential geometry Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2012-9783034801546:ONLINE Show nearby items on shelf Title: Frames and Locales [electronic resource] : Topology without points Author(s): Jorge Picado Ale Pultr Date: 2012 Publisher: Basel : Springer Basel Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the 1920s offered an expression of the geometric intuition of arealistic place (spot, gr ain) of non-trivial extent. Imitating the behaviour of open sets and their relationsled to a new approach to topology flourishing since the end of the 1950s. It has proved to be beneficial in manyrespects. Neglecting points, only little information was lo st, while deeper insights have been gained moreover, many results previously dependent onchoice principles became constructive. The result is often a smoother, rather than amore entangled, theory. No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book formfor the first time or is presented from new points of view. Two appendices provide an introduction to s ome requisite concepts from order and category theories Note: Springer eBooks Contents: Preface Introduction I. Spaces and lattices of open sets II. Frames and locales. Spectra III. Sublocales IV. Structure of localic morphisms. The categories Loc and Frm V. Separation axioms VI. More on sublocales VII. Compactness and local compactness VIII. (Symmetric) uniformity and nearness IX. Paracompactness X. More about completion XI. Metric frames XII. Entourages, non symmetric uniformity XIII. Connectedness XIV. The frame of reals and real functions XV. Localic groups Appendix I: Posets Appendix II: Categories Bibliography Index ISBN: 9783034801546 Series: e-books Series: SpringerLink (Online service) Series: Frontiers in Mathematics, 1660-8046 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Topology Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2012-9781447122944:ONLINE Show nearby items on shelf Title: Syzygies and Homotopy Theory [electronic resource] Author(s): F.E.A Johnson Date: 2012 Publisher: London : Springer London Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: The most important invariant of a topological space is its fundamental group. When this is trivial, the resulting homotopy theory is well researched and familiar. In the general case, however, homotopy theory over nontrivialfundamental groups is much more problematic and far less well understood. Syzygies and Homotopy Theory explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents, for the first time, a systematicrehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory. The first part of the book is theoretical, formulated to allow a general finitely presented group as a fundamental group. Theinnovation here is to regard syzygies as stable modules rather than minimal modules. Inevitably this forces a reconsideration of the problems of noncancellation these are confronted in the second, practical, part of the book. Inparticular, the second part of the book considers how the theory works out in detail for the specific e xamples Fn F where Fn is a free group of rank n and F is finite. Another innovation is to parametrize the first syzygy in terms ofthe more familiar class of stably free modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares. The theory developed within this book has potentialapplications in various branches of algebra, including homological algebra, ring theory and K-theory. Syzygies and Homotopy Theory will be of inte rest to researchers and also to graduate students with a background in algebra andalgebraic topology Note: Springer eBooks Contents: Preliminaries The restricted linear group The calculus of corners and squares Extensions of modules The derived module category Finiteness conditions The Swan mapping Classification of algebraic complexes Rings with stably free cancellation Group rings of cyclic groups Group rings of dihedral groups Group rings of quaternionic groups Parametrizing W1 (Z) : generic case Parametrizing W1 (Z) : singular case Generalized Swan modules Parametrizing W1 (Z) : G = C F Conclusion ISBN: 9781447122944 Series: e-books Series: SpringerLink (Online service) Series: Algebra and Applications, 1572-5553 : v17 Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Algebra , Group theory Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE
Call number: SPRINGER-2012-9780817683436:ONLINE Show nearby items on shelf Title: Singularities of Differentiable Maps, Volume 2 [electronic resource] : Monodromy and Asymptotics of Integrals Author(s): V.I Arnold S.M Gusein-Zade A.N Varchenko Date: 2012 Publisher: Boston : Birkhuser Boston : Imprint: Birkhuser Size: 1 online resource Note: Springer e-book platform Note: Springer 2013 e-book collections Note: Originally published inthe 1980s, Singularities of DifferentiableMaps: Monodromy and Asymptotics of Integrals was thesecond oftwovolumes that togetherformed a translation of the authors'influential Russian monographon singularity theory.This uncorrec ted softcover reprint of the work brings its still-relevant content back into the literature, making it availableand affordableto a global audience of researchers and practitioners. Whilethe first volume of this title, subtitled Classification of Critical Points, Caustics and Wave Fronts, contained the zoology of differentiable mapsthat is, was devoted to a description of what, where, and how singularities could beencounteredthis second volume concentrates on elements of theanatomy and physiology of singu larities of differentiable functions. The questions considered here are about the structure of singularities and how they function. Inthe first part the authors consider the topological structure of isolated critical points of holomorphic functions: vanis hing cycles distinguished bases intersection matricesmonodromy groupsthe variation operatorand theirinterconnections and method of calculation. The second part is devoted to the study of the asymptotic behavior of integrals of the method of stationary pha se, which is widely met within applications. The third and last part dealswith integrals evaluated over level manifolds in a neighborhood of the critical point of a holomorphic function. Thismonograph is suitablefor mathematicians, researchers, postgradua tes, and specialists in the areas ofmechanics, physics, technology, and other sciences dealing with the theory of singularities of differentiable maps Note: Springer eBooks Contents: Part I. The topological structure of isolated critical points of functions Introduction Elements of the theory of Picard Lefschetz The topology of the non singular level set and the variation operator of a singularity The bifurcation sets and the monodromy group of a singularity The intersection matrices of singularities of functions of two variables The intersection forms of boundary singularities and the topology of complete intersections Part II. Oscillatory integrals Discussion of results Elementary integrals and the resolution of singularities of the phase As ISBN: 9780817683436 Series: e-books Series: SpringerLink (Online service) Series: Modern Birkhuser Classics Series: Mathematics and Statistics (Springer-11649) Keywords: Mathematics , Geometry, algebraic , Topological Groups , Global analysis (Mathematics) , Global differential geometry , Cell aggregation Mathematics Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. More info: Amazon.com More info: Barnes and Noble Full Text: Click here Location: ONLINE | 2019-04-25 00:10:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38380005955696106, "perplexity": 3787.97944286764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578675477.84/warc/CC-MAIN-20190424234327-20190425020327-00181.warc.gz"} |
http://crypto.stackexchange.com/questions?page=3&sort=newest&pagesize=30 | # All Questions
15 views
### How to decrypt the PBKDF2 Algorithm [closed]
can you please brief me about PBKDF2,how to decrypt the PBKDF2 algorithm.
31 views
### Elliptic Curve Blind Signature Implementation
I have seen this prior post: Elliptic Curve based blind signature implementation Currently I'm sizing up how difficult it would be to attain Elliptic Curve Blind signatures for an application I'm ...
28 views
### Execution time RSA [closed]
I am doing a paper focusing on the public key exponent in textbook RSA. However, I need to know the time complexity for encryption time. I have heard both $O(\hspace{.02 in}\log_2(x))$ and $O(x^2)$ as ...
45 views
### Decoding base64 encoded message with a key
I am pentesting a database that where I found data which was base64 encoded. I tried the usual method of decoding base64, which didn't work. Later, I found in ...
112 views
### Memory hard key derivation (password hash) using AES encryption
I am restricted on a certain environment involving PHP and am currently unable to implement new memory hard hashes such as scrypt (and I am not trying to compete with the likes of scrypt). My ...
74 views
### Why can you reverse a modulo function when knowing its primes
We are dealing with cryptography in school right now and superficially went over the Rabin cryptosystem with the (apparently usual) example of p=7 and q=11 etc (we didn't do RSA). I understand that ...
9 views
### Decipher APDU for OpenPGP smart card applet [migrated]
I'm implementing data deciphering into my Java application using javax.smartcardio APIs. I'm using Yubikey NEO smart card element. I managed to: Select OpenPGP applet ...
85 views
5 views
### Looking for zip-like file archiver with strong asymmetric encryption [migrated]
I need a mechanism that in essence locks a file so it cannot be modified. It's contents must be read-only and one must be able to view, just like an http connection, the SSL key-chain. Some kind of ...
27 views
### Key derivation function in theory & practice
I am implementing a cryptographic protocol and I'd like to generate a set of seeds using a seed for a pseudo-random function. The reason is that I can pass on only one seed and it can be used to ...
42 views
### How does hacking work exactly? [closed]
I've seen alot of posts on the web asking the same thing or "how to hack" but the answers all say that you need to know programming, this, that, etc. It's all very generic. Boiling it down to the ...
205 views
### Keyless integrity checking with SHA-256
Currently a program is loading some files from an untrustworthy source (e.g. a CDN) which could have been tampered with. It has a known SHA-256 hash of the file stored locally, then it downloads the ...
30 views
### Signing/verification between Java and the OpenSSL
I'm trying to sign/verify data between a C application and a Java application. In java for now I was using the built-in JCE provider with SHA256withRSA as ...
62 views
### How long does it take to decrypt an RSA/ECB encrypted message?
I am working on assignment where I have specific scenarios and I am little bit struggling with this one. Alice sends and email to Bob.For this e-mail, she uses the following method of encryption: ...
30 views
### Using AWS KMS with AES GCM
I am implementing client-side encryption for data stored in AWS DynamoDB, and was wondering about the correct use of AWS KMS for key management in addition to using AES/GCM. I am using the master key ...
45 views
### How to compare the efficiency of public key cryptosystems, i.e., RSA vs El Gamal?
As part of my Mathematics degree I'm taking a Discrete Mathematics module which partially covers Public Key Cryptography but does not at all enter it in depth. I'm currently working on a project that ...
15 views
### Are there any real-world E-voting systems in use with the Paillier cryptosystem?
There are a lot of theories of Paillier cryptosystem with references to e-voting. Are there any real-world E-voting systems in use with the Paillier cryptosystem?
22 views
### Are there any FHE-MPC schemes implemented?
I want to know if there are any publicly available multiparty computation schemes derived from fully homomorphic encryption schemes. An example would be the implementation of this scheme ...
336 views
### Is there a scheme where two cipher texts can be proven to have the same (unencrypted) content?
Is there a scheme where two ciphertexts (encrypted with different keys) can be proven to have the same (plaintext) content without disclosing the plaintext or any private keys? I assume that the ...
33 views
### What part of an RSA key is referred to by the number of bits? [duplicate]
When we speak about 4096- or 2048-bit RSA keys, what part of the key is this number of bits? The public key comprises both the modulus and the public exponent, and the strength of the key can also be ...
97 views
### HMAC vs ECDSA for JWT
I will be implementing JSON web tokens into my website and have a question about implementing them. I have a choice of using two algorithms, HMAC-SHA256 and ECDSA-SHA256. I have used HMAC-SHA256 in ...
249 views
### Using a SHA512 hash to encrypt data
How can I judge the level of security with the following algorithm: I create a 64 byte hash using SHA512 via some input. I use this hash to iterate over the plaintext, byte by byte, and similarly ...
53 views
### Is it OK to use a secret IV as key when creating keyed MD5 checksum for fixed-size data?
Is it OK to use a secret IV as key when creating keyed MD5 checksum for fixed-size data? Because data size is known, attacker cannot append anything. And I think it's hard to raise chosen-plaintext ...
61 views
### Securely generating auth tokens
(Disclaimer: I'm an experienced server-side Java programmer, but with little crypto experience.) Objective: I've got a login server which, given a login+password pair, should issue a unique auth ...
50 views
### Improving security of (authentication) protocol
I have a basic protocol which tries to authenticate messages going from client to server. Basically it is like this: Imagine I want to send message M to server. Client computes a MAC over the ... | 2015-12-01 04:10:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5219389796257019, "perplexity": 3296.5770860344733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398464396.78/warc/CC-MAIN-20151124205424-00318-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://www.ask.com/question/how-are-the-numerator-and-denominator-determined-when-converting-decimal-to-a-fraction | # How Are the Numerator and Denominator Determined When Converting Decimal to a Fraction?
You can convert a decimal to a fraction easily by just doing a few simple steps. First, you need to write down the decimal divided by 1. for example, the decimal is .80 so .80/1. Then you need to multiply the top and bottom of the fraction by 100. So, .80 times 100 is 80, 1 times 100 is 100, the fraction would become 80/100. After that, you need to simplify the fraction. So if we simplify 80/100, it would become 4/5. The 80/100 is called a decimal fraction and the 4/5 is called a common fraction.
Q&A Related to "How Are the Numerator and Denominator Determined..."
1. Write the decimal number with a denominator of 1. For example, take the decimal number .25 and write it thus: .25/1. 2. Count how many digits there are after the decimal point. http://www.ehow.com/how_8517289_convert-decimals-f...
Maybe an example will make it clear. Example: 13/3. 3 goes into 13 4 times; so start by writing the 4. 4. Now put the decimal place after the 4. 4. Now there is 1 remainder, so add http://wiki.answers.com/Q/How_do_you_divide_the_de...
You may just notice that. 1/99 = 0.010101010101010101. So: 4700/9900 = 0.47474747474747. 53/9900 = 0.005353535353. 4753/9900 = 0.480101010101. Or simply notice that 4753/9900 = 4752 http://answers.yahoo.com/question/index?qid=201312...
When you convert a mixed number in lowest terms to a decimal (mixed) number, you need only convert the fractional part. The whole number part is unchanged in the two representations http://www.blurtit.com/q7035205.html
Similar Questions
Top Related Searches
Explore this Topic
1. Write the decimal number with a denominator of 1. For example, take the decimal number .25 and write it thus: .25/1. 2. Count how many digits there are after ...
1. Write the decimal number with a denominator of 1. For example, take the decimal number .25 and write it thus: .25/1. 2. Count how many digits there are after ...
1. Write the decimal number with a denominator of 1. For example, take the decimal number .25 and write it thus: .25/1. 2. Count how many digits there are after ... | 2014-04-20 14:00:29 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9065032005310059, "perplexity": 627.4024493135183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00494-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://socratic.org/questions/what-is-the-z-score-of-sample-x-if-n-4-mu-32-sd-10-and-e-x-31 | What is the z-score of sample X, if n = 4, mu= 32, SD =10, and E[X] =31?
$Z = \frac{E \left(X\right) - \mu}{\frac{s d}{\sqrt{n}}} = \frac{31 - 32}{\frac{10}{2}} = \frac{1}{5}$
$Z = \frac{E \left(X\right) - \mu}{\frac{s d}{\sqrt{n}}} = \frac{31 - 32}{\frac{10}{2}} = \frac{1}{5}$ | 2023-03-29 06:56:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.475961297750473, "perplexity": 6096.407968679193}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948951.4/warc/CC-MAIN-20230329054547-20230329084547-00635.warc.gz"} |
https://lists.macromates.com/textmate/2007-May/019706.html | # [TxMt] Re: Problem with snippet
rowkajjh spamhalde at fastmail.fm
Mon May 7 10:43:14 UTC 2007
Hans-Joerg Bibiko <bibiko at eva.mpg.de>
wrote:
> To get a dollar sign you have to escape it by \ not by $. > \$\mu m\$Aaah. And how do I get one space back? > But two questions in general: > What do you want to do with it? Not alwyas typing "$\mu m\$". "mü" "tab" is easier.
> And why you are not using a command with output 'Insert Text'?
Dont know. What command? cat file or somehing like that? | 2019-01-18 01:23:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6127108335494995, "perplexity": 6154.434002606823}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583659654.11/warc/CC-MAIN-20190118005216-20190118031216-00270.warc.gz"} |
https://chemistry.tutorvista.com/analytical-chemistry/chromophore.html | Top
# Chromophore
When light applied to target tissue, the chromophore helps in absorbing these light energy within cellular organs especially mitochondria. A chromophore is part of the molecular structure responsible for colour. So anything that changes colour when exposed to light or responds to light in a different manner than its original form definitely shows the presence of chromophore in the tissue. The chromophore polymer composite are chromophores physically incorporated into commercially available polymer materials such as amorphous polycarbonate etc.
Haemoglobin lycopene and beta carotene are good examples of chromophores and the physiological effects of these chromophore activation are mostly related to cellular level conditions. According to IUPAC chromophores are group of atoms or groups within a molecule which provides specific colors to the external features of the molecule.
## Chromophore Definition
Apart from the IUPAC definition where the chromophore has been described as specialised cells or atoms which helps in pertaining certain colour to the molecule, this also covers two important aspects.
One is related to the system’s response to an external perturbation or the spectrum, through which the chromophore is linked to the experimental behaviour of molecular system. Secondly, it also helps in understanding the localised excitement states which helps controlling the tentative response to the spectrum.
The modern day theory of McWeeny helps in getting a better picture of electronic structure of the chromophore where electron groups are involved.
If the basic description of the electronic absorption process is to be understood, the electromagnetic radiation or spectrum of wavelengths in the range of 180 nm to approximately 800 nm, interact through rapid varying electric field with the electronic charge distribution that defines shape, size and energy of the chromophore.
The magnetic field of light can also interact with a chromophore which is typically more than 106 times weaker than the interaction with the electric field and has a critical role to play in circular dichroism spectroscopy.
The influence can be seen directly in absorption spectra when the electric interaction is weak. The parity of the electric and magnetic dipole operators are found to be odd and even respectively which helps them segregating properly.
The chromophore can also be defined as covalently unsaturated group which are responsible for electronic absorption. The colours usually appears in an organic compound if it contains certain unsaturated groups. These unsaturated groups are called chromophores and any compound having these chromophore are called chromogen.
These are basically saturated groups with the non-bonded electrons which when gets bonded to a chromophore leads to alteration of both the wavelength as well as the intensity of the absorption. But they themselves cannot impart colour to any compound.
The auxochromes are either acidic or basic and usually are salt forming groups such as $–NH_{2}$, -OH or even soluble radicals like –COOH or $-SO_{3}H$. These auxochromes generally deepen the colour of a chromogen but they themselves have any colour of their own.
Any functional group which is responsible for imparting colour to the compound was originally called as chromophore, like the nitro compounds are yellow due to presence of $–NO_{2}$ group as chromophore.
## Example of Chromophore Group
The concept of a chromophore is analogous to that of group vibration and just as the wavenumber of a group vibration is treated as transferable from one molecule to another, so is the wave number at which the electronic transition occurs in a particular group. Such a group is called a chromophore since it results in a characteristics colour of the compound due to absorption of visible or widening the use of ultra violet radiation.
Ethylene group: whatever molecule contains a group such as $H_{2}C$ = $CH_{2}$ or even cyclohexene would show an intense absorption system with a maximum intensity at about 180 nm. Acetylene group: the acetylene group shows an intense absorption system at about 190 nm and allylic group absorbs very strongly 225 nm.
A transition involving π* --n promotion is useful in identifying a chromophore as it gives a characteristically weak absorption system which is usually of high wavelength of system due to π* --n promotion and may be because it is interfered with by them.
Aldehyde group: the –CHO or aldehyde is useful chromophore showing π* --n absorption system at about 280 nm quite like formaldehyde itself. Benzaldehyde group: the aldehyde part of a conjugated π electron system and cannot be treated as a chromophore.
The chromophore groups are.
Chromophore Example solvent Ethylene Vapour Acetylene Hexane vapour Acetaldehyde Vapour and hexane Acetoxime Water Acetonitrile Vapour -COOH Acetic acid Water -NO2 Nitro methane Methanol -COOR Ethyl acetate Ethanol -CONH2 Acetamide Hexane and water -N=N- Azomethane Ethanol
## Example of Basic Chromophore
The basic chromophore include the azo compounds, the azine group and the indamine group. The azo group –N=N- is found in all azo dyes and in these compounds a benzene ring is attached to each nitrogen atom. The dyes in this group can be considered as the derivative as azobenzene.
The examples of biological stains having this chromophore are Bismarck brown, methyl red and methyl orange.
The azine group is mainly found in phenazines. The neutral red and the safranin are good example of azine stains.
The indamine group –N= is mainly found in indamines, the thiazines and other types of dyes. Many of the dyes have two benzene rings attached to a nitrogen atom.
The quinonoid structure is as follows.
The thiazines have the two benzene rings getting attached further together by sulphur atom apart from nitrogen atom. The most common and well known of these thiazine stains is methylene blue.
## Types of Chromophores
Overall there are four types of chromophores and they are all open tetra pyrroles better known as phycobilins. These are synthesised by heme synthetic pathways which are common to heme and Chl synthesis. A close tetra pyrrole ring is synthesised and then modified into side chains which yields four types of chromophores.
• Blue coloured phyco-cyano-bilin
• Red coloured phyco-erythro-bilin
• Yellow coloured phyco-uro-bilin
• The purple coloured phyco-violo-bilin
These compounds differ in the number of conjugated double bonds and these are directly related to their absorption capabilities. Apart from this the number of covalent bonds through which the chromophores are attached to apo-proteins also differ. Normally the apo-proteins attach at one site and linkage through two covalent bonds at specific binding sites.
Identical functional groups in different molecules do not absorb at exactly same wavelenght and the energy change for a particular transition determines the position of given group. The energy change depend upon the structural environment of the molecule and for qualitative interpretation of the spectra only the region above 200 nm are important.
The position and intensity of an absorption band of a simple chromophore can be modified by the attachment of certain groups in basic chromophore system known as auxochromes.
Independent chromophore:
When a single chromophore is sufficient to impart colour to the compound like azo group, nitroso group, quinonoid group, they are termed as independent chromophores.
Dependent chromophores:
When more than one chromophore is essential to bring about a specific colour in the chromogen then these are grouped under dependent chromophores.
Examples are >C =O group, >C = C< group etc. The acetone having one ketone group is colourless but diacetyl having two ketones is yellow while triketopentane with three ketonic groups is orange.
The series of diphenyl polyenes, having first three members are colourless while the tri form is yellow. The penta series has orange colour and with n = 11 the colour is found to be deep violet.
The intensity of colour increases because of the auxochromes and also make the chromogen a dye by fixing these to fibre by associating and by salt formation. The fixation happens due to the bond formation of chemical bond between fibre and auxochromes.
## Function of Chromophore
The chromophores are responsible for many functions apart from the molecular colour. | 2019-06-19 19:18:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4389328956604004, "perplexity": 3212.3499820858974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999040.56/warc/CC-MAIN-20190619184037-20190619210037-00008.warc.gz"} |
http://bktsupply.com/queen-bed-vnc/2a09d5-a-geometric-series-is-a-%2B-ar-ar-2 | a+ar+ar^2 + ...The term of the series is ar^n.This series converges if |r|<1 and diverges otherwise.If it converges, it converges to a/(1-r). A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence i.e a sequence of numbers in which the ratio between consecutive terms is constant. 16. Partial sums of geometric series Start (how else?) with partial sums: A nite geometric sum is of the form: S N = a + ar + ar2 + ar3 + + arN Multiply both sides by r to get: rS N = ar + ar2 + ar3 + ar4 + + arN+1 Now subtract the second equation from the rst (look at all the cancellation on the right side!) For this series nd, (b) the common ratio, [2] (c) the rst term, [2⦠A new series, obtained by squaring each term of the original series, has sum 16 times the sum of the original series. I'm not too sure how to go about answering this question. To determine the long-term effect of Warfarin, we considered a finite geometric series of $$n$$ terms, and then considered what happened as $$n$$ was allowed to grow without bound. For example, the series For any two successive terms in the geometric series Σar^(n-1), the ratio of the two terms, (ar^n) / ar^(n-1), simplifies into an algebraic expression given by? A geometric series is any series that we can write in the form $a+ar+ar^2+ar^3+â¯=\sum_{n=1}^âar^{nâ1}.$ Because the ratio of each term in this series to the previous term is r, the number r is called the ratio. $$\{a, ar, ar^2, ar^3, ar^4, \ldots\}$$ The sum of all the terms, is called the summation of the sequence. Before we can learn how to determine the convergence or divergence of a geometric series, we have to define a geometric series. Geometric Series. Once you determine that youâre working with a geometric series, you can use the geometric series test to determine the convergence or divergence of the series. Definition 8.2.2. geometric sequence: An ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. A geometric series is a+ ar + ar2 + ::: (a) Prove that the sum of the rst n terms of this series is given by S n = a(1 rn) 1 r [4] The third and fth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. 9 - 11 + 121/9 ... is a geometric series. Each term of a geometric series, therefore, involves a higher power than the previous term. Series List Geometric Series. In this sense, we were actually interested in an infinite geometric series (the result of letting $$n$$ go to infinity in the finite sum). An infinite geometric series has sum 2000. The general form of a geometric sequence is: $a, ar, ar^2, ar^3, ar^4, \cdots$ ... Key Terms. The common ratio of the original series is m/n, where m and n are relatively prime positive integers. We refer to a as the initial term because it is the first term in the series. Also known as a geometric ⦠The summation of an infinite sequence of values is called a series . Geometric Series A pure geometric series or geometric progression is one where the ratio, r, between successive terms is a constant. | 2021-04-18 01:36:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8986406326293945, "perplexity": 212.280994748335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038464146.56/warc/CC-MAIN-20210418013444-20210418043444-00075.warc.gz"} |
https://brilliant.org/practice/taylor-series/ | Calculus
# Taylor Series
Determine the Taylor series for the function $f(x) = \sin(x)\cos(x) \text{ centered at } x = \pi.$
At $a = 0$, what is the Taylor series expansion of
$\ln ( 1 + x ) ?$
At $a = 0$, what is the Taylor series expansion of
$\frac{1}{(1-x)^2} ?$
Given the Maclaurin series expansion of $\exp(x^2)$ as $a_0 + a_1 x^1 + a_2 x^2 + \cdots ,$ what is the value of $a_0 + a_1 + a_2$?
Determine the first three non-zero terms of the Taylor series for $f(x) = \tan(x) \text{ centered at } x = \frac{\pi}{4}.$
× | 2020-07-09 22:11:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 25, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9675086736679077, "perplexity": 197.96089031905953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655901509.58/warc/CC-MAIN-20200709193741-20200709223741-00264.warc.gz"} |
http://www.caopt.com/LION12/tutorials.php | Invited and Tutorials
# Invited talks
1. Algorithmic aspects of the Lovász local lemma
Can a needle be efficiently located in a haystack? Sometimes!
Department of Mathematics, National and Kapodistrian University of Athens, Greece
Let $\inline&space;\small&space;X_1,&space;\dots,X_n$ be random variables and let $\inline&space;\small&space;E_1,\dots,E_m$ be "undesirable" events, each depending on a subset of the variables. Suppose that the probabilities of all events are bounded by a number p<1. Obviously, if the events are mutually independent, the probability of an assignment to the variables avoiding all undesirable events (i.e. none occurring) is at least $\inline&space;\small&space;(1-p)^n>0$. Therefore, as the latter probability is strictly positive, we can conclude that there is at least one assignment for which none of the events occur. By an old and much used result of Lovász, the same is true if the events are not independent, but the number of events that each individual event is correlated with is inversely proportional to p, and with constant of proportionality not larger than Euler’s number e. This sufficient condition of Lovász is fortunately weak enough to hold in several cases where the probability of an assignment avoiding all undesirable events is very small (but positive), smaller than $\inline&space;\small&space;(1-p)^n$ (needle in a haystack). So rightly, the extremely simple randomized algorithm designed by Moser (thirty-five years after Lovász' non-algorithmic proof) to locate such an assignment (locate the needle) was hailed as a major algorithmic success. I am going to discuss a very simple direct approach to proving the correctness of Moser’s algorithm and various variants of it. More importantly, I will formulate a condition weaker than that of Lovász that is still sufficient to show the existence of a good assignment (one that avoids all undesirable events). However under only this condition, it seems that it is indeed inherently hard to locate a good assignment. Nevertheless, under this condition, the good assignment will be constructed efficiently, but alas, only through an algorithm that utilizes two interacting computing agents.
2. George Michailidis, Professor and Director of the Informatics Institute, University of Florida, United States | 2017-12-15 06:06:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7060068249702454, "perplexity": 513.601490831678}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948567042.50/warc/CC-MAIN-20171215060102-20171215080102-00284.warc.gz"} |
https://electronics.stackexchange.com/questions/19048/cheap-effective-proximity-sensors-for-detecting-people | # Cheap effective proximity sensors for detecting people?
I'm building a small device that will be mounted on the ceiling (along with many other identical ones), facing downwards. I'd like to detect when someone walks underneath it. Due to the constraints of my system, the sensor needs to be:
• Cheap - the whole device needs to cost about $5, so I'd rather not spend more than about$1 on the sensor.
• Compact - the whole device is about 3cm diameter, so the sensor needs to be significantly smaller than that.
• Reasonable range - when someone is standing under the device, their head may be 50cm - 1 meter away.
The sensor doesn't need to be fast - a few checks a second ought to be sufficient.
Cheap pretty much rules out 'passive IR' pyrometers, and compact and cheap rule out ultrasonic transponders. I tried an eletrostatic detector, but while it does a great job of detecting charged plastic objects, it doesn't respond to a person standing on a wooden floor at all.
Thus far, the best option seems to be an IR LED as a photodiode (I tried an actual phototransistor, but weirdly got a poorer signal than the LED). Using a setup with one LED emitting IR, and another reverse biased LED connected to an Arduino's analog in, I'm able to discern a useful return reflecting off my hand at anything up to half a meter. While this is usable, it's right at the edge of the range, and I'm concerned it may not work in the finished system. It does have the major advantage that I can put just one LED on each board, and use one board in sensing mode while another provides the illumination.
Can anyone suggest a better option for proximity detection, or a refinement to the active IR option to extend its range?
• See futurlec.com/PIR_Sensors.shtml for some well priced PIR components. – Russell McMahon Sep 6 '11 at 17:20
• @Russell Thanks, but they're still at least twice my budget per sensor. – Nick Johnson Sep 6 '11 at 23:25
• You say total budget of $5 - what is covered by that? Sensor control electronics and PCB can be under$1. What else is in each unit and what do they do? – Russell McMahon Sep 7 '11 at 0:15
• @Russell The PCB (about $1), the microcontroller (about$2 for an ATTiny45), the LED ($0.3 -$1 depending on parts), the cable to connect it to neighbouring devices (~$1, probably less in quantities), and misc passive components ($0.1-ish). – Nick Johnson Sep 7 '11 at 3:44
• ATTiny45 - in stock Digikey. $2.20/1.$1.38/25. $1.23/100. Knowing the whole spec and what was required a cheaper processor would probably be possible. Whole project as decribed is almost certainly doable for$5 for materials with up to say $2 for sensor(s). ATTiny25 probably OK$1.89 / $1.19 / 1.05 volumes as above. There are cheaper parts that would probably work for you. What's the total spec? What volume? – Russell McMahon Sep 7 '11 at 5:03 ## 3 Answers From your explanation and other question, it seems you are connecting your IR LED directly to the ADC input. I don't think this will work too well at a distance, the ADC input will probably have quite a low impedance that will attenuate your signal. Photodiodes have a very large impedance so you need a transimpedance amplifier to convert the current to voltage. I would use something that is designed for sensing rather than emitting, like a photodiode or your IR phototransistor (if it didin't work then you are probably not using it correctly), and feed this into an opamp, then into the ADC. In the app note you link to, there are plenty of example circuits, all of which involve a transistor or opamp to amplify/buffer the signal. Try one of these and see how it performs. • I'm really trying to cut down on the component count and cost, and using the same device alternately for sensing and emitting seemed like a good way to do that. Not all of the sample circuits involve an opamp, and the voltages I'm seeing on the AVR are in a range that's usable without one - I'm just trying to figure out what works best. – Nick Johnson Sep 5 '11 at 0:36 • Which sample circuit are you looking at? All the ones I can see involve either a transistor or opamp. I think it may be pretty flaky without one at any reasonable distance, as the irradiance will be proportional to distance squared and the 20 Meg used may cause issues. If you are trying to keep component count down then I would use the phototransistor, it should be far more sensitive. I understand the desire to keep cost down, but$0.02 for a jellybean transistor would seem worth it to ensure things work as they should. – Oli Glaser Sep 5 '11 at 2:13
• 2A, which is actually a 'fundamental circuit', but as I've observed, the voltage range is reasonable. The irradiance will be proportional to distance squared even with an op-amp, and I think the difference between it and background levels may be impossible to measure over about 50cm anyway, but I will try an op-amp. An IR phototransistor is more like $0.5 than$0.02, unfortunately. – Nick Johnson Sep 5 '11 at 2:33
• Looking at the datasheet of my chosen microcontroller (the ATTiny45), it actually already supports 20x gain on differential ADC, so hopefully I can skip the external opamp, if I just know the right parameters for the resistor part of the voltage divider - about which I'm still in the dark, and it still sounds like experimentation is the only real solution. – Nick Johnson Sep 5 '11 at 3:10
• So, I tried an opamp with circuit 6A from the appnote - with a 10M resistor it was actually less sensitive than the simple 10M voltage divider. – Nick Johnson Sep 5 '11 at 8:00
IR
(1) LEDs used to wash area from above with modulated IR with cheapest IR detectors that work for you. (LED,photodiode,...)
(2) IF you are able to provide emitters at floor level facing up you can use beam interruption with sensors above. IR emitters can easily be very inobvious (IR filter can be black and opaque to visible light.) Emitters should be able to be made "walk on" damage resistant.
(3) Emitters above with floor level reflectors - need not be visibly reflective. More liable to be subject to damage than low level emitters.
(4) Alibaba India has active IR sensor boards at Rs157 =~~~ $US3. This is whole PCB and volume unknown. Gives an idea of bottom order of cost for completed units. (5) Capacitive may be able to be made to work at your range. Philips PCF8883 sounds promising, priced at Digikey at$1.08/2500 or $2.80/1. If you can install the sensor plates at floor level rather than above the targets then capacitive sensing should be very suitable. Many capacitive sensor circuits of variable merit here via Google images Some well priced PIR sensor components Ultrasonic sensor pairs From$US2.90/pair 1's, $US2.30/100's. • Thanks for the suggestions. Idea 1 is more or less what I'm thinking of as my leading option, only I can use an IR LED as both sensor and detector; one module emits while the other detects. I'd rather not require placing and powering anything at floor level, though. I've seen pyroelectric sensors as cheap as$1 on Aliexpress, but they tend to be too wide angle, as well as requiring interface circuitry. – Nick Johnson Sep 4 '11 at 11:05
you can use light and photoresistor.which will be cheapest.But photoresistors are sensitive to all lights.The idea is that the light will be reflected by the ground and measured by the ldr.If the is obstacle the value read will not be the same:so detection.But ldr as I said is sensitive so where it is beeing used is important | 2020-04-05 21:19:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4166809916496277, "perplexity": 2209.996021969479}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371609067.62/warc/CC-MAIN-20200405181743-20200405212243-00164.warc.gz"} |
https://de.zxc.wiki/wiki/Str%C3%B6mungen_in_offenen_Gerinnen | # Currents in open channels
Natural or near-natural open channel
Artificial channel
(channeling the line in Göttingen)
Currents in open channels and standing waters are a type of physical flow that is important for many areas of hydrology . The subject is also known as channel hydraulics .
Flumes are natural or artificially created drainage possibilities with a free water level, the hydraulics of which differ from flows in pipelines . The natural channels include z. B. rivers and streams . Examples of artificial flume are inflow and outflow channels , irrigation ditches and channelization of naturally formed clotting.
## Open channel as a flow guide
Like rivers, channels have a water level, here called a gauge . Open channels are - at water level - always under atmospheric pressure (with closed channels there can be overpressure above the liquid level). Flow characteristics are, for example, flow profile and flow velocity .
The opposite are currents in pipelines (e.g. in water pipes and pressure tunnels ). The difference is that in the classic case the line cross-section is constant. A larger amount of water ( flow , hydrological discharge ) increases the pressure and the flow speed in the closed pipe. The level also rises in open water. In the natural bed of rivers, complex changes in the cross-sectional areas and the local flow directions of the water result.
A third type of flow is the seepage flow of the groundwater in a porous medium.
## application areas
In hydrology , the models and solutions developed for currents in open channels or stagnant waters help, for example, to clarify the following questions about the flow behavior of waters :
## Types of flow processes
Currents in open channels and standing water are usually unsteady (at a certain point dependent on time) and must also be viewed in all three spatial directions. Such calculations are extremely complex. In many cases, however, simplifications are permitted. A model test is also often required.
For the calculation in channels, stationary, one-dimensional calculation methods are mostly used. A constant discharge over time along the channel axis is considered. Is provided in most cases - as in other fluid mechanical problems - friction freedom and laminar flow , so irrotationality.
Due to the increasing demands on the calculation accuracy and the continuously improved performance of the computing systems, however, transient, two- and three-dimensional calculations have also been carried out in the recent past. So that the timing z. B. floods can also be displayed in complex runoff situations (e.g. flat, wide valleys, dam breaks). This also applies to the calculation of currents in shallow lakes or areas of the seas close to the coast.
## One-dimensional drainage in open channels
### Stream and shoot
Nature observations show that there are (small) disturbances in surface waters (e.g. due to installations, stones on the ground, branches protruding into the water)
• at high flow velocity only have a downward or downward effect, i.e. in the direction of flow: rapid or supercritical discharge
• at low flow speeds, however, also have an upward or upward effect, i.e. also against the direction of flow: flowing or subcritical discharge .
Mathematically, this can be derived from Bernoulli's energy equation. As a quadratic equation , this has a minimum energy level at constant discharge, at which the critical velocity or critical discharge occurs. This condition lies exactly between the two above. States.
The mathematical criterion for the exact state of the flow is the Froude number of the channel, which describes the ratio of the flow velocity to the propagation velocity of a shallow water wave: ${\ displaystyle Fri}$ ${\ displaystyle v_ {fl}}$ ${\ displaystyle v_ {ausbr}}$
• ${\ displaystyle Fr> 1 \ Leftrightarrow v_ {fl}> v_ {ausbr}}$: supercritical condition / shooting
• ${\ displaystyle Fr = 1 \ Leftrightarrow v_ {fl} = v_ {ausbr}}$: critical condition
• ${\ displaystyle Fr <1 \ Leftrightarrow v_ {fl} : subcritical state / currents.
This is of great importance for the calculation of channels:
• If the discharge is running fast, the calculation of the energy line has to be done downstream
• in the case of flowing outflow, the energy line must be calculated upstream.
• At the intermediate point of the flow change (e.g. at weirs ), the initial conditions for a discharge calculation can be obtained.
The change from flowing to flowing discharge (e.g. when the gradient increases along the flow path or when there are large constrictions) takes place continuously, whereas the change from flowing to flowing discharge occurs abruptly ( alternating jump ), combined with high energy dissipation . The latter is used in the stilling basins of hydropower plants for targeted energy conversion.
### Uniform and uneven discharge
• In the case of uniform discharge , the flow velocity does not change along a streamline .
• In the case of stationary (constant over time) and uniform discharge , the water level is parallel to the channel bottom .
• Constrictions, expansions, thresholds and the like lead to discharge conditions in which the discharge is uneven and the water level is no longer parallel to the channel bottom.
### Flood and sink
If the discharge changes over time, one speaks of unsteady conditions . This occurs particularly clearly with sudden changes in discharge z. B. by opening and closing weirs or in the event of disasters such as dams breaking. A surge is understood to mean a sudden increase in the discharge and sink to mean a sudden decrease in the discharge.
### calculation
Under steady-state conditions , the calculation is carried out either according to simple formulas for given discharge cross-sections or in sections from profile to profile.
• In practice, the average flow velocity is usually calculated using empirical formulas (e.g. the flow formula according to Gauckler-Manning-Strickler or Darcy-Weisbach ) if the cross-section is known .
• An exact mathematical calculation of the water level is only possible for a rectangular channel after solving the one-dimensional equation of motion . The calculations using structured transverse profiles require a calibration of the numerical model on the basis of natural measurements or, alternatively, a higher dimensional calculation approach is chosen to calculate the water level.
• The course of the water level along the flow path with a known discharge is based on Bernoulli's energy equation and occurs with flowing discharge against the direction of flow and with swift discharge with the direction of flow, starting in an initial cross-section with a known level.
The calculation against the direction of flow with flowing discharge means that a possible misjudgment of the water level (both too high and too low) is compensated for in the following section. An overestimated water height results in a lower speed; which has a flatter energy line gradient in the next section and, as a result, a lower water height. The tolerance deviation from the first section is thus compensated for.
Surge and sink, as unsteady flow processes, can only be calculated with more complex formulas.
## sport and freetime
For river surfing and paddling acrobatics , natural or artificial waterfalls with recesses are used. Both in natural rivers and in artificial channels in the course of the use of water power, irrigation or drainage, as well as waterways built purely for sports, which sometimes pump the water in a circuit. Examples are Almkanal south of the city of Salzburg, Mur (until 2016) and Mühlgang in Graz, Paddelkanal on the Danube Island Vienna, Eiskanal in Augsburg and the Eisbach in Munich.
Counter-current systems and wave pools are used for swimming training in confined spaces and for experiencing water.
## Flood of wood, rafting
The Schwarzenbergsche Schwemmkanal , the Holztrift in the Reichraminger Hintergebirge are examples of the use of channels for timber transport without being occupied by people. The rafting down the river was done with rafts working on the rafts. In the past, bays on the west coast of North America were also used to collect and store delimbed logs, especially for processing into paper . Keeping it moist with water prevents insect infestation on logs.
## Flooding
Small channels for drainage with and without the floating of floating and / or sinking substances are used in many ways:
• Gutters diagonally across steep, especially unpaved roads to prevent their erosion by the flow of water
• Paved pointed ditch between the asphalt road and the sidewalk edge.
• on the streets of Freiburg im Breisgau , occasionally also in Villingen and historically in Bern (CH) for cleanliness, irrigation and irrigation
• Gutters under the eaves of a roof area
• Flumes in washrooms
• Flood protection structures on mountain slopes as barriers with small openings to catch debris that has swept away
• Deposition of gravel where a river becomes shallower and wider
• in cattle sheds for the drainage of excrement and urine including litter, sometimes with the support of pushing elements
## Ferries
Running waters can drive cable ferries .
## Flow tests
Technical-physical experiments, for example with boat models, can be carried out in flow or towing channels . | 2022-06-25 17:08:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6521073579788208, "perplexity": 2302.5605694285528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036077.8/warc/CC-MAIN-20220625160220-20220625190220-00624.warc.gz"} |
https://stats.stackexchange.com/questions/520276/survival-analysis-how-to-handle-data-where-i-know-the-event-occurred-but-i-don | # Survival analysis: How to handle data where I know the event occurred, but I don't know when it occurred?
I am trying to do a survival analysis on some cancer data. I plan on doing Kaplan-Meier and Cox proportional hazards regression. I am interested in looking at the impact of various variables on overall survival. I have three potentially relevant pieces of data for each sample:
• Status (binary variable; whether the patient died or is alive)
• Days to death (elapsed time between when the study began and when the patient died)
• Days to final follow up (elapsed time between when the study began and the last time the patient was followed up with; from what I can tell, this is <= to days to death)
For the patients who are marked as "alive" in the data, I plan to use the 'days to final follow up' as the right censored survival value. The event (death) did not occur for these patients, so event = 0.
For the patients who are marked as "dead" in the data, I plan to use 'days to death', when available, as the survival value. The event occurred for these patients, so event = 1.
I am confused what to do when patients are marked as "dead" in the data, but I do not have a 'days to death' value for them. I only have a 'days to final follow up' value for these patients. The event occurred for these patients, but it may have occurred anytime on or after 'days to final follow up' (since they had to have been alive at the follow up). I am assuming that what happened in these cases was that the patient died sometime after their final follow up or maybe after the conclusion of the study, so they were marked as 'dead' but their 'days to death' value may have been not recorded.
For these "dead" patients, should I assume that they were alive at the 'days to final follow up' date, and mark them as "alive" (event = 0) and use the 'days to final follow up' as right-censored survival time? Or should I keep them as 'dead' (event = 1) and use 'days to final follow up' as right-censored survival time? Does it make sense to have censoring for patients for whom the event occurred?
I would like to avoid dropping data, if possible - around half of the events are patients marked as 'dead' but for whom 'days to death' is not available.
Thank you so much!
Such cases are called interval censored: unlike right censoring, where you have only a lower bound for the time-to-event, you have both a lower and an upper limit. That is best analyzed by specifying both the left and right limits in time from study start between which the event happened (those can be the same values, to specify uncensored event times) and using a program (unlike the standard coxph() function in R) that is designed to perform the calculations needed to handle this sort of data.
The R icenReg package has a vignette that explains the issues with interval-censored survival times and ways to analyze such data. A web search on interval censoring will suggest many more sources of information.
• Thank you. For patients who do not experience the event (i.e., they do not die), I use the 'days to final follow up' as the right censored data. But some set, P, of patients does experience the event (death), but I don't know when they died; only that they must have experienced the event sometime after 'days to final follow up.' So I know only the lower limit - I don't see how to get the upper limit. Should I label the patients in P as 'alive' and right-censor 'days to final follow up', since they were technically alive on that day and died later? Or should I keep them as 'dead'? Apr 19, 2021 at 6:10
• @user318967 for that set “P” of patients, it might be simplest to right-censor as of last follow-up date. Alternatively, you could interval-censor between the last follow-up date and the date that you received information that they had died. With icenReg and the (leftLimit, rightLimit] coding of event times I think there is no need for a separate alive/dead variable; you use $\infty$ or NA for the rightLimit for right-censored event times.
– EdM
Apr 19, 2021 at 11:57 | 2022-05-27 22:50:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3054807484149933, "perplexity": 1412.7453055042572}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663006341.98/warc/CC-MAIN-20220527205437-20220527235437-00672.warc.gz"} |
https://math.stackexchange.com/questions/2521310/p-adic-integers-and-projective-limit-of-u-u-n | # $p$-adic integers and projective limit of $U/U_n$
Context: I am currently reading through Serge Lang's Algebraic Number Theory without much knowledge of category theory or advanced algebra.
In the book, $\mathbb{Z}_p$ is defined as a subgroup of infinite direkt product $$\mathbb{Z}_p = \cdots \times \mathbb{Z}/p^n\mathbb{Z}\times\mathbb{Z}/p^{n-1}\mathbb{Z}\times\cdots\times \mathbb{Z}/p\mathbb{Z}$$ where the $n$-th "component of the sequence" modulo $p^{n-1}$ equals the $n-1$-th component, i.e. $x_n \equiv x_{n-1}~~(\text{mod}~p^{n-1})$. This process of successively taking products is formalized as the (projective) limit $$\mathbb{Z}_p = \lim_\leftarrow ~\mathbb{Z}/p^n\mathbb{Z}~.$$
Question: Later on the groups $U = \mathbb{Z}_p^\times$ and $U_n = 1+p^n\mathbb{Z}_p$ are introduced and it is shown that $U/U_n \cong (\mathbb{Z}/p^n\mathbb{Z})^\times$.
Then the book states (without further explanation) that $$U = \lim_\leftarrow~U/U_n~.$$ If have trouble understanding why this is true. It feels like it should be true but from a formal point of view I need that $$\lim_\leftarrow~(\mathbb{Z}/p^n\mathbb{Z})^\times~=~\big(\lim_\leftarrow~\mathbb{Z}/p^n\mathbb{Z}\big)^\times$$ to be able to resort to the definition of $\mathbb{Z}_p$. Is there an elementary way to explain why this or the above statement about $U$ is true without using much theory about projective limits, profinite groups and related concepts? | 2019-05-21 02:52:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8930814266204834, "perplexity": 133.74707145175358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256215.47/warc/CC-MAIN-20190521022141-20190521044141-00149.warc.gz"} |
http://www.dlyj.ac.cn/CN/10.11821/yj2002040010 | • 论文 •
### 阿拉善东南部自然环境演变与地面流沙路径的分析
1. 1. 安徽师范大学国土资源学院遥感实验室,芜湖 241000;
2. 中国科学院遥感信息科学开放实验室,北京 100101;
3. 中煤航测遥感局遥感应用研究院,西安 710054;
4. 皖西学院地理系,六安 237012
• 收稿日期:2002-01-05 修回日期:2002-05-26 出版日期:2002-08-15 发布日期:2002-08-15
• 作者简介:王心源(1964-),男,安徽六安人,教授,博士。从事遥感应用与环境变化等研究。
• 基金资助:
国家自然科学基金重大项目(49989001-4);中国科学院遥感信息科学开放实验室开放基金(SK010004);安徽省教委自然科学基金(01JL0091);安徽省自然科学基金项目(01045406)
### The physical environment evolution and trajectories of drift sand in southeastern Alxa of China
WANG Xin-yuan1,2, WANG Fei-yue3, DU Fang-ming4, ZHOU Bing-gen1, CHANG Yue-ming1, HU Wei1
1. 1. Laboratory of Remote Sensing,College of National Territorial Resource of Anhui Normal University,Wuhu 241000,China;
2. Laboratory of Remote Sensing Information Sciences,CAS,Beijing 100101,China;
3. Institute of Remote Sensing Application,Chinese Coal Remote Sensing Bureau,Xi’an 710054,China;
4. Department of Geography,Wanxi College,Lu’an 237012,China
• Received:2002-01-05 Revised:2002-05-26 Online:2002-08-15 Published:2002-08-15
Abstract:
The research area, the southeastern part of Alxa, lies in the west of Inner Mongolia Plateau in China. It bounds Mt. Yabrai and Mt.Bayan Ul to the west and Helan Mountains and Yellow River to the east,connects Hexi Corridor to the south and Langshan Mountains to the north. The area includes Tengger Desert and Ulan Buh Desert. The well-known Jartai Salt Lake lies in Ulan Buh Desert. Topographically it is the connection part of Alxa Plateau, Loess Plateau and Qinghai-Tibet Plateau. It is also the transitional zone between the semiarid dry-grassland areas and arid-hyperarid areas.Its climate is arid and semiarid. This is the sensitive region of global change and is one of the sand storms most frequently hit areas in China. In the late Pleistocene, Tengger Desert and Jartai Lake were large broad fresh water lakes separately. In Tengger Desert, there were more than 400 lakes of different sizes. The Alxa Plateau had ever been such a geographical environment with many rivers and lakes. Due to the effect of the uplifting of the Qinghai-Tibet Plateau, the moisture laden ocean air current was blocked from entering in. This turns central-Asia into inland arid climate region, and its environment became deteriorated.The cause of the aridity is due to two aspects:First, the climate became dry, and the water of the lakes was strongly vaporized. So, the areas of the lakes turned smaller and smaller, and finally into dry basin and lacustrine sediments exposed to the ground to turn into desert under the wind erosion. Secondly, due to the funnelling effect, drifting sand from Yamaleike Desert in the west of Jartai and from Badain Jaran Desert in the west of Tengger Desert invades along the narrow passageway.That is a very important factor leading to desertification of the lake basin.In the arid and semiarid regions, the structure of the eco-system is very simple,which is liable to induce ecological calamities.Today the main problem in these areas is the invasion of the drifting sand and sandstorms.Therefore, based on studies of remote sensing images,relevant measures for improving the research area's ecological environment are identified as to block and fix the sand in the west, and to establish ecological protective belt in the east, to renovate and control sand encroachment in divided blocks. With these measures to harness ground surface environment, the invasion and expansion of the drifting sand, the occurrence and the intensity of the sandstorms from the area's surface can be controlled effectively. The prevention and control of the sandstorms of the area is of very important to the mitigation of the sandstorm calamity in Beijing. | 2023-03-25 23:42:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1776038557291031, "perplexity": 11571.960546025857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945376.29/warc/CC-MAIN-20230325222822-20230326012822-00060.warc.gz"} |
http://math.stackexchange.com/questions/73457/the-convergence-of-fourier-series | # the convergence of Fourier series
Assume now we have $f(x)\in L^1([0,1])$, then we don't necessarily have the convergence of the partial sum of the Fourier series, moreover, by the theorem of kolmogorov, we can even have a.e. divergence of the partial sum.
Now my question is, for $f(x)\in L^1([0,1])$, denote the Fourier Transform as $\{a_n\}_{n=-\infty}^{\infty}$, and assume that the partial sum $S_n(x)=\sum_{k=-n}^{n}a_ke^{ikx}$ converges pointwise almost everywhere in $[0,1]$, then can we expect that the partial sum will converge back to the original function $f(x)$?
-
A Fourier transform is different from a Fourier sequence. – Thomas Andrews Oct 17 '11 at 21:57
@Thomas Andrews: but I think in some general sense, there's no problem to call it like this, or? – bonnnnn2010 Oct 17 '11 at 22:00
No, in the "general sense," the Fourier transform of a function is different from the sequence of coefficients of the Fourier series of that function. In mathematics, words are precise things. You can argue, perhaps, that the Fourier series of a function is somehow akin to the Fourier transform, but then the Fourier series is "a (generalized) Fourier transform" of the function, not "the Fourier transform" of that function. – Thomas Andrews Oct 18 '11 at 17:00
The Cesaro means $\sigma_n(x)$ of the Fourier series of an $L^1$ function $f(x)$ converge almost everywhere to $f(x)$. At any point where the partial sums $S_n(s)$ converge, the Cesaro means $\sigma_n(x)$ converge to the same value. So if the partial sums converge pointwise almost everywhere, the limit must almost everywhere be $f(x)$. Of course, as user16892 noted, the limit might not be $f(x)$ everywhere (but that's obvious, because you can change an $L^1$ function on a set of measure 0 and not affect the Fourier series). | 2014-08-28 15:39:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874764084815979, "perplexity": 178.36545285345633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830903.34/warc/CC-MAIN-20140820021350-00019-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.gamedev.net/forums/topic/479357-vector-class-overriding-the---operator/ | • 13
• 27
• 9
• 9
• 20
# Vector class, overriding the - operator
This topic is 3713 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi, I'm still trying to find the problem with my messed up normals, and I've managed to trace the problem to my vector class. I'm using the following code to override the minus operator:
SYM_VECTOR SYM_VECTOR::operator -(SYM_VECTOR Vector)
{
SYM_VECTOR Temp;
Temp.i = i - Vector.i;
Temp.k = j - Vector.j;
Temp.j = k - Vector.k;
return Temp;
}
Very simple. However, the following two expressions give me different results, yet in practice they are exactly the same:
SYM_VECTOR v1, v2, v3, b1, b2;
b1 = v2 - v1;
b2 = v3 - v1;
SYM_VECTOR v1, v2, v3, b1, b2;
b1.i = v2.i - v1.i;
b1.j = v2.j - v1.j;
b1.k = v2.k - v1.k;
b2.i = v3.i - v1.i;
b2.j = v3.j - v1.j;
b2.k = v3.k - v1.k;
Any help appreciated.
##### Share on other sites
Temp.j and Temp.k are transposed in your operator-() function. Perhaps that's the problem.
##### Share on other sites
Temp.k = j - Vector.j;Temp.j = k - Vector.k;
##### Share on other sites
Thank you both.
I think I need sleep. I've scanned over that function over 20 times :/
##### Share on other sites
Quote:
Original post by deadstarI think I need sleep. I've scanned over that function over 20 times :/
Haha. Been there. | 2018-03-17 18:56:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20099233090877533, "perplexity": 6892.082682392694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645280.4/warc/CC-MAIN-20180317174935-20180317194935-00789.warc.gz"} |
https://tex.stackexchange.com/questions/264260/matrix-product-illustration | # Matrix product illustration
I would like to write the presentation of the matrix product by putting the two matrices in diagonal, giving an easy-to-remember method to compute the coefficients. Wikipedia has the following presentation:
I am not necessarily aiming at something that fancy, but still I am not sure which tools to use... arrays? matrices? tikz?
Suggestions welcome.
You can use TikZ and matrix of math nodes:
The code:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows.meta,positioning}
\definecolor{myyellow}{RGB}{240,217,1}
\definecolor{mygreen}{RGB}{143,188,103}
\definecolor{myred}{RGB}{234,38,40}
\definecolor{myblue}{RGB}{53,101,167}
\begin{document}
\begin{tikzpicture}[
mymatrix/.style={
matrix of math nodes,
outer sep=0pt,
nodes={
draw,
text width=2.5em,
align=center,
minimum height=2.5em,
text=gray
},
nodes in empty cells,
column sep=-\pgflinewidth,
row sep=-\pgflinewidth,
left delimiter=[,
right delimiter=],
},
mycircle/.style 2 args={
draw=#1,
circle,
fill=#2,
line width=2pt,
inner sep=5pt
},
arr/.style={
line width=4pt,
-{Triangle[angle=60:1.5pt 3]},
#1,
shorten >= 3pt,
shorten <= 3pt
}
]
%the matrices
\matrix[mymatrix] (A)
{
|[text=black]|a_{11} & |[text=black]|a_{12} \\
a_{21} & a_{22} \\
|[text=black]|a_{31} & |[text=black]|a_{32} \\
a_{41} & a_{42} \\
};
\matrix[mymatrix,right=of A.north east,anchor=north west] (prod)
{
& & \\
& & \\
& & \\
& & \\
};
\matrix[mymatrix,above=of prod.north west,anchor=south west] (B)
{
b_{11} & |[text=black]|b_{12} & |[text=black]|b_{13} \\
b_{21} & |[text=black]|b_{22} & |[text=black]|b_{23} \\
};
%the labels for the matrices
\node[font=\huge,left=10pt of A] {$A$};
\node[font=\huge,above=2pt of B] {$B$};
%the frames in both matrices
\draw[myyellow,line width=2pt]
([shift={(1.2pt,-1.2pt)}]A-1-1.north west)
rectangle
([shift={(-1.2pt,1.2pt)}]A-1-2.south east);
\draw[myyellow,line width=2pt]
([shift={(1.2pt,-1.2pt)}]B-1-2.north west)
rectangle
([shift={(-1.2pt,1.2pt)}]B-2-2.south east);
\draw[mygreen,line width=2pt]
([shift={(1.2pt,-1.2pt)}]A-3-1.north west)
rectangle
([shift={(-1.2pt,1.2pt)}]A-3-2.south east);
\draw[mygreen,line width=2pt]
([shift={(1.2pt,-1.2pt)}]B-1-3.north west)
rectangle
([shift={(-1.2pt,1.2pt)}]B-2-3.south east);
%the filled circles in the product
\node[mycircle={myblue}{mygreen}]
at (prod-3-3) (prod33) {};
\node[mycircle={myred}{myyellow}]
at (prod-1-2) (prod12) {};
%the arrows
\draw[arr=myred]
(A-1-2.east) -- (prod12);
\draw[arr=myred]
(B-2-2.south) -- (prod12);
\draw[arr=myblue]
(A-3-2.east) -- (prod33);
\draw[arr=myblue]
(B-2-3.south) -- (prod33);
%the legend
\matrix[
matrix of math nodes,
nodes in empty cells,
column sep=10pt,
anchor=north,
nodes={
minimum height=2.2em,
minimum width=2em,
anchor=north west
},
below=5pt of current bounding box.south
]
(legend)
{
& a_{11}b_{12} + a_{12}b_{22} \\
& a_{31}b_{13} + a_{32}b_{23} \\
};
\node[mycircle={myblue}{mygreen}]
at (legend-2-1) {};
\node[mycircle={myred}{myyellow}]
at (legend-1-1) {};
\end{tikzpicture}
\end{document}
• not only fix, also improved! I'm just curious, why you use \node[font=\huge,xshift=-20pt] at (A.west) {$A$} ; instead shorter \node[font=\huge,left=20pt of A] {$A$};. And wondering if it is possible to define outer sep for matrix? If it is, than I suspect that code can be further simplified. But anyway, very nice solution! I already vote before for it :-) Sep 1 '15 at 23:47
• @Zarko Thanks. Yes, your proposal for the simplification is valid; I introduced it in an edit. Regarding outer sep`, yes, you can also set its value for the matrix. Sep 2 '15 at 1:38 | 2021-09-21 11:37:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9522625803947449, "perplexity": 14865.802833565109}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00193.warc.gz"} |
http://pythonhosted.org/DAGPype/numpy_chunks.html | # NumPy And High-Performance Chunking¶
• NumPy Array Sinks describes how to transform the output of a pipeline to a NumPy array, and some points regarding this.
• High Performance Chunking describes stages that chunk the data internally for high performance. Each such stage uses NumPy arrays internally.
The relevant stages are in the dagpype.np sub-package.
## NumPy Array Sinks¶
Much of the purpose of this library is for preprocessing data for further processing using other Python libraries.
To build a pipe resulting in a NumPy array, we can do something like one of the following:
>>> a = stream_vals('rain.txt') | np.to_array()
>>> type(a)
<type 'numpy.ndarray'>
>>> a.shape
(61,)
>>> a = stream_vals('meteo.csv', ('wind', 'rain')) | np.to_array()
>>> type(a)
<type 'numpy.ndarray'>
>>> a.shape
(60, 2)
>>> a = stream_vals('meteo.csv', ('wind', 'rain')) | \
... filt(pre = lambda (wind, rain) : wind < 10 and rain < 10) | \
... np.to_array()
>>> type(a)
<type 'numpy.ndarray'>
>>> a.shape
(48, 2)
If the the sole reason we’re creating an array is for applying a NumPy function, we can chain sinks:
>>> print stream_vals('meteo.csv', 'rain') | (to_array() | sink(lambda a : numpy.median(a)))
and, of course, we can apply more than a single function to the array, like this:
>>> print stream_vals('meteo.csv', 'rain') | \
... (to_array() | sink(lambda a : (numpy.median(a), numpy.kurtosis(a))))
or like this:
>>> print stream_vals('meteo.csv', 'rain') | \
... (to_array() | sink(lambda a : numpy.median(a)) + sink(lambda a : numpy.kurtosis(a)))
Some aggregates, e.g., the median, cannot be calculated (or even approximated) using constant memory. This might cause a problem if the dataset is large. In such cases, we can use sub-sampling. The following samples approximately 1% of the elements, and uses them to find the median:
>>> stream_vals('meteo.csv', 'rain') | prob_rand_sample(0.01) | (to_array() | sink(lambda a : numpy.median(a))
The following samples (with replacement) 100 of the elements, then uses them to find the median:
>>> stream_vals('meteo.csv', 'rain') | size_rand_sample(100) | (to_array() | sink(lambda a : numpy.median(a))
Other aggregates can be calculated using constant memory. In this case, using a DAGPype stage is more efficient than first streaming into a NumPy array, then calculating the aggregate. E.g., finding the mean through an array, like this:
>>> print stream_vals('meteo.csv', 'rain') | (to_array() | sink(lambda a : numpy.mean(a)))
can use much more memory than this version
>>> print stream_vals('meteo.csv', 'rain') | ave()
## High Performance Chunking¶
Modern numeric libraries process data more efficiently in chunks. Even if the original data is logically a sequence of individual elements, we can utilize stages that chunk it, then process these chunks. The size of the chunks depends on the system: they should be large enough to take advantage of the chunk performance of the numerical library, but not so large that they overburden system memory.
See the Performance page for the effect.
E.g., the following code snippet shows how to calculate the correlation between two variables stored in a binary format:
print np.chunk_stream_bytes(_f_name, num_cols = 2) | np.corr()
The first stage streams chunks of data into arrays, in this case of 2 columns. The second stage calculates their correlation. If the file is in CSV format, we can do the following:
np.chunk_stream_vals('meteo.csv', ('day', 'wind')) | np.corr()
The first stage reads the ‘day’ and ‘wind’ columns from the CSV file, and emits tuples of chunks.
A stream of individual elements can be chunked to a stream of NumPy arrays using the dagpype.np.chunk() stage, then processed using other dagpype.np stages:
>> source([1, 2, 3, 4]) | np.chunk() | np.mean()
its complementary stage is dagpype.np.unchunk().
The stages that actively chunk data from a stream in dagpype.np take the required chunk size as a parameter. For example, dagpype.np.chunk_stream_bytes() has the following interface:
def chunk_stream_bytes(stream, max_elems = 8192, dtype = numpy.float64, num_cols = 1):
"""
Reads a binary file containing a numpy.array, and emits a series of chunks. Each chunk
is a numpy array with num_cols columns.
Arguments:
stream -- Either the name of a file or a *binary* stream.
Keyword Arguments:
max_elems -- Number of rows per chunk (last might have less) (default 8192).
dtype -- Underlying element type (default numpy.float64)
num_cols -- Number of columns in the chunks' arrays (default 1).
np.chunk_stream_vals
np.chunks_to_stream_bytes
Example:
>>> # Reads from a binary file, and writes the cumulative average to a different one.
>>> np.chunk_stream_bytes('foo.dat') | np.cum_ave() | np.chunks_to_stream_bytes('wind_ave.dat')
"""
A stream of chunks can be processed by either stages in dagpype.np or dagpype, however, the stages in dagpype.np semantically deal with the elements composing the array, whereas those in dagpype consider the arrays the elements themselves. For example:
>>> source([1, 2, 3, 4]) | np.chunk() | np.count()
4
>>> source([1, 2, 3, 4]) | np.chunk() | count()
1
In the above two examples, dagpype.np.chunk() happened to chunk the 4 elements into a single chunk. The first pipeline counted the total number of elements in the chunks as 4, and the second pipeline counted a single chunk.
Given NumPy’s wealth of ways to manipulate arrays, it is often possible to manipulate a chunked stream by using dagpype‘s dagpype.filt() function with NumPy constructs, instead of writing specialized chunk-aware stages. For example, to calculate the correlation, pruning out values greater than 10 in each of some data, we can use:
np.chunk_stream_bytes(_f_name, num_cols = 2) | \
filt(lambda a : a[logical_and(a[:, 0] < 10, a[:, 1] < 10), :]) | \
np.corr()
and to truncate outliers to 10, we can use:
np.chunk_stream_bytes(_f_name, num_cols = 2) | \
filt(lambda a : where(a, a < 10, a, 10)) | \
np.corr() | 2013-05-25 08:46:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3657238781452179, "perplexity": 6598.255658045184}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705790741/warc/CC-MAIN-20130516120310-00039-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://golem.ph.utexas.edu/forum/posts?page=5 | # Recent Posts
408 posts found
Forum: Instiki – Topic: Windows installation not working
Thanks for the response.. I have installed sqlite3 per your direction and looks like it has passed the sqlite3 portion of the install.. However looks like every gem listed on the gem file i have to install manually including “rake, nokogiri, and itextomml” however im getting errors now with itextomml so that would be a bit to troubleshoot.
Never had so much trouble for something that is supposed to be so simple.. ugh..
Forum: Instiki – Topic: Windows installation not working
The sqlite3 rubygem won’t compile without thelibsqlite3 C-library. (Unlike many rubygems, which are pure Ruby, this one contains a C extension that links to the aforementioned library.)
Windows (unlike other operating systems) doesn’t come with that library installed. Some Ruby installers, for Windows, install it; evidently, some don’t. For those, you’ll have to install it yourself.
If you google around, you’ll find plenty of useful advice on this topic. Unfortunately (since i don’t have any familiarity with Windows), I’m not a useful source for such advice. But, yes, as far as I can tell, installing the Windows SQLite3 package is a prerequisite for getting the sqlite3 rubygem installed.
How does this installation work? What i gather you use “GIT” to download all the packages or i think bundles is the term and then ruby installs them on the system?
Bundler lets you manage/install rubygems, without installing them on the system. Instead, they are installed in your application’s vendor/bundle directory.
But that’s not where (as far as I can tell) your problem lies.
Forum: Instiki – Topic: Windows installation not working
thanks for the reply. I currently have Ruby 1.8.7-p371 installed with the dev kit.
So are you saying i need to download and install that package then attempt to run “ruby bundle install –path vendor/bundle” again?
Im completly new to everything here.. I have aquired this outdated windows server that i had to upgrade to 2008r2. A week ago i just learned what ruby is so you really have to bear with me.
How does this installation work? What i gather you use “GIT” to download all the packages or i think bundles is the term and then ruby installs them on the system? If thats the case, then my problem is when ruby is tries downloading the package from rubygems.org repository as it cant locate sqlite3. Even though i could manually go to the rubygems website and locate sqlite3.
Forum: Instiki – Topic: Windows installation not working
I don’t know what Ruby installation you have for Windows, but presumably, you are missing the SQLite3 precompiled binaries for Windows.
Presumably, the Windows Installation Instructions could be improved.
Forum: Instiki – Topic: Windows installation not working
I have figured out the above error.. however, now i recieve an error from the gemfile when trying to download the bundles:
D:\instiki-0.19.6>ruby bundle install –path vendor/bundle Updating http://github.com/distler/file_signature.git Updating http://github.com/distler/maruku.git Fetching source index for http://rubygems.org/ Could not reach rubygems repository http://rubygems.org/ Could not find gem ‘sqlite3 (>= 0)’ in any of the gem sources listed in your Gemfi
It cant find sqlite3 in the repository however if i try to modify the gemfile in anyway and give sqlite3 a version number or somthing i receive an syntax error. How should I contruct the gemfile and maybe force it to find the specific version thats in rubygems.org
This is the sqlite portion of the gemfile now:
“source “http://rubygems.org”gem “sqlite3”, :require => “sqlite3”
If i follow the syntax from the other bundles ie.. :require => “sqlite3” , “~> 1.3.7”
Forum: Instiki – Topic: instiki without database?
Maybe You’re right, I’m not a programmer or linux specialist. But I’ve minimum to install the sqlite libraries and perhaps other dependent packages, too. And because of older packages, maybe the need of compiling…
Keep it simple :)
antonio
Forum: Instiki – Topic: instiki without database?
I’m not sure why you think the Madeleine Persistence Layer (which, I believe, is what 0.9.2 uses) is lighter-weight than Sqlite3.
You do need to store the data somewhere. And, with Sqlite3,
• there’s no separate database process
• the data is stored in a single file, db/production.db.sqlite3.
Forum: Instiki – Topic: Feature Requests
Thanks.
Forum: Instiki – Topic: instiki without database?
what a pity; well, version 0.9.2 works ok for my usage.
I’m using a small and older linux installation (puppylinux 4.1.2) on a thin client with CF-Card. I tried some perl-wikis without database and QuickiWiki (the original from Ward Cunningham) but it’s a bit outdated :) . Newer perl-wikis have problems with hiawatha (webserver security) and/or maybe perl 5.8.8. So I give ruby-wikis a try and instiki works nice
antonio
Forum: Instiki – Topic: instiki without database?
No.
You need some database. But the default sqlite3 is as lightweight as humanly possible.
Forum: Instiki – Topic: instiki without database?
Dear,
is there a possibility to run the latest release without database? like 0.9.2?
I’m running version 0.9.2 on a small thin client and don’t want to bloat the system.
Thx antonio
Forum: Instiki – Topic: Feature Requests
Forum: Instiki – Topic: Feature Requests
Okay, so that was a pretty dubious feature request!
How about this one: if a page exists (meaning, really exists - not just a redirect) then a request to <web>/new/page should redirect either to <web>/edit/page or to <web>/show/page. If the page does exist then the effect of going to <web>/new/page and submitting stuff is the same as submitting an edit except that you don’t get the previous edit in the text box so there’s nothing to show that you’re replacing something already there. The argument for <web>/show/page being that if a page exists and you didn’t know it then you should probably have a good look at what’s already there before writing something new.
(This came up most recently because a Google search for a page led to the /new/ link even though the page exists - Google had clearly found the link somewhere and added it to its list, it does this even if a robots.txt file exists since the link exists on a page that it can read.)
Forum: Instiki – Topic: Windows installation not working
Im trying to migrate an existing wiki running on a old outdated server onto a new 2008 r2 x64 server. I have installed Ruby 1.9.3 onto the new computer and ran the Ruby bundle install –path vendor/bundle command as stated on the website however it looks like im getting a no such file or directory. I have verified that the source.rb file is located in the correct location.
D:\instiki-0.19.6\instiki-0.19.6>ruby bundle install –path vendor/bundle Fetching http://github.com/distler/file_signature.git D:/instiki-0.19.6/instiki-0.19.6/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/source.rb:57 8:in “’: No such file or directory - git clone “http://github.com/distler/file_signature.git” “D:/i nstiki-0.19.6/instiki-0.19.6/vendor/bundle/ruby/1.9.1/cache/bundler/git/file_signature-9e3f3d6fbf544 b2242ffca379f5f61bb2971e94e” –bare –no-hardlinks (Errno::ENOENT)
Also If there is any documentation or help on migrating the old wiki to the new server would be great.
Forum: Instiki – Topic: problem running instiki after installation
Probably, those instructions are out-of-date. (Help updating them would be appreciated.)
It would be best to ensure that you have a fully-functioning Ruby (1.9.3 is preferable) installation before proceeding with getting Instiki running. Perhaps these instructions might help.
Forum: Instiki – Topic: problem running instiki after installation
http://golem.ph.utexas.edu/wiki/instiki/show/Installation, the part that talked about ubuntu
Forum: Instiki – Topic: problem running instiki after installation
I followed all the instructions on your installation page for ubuntu.
What page are you talking about?
Forum: Instiki – Topic: problem running instiki after installation
Hi all, first time poster. Sorry if this has been discussed before, but I didn’t find anything on a quick search. I am running ubuntu 10.04, I followed all the instructions on your installation page for ubuntu. Everything seemed to install ok, I did all the apt-get and bundle install, but when I tried to run “./instiki –daemon” I get the following errors, fyi, my version of ruby is 1.8.7, patchlevel 249).
/usr/lib/ruby/1.8/rubygems/dependency.rb:52:in initialize': Valid types are [:development, :runtime], not nil (ArgumentError) from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:352:in new' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:352:in search' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:346:in gems_size' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:179:in resolve' from /usr/lib/ruby/1.8/rubygems/source_index.rb:95:in sort_by' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:175:in each' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:175:in sort_by' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:175:in resolve' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:160:in start' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:128:in resolve' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:127:in catch' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/resolver.rb:127:in resolve' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/definition.rb:151:in resolve' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/definition.rb:90:in specs' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/definition.rb:135:in specs_for' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/definition.rb:124:in requested_specs' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/environment.rb:23:in requested_specs' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler/runtime.rb:11:in setup' from /home/riovine/instiki/instiki-svn/vendor/plugins/bundler/gems/bundler-1.0.18/lib/bundler.rb:107:in setup' from ./config/../config/preinitializer.rb:18 from ./config/boot.rb:28:in load' from ./config/boot.rb:28:in preinitialize' from ./config/boot.rb:10:in boot!' from ./config/boot.rb:124 from ./script/server:3:in require' from ./script/server:3 from ./instiki:6:in load' from ./instiki:6
Forum: Instiki – Topic: migration to Rails3.
Here are the steps to follow:
Forum: Instiki – Topic: Feature Requests
It’s been suggested that this is because that is in a div with class name byline.
That seems a pretty thin reed on which to base a request for changing the class names we use.
Forum: Instiki – Topic: Bugs
I am fairly certain that none of my recent updates would affect this scenario..
But I’m happy to hear that it fixed itself.
Forum: Instiki – Topic: Bugs
… and in the time since you asked for clarification, it would appear that you’ve fixed it anyway as it no longer appears having just updated instiki.
Thanks.
Forum: Instiki – Topic: Bugs
Distillation for Distler (sorry, …)
1. Create a page with an apostrophe in the title, say apostrophe's.
2. Edit page.
3. Page magically becomes apostrophe's.
4. Edit page, changing its name back to apostrophe's.
5. Page name is now correct.
6. Edit page.
7. Page name is mangled again.
Forum: Instiki – Topic: Feature Requests
Not sure if this is a bug or a feature request …
Google searches now include author information which it tries to glean from the page. It would appear that it uses the “Revised by XYZ” information to do this. It’s been suggested that this is because that is in a div with class name byline. I’m going to try changing this to see if it stops Google from assuming that to be the author. I don’t yet know how to override Google’s ad hoc method (which really does seem ad hoc if it uses a CSS class name as evidence).
I’ll report back on whether or not it works. If it does, consider this a feature request for changing byline to something like revisedby.
Forum: Instiki – Topic: migration to Rails3.
Hi,
I have it working (mostly), today spend some time fixing
• asset pipeline for rails3
• further clean-up
• big issue running with passenger, rvm etc…
so, the test site is running here:
http://test.netxforge.com => This is the front site, which “presents” the wiki pages. http://test.netxforge.com/netxforge/list => The actual wiki
There are a some pending issues, with URL generator, making some links not working.. etc.. so still a bit of work….
Cheers Christophe
Forum: Instiki – Topic: migration to Rails3.
This is extremely interesting.
Porting Instiki to Rails3 has long been on my TODO list. But (as you’ve seen), it’s not a small job. So it keeps getting pushed back in favour of other things. So I’m really happy you’re working on this!
Forum: Instiki – Topic: migration to Rails3.
Ah, found the problem for the issue, I reported earlier. … (took me the better part of the morning grrrr…). Ok, so WikiContent extends ActiveSupport::SafeBuffer. Now this class changed in Rails 3, overriding (Or opening up in Ruby Parlence) with the following code:
def html_safe ActiveSupport::SafeBuffer.new(self) end
So, this produces a different Class type, hence the mixed-in methods get lost! and causes a problem in PageRenderer trying to call this method. Wow, I am happy I found this one.
the file holding this method is called: output_safety.rb in lib/activesupport/kernel. I couldn’t find the alternative for Rails 2.3, to prove my point. but regardless it now works, by commenting out this line in WikiContenr.render!
# self.html_safe
BTW: The teaser screenshot is now shown, in my previous post.
Forum: Instiki – Topic: migration to Rails3.
Hi, (Jacques, I hope you are reading this!)
I am running into a tricky problem in the Rails3 migration. As you likely know, Rails 3 doesn’t load the /lib directory when starting. There are 2 options to auto-load. 1) In application.rb (which is required by Rails 3) put in a config parameter to load. There are some caveats as the names of the files versus the names of the class. 2) there is now a /config/initializers directory, and .rb is executed here.
What I have done now is:
1) config/application.rb
This will load most files, except InstikiErrors and WikiContent as the naming of the files doesn’t fit the rails loading algorithm. To load these I use:
2) config/instiki_init.rb
require ‘instiki_errors’ require ‘wiki_content’
(I also had to rename the wiki.rb class in /chunks/wiki as rails would complain, the file should declare the Wiki class ) “Expected /Users/Christophe/Documents/Spaces/netxforge_aptana/com.netxforge.store/lib/chunks/wiki.rb to define Wiki
This /lib classes are loaded, however I run into a problem when saving a page. What happens is that somehow the methods in WikiChunck Module are not available .. so it seems the mixin of this module into the WikiContent class is not working well… I know it’s a fuzzy description but I get for example the following error. You see this occurs in the PageRenderer, when calling ‘update_references’ while prior to that the method ‘render(…)’ is called which creates the variable ‘rendering_result’ correctly created. (Also some of the mixin methods should have been called by then). I am really puzzled about this. I even consider rewritting it, so that mixin is not used. (I actually wonder why a mixin solution was chosen here, it’s not code re-use is it?).
Can you help me fix this?
Here is the error:
NoMethodError in WikiController#save
undefined method find_chunks’ for #<ActiveSupport::SafeBuffer:0x007ff4863dc368>
Rails.root: /Users/Christophe/Documents/Spaces/netxforge_aptana/com.netxforge.store Application Trace | Framework Trace | Full Trace
lib/page_renderer.rb:122:in wiki_links' lib/page_renderer.rb:102:in find_wiki_words’ lib/page_renderer.rb:150:in update_references' lib/page_renderer.rb:141:in render’ lib/page_renderer.rb:29:in display_content' app/models/page.rb:30:in revise’ app/models/wiki.rb:79:in revise_page' app/controllers/wiki_controller.rb:325:in save’
Teaser below :-), Instiki on rails3.
Forum: Instiki – Topic: migration to Rails3.
Hi,
I am using Instiki, as a sort of content backend for my website. It’s a base for the website application, which needs additional functionality, which I would like to build on the latest rails 3.x. So I rolled up my sleeves and started to migrate Instiki to Rails 3, which is now almost complete. The approach is pragmatic, fixing things when they break in Rails3. I have recorded all the adaptation, and this could be a check list to ‘replay’.
So here is the thought. I would like to share this experience and help out if there is a interrest to push the original instiki to Rails3. I would however need some help from the original developers.
I can clone the original, but then would like to push it back to the original git, of course in another branch, and when all ok, that could become the new official version… what do you say?
Cheers Christophe
Forum: Instiki – Topic: Bugs
Fixed. Thanks! | 2016-08-25 16:38:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3528488874435425, "perplexity": 6328.7980329649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982293692.32/warc/CC-MAIN-20160823195813-00166-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41467-019-09187-w?error=cookies_not_supported&code=7b03f2ef-f889-4770-bf6f-4e8a8ed94f48 | Article | Open | Published:
# Phase transitions in few-monolayer spin ice films
## Abstract
Vertex models are an important class of statistical mechanical system that admit exact solutions and exotic physics. Applications include water ice, ferro- and antiferro-electrics, spin ice and artificial spin ice. Here we show that it is possible to engineer spin ice films with atomic-layer precision down to the monolayer limit. Specific heat measurements show that these films, which have a fundamentally different symmetry to bulk spin ice, realise systems close to the two-dimensional F-model, with exotic phase transitions on topologically-constrained configurational manifolds. Our results show how spin ice thin films can release the celebrated Pauling entropy of spin ice without an anomaly in the specific heat. They also significantly expand the class of vertex models available to experiment.
## Introduction
Vertex models were first introduced as statistical mechanical models of water ice and hydrogen bonded ferro- and antiferro-electrics1,2, but they rapidly assumed a much broader importance as model many-body systems that admit exact analysis3,4,5,6,7. Their theoretical study8 has led to many insights, including phase transitions on constrained manifolds2,3, continuously variable critical exponents7 and quantum to classical mappings9. Experimental realisations of these theoretical models were slower to follow, but the discovery10,11,12,13 of spin ice and the development of artificial spin ice micromagnetic arrays14,15,16 have significantly enlarged the set of real systems that closely approximate vertex models17,18. Such systems also have a general relevance to the physics of highly correlated spin liquid19 and quantum spin ice20 states, as they provide experimental examples of fractionalised excitations21 and Coulomb phases22.
The defining property of a vertex model is a set of vertex weights, which in experiment is determined by symmetry and tuned by temperature, pressure or applied fields. Spin ice, which consists of a cubic array of corner-linked spin tetrahedra, maps to a 16-vertex model, with the vertices represented by spin configurations on a tetrahedron11. Low temperature (in zero applied field) selects six degenerate vertices, making spin ice a realisation of Pauling’s model of water ice1, with extensive residual entropy12. In fact, the physics of bulk spin ice is even richer than that of a conventional 16-vertex model: the long-range part of the dipole–dipole interaction, which is significant in spin ice, is largely self-screened in the six-vertex spin ice state23,24, but manifests as a magnetic Coulomb interaction between excited vertices to give the fractionalised magnetic monopoles of spin ice21,25. The discovery21,23,24,25 that the addition of the long-ranged and conditionally convergent dipole–dipole interaction to a conventional vertex model resulted in such sharply -defined, exotic and realisable physics was a surprise that has stimulated considerable interest and activity in the field22,26,27,28,29,30,31,32,33.
Recently, the first thin films of spin ice have been prepared34,35. We introduced ultra-thin epitaxial films of the spin ice material Dy2Ti2O7 (DTO) grown on the isomorphic pyrochlore substrate Y2Ti2O7 (YTO)34. This choice of substrate led to a very high degree of epitaxy with compressive strain perpendicular to the growth direction. The strain was found to remain homogenous even at relatively high thicknesses—certainly up to 60 nm34. Spin ice properties were observed at temperatures T > 2 K. However, at lower temperatures (down to 0.35 K) specific heat and magnetisation measurements showed that the Pauling entropy was completely released, although no anomaly was observed in the specific heat34.
In the present work, we have further improved the fabrication of the thin films, such that the films are grown with atomic-layer precision along different crystallographic directions and have negligible interface effects. Owing to the homogenous strain, these films may realistically be regarded as low-dimensional variants of spin ice that have a reduced symmetry compared to that of the bulk. This raises the prospect of engineering and experimentally studying more general microscopic vertex systems than those afforded by bulk spin ice.
## Results
### Growth and characterisation of spin ice thin films
Different schemes for altering vertex weights may be realised by growing spin ice films along different crystallographic directions: here we choose the cubic [100], [110] and [111] directions, respectively. Table 1 reports the samples that were prepared, separated into the three sets DTO||YTO(100), DTO||YTO(110) and DTO||YTO(111); within each set several different layer thicknesses were prepared. Note that, in the bulk cubic structure of pyrochlores (space group $$Fd\bar 3m$$, lattice constant a = 1.013 nm for DTO, 1.0083 nm for YTO), 1 nm corresponds to ca. 1 monolayer (ML).
A layer-by-layer growth mechanism was confirmed by following the presence of a characteristic intensity oscillation in RHEED (reflection high-energy electron diffraction). This mechanism is similar to that we recently reported for the pyrochlore magnet Tb2Ti2O736. Figure 1a is an example of three RHEED patterns that were recorded during the deposition of thin films with different out-of-plane crystallographic directions. In the case of DTO||YTO(110) and DTO||YTO(111), the oscillation period measured at the specular reflection turns out to be compatible with the formation of a quarter of the unit cell along the out-of-plane direction, [110] and [111], respectively: here one RHEED oscillation corresponds to the formation of one tetrahedral layer of the pyrochlore structure. The thickness calculated from this model is in excellent agreement with the X-ray powder diffraction (XRD) analysis described below and reported in Table 1. In contrast, samples of DTO||YTO(100) present a different RHEED pattern with only one big oscillation of period roughly four times bigger than the other two sets; this is compatible with the creation of a complete unit cell. This tendency to grow in layers of unit cells rather than tetrahedra suggests that 〈100〉 faces grow relatively quickly, as implied by the fact (see ref. 13, Fig. 2) that flux-grown crystals show 〈111〉 faces (i.e. when grown in the absence of significant external forces crystals tend to eliminate their fastest growing faces). It also seems consistent with the observation that crystals grown by the floating-zone technique (which encourages a particular growth direction) tend to show 〈110〉 preferential growth axes37,38.
Figure 1b displays X-ray reflectivity curves measured for each sample, as well as their relative fitting, the results of which are summarised in Table 1. The reflectivity analysis was used to estimate the thickness and the density of the film, both of which are expected to be reliable numbers in our analysis. The estimated total thicknesses are in close agreement with the values calculated on the basis of the number of laser pulses used during the growth. Furthermore, the density of the film is close to the tabulated bulk density of DTO (Table 1).
Epitaxy and compressive strain in the films were assessed by measuring the out-of-plane reflections (400), (440) and (444), as shown in Fig. 2. The out-of-plane lattice parameters are all compatible with 1.018(1) and 1.009(1) nm for the films and substrate, respectively. The thickness of each sample was determined from the fringes observed in the XRD pattern, with related values reported in Table 1. Two orthogonal off-specular reflections were measured as illustrated in Fig. 2 (panels b and c) for the thickest sample of each series. These measurements show that the films are very uniform (clear fringes appear in both spectra) and that they have a compressive strain, consistent with the smaller lattice constant of the substrate. In all three film orientations, analysis of the diffraction data indicates that the epitaxial strain is homogenous, the film structure being compressed equally within the in-plane directions, with the average density maintained by elongation along the out-of-plane direction. All our studies have indicated that the distortion does not depend on film thickness.
The epitaxial strain-induced lattice distortion affects the distance between the Dy ions and their oxide environment (hence the magnetic moments, and spin hamiltonian). The distortion is different depending on the chosen out-of-plane orientations. Figure 2d shows a schematic of one Dy tetrahedron and its orientation with respect to the cubic pyrochlore axis. In the cubic phase, all Dy–Dy distances are equal: dTb = 3.5815 Å for a lattice constant of 1.013 nm. In the idealised film structure, not all the Dy–Dy distances are equivalent, as detailed in Table 2 and depicted in Fig. 2d. The angles between the tetrahedron edges are perturbed accordingly.
With homogenous strain, the space group symmetry is reduced to orthorhombic, A1, with a six-fold increase in the number of atoms per unit cell. A complete analysis of the crystal structure would be a challenging project for synchrotron X-ray techniques. However, despite the large unit cell, it is reasonable to expect a single local environment for the Dy3+ ions, and our analysis of the magnetic and thermal properties below makes this approximation.
A commercial Physical Properties Measurement System (PPMS) (see Methods) was used to measure the heat capacity of the DTO thin films. This takes advantage of our previous demonstration34,36,39 that the PPMS can be used to accurately measure the heat capacity of the thin films. To estimate the magnetic specific heat of the film, CM, the measured heat capacity was corrected for the contributions from substrate and phonons using the same method described at length in ref. 34.
The thickest sample of each series, with 44 or 60 ML, that is, 44DTO||YTO(100), 44DTO||YTO(110) and 60DTO||YTO(111), were chosen as a prototype to present the specific heat behaviour of the epitaxial layers. The thinner films are discussed subsequently and also reported in Supplementary Figure 1 and Supplementary Figure 2. The heat capacity was found to scale closely with thickness of the films, as previously reported34. However, it must be noted that certain small features described below (hysteresis) were not as clearly seen, possibly due to the difficulties in quantitatively separating the contribution of the active material (DTO) from that of the substrate (YTO) for such thin layers.
For each series of samples, the magnetic specific heat, CM, divided by temperature is plotted vs. temperature in Fig. 3. A bulk single crystal DTO has also been measured and reported in Fig. 3, for comparison. The behaviour of all thin films is close to that of bulk spin ice at T > 2 K, but starts to deviate from it at lower temperatures, as previously reported34. Furthermore, as shown in Fig. 3d, curves for samples grown along different crystallographic orientation are coincident for T > 0.75 K, whereas at lower temperature each set of samples behaves differently.
For 44DTO||YTO(110) and 60DTO||YTO(111) (Fig. 3b, c, respectively), a small hysteresis loop opens up at low enough temperatures, below T = 0.75 K. During cooling, a small but clear and reproducible discontinuity is observed at T = 0.510(2) K (44DTO||YTO(110)) and T = 0.520(2)K (60DTO||YTO(111)). In both samples, the width of the hysteresis loop does not vary for repetitive cycles. In contrast, for 44DTO||YTO(100), corresponding to Fig. 3a, data measured during cooling and heating do not show any hysteresis or discontinuity. In the Supplementary Figure 1 we show how the present data are fully consistent with that of ref. 23, even though the latter failed to resolve these transitions. The observed hysteresis and orientation-dependent specific heat below the peak in the specific heat suggests an ordering transition. This transition must be of a very unusual sort as the anomaly is extremely weak. The 44 ML (100) film does not show any obvious transition, but as discussed below, a transition is visible in a much thinner film of this orientation.
### Comparison with vertex models
These results may be compared with expectations for vertex models. Theoretical results that are broadly relevant to this situation include exact solutions for two-dimensional (2D) vertex models like the Slater2 and F-models3,5, in which the degeneracy of the ice-like six-vertex manifold is reduced such that a pair of vertices lies lowest in energy, see inset of Fig. 4a. In addition, a recent study40 of the undistorted dipolar spin ice model in slab geometry identified a tendency to the F-model, with the complication of a surface phase transition. Slater and F-type models are of particular interest as they are defined on a topologically constrained configurational manifold and exhibit ordering transitions from a critical phase. In the spin ice context, this is a Coulomb phase with monopole excitations22, and transitions out of this state caused by perturbations may be viewed as Higgs transitions41.
In detail, the cubic spin ice structure is homogeneously compressed within the plane causing differences in bond length on each spin tetrahedron. A film grown along [100] of the parent cubic structure has two shorter in-plane bonds and four longer out-of-plane bonds. In terms of vertex weights, this produces either an F- or a Slater-type model, but the former is the more likely for the following reasons. The near-neighbour coupling in spin ice consists of a dominant ferromagnetic dipolar term and a weaker antiferromagnetic exchange term. The latter will vary with distance much more rapidly than the former; hence, it is reasonable to expect that compressed (in-plane) bonds will be weaker than relaxed (out-of-plane) ones. The result is a splitting of the six-fold degenerate ground state of a spin tetrahedron in the manner of the F-model, rather than the Slater model. Furthermore, the distortions expected for films grown along [110] and [111] (surprisingly) do not break the six-fold degeneracy of spin ice at the level of nearest-neighbour interactions. Hence, we might expect the rough applicability of the results of ref. 40 to this case, with this time an effective F-model modified by dipole interactions.
Given that we expect the 2D F-model to be the basic model of a spin ice thin film at low temperature, it is worth recalling its properties. The basic model admits only six vertices and the high temperatures state tends to the Pauling state. The transition to an ordered phase5,7,42, although ostensibly an order–disorder transition, is formally classified as an unusual representative of the Berezinskii–Kosterlitz–Thouless (BKT) class. The BKT transition, driven by the unbinding of topological defects (spin vortices or effective charges), more normally separates a high-temperature paramagnetic phase, with exponentially decaying correlations, from a low-temperature critical phase, with algebraically decaying correlations43. In the case of the F-model, the high-temperature six-vertex phase is critical, or algebraic, while the low temperature ordered phase has exponential correlations. At low temperature the entropy goes to zero, but the specific heat—which can be exactly calculated5,7—shows no anomaly. This ‘non-scaling’ behaviour has even been recommended as a surprising diagnostic of the transition42. The addition of extra vertices can change the nature of the transition, depending on the energy scales involved44. We note that ‘non-scaling’ and zero entropy are indeed properties of spin ice films34.
In order to probe the relevance of the 2D F-model to the real films, we observe that in the limit of a weak perturbation to the Pauling manifold of the spin ice (16-vertex) model, the partition function of the system may be approximated by unperturbed (dipolar) spin ice at high temperature and by the F-model at low temperature. In this limit the specific heats of the F-model and spin ice model become additive (see Methods and Fig. 5). This approximation is not properly controlled for the actual films, where the perturbation is relatively strong. However, it would be expected to be accurate at low temperature where spin ice excited states are not yet thermally populated and cannot be too inaccurate at higher temperature as it correctly accounts for the total entropy increment (the integral of CM/T—see Methods). Hence, we may assess the relevance of the 2D F-model by subtracting the contribution of the spin ice excited states (the peak measured for bulk DTO) to the magnetic specific heat (divided by temperature) of the spin ice films and plotting the residual, as in Fig. 4.
Figure 4a compares the experimental result for films of varying thickness with the exact result for the 2D F-model with splitting parameter ε = 0.55 K (see Methods for the formula). It is seen that the 4 and 8 ML films closely approximate the 2D F-model prediction, especially in the low-temperature side of the peak in the specific heat, while the thicker (44 ML) film is slightly less well described (we believe that the slight vertical offset between the theory and experiment in the 44 ML data set may be attributed to small amounts of impurity in the substrate which give a finite contribution to the entropy). Figure 4b shows the effect of varying the splitting parameter ε: all the data are captured with the parameter in the narrow temperature range between 0.5 and 0.6 K; hence, we may quote ε = 0.55 ± 0.05 K as an estimate of the splitting of the Pauling manifold. The small variation in estimated splitting could reflect our approximations or it could have a physical cause, such as a variation of strain, but we cannot distinguish these possibilities at present.
Both panels of Fig. 4 confirm our expectation that the 2D F-model describes the low temperature properties of the spin ice thin films to a close approximation. They suggest that a single-monolayer film (of tetrahedra) would indeed realise the 2D F-model. Our results show that such a monolayer film is available to experiment, although to measure it will be a challenge as thermodynamic probes will need to be extremely sensitive. More generally, Fig. 4 explains the puzzle implied by ref. 34: how the films can release the spin ice entropy without an anomaly in the specific heat. It now seems clear that the mechanism is related to that of the 2D F-model, which orders via an unusual BKT transition (in Fig. 4 this is indicated by a slight kink in the theoretical curve near 0.8 K, which is an artefact of truncating a summation: see Methods).
In the thinnest (4 ML) (100) film, a very striking jump in the specific heat is observed near to the lowest temperature that we can access (a similar jump is observed for the (111) film—see Supplementary Figure 2). We believe that this is a real effect, although given its unusual nature, we interpret it only with caution. Given that this feature is not observed clearly in the thicker films, and given that it has only a very small effect on the observed entropy (see Supplementary Figure 3), we might speculate that it signifies a surface phase transition; indeed, the very weak anomalies observed in some of the thicker films may also be in this class. The simple F-model envisaged by the stacking of identical spin tetrahedra puts a ferromagnetic moment perpendicular to the film direction that is likely to be removed by the re-ordering of surface charge to minimise magnetostatic energy. A tangible example of this interesting physics has been given in ref. 40, but we cannot compare this directly with the experiment as the model analysed in ref. 40 has a different effective symmetry (quasi-cubic) to the epitaxial spin ice films discussed here (orthorhombic, A1). Nevertheless, our results are not inconsistent with the general implications of ref. 40.
In conclusion, thin film epitaxy allows the possibility of realising theoretical models of spin ice in confined dimensionality40,45, at interfaces46,47 and in reduced symmetry. One route to realising the exotic physics of low symmetry vertex models is to apply very high pressure to bulk spin ice48, but in practical terms, the very large homogenous distortions offered by epitaxial thin films offer a much more promising route. The films have the experimental advantage of being mechanically stable, which allows for diverse experimental probes, and given that their growth can be monitored layer by layer, they are less susceptible to uncontrolled defects and disorder than are bulk pyrochlore materials33. Our results show that it is possible to prepare single crystal spin ice films right down to the 2D limit of an ML of tetrahedra. The development of experimental methods to probe this limit would be of great interest. Similarly, we have presented evidence to suggests the F-model is the minimal theoretical model of the spin ice films and should be the starting point for any theoretical description. It would be of great interest to study such a model in the slab geometry of the real films and with perturbations appropriate to experiment.
## Methods
### Single crystal growth for substrates
Single crystals of YTO were grown using optical floating-zone technique26. First stoichiometric powder sample was prepared using high-purity Y2O3 and TiO2 chemicals, sintered at 1200 °C for 48 h with intermediate grinding. A cylindrical rod of diameter 12 mm was placed in an optical floating-zone furnace and grown into a single crystal at a rate of 3.5 mm h−1 in Ar/O2 mixed gas flow atmosphere.
### Pulsed laser deposition
Single crystals of Y2Ti2O7 were cut and epi-polished on one side (SurfaceNet GmbH, http://www.surfacenet.de). Three sets of fully oriented YTO substrates were prepared: YTO110K111, YTO111K110 and YTO100K010 where the first crystallographic direction (hkl) is the out-of-plane orientation and the second (Khkl) identify one of the in-plane edges of the square substrate. Epitaxial DTO thin films (between 65 and 5 nm in thickness) were grown on YTO substrates by pulsed laser epitaxy (KrF, λ = 248 nm) at 750 °C in 113 mTorr O2. The laser fluence at the target was fixed at 1.97 J cm−2; the laser repetition rate was initially set at 1 Hz for the first 200 shots and increase to 5 Hz for the remaining growth. Samples were subsequently post-annealed for 1 h at 750 °C in 400 Torr O2 before cooling down to room temperature. RHEED was used to monitor the surface structure and to control the film thickness with atomic-layer precision.
### Structural characterisation
Lattice parameters and film epitaxy were studied at room temperature by XRD using Cu Kα1 radiation in a Rigaku high-resolution diffractometer. Film thickness was determined by X-ray reflectivity; the fits were performed with the Integrated Thin Film Analysis Software GlobalFit 1.3 (Rigaku Corporation). High-resolution reciprocal space maps were collected using the same machine.
### Specific heat
The specific heat of each sample was measured from 50 to 0.4 K with a Quantum Design PPMS (Physical Properties Measurement System) with a 3He option. The data were corrected for the addenda (sample holder, attaching grease) heat capacity, which was measured in a separate run. Furthermore, each sample was measured during both cooling and warming to check if any hysteretic behaviour could be observed.
### 2D F-model
The specific heat per mole of Dy is given by:
$$C_{\mathrm{M}} = \frac{{\mathrm{RT}}}{2}\left\{ {\frac{{ - \partial ^2G_{{\mathrm{low}}}}}{{\partial T^2}}\left[ {1 - \theta \left( {T - \frac{\varepsilon }{{{\mathrm{log}}(2)}}} \right)} \right] + \frac{{ - \partial ^2G_{{\mathrm{high}}}}}{{\partial T^2}}\theta \left( {T - \frac{\varepsilon }{{{\mathrm{log}}(2)}}} \right)} \right\},$$
(1)
where θ is the Heaviside theta, ε is the splitting parameter and G’s are Gibbs free energy contributions given by:
$$G_{{\mathrm{low}}} = - {\mathrm{RT}}\left[ {\frac{\lambda }{2} + \mathop {\sum }\limits_{m = 1}^\infty \frac{{{\mathrm{exp}}\left( { - \lambda m} \right){\mathrm{tanh}}\left( {\lambda m} \right)}}{m}} \right],$$
(2)
$$G_{{\mathrm{high}}} = \frac{{ - {\mathrm{RT}}}}{{4\mu }}{\kern 1pt} \mathop {\int }\nolimits_{\!\!\!0}^\infty {\kern 1pt} \frac{1}{{{\mathrm{cosh}}\left( {\frac{{\pi x}}{{2\mu }}} \right)}}\frac{{{\mathrm{cosh}}(x) - {\mathrm{cosh}}(2\mu )}}{{{\mathrm{cosh}}(x) - 1}}{\mathrm{d}}x,$$
(3)
where
$$\lambda = {\mathrm{arccosh}}\left( {\frac{1}{2}{\mathrm{exp}}\left( {\frac{{2\varepsilon }}{T}} \right) - 1} \right),$$
(4)
$$\mu = {\mathrm{arccos}}\left( {\frac{1}{2}{\mathrm{exp}}\left( {\frac{{2\varepsilon }}{T}} \right) - 1} \right).$$
(5)
These expressions were evaluated numerically to give the different curves in Fig. 4.
### Subtraction procedure for specific heat
The system approximates an F-model (2 + 4 vertex) at low temperature and a spin ice (6 + 8 + 2-vertex) model at high temperature. We argue that the specific heat divided by temperature is approximately the sum of these contributions. The accuracy of this approximation is perhaps best demonstrated by means of an example. We choose a single tetrahedron model of spin ice with energy levels at 0 K (6 levels), 4 K (8 levels) and 16 K (2 levels). The 6-fold degenerate ground term is then split such that two states are at 1 K lower than the other four. The partition function is (with energies measured in kelvin).
$$Q_0 = 2{\mathrm{exp}}\left( {\frac{1}{T}} \right) + 4 + 8{\mathrm{exp}}\left( {\frac{4}{T}} \right) + 2{\mathrm{exp}}\left( { - \frac{{16}}{T}} \right).$$
(6)
We also define the partition function of the unperturbed spin ice-like model,
$$Q_{{\mathrm{SI}}} = 6 + 8{\mathrm{exp}}\left( {\frac{4}{T}} \right) + 2{\mathrm{exp}}\left( { - \frac{{16}}{T}} \right)$$
(7)
and the partition function of an F-like model,
$$Q_{\mathrm{F}} = 2{\mathrm{exp}}\left( {\frac{1}{T}} \right) + 4$$
(8)
From these we compute the corresponding specific heats $$(c/T)_0$$, $$(c/T)_{{\mathrm{SI}}}$$ and $$(c/T)_{\mathrm{F}}$$.
In Fig. 5 we show how the sum $$(c/T)_{{\mathrm{SI}}} + (c/T)_{\mathrm{F}}$$ compares to the exact $$(c/T)_0$$ and how the difference $$(c/T)_0 - (c/T)_{{\mathrm{SI}}}$$ compares to the exact $$(c/T)_{\mathrm{F}}$$. It can be seen that the differences are always small. This roughly reproduces the experimental procedure which can be seen to be trustworthy for the values of the energy splitting we observe.
## Data availability
The datasets generated and/or analysed during the current study are available in the main text, the Supplementary information and from the co-corresponding author L.B. on reasonable request.
Journal peer review information: Nature Communications thanks Claudio Castelnovo, Hans Hilgenkamp and the other anonymous reviewer for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## References
1. 1.
Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935).
2. 2.
Slater, J. C. Theory of the transition in KH2PO4. J. Chem. Phys. 9, 16–33 (1941).
3. 3.
Rys, F. Über ein zweidimensionales klassisches Konfigurationsmodell. Helv. Phys. Acta 36, 537–559 (1963).
4. 4.
Lieb, E. H. Residual entropy of square ice. Phys. Rev. 162, 162–172 (1967).
5. 5.
Lieb, E. H. Exact solution of the F model of an antiferroelectric. Phys. Rev. Lett. 18, 1046–1048 (1967).
6. 6.
Fan, C. & Wu, F. Y. General lattice model of phase transitions. Phys. Rev. B 2, 723–733 (1970).
7. 7.
Baxter, R. J. Exactly Solved Models in Statistical Mechanics (Academic Press, New York, 1982).
8. 8.
Baxter, R. J. Eight-vertex model in lattice statistics. Phys. Rev. Lett. 26, 832–833 (1971).
9. 9.
Baxter, R. J. One-dimensional anisotropic Heisenberg chain. Ann. Phys. 70, 323–337 (1972).
10. 10.
Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K.W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7. Phys. Rev. Lett. 79, 2554–2557 (1997).
11. 11.
Bramwell, S. T. & Harris, M. J. Frustration in Ising-type spin models on the pyrochlore lattice. J. Phys. Condens. Matter 10, L215–L220 (1998).
12. 12.
Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zero-point entropy in ‘spin ice’. Nature 399, 333–335 (1999).
13. 13.
Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore. Mater. Sci. 294, 1495–1501 (2001).
14. 14.
Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006).
15. 15.
Perrin, Y., Canals, B. & Rougemaille, N. Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice. Nature 540, 410–413 (2016).
16. 16.
Östman, E. et al. Interaction modifiers in artificial spin ices. Nat. Phys. 14, 375–379 (2018).
17. 17.
Bramwell, S. T., Holdsworth, P. C. W., Gingras, M. J. P. in Frustrated spin systems (ed. Diep, H. T.) Ch. 7 (World Scientific, Singapore, 2004).
18. 18.
Cugliandolo, L. F. Artificial spin-ice and vertex models. J. Stat. Phys. 167, 499–514 (2017).
19. 19.
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
20. 20.
Benton, O., Sikora, O. & Shannon, N. Classical and quantum theories of proton disorder in hexagonal water ice. Phys. Rev. B 93, 125143 (2016).
21. 21.
Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).
22. 22.
Fennell, T. et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7. Science 326, 415–417 (2009).
23. 23.
Melko, R. G., Gingras, M. J. P. & M. J. P. Monte Carlo studies of the dipolar spin ice model. J. Phys. Condens. Matter 16, R1277–R1319 (2004).
24. 24.
Isakov, S. V., Moessner, R. & Sondhi, S. L. Why spin ice obeys the ice rules. Phys. Rev. Lett. 95, 217201 (2005).
25. 25.
Ryzhkin, I. A. Magnetic relaxation in rare-earth pyrochlores. J. Exp. Theor. Phys. 101, 481–486 (2005).
26. 26.
Jaubert, L. D. C. & Holdsworth, P. D. W. Signature of magnetic monopole and Dirac string dynamics in spin ice. Nat. Phys. 5, 258–261 (2009).
27. 27.
Morris, D. J. P. et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7. Science 326, 411–414 (2009).
28. 28.
Kadowaki, H. et al. Observation of magnetic monopoles in spin ice. J. Phys. Soc. Jpn. 78, 103706 (2009).
29. 29.
Revell, H. M. et al. Evidence of impurity and boundary effects on magnetic monopole dynamics in spin ice. Nat. Phys. 9, 34–37 (2013).
30. 30.
Kaiser, V., Bramwell, S. T., Holdsworth, P. C. W. & Moessner, R. ac Wien effect in spin ice, manifest in nonlinear, nonequilibrium susceptibility. Phys. Rev. Lett. 115, 037201 (2015).
31. 31.
Paulsen, C. et al. Experimental signature of the attractive Coulomb force between positive and negative magnetic monopoles in spin ice. Nat. Phys. 12, 661–666 (2016).
32. 32.
Kaiser, V. et al. Emergent electrochemistry in spin ice: Debye–Hückel theory and beyond. Phys. Rev. B 98, 144413 (2018).
33. 33.
Sala, G. et al. Vacancy defects and monopole dynamics in oxygen-deficient pyrochlores. Nat. Mater. 13, 488–493 (2014).
34. 34.
Bovo, L. et al. Restoration of The Third Law in spin ice thin films. Nat. Commun. 5, 3439 (2014).
35. 35.
Leusink, D. P. et al. Thin films of the spin ice compound Ho2Ti2O7. APL Mater. 2, 032101 (2014).
36. 36.
Bovo, L., Rouleau, C. M., Prabhakaran, D. & Bramwell, S. T. Layer-by-layer epitaxial thin films of the pyrochlore Tb2Ti2O7. Nanotechnology 28, 055708 (2017).
37. 37.
Prabhakaran, D. & Boothroyd, A. T. Crystal growth of spin-ice pyrochlores by the floating-zone method. J. Cryst. Growth 318, 1053–1056 (2011).
38. 38.
Balakrishnan, G., Petrenko, O. A., Lees, M. R. & Paul, D. Mc. K. Single crystal growth of rare earth titanate pyrochlores. J. Phys. Condens. Matter 10, L723–L725 (1998).
39. 39.
Bovo, L. & Bramwell, S. T. Determination of the entropy via measurement of the magnetization: application to the spin ice Dy2Ti2O7. J. Phys. Condens. Matter 25, 356003 (2013).
40. 40.
Jaubert, L. D. C., Lin, T., Opel, T. S., Holdsworth, P. C. W. & Gingras, M. J. P. Spin ice thin film: surface ordering, emergent square ice, and strain effects. Phys. Rev. Lett. 118, 207206 (2017).
41. 41.
Powell, S. Higgs transitions of spin ice. Phys. Rev. B 84, 094437 (2011).
42. 42.
Weigel, M. & Janke, W. The square-lattice F model revisited: a loop-cluster update scaling study. J. Phys. A 38, 7067–7092 (2005).
43. 43.
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973).
44. 44.
Wu, F. Y. Critical behavior of two-dimensional hydrogen-bonded antiferroelectrics. Phys. Rev. Lett. 22, 1174–1176 (1969).
45. 45.
Lantagne-Hurtubise, É., Rau, J. G. & Gingras, M. J. P. Spin-ice thin films: large-N theory and Monte Carlo simulations. Phys. Rev. X 8, 021053 (2018).
46. 46.
She, J.-H., Kim, C. H., Fennie, C. J., Lawler, M. J. & Kim, E.-A. Topological superconductivity in metal/quantum-spin-ice heterostructures. npj Quantum Mater. 2, 64 (2017).
47. 47.
Sasaki, T., Imai, E. & Kanazawa, I. Witten effect and fractional electric charge on the domain wall between topological Insulators and spin ice compounds. J. Phys. Conf. Ser. 568, 052029 (2014).
48. 48.
Jaubert, L. D. C., Chalker, J. T., Holdsworth, P. C. W. & Moessner, R. Spin ice under pressure: symmetry enhancement and infinite order multicriticality. Phys. Rev. Lett. 105, 087201 (2010).
## Acknowledgements
We would like to thank R. Thorogate for technical assistance. We also thank M.J.P. Gingras and H. Kurebayashi for a related collaboration. Pulsed laser deposition of the thin films was conducted by L.B. and with assistance from CMR at the Centre for Nanophase Materials Sciences, which is a DOE Office of Science User Facility (CNMS2015-251). L.B. was supported by The Leverhulme Trust through the Early Career Fellowship programme (ECF2014-284). D.P. acknowledges support from the EPSRC grant EP/K028960/1. L.B. and S.T.B. acknowledge additional support from Leverhulme Trust grant RPG-2016-391.
## Author information
L.B. and S.T.B. conceived the project. D.P. prepared the target for PLD deposition and grew the crystals used to obtain substrates. L.B. grew the films with assistance from C.M.R. L.B. performed all the experiments and analysed the data. S.T.B. conceived the theoretical framework and calculated the 2D F-model. L.B. and S.T.B. wrote the paper and incorporated suggestions from the co-authors.
### Competing interests
The authors declare no competing interests.
Correspondence to L. Bovo or S. T. Bramwell. | 2019-06-16 11:42:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7147706151008606, "perplexity": 2240.2146559296366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998100.52/warc/CC-MAIN-20190616102719-20190616124719-00354.warc.gz"} |
https://quant.stackexchange.com/tags/heston/new | # Tag Info
4
The numerical approximation of the call option price in the Heston model is notoriously unstable and can easily lead to imprecise answers for extreme parameter. Several different formulas exist for computing the price with some being more stable than others. The formula you are using is arguably one of the worst ones. The most precise algorithm I know of is ...
1
The Feller condition is not verified in your case: $2\kappa\theta>\xi ^ {2}$ If this condition is not verified, you can get negative variance as explained in this wikipedia article: https://en.wikipedia.org/wiki/Heston_model
5
Let \begin{align*} \mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\ \mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t}, \end{align*} where $\mathrm{d}B_{S,t}\mathrm{d}B_{v,t}=\rho\mathrm{d}t$. The market price of risk (or Girsanov kernel or Sharpe ratio) is ${\varphi}_t=\left(\frac{\mu-r}{\sqrt{v_t}},\... 2 Under Heston LSV (HLSV) dynamics, Gatheral's equality is: $$\sigma_{LV}^{HLSV}(S_t,t) = \sqrt{E^{HSLV}\left[V_tL(S_t,t)^2 | S_t \right]} = L(S_t,t)\sqrt{E^{HSLV}\left[V_t | S_t \right]},$$ as$L(S_t,t)$is$\sigma(S_t)$-measurable, where superscript$HSLV$is meant to remind us what is our dynamics we started with (in particular the joint probability ... 0 First, couple of corrections (I am not sure, just guessing):$X_T$- is it strike or price of forward underlying? Let it be strike,$X$and the underlying is$S_t$with forward:$Fwd_T=S_t/D_T$. Breeden and Litzenberger formula: No, B&L formula is this:$PDF(S_T)=D_T\cdot\frac{d^2C(X)}{dX^2}$, where$D_T$is discount factor. Finally, my recipe to ... 3 The replicate function works best when you fully define your discretization scheme within a function. Then you can simply replicate the function-call x amount of times. Also, try and keep code duplication to a minimum and improve your general syntax. This will help you and your peers that might need to review and/or change your code in the future. ... 8 Bad news: Your calculation is not quite correct As you say, the initial price of a European call option is $$C(S_0;K,T)= S_0e^{-qT}\Pi_1-Ke^{-rT}\Pi_2. \tag{\star}$$ However, the exercise probabilities$\Pi_1$and$\Pi_2$depend on the stock price$S_0\$ too! Thus, you need the product rule and the chain rule to differentiate the option price with respect ...
Top 50 recent answers are included | 2021-04-12 01:53:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9928703904151917, "perplexity": 969.2019518211113}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038065903.7/warc/CC-MAIN-20210411233715-20210412023715-00184.warc.gz"} |
https://www.numerade.com/books/chapter/differentiation-rules/?section=17 | 💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
Educators
AM
+ 53 more educators
Problem 1
Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \sqrt[3]{1 + 4x}$
Heather Z.
Problem 2
Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = (2x^3 + 5)^4$
Heather Z.
Problem 3
Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \tan \pi x$
Heather Z.
Problem 4
Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \sin(\cot x)$
Heather Z.
Problem 5
Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = e^{\sqrt{x}}$
Heather Z.
Problem 6
Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \sqrt{2 - e^x}$
Heather Z.
Problem 7
Find the derivative of the function.
$F(x) = (5x^6 + 2x^3)^4$
dd
Deepak D.
Problem 8
Find the derivative of the function.
$F(x) = (1 + x + x^2)^{99}$
PC
Partha Sarathi C.
Problem 9
Find the derivative of the function.
$f(x) = \sqrt{5x + 1}$
Heather Z.
Problem 10
Find the derivative of the function.
$f(x) = \frac {1}{\sqrt [3]{x^2 - 1}}$
Heather Z.
Problem 11
Find the derivative of the function.
$f(\theta) = \cos (\theta^2)$
Heather Z.
Problem 12
Find the derivative of the function.
$g(\theta) = \cos^2 \theta$
Heather Z.
Problem 13
Find the derivative of the function.
$y = x^2 e^{-3x}$
Heather Z.
Problem 14
Find the derivative of the function.
$f(t) = t \sin \pi t$
Heather Z.
Problem 15
Find the derivative of the function.
$f(t) = e^{at} \sin bt$
Heather Z.
Problem 16
Find the derivative of the function.
$g(x) = e^{x^2 - x}$
Heather Z.
Problem 17
Find the derivative of the function.
$f(x) = (2x - 3)^4 (x^2 + x + 1)^5$
Heather Z.
Problem 18
Find the derivative of the function.
$g(x) = (x^2 + 1)^3 (x^2 + 2)^6$
Heather Z.
Problem 19
Find the derivative of the function.
$h(t) = (t +1)^{2/3} (2t^2 - 1)^3$
Heather Z.
Problem 20
Find the derivative of the function.
$F(t) = (3t - 1)^4 (2t + 1)^{-3}$
Heather Z.
Problem 21
Find the derivative of the function.
$y = \sqrt \frac {x}{x + 1}$
Heather Z.
Problem 22
Find the derivative of the function.
$y = (x + \frac {1}{x})^5$
Heather Z.
Problem 23
Find the derivative of the function.
$y = e^{\tan \theta}$
Heather Z.
Problem 24
Find the derivative of the function.
$f(t) = 2^{t^3}$
Heather Z.
Problem 25
Find the derivative of the function.
$g(u) = ( \frac {u^3 - 1}{u^3 +1})^8$
Heather Z.
Problem 26
Find the derivative of the function.
$s(t) = \sqrt \frac {1 + \sin t}{1 + \cos t}$
Heather Z.
Problem 27
Find the derivative of the function.
$r(t) = 10^{2 \sqrt {t}}$
AK
Ayush K.
Problem 28
Find the derivative of the function.
$f(z) = e^{z/(z - 1)}$
Heather Z.
Problem 29
Find the derivative of the function.
$H(r) = \frac {(r^2 - 1)^3}{(2r + 1)^5}$
Heather Z.
Problem 30
Find the derivative of the function.
$J(\theta) = \tan^2 (n \theta)$
Heather Z.
Problem 31
Find the derivative of the function.
$F(t) = e^{t \sin 2t}$
Heather Z.
Problem 32
Find the derivative of the function.
$F(t) = \frac {t^2}{\sqrt {t^3 + 1}}$
Heather Z.
Problem 33
Find the derivative of the function.
$G(x) = 4^{C/x}$
Heather Z.
Problem 34
Find the derivative of the function.
$U(y) = (\frac {y^4 + 1}{y^2 + 1})^5$
Heather Z.
Problem 35
Find the derivative of the function.
$y = \cos (\frac {1 - e^{2x}}{1 + e^{2x}})$
Heather Z.
Problem 36
Find the derivative of the function.
$y = x^2 e^{-1/x}$
Heather Z.
Problem 37
Find the derivative of the function.
$y = \cot^2 (\sin \theta)$
Heather Z.
Problem 38
Find the derivative of the function.
$y = \sqrt {1 + xe^{-2x}}$
Heather Z.
Problem 39
Find the derivative of the function.
$f(t) = \tan (\sec(\cos t))$
Heather Z.
Problem 40
Find the derivative of the function.
$y = e^{\sin 2x} + \sin (e^{2x})$
Heather Z.
Problem 41
Find the derivative of the function.
$f(t) = \sin^2 (e^{\sin^2 t})$
Heather Z.
Problem 42
Find the derivative of the function.
$y = \sqrt {x + \sqrt {x + \sqrt {x}}}$
Heather Z.
Problem 43
Find the derivative of the function.
$g(x) = (2ra^{rx} + n)^P$
Heather Z.
Problem 44
Find the derivative of the function.
$y = 2^{3^{4^{x}}}$
Heather Z.
Problem 45
Find the derivative of the function.
$y = \cos \sqrt {\sin (\tan \pi x)}$
Heather Z.
Problem 46
Find the derivative of the function.
$y = [x + (x + \sin^2 x)^3]^4$
Heather Z.
Problem 47
Find $y^{\prime}$ and $y^{\prime \prime}$
$$y=\cos (\sin 3 \theta)$$
Heather Z.
Problem 48
Find $y'$ and $y".$
$y = \frac {1}{(1 + \tan x)^2}$
Frank L.
Problem 49
Find $y^{\prime}$ and $y^{\prime \prime}$
$$y=\sqrt{1-\sec t}$$
Frank L.
Problem 50
Find $y'$ and $y".$
$y = e^{e^x}$
Heather Z.
Problem 51
Find an equation of the tangent line to the curve at the given point.
$y = 2^x, (0, 1)$
Heather Z.
Problem 52
Find an equation of the tangent line to the curve at the given point.
$y = \sqrt {1 + x^3}, (2, 3)$
Heather Z.
Problem 53
Find an equation of the tangent line to the curve at the given point.
$y = \sin (\sin x), (\pi, 0)$
Heather Z.
Problem 54
Find an equation of the tangent line to the curve at the given point.
$y = xe^{-x^2}, (0, 0)$
Heather Z.
Problem 55
(a) Find an equation of the tangent line to the curve $y = 2/(1 + e^{-x})$ at the point (0, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
Heather Z.
Problem 56
(a) The curve $y = \mid x \mid /\sqrt {2 - x^2}$ is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
Heather Z.
Problem 57
(a) If $f(x) = x \sqrt {2 - x^2},$ find $f'(x).$
(b) Check to see that your answer to part (a) is reasonable by comparing the graph of $f$ and $f'$.
Heather Z.
Problem 58
The function $f(x) = \sin (x + \sin 2x), 0 \le x \le \pi,$ arises in applications to frequency modulation (FM) synthesis.
(a) Use a graph of $f$ produced by a calculator lo make a rough sketch of the graph of $f'.$
(b) Calculate $f'(x)$ and use this expression, with a calculator, to graph $f'.$ Compare with your sketch in part (a).
Heather Z.
Problem 59
Find all points on the graph of the function $f(x) = 2 \sin x + \sin^2 x$ at which the tangent line is horizontal.
Heather Z.
Problem 60
At what point on the curve $y = \sqrt {1 + 2x}$ is the tangent line perpendicular to the line $6x + 2y = 1?$
Heather Z.
Problem 61
If $F(x) = f(g(x)),$ where $f(-2) = 8, f'(-2) =4, f'(5) = 3, g(5) = -2,$ and $g'(5) = 6,$ find $F'(5).$
Heather Z.
Problem 62
If $h(x) = \sqrt {4 + 3f(x)},$ where $f(1) = 7$ and $f'(1) = 4,$ find $h'(1).$
Heather Z.
Problem 63
A table of values for $f, g, f' ,$ and $g'$ is given.
(a) If $h(x) = f(g(x)),$ find $h'(1).$
(b) If $H(x) = g(g(x)),$ find $H(1).$
Heather Z.
Problem 64
Let $f$ and $g$ be the function in Exercise 63.
(a) If $F(x) = f(f(x)),$ find $F'(2).$
(b) If $G(x) = g(g(x)),$ find $G'(3).$
Heather Z.
Problem 65
If $f$ and $g$ are the functions whose graphs are shown, let $u(x) = f(g(x)), v(x) = g(f(x)),$ and $w(x) = g(g(x)).$ Find each derivative, if it exists. If it does not exist, explain why.
(a) $u'(1)$
(b) $v'(1)$
(c) $w'(1)$
Carson M.
Problem 66
If $f$ is the function whose graph is shown, let $h(x) = f(f(x))$ and $g(x) = f(x^2).$ Use the graph of $f$ to estimate the value of each derivative,
(a) $h'(2)$
(b) $g'(2)$
Heather Z. | 2021-10-21 10:45:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7562199234962463, "perplexity": 1995.7438863328823}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585405.74/warc/CC-MAIN-20211021102435-20211021132435-00384.warc.gz"} |
http://mathhelpforum.com/calculus/20609-describing-path-steepest-ascent-w-parametric-equations.html | ## Describing the path of steepest ascent w/ parametric equations
I have a function f(x,y) = y / x^2 , starting at the pt. (4,0) where z = 0.
I want to find parametric equations that describe the path of steepest ascent to a point on the surface z = 3.
I have no problem sketching the curve and finding the gradients at each level curve, but I can't figure out how to turn that info into parametric equations.
Thanks for any help | 2015-09-03 22:34:21 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8526814579963684, "perplexity": 169.80487104715115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645328641.76/warc/CC-MAIN-20150827031528-00092-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://mathhelpboards.com/threads/sturm-liouville-operator.709/ | # [SOLVED]Sturm-Liouville operator
#### Poirot
##### Banned
What does it mean for Sturm-Liouville operator to be symmetric w.r.t an inner product?
I was reading in a book that it is symmetric but that was about a certain integral being zero and inner products had not even been mentioned.
#### Ackbach
##### Indicium Physicus
Staff member
You have to generalize the concept of "inner product". In normal 3D-space, it's called the dot product:
$$\langle \vec{x}|\vec{y}\rangle=\vec{x}\cdot\vec{y}=\sum_{j=1}^{3}x_{j}y_{j}.$$
But there are other "spaces" out there: metric spaces, normed spaces, inner product spaces, Banach spaces, Hilbert spaces, Sobolev spaces. They each have different axioms with which you start. The inner product space is fairly general, and the "vectors" can be the usual vectors in 3D space, or they could be functions in a function space. The usual inner product defined in a function space is
$$\langle f|g\rangle:=\int_{A} \overline{f} \,g\,d\mu.$$
Here $\overline{f}$ indicates the complex conjugate of $f$, and the $d\mu$ indicates that we've defined this integral to be a Lebesgue integral. $A$ is the set over which the function space is defined.
Now we have the background to answer your question. A Sturm-Liouville operator
$$L=\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\,\frac{d}{dx}\right]+q(x)\right)$$
is symmetric (more properly, Hermitian) w.r.t. the inner product
$$\langle f|g\rangle:=\int_{A}\overline{f}\,g\,w(x)\,d\mu,$$
if and only if for every $f, g$ in the inner product space, it is true that
$$\langle Lf|g\rangle=\langle f|Lg\rangle.$$
With the Sturm-Liouville operator, you need to show that
$$\int_{A}\overline{\left\{\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\,\frac{df}{dx}\right]+q(x)f\right)\right\}}\,g\,w(x)\,d\mu=\int_{A} \overline{f} \left\{\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\,\frac{dg}{dx}\right]+q(x)g\right)\right\}\,w(x)\,d\mu.$$
You can do this using simple integration by parts twice. The boundary terms vanish because of the conditions on them. (Note that the boundary conditions are considered to be part of the operator.)
I note you marked this thread as solved. That is good. Perhaps this post will throw in a few helpful concepts.
#### Poirot
##### Banned
Sorry you took your time. I should perhaps do a bit more reasearch before asking here.
#### Ackbach
##### Indicium Physicus
Staff member
Sorry you took your time. I should perhaps do a bit more reasearch before asking here.
No, that's all right. Incidentally, there are some concepts here that might help you with your other problem. Take a look at how $w(x)$ appears in the Sturm-Liouville operator, as well as how it shows up in the inner product w.r.t. which the Sturm-Liouville operator is symmetric. | 2021-03-05 13:21:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8892739415168762, "perplexity": 390.9417133752302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178372367.74/warc/CC-MAIN-20210305122143-20210305152143-00225.warc.gz"} |
https://blender.stackexchange.com/questions/909/how-can-i-set-and-get-the-vertex-color-property/911 | # How can I set and get the vertex color property?
I can't find a way to display RGB values (or equivalent) without using the python shell.
I would like to get and set color values numerically (ie. not painting), I can't find it for weight painting either (I expected it to appear in the "n" panel with something like "vertex" as a title).
• Are you asking how to get the RGB of a color in weight paint mode? – CharlesL Jun 11 '13 at 19:44
• Sorry for the late answer ! Yes, I needed a way to edit or read the color's data of a vertex, thanks for the nice answer @zeffii , but I'm looking for something in the UI, without Python. At the UI level why is it at face level ? – user1759333 Mar 29 '16 at 21:43
• i'll edit my answer. – zeffii Mar 30 '16 at 9:10
I would like to get and set color values numerically (ie. not painting),
There may be add-ons that cater to this, but at present (March 2016)
• Blender doesn't have a (non-Python) way to set the Vertex Colors per selected vertex/vertices. Per face is however, as stated below, not a problem.
• There is no built-in interface (yet) to get the rgb value of a selected Vertex. Each vertex can be shared by a number of faces, therefore a vertex doesn't necessarily have one color associated with it. Behind the scenes the Vertex Colors are not stored in the data per vertex but in a Vertex Color layer, which stores vertex colors for each face of the mesh.
Possibly this explains that a bit more clear. There are 9 verts in this subdivided plane, and the middle vertex is used in 4 faces and therefore has 4 different colors associated.
### Vertex Color Map
You can set a collection of faces to one colour in vertex paint mode, by selecting them and setting the colour from the colour picker, then shift+K will fill the selection with that colour. You can enter numericals into the Blender colour picker.
You can pick faces with face selection masking
If you want to do it by script:
import bpy
import random
# start in object mode
obj = bpy.data.objects["Cube"]
mesh = obj.data
if not mesh.vertex_colors:
mesh.vertex_colors.new()
"""
let us assume for sake of brevity that there is now
a vertex color map called 'Col'
"""
color_layer = mesh.vertex_colors["Col"]
# or you could avoid using the color_layer name
# color_layer = mesh.vertex_colors.active
i = 0
for poly in mesh.polygons:
for idx in poly.loop_indices:
rgb = [random.random() for i in range(3)]
color_layer.data[i].color = rgb
i += 1
# set to vertex paint mode to see the result
bpy.ops.object.mode_set(mode='VERTEX_PAINT')
Also a small blogpost about vertex colours here and here, if you are so inclined.
For Weight Painting, i'm not sure -- i'll let someone more familiar with that give an answer
• How would you change this script to set the vertex colors equal to the value of the vertices themselves? I.e. each vertex RGB = XYZ? – twerdster Feb 1 '14 at 16:21
• How could I set the color to the value under the normal of a face or the approximate blended color if the vertices of that face are different? – iKlsR Feb 23 '15 at 12:39
• @iKlsR using vertex_normals then, might be worth asking a separate question. – zeffii Feb 23 '15 at 14:35
• @iKlsR gist.github.com/zeffii/90399b67820dbf64628b vertex.normal – zeffii Feb 23 '15 at 14:44
• @zeffii The color under the normal or the color in the middle of a polygon. I've never used vertex paint programmatically before so not even sure the api allows this. – iKlsR Feb 23 '15 at 14:56
You can try this code to apply color to vertices selected in edit mode:
import bpy
def color_to_vertices(color):
mesh = bpy.context.active_object.data
bpy.ops.object.mode_set(mode = 'VERTEX_PAINT')
selected_verts = []
for vert in mesh.vertices:
if vert.select == True:
selected_verts.append(vert)
for polygon in mesh.polygons:
for selected_vert in selected_verts:
for i, index in enumerate(polygon.vertices):
if selected_vert.index == index:
loop_index = polygon.loop_indices[i]
mesh.vertex_colors.active.data[loop_index].color = color
bpy.ops.object.mode_set(mode = 'EDIT')
import random
RGB = [random.uniform(0,1) for i in range(3)]
color_to_vertices(RGB)
The result will be like this:
Don't forget about proper material node setup if you want to see the result in edit mode (material viewport shading) and render:
For convenience sake you maybe want to create some panel and an operator.
# Setting weight numerically
Enter edit mode and you'll find it in the Properties window under, Object Data, Vertex Groups. | 2019-12-09 08:21:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23881784081459045, "perplexity": 2398.2382451433523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00496.warc.gz"} |
https://plainmath.net/45896/suppose-that-we-roll-a-fair-die-until-a-6 | # Suppose that we roll a fair die until a 6
Suppose that we roll a fair die until a 6 comes up.
a) What is the probability that we roll the die n times?
b) What is the expected number of times we roll the die?
You can still ask an expert for help
## Want to know more about Probability?
• Questions are typically answered in as fast as 30 minutes
Solve your problem for the price of one coffee
• Math expert for every subject
• Pay only if we can solve it
Wendy Boykin
Step 1
a) If we roll a fair die, then we have 1 chance in 6 of rolling a six and 5 chances in 6 of not rolling a 6.
$P\left(6\right)=\frac{1}{6}$
We roll the die until a 6 is rolled.
Let X be the number of rolls of the die.
When we roll the die n times, then the first $n-1$ rolls cannot be a 6, while the n-th roll has to be a 6:
Note: The variable X has a geometric distribution with $p=\frac{1}{6}$.
Step 2
b) The expected value of a random variable X with a geometric distribution is the reciprocal of the constant probability of success $p=P\left(6\right)$
$E\left(x\right)=\frac{1}{p}=\frac{1}{P\left(6\right)}=\frac{1}{\frac{1}{6}}=6$
Heather Fulton
a) We would have to roll $n-1$ numbers that are not 6 followed by a 6, so the probability is ${\left(\frac{5}{6}\right)}^{n-1}\cdot \left(\frac{1}{6}\right)$
b) $E\left(N\right)=\sum _{n=1}^{\mathrm{\infty }}n\cdot {\left(\frac{5}{6}\right)}^{n-1}\left(\frac{1}{6}\right)=\left(\frac{1}{6}\right){\left(\frac{1}{1-\left(\frac{5}{6}\right)}\right)}^{2}=6$.
This is a nice answer since after 6 rolls we would expect to have rolled exactly one 6.
nick1337
a. If we roll the die n times (assuming $n\ge 1$ and the dice is 6-sided and fair), then we must roll n-1 "not 6" rolls followed by 1 "6" roll. The probability of that is:
$\left(\frac{5}{6}{\right)}^{n-1}\left(\frac{1}{6}{\right)}^{1}$
b. We could roll the die any number of times from 1 to infinite. Consider a random variable R which is the number of rolls to roll a 6. We could determine the expected number in two ways:
i. We could say that R is a geometric RV with a chance of success of $\frac{1}{6}$, so $E\left(R\right)=\frac{1}{\left(\frac{1}{6}\right)}=6$
ii. We could show i. explicitly, using the formula for expected value:
$E\left(R\right)=\sum _{s\in S}R\left(s\right)p\left(s\right)$
$=\sum _{i=1}^{\mathrm{\infty }}\left(i\right)\left(\left(\frac{5}{6}{\right)}^{i-1}\left(\frac{1}{6}{\right)}^{1}\right)$
$=\frac{1}{6}\left(\sum _{i=1}^{\mathrm{\infty }}\left(i\right)\left(\left(\frac{5}{6}{\right)}^{i-1}\right)$
This summation takes the form
$\sum _{i=1}^{\mathrm{\infty }}\left(i\right)\left({r}^{i-1}\right)$, where $|r|<1$
$\sum _{i=1}^{\mathrm{\infty }}\left(i\right)\left({r}^{i-1}\right)=\sum _{i=1}^{\mathrm{\infty }}\frac{d}{dr}\left({r}^{i}\right)$
$=\frac{d}{dr}\sum _{i=1}^{\mathrm{\infty }}\left({r}^{i}\right)$
$=\frac{d}{dr}\frac{1}{1-r}$ (since $|r|<1$)
$=\frac{1}{\left(1-r{\right)}^{2}}$
Let $r=\frac{5}{6}\left(|r|=\frac{5}{6}<1$.) Then,
$E\left(R\right)=\frac{1}{6}\left(\frac{1}{\left(1-\frac{5}{6}{\right)}^{2}}\right)$
$⇒E\left(R\right)=6$
Either way, we get the same answer: 6 rolls. | 2022-10-01 17:01:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 46, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7851998209953308, "perplexity": 321.4887718543114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030336880.89/warc/CC-MAIN-20221001163826-20221001193826-00001.warc.gz"} |
https://gomathanswerkey.com/texas-go-math-grade-4-lesson-1-1-answer-key/ | # Texas Go Math Grade 4 Lesson 1.1 Answer Key Place Value and Patterns
Refer to our Texas Go Math Grade 4 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 4 Lesson 1.1 Answer Key Place Value and Pattern.
## Texas Go Math Grade 4 Lesson 1.1 Answer Key Place Value and Patterns
Investigate
Materials base-ten blocks
You can use base-ten blocks to understand the relationships among place-value positions. Use a large cube for 1,000, a flat for 100, along with the cube for 10, and a small cube for 1.
Complete the comparisons below to describe the relationship from one place-value position to the next place-value position.
A.
Look at the long and compare it to the small cube.
The long is ________ times as much as the small cube.
The given table is:
Now,
From the given table,
We can observe that
The length is 10 times as much as the small cube.
Hence, from the above,
We can conclude that
The length is 10 times as much as the small cube.
Look at the flat and compare it to the long.
The flat is _________ times as much as the long.
The given table is:
Now,
From the given table,
We can observe that
The flat is 10 times as much as the long.
Hence, from the above,
We can conclude that
The flat is 10 times as much as the long.
Look at the large cube and compare it to the flat.
The large cube is ___________ times as much as the flat.
The given table is:
Now,
From the given table,
We can observe that
The large cube is 10 times as much as the flat.
Hence, from the above,
We can conclude that
The large cube is 10 times as much as the flat.
B.
Look at the flat and compare it to the large cube.
The flat is ___________ of the large cube.
The given table is:
Now,
From the given table,
We can observe that
The flat is $$\frac{1}{10}$$ of the large cube.
Hence, from the above,
We can conclude that
The flat is $$\frac{1}{10}$$ of the large cube.
Look at the long and compare it to the flat.
The long is ___________ of the flat.
The given table is:
Now,
From the given table,
We can observe that
The long is $$\frac{1}{10}$$ of the flat.
Hence, from the above,
We can conclude that
The long is $$\frac{1}{10}$$ of the flat.
Look at the small cube and compare it to the long.
The small cube is ___________ of the long.
The given table is:
Now,
From the given table,
We can observe that
The small cube is $$\frac{1}{10}$$ of the long.
Hence, from the above,
We can conclude that
The small cube is $$\frac{1}{10}$$ of the long.
Make Connections
You can use your understanding of place-value patterns and a place-value chart to write numbers that are 10 times as much as or $$\frac{1}{10}$$ of any given number.
3,000 is 10 times as much as 300.
30 is $$\frac{1}{10}$$ of 300.
Use the steps below to complete the below table.
STEP 1. Write the given number in a place-value chart.
STEP 2. Use the place-value chart to write a number that is 10 times as much as the given number.
STEP 3. Use the place-value chart to write a number that is of the given number.
The given steps are:
STEP 1. Write the given number in a place-value chart.
STEP 2. Use the place-value chart to write a number that is 10 times as much as the given number.
STEP 3. Use the place-value chart to write a number that is of the given number.
Hence, from the above,
We can conclude that
The complete table by using the given steps is:
Share and Show
Complete the sentences.
Question 1.
500 is 10 times as much as ____________ .
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above definition,
x × 10 = 500
x = $$\frac{500}{10}$$
x = 50
Hence, from the above,
We can conclude that
500 is 10 times as much as 50
Question 2.
20,000 is $$\frac{1}{10}$$ of ____________ .
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above definition,
x = 20,000 × $$\frac{1}{10}$$
x = 2,000
Hence, from the above,
We can conclude that
20,000 is $$\frac{1}{10}$$ of 2,000
Use the place-value patterns to complete the table.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
Hence, from the above,
We can conclude that
The complete table is:
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
Hence, from the above,
We can conclude that
The complete table is:
Complete the sentence with 100 or 1,000
Question 7.
200 is _______________ times as much as 2.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above,
2 × a = 200
a = $$\frac{200}{2}$$
a = 100
Hence, from the above,
We can conclude that
200 is 100 times as much as 2.
Question 8.
7,00,000 is ___________ times as much as 700.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above,
700 × a = 700,000
a = $$\frac{700,000}{700}$$
a = 1,000
Hence, from the above,
We can conclude that
7,00,000 is 1,000 times as much as 700.
Question 9.
4,000 is ___________ times as much as 4.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above,
4 × a = 4,000
a = $$\frac{4,000}{4}$$
a = 1,000
Hence, from the above,
We can conclude that
4,000 is 1,000 times as much as 4.
Question 10.
600 is _____________ times as much as 6.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above,
6 × a = 600
a = $$\frac{600}{6}$$
a = 100
Hence, from the above,
We can conclude that
600 is 100 times as much as 6.
Problem Solving
Question 11.
Mark and Robyn used base-ten blocks to show that 300 in 100 times as much as 3. Whose model makes sense? Whose model is nonsense? Explain your reasoning.
It is given that
Mark and Robyn used base-ten blocks to show that 300 in 100 times as much as 3
Now,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
Now,
According to the given information,
Hence, from the above,
We can conclude that
Mark’s model is nonsense
Question 12.
Explain How you would help Mark understand why he should have used small cubes instead of longs.
We know that,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
So,
We considered “Cubes” as 1 unit
We considered “Longs” as a combination of “10 cubes” i.e., 10 units
Hence, from the above,
We can conclude that
Since Mark’s model has a single digit, he considered using “Cubes”
Question 13.
Melinda has 500 pennies in a jar. Brenda has 10 times as many pennies as Melinda does. How many pennies does Brenda have?
A. 50,000
B. 5
C. 5,000
D. 50
It is given that
Melinda has 500 pennies in a jar. Brenda has 10 times as many pennies as Melinda does
Now,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
Now,
Let the number of pennies Brenda has to be: y
Now,
According to the given information,
y = 500 × 10
= 5,000 pennies
Hence, from the above,
We can conclude that
The number of pennies does Brenda have is:
Question 14.
Which statement is true?
A. 8,000 is 10 times as much as 80.
B. 80 is $$\frac{1}{10}$$ of 8.
C. 800 is 10 times as much as 80.
D. 8 is $$\frac{1}{10}$$ of 800.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
Hence, from the above,
We can conclude that
The statement that is true is:
Question 15.
Multi-Step Harry has a collection of comic books. He currently has 1,000 comic books. His friend George has $$\frac{1}{10}$$ the number of comic books that Harry does. Mike has $$\frac{1}{10}$$ the number of comic books that George has. How many comic books does Mike have?
A. 1000
B. 100
C. 10,000
D. 10
It is given that
Harry has a collection of comic books. He currently has 1,000 comic books. His friend George has $$\frac{1}{10}$$ the number of comic books that Harry does. Mike has $$\frac{1}{10}$$ the number of comic books that George has
Now,
Let y be the number of comic books George has
Let z be the number of comic books Mike has
Now,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the given information,
The number of books George has = $$\frac{1,000}{10}$$
= 100 comic books
The number of books Mike has = $$\frac{100}{10}$$
= 10 comic books
Hence, from the above,
We can conclude that
The number of books Mike has:
Texas Test Prep
Question 16.
Sam has 1,300 dimes. Anna has $$\frac{1}{10}$$ the number of dimes that Sam does. How many dimes does Anna have?
A. 13,000
B. 130
C. 13
D. 3
It is given that
Sam has 1,300 dimes. Anna has $$\frac{1}{10}$$ the number of dimes that Sam does
Now,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the given information,
The number of dimes does Anna have = $$\frac{1,300}{10}$$
= 130 dimes
Hence, from the above,
We can conclude that
The number of dimes does Anna have:
### Texas Go Math Grade 4 Lesson 1.1 Homework and Practice Answer Key
Question 1.
Emma and Jamie used base-ten blocks to show that 40 is one-tenth of 400. Whose model makes sense? Whose model is nonsense? Explain your reasoning.
Describe the relationship between the value of a dollar and the value of a dime.
It is given that
Emma and Jamie used base-ten blocks to show that 40 is one-tenth of 400
Now,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
Now,
According to the given information,
Hence, from the above,
We can conclude that
Emma’s model makes sense
Jamie’s model is nonsense
Problem Solving
Question 2.
Lisa had 3 dollars. She went to the bank and exchanged the 3 dollars for 30 dimes.
Describe the relationship between the value of a dollar and the value of a dime.
It is given that
Lisa had 3 dollars. She went to the bank and exchanged the 3 dollars for 30 dimes.
Now,
We know that,
1 dollar = 10 dimes
So,
3 dollars = 30 dimes
Now,
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
So,
According to the above information,
(The number of dimes) × 10 = (The number of dollars)
So,
The dollar is 10 times as many as 1 dimes
Hence, from the above,
We can conclude that
The relationship between the value of a dollar and the value of a dime is:
1 dollar = 10 dimes
Lesson Check
Question 3.
Which statement is true?
A. 500 is 10 times as much as 50.
B. 500 is $$\frac{1}{10}$$ much as 50.
C. 50,000 is 1,000 times as much as 5.
D. 5 is $$\frac{1}{10}$$ as much as 500.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
Hence, from the above,
We can conclude that
The statement that is true is:
Question 4.
7,000 is ten times as much as what number?
A. 70
B. 7
C. 70,000
D. 700
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the above statements,
x × 10 = 7,000
x = $$\frac{7,000}{10}$$
x = 700
Hence, from the above,
We can conclude that
7,000 is ten times as much as:
Question 5.
Which statement is true?
A. 90 is $$\frac{1}{10}$$ of 100.
B. 900 is 100 times as much as 9.
C. 9,000 is 1,000 times as much as 90.
D. 9 is $$\frac{1}{10}$$ of 900.
We know that,
If a number x is as many times as much as the same number and the result is represented as y, then the representation of that operation is:
x × a = y
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
Hence, from the above,
We can conclude that
The statement that is true is:
Question 6.
720 is $$\frac{1}{10}$$ of what number?
A. 7,200
B. 72
C. 7
D. 72,000
We know that,
If a number y is $$\frac{1}{a}$$ the same number and the result is represented as x, then the representation of that operation is:
y × $$\frac{1}{a}$$ = x
So,
According to the given information,
y ×$$\frac{1}{10}$$ = 720
y = 720 × 10
y = 7,200
Hence, from the above,
We can conclude that
720 is $$\frac{1}{10}$$ of:
Question 7.
Multi-Step The owner of Pattie’s Party Shop ordered 4 cartons of balloons. How many balloons did she order?
1 carton = 10 boxes
1 box = 10 packages
1 packages = 10 balloons
A. 40
B. 400
C. 40,000
D. 4,000
It is given that
The owner of Pattie’s Party Shop ordered 4 cartons of balloons
Now,
We know that,
1 carton = 10 boxes
1 box = 10 packages
1 packages = 10 balloons
So,
The number of balloons did Pattie ordered = 4 × 10 boxes
= 4 × 10 × 10 packages
= 4 × 10 × 10 × 10 balloons
= 4 × 1,000
= 4,000 balloons
Hence, from the above,
We can conclude that
The number of balloons did Pattie ordered is:
Question 8.
Multi-Step Greg bought 2 boxes of balloons. He used half of them to decorate his yard. He used 40 to decorate his porch. He used the rest inside his house. How many balloons did he use inside?
A. 6
B. 6,000
C. 60
D. 600
It is given that
Greg bought 2 boxes of balloons. He used half of them to decorate his yard. He used 40 to decorate his porch. He used the rest inside his house
Now,
We know that,
1 carton = 10 boxes
1 box = 10 packages
1 packages = 10 balloons
So,
The total number of balloons did Greg bought = 2 × 10 packages
= 2 × 10 × 10 balloons
= 200 balloons
So,
The number of balloons Greg used inside = 200 – ($$\frac{200}{2}$$ + 40)
= 200 – 140
= 60 balloons
Hence, from the above,
We can conclude that
The number of balloons did Greg used inside is:
Scroll to Top | 2022-01-24 23:44:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7095181941986084, "perplexity": 1403.7861994293448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304686.15/warc/CC-MAIN-20220124220008-20220125010008-00704.warc.gz"} |
https://blog.myrank.co.in/cramers-rule-example/ | # Cramer’s Rule Example
### Cramer’s Rule Example
1) Solve the system of equation X – 2Y + 3Z = 2, 2X + 4Y + 2Z = -1, X + 2Y – 2Z = 5.
Solution: Given that
X – 2Y + 3Z = 2,
2X + 4Y + 2Z = -1,
X + 2Y – 2Z = 5
We can written as matrix form
$$\Delta =\left( \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & 2 \\ 1 & 2 & -2 \\\end{matrix} \right)$$ ,
= 1(-8 – 4) + 2(-4 – 2) + 3(4 – 4)
= -12-12 + 0
= -24 ≠ 0
$${{\Delta }_{1}}=\left( \begin{matrix} 2 & -2 & 3 \\ -1 & 4 & 2 \\ 5 & 2 & -2 \\\end{matrix} \right)$$,
= 2(-8 – 4) + 2(2 – 10) + 3(-2 – 20)
= -24 – 16 – 66
= -106
$${{\Delta }_{2}}=\left( \begin{matrix} 1 & 2 & 3 \\ 2 & -1 & 2 \\ 1 & 5 & -2 \\\end{matrix} \right)$$,
= 1(2 – 10) – 2(-4 – 2) + 3(10 + 1)
= -8 + 12 + 33
= 37
$${{\Delta }_{3}}=\left( \begin{matrix} 1 & -2 & 2 \\ 2 & 4 & -1 \\ 1 & 2 & 5 \\\end{matrix} \right)$$,
= 1(20 + 2) + 2(10 + 1) + 2(4 – 4)
= 22 + 22 + 0
= 44
∴ $$X=\frac{{{\Delta }_{1}}}{\Delta }\,=\,\frac{-106}{-24}\,=\,\frac{53}{12}$$.
$$Y\,=\,\frac{{{\Delta }_{2}}}{\Delta }\,=\,\frac{37}{-24}\,=\,\frac{-37}{24}$$.
$$Z=\frac{{{\Delta }_{3}}}{\Delta }\,=\,\frac{44}{-24}\,=\,\frac{-11}{6}$$.
Solution is x = $$\frac{53}{12}$$, y = $$\frac{-37}{12}$$ and z = $$\frac{-11}{6}$$ unique solution. | 2022-10-03 11:21:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.652239203453064, "perplexity": 812.1801768339726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337415.12/warc/CC-MAIN-20221003101805-20221003131805-00562.warc.gz"} |
http://math.stackexchange.com/questions/63785/non-principal-prime-ideals-of-mathbbzx | # Non-principal prime ideals of $\mathbb{Z}[x]$ [duplicate]
This question already has an answer here:
How can you show that the non-principal prime ideals of $\mathbb{Z}[x]$ can be generated by only two elements, a prime number $p$ and an irreducible polynomial not in $p\mathbb{Z}[x]$?
I can get to the point in the proof that a prime ideal with more than one generator must contain some $p$, but I can't prove that appending the polynomial can generate the prime ideal itself.
-
## marked as duplicate by Najib Idrissi, SHOBHIT GAUTAM, Jeremy Rickard, David K, Mike MillerFeb 8 '15 at 17:15
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
## 1 Answer
HINT $\$ The image of the ideal in $\rm\:\mathbb F_p[x]$ is principal. Pull this information back to $\rm\:\mathbb Z[x]\:.$
- | 2016-06-29 13:14:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5665189027786255, "perplexity": 308.25920959491435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397744.64/warc/CC-MAIN-20160624154957-00026-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.neetprep.com/ncert-question/228605 | 7.60 The pH of 0.1M solution of cyanic acid (HCNO) is 2.34. Calculate the ionization constant of the acid and its degree of ionization in the solution.
c = 0.1 M
pH = 2.34
$-\mathrm{log}\left[{\mathrm{H}}^{+}\right]=\mathrm{pH}\phantom{\rule{0ex}{0ex}}-\mathrm{log}\left[{\mathrm{H}}^{+}\right]=2.34\phantom{\rule{0ex}{0ex}}\left[{\mathrm{H}}^{+}\right]=4.5×{10}^{-3}\phantom{\rule{0ex}{0ex}}\mathrm{Also},\phantom{\rule{0ex}{0ex}}\left[{\mathrm{H}}^{+}\right]=\mathrm{c\alpha }\phantom{\rule{0ex}{0ex}}4.5×{10}^{-3}=0.1×\mathrm{\alpha }\phantom{\rule{0ex}{0ex}}\frac{4.5×{10}^{-3}}{0.1}=\mathrm{\alpha }\phantom{\rule{0ex}{0ex}}\mathrm{\alpha }=45×{10}^{-3}=.045\phantom{\rule{0ex}{0ex}}\mathrm{Then},\phantom{\rule{0ex}{0ex}}{\mathrm{K}}_{\mathrm{\alpha }}={\mathrm{c\alpha }}^{2}\phantom{\rule{0ex}{0ex}}=0.1×{\left(45×{10}^{-3}\right)}^{2}\phantom{\rule{0ex}{0ex}}=202.5×{10}^{-6}\phantom{\rule{0ex}{0ex}}=2.02×{10}^{-4}$ | 2023-02-01 16:53:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5298156142234802, "perplexity": 3201.005756499978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499946.80/warc/CC-MAIN-20230201144459-20230201174459-00543.warc.gz"} |
https://sinews.siam.org/Details-Page/exploring-the-dynamics-of-honeybee-mortality-and-colony-collapse-in-winter | SIAM News Blog
SIAM News
# Exploring the Dynamics of Honeybee Mortality and Colony Collapse in Winter
Global honeybee populations are declining at a troubling rate. Scientists attribute this continued drop to a variety of phenomena, including industrial agriculture, colony collapse disorder, parasitic/pathogenic infections, and climate change. Given that one-third of our food production depends on bee pollination, a continued decline could have deadly effects on agricultural practices. Honeybee colonies are particularly vulnerable during the winter, when both outside temperatures and colony temperatures drop well below the bees’ preferred temperature of 30 degrees Celsius. When colony temperature surpasses a certain threshold, the entire colony dies off (see Figure 1). During a minisymposium presentation at the 2019 SIAM Conference on Applications of Dynamical Systems, currently taking place in Snowbird, Utah, Vivi Rottschäfer of Leiden University examined thermoregulation in honeybee colonies to better understand the consequences of honeybee mortality in the winter.
Figure 1. Comparison of temperature in the colony versus ambient temperature. When colony temperature surpasses a certain threshold, the entire colony collapses and dies.
“The key to surviving the winter is keeping the temperature in the hive high enough,” Rottschäfer said. “No new bees are produced during this time, so it’s important for both them and for us that they survive.” Existing studies of honeybees indicate that bees have no centralized mechanism, implying that they rely on a self-monitoring thermoregulation system that allows them to regulate their body temperatures and sense the ambient temperature around them. This mechanism consists of two processes. First, honeybees shiver with their flight muscles to produce heat when the colony falls below the preferred temperature. Second, bees adjust their positioning in the colony based on colony temperature. “If it’s too hot, they move towards a lower temperature” Rottschäfer said. “If it’s too cold, they move towards a higher temperature.” This is a form of thermotactic movement; although slightly different than chemotaxis, one can model bees’ motion with a type of chemotactic term.
Rottschäfer’s model for thermotactic movement includes variables for temperature and density, incorporates the presence of diffusion, and accounts for heat production from the bees’ shivering flight muscles. This is a generalized Keller-Segel model based on James Watmough and Scott Camazine’s 1995 work. Rather than analyze a three-dimensional structure, Rottschäfer examines a cross-section of the cluster of bees in a honeycomb. “We take a cross-section through the middle, with axis 0 as the middle of the beehive,” she said. The boundary conditions are thus symmetric.
Model analysis distinguished two states of honeybee colonies: (i) one in which the colony size is above a critical population number, thus allowing the bees to maintain a core temperature above the temperature threshold, and (ii) one in which the colony’s core temperature drops below the critical threshold, thus increasing bee mortality and ultimately resulting in sudden death of the colony. “If the ambient temperature is lower, the critical total bee concentration must be higher,” Rottschäfer said.
Next she included mortality in the model, as this factor specifically explores the reason for honeybee death. Rottschäfer explained that bees have to work quite hard if the local temperature is low. Honeybees can only work about 30 minutes at a time before they have to rest. When they reach their work limit, they move towards higher temperatures at colony’s core and are replaced in the cooler parts by other bees; this is called the refresh rate. “If the total bee population is low, they have to work harder because they have to work more often,” Rottschäfer said. “If the temperature is high enough, they don’t have to work and there’s no influence on mortality.”
Figure 2. Colonies with the highest population density at t=0 have the greatest chance of surviving the winter.
Rottschäfer then expanded her model to include the effect of mites that sicken bees, make them lethargic, and ultimately reduce their lifespan. The presence of enough mites affects honeybee mortality rates, as the mites do not die with their hosts — they simply move on to another bee. She presented the collective results of her simulation with a graph depicting the sudden death of colonies with different population densities (see Figure 2). The schematic plots time on the horizontal axis and population density on the vertical axis, and indicates that a higher colony density at $$t=0$$ correlates with the increased likelihood of a colony making it to spring without dying off. Rottschäfer also experimented with other parameters—like ambient temperature or quantity of mites—which reconfirmed that honeybee populations die off more quickly in colder weather and in the presence of more mites.
In light of these results, Rottschäfer wondered whether scientists can actively help colonies survive the winter. At this point she had only examined the cross-section of one honeycomb, when in reality multiple combs comprise a colony. So she decided to look at two combs with the potential for interaction. If the core temperature dips below the critical threshold in one comb, all of those bees move to the other comb. This observation reveals a potential means of human interference. “It might be an option to put two colonies together towards the end of winter,” Rottschäfer said. “But we still need to look into this in much more detail.”
Ultimately, Rottschäfer’s model reveals that the density of honeybees populating a colony is crucial to that colony’s survival through the winter. In the future, she hopes to analyze multiple honeycombs rather than just one or two, and study the relationship between pre-winter colony size and colony collapse. Possessing a better understanding of winter colony collapse can help researchers better comprehend the consequences of collapse, preserve colonies, and prevent further loss of worldwide honeybee populations.
Lina Sorg is the associate editor of SIAM News. | 2020-02-22 17:27:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.436234712600708, "perplexity": 2578.4294430551113}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145708.59/warc/CC-MAIN-20200222150029-20200222180029-00431.warc.gz"} |
http://psjd.icm.edu.pl/psjd/element/bwmeta1.element.ojs-nameId-6e3e8ea9-5a94-37a7-828b-e6cd5da23db6-year-2015-article-2097 | PL EN
Preferences
Language
enabled [disable] Abstract
Number of results
Journal
## Czasopismo Techniczne
- |
Article title
### Energy security aspects in the context of the climate change package 3×20
Authors
Content
Title variants
Languages of publication
PL
Abstracts
PL
This paper presents the problem of energy security in the context of the EU Directive concerning the energy and climate package 3 × 20%. The Directive assumes particulary other limitations in energy consumption and carbon emission. As proved this will translate into the increase in energy prices. It was also revealed the negative impact on this Directive on the economic growth. It is also important that Poland will feel the greates impact of the introduction this Directive into practice. This is due to the carvon structure of fuel and energy balance.
Keywords
Publisher
Journal
Year
Physical description
Dates
online
2015-05-07
Contributors
author
References
Document Type
Publication order reference
Identifiers | 2021-06-18 13:10:16 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8199145197868347, "perplexity": 3492.0158208984085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487636559.57/warc/CC-MAIN-20210618104405-20210618134405-00427.warc.gz"} |