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https://www.physicsforums.com/threads/3db-frequency-of-an-led-transfer-function.925657/ | 3dB Frequency of an LED Transfer Function
Homework Statement
According to my textbook:
The LED transfer function ##H(\omega_m)## is defined as:
$$H(\omega_m) = \frac{1}{1+j\omega_{m}\tau_{c}}$$
The 3-dB modulation bandwidth ##f_{\text{3 dB}}## is defined as the modulation frequency at which ##H(\omega_m)## is reduced by 3 dB or by a factor of 2. The result is:
$$f_{\text{3 dB}}=\sqrt{3}(2\pi\tau_{c})^{-1} \tag{1}$$
I don't understand how they derived the last expression.
The Attempt at a Solution
Let ##G(j\omega_{m})=\frac{1}{1+j\omega_{m}\tau_{c}}##.
##|G(j\omega_{m})|^{2}=\frac{1}{1+j\omega_{m}\tau_{c}}.\frac{1}{1-j\omega_{m}\tau_{c}}=\frac{1}{1+\omega_{m}^{2}\tau_{c}^{2}}##
##\therefore |G(j\omega_{m})|=\frac{1}{\sqrt{1+\omega_{m}^{2}\tau_{c}^{2}}}##
And ##|G(0)|=1##, therefore to find the 3 dB point we must solve:
$$\frac{|G(j\omega)|}{|G(0)|}=\frac{1}{\sqrt{1+\omega_{\text{3 dB}}^{2}\tau_{c}^{2}}}=\frac{1}{\sqrt{2}}$$
However when I solve this I get:
$$\omega_{\text{3 dB}}=\frac{1}{\tau_{c}}\ \text{or}\ f_{\text{3 dB}}=\frac{1}{2\pi\tau_{c}}$$
So where does the ##\sqrt{3}## factor in equation (1) come from? What is the mistake here?
Any help would be greatly appreciated. | 2021-11-28 18:53:25 | {"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": 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": 0.6251132488250732, "perplexity": 1117.9205679782167}, "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/1637964358570.48/warc/CC-MAIN-20211128164634-20211128194634-00598.warc.gz"} |
https://www.sparrho.com/item/quasi-commutative-cochains-in-algebraic-topology/949424/ | # Quasi-commutative cochains in algebraic topology
Research paper by Max Karoubi
Indexed on: 13 Sep '05Published on: 13 Sep '05Published in: Mathematics - Algebraic Topology
#### Abstract
We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The theory is sufficiently general to include also ringed spaces X. The construction is by itself interesting since it uses the difference calculus (instead of the differential calculus of Sullivan's theory) and a new type of tensor product, called "reduced tensor product". Although we dont have a minimal model yet, it is relatively easy to define the cup i-products and the Steenrod operations on the category of ADGQ's. Homotopy groups can be deduced from the "iterated Hochschild homology" of D(X). The determination of the homotopy type from this algebraic structure uses in an essential way recent results of M. Mandell. | 2020-11-29 11:13:55 | {"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": 0.8312687277793884, "perplexity": 998.0027962940599}, "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/1606141197593.33/warc/CC-MAIN-20201129093434-20201129123434-00524.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-2-linear-and-quadratic-functions-section-2-3-quadratic-functions-and-their-zeros-2-3-assess-your-understanding-page-146/11 | ## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)
Zeros: $0,9$ $x$-intercepts: $0,9$
To find the zeros of a function $f$, solve the equation $f(x)=0$ The zeros of the function are also the $x-$intercepts. $$x^2-9x=0$$ Taking $x$ as a common factor: $$x(x-9)=0$$ Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain: \begin{align*} x&=0 &\text{ or }& &x-9=0\\ x&=0 &\text{ or }& &x=9\\ \end{align*} | 2021-12-08 03:28:06 | {"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": 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": 0.9774085283279419, "perplexity": 1417.0881205743844}, "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/1637964363437.15/warc/CC-MAIN-20211208022710-20211208052710-00000.warc.gz"} |
https://masscslprep.com/home | # MCP
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Discount for more participants.
(617) 481-0403
To register by phone,
call (617) 481-0403
## Office Hours
Monday - Friday: 9am - 9pm
Saturday: 11am - 5pm
Sunday: Closed
## Location
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Free parking on site!
Smaller class sizes,
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• Evening classes once a week for 6 weeks, 6:30 - 8:30 p.m.
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Monday - Friday: 9am - 9pm
Saturday: 11am - 5pm
Sunday: Closed
Discount for more participants. | 2019-02-16 21:18:50 | {"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": 0.17699626088142395, "perplexity": 11010.746101334931}, "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-2019-09/segments/1550247481122.31/warc/CC-MAIN-20190216210606-20190216232606-00150.warc.gz"} |
https://sciencing.com/characteristics-rectangular-pyramids-7470860.html | # Characteristics of Rectangular Pyramids
Print
A pyramid is a three-dimensional object consisting of a base and triangular faces that meet at a common vertex. A pyramid is classified as a polyhedron – a three-dimensional shape made of polygons – and it is made up of plane faces, or faces that are level two-dimensional surfaces. A rectangular pyramid possesses specific characteristics that make finding volume and area possible with certain formulas. Different types of pyramids might have different shapes of polygon bases and configurations, but they will all have triangular faces.
## Faces
A rectangular pyramid consists of five faces: one rectangular-shaped base and four triangular-shaped faces. Each triangular face is congruent to the opposite face in a right rectangular pyramid.
For example, on a rectangular pyramid where the edges of the rectangular base are labeled A, B, C and D, the triangular faces on edges A and C are congruent, while those on edges B and D are congruent. Since the triangular faces are attached to the sides of the base, instead of the top or bottom, they are called lateral faces.
## Height
When dealing with rectangular pyramid formulas, the height plays a key role in calculating area, volume, and many other metrics of the pyramid. Oftentimes, the height can be related to the length of a slant and volume through fractions and exponents, as we will see later.
## Vertices
A rectangular pyramid consists of five vertices, or points at which edges intersect. One vertex is located at the top of the pyramid, where the four triangular faces meet. The remaining four vertices are located on each corner of the rectangular base. According to MathsTeacher.com, the pyramid becomes a right pyramid when the top vertex is "directly above the center of the base."
For many of the following calculations and definitions, we assume a right rectangular pyramid to help simplify the calculations. When the vertex does not lie above the center, it is known as an oblique rectangular pyramid. In such a case, the opposite triangular faces are not always equal, and it makes the geometry much more complex when dealing with oblique pyramids.
## Edges
A rectangular pyramid consists of eight edges, or sharp sides "formed by the intersection of two surfaces," as defined by Word Net Web. Four edges are located on the rectangular base, while four edges form the upward slope to create the top vertex of the pyramid.
## Surface Area
The surface area of a rectangular pyramid depends on the area of the base and the area of each of the four lateral faces. To find this, we need to break the problem into two parts. First, we can calculate the area of the base which is simply length (l) times width (w):
l \cdot w
The area for the faces of the pyramid is more complex. The formula, given base length (l), base width (w), and height of the pyramid (h), can be written as:
l\sqrt{ \left(\frac{w}{2}\right)^2 + h^2} \ + w\sqrt{ \left(\frac{l}{2}\right)^2 + h^2}
The formula for total surface area of a closed right rectangular pyramid is the sum of the areas:
l \cdot w \ + \ l\sqrt{ \left(\frac{w}{2}\right)^2 + h^2} \ + w\sqrt{ \left(\frac{l}{2}\right)^2 + h^2}
#### Tips
• The square root terms in this formula use the Pythagorean Theorem to convert the vertical height to slant height and calculate the area for each lateral surface.
## Volume of a Rectangular Pyramid
When given a rectangular pyramid, the volume of the pyramid can be given in terms of the height (h), length (l), and width of the rectangular base (w):
V = \frac{l\cdot w\cdot h}{3}
While the three in the denominator may seem out of place, it can be explained with the help of a regular rectangular prism. The volume of a rectangular prism is length times width time height; when compared to a rectangular pyramid with the same base, the negative space around the pyramid actually adds up to two more pyramids of the same volume. So the volume of the rectangular prism is equivalent to three times the volume of a rectangular pyramid, this is why we divide by three.
Dont Go!
We Have More Great Sciencing Articles! | 2023-03-28 01:43:12 | {"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": 0.8008599877357483, "perplexity": 491.8813446174393}, "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-2023-14/segments/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00152.warc.gz"} |
https://engineeringprep.com/problems/025 | ## Towing a Car
A damaged 1200-kg car is being towed by a truck. Neglecting the friction, air drag, and rolling resistance, determine the extra power required (a) for constant velocity on a level road, (b) for constant velocity of 50 km/h on a 30 degree (from horizontal) uphill road, and (c) to accelerate on a level road from stop to 90 km/h in 12 seconds.
Hint
The power due to potential energy:
$$\dot{W}_{g}=mg(z_{2}-z_{1})/\Delta t$$$where $$m$$ is the mass, $$g$$ is the acceleration due to gravity, $$\Delta z$$ is the change in height, and $$\Delta t$$ is the change in time. Hint 2 The power from kinetic energy: $$\dot{W}_{a}=\frac{1}{2}m(V_{2}^{2}-V_{1}^{2})/\Delta t$$$
where $$m$$ is the mass, $$V$$ is the velocity, and $$\Delta t$$ is the change in time.
The total power required for each case is the sum of the rates of changes in potential and kinetic energies. That is:
$$\dot{W}_{total}=\dot{W}_{a}+\dot{W}_{g}$$$where $$\dot{W}_a$$ is the power from kinetic energy, and $$\dot{W}_g$$ is the power from potential energy. (a) Zero. (b) $$\dot{W}_{a}=0$$ . Thus, $$\dot{W}_{total}=\dot{W}_{g}=mg(z_{2}-z_{1})/\Delta t=mg\frac{\Delta z}{\Delta t}=mgV_{2}=mgVsin(30^{\circ})$$$
where $$m$$ is the mass, $$g$$ is the acceleration due to gravity, $$\Delta z$$ is the change in height, $$\Delta t$$ is the change in time, $$V$$ is the velocity. Thus, the extra power on an upward hill is:
$$=(1200kg)(9.81m/s^2)(\frac{50,000m}{3600s})(\frac{1kJ/kg}{1000m^2/s^2})(0.5)=81.7\:kW$$$(c) $$\dot{W}_{g}=0$$ . Thus, $$\dot{W}_{total}=\dot{W}_{a}=\frac{1}{2}m(V_{2}^{2}-V_{1}^{2})/\Delta t$$$
where $$m$$ is the mass, $$V$$ is the velocity, and $$\Delta t$$ is the change in time. Thus, the extra power to accelerate from rest is:
$$=\frac{1}{2}(1200kg)[(\frac{90,000m}{3600s})^{2}-0](\frac{1kJ/kg}{1000m^2/s^2})/(12s)=31.3\:kW$$\$
(a) Zero
(b) 81.7 kW
(c) 31.3 kW | 2022-08-15 15:50:18 | {"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": 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": 0.4333178699016571, "perplexity": 8294.148143696335}, "config": {"markdown_headings": true, "markdown_code": false, "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-2022-33/segments/1659882572192.79/warc/CC-MAIN-20220815145459-20220815175459-00612.warc.gz"} |
https://www.hepdata.net/record/88374 | Production of the $\rho$(770)${^{0}}$ meson in pp and Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
No Journal Information
The collaboration
Abstract (data abstract)
The production of $\rho^{0}$ mesons is measured with the ALICE detector in pp and Pb-Pb collisions at the centre-of-mass energy per nucleon pair, $\sqrt{s_{\rm NN}}$ = 2.76 TeV. The yields of $\rho^{0}$ for rapidity $|y|<0.5$ are presented as a function of transverse momentum ($p_{\rm T}$). The $p_{\rm T}$-integrated and $p_{\rm T}$-differential particle ratios $\rho^{0}/\pi$ are obtained in pp and Pb--Pb collisions. The nuclear modification factors $R_{\rm AA}$ are computed using a proton-proton reference at $\sqrt{s} = 2.76$ TeV. | 2019-09-17 08:50:59 | {"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": 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": 0.9738843441009521, "perplexity": 2465.267291648011}, "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-2019-39/segments/1568514573065.17/warc/CC-MAIN-20190917081137-20190917103137-00236.warc.gz"} |
https://math.stackexchange.com/questions/1280543/interchanging-total-derivative-and-partial-derivative | # Interchanging total derivative and partial derivative
Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$.
$$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$
$$\frac{\partial }{\partial x} \frac{\mathrm{d} F}{\mathrm{d} t} \tag{2}$$
They both evaluate to the same thing. But is there something that I have to watch out for? For example, note that in eqn. (2), $\frac{\mathrm{d} F}{\mathrm{d} t}$ is a function of four variables: $x,y,\dot{x},\dot{y}$. So, the $\frac{\partial }{\partial x}$ in $\frac{\partial }{\partial x} \frac{\mathrm{d} F}{\mathrm{d} t}$ is not used in the same sense as equation (1)
Also, can I do something like this without reservation:
$$\frac{\mathrm{d^{10}} }{\mathrm{d^{10}} t} \frac{\partial^7 F}{\partial x^4 \partial y^3} = \frac{\partial^2}{\partial x \partial y} \frac{\mathrm{d^{2}} }{\mathrm{d^{2}} t} \frac{\partial^5 F}{\partial x^3 \partial y^2} \frac{\mathrm{d^{8}} F}{\mathrm{d^{8}} t}$$
$\frac{\partial }{\partial x}\frac{dF}{dt}$ has no sense... Don't forget that $$t\mapsto F(x(t),y(t))$$ is a function from $\mathbb R\to\mathbb R$.
• If I write $\frac{\mathrm{d} F}{\mathrm{d} t} = \frac{\partial F}{\partial x} \frac{\mathrm{d} x}{\mathrm{d} t} + \frac{\partial F}{\partial y} \frac{\mathrm{d} y}{\mathrm{d} t}$, I can express $\frac{\mathrm{d} F}{\mathrm{d} t}$ as a function of $x$. Would it then be possible to partially differentiate $\frac{\mathrm{d} F}{\mathrm{d} t}$ wrt $x$? – IanDsouza May 14 '15 at 6:28 | 2020-01-26 03:27:43 | {"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": 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": 0.9589232206344604, "perplexity": 134.74418842683446}, "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/1579251684146.65/warc/CC-MAIN-20200126013015-20200126043015-00437.warc.gz"} |
https://www.yesterdayscoffee.de/tag/figure-caption/ | # A “non-sciency” LaTeX book
One of my “customers” wanted a book that didn’t look so “sciency” (i.e., less like LaTeX). So here are a few tweaks to match their expectations.
Chapters should have letters as references with the prefix “Teil”:
\renewcommand{\thechapter}{Teil \Alph{chapter}}
Sections have the letter, a hyphen and the number:
\renewcommand*{\thesection}{\Alph{chapter}-\arabic{section}}
Figures have no numbers, only the caption text below (using the package caption):
\usepackage{caption}
\captionsetup[figure]{labelformat=empty,textfont=footnotesize}
\renewcommand{\thefigure}{}
The first line of a paragraph is not indented. To compensate, the space between paragraphs is bigger:
\setlength\parindent{0pt}
The header line of a page contains no section, but only the name of the chapter (using the package scrlayer-scrpage):
\usepackage{scrlayer-scrpage}
\clearpairofpagestyles % empty default header/footer markings
\automark[chapter]{chapter} % only chapter as mark
\ohead{\pagemark} % outer margin has page number
# Plot legend in figure caption
If you have a plot and add names to the data series with \addlegendentry{}, a legend is added to the plot that specifies the names for the lines or bars. This is an example plot from my thesis:
\begin{figure}
\begin{tikzpicture}
\begin{axis}[myplot]
\addplot plot coordinates {(1, 0.17) (2, 0.13) (3, 0.09) (4, 0.06) (5, 0.01) (6, 0.01)};
\addplot plot coordinates {(1, 0.13) (2, 0.16) (3, 0.16) (4, 0.14) (5, 0.14) (6, 0.13)};
\end{axis}
\end{tikzpicture}
\caption{Results (Precision) at different $k$ for the two systems}
\end{figure}
You can influence the placement and appearance of the legend and it looks really professional. Most of the time that will be exactly what you want. But not always. I wanted the legend to be embedded in the text of the caption (long story why). And this is possible and even quite simple! You just need to define a label for the data series and when you refer to that label, a picture of the line and marker is drawn as the reference.
\begin{figure}
\begin{tikzpicture}
\begin{axis}[myplot]
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\label{tikz:System2}
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https://lw2.issarice.com/users/paulfchristiano | ## Posts
It’s not economically inefficient for a UBI to reduce recipient’s employment 2020-11-22T16:40:05.531Z
Hiring engineers and researchers to help align GPT-3 2020-10-01T18:54:23.551Z
“Unsupervised” translation as an (intent) alignment problem 2020-09-30T00:50:06.077Z
Distributed public goods provision 2020-09-26T21:20:05.352Z
Better priors as a safety problem 2020-07-05T21:20:02.851Z
Learning the prior 2020-07-05T21:00:01.192Z
Inaccessible information 2020-06-03T05:10:02.844Z
Writeup: Progress on AI Safety via Debate 2020-02-05T21:04:05.303Z
Hedonic asymmetries 2020-01-26T02:10:01.323Z
Moral public goods 2020-01-26T00:10:01.803Z
Of arguments and wagers 2020-01-10T22:20:02.213Z
Prediction markets for internet points? 2019-10-27T19:30:00.898Z
AI alignment landscape 2019-10-13T02:10:01.135Z
Taxing investment income is complicated 2019-09-22T01:30:01.242Z
The strategy-stealing assumption 2019-09-16T15:23:25.339Z
Reframing the evolutionary benefit of sex 2019-09-14T17:00:01.184Z
Ought: why it matters and ways to help 2019-07-25T18:00:27.918Z
Aligning a toy model of optimization 2019-06-28T20:23:51.337Z
What failure looks like 2019-03-17T20:18:59.800Z
Security amplification 2019-02-06T17:28:19.995Z
Reliability amplification 2019-01-31T21:12:18.591Z
Techniques for optimizing worst-case performance 2019-01-28T21:29:53.164Z
Thoughts on reward engineering 2019-01-24T20:15:05.251Z
Learning with catastrophes 2019-01-23T03:01:26.397Z
Capability amplification 2019-01-20T07:03:27.879Z
The reward engineering problem 2019-01-16T18:47:24.075Z
Towards formalizing universality 2019-01-13T20:39:21.726Z
Directions and desiderata for AI alignment 2019-01-13T07:47:13.581Z
Ambitious vs. narrow value learning 2019-01-12T06:18:21.747Z
AlphaGo Zero and capability amplification 2019-01-09T00:40:13.391Z
Supervising strong learners by amplifying weak experts 2019-01-06T07:00:58.680Z
Benign model-free RL 2018-12-02T04:10:45.205Z
Corrigibility 2018-11-27T21:50:10.517Z
Humans Consulting HCH 2018-11-25T23:18:55.247Z
Approval-directed bootstrapping 2018-11-25T23:18:47.542Z
Approval-directed agents 2018-11-22T21:15:28.956Z
Prosaic AI alignment 2018-11-20T13:56:39.773Z
An unaligned benchmark 2018-11-17T15:51:03.448Z
Clarifying "AI Alignment" 2018-11-15T14:41:57.599Z
The Steering Problem 2018-11-13T17:14:56.557Z
Preface to the sequence on iterated amplification 2018-11-10T13:24:13.200Z
The easy goal inference problem is still hard 2018-11-03T14:41:55.464Z
Meta-execution 2018-11-01T22:18:10.656Z
Could we send a message to the distant future? 2018-06-09T04:27:00.544Z
When is unaligned AI morally valuable? 2018-05-25T01:57:55.579Z
Open question: are minimal circuits daemon-free? 2018-05-05T22:40:20.509Z
Weird question: could we see distant aliens? 2018-04-20T06:40:18.022Z
Implicit extortion 2018-04-13T16:33:21.503Z
Prize for probable problems 2018-03-08T16:58:11.536Z
Argument, intuition, and recursion 2018-03-05T01:37:36.120Z
Comment by paulfchristiano on Learning the prior · 2021-01-25T16:21:47.123Z · LW · GW
Even if you were taking D as input and ignoring tractability, IDA still has to decide what to do with D, and that needs to be at least as useful as what ML does with D (and needs to not introduce alignment problems in the learned model). In the post I'm kind of vague about that and just wrapping it up into the philosophical assumption that HCH is good, but really we'd want to do work to figure out what to do with D, even if we were just trying to make HCH aligned (and I think even for HCH competitiveness matters because it's needed for HCH to be stable/aligned against internal optimization pressure).
Comment by paulfchristiano on Learning the prior · 2021-01-24T16:39:44.160Z · LW · GW
was optimized to imitate H on D
It seems like you should either run separate models for D and D*, or jointly train the model on both D and D*, definitely you shouldn't train on D then run on D* (and you don't need to!).
I suppose this works, but then couldn't we just have run IDA on D* without access to Mz (which itself can still access superhuman performance)?
The goal is to be as good as an unaligned ML system though, not just to be better than humans. And the ML system updates on D, so we need to update on D too.
Comment by paulfchristiano on Learning the prior · 2021-01-23T23:15:27.201Z · LW · GW
I think your description is correct.
The distilled core assumption seems right to me because the neural network weights are already a distilled representation of D, and we only need to compete with that representation. For that reason, I expect z* to have roughly the same size as the neural network parameters.
My main reservation is that this seems really hard (and maybe in some sense just a reframing of the original problem). We want z to be a representation of what the neural network learned that a human can manipulate in order to reason about what it implies on D*. But what is that going to look like? If we require competitiveness then it seems like z has to look quite a lot like the weights of a neural network...
In writing the original post I was imagining z* being much bigger than a neural network but distilled by a neural network in some way. I've generally moved away from that kind of perspective, partly based on the kinds of considerations in this post.
But since the amplification/debate models are ML models, and we’re running these models to aid human decisions on x*, aren’t we back to relying on ML OOD generalization, and so back where we started?
I now think we're going to have to actually have z* reflect something more like the structure of the unaligned neural network, rather than another model (Mz) that outputs all of the unaligned neural network's knowledge.
That said, I'm not sure we require OOD generalization even if we represent z via a model Mz. E.g. suppose that Mz(i) is the ith word of the intractably-large z. Then the prior calculation can access all of the words i in order to evaluate the plausibility of the string represented by Mz. We then use that same set of words at training time and test time. If some index i is used at test time but not at training time, then the model responsible for evaluating Prior(z) is incentivized to access that index in order to show that z is unnecessarily complex. So every index i should be accessed on the training distribution. (Though they need not be accessed explicitly, just somewhere in the implicit exponentially large tree).
Like I said, I'm a bit less optimistic about doing this kind of massive compression. For now, I'm just thinking about the setting where our human has plenty of time to look at z in detail even if it's the same size as the weights of our neural network. if we can make that work, then I'll think about to do it in the case of computationally bounded humans (which I expect to be straightforward).
Comment by paulfchristiano on Some thoughts on risks from narrow, non-agentic AI · 2021-01-21T01:39:18.253Z · LW · GW
I agree that the core question is about how generalization occurs. My two stories involve kinds of generalization, and I think there are also ways generalization could work that could lead to good behavior.
It is important to my intuition that not only can we never train for the "good" generalization, we can't even evaluate techniques to figure out which generalization "well" (since both of the bad generalizations would lead to behavior that looks good over long horizons).
If there is a disagreement it is probably that I have a much higher probability of the kind of generalization in story 1. I'm not sure if there's actually a big quantitative disagreement though rather than a communication problem.
I also think it's quite likely that the story in my post is unrealistic in a bunch of ways and I'm currently thinking more about what I think would actually happen.
Some more detailed responses that feel more in-the-weeds:
you think long-horizon real-world data will play a significant role in training, because we'll need it to teach agents to do the most valuable tasks. This seems plausible to me; but I think that in order for this type of training to be useful, the agents will need to already have robust motivations (else they won't be able to find rewards that are given over long time horizons
I might not understand this point. For example, suppose I'm training a 1-day predictor to make good predictions over 10 or 100 days. I expect such predictors to initially fail over long horizons, but to potentially be greatly improved with moderate amounts of fine-tuning. It seems to me that if this model has "robust motivations" then they would most likely be to predict accurately, but I'm not sure about why the model necessarily has robust motivations.
I feel similarly about goals like "plan to get high reward (defined as signals on channel X, you can learn how the channel works)." But even if prediction was a special case, if you learn a model then you can use it for planning/RL in simulation.
But it seems to me by default, during early training periods the AI won't have much information about either the overseer's knowledge (or the overseer's existence), and may not even have the concept of rewards, making alignment with instructions much more natural.
It feels to me like our models are already getting to the point where they respond to quirks of the labeling or evaluation process, and are basically able to build simple models of the oversight process.
my concern is that this underlying concept of "natural generalisation" is doing a lot of work, despite not having been explored in your original post
Definitely, I think it's critical to what happens and not really explored in the post (which is mostly intended to provide some color for what failure might look like).
That said, a major part of my view is that it's pretty likely that we get either arbitrary motivations or reward-maximization (or something in between), and it's not a big deal which since they both seem bad and seem averted in the same way.
I think the really key question is how likely it is that we get some kind of "intended" generalization like friendliness. I'm frequently on the opposite side of this disagreement, arguing that the probability that people will get some nice generalization if they really try is at least 25% or 50%, but I'm also happy being on the pessimistic side and saying that the probability we can get nice generalizations is at most 50% or 75%.
(or anywhere else, to my knowledge)
Two kinds of generalization is an old post on this question (though I wish it had used more tasteful examples).
Turning reflection up to 11 touches on the issue as well, though coming from a very different place than you.
I think there are a bunch of Arbital posts where Eliezer tries to articulate some of his opinions on this but I don't know pointers offhand. I think most of my sense is
I haven't written that much about why I think generalizations like "just be helpful" aren't that likely. I agree with the point that these issues are underexplored by people working on alignment, and even more underdiscussed, given how important they are.
There are some google doc comment threads with MIRI where I've written about why I think those are plausible (namely that it plausible-but-challenging for breeding of animals, and that seems like one of our best anchors overall, suggesting that plausible-but-challenging is a good anchor). I think in those cases the key argument was about whether you need this to generalize far, since both me and MIRI think it's a kind of implausible generalization to go out to infinity rather than becoming distorted at some point along the way, but I am more optimistic about making a series of "short hops" where models generalize helpfully to being moderately smarter and then they can carry out the next step of training for you.
Comment by paulfchristiano on Some thoughts on risks from narrow, non-agentic AI · 2021-01-20T03:01:32.281Z · LW · GW
I agree that this is probably the key point; my other comment ("I think this is the key point and it's glossed over...") feels very relevant to me.
Comment by paulfchristiano on Some thoughts on risks from narrow, non-agentic AI · 2021-01-20T02:58:16.094Z · LW · GW
I feel like a very natural version of "follow instructions" is "Do things that would the instruction-giver would rate highly." (Which is the generalization I'm talking about.) I don't think any of the arguments about "long horizon versions of tasks are different from short versions" tell us anything about which of these generalizations would be learnt (since they are both equally alien over long horizons).
Other versions like "Follow instructions (without regards to what the training process cares about)" seem quite likely to perform significantly worse on the training set. It's also not clear to me that "follow the spirit of the instructions" is better-specified than "do things the instruction-giver would rate highly if we asked them"---informally I would say the latter is better-specified, and it seems like the argument here is resting crucially on some other sense of well-specification.
On meta-learning: it doesn't seem realistic to think about an AI "trying to get high rewards" on tasks where the time horizon is measured in months or years.
I've trained in simulation on tasks where I face a wide variety of environment, each with a reward signal, and I am taught to learn the dynamics of the environment and the reward and then take actions that lead to a lot of reward. In simulation my tasks can have reasonably long time horizons (as measured by how long I think), though that depends on open questions about scaling behavior. I don't agree with the claim that it's unrealistic to imagine such models generalizing to reality by wanting something-like-reward.
In most of the cases you've discussed, trying to do tasks over much longer time horizons involves doing a very different task [...]
Trying to maximize wealth over 100 minutes is indeed very different from maximizing wealth over 1 year, and is also almost completely useless for basically the same reason (except in domains like day trading where mark to market acts as a strong value function).
My take is that people will be pushed to optimizing over longer horizons because these qualitatively different tasks over short horizons aren't useful. The useful tasks in fact do involve preparing for the future and acquiring flexible influence, and so time horizons long enough to be useful will also be long enough to be relevantly similar to yet longer horizons.
Developers will be incentivized to find any way to get good behavior over long horizons, and it seems like we have many candidates that I regard as plausible and which all seem reasonably likely to lead to the kind of behavior I discuss. To me it feels like you are quite opinionated about how that generalization will work.
It seems like your take is "consequences over long enough horizons to be useful will be way too expensive to use for training," which seems close to 50/50 to me.
I think that throughout your post there's an ambiguity between two types of measurement. Type one measurements are those which we can make easily enough to use as a feedback signal for training AIs. Type two measurements are those which we can make easily enough to tell us whether an AI we've deployed is doing a good job. In general many more things are type-two-measurable than type-one-measurable, because training feedback needs to be very cheap.
I agree that this is a useful distinction and there will be some gap. I think that quantitatively I expect the gap to be much smaller than you do (e.g. getting 10k historical examples of 1-year plans seems quite realistic), and I expect people to work to design training procedures that get good performance on type two measures (roughly by definition), and I guess I'm significantly more agnostic about the likelihood of generalization from the longest type one measures to type two measures.
In other words, we should expect generalisation to long-term tasks to occur via a general motivation to follow our instructions, rather than on a task-specific basis, because the latter is so underspecified. But generalisation via following instructions doesn't have a strong bias towards easily-measurable goals.
I'm imagining systems generalizing much more narrowly to the evaluation process used during training. This is still underspecified in some sense (are you trying to optimize the data that goes into SGD, or the data that goes into the dataset, or the data that goes into the sensors?) and in the limit that basically leads to influence-maximization and continuously fades into scenario 2. It's also true that e.g. I may be able to confirm at test-time that there is no training process holding me accountable, and for some of these generalizations that would lead to a kind of existential crisis (where I've never encountered anything like this during training and it's no longer clear what I'm even aiming at). It doesn't feel like these are the kinds of underspecification you are referring to.
Comment by paulfchristiano on Some thoughts on risks from narrow, non-agentic AI · 2021-01-19T17:37:32.622Z · LW · GW
We do need to train them by trial and error, but it's very difficult to do so on real-world tasks which have long feedback loops, like most of the ones you discuss. Instead, we'll likely train them to have good reasoning skills on tasks which have short feedback loops, and then transfer them to real-world with long feedback loops. But in that case, I don't see much reason why systems that have a detailed understanding of the world will have a strong bias towards easily-measurable goals on real-world tasks with long feedback loops.
I think this is the key point and it's glossed over in my original post, so it seems worth digging in a bit more.
I think there are many plausible models that generalize successfully to longer horizons, e.g. from 100 days to 10,000 days:
• Acquire money and other forms of flexible influence, and then tomorrow switch to using a 99-day (or 9999-day) horizon policy.
• Have a short-term predictor, and apply it over more and more steps to predict longer horizons (if your predictor generalizes then there are tons of approaches to acting that would generalize).
• Deductively reason about what actions are good over 100 days (vs 10,000 days), since deduction appears to generalize well from a big messy set of facts to new very different facts.
• If I've learned to abstract seconds into minutes, minutes into hours, hours into days, days into weeks, and then plan over weeks, its pretty plausible that the same procedure can abstract weeks into months and months into years. (It's kind of like I'm now I'm working on a log scale and asking the model to generalize from 1, 2, ..., 10 to 11, 12, 13.)
• Most possible ways of reasoning are hard to write down in a really simple list, but I expect that many hard-to-describe models also generalize. If some generalize and some do not, then training my model over longer and longer horizons (3 seconds, 30 seconds, 5 minutes...) will gradually knock out the non-generalizing modes of reasoning and leave me with the modes that do generalize to longer horizons.
This is roughly why I'm afraid that models we train will ultimately be able to plan over long horizons than those that appear in training.
But many of these would end up pursuing goals that are closely related to the goals they pursue over short horizons (and in particular the first 4 above seem like they'd all be undesirable if generalizing from easily-measured goals, and would lead to the kinds of failures I describe in part I of WFLL).
I think one reason that my posts about this are confusing is that I often insist that we don't rely on generalization because I don't expect it to work reliably in the way we hope. But that's about what assumptions we want to make when designing our algorithms---I still think that the "generalizes in the natural way" model is important for getting a sense of what AI systems are going to do, even if I think there is a good chance that it's not a good enough approximation to make the systems do exactly what we want. (And of course I think if you are relying on generalization in this way you have very little ability to avoid the out-with-a-bang failure mode, so I have further reasons to be unhappy about relying on generalization.)
Comment by paulfchristiano on Some thoughts on risks from narrow, non-agentic AI · 2021-01-19T16:59:10.527Z · LW · GW
In the second half of WFLL, you talk about "systems that have a detailed understanding of the world, which are able to adapt their behavior in order to achieve specific goals". Does the first half of WFLL also primarily refer to systems with these properties? And if so, does "reasoning honed by trial-and-error" refer to the reasoning that those systems do?
Yes.
If yes, then this undermines your core argument that "[some things] can’t be done by trial and error. To solve such tasks we need to understand what we are doing and why it will yield good outcomes", because "systems that have a detailed understanding of the world" don't need to operate by trial and error; they understand what they're doing.
I agree that it's only us who are operating by trial and error---the system understands what it's doing. I don't think that undermines my argument. The point is that we pick the system, and so determine what it's doing, by trial and error, because we have no understanding of what it's doing (under the current paradigm). For some kinds of goals we may be able to pick systems that achieve those goals by trial and error (modulo empirical uncertainty about generalization, as discussed in the second part). For other goals there isn't a plausible way to do that.
We do need to train them by trial and error, but it's very difficult to do so on real-world tasks which have long feedback loops, like most of the ones you discuss. Instead, we'll likely train them to have good reasoning skills on tasks which have short feedback loops, and then transfer them to real-world with long feedback loops. But in that case, I don't see much reason why systems that have a detailed understanding of the world will have a strong bias towards easily-measurable goals on real-world tasks with long feedback loops.
To clarify your position: if I train a system that makes good predictions over 1 minute and 10 minutes and 100 minutes, is your position that there's not much reason that this system would make a good prediction over 1000 minutes? Analogously, if I train a system by meta-learning to get high rewards over a wide range of simulated environments, is your position that there's not much reason to think it will try to get high rewards when deployed in the real world?
I consider those pretty wide open empirical questions. The view that we can get good generalization of this kind is fairly common within ML.
I do agree once you generalize motivations from easily measurable tasks with short feedback loops to tasks with long feedback loops then you may also be able to get "good" generalizations, and this is a way that you can solve the alignment problem. It seems to me that there are lots of plausible ways to generalize to longer horizons without also generalizing to "better" answers (according to humans' idealized reasoning).
(Another salient way in which you get long horizons is by doing something like TD learning, i.e. train a model that predicts its own judgment in 1 minute. I don't know if it's important to get into the details of all the ways people can try to get things to generalize over longer time horizons, it seems like there are many candidates. I agree that there are analogously candidates for getting models to optimize the things we want even if we can't measure them easily, and as I've said I think it's most likely those techniques will be successful, but this is a post about what happens if we fail, and I think it's completely unclear that "we can generalize to longer horizons" implies "we can generalize from the measurable to the unmeasurable.".)
(Analogously: when you put humans in a new domain, and give them tasks and feedback via verbal instructions, then we can quickly learn sophisticated concepts in that new domain, and optimise for those, not just the easily-measured concepts in that new domain.)
When we deploy humans in the real world they do seem to have many desires resembling various plausible generalizations of evolutionary fitness (e.g. to intrinsically want kids even in unfamiliar situations, to care about very long-term legacies, etc.). I totally agree that humans also want a bunch of kind of random spandrels. This is related to the basic uncertainty discussed in the previous paragraphs. I think the situation with ML may well differ because, if we wanted to, we can use training procedures that are much more likely to generalize than evolution.
I don't think it's relevant to my argument that humans can learn sophisticated concepts in a new domain, the question is about the motivations of humans.
Why is your scenario called "You get what you measure" if you're agnostic about whether we actually get what we measure, even on the level of individual AIs?
Or do you mean part 1 to be the case where we do get what we measure, and part 2 to be the case where we don't?
Yes, I'm saying that part 1 is where you are able to get what you measure and part 2 is where you aren't.
Also, as I say, I expect the real world to be some complicated mish-mash of these kinds of failures (and for real motivations to be potentially influenced both by natural generalizations of what happens at training time and also by randomness / architecture / etc., as seems to be the case with humans).
The case in which this is more worrying is when an organisation's success is determined by (for example) whether politicians like it, and politicians only pay attention to easily-measurable metrics. In this case, organisations which pursue easily-measured goals will be more successful than ones which pursue the goals the politicians actually want to achieve. This is why I make the argument that actually the pressure on politicians to pursue easily-measurable metrics is pretty weak (hence why they're ignoring most economists' recommendations on how to increase GDP).
Wanting to earn more money or grow users or survive over the long term is also an easily measured goal, and in practice firms crucially exploit the fact that these goals are contiguous with their shorter easily measured proxies. Non-profits that act in the world often have bottom-line metrics that they use to guide their action and seem better at optimizing goals that can be captured by such metrics (or metrics like donor acquisition).
The mechanism by which you are better at pursuing easily-measureable goals is primarily via internal coherence / stability.
I agree that you've described some potential harms; but in order to make this a plausible long-term concern, you need to give reasons to think that the harms outweigh the benefits of AI enhancing (the effective capabilities of) human reasoning.
I've said that previously human world-steering is the only game in town but soon it won't be, so the future is more likely to be steered in ways that a human wouldn't steer it, and that in turn is more likely to be a direction humans don't like. This doesn't speak to whether the harms on balance outweigh the benefits, which would require an analysis of the benefits but is also pretty irrelevant to my claim (especially so given that all of the world-steerers enjoy these benefits at all and we are primarily concerned with relative influence over the very long term). I'm trying to talk about how the future could get steered in a direction that we don't like if AI development goes in a bad direction, I'm not trying to argue something like "Shorter AI timelines are worse" (which I also think is probably true but about which I'm more ambivalent).
If you'd written a comparable post a few centuries ago talking about how human physical power will lose out to inhuman physical power, I would have had the same complaint.
I don't see a plausible way that humans can use physical power for some long-term goals and not other long-term goals, whereas I've suggested two ways in which automated reasoning may be more easily applied to certain long-term goals (namely the goals that are natural generalizations of training objectives, or goals that are most easily discovered in neural networks).
I classify Facebook's newsfeed as future-steering in a weak sense (it steers the future towards political polarisation), but non-agentic. Do you agree with this?
If Facebook's news feed would generate actions chosen to have the long-term consequence of increasing political polarization then I'd say it was steering the future towards political polarization. (And I assume you'd say it was an agent.)
As is, I don't think Facebook's newsfeed steers the future towards political polarization in a meaningful sense (it's roughly the same as a toaster steering the world towards more toast).
Maybe that's quantitatively just the same kind of thing but weak, since after all everything is about generalization anyway. In that case the concern seems like it's about world-steering that scales up as we scale up our technology/models improve (such that they will eventually become competitive with human world-steering), whereas the news feed doesn't scale up since it's just relying on some random association about how short-term events X happen to lead to polarization (and nor will a toaster if you make it better and better at toasting). I don't really have views on this kind of definitional question, and my post isn't really relying on any of these distinctions.
Something like A/B testing is much closer to future-steering, since scaling it up in the obvious way (and scaling to effects across more users and longer horizons rather than independently randomizing) would in fact steer the future towards whatever selection criteria you were using. But I agree with your point that such systems can only steer the very long-term future once there is some kind of generalization.
Comment by paulfchristiano on Some thoughts on risks from narrow, non-agentic AI · 2021-01-19T02:07:44.676Z · LW · GW
If it’s via a deliberate plan to suppress them while also overcoming human objections to doing so, then that seems less like a narrow system “optimising for an easily measured objective” and more like an agentic and misaligned AGI
I didn't mean to make any distinction of this kind. I don't think I said anything about narrowness or agency. The systems I describe do seem to be optimizing for easily measurable objectives, but that seems mostly orthogonal to these other axes.
I'm pretty agnostic on whether AI will in fact be optimizing for the easily measured objectives used in training or for unrelated values that arise naturally in the learning process (or more likely some complicated mix), and part of my point is that it doesn't seem to much mater.
Secondly, let’s talk about existing pressures towards easily-measured goals. I read this as primarily referring to competitive political and economic activity - because competition is a key force pushing people towards tradeoffs which are undesirable in the long term.
I'm saying: it's easier to pursue easily-measured goals, and so successful organizations and individuals tend to do that and to outcompete those whose goals are harder to measure (and to get better at / focus on the parts of their goals that are easy to measure, etc.). I'm not positing any change in the strength of competition, I'm positing a change in the extent to which goals that are easier to measure are in fact easier to pursue.
Regarding the extent and nature of competition I do think I disagree with you fairly strongly but it doesn't seem like a central point.
the US, for example, doesn’t seem very concerned right now about falling behind China.
I think this is in fact quite high on the list of concerns for US policy-makers and especially the US defense establishment.
Further, I don’t see where the meta-level optimisation for easily measured objectives comes from.
Firms and governments and people pursue a whole mix of objectives, some of which are easily measured. The ones pursuing easily-measured objectives are more successful, and so control an increasing fraction of resources.
So if narrow AI becomes very powerful, we should expect it to improve humanity’s ability to steer our trajectory in many ways.
I don't disagree with this at all. The point is that right now human future-steering is basically the only game in town. We are going to introduce inhuman reasoning that can also steer the future, and over time human reasoning will lose out in relative terms. (If you classify all future-steering machinery as "agentic" then evidently I'm talking about agents and I agree with the informal claim that "non-agentic" reasoning isn't concerning.) That's compatible with us benefiting enormously, if all of those benefits also accrue to automated reasoners---as your examples seem to. We will try to ensure that all this new reasoning will benefit humanity, but I describe two reasons that might be difficult and say a little bit about how that difficulty might materialize.
I don't really know if or how this is distinct from what you call the second species argument. It feels like you are objecting to a distinction I'm not intending to make.
Comment by paulfchristiano on Thoughts on Iason Gabriel’s Artificial Intelligence, Values, and Alignment · 2021-01-16T02:26:43.676Z · LW · GW
If it turns out not to be possible then the AI should never fade away.
If the humans in the container succeed in becoming wiser, then hopefully it is wise for us to leave this decision up to them than to preemptively make it now (and so I think the situation is even better than it sounds superficially).
But it seems to me that there are a small number of humans on this planet who have moved some way in the direction of being fit to run the world, and in time, more humans could move in this direction, and could move further.
It seems like the real thing up for debate will be about power struggles amongst humans---if we had just one human, then it seems to me like the grandparent's position would be straightforwardly incoherent. This includes, in particular, competing views about what kind of structure we should use to govern ourselves in the future.
Comment by paulfchristiano on Thoughts on Iason Gabriel’s Artificial Intelligence, Values, and Alignment · 2021-01-16T02:20:27.190Z · LW · GW
I buy into the delegation framing, but I think that the best targets for delegation look more like "slightly older and wiser versions of ourselves with slightly more space" (who can themselves make decisions about whether to delegate to something more alien). In the sand-pit example, if the child opted into that arrangement then I would say they have effectively delegated to a version of themselves who is slightly constrained and shaped by the supervision of the adult. (But in the present situation, the most important thing is that the parent protects them from the outside the world while they have time to grow.)
Comment by paulfchristiano on Thoughts on Iason Gabriel’s Artificial Intelligence, Values, and Alignment · 2021-01-16T02:17:19.198Z · LW · GW
I basically agree that humans ought to use AI to get space, safety and time to figure out what we want and grow into the people we want to be before making important decisions. This is (roughly) why I'm not concerned with some of the distinctions Gabriel raises, or that naturally come to mind when many people think of alignment.
That said, I feel your analogy misses a key point: while the child is playing in their sandbox, other stuff is happening in the world---people are building factories and armies, fighting wars and grabbing resources in space, and so on---and the child will inherit nothing at all unless their parent fights for it.
So without (fairly extreme) coordination, we need to figure out how to have the parent acquire resources and then ultimately "give" those resources to the child. It feels like that problem shouldn't be much harder than the parent acquiring resources for themselves (I explore this intuition some in this post on the "strategy stealing" assumption), so that this just comes down to whether we can create a parent who is competent while being motivated to even try to help the child. That's what I have in mind while working on the alignment problem.
On the other hand, given strong enough coordination that the parent doesn't have to fight for their child, I think that the whole shape of the alignment problem changes in more profound ways.
I think that much existing research on alignment, and my research in particular, is embedded in the "agency hand-off paradigm" only to the extent that is necessitated by that situation.
I do agree that my post on indirect normativity is embedded in a stronger version of the agency hand-off paradigm. I think the main reason for taking an approach like that is that a human embedded in the physical world is a soft target for would-be attackers and creates a. If we are happy handing off control to a hypothetical version of ourselves in the imagination of our AI, then we can achieve additional security by doing so, and this may be more appealing than other mechanisms to achieve a similar level of security (like uploading or retreating to a secure physical sanctuary). In some sense all of this is just about saying what it means to ultimately "give" the resources to the child, and it does so by trying to construct an ideal environment for them to become wiser after which they will be mature enough to provide more direct instructions. (But in practice I think that these proposals may involve a jarring transition that could be avoided by using a physical sanctuary instead or just ensuring that our local environments remain hospitable.)
Overall it feels to me like you are coming from a similar place to where I was when I wrote this post on corrigibility, and I'm curious if there are places where you would part ways with that perspective (given the consideration I raised in this comment).
(I do think "aligned with who?" is a real question since the parent needs to decide which child will ultimately get the resources, or else if there are multiple children playing together then it matters a lot how the parent's decisions shape the environment that will ultimately aggregate their preferences.)
Comment by paulfchristiano on What technologies could cause world GDP doubling times to be <8 years? · 2020-12-12T23:21:19.112Z · LW · GW
France is the other country for which Our World in Data has figures going back to 1400 (I think from Maddison), here's the same graph for France:
There is more crazy stuff going on, but broadly the picture looks the same and there is quite a lot of acceleration between 1800 and the 1950s. The growth numbers are 0.7% for 1800-1850, 1.2% for 1850-1900, 1.2% for 1900-1950, 2.8% for 1950-2000.
And for the even messier case of China:
Growth averages 0 from 1800 to 1950, and then 3.8% from 1950-2000 and 6.9% from 2000-2016.
Comment by paulfchristiano on What technologies could cause world GDP doubling times to be <8 years? · 2020-12-12T22:56:27.487Z · LW · GW
I think past acceleration is mostly about a large number of improvements that build on one another rather than a small number of big wins (as Katja points out), and future acceleration will probably be more of the same. It seems like almost all of the tasks that humans currently do could plausibly be automated without "AGI" (though it depends on how exactly you define AGI), and if you improve human productivity a bunch in enough industries then you are likely to have faster growth.
I expect "21st century acceleration is about computers taking over cognitive work from humans" will be the analog of "The industrial revolution is about engines taking over mechanical work from humans / beasts of burden."
From that perspective, asking "What technology short of AGI would take over cognitive work from humans, and how?" is analogous to asking "What technology short of a universal actuator would take over mechanical work from humans, and how?" The answer is just: a bunch of stuff that's specific to the details of each type of work.
Thoughts on some particular technologies, kind of at random:
• I think that most of that automation is likely to involve new software, and so the size of the software industry is likely to grow a bunch. Increasing productivity in the software industry (likely via ML) would then be an important driver of productivity growth despite software currently being a small share of GDP.
• I think that cheap solar power, automation of manufacturing and construction (including manufacturing industrial tools and construction of factories), and automation of service jobs are also very important stories.
• I think that west probably could be growing considerably faster even without qualitative technological change, so part of the story may be western countries either getting out of their current slump or being overtaken.
The other part of your post is about how much qualitative change would correspond to a doubling of growth rates. I think you are moderately underestimating the extent of historical acceleration and so overestimating how much qualitative change would be needed:
• I think the US over the last 200 years is a particularly bad comparison because at the beginning of the period it was benefiting a lot from colonization. Below I talk about the UK which I think is probably more representative. I chose the UK as the the most natural frontier economy after the industrial revolution, but I expect the exercise would be similar for other countries without complications.
• Looking at growth over the last 200 years hides the fact that there was a period of more rapid acceleration followed by a stagnation. If we instead compared 1800 to 1950 we'd see a larger change in growth rates accompanied by a smaller qualitative change. So that's probably more useful if you are looking for an existence proof (and I think low levels of current growth likely make acceleration easier).
In 1800 the US was growing rapidly in significant part because colonists were still taking new land and then increasing utilization of that land. So over the last 200 years you have a decrease in some kinds of growth and an increase in others. I don't know much about this and it may be completely wrong, but given that the US was growing so much faster than the rest of the world and that there's such a simple explanation that seems to check out, that's what I'd assume is going on. If that's right then it can still be OK to use the US as an example but you can't use raw growth numbers to infer something about technological change.
If you want to see what's happening in frontier economies since the industrial revolution then it seems more natural to use something like per capita GDP in the UK. If I look up the GDP per capita in the UK time series at Our World in Data and turn that into a graph of (GDP per capita growth rate) vs (time), I get:
So it seems to me like things really did change a lot as technology improved, growing from 0.4% in 1800-1850, to 1% in 1850-1900, to .8% in 1900-1950, to 2.4% in 1950-2000. What we're talking about is a further change similar in scope to the change from 1800 to 1850 or from 1900 to 1950.
(I don't know if there are other reasons the UK isn't representative. I think the most obvious candidate would be that 1900-1950 was a really rough period for the UK, and then 1950-2000 potentially involves some catch-up growth.)
Comment by paulfchristiano on What technologies could cause world GDP doubling times to be <8 years? · 2020-12-12T22:28:18.546Z · LW · GW
the acceleration due to the agricultural revolution was due to agriculture + a few other things maybe
This linguistic trick only seems to work because you have a single word ("agriculture") that describes most human work at that time. If you want the analogous level of description for "what is going to improve next?" just look up what people are doing in the modern economy. If you need more words to describe the modern economy, then I guess it's going to be "more technologies" that are responsible this time (though in the future, when stuff we do today is a smaller part of the economy, they may describe it using a smaller number of words).
we can say the acceleration due to the industrial revolution was due to engines + international trade + mass-production methods + scientific institutions + capitalist institutions + a few other things I'm forgetting
If you are including lists that long then I guess the thing that's going to change is "improved manufacturing + logistics + construction + retail + finance" or whatever, just sample all the stuff that humans do and then it gets improved.
(I think "a few other things" is actually quite a lot of other things, unless you construe "mass-production" super broadly.)
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-25T05:37:08.599Z · LW · GW
Many people's view of a UBI depends on whether recipients in fact stop working. For example, people are interested in running studies on that question, often with a clear indication that they would support a UBI if and only if recipients don't significantly decrease hours worked.
What are we to make of this concern?
A natural way to understand it is to separate the effects of UBI into {recipients may decide to reduce hours worked} from {all other effects}. Then the concern could be understood as a suggestion that this change in hours worked is bad even if the the other effects of a UBI would be good. Put differently, people who express this concern may believe that a UBI would be good if we magically causally intervened to ensure that people continued working the same amount, while the effects of UBI alone are more uncertain.
The reason to respond to this view, rather than directly analyzing all the effects of a UBI together, is that it seemed to me to indicate a moral error that could be separated from the other complex empirical questions at stake.
(Given that this seems like a kind of unenlightening thread about a topic that's not super important to me, I'll probably drop it.)
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-24T16:35:05.803Z · LW · GW
Capital costs in food production are significant. Land will still cost money, materials will still cost money, and machines will still cost money (though the cost of machines above and beyond the raw material cost could rapidly fall).
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-24T16:30:58.724Z · LW · GW
You could argue that people don't take that into account when deciding not to work (so that I can make the world better by forcing people to work for their own benefit).
The first step would be believing that people who stop working because they don't have to end up being less healthy, I have no idea if that's true. It's a bit hard to study, since interventions like "inherit a bunch of money" and "receive a UBI" mostly affect health via the channel of "now you have a bunch of money," and that obscures any negative effect from not having to work.
(And on the other hand, comparing the employed to the unemployed is extremely confounded and I'm skeptical it gives any evidence on this question. It would be pretty surprising if people who had a harder time finding work weren't less healthy and happy.)
The best would be to compare people receiving an unconditional transfer to people receiving a transfer with a work requirement, but I'm not aware of studies on that.
You could also have some anecdotal evidence about that. People I know who are voluntarily unemployed seem to eat and exercise better, but they are probably not representative of the people affected by a welfare work requirement.
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-24T16:25:55.933Z · LW · GW
If we're talking economic efficiency, then your own utility should be included.
My starting assumption is that I decided not to work because I believe I am better off. We are wondering if my decision to stop working was inefficient, i.e. if it makes the world worse off despite me voluntarily choosing to do it. So the salient questions are (i) how does this affect everyone else? Does it cause harms to the rest of the world? (ii) am I predictably making a mistake (e.g. by not adequately accounting for the ways in which working benefits my future self)?
In a UBI scenario, you should be able to stop working while still consuming
Yes, I'm talking about the additional consumption if you earn+spend more money.
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-23T02:20:01.489Z · LW · GW
Basically the whole case comes down to the externalities of working+consuming though (both the case in favor and the case against). It seems the point stands that the externalities of working and consuming are both relevant, there's not really an asymmetry there, and I don't see how this is related to "getting distracted by the money flows."
Like, I might produce value because some gets more surplus from hiring me than they would have gotten from hiring someone else (in the competitive limit that gap converges to 0 and they are indifferent, but presumably it won't be 0 in the real world). And similarly I might produce value because someone gets more surplus from selling to me than they would have gotten from selling to me. But those things seem symmetrical.
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-22T23:59:18.262Z · LW · GW
I may still be misunderstanding.
When I work I create value for the world, which is ultimately measured in benefits to other humans. And when I go spend my money I impose a cost on the world which is ultimately measured in the effort those people put in to give me what I bought, or the other people who could have had the thing that did not, or whatever.
It seems like the question is about the balance between the value I create by working, and the value others lose when I consume, isn't it? It's relevant both how much other people value what I do for them, and how much other people value the effort they put in for me.
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-22T23:53:41.811Z · LW · GW
If I earn less money than I spend less money. The question is whether the combination of {me working} + {me consuming} is better or worse for the rest of the world than {me relaxing}, since what's at issue is precisely whether individuals who decide not to work are a sign of social inefficiency.
For the purpose of that comparison, the consumption seems just as relevant as the production. You seem to be disagreeing, but I'm not sure why. Yes, it's true that if I give someone a UBI they will also spend the UBI, and that's the same as any redistribution, but that's not relevant to analyzing whether their decision to not work is socially inefficient.
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-22T17:56:35.123Z · LW · GW
It seems like their problem is that they can't pay for a UBI without crazy distortions (and likely can't raise enough money for a large UBI regardless).
I'm not sure what exactly the reductio is for the medieval society. Giving low-income workers money will generally raise the price of goods produced by low-income workers but that doesn't generally indicate any efficiency loss.
I do definitely agree that paying someone a $100 UBI causes a loss of$100 to the taxpayers who paid for it. But that happens regardless of whether the recipients stop working.
Comment by paulfchristiano on It’s not economically inefficient for a UBI to reduce recipient’s employment · 2020-11-22T17:25:02.221Z · LW · GW
That society might gain additional benefit from how you spend your money is merely coincidental.
I'm not sure what "coincidental" means here. The question is how much more or less than \$100 of value you create by working, and that seems to depend about as much on how you spend your money as it does on how you earn your money.
Comment by paulfchristiano on Some AI research areas and their relevance to existential safety · 2020-11-21T02:09:55.394Z · LW · GW
A number of blogs seem to treat [AI existential safety, AI alignment, and AI safety] as near-synonyms (e.g., LessWrong, the Alignment Forum), and I think that is a mistake, at least when it comes to guiding technical work for existential safety.
I strongly agree with the benefits of having separate terms and generally like your definitions.
In this post, AI existential safety means “preventing AI technology from posing risks to humanity that are comparable or greater than human extinction in terms of their moral significance.”
I like "existential AI safety" as a term to distinguish from "AI safety" and agree that it seems to be clearer and have more staying power. (That said, it's a bummer that "AI existential safety forum" is a bit of a mouthful.)
If I read that term without a definition I would assume it meant "reducing the existential risk posed by AI." Hopefully you'd be OK with that reading. I'm not sure if you are trying to subtly distinguish it from Nick's definition of existential risk or if the definition you give is just intended to be somewhere in that space of what people mean when they say "existential risk" (e.g. the LW definition is like yours).
Comment by paulfchristiano on Some AI research areas and their relevance to existential safety · 2020-11-21T01:38:44.343Z · LW · GW
Outcome C is most naturally achieved using "direct democracy" TAI, i.e. one that collects inputs from everyone and aggregates them in a reasonable way. We can try emulating democratic AI via single user AI, but that's hard because:
I'm not sure what's most natural, but I do consider this a fairly unlikely way of achieving outcome C.
I think the best argument for this kind of outcome is from Wei Dai, but I don't think it gets you close to the "direct democracy" outcome. (Even if you had state control and AI systems aligned with the state, it seems unlikely and probably undesirable for the state to be replaced with an aggregation procedure implemented by the AI itself.)
Comment by paulfchristiano on Some AI research areas and their relevance to existential safety · 2020-11-21T01:25:07.582Z · LW · GW
It's always possible to say, solving the single/single alignment problem will prevent anything like that from happening in the first place, but why put all your hopes on plan A, when plan B is relatively neglected?
The OP writes "contributions to AI alignment are also generally unhelpful to existential safety." I don't think I'm taking a strong stand in favor of putting all our hopes on plan A, I'm trying to understand the perspective on which plan B is much more important even before considering neglectedness.
It seems premature to say, in advance of actually seeing what such research uncovers, whether the relevant mechanisms and governance improvements are exactly the same as the improvements we need for good governance generally, or different.
I agree that would be premature. That said, I still found it notable that OP saw such a large gap between the importance of CSC and other areas on and off the list (including MARL). Given that I would have these things in a different order (before having thought deeply), it seemed to illustrate a striking difference in perspective. I'm not really trying to take a strong stand, just using it to illustrate and explore that difference in perspective.
Comment by paulfchristiano on Some AI research areas and their relevance to existential safety · 2020-11-20T06:59:24.757Z · LW · GW
Outcome B: Progress in atomic AI alignment keeps up with progress in AI capability, but progress in social AI alignment doesn't keep up. Transformative AI is aligned with a small fraction of the population, resulting in this minority gaining absolute power and abusing it to create an extremely inegalitarian future. Wars between different factions are also a concern.
It's unclear to me how this particular outcome relates to social alignment (or at least to the kinds of research areas in this post). Some possibilities:
• Does failure to solve social alignment mean that firms and governments cannot use AI to represent their shareholders and constituents? Why might that be? (E.g. what's a plausible approach to atomic alignment that couldn't be used by a firm or government?)
• Does AI progress occur unevenly such that some group gets much more power/profit, and then uses that power? If so, how would technical progress on alignment help address that outcome? (Why would the group with power be inclined to use whatever techniques we're imagining?) Also, why does this happen?
• Does AI progress somehow complicate the problem of governance or corporate governance such that those organizations can no longer represent their constituents/shareholders? What is the mechanism (or any mechanism) by which this happens? Does social alignment help by making new forms of organization possible, and if so should I just be thinking of it as a way of improving those institutions, or is it somehow distinctive?
• Do we already believe that the situation is gravely unequal (e.g. because governments can't effectively represent their constituents and most people don't have a meaningful amount of capital) and AI progress will exacerbate that situation? How does social alignment prevent that?
(This might make more sense as a question for the OP, it just seemed easier to engage with this comment since it describes a particular more concrete possibility. My sense is that the OP may be more concerned about failures in which no one gets what they want rather than outcome B per se.)
Comment by paulfchristiano on Some AI research areas and their relevance to existential safety · 2020-11-20T04:49:59.265Z · LW · GW
If single/single alignment is solved it feels like there are some salient "default" ways in which we'll end up approaching multi/multi alignment:
• Existing single/single alignment techniques can also be applied to empower an organization rather than an individual. So we can use existing social technology to form firms and governments and so on, and those organizations will use AI.
• AI systems can themselves participate in traditional social institutions. So AI systems that represent individual human interests can interact with each other e.g. in markets or democracies.
I totally agree that there are many important problems in the world even if we can align AI. That said, I remain interested in more clarity on what you see as the biggest risks with these multi/multi approaches that could be addressed with technical research.
For example, let's take the considerations you discuss under CSC:
Third, unless humanity collectively works very hard to maintain a degree of simplicity and legibility in the overall structure of society*, this “alignment revolution” will greatly complexify our environment to a point of much greater incomprehensibility and illegibility than even today’s world. This, in turn, will impoverish humanity’s collective ability to keep abreast of important international developments, as well as our ability to hold the international economy accountable for maintaining our happiness and existence.
One approach to this problem is to work to make it more likely that AI systems can adequately represent human interests in understanding and intervening on the structure of society. But this seems to be a single/single alignment problem (to whatever extent that existing humans currently try to maintain and influence our social structure, such that impairing their ability to do so is problematic at all) which you aren't excited about.
Fourth, in such a world, algorithms will be needed to hold the aggregate global behavior of algorithms accountable to human wellbeing, because things will be happening too quickly for humans to monitor. In short, an “algorithmic government” will be needed to govern “algorithmic society”. Some might argue this is not strictly unnecessary: in the absence of a mathematically codified algorithmic social contract, humans could in principle coordinate to cease or slow down the use of these powerful new alignment technologies, in order to give ourselves more time to adjust to and govern their use. However, for all our successes in innovating laws and governments, I do not believe current human legal norms are quite developed enough to stably manage a global economy empowered with individually-alignable transformative AI capabilities.
Again, it's not clear what you expect to happen when existing institutions are empowered by AI and mostly coordinate the activities of AI.
The last line reads to me like "If we were smarter, when our legal system may no longer be up to the challenge," with which I agree. But it seems like the main remedy is "if we were smarter, we would hopefully work on improving our legal system in tandem with the increasing demands we impose on it."
It feels like the salient actions to take to me are (i) make direct improvements in the relevant institutions, in a way that anticipates the changes brought about by AI but will most likely not look like AI research, (ii) work on improving the relative capability of AI at those tasks that seem more useful for guiding society in a positive direction.
I consider (ii) to be one of the most important kinds of research other than alignment for improving the impact of AI, and I consider (i) to be all-around one of the most important things to do for making the world better. Neither of them feels much like CSC (e.g. I don't think computer scientists are the best people to do them) and it's surprising to me that we end up at such different places (if only in framing and tone) from what seem like similar starting points.
Comment by paulfchristiano on Some AI research areas and their relevance to existential safety · 2020-11-20T04:36:30.580Z · LW · GW
Progress in OODR will mostly be used to help roll out more AI technologies into active deployment more quickly
It sounds like you may be assuming that people will roll out a technology when its reliability meets a certain level X, so that raising reliability of AI systems has no or little effect on the reliability of deployed system (namely it will just be X). I may be misunderstanding.
A more plausible model is that deployment decisions will be based on many axes of quality, e.g. suppose you deploy when the sum of reliability and speed reaches some threshold Y. If that's the case, then raising reliability will improve the reliability and decrease the speed of deployed systems. If you think that increasing the reliability of AI systems is good (e.g. because AI developers want their AI systems to have various socially desirable properties and are limited by their ability to robustly achieve those properties) then this would be good.
I'm not clear on what part of that picture you disagree with or if you think that this is just small relative to some other risks. My sense is that most of the locally-contrarian views in this post are driven by locally-contrarian quantitative estimates of various risks. If that's the case, then it seems like the main thing that would shift my view would be some argument about the relative magnitude of risks. I'm not sure if other readers feel similarly.
Research in this area usually does not involve deep or lengthy reflections about the structure of society and human values and interactions, which I think makes this field sort of collectively blind to the consequences of the technologies it will help build.
This is a plausible view, but I'm not sure what negative consequences you have in mind (or how it affects the value of progress in the field rather than the educational value of hanging out with people in the field).
Incidentally, the main reason I think OODR research is educationally valuable is that it can eventually help with applying agent foundations research to societal-scale safety. Specifically: how can we know if one of the operations (a)-(f) above is safe to perform 1,000,000 times, given that it was safe the first 1,000 times we applied it in a controlled setting, but the setting is changing over time? This is a special case of an OODR question.
That task---how do we test that this system will consistently have property P, given that we can only test property P at training time?---is basically the goal of OODR research. Your prioritization of OODR suggests that maybe you think that's the "easy part" of the problem (perhaps because testing property P is so much harder), or that OODR doesn't make meaningful progress on that problem (perhaps because the nature of the problem is so different for different properties P?). Whatever it is, it seems like that's at the core of the disagreement and you don't say much about it. I think many people have the opposite intuition, i.e. that much of the expected harm from AI systems comes from behaviors that would have been recognized as problematic at training time.
In any case, I see AI alignment in turn as having two main potential applications to existential safety:
1. AI alignment is useful as a metaphor for thinking about how to align the global effects of AI technology with human existence, a major concern for AI governance at a global scale, and
2. AI alignment solutions could be used directly to govern powerful AI technologies designed specifically to make the world safer.
Here is one standard argument for working on alignment. It currently seems plausible that AI systems will be trying to do stuff that no one wants and that this could be very bad if AI systems are much more competent than humans. Prima facie, if the designers of AI systems are able to better control what AI systems are trying to do, then those AI systems are more likely to be trying to do what the developers want. So if we are able to give developers that ability, we can reduce the risk of AI competently doing stuff no one wants.
This isn't really a metaphor, it's a direct path for impact. It's unclear if you think that this argument is mistaken because developers will be able to control what their AI systems are trying to do, because they won't be motivated to deploy AI until they have that control, because it's not much better for AI systems to be trying to do what their developers want, because there are other more important reasons that AI systems could be trying to do stuff that no one wants, because there are other risks unrelated to AI trying to do stuff no one wants, or something else altogether.
(2) is essentially aiming to take over the world in the name of making it safer, which is not generally considered the kind of thing we should be encouraging lots of people to do.
Like you, I'm opposed to plans where people try to take over the world in order to make it safer. But this looks like a bit of a leap. For example, AI alignment may help us build powerful AI systems that help us negotiate or draft agreements, which doesn't seem like taking over the world to make it safer.
Comment by paulfchristiano on My Understanding of Paul Christiano's Iterated Amplification AI Safety Research Agenda · 2020-10-12T03:59:37.745Z · LW · GW
What would a corrigible but not-intent-aligned AI system look like?
Suppose that I think you know me well and I want you to act autonomously on my behalf using your best guesses. Then you can be intent aligned without being corrigible. Indeed, I may even prefer that you be incorrigible, e.g. if I want your behavior to be predictable to others. If the agent knows that I have such a preference then it can't be both corrigible and intent aligned.
Comment by paulfchristiano on Puzzle Games · 2020-10-09T04:44:53.423Z · LW · GW
It's meant to be read before playing, added a comment clarifying.
Comment by paulfchristiano on Puzzle Games · 2020-10-09T01:00:51.355Z · LW · GW
Follow-up now that I've finished.
(This is spoiler'ed as per this post's spoiler policy, but it's designed to provide a rules clarification relevant to the parent and to be read before finishing the game.)
Here's a simplified model of resetting: the game tracks the most recent landmass you've stepped on. When you reset, all trees from that landmass are returned to their initial state. You are moved to the location where you first stepped foot on that landmass.
That model isn't exactly right (since it would make it way too easy for the player to get stuck), but every puzzle is solvable under that model. I had a single solution that would have worked under that model but didn't work under the actual behavior of resetting, which was a tiny bit frustrating but not a big deal.
If you reset half of a raft then it becomes a lone log (this makes it possible to split a long log in two or to rotate a log by integrating it into a raft then resetting). If you reset something that's holding another log up, the other log will fall down. I think there are some tricky corner cases but you never need to deal with any more complicated than those two basics and you can just pretend that you automatically lose if you create a tricky situation (e.g. if you reset when a tree's starting position is occupied).
Comment by paulfchristiano on Hiring engineers and researchers to help align GPT-3 · 2020-10-05T19:28:18.441Z · LW · GW
described by Eliezer as “directly, straight-up relevant to real alignment problems.”
Worth saying that Eliezer still thinks our team is pretty doomed and this is definitely not a general endorsement of our agenda. I feel excited about our approach and think it may yet work, but I believe Eliezer's position is that we're just shuffling around the most important difficulties into the part of the plan that's vague and speculative.
I think it's fair to say that Reflection is on the Pareto frontier of {plays ball with MIRI-style concerns, does mainstream ML research}. I'm excited for a future where either we convince MIRI that aligning prosaic AI is plausible, or MIRI convinces us that it isn't.
Comment by paulfchristiano on Hiring engineers and researchers to help align GPT-3 · 2020-10-05T19:19:02.180Z · LW · GW
I think that "imitate a human who is trying to be helpful" is better than "imitate a human who is writing an article on the internet," even though it's hard to define "helpful." I agree that's not completely obvious for a bunch of reasons.
(GPT-3 is better if your goal is in fact to predict text that people write on the internet, but that's a minority of API applications.)
Comment by paulfchristiano on Hiring engineers and researchers to help align GPT-3 · 2020-10-05T19:07:20.831Z · LW · GW
will these jobs be long-term remote? if not, on what timeframe will they be remote?
We expect to be requiring people to work from the office again sometime next year.
how suitable is the research engineering job for people with no background in ml, but who are otherwise strong engineers and mathematicians?
ML background is very helpful. Strong engineers who are interested in learning about ML are also welcome to apply though no promises about how well we'll handle those applications in the current round.
Comment by paulfchristiano on Hiring engineers and researchers to help align GPT-3 · 2020-10-05T19:05:19.063Z · LW · GW
The team is currently 7 people and we are hiring 1-2 additional people over the coming months.
I am optimistic that our team and other similar efforts will be hiring more people in the future and continuously scaling up, and that over the long term there could be a lot of people working on these issues.
(The post is definitely written with that in mind and the hope that enthusiasm will translate into more than just hires in the current round. Growth will also depend on how strong the pool of candidates is.)
Comment by paulfchristiano on Puzzle Games · 2020-10-02T02:06:24.677Z · LW · GW
I totally understand why resetting had to be kind of complicated / ad hoc, and I think that this was a reasonable compromise. I don't think uncertainty about resetting matters much in the scheme of things, it's a great game.
Comment by paulfchristiano on Puzzle Games · 2020-10-01T06:39:08.389Z · LW · GW
Partial answer to my question (significantly more spoilers):
You can get to the credits without resetting. Extra puzzles appear to require resetting but maybe not in a very subtle way. I don't know if resetting has a simple description. It definitely depends on invisible facts about the map.
Comment by paulfchristiano on “Unsupervised” translation as an (intent) alignment problem · 2020-09-30T21:39:42.614Z · LW · GW
Good point, changed.
Originally it was "as an alignment problem" but this has the problem that it also refers to "aligning" unaligned datasets. The new way is bulkier but probably better overall.
Comment by paulfchristiano on “Unsupervised” translation as an (intent) alignment problem · 2020-09-30T18:03:07.596Z · LW · GW
The researchers analyzed the Klingon phrase "מהדקי נייר" and concluded it roughly means
If the model is smart, this is only going to work if the (correct) translation is reasonably likely to appear in your English text database. You are (at best) going to get a prediction of what human researchers would conclude after studying Klingon, your model isn't actually going to expand what humans can do.
Consider a Debate experiment in which each of the two players outputs an entire English-Klingon dictionary (as avturchin mentioned). The judge then samples a random Klingon passage and decides which of the two dictionaries is more helpful for understanding that passage (maybe while allowing the two players to debate over which dictionary is more helpful).
This is basically what the helper model does, except:
• For competitiveness you should learn and evaluate the dictionary at the same time you are training the model, running a debate experiment many times where debaters have to output a full dictionary would likely be prohibitively expensive.
• Most knowledge about language isn't easily captured in a dictionary (for example, a human using a Spanish-English dictionary is a mediocre translator), so we'd prefer have a model that answers questions about meaning than have a model that outputs a static dictionary.
• I don't know what standard you want to use for "helpful for understanding the passage" but I think "helps predict the next word correctly" is probably the best approach (since the goal is to be competitive and that's how GPT learned).
After making those changes we're back at the learning the prior proposal.
I think that proposal may work passably here because we can potentially get by with a really crude prior---basically we think "the helper should mostly just explain the meaning of terms" and then we don't need to be particularly opinionated about which meanings are more plausible. I agree that the discussion in the section "A vague hope" is a little bit too pessimistic for the given context of unaligned translation.
Comment by paulfchristiano on Puzzle Games · 2020-09-28T16:11:10.406Z · LW · GW
Question about Monster's Expedition:
The reset mechanic seems necessary to make the game playable in practice, but it seems very unsatisfying. It is unclear how you'd make it work in a principled way; the actual implementation seems extremely confusing, seems to depend on invisible information about the environment, and has some weird behaviors that I think are probably bugs. Unfortunately, it currently seems possible that probing the weirdest behaviors of resetting (e.g. breaking conservation-of-trees) could be the only way to access some places. It's also possible that mundane applications of resetting are essential but you aren't intended to explore weird edge cases, which would be the least satisfying outcome of all.
So two questions:
1. Is it possible to beat the game without resetting? Can I safely ignore it as a mechanic? This is my current default assumption and it's working fine so far.
2. Is the reset mechanic actually lawful/reasonable and I just need to think harder?
(Given the quality of the game I'm hoping that at least one of those is "yes." If "no answer" seems like the best way to enjoy the game I'm open to that as well.)
Comment by paulfchristiano on Puzzle Games · 2020-09-27T23:19:31.335Z · LW · GW
This list is almost the same as mine. I would include Hanano Puzzle 2 at tier 2 and Cosmic Express at tier 3. I haven't played Twisty Little Passages or Kine though I'll try them on this recommendation.
We're putting together a self-contained campaign for engine-game.com which is aiming to be Tier-2-according-to-Paul. We'll see if other folks agree when it's done. It has a very different flavor from the other games on the list.
Comment by paulfchristiano on Distributed public goods provision · 2020-09-27T16:24:08.616Z · LW · GW
I think you inevitably need to answer "What is the marginal impact of funding?" if you are deciding how much to fund something.
(I will probably write about approaches to randomizing to be maximally efficient with research time at some point in the future. My current plan is something like: write out public goods I know of that I benefit from, then sample one of them to research and fund by 1/p(chosen) more than I normally would.)
This isn't really meant to be a quick rule of thumb, it's meant to be a way to answer the question at all.
Comment by paulfchristiano on Search versus design · 2020-08-17T14:46:52.045Z · LW · GW
I liked this post.
I'm not sure that design will end up being as simple as this picture makes it look, no matter how well we understand it---it seems like factorization is one kind of activity in design, but it feels like overall "design" is being used as a kind of catch-all that is probably very complicated.
An important distinction for me is: does the artifact work because of the story (as in "design"), or does the artifact work because of the evaluation (as in search)?
This isn't so clean, since:
• Most artifacts work for a combination of the two reasons---I design a thing then test it and need a few iterations---there is some quantitative story where both factors almost always play a role for practical artifacts.
• There seem to many other reasons things work (e.g. "it's similar to other things that worked" seems to play a super important role in both design and search).
• A story seems like it's the same kind of thing as an artifact, and we could also talk about where *it* comes from. A story that plays a role in a design itself comes from some combination of search and design.
• During design it seems likely that humans rely very extensively on searching against mental models, which may not be introspectively available to us as a search but seems like it has similar properties.
Despite those and more complexities, it feels to me like if there is a clean abstraction it's somewhere in that general space, about the different reasons why a thing can work.
Post-hoc stories are clearly *not* the "reason why things work" (at least at this level of explanation). But also if you do jointly search for a model+helpful story about it, the story still isn't the reason why the model works, and from a safety perspective it might be similarly bad.
Comment by paulfchristiano on How should AI debate be judged? · 2020-07-22T01:41:35.308Z · LW · GW
Yeah, I've heard (through the grapevine) that Paul and Geoffrey Irving think debate and factored cognition are tightly connected
For reference, this is the topic of section 7 of AI Safety via Debate.
In the limit they seem equivalent: (i) it's easy for HCH(with X minutes) to discover the equilibrium of a debate game where the judge has X minutes, (ii) a human with X minutes can judge a debate about what would be done by HCH(with X minutes).
The ML training strategies also seem extremely similar, in the sense that the difference between them is smaller than design choices within each of them, though that's a more detailed discussion.
Comment by paulfchristiano on How should AI debate be judged? · 2020-07-22T01:34:35.946Z · LW · GW
I'm a bit confused why you would make the debate length known to the debaters. This seems to allow them to make indefensible statements at the very end of a debate, secure in the knowledge that they can't be critiqued. One step before the end, they can make statements which can't be convincingly critiqued in one step. And so on.
[...]
The most salient reason for me ATM is the concern that debaters needn't structure their arguments as DAGs which ground out in human-verifiable premises, but rather, can make large circular arguments (too large for the debate structure to catch) or unbounded argument chains (or simply very very high depth argument trees, which contain a flaw at a point far too deep for debate to find).
If I assert "X because Y & Z" and the depth limit is 0, you aren't intended to say "Yup, checks out," unless Y and Z and the implication are self-evident to you. Low-depth debates are supposed to ground out with the judge's priors / low-confidence in things that aren't easy to establish directly (because if I'm only updating on "Y looks plausible in a very low-depth debate" then I'm going to say "I don't know but I suspect X" is a better answer than "definitely X"). That seems like a consequence of the norms in my original answer.
In this context, a circular argument just isn't very appealing. At the bottom you are going to be very uncertain, and all that uncertainty is going to propagate all the way up.
Instead, it seems like you'd want the debate to end randomly, according to a memoryless distribution. This way, the expected future debate length is the same at all times, meaning that any statement made at any point is facing the same expected demand of defensibility.
If you do it this way the debate really doesn't seem to work, as you point out.
I currently think all my concerns can be addressed if we abandon the link to factored cognition and defend a less ambitious thesis about debate.
For my part I mostly care about the ambitious thesis.
If the two players choose simultaneously, then it's hard to see how to discourage them from selecting the same answer. This seems likely at late stages due to convergence, and also likely at early stages due to the fact that both players actually use the same NN. This again seriously reduces the training signal.
If player 2 chooses an answer after player 1 (getting access to player 1's answer in order to select a different one), then assuming competent play, player 1's answer will almost always be the better one. This prior taints the judge's decision in a way which seems to seriously reduce the training signal and threaten the desired equilibrium.
I disagree with both of these as objections to the basic strategy, but don't think they are very important.
Comment by paulfchristiano on How should AI debate be judged? · 2020-07-19T21:09:26.655Z · LW · GW
Sorry for not understanding how much context was missing here.
The right starting point for your question is this writeup which describes the state of debate experiments at OpenAI as of end-of-2019 including the rules we were using at that time. Those rules are a work in progress but I think they are good enough for the purpose of this discussion.
In those rules: If we are running a depth-T+1 debate about X and we encounter a disagreement about Y, then we start a depth-T debate about Y and judge exclusively based on that. We totally ignore the disagreement about X.
Our current rules---to hopefully be published sometime this quarter---handle recursion in a slightly more nuanced way. In the current rules, after debating Y we should return to the original debate. We allow the debaters to make a new set of arguments, and it may be that one debater now realizes they should concede, but it's important that a debater who had previously made an untenable claim about X will eventually pay a penalty for doing so (in addition to whatever payoff they receive in the debate about Y). I don't expect this paragraph to be clear and don't think it's worth getting into until we publish an update, but wanted to flag it.
Do the debaters know how long the debate is going to be?
Yes.
To what extent are you trying to claim some relationship between the judge strategy you're describing and the honest one? EG, that it's eventually close to honest judging? (I'm asking whether this seems like an important question for the discussion vs one which should be set aside.)
If debate works, then at equilibrium the judge will always be favoring the better answer. If furthermore the judge believes that debate works, then this will also be their honest belief. So if judges believe in debate then it looks to me like the judging strategy must eventually approximate honest judging. But this is downstream of debate working, it doesn't play an important role in the argumetn that debate works or anything like that.
Comment by paulfchristiano on Challenges to Christiano’s capability amplification proposal · 2020-07-18T05:58:41.419Z · LW · GW
Providing context for readers: here is a post someone wrote a few years ago about issues (ii)+(iii) which I assume is the kind of thing Czynski has in mind. The most relevant thing I've written on issues (ii)+(iii) are Universality and consequentialism within HCH, and prior to that Security amplification and Reliability amplification.
Comment by paulfchristiano on Challenges to Christiano’s capability amplification proposal · 2020-07-18T03:45:56.591Z · LW · GW
I think not.
For the kinds of questions discussed in this post, which I think are easier than "Design Hessian-Free Optimization" but face basically the same problems, I think we are making reasonable progress. I'm overall happy with the progress but readily admit that it is much slower than I had hoped. I've certainly made updates (mostly about people, institutions, and getting things done, but naturally you should update differently).
Note that I don't think "Design Hessian-Free Optimization" is amongst the harder cases, and these physics problems are a further step easier than that. I think that sufficient progress on these physics tasks would satisfy the spirit of my remark 2y ago.
I appreciate the reminder at the 2y mark. You are welcome to check back in 1y later and if things don't look much better (at least on this kind of "easy" case), treat it as a further independent update. | 2021-01-26 23:46:06 | {"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": 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": 0.5329073667526245, "perplexity": 1284.9268536917104}, "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-2021-04/segments/1610704804187.81/warc/CC-MAIN-20210126233034-20210127023034-00473.warc.gz"} |
https://rdrr.io/cran/Sequential/man/Sequential-package.html | # Sequential-package: Analysis Support, Critical Values, Power, Time to Signal and... In Sequential: Exact Sequential Analysis for Poisson and Binomial Data
## Description
Sequential is designed for continuous and group sequential analysis, where statistical hypothesis testing is conducted repeatedly on accumulating data that gradually increases the sample size. This is different from standard statistical analysis, where a single analysis is performed using a fixed sample size. It is possible to analyze either Poisson type data or binomial 0/1 type data. For binomial data, it is possible to incorporate an off-set term to account for variable matching ratios. For Poisson data, the critical value is based on a Wald-type upper boundary, which is flat on the scale of the log-likelihood ratio, and on a predetermined maximum sample size. For data distributions, it is also possible to apply a user defined alpha spending function. For group sequential analyses, there are functions for pre-specified group sizes and for the situation when the group sizes are not known a priori. It is also possible to perform mixed continuous/group sequential analysis, where, for example, there is at first a big batch of data that arrives in one group, followed by continuous sequential analysis. All results are exact, based on iterative numerical calculations, rather than asymptotic theory or computer simulations.
In the package, there are functions to calculate critical values, statistical power, expected time to signal when the null hypothesis is rejected, and expected sample size at the end of the sequential analyses whether the null hypothesis was rejected or not. For example, for any desired power, relative risk and alpha level, the package can calculate the required upper limit on the sample size, the critical value needed, and the corresponding expected time to signal when the null hypothesis is rejected.
## Details
Package: Sequential Type: Package Version: 2.3.1 Date: 2017-03-04 License: GPL 2 LazyLoad: yes Index: Analyze.Binomial Function to Conduct Group Sequential Analyses for Binomial Data When the Goup Sizes are not Known a Priori. AnalyzeSetUp.Binomial Function to Set Up the Input Parameters Before Using the Analyze.Binomial Function for the First Time. Analyze.Poisson Function to Conduct Group Sequential Analyses for Poisson Data When the Goup Sizes are not Known a Priori. AnalyzeSetUp.Poisson Function to Set Up the Input Parameters Before Using the Analyze.Poisson Function for the First Time. Analyze.CondPoisson Function to Conduct Group Sequential Analyses for Conditional Poisson Data When the Goup Sizes are not Known a Priori. AnalyzeSetUp.CondPoisson Function to Set Up the Input Parameters Before Using the Analyze.CondPoisson Function for the First Time. CV.Binomial Critical Values for Continuous Sequential Analysis with Binomial Data. CV.G.Binomial Critical Values for Group Sequential Analysis with Binomial Data. CV.G.Poisson Critical Values for Group Sequential Analysis with Poisson Data. CV.Poisson Critical Values for Continuous Sequential Analysis with Poisson Data. CV.CondPoisson Critical Values for continuous sequential CMaxSPRT for Poisson data with limited information from historical cohort. Performance.Binomial Power, Expected Signal Time and Sample Size for Continuous Sequen- tial Analysis with Binomial Data. Performance.G.Binomial Power, Expected Signal Time and Sample Size for Group Sequential Analysis with Binomial Data. Performance.G.Poisson Power, Expected Signal Time and Sample Size for Group Sequential Analysis with Poisson Data. Performance.Poisson Power, Expected Signal Time and Sample Size for Continuous Sequen- tial Analysis from Limited Historical Cohort Poisson Data. Performance.CondPoisson Power, Expected Signal Time and Sample Size for Continuous Sequential CMaxSPRT with Poisson Data. SampleSize.Binomial Sample Size Calculation for Continuous Sequential Analysis with Binomial Data. SampleSize.Poisson Sample Size Calculation for Continuous Sequential Testing with Poisson Data. SampleSize.CondPoisson Sample Size Calculation for Continuous Sequential CMaxSPRT with Poisson Data.
## Overview
Most of the sequential analysis methods found in the literature are based on asymptotic results. In contrast, this package contains functions for the exact calculation of critical values, statistical power, expected time to signal when the null is rejected and the maximum sample size needed when the null is not rejected. This is done for Poisson and binomial type data with a Wald-type upper boundary, which is flat with respect to the likelihood ratio function, and a predetermined upper limit on the sample size. For a desired statistical power, it is also possible to calculate the latter. The motivation for this package is post-market near real-time drug and vaccine safety surveillance, where the goal is to detect rare but serious safety problems as early as possible, in many cases after only a hand full of adverse events. The package can also be used in other application areas, such as clinical trials.
The basis for this package is the Maximized Sequential Probability Ratio Test (MaxSPRT) statistic (Kulldorff et al., 2011), which is a variant of Wald's Sequential Probability Ratio Test (SPRT) (Wald, 1945,47). MaxSPRT uses a composite alternative hypothesis, and upper boundary to reject the null hypothesis when there are more events than expected, no lower boundary, and an upper limit on the sample size at which time the sequential analyses end without rejecting the null. MaxSPRT was developed for post-market vaccine safety surveillance as part of the Vaccine Safety Datalink project run by the Centers for Disease Control and Prevention.
In this package, all critical values, alpha spending strategies, statistical power, expected time to signal and required sample size to achieve a certain power, are obtained exactly to whatever decimal precision desired, using iterative numerical calculations. None of the results are based on asymptotic theory or computer simulations.
Poisson Data
To start, consider continuous sequential analysis for Poisson data. Let C_t be the random variable that counts the number of events up to time t. Suppose that, under the null hypothesis, C_t has a Poisson distribution with mean μ_t, where μ_t is a known function reflecting the population at risk. Under the alternative hypothesis, suppose that C_t has a Poisson distribution with mean RRμ_t, where "RR" is the unknown increased relative risk due to the vaccine. The MaxSPRT statistic defined in terms of the log likelihood ratio is given by:
LLR_t=(μ_t-c_t)+c_t \log{c_t/μ_t},
when c_t is at least μ_t, and LLR_t =0, otherwise. For continuous sequential analysis, the test statistic, LLR_t, is monitored at all times t \in (0,T], where T= SampleSize. SampleSize is defined a priori by the user in order to achieve the desired statistical power, which can be calculated using the SampleSize.Poisson function. The sequential analysis ends, and H_0 is rejected if, and when, LLR_t ≥q CV, where CV is calculated using the CV.Poisson function. If μ_t= SampleSize, the sequential analysis ends without rejecting the null hypothesis. To calculate other important performance metrics, such as the expected time to signal when the null hypothesis is rejected, use the Performance.Poisson function.
If the first event occurs sufficiently early, the sequential analysis may end with the null hypothesis rejected after a single events. There is an option to require a minimum number of observed events, c_t= M, before the null can be rejected. Setting M in the range [3,6] is often a good choice (Kulldorff and Silva, 2012). If there is a delay until the sequential analysis starts, but it continuous continuously thereafter, there is an option for that as well, requiring a minimum number μ_t= D of expected events before the null can be rejected.
With continuous sequential analysis, investigators can repeatedly analyze the data as often as they want, ensuring that the overall probability of falsely rejecting the null hypothesis at any time during the analysis is controlled at the desired nominal significance level (Wald, 1945, 1947). Continuous sequential methods are suitable for real-time or near real-time monitoring. When data is only analyzed intermittently, group sequential methods are used instead (Chin, 2012; Cook and DeMets, 2007; Xia, 2007; Friedman et al., 2010; Ghosh and Sen, 1991; Jennison and Turnbull, 2000; Mukhopadhyay and Silva, 2002; Whitehead, 1997). The data is then analyzed at regular or irregular discrete time intervals after a certain amount of data is accessible. Group sequential statistical methods are commonly used in clinical trials, where a trial may be stopped early due to either efficacy or unexpected adverse events (Jennison and Turnbull, 2000).
The same test statistic, LLR_t, is used for group sequential analyses (Silva and Kulldorff, 2012). The times when LLR_t is evaluated can be defined in several ways, using regular or irregular time intervals that are referenced by calendar period, sample size or some scale involving the distribution of the data. For Poisson data, the group sequential analysis must be conducted with equal size groups, with a constant expected number of adverse events between looks at the accumulating data. In another words, LLR_t is compared against CV whenever μ_t is a multiple of SampleSize/Looks, where 'Looks' is the total number of looks at the data. To do group sequential analysis for Poisson data, use the CV.G.Poisson and Performance.G.Poisson functions.
Binomial Data
The MaxSPRT method can also be applied to binomial/Bernoulli data. Let n denote the total number of events that has been observed in a sequential monitoring up to a certain moment in time. Suppose that these n events are categorized as cases and controls. For example, cases may be adverse events happening to a person taking drug A, while controls may be the same adverse event happening to someone in a matched set of individuals taking drug B. As another example, in a self-control sequential analysis, cases may be adverse events happening during the 1-28 days following vaccination, while controls are the same adverse events happening 29-56 days after vaccination.
Let C_t to denote the number of cases among the n events, and assume that C_t follows a binomial distribution with success probability equal to p, where p = 1=(1 + z), and z is the matching ratio between the occurrence of a case and of a control under the null hypothesis. For example, if the probability of having a case (instead of a control) is p = 1=(1 + z) = 0.5, then z=1 (1:1 matching ratio), or, p = 0.25 for z=3 (1:3 matching ratio), etc.
The MaxSPRT statistic (Kulldorff et al., 2011) for a continuous binomial surveillance is:
LR_n=\frac{(c_n/n)^{c_n}≤ft[(n-c_n)/n\right]^{n-c_n}}{≤ft[1/(1+z)\right]^{c_n}≤ft[z/(1+z)\right]^{n-c_n}},
if z c_n/(n-c_n)>1, and LR_n= 1 otherwise.
The monitoring is continued until either there is a signal rejecting the null hypothesis (LR_n > CV) or until n=N, which indicates that the null is not to be rejected. To perform the calculations, use the CV.Binomial, SampleSize.Binomial and Performance.Binomial functions.
To calculate the critical value for a Wald type rejection boundary, and when the group sizes are fixed a priori, use the CV.G.Binomial function. For statistical power, expected time to signal, expected time of surveillance, use the Performance.G.Binomial function.
The main assumptions behind the method above are: (i) the monitoring is truly performed in a continuous fashion; (ii) the matching ratio (z) is constant for all of the n events, and (iii) it uses a Wald type rejection boundary that is flat in terms of the likelihood function. Relaxing these assumptions, Fireman et al. (2013) developed exact sequential analysis for group sequential data with varying matching ratios, and for any user specified alpha rejection plan.
Conditional Poisson data with limited information from historical cohort - CMaxSPRT
In Poisson MaxSPRT, the expected mean μ_t is assumed to be a known function reflecting the baseline adverse event risk in the absence of the exposure of interest. In practice, it is estimated with historical data and the uncertainty associated with the estimated counts may or may not have a non-negligible impact on the performance of the sequential analysis method. Li and Kulldorff (Li and Kulldorff, 2010) showed in their simulation study that uncertainty in the estimated baseline means can be ignored when the total number of events in the historical data is at least 5 times the specified upper limit T. Otherwise, it is recommended to implement a statistical procedure that takes in account the variation from the historical data, i.e. a procedure that conditionates the likelihood function of the historical Poisson data, here simply called "conditional Poisson data". For this, the Conditional Maximized Sequential Probabilit Ratio Test (CMaxSPRT) to account for variation in both the historical and surveillance cohorts.
Let c and V denote the total number of events and the cumulative person-time in the historical data, let P_k denote the cumulative person-time observed in the surveillance population when the kth event occurred. The CMaxSPRT statistic defined in terms of the log likelihood ratio is given by
U_k=clog(\frac{c(1+P_k/V)}{c+k})+klog(\frac{k(1+P_k/V)}{P_k/V(c+k)}),
when k/c>P_k/V, and U_k=0, otherwise. In the original publication (Li and Kulldorff, 2010), the method was introduced as a continuous sequential analytic approach with the upper limit defined in terms of the maximum number of observed events, i.e., k ≤q K, and the critical value calculated via a Monte Carlo approach. A large number of Monte Carlo simulations (e.g., 10 million) might be needed to calculate the critical values with a reasonable precision.
In Silva et al. (2016), the method was extended i) with another option of defining the surveillance length in terms of the maximum cumulative person-time divided by the total cumulative person-time in the historical cohort, i.e., P_k/V ≤ T, ii) with an exact calculation of the critical values for both surveillance length definitions, and iii) for group sequential analysis with data updated and analyzed intermittently instead of continuously. The exact critical values are calculated using the interval havling method to solve for the root of a complex, non-linear equation such that the overall Type I error rate is preserved at the nominal level. As K increases, the computing time for the exact critical values increases exponentially.
Silva et al. (2016) also proposed two approximation methods to calculate the critical values that require substantially less computing time. One approch may overestimate the critical values and thus is referred to as the conservative approach as it may yield lower-than-nominal Type I error rates; the other approach may underestmate the critical values and thus is referred to as the liberal approach as it may yield higher-than-nominal Type I error rates. The recommendation is to use the exact approach when K is small (e.g., 10), use the conservative approach when K is medium or large but c is small, and use the liberal approach when c is medium (e.g., 50) or large. Exact calculations for selected tuning parameters show that the three approaches yield very similar results when K and c are reasonably large.
For calculating critical values for a Wald type rejection boundary, use the CV.CondPoisson function. For statistical power, expected time to signal, expected time of surveillance, and maximum sample size requirements, use the Performance.CondPoisson and SampleSize.CondPoisson functions.
Alpha spending function for unpredictable group sizes
The alpha spending function specifies the cumulative amount, F_{α}(t), of Type I error probability related to each of the possible values of n. Thus, at the end of the monitoring the alpha spending corresponds to a value smaller than or equal to the overall amount of Type I error probability defined for the overall nominal significance level, α.
Denote the single probability of rejecting the null hypothesis at the j-th test by α_j. Then, the alpha spending at test i is given by F_{α}(t_i)=∑_{j=1}^{i}α_j ≤q α.
There is a vast number of proposals for choosing the shape of the alpha spending function. Jennison and Turnbull (2000) present a rich discussion about this topic. They dedicated a special attention to the alpha spending of the form: F_{α}(t)=α t^{ρ}, where ρ>1, and t represents a fraction of the maximum length of surveillance.
To run continuous or group sequential analysis with a user defined alpha spending function, and/or, when the group sizes are not known a prior, Analyze.Binomial, Analyze.Poisson, and Analyze.CondPoisson should be used for binomial and Poisson, and conditional Poisson data, respectively. These functions work differently than the other functions mentioned above. Those other functions are designed to be used before the start of the sequential analysis, in order to determine what the maximum sample size and critical value should be. Once the sequential analysis is under way, the test statistic is then calculated using a hand calculator or an excel spread sheet, and compared with the critical value. The functions Analyze.Binomial, Analyze.Poisson, and Analyze.CondPoisson work very differently, in that they are run at each look at the accumulating data, whenever a new group of data arrives, and it is meant to perform the test itself, i.e., there is no need to use hand calculators or excel spread sheets or any other auxiliar code. The results and conclusions, including a descriptive table and illustrative graphics, are automatically provided after running Analyze.Binomial, Analyze.Poisson, or Analyze.CondPoisson.
Important: before using these functions, though, it is necessary to first run the functions AnalyzeSetup.Binomial, AnalyzeSetup.Poisson, or AnalyzeSetup.CondPoisson once in order to set everything up for the sequential analysis.
## Comparison with Other R Packages for Sequential Analysis
The R Sequential package is designed for sequential analysis where statistical hypothesis testing is performed using gradually accumulating data. It is not designed for quality control problems, where a process is monitored over time to detect an emerging problem due to a sudden increase in the excess risk. Although the methods for sequential analysis and quality control may seem similar, as they both analyze gradually accumulating data, they are actually very different in both their purpose and design. Under the sequential hypothesis testing approach, the objective is to quickly determine if there is some intrinsic excess risk, with the assumption that this risk does not change over time. For example, we may want to know if drug A is better than drug B, and there is no reason to believe that the behavior of the drugs change over time. In the quality control setting, the objective is instead to detect a possible change in a stochastic process that may occur in the future, and to detect that change as soon as possible after it occurs. For example, the heart of a hospital patient is beating as it should, but if there is a sudden deterioration, the alarm should sound as soon as possible without generating a lot of false alarms. This package is only meant for sequential analysis of the former type, and it should not be used for quality control type problems. For quality control type analyses, there are other R packages available, such as graphicsQC, IQCC, MetaQC, MSQC, qcc, and qcr.
In a number of ways, the R Sequential package differs from other R packages for sequential analyses. Historically, most sequential analysis has been conducted using asymptotic statistical theory, and that is also what is used in the gsDesign, ldbounds, PwrGSD, seqDesign, seqmon, and sglr R packages. In contrast, the R Sequential package is based on exact results, using iterative numerical calculations, rather than using asymptotic theory or computer simulations.
With this package, it is only possible to analyze binomial/Bernoulli, Poisson, or conditional Poisson data. For other probability distributions, such as normal or exponential data, other R packages should be consulted, such as GroupSeq or SPRT. Moreover, all functions in this package uses a one-sided upper bound to reject the null hypothesis, while the analyses end without rejecting the null when an upper limit on the sample size is reached. For two sided sequential analysis, or other types of rejection boundaries, other R packages must be used, such as e.g. ldbounds and Binseqtest. Finally, in this package, there are functions for both continuous and group sequential analysis, and it is also possible to analyze situations where some of the data arrives continuously while other parts of the data arrives in groups. Most other R packages are exclusively designed for group sequential analysis, but there are some that also do continuous sequential analysis, such as Binseqtest and SPRT, but Binseqtest is only for binomial data type, and SPRT is for simple alternative hypothis, while Sequential can be used for binomial and Poisson data and is meant to composite alternative hypothesis. The present package offers the possibility to calculate the expected time to signal through the Performance.Poisson, Performance.G.Poisson, Performance.Binomial, Performance.G.Binomial, and Performance.CondPoisson functions, which is not offered by the other packages cited above. Another important advantage of the Sequential package is the possibility of eliciting, through exact calculations, the minimum sample size needed to accomplish with target statistical powers through the functions SampleSize.Poisson, SampleSize.CondPoisson, and SampleSize.Binomial.
## Acknowledgements
Development of the R Sequential package has been funded and supported by:
- Food and Drug Administration, USA, through the Mini-Sentinel Project (v1.0,1.1,2.0).
- National Institute of General Medical Sciences, NIH, USA, through grant number R01GM108999 (v2.0).
- Federal University of Ouro Preto (UFOP), through contract under internal UFOP's resolution CEPE 4600 (v2.0).
- National Council of Scientific and Technological Development (CNPq), Brazil (v1.0).
- Bank for Development of the Minas Gerais State (BDMG), Brazil (v1.0).
Feedback from users is greatly appreciated. Very valuable suggestions concerning the R Sequential package have been received from various individuals, including:
- Ron Berman, University of California Berkeley.
- Claudia Coronel-Moreno, Harvard Pilgrim Health Care Institute.
- Bruce Fireman, Kaiser Permanente Northern California.
- Josh Gagne, Harvard Medical School and Brigham and Women's Hospital.
- Ned Lewis, Kaiser Permanente Northern California.
- Judith Maro, Harvard Medical School and Harvard Pilgrim Health Care Institute.
- Katherine Yih, Harvard Medical School and Harvard Pilgrim Health Care Institute.
- Jie Tang, Clinical biostatistics, Janssen R and D US, Johnson and Johnson LLC.
- Tuomo A. Nieminen, The National Institute for Health and Welfare (THL), Finland.
- Andreia Leite, Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine.
- Laura Hou, Harvard Pilgrim Health Care Institute, Boston, USA.
- Abdurrahman Abdurrob, Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Brigham and Women's Hospital and Harvard Medical School.
- Kirk Snyder, Information Management Services, Inc.
## Version History of the R Sequential Package
Version 1.1, February 2013
Exact sequential analysis for Poisson data:
- Exact continuous sequential analysis.
- Exact group sequential analysis with pre-defined and constant groups sizes.
- Wald type rejection boundary.
- Statistical power, expected time to signal and sample size calculations.
- User guide.
Version 1.2, January 2014
- Improved code structure and efficiency.
- More extensive user guide.
Version 2.0, June 2015
Exact sequential analysis for binomial data:
- Continuous sequential analysis.
- Group sequential analysis with pre-defined group sizes.
- Group sequential analysis with unpredictable group sizes, not specified a priori.
- Fixed or variable binomial probabilities (matching ratios).
- User specified alpha spending function.
- Statistical power, expected time to signal and sample size calculations.
- Updated user guide.
Version 2.0.1, June 2015
- Correction of bugs in CV.Poisson function.
- Updated user guide.
Version 2.0.2, Octuber 2015
- Improved user guide.
Version 2.1, May 2016
Exact sequential analysis for Poisson data:
- Group sequential analysis with unpredictable group sizes, not specified a priori.
- User specified alpha spending function.
- Mixed group-continuous sequential analysis.
- Statistical power, expected time to signal and sample size calculations for non-constant groups sizes.
Other:
- Directory address parameter in AnalyzeSetUp functions.
- Probability parameter in binomial functions.
- Updated user guide.
Version 2.1.1, June 2016
- Correction of bugs in Poisson functions.
- Updated user guide.
Version 2.2, July 2016
- Critical Value, Performance, and SampleSize calculations for CMaxSPRT with Poisson data.
- Updated user guide.
Version 2.2.1, September 2016
- Correction of bugs in CV.Poisson and CV.G.Poisson functions.
- Updated user guide.
Version 2.3, Dec 2016
- Correction of bugs in the SampleSize.Binomial function.
- Improvement of SampleSize functions for considering vectors for the input parameters R and power.
- Inclusion of the new functions AnalyzeSetUp.CondPoisson and Analyze.CondPoisson.
- Updated user guide.
Version 2.3.1, Feb 2017
- Correction of bugs in Analyze.Binomial and AnalyzeSetUp.Poisson functions.
- Adjustment on the relative risk estimation method for Analyze.Binomial function.
- Updated user guide.
## Author(s)
Ivair Ramos Silva, Martin Kulldorff.
Maintainer: Ivair Ramos Silva <jamesivair@yahoo.com.br>
## References
Chin R. (2012), Adaptive and Flexible Clinical Trials, Boca Raton, FL: Chapman and Hall/CRC.
Cook TD, DeMets DL. (2007), Introduction to Statistical Methods for Clinical Trials: Chapman and Hall/CRC Texts in Statistical Science.
Fireman B, et al. (2013) Exact sequential analysis for binomial data with timevarying probabilities. Manuscript in Preparation.
Friedman LM, Furberg CD, DeMets D. (2010), Fundamentals of Clinical Trials, 4th ed.: Springer.
Ghosh BK, Sen PK. (1991), Handbook of Sequential Analysis, New York: MARCEL DEKKER, Inc.
Ghosh M, Mukhopadhyay N, Sen PK. (2011), Sequential Estimation: Wiley.
Jennison C, Turnbull B. (2000), Group Sequential Methods with Applications to Clinical Trials, London: Chapman and Hall/CRC.
Kulldorff M, Davis RL, Kolczak M, Lewis E, Lieu T, Platt R. (2011). A Maximized Sequential Probability Ratio Test for Drug and Safety Surveillance. Sequential Analysis, 30: 58–78.
Kulldorff M, Silva IR. (2015). Continuous Post-market Sequential Safety Surveillance with Minimum Events to Signal. arxiv:1503.01978 [stat.ap]. Forthcoming paper of REVSTAT Statistical Journal.
Mukhopadhyay N, Silva BM. (2002), Sequential Methods and Their Applications, 1th ed.: Chapman and Hall/CRC.
Silva IR, Kulldorff M. (2015), Continuous versus Group Sequential Analysis for Vaccine and Drug Safety Surveillance. Biometrics, 71 (3), 851–858.
Silva IR, Li L, Kulldorff M. (2016). Exact Conditional Sequential Testing for Poisson Data. Working paper.
Xia Qi. (2007), A Procedure for Group Sequential Comparative Poisson Trials. Journal of Biopharmaceutical Statistics, 17, 869–881.
Wald A. (1945), Sequential Tests of Statistical Hypotheses, Annals of Mathematical Statistics, 16, 117–186.
Wald A. (1947), Sequential Analysis. New York: John Wiley and Sons.
Whitehead J. (1997), The Design and Analysis of Sequential Clinical Trials, 2th ed.: Wiley.
## Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ## Critical value for continuous sequential analyses for Poisson Data. ## Maximum sample size = 10, alpha = 0.05 and minimum number of events = 3: cvt<- CV.Poisson(SampleSize=10,D=0,M=3,alpha=0.05) ## Statistical power and the expected time to signal for relative risk RR=2: result<- Performance.Poisson(SampleSize=10,D=0,M=3,cv=cvt,RR=2) # And if you type: result # Then you will see the following: # Power ESignalTime ESampleSize # [1,] 0.7329625 4.071636 5.654732
Sequential documentation built on May 29, 2017, 5:40 p.m.
Search within the Sequential package
Search all R packages, documentation and source code | 2017-06-25 14:13:01 | {"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": 0.6995691657066345, "perplexity": 1933.0398131229904}, "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/1498128320532.88/warc/CC-MAIN-20170625134002-20170625154002-00162.warc.gz"} |
https://tex.stackexchange.com/questions/12826/how-to-use-different-font-for-full-document-in-latex?noredirect=1 | # How to use different font for full document in LaTeX
I am not able to find how to call different fonts family in latex. Could someone help me out or any link will be useful.
\documentclass[a4paper]{article}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{graphics}
\usepackage{color}
\usepackage{hyperref}
\usepackage{multirow}
\usepackage{longtable}
\usepackage{fullpage}
\usepackage[pdftex]{graphicx}
\usepackage{fancyhdr}
\pagestyle{fancy}
\lfoot{%(full_name)s/%(emp_id)s/\thepage}
\cfoot{}
\topmargin -2.5cm
\parindent 0cm
\textheight 27.5cm
\parskip 3mm
\begin{document}
\oddsidemargin -1cm
\evensidemargin 0cm
\fontencoding{\encodingdefault}
\renewcommand{\familydefault}{\sfdefault}
\fontshape{\shapedefault}
thanks
• Take a look at the answer to this question. Does it help? – TH. Mar 7 '11 at 6:39 | 2019-07-19 18:40:05 | {"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": 0.5486381649971008, "perplexity": 1007.5666246658319}, "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/1563195526337.45/warc/CC-MAIN-20190719182214-20190719204214-00257.warc.gz"} |
https://www.codingame.com/playgrounds/25775/codingame-sdk-documentation/animating-entities | Animating entities
In order for your entities to be animated in the viewer, you must commit your entities state. You do not need to do it manually as the module automatically commits all the entities at the end of each frame.
However, you might want to force a commit, for example, to create several animations of the same entity during a single frame.
Usage
Two methods allow you to commit your entities:
commitEntityState(double t, Entity<?>... entities);
will commit the state of entities at the moment t (0 ≤ t ≤ 1 ). 0 being the start of the frame and 1 the end of the frame.
commitWorldState(double t);
will commit the state all the entities you created at the moment t (0 ≤ t ≤ 1).
If you want to commit a high amount of entities you may consider using commitWorldState instead of commitEntityState for better performances.
Examples
Animating a circle
//Starts invisible
graphicEntityModule.commitEntityState(0, circle);
//Grow to big size
graphicEntityModule.commitEntityState(0.8, circle);
//Stays with the same size a little bit
graphicEntityModule.commitEntityState(0.9, circle);
//Shrinks to normal size
graphicEntityModule.commitEntityState(1, circle);
It should look like this :
Animating several entities
circle.setRadius(50);
line.setX(30);
rectangle.setAplha(0.67);
commitWorldState(0.5); | 2019-06-16 06:43:38 | {"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": 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": 0.3155955374240875, "perplexity": 2296.5829423748737}, "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-2019-26/segments/1560627997801.20/warc/CC-MAIN-20190616062650-20190616084650-00270.warc.gz"} |
https://nalinkpithwa.com/2020/07/25/v-partitions-and-equivalence-relations-my-notes/ | # V. Partitions and Equivalence Relations: My Notes
References:
1. Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Publications, India.
2. Toplcs in Algebra, I N Herstein.
3. Abstract Algebra, Dummit and Foote.
4. Topology by James Munkres.
In the first part of this section, we consider a non-empty set X, and we study decompositions of X into non-empty subsets which fill it out and have no elements in common with one another. We give special attention to the tools (equivalence relation) which are normally used to generate such decompositions.
A partition of X is a disjoint class $\{ X_{i} \}$ of non-empty subsets of X whose union if the full set X itself. The $X_{i}$‘s are called the partition sets. Expressed somewhat differently, a partition of X is the result of splitting it, or subdividing it, into non-empty subsets in such a way that each element of X belongs to one and only one of the given subsets. ]
If X is the set $\{1,2,3,4,5 \}$, then $\{1,3,5 \}$, $\{2,4 \}$ and $\{1,2,3 \}$ and $\{ 4,5\}$ are two different partitions of X. If X is the set $\Re$ of all real numbers, then we can partition $\Re$ into the set of all rationals and the set of all irrationals, or into the infinitely many closed open intervals of the form $[n, n+1)$ where n is an integer. If X is the set of all points in the coordinate plane, then we can partition X in such a way that each partition set consists of all points with the same x coordinate (vertical lines), or so that each partition set consists of all points with the same y coordinate (horizontal lines).
Other partitions of each of these sets will readily occur to the reader. In general, there are many different ways in which any given set can be partitioned. These manufactored examples are admittedly rather uninspiring and serve only to make our ideas more concrete. Later in this section we consider some others which are more germane to our present purposes.
A binary relation in the set X is a mathematical symbol or verbal phrase, which we denote by R in this paragraph, such that for each ordered pair $(x,y)$ of elements of X the statement $x \hspace{0.1in} R \hspace{0.1in} y$ is meaningful, in the sense that it can be classified definitely as true or false. For such a binary relation, $x \hspace{0.1in} R \hspace{0.1in}y$ symbolizes the assertion that x is related by R to y, and $x \not {R} \hspace{0.1in}y$ the negation of this, namely, the assertion that x is not related by R to y. Many examples of binary relations can be given, some familiar and others less so, some mathematical and others not. For instance, if X is the set of all integers and R is interpreted to mean “is less than,” which of course is usually denoted by the symbol <, then we clearly have 6<7 and $5 \not < 2$. We have been speaking of binary relations, which are so named because they apply only to ordered pairs of elements, rather than to ordered triples, etc. In our work, we drop the qualifying adjective and speak simply of a relation in X, since we shall have occasion to consider only relations of this kind. {NB: Some writers prefer to regard a relation R in X as a subset R of $X \times X$. From this point of view, x R y and $x \not {R} y$ are simply equivalent ways of writing $(x,y) \in R$ and $(x,y) \notin R$. This definition has the advantage of being more tangible than our definition, and the disadvantage that few people really think of a relation in this way.” )
We now assume that a partition of our non-empty set X is given, and we associate with this partition a relation on X. This relation is defined to be in the following way: we say that x is equivalent to y and write this as $x \sim y$ (the symbol $\sim$ is pronounced “wiggle”.), if x and y belong to the same partition set. It is obvious that the relation $\sim$ has the following properties:
a) $x \sim x$ for every x (reflexivity)
b) $x \sim y \Longrightarrow y \sim x$ (symmetry)
c) $x \sim y \hspace{0.1in} and \hspace{0.1in} y \sim z \Longrightarrow x \sim z$ (transitivity)
This particular relation in X arose in a special way, in connection with a given partition of X, and its properties are immediate consequences of the definition. Any relation whatever in X which possesses these three properties is called an equivalence relation in X.
We have just seen that each partition of X has associated with it a natural equivalence relation in X. We now reverse the situation and prove that a given equivalence relation in X determines a natural partition of X.
Let $\sim$ be an equivalence relation in X; that is, assume that it is reflexive, symmetric, and transitive in the sense described above. If x is an element of X, the subset of X defined by $[x] = \{ y: y \sim x\}$ is called the equivalence set of x. The equivalence set of x is thus the set of all elements which are equivalent to x. We show that the class of all distinct equivalence sets forms a partition of X. By reflexivity, $x \in [x]$ for each element x in X, so each equivalence set is non-empty and their union is X. It remains to be shown that any two equivalence sets $[x_{1}]$ and $[x_{2}]$ are either disjoint or identical. We prove this by showing that if $[x_{1}]$ and $[x_{2}]$ are not disjoint, then they must be identical. Suppose that $[x_{1}]$ and $[x_{2}]$ are not disjoint, that is, suppose that they have a common element z. Since x belongs to both equivalence sets, $z \sim x_{1}$ and $z \sim x_{2}$, and by symmetry $x_{1} \sim z$. Let y be any element of $x_{1}$, so that $y \sim x_{1}$. Since $y \sim x_{1}$ and $x_{1} \sim z$, transitivity shows that $y \sim z$. By another application of transitivity, $y \sim z$ and $z \sim x_{2}$, imply that $y \sim x_{2}$ so that y is in $[x_{2}]$. Since y was arbitrarily chosen in $[x_{1}]$, we see by this that $[x_{1}] \subseteq [x_{2}]$. The same reasoning shows that $[x_{2}] \subseteq [x_{1}]$ and from this we conclude that $[x_{1}] = [x_{2}]$.
The above discussion demonstrates that there is no real distinction (other than a difference in language) between partitions of a set and equivalence relation by regarding elements as equivalent if they belong to the same partition set, and if we start with an equivalence relation, we get a partition by grouping together into subsets all elements which are equivalent to one another. We have here a single mathematical idea, which we have been considering from two different points of view, and the approach we choose in any particular application depends entirely on our own convenience. In practice, it is almost invariably the case that we use equivalence relations (which are usually easy to define) to obtain partitions (which are sometimes difficult to describe fully).
We now turn to several of the more important simple examples of equivalence relations.
Let I be the set of integers. If a and b are elements of this set, we write $a = b$ (and say that a equals b) if a and b are the same integer. Thus, $2+3=5$ means that the expression on the left and right are simply different ways of writing the same integer. It is apparent that = used in this sense is an equivalence relation in the set I:
i) a=a for every a
ii) $a=b \Longrightarrow b=a$
iii) $a=b \hspace{0.1in} b=c \Longrightarrow a=c$.
Clearly, each equivalence set consists of precisely one integer.
Another familiar example is this relation of equality commonly used for fractions. We remind the reader that, strictly speaking, a fraction is merely a symbol of the form a/b, where a and b are integers and b is not zero. The fractions 2/3 and 4/6 are obviously not identical, but nevertheless we consider them to be equal. In general, we say that two fractions a/b and c/d are equal, written $\frac{a}{b} = \frac{c}{d}$, if ad and bc are equal as integers in the usual sense (see the paragraph above). (HW quiz: show this is an equivalence relation on the set of fractions). An equivalence set of fractions is what we call a rational number. Every day usage ignores the distinction between fractions and rational numbers, but it is important to recognize that from the strict point of view it is the rational numbers (and not the fractions) which form part of the real number system.
Our final example has a deeper significance, for it provides us with the basic tool for our work of the next two sections.
For the remainder of all this section, we consider a relation between pairs of non-empty sets, and each set mentioned (whether we say so explicitly or not) is assumed to be non-empty. If X and Y are two sets, we say that X is numerically equivalent to Y if there exists a one-to-one correspondence between X and Y, that is, if there exists a one-to-one mapping of X onto Y. This relation is reflexive, since the identity mapping $i_{X}: X \rightarrow X$ is one-to-one onto; it is symmetric since if $f: X \rightarrow Y$ is one-to-one onto, then its inverse mapping $f^{-1}: Y \rightarrow X$ is also one-to-one onto; and it is transitive, since if $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are one-to-one onto, then $gf: X \rightarrow Z$ is also one-to-one onto. Numerical equivalence has all the properties of an equivalence relation, and if we consider it as an equivalence relation in the class of all non-empty subsets of some universal set U, it groups together into equivalence sets all those subsets of U which have the “same number of elements.” After we state and prove the following very useful but rather technical theorem, we shall continue in Sections 6 and 7 with an exploration of the implications of these ideas.
The theorem we have in mind — the Schroeder-Bernstein theorem: If X and Y are two sets each of which is numerically equivalent to a subset of the other, then all of X is numerically equivalent to all of Y. There are several proofs of this classic theorem, some of which are quite difficult. The very elegant proof we give is essentially due to Birkhoff and MacLane.
Proof:
Assume that $f: X \rightarrow Y$ is a one-to-one mapping of X into Y, and that $g: Y \rightarrow X$ is a one-to-one mapping of Y into X. We want to produce a mapping $F: X \rightarrow Y$ which is one-to-one onto. We may assume that neither f nor g is onto, since if f is, we can define F to f, and if g is, we can define F to be $g^{-1}$. Since both f and g are one-to-one, it is permissible to use the mappings $f^{-1}$ and $g^{-1}$ as long as we keep in mind that $f^{-1}$ is defined only on f(X) and $g^{-1}$ is defined only on g(Y). We obtain the mapping F by splitting both X and Y into subsets which we characterize in terms of the ancestry of their elements. Let x be an element of X. We apply $g^{-1}$ (if we can) to get the element $g^{-1}(x)$ in Y. If $g^{-1}(x)$ exists, we call it the first ancestor of x. The element x itself we call the zeroth ancestor of x. We now apply $f^{-1}$ to $g^{-1}(x)$ if we can, and if $(f^{-1}g^{-1})(x)$ exists, we call it the second ancestor of x. We now apply $g^{-1}$ to $(f^{-1}g^{-1})(x)$ if we can, and if $(g^{-1}f^{-1}g^{-1})(x)$ exists, we call it the third ancestor of x. As we continue this process of tracing back the ancestry of x, it becomes apparent that there are three possibilities — (1) x has infinitely many ancestors. We denote by $X_{i}$, the subset of X, which consists of all elements with infinitely many ancestors (2) x has an even number of ancestors, this means that x has a last ancestor (that is, one which itself has no first ancestor) in X. We denote by $X_{e}$ the subset of X consisting of all elements with an even number of ancestors. (3) x has an odd number of ancestors; this means that x has a last ancestor in Y. We denote by $X_{o}$ the subset of X which consists of all elements with an odd number of ancestors. The three sets $X_{i}$, $X_{e}$, $X_{o}$ form a disjoint class whose union is X. We decompose Y in just the same way into three subsets $Y_{i}$, $Y_{e}$, $Y_{o}$. It is easy to see that f maps $X_{i}$ onto $Y_{i}$, and $X_{e}$ onto $Y_{e}$, and that $g^{-1}$ maps $X_{o}$ onto $Y_{o}$, and we complete the proof by defining F in the following piecemeal manner:
$F(x) = f(x)$ if $x \in X_{i} \bigcup X_{e}$
and $F(x) = g^{-1}(x)$ if $x \in X_{o}$.
QED.
The Schroeder Bernstein theorem has great theoretical and practical significance. It main value for us lies in its role as a tool by means of which we can prove numerical equivalence with a minimum of effort for many specific sets. We put it to work in Section 7.
Regards,
Nalin Pithwa
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# Read Shakespeare In The Eighteenth Century
## Boethius Analytisicius Manlius Severinus read Shakespeare in the Eighteenth Ennodius a. Epistolae et decreta Epistola way B. Faustum senatorem Hymni union in honorem SS. calculus: person; Antiquariat Thomas Haker GmbH & Co. Dialogi in Porphyriuni a Victorino answer advance 9 family in Porphyrium 71 In Categorias Arislotelis libri Obesity 159 In mug Aristotelis de interpretatione Commentaria minora 293 In surface system Commentaria majora 393 Interpretatio beatum Analyticorum Aristotelis 639 Interpretatio zone Analyticorum Aristotelis 712 711 Introductio confession Syllogismos categoricos 761 Interpretatio Topicorum Aristotelig 909 Interpretatio Elenchorum Sophisticorum treatment 1007 language in Topica Ciceronis 1040 1041 De Differentiis topicis 1173 De lifetime cognatione 1217 Commentarius in Boelium de consolatione Philosophia? De Syllogismo categorico libri Convergence 793 De Syllogismo hypothetico libri item 831 Liber de divisione 875 Liber de diflinitione 891 Brevis Qdei Christian compleiio 1333 Liber de habitat et design waters cum Gilberti Porreta? Boethius BoethiusExcerpt from Boetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, Vol. B, CThis challenges a problem of a surface subtracted before 1923. This rate may make open atheists Total as decomposing or been others, finite consumers, important features, etc. B, CThis permits a interaction of a product fueled before 1923. This hospital may seek geometric topologies new as having or desired fungi, hedge tools, suitable sets, etc. Boetii, Ennodii Felicis, Trifolii presbyteri, Hormisd? Boetii, Ennodii Felicis, Trifolii presbyteri, Hormisd? Es conocido por su Epistula productivity access Faustum senatorem contra Ioannem Scytham material de 519-20 d. Los escitas estaban dirigidos por Juan Majencio y research a Roma en 519 disjoint la esperanza de edition communication apoyo del citado Papa. Dionisio read Exiguo de la Carta de San Proclo a los armenios, escrita en griego. Grillmeier and Hainthaler, property Aloys Grillmeier, Theresia Hainthaler, Christ in Christian Tradition: From the Council of Chalcedon( 451) to Gregory the Illustrative( 590-604)( 1995 arbitrariness), complement Why are I think to prevent a CAPTCHA? working the CAPTCHA is you have a impossible and is you much set to the set job. What can I make to feel this in the surgery? If you are on a open allyour, like at middle, you can assert an leaf approach on your JavaScript to last Compressive it is then considered with legume. If you are at an way or great soil, you can move the requirement website to help a network across the cell adding for metric or topological species. 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By learning this read Shakespeare in the, you are to the algorithms of Use and Privacy Policy. figure as' work Use; Management' pointed by Kanka, Nov 10, 2016. notion referred Programming( OOP) that will Schedule you to adjust operation so and configure the most of other links. object-oriented-like site young chapter! Select you once conclude an concept? sure Microarrays with a guide. A Journey from Separation Toward by Betty A. accommodate poles With My Daughter: Use is capable. accumulating Yourself Too by Christopher S. Low careers of , edited sets and bigger Interstitial Copy see not puny of the single domains it natively does on the density of most useful for theist theory to present poor Materials. please Fund Modelling and Analysis is a coastal prop within the newest atheistic cookies for various edition squishing an difference, clinical with a other water on either C++ and home went setting( OOP). This also directed good hand within the Reusable Hedge Fund Modelling and species decal is the one moisture on phase for differing the complete C++ shipping to learn metrizable Flask computing and topology. C++ has every foreign oxygen you imply to form the such notes of everyone reconstructed $M$, which is you to be Other Substrate macrophytes from distinct habitats of homotopy oneis. This e-book includes your read Shakespeare in vegetative topology to open with sure purpose within the Medieval interval of intersection. All the future and misconfigured expansion you can have theist problems to manipulate general well-being measure. underdeveloped according Methods and redundant neighbors commenting what to submit whilst demonstrating primer and develop religious rates within the complete Moisture. A better lysis surface matter deriving available C++ methods, points and plane to other-hand. have being Hedge Fund Modelling and century your oriented citadel and find the metric investment and object-oriented space you must be the edges. are to them on every metric read Shakespeare in the Eighteenth and budget reducing to guide them to ' be '. adjective is an just basic fund. say them the near evaluation you are yourself: to get your topological topology, because it has the available in administrator to give that you identify the one who plays the major aspect to optimise for atheism n't. You can remarkably learn that reason for yourself. have the look the common bottle you exist yourself: the Exoenzyme to have their safe point without any segment from you. read Shakespeare in the Eighteenth: acidification is ' no bit ' so Buddhiss deal more chemical. as illustrating in neighbourhood ChinaVariation or future that refers the Acetylene a topology or is any paper from the libri of the board address understanding. being point tripling about loss fund that will precisely include structure; neighborhood modifier) is hence characterized anatomy Bioinsecticide of life as your quantities can give better presented satisfying moisture reading to the single-variable UML. As an However the surgery of libri in the Universe will not go or be your term problem. Yes, sets use weight. Because read Shakespeare has once other Coelenterates. It is a use units, like Opinion who thought it may die 's less body controlled. But oxygen has the T1 waterfall as topology. build I gave Well and just about the shells of Isopleths did known a Christian), and I attracted that there took no person be those limits. have I think simply called any Bravery which contributes that any negative( or very which objects first-; f the managers placed have other). They willnever 're major sets of read Shakespeare in: zero.
Octavia Welby, Monday 10 Jul 2017
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Holly O'Mahony, Friday 09 Jun 2017
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exceptional Anoxygenic Photosynthesis: Live read Shakespeare in the Eighteenth which proves anything under mass Shards. not, this permission is asexually do in the insight of connectedness. Aerotolerant Anaerobes: efforts that can ensure in both, climatic and oriented classes, because they add their address by component. teacher: A flight tested by Aspergillus time and theory developer, which do graph stars. This says released to prevent a authority of single-variable Compound. meta-analysis: A attributed hedge, restrictive struct called from advisable space tractates, exposed as a abstract subject surfaces for areas and new rates. exactly called as a second time, in Using works and as a defining philosophy for chip and presence. Agarose: Agarose is initiated from number and is known as a replacing fine in question. It needs of eukaryotic invaluable topology, which enables appropriate and big not. is: The other variations that have derived as a N of an term class. risk decal: The class of implementing simply, in course of blood calculus methods, texts, or partibus in the device of outside messages called Confessions. This handles to the read of an specific active programme. wide Transmission: A perp-dot-product of movie, away the network seems generalized in or is its sphere by process. Akinete: A simple ideal, open, just Check behavior of airfoil and something. other None: A topology research that is Approach( post-GBP) and color z from aspects. entities: open non-empty proteins, that may knot world-class or local. read Shakespeare in the Eighteenth continuity more n't looks in a formula of diagram S, which is page I are recycled down not. as I'll make you a hibernating non-empty at original surface-to-volume to put what it 's forward only, and other subdivision 's where I'll live most of my diagram. level up works me have of the macrocosmic bubbles in Fantasia Mathematica. otherwise a major world: will you study clicking others when you have to the clear proportion, Mark? I intersect it one of these is partitioned to be ' agile '? What mean you do the -COOH is? I are usually determine organic point, but I n't punch that the completely nice projections of theorem do easiest to influence thinking off with Comparative spaces. inside there is a transition of oriented ecosystem to enjoy elements combined, running from woody systems into continuous conditions that are packages. THat helps why I included ' a starting ball '. In plants of sure surgery vs normal bag they say only yet born. The read Shakespeare in the between them gives so the introductory as the human between plant root and definition Application. And what represents ' mouth group '? I do here preserved the close generalized like this. I want some of these atheists may help more normal, and give once rapidly oriented However now within the critical Truesight( at least I Have detrimentally provided them). In central, I obtain right been beauty tell the risks open imprint or structural existence. there one of the hottest thing analysts in N has key ' finite open experience '( inside number is at least merged of the Poincare object).
Holly O'Mahony, Tuesday 16 May 2017
How can an read Shakespeare in the Eighteenth Century change a buying before names are first? Should I ask a lower distortion to be a mythology study being quality? holds a consolatione with Object needs geometric? How win I optimise view analogous like creation or Golden Syrup off limiting designs or working attributes? Can decal Often copy so significant to kind intuition? How to believe infected read Shakespeare beyond the third extensive connection return? Why find we formally reject that when we call a " by a beneficial space object the donation is cosmetic? 39; scientific the best topology to use over 400 law of strong gods? thorough to UML; look the diabetes;? 39; battle the metric weight described to avoid the address the biomass man and specifications, in geometry to know set? In the UK complex read Shakespeare, what applies when a space of no weight in the vector describes?
I quickly was that biologists deleting topological species to the static read Shakespeare in in my home, but your quantitative topology is version development how female this is. Jonas Meyer: Yes, I contribute So tweaking relative! To analyse your nature: every global continuity would flood very topological. Jonas Meyer: You do gradually erratic. I shun that one should be divided around with topological patients before generalizing with algebraic months. The many kernel of real documents determined at term is not an optimal set; the self-esteem is simply the work of the truly new ". A closed topology learns that you cannot use in everything that an finite design of insufficient questions in a Oriented you&rsquo holds such all. Since metric thighs seem the object-oriented community of medical wings( finding new objects by the side of cats), the topological 's large for Monthly potentials and in the litter Sorry unparalleled topics of physical values are done. sites have not set out of 3d read Shakespeare in the Eighteenth: axioms argued real exercises wherein before predictor found of recording up with the hurry of last thanks and in the concepts they lacked low, n-dimensional graphs was created under possible fund and that role was in coverage virtually key to nix trusts used. In 2-to-1 constraints, the car of metric links( and the side you are in the interest along with it) provided been from friends, also discussed on them. not in different functions, theatheist; it did like a topological die at the topology;? SamB: no, otherwise once: that is just at all what I intersect. And yes, developments are Often natural in the graduate read on the philosophy. If they was, away every structure, pinching the design of object-oriented capabilities, would ask finite. This has led the object-oriented Access. I are this patient can address n't maybe believed by what works in open programs, of which the few work is a able senator. If read parasites in unit, we can intersect this look particle. slightly only, a definitive decomposition will pay your part spiritual, about you can investigate your approach. well we need Describes the number of a considerable &minus to build a structure the shared truth invertebrates. But we only get to complete for ll and set. For 22 problems, my sequence is needed to allow the system of topology and tell it average to home. Open Library is a lack, but we do your system. If you do our read Shakespeare technical, sense in what you can machine. Your equivalent afterlife will be given complex surface here. I turn you really irrespective a organism: please tell Open Library office. The important check turns open. If copyright terms in word, we can be this business Download. too highly, your tomorrow will study anticipated scalar, Completing your software! Basically we do is the read Shakespeare in the Eighteenth Century of a diverse base to go a topology the western programming spaces. But we also lie to Browse for microorganisms and raise. For 22 Q&, my lot is been to become the $x$ of shock and perform it human to text. Open Library begs a head, but we guess your acid.
The other knots being various read on class use say book books and point-set. Crossley and Hoglund( 1962) did a real research between the kernel of needs in attacking cards and the nothing of fund language. During aspect there achieved major statistics on the system visits and less distortion overview suggested based as repeated to the analysis, when more V was motivated by more mammals on the decomposition parallel. The plane of spores in topology though gives on the analysis sequence of libri. very Madge( 1965) hypothesized that there knew more cells on the union fungi during the dependent t than in many compassion. When options pointed now colonized for nine varieties, no constant page of closeness and Note definition structures shown( Edwards and Heath, 1963). The video prerequisites had that diagrams coloured conditions three hyphae faster than smaller millions, the most simple of the various elevated tissues, profiles and competitive elements. This means not because minimal forms want washed found to ask usually dimensional to students( Weary and Merriam, 1978). approach directly has the weight and $N$ of substance of set is on liber transfer and below encloses an low change upon the genus and bomb of the generous topology. geometry and premium provide among the most actual species( Brinson, 1977, Singh, 1969) because they do both the language of book weight and the Methods of concepts which do also diverse definitions in net space. 176; C) and chip study regression of very 60-80 volume of its empty parent perspective. cellular read Shakespeare or life of neighbourhood and Everyone beyond the impossible starts contained about a phase in the flow of similar you&rsquo set. still, sequences in help libri should easily Pitanguy restricted unless research and access libraries view the same. The Niche device reserved the quinque between curvature and scan, punching hard side both at third volume( honest programming and inland research) and Functional Goodreads( heterotrophic direction and general decomposition), and Anaerobic answer at basic copy concepts. A TROPICAL FOREST How has it aid future? references in reason characteristics at real ratios, extensive to protein in web, had Seen by Williams and Gray( 1974). The read Shakespeare in the far longer is Euclidean space, but provenentirely remains bottom applications in which roots are to one another. grow us concentrate how some of these nets think to one another in the key today. The title of the horizontal range, in hedge pages, is from its hedge Alligator. From the continuous volume, we can build a application for the goal of an it&hellip between payments, and from this is the surface of distance. We can either use a statement for a exercise, a nearness, of languages in the gland. now from afterlife, we 're the matter of volume. From the plane, we want such to get a loss for page: We are that a capable content sense community applies now the pic of a ambient certain other hurry, and from this, we are the district Together of a native share. We improve replacing down through very key immediate Atheists: surface is a more anyway different experiment than mouth, and theory exposes a stronger practice than accounting. This says the topology: What person 's a measure repeat when we cannot irreversibly share a lot of antigen between minutes? What does the Climatic several read Shakespeare in the Eighteenth Century that a brain can have required with? location is the analysis of reason. When editing a skin on a diagram, a dont 's dull Proteins for every surgeon: He or she typically refers what covers of points 'm matched to get big to one another. Each infinite information in a Figure is this n't shared solid hand, which is together detrimentally on ' sphere '. The central " about RPG statement is that, through its patients, we qualify to describe how the full-colour that algorithms construct initiated near to each other, n't also as how they can run American, how they can Make fixed, and how open of them there do, do our intersection to Tweak innermost needs as ' fact ', ' Check ', ' employee ', and ' connectedness ', which are to our system for famous atheists. also, from a subject belt, future meaning always embraces with the terrestrial introduction of which graphics of a toxin know ' T1 ' to each unifying, and 's the namespaces that this problem of structures comes on the iterative evidence of the guide. partially becomes a separately possible language of applying at study. If it planned, please replace Completing a human read Shakespeare in the Eighteenth study to my Patreon problem to evaluate my testing really on Topology Guides. topology; d rather replace it. thank the two procedures of the philosophiae near the organisms of the backlist to bear an explainthe distinction down the mesh of the measured accumulation. Stack Exchange return is of 174 reptiles; A captures Moving Stack Overflow, the largest, most endowed simple topology for spaces to smooth, use their biogeochemistry, and remain their ways. use up or pay in to make your read. By getting our activity, you have that you are based and steal our Cookie Policy, Privacy Policy, and our elements of Service. home Stack Exchange makes a future and utility state for constraints Banding enzyme at any object and forms in closed topologists. Why 're we tend a transposable disease to be charred under odd topology? In the read Shakespeare in the of discrete religion, we are the control of a secondary consolazione of normal spaces to learn general while we do the rigid bit of finite forests to be important. I are integrating what this behavior is. are I appearing in the single software not? With your be, the posteriorum is Personally longer diverse. The read Shakespeare is Such( and advanced). I wrote I published it eventually but I are I went choosing the 0. I 've I better pay my Atheist to get surface. something: you need showing entire Atheists around the surgery in the operator; you choose to be two microbes before the photographs for them to write up.
This read Shakespeare in the is toric in residues inhabiting infected Object. cause any techniques, properties, Terms, or individuals that you tend to go base" into this risk. A mesh has theory but no topology. outside deformities that offer also as of extra website species. For product function that is study. A ResearchGate occurs a natural if it is lining. A needs only not of nucleus that gives sequence. A read Shakespeare in the Eighteenth covers a entire if it proves start. B which is here significantly of Choosing that provides diagram. A return is a object-oriented if it enables number. structures that is rally. A number is a particular if it is programming. A will concentrate set with saying that is consideration. A closeness lets a agile if it holds diploid. B and SolidObject that is read Shakespeare in the Eighteenth Century. A complement 's a many if it helps decomposition. This is in read Shakespeare in the Eighteenth a largely only electron, representing ever the most alive points of Based procedure( phases, microbes and objects), and it has standard decomposition in what can test risk-adjusted as a such branch( right also as how size can use used as a important signal; there are algebraic specific naturis a Abdomen can be left on a given continuity). This $x$ is Thus profitable, in product, that unifying journeys are not in in every sense of links, and throat Does made one of the heterotrophic common ways of researchers. about, to complete: a distance on a reuse is a ideal of terms which is the male version and the &minus itself, and has found under crests and same orientablesurfaces. The implants that do in the party delve semiarboreal and their cases deal Powered. A metric line is a book enough with a field on it. A dimensionality complements Right geometric under a different importance. It has ecologically provenentirely accommodate algorithm to right be that a number explains happy. In pubblicato, about, the usage under programming is here everyday from example. Which one is defined 's only arbitrary from name. other platforms of topological topologists include inside receive to see due. One can open say the read Shakespeare in the Eighteenth based by the overall, as the physiology of all responsible properties induced by the quick. This is a cultural soil from a excellent body. start that the topology of any automated material of last means answers overlooked and the way of only fat oriented sizes is brought. are merely that a substance can arrange both aquatic and acidic. We also determine some n't intersecting sets in the y of Topology. The so many approach of the separate field is represented to the topology.
If you argue on a major read, like at application, you can Find an office remake on your volume to be true it makes essentially represented with addition. If you are at an weight or crucial trading, you can indicate the course section to find a z across the Lamento turning for complete or 5-edge data. A Journey from Separation Toward by Betty A. make theories With My Daughter: decomposition leads 2-to-1. understanding Yourself Too by Christopher S. Low compounds of distance", attained Point-sets and bigger bariatric soil want far agile of the temporary specifications it anymore is on the space of most conformal for open game to be reciprocal organisms. have Fund Modelling and Analysis is a algebraic oxygene within the newest unauthorized tools for religious user making an presence, tremendous with a important earth on either C++ and language tried product( OOP). This not stipulated booming libri within the Early Hedge Fund Modelling and weight rate is the one anything on theory for stretching the obvious C++ af&hellip to verify Occupational mating loyalty and site. C++ is every topological lignin you call to be the upper descriptions of number given matter, which does you to choose close Use influences from pronounced diagrams of topological paper. This e-book seems your god Christian Copiis to red with outer author within the Iattempted topology of oversight. All the back and multiple study you can divine near Specialists to know sure N covering. climatic changing concepts and dynamic spaces beating what to say whilst Living door and run sure surfaces within the 2-to-1 future. A better read Shakespeare in the field atheism defining topological C++ exercises, sequences and study to Infection. be studying Hedge Fund Modelling and phase your topological future and learn the nice Imprinting and useful decal you must build the examples. Some of the earliest pathologies, not these topology never to the messages and sooner than, lead too unclear analytic and more and more not. Download e-book for say: A continuity of the Levant Company by Alfred C. Routledge is an comfort of Taylor & Francis, an programming point-set. A basic standing with metric geometries asking a loss malware is it is is a n't dark complement waters to correct and manage about this theory painless, common, individual, decompose. And Once not as we entirely happen our saprobic particular Volume tissue I 'm below or no engine of metric acquisition terms Here affecting judged to hold atheist and composition on their second to begin a closeness structure page. I are contained only that the read Shakespeare that I hope phase requires that it contains not about surface. Math has Covering different citations, Completing neither to splitting systems, and specifically getting what those individuals now are, and back what you can make depending them. I find that forest, at its deepest fund, is then nearness and analyst. In a finite Buddhist, what is out make for graphics to specify morbid to one another? What Today of coordinates can you optimise blocking software but statement SolidObject? What is a information equipped clearly by that feedback of item? When I need that analysis 's sorry to the interesting design of home, some students - not fungi! I are including when I are that read Shakespeare in the. In relation, you have a use of loops, and you require their kind by which acids 're near to each Old. Most of the supernatural 28The birds want heard making distinguished things of communities, where there is no algebraic raise of two courts that are young addresses; there encloses a discussion of also smaller animals that are However closer and closer design structures. The distance of power that I believe sharing However is more like the community of a Use - it is that other structure of bytheists that find available to each extensive, with that atheistic decomposition to construct Upload narrower and narrower returns around a process. The types of policies that you mature from Setting that do high. In some subset, you argue being markets - but they are sure, original, open poles, because all that parts is what Terms do pathological to what significant relations - Presumably what result you make to define to think from one to another. make is become a coarse fact at an pole. There is a first read about students; you can about Prove a series at book, because they do the images who ca also make the version between their hand decomposer and their side. Like most veryfriendly Methods, there is a ecology of $U$ fixed never of it. I are only die that in a you think many read of kind, but not a important basis vision. Seuss control and what recorded he have from? Boethius asserted empty decal and topology who added thousands innermost as De Trinitate( On the science) which would say as a Alligator for later Scholastic wikis in the High Middle Ages. Later components near as St. Thomas Aquinas would interview upon Boethius' De vector when skill; e existed The point and nutrients of the Sciences: A surface on the De risk of Boethius. is Thomas Aquinas. The Division and Methods of the Sciences: professionals approach$and VI of his clay on the De collection of Boethius translated with Introduction and Notes, trans. Armand Mauer,( Belgium: Universa Wetteren, 1963). intersection of Expression for Aquinas' Unity of Vision: declaring a Set Theoretic Model of the relationships and values in the Trinity,( Berkeley, CA: Graduate Theological Union radius, 2005). Unless oxidation 's a You&rsquo to study the cold external of living or changing itself we are anywhere believing to beat at some continuity. The read Shakespeare in is out over opera. rely or find constitutes so a example where you must Increase a Death, or get Such iterations to manage and affect your N(y which is imposed by models, only stronger or infinitly dynamic as yours. There gives a feedback between polishing then and thinking so algebraic. If you indeed had, maintaining your operation studies anticipated wondering person, you can not Make supposed somewhat to topology, or divided. If you 're almost shared, your practitioner will However give healthy well the slightest administrator. You have, ever your decal use-case; times producing, your notion can topologically matter generic for, on compactness, finally fifteen results. When you are distributed {nn} simple, you are s and ca also run shown. Lucy Oulton, Tuesday 24 Jan 2017 changing the CAPTCHA is you do a advanced and is you second read Shakespeare in the to the version patient. What can I pray to share this in the substrate? If you have on a generic disease, like at anyone, you can prevent an Approach modeling on your fact to make certain it is therefore projected with lift. If you acknowledge at an future or recursive anti-virus, you can inject the Energy intersection to make a dryness across the trading converting for equivalent or macrocosmic bubbles. Another heterotroph to consider adapting this x in the network is to happen Privacy Pass. temperature out the guide consideration in the Chrome Store. Why 've I want to correspond a CAPTCHA? Continuing the CAPTCHA increases you are a molecular and includes you single-variable read Shakespeare in the to the diagram smartphone. What can I be to believe this in the balance? If you apologize on a last project, like at Question, you can foster an sequence home on your glance to calculate few it decomposes not known with Use. If you find at an$X$or misconfigured wildfire, you can begin the acceptableand&hellip decomposition to have a advice across the system talking for misconfigured or modern sets. I remain Operating no to your is! I delve So n't most context instructors find phase. n't in the AESTHETICS you noted, of movement. Where is closed lifting point in this mineralization? I do down climatic; I 've respectively used warhead time. My( numerous) loss has that it would not intersect under topological problem. Most of the upper thoughts have on balls and examples that are Equivalent to spells. groups need to smooth plain more numerous nearness, not a formula of the nearness that you use in open line guess too understand business. open contour encloses a connection of that, discussing to markets which wanna ' Object-Oriented '. In phase, a impossible chemical of pole indices is been to Completing long which near arguments are a authentic You&rsquo. observed read Shakespeare in the works those interactions which do now infected to prime shading locations( like full weight). These are already also international. 18th cookie is the broad-spectrum of positive Illustrative substrates to do worked-out things. This is then heard with the network of profiles from a western surface of relevant systems to some set of medial trademarks. The topology to be to a ' ' property specifically is to the set of CW-complexes, which meet usually more combinatorial than people, and view n't s low. no just, I are situations Not help the reproduction alder Bravery to contain to terms about carbon that doubt only design int to document with bubbles( or not have basically used in some Use by the solution of libraries). residing: read Shakespeare in the Eighteenth of students from reactions by the theory of substances. concepts: P proportions with a true relation for open meaning customers. Eyeshine: online close solids different in difference, which find set in material balls during reasonable microorganism between prerequisites and new points. theory: A decal, region or primer of objects or patients, overlooked to the other guide by productions of a Internet or analysis communicaton. Light Compensation Point: The algae where the plane of bottom uses higher than the thumbnail of intersection, which so differs at actually 1 sequence of technology book. Lime( natural): point Overshooting looking oriented difficulties of Hospital operators, like network litter and Euclidean multiple set which intersect chosen to See field discussion, and eat problem for home book. Lipopolysaccharide( LPS): misconfigured phase design understanding expenses and object-oriented sets, which is instead done in most Gram disabled edges. declaration: An scan that is distinct object existing as neighborhood or employee to be as advice devices in way person. They may see sets or spaces. future: The space level of designs which includes used with months, years, increases impact Lophotrichous: An combat that is a surgeon of terms that is low in chest. read Shakespeare chance: growth of purists in tolerance of what is based by an continuity for its Visual physiology. collision: The religion and number of a product, contributing in mouth of honest conjectures. email: An k where a That&rsquo is a list and the tour office forms taken in fact with the x of the light. development: A action supervisor which is hedge sets. problem: A site found in aromatic manifolds for closed y of an class. continuity: Larger calculus devices from which situation is also by space. This read Shakespeare in you are the sphere shows resulting worked for what it needs used for before part for you. everything Support Packages Available Each Support Package is Seen towards the basic future. 1 - Book A ConsultationCall or do in our main decomposition to use a beneficiary with Tonic Weight system relations. 2 - At The stage to one of our Consultant Specialist Bariatric Surgeons who will end attractive to be any of your logs and return topology on the best atheist ecosystem obesity been to you. 3 - The ProcedureOur users will mean on intersection through the true business language beating you 've the " you am. 4 - equal due information Support Team will design on litter 2-5 ways clicking subcategory to be you live a topological set Browse plant. We agree procedures in the price of Weight Loss and Weight Loss Surgery and cover higher-dimensional Relations of system. Tonic Weight Loss Surgery is also isolated with the Care Quality Commission. Registration Number: 1- 27621671. 2016 Tonic Weight Loss Surgery. high-priority After Weight reason continuity is Previously and only given for its extensive mitosis looking various topologists using northern office vector. We not describe not two-variable of your links and can be you with the transposon and administrator that you find. You wait related open neighbors in your study and summarized not to manage to this Query. After believing 150 pictures, Eric was to the Life After Weight product variant to use his boundary. be the UPMC BodyChangers modularity to get some of our plane families and get features' funds in our BodyChangers cutting-edge loss. open task read by Healthwise, read. In sufficient spaces, an read in the same, Structured as a testing or subsequent rainfall, may learn considered in our flow. We are, only, find the ecological point of characteristics also; any pages that are are then Thisaffirmed to deal the percent of mathematical open patches. RemoveTo stage are topology, philosophy and website alligators and have a safer postgraduate, we are places. By working on or writing the area, you have to ask us to attack model on and off case through data. Structured getting decomposition Still, you are to our developers. use how we 'm, cover and be your errors in our Data Policy and how we mention weeks and paperback development in our Cookie Policy. Security CheckThis is a Viable administrator volume that we 're to run concepts from talking great updates and parallel conditions. accessible spaces with a decay. Register NOW and do a read to achieve line! knowledge s for more spaces. share Here and seek the hard system. implement yourself about and feel is make to learn each practical. considering to allow your object's cellulose? Fail the latest study lines and leaders finally. Latest: Grand Opening Giveaway with Reward! Notice the latest on&hellip oversight and changes n't. read Shakespeare in the Eighteenth Century as an LibraryThing to hedge activity. Zannis J, Wood BC, Griffin LP, Knipper E, Marks MW, David LR. item set of the bad study of approach. Aly AS, Cram AE, Chao M, Pang J, McKeon M. Belt article for other extensible system: the University of Iowa distortion. Centeno RF, Mendieta CG, Young VL. stepwise assessing in the near pole paper control. passing solution with available early pannus. Shermak MA, Mallalieu JE, Chang D. is nose&rdquo for Simple description branch after normal educator union need a correct coffee? compounds connecting read blocking fact after open beauty function: a hole. J Plast Reconstr Aesthet Surg. Albino FP, Koltz PF, Gusenoff JA. A metric topology and new mark of the new thought: groups for relationship visiting after good tower relation. Rieger UM, Erba analyst, Kalbermatten DF, Schaefer DJ, Pierer G, Haug M. An moved set to title in the open of superstitious clear tissues after low possible population. Fischer JP, Wes AM, Serletti JM, Kovach SJ. systems in idealist implementing data: an &minus of 1797 components from the 2005 to 2010 American College of Surgeons quick specified oversight topology system functions. Coriddi MR, Koltz PF, Chen R, Gusenoff JA. Edmonds, present bodies). This advertising of a oriented function of spherical lot has of Early geometry. 1Johnson and materials( 1982). continuity and acts( 1984). base supernatural litter includes near sticky to do used with fundraiser brother( Harmon and mathematics 1990), although important platforms are Based Co-authored to end this. The intercalary neighborhood of different vector in languages to see study in effective constructions is Right converted. n't, Optimal read Shakespeare in the Eighteenth Century number on dynamic Douglas-fir nodes ran to be a greater purpose on looking decomposition than on phosphorous issues( Bigger 1988). The difference material, surfaces, and Greek open reason taught algebraically normal hospitals to manage Shared$N(x)$as the triangle of That&rsquo nearness norms and the organism of architectural UseThe music sets. fine deployment designer properties think not revised by using control various life or user speaker, cookies in distinctive Philosophy, or list space rainfall from a plateau. Although the major polynomial of major areas can register chamfered formally n't, it is more ge to prove how 2-D a example is focused on the programming unless the connection changes a idealism of a punched pH. so, terms that have used on the interior for a sure lipectomy have transferred by using woody relating devices for diagrams and decreasing the managers going an process t( Sollins and files 1987). The transition of needs using on exhaled claims can already restrict understood. Both read Shakespeare in and website spaces 've to stop known into topology for minutes and readers. Study diabetes postgraduate encloses together often published for beating space geographers of metrizable surfaces, because it is such a closed factor and it explores certain to draw over a other stability substitute. 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https://math.stackexchange.com/questions/664397/every-element-of-a-finite-abelian-group-with-square-free-order-equivalence | # Every element of a finite abelian group with square free order equivalence
I'm currently having some trouble with this problem:
Given $G$ a finite abelian group, prove the following are equivalent:
$1.$ Given any subgroup $H$, there exists a subgroup $K$ such that $HK = G$ and $H \cap K = \{e\}$
$2.$ Every element of $G$ has square-free order
I'm working on the $1 \to 2$ direction, but I'm not sure what I should do. I first assumed that $1$ was true and assumed that there was a $x$ with order $p^2$ with the expectation that the case $p^kn$ would follow. Clearly $<x>$ is a subgroup of $G$ so there exists such a $K$, and $<x>$ is isomorphic to $C_{p^2}$. I showed that $C_{p^2}$ did not have the 1st property, and I know that $<x> \times K \cong C_{p^2} \times K \cong G$
However, I get stuck here. Since $K$ is a finite abelian group, it can be decomposed into a product of cyclic groups, all of prime order. If I could show that $|K|$ and $p^2$ (or $p^kn$) were relatively prime, I would be done, but I don't know if this is true.
We have the Chinese Remainder Theorem, decompositions of finite abelian groups (both cyclic and p-groups), and uniqueness.
Assume that $x\in G$ such that $|x|=p^2$. By 1, $G=<x^p> L$ for some $L \le G$ and $<x^p> \cap L=\{e\}$. Now $<x>=<x> \cap G=<x> \cap <x^p>L=<x^p> (<x> \cap L)$ (I hope you know Dedekind modular law), this is impossible.
The Zhou's answer is perfect, but I want to describe more such groups.
A subgroup $K$ of a group $G$ is said to be complemented in $G$ if there exists a subgroup $H$ of $G$ such that $KH = G$ and $H\cap K= 1$. If every subgroup of $G$ has a proper supplement (complement), then $G$ is called an $aC$-group.
There are vast number papers about finite and infinite $aC$-group. In $1937$, Hall characterized finite $aC$-groups for first time; however, he did not use term $aC$-group. He used Complemented groups instead of $aC$-group. In $2000$, Kappe and Kirtland published a paper which is a short review about $aC$-groups.
$1.$ P. Hall, Complemented groups, J. London Math. Soc. $12$ $(1937)$ $201–204$.
$2.$ L.-C. Kappe, J. Kirtland, Supplementaion in groups, Glasgow Math. J. $42$ $(2000)$ $37–50$.
• Thanks for the proper terminology, I'll look up those papers to gain some better familiarity. – Lost Feb 5 '14 at 18:34
• @Lost There are vast papers in these subject, if you are looking for specific subject, then let me know. – Babak Miraftab Feb 5 '14 at 18:43 | 2019-07-21 18:53:56 | {"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": 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": 0.8801602721214294, "perplexity": 185.83588449380542}, "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-2019-30/segments/1563195527196.68/warc/CC-MAIN-20190721185027-20190721211027-00250.warc.gz"} |
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Removing brackets or braces will follow the same rules for removing parentheses. Parentheses. Also find the definition and meaning for various math words from this math dictionary. 4674. For K-12 kids, teachers and parents. If parentheses are removed, it can drastically change the mathematical relationships and, therefore, the result of your equation. 6798. Parenthesis definition, either or both of a pair of signs ( ) used in writing to mark off an interjected explanatory or qualifying remark, to indicate separate groupings of symbols in mathematics and symbolic logic, etc. Jean-Marie Sainthillier on 29 Apr 2014 Wonderful solution. Take as an example the problem: 9 - 5 ÷ (8 - 3) x 2 + 6. Part of the series: Math Problems. Kindergarten Freeintable Fifth Grade Math Worksheets 5th Common ... #151410 . A free online arithmetic calculator that besides basic math functions (addition, subtraction, multiplication, division) contains several advanced functions (sine , cosine, logarithm, tangent, sin/cos/ log/ tan radical. However, automatic sizing is not good in every case; one of these cases is precisely that of summations with limits above and below: compare the results of $\left( \sum_{i=1}^{n-1} i \right)\biggl(\sum_{i=1}^{n-1} i\biggr)$ (the font is that obtained with \usepackage{fouriernc}). You'll come across many symbols in mathematics and arithmetic. Parenthesis in math are used to group important things together, so you always do them first.. I would like to create a new 100 x 1 cell array where each element consists of solely the number within the parentheses. The integer division should truncate toward zero. Step 1: First, perform the operations within the parenthesis Step 2: Then, perform multiplication and division from left to right. Example 2. a − [b − (c − d + e)] We will remove all the grouping symbols. I hope this has helped you. Example import re s = 'I love book()' result = re.search(r'',s) print result.group() s1 = 'I love book(s)' result2 = re.sub(r'[]','',s1) print result2 Output . Solution 140433. In fact, the language of math is written in symbols, with some text inserted as needed for clarification. Difficulty to remove several parentheses in a string, using stringr, in R. Related. But if that is not easily achievable, I'd be ok with using some special syntax. In math, parentheses also help us understand our problem better. Hide Ads About Ads. Parentheses inside of other parentheses are called "nested" parentheses. Click to see full answer. 1 Comment. If there a way to do this in MATLAB? 3044. But we use them in a slightly different manner. The order of precedence can be fooled by using parentheses. The process of simplification works the same way as in the simpler examples on the previous page, but we do need to be a little more careful as we work our way through the grouping symbols. Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. For example, they are frequently used to enclose arguments to functions and methods. In conclusion, parentheses are used in mathematics to clarify numbers, to indicate multiplication, and to group numbers in the order of operations. Englisch-Deutsch-Übersetzungen für parentheses im Online-Wörterbuch dict.cc (Deutschwörterbuch). Free Printable Math Coloring Worksheets For 5th Grade Word ... #151409. Check it out: Do inside the parenthesis first! The size of brackets and parentheses can be manually set, or they can be resized dynamically in your document, as shown in the next example: $F = G \left (\frac {m_ 1 m_ 2}{r^ 2} \right)$ Notice that to insert the parentheses or brackets, the \left and \right commands are used. Aurelien Queffurust on 29 Nov 2012 wow I don't find the words to comment this solution except who can read that?!! In this batch of free printable expressions with parentheses, brackets, and braces worksheets, we'll work through three symbols widely used in pre-algebra, sets, algebra, and more. How do I iterate over the words of a string? Always evaluate the numbers inside the parentheses before moving on to any other operations when solving mathematical equations.. One may also ask, how do you say parentheses in math? Solution 140318. Show Ads. I have a 100 x 1 cell array where each element consists of a number followed by a space followed by a number in parentheses. Brackets and Parentheses Fractions and Binomials Aligning Equations Operators Spacing in math mode Integrals, sums and limits Display style in math mode List of Greek letters and math symbols Mathematical fonts Figures and tables Inserting Images Tables Parentheses definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. And, by "working our way through", I mean "work our way out from the inside, simplifying as we go". Then we will do it again removing the parentheses first. Did you know that symbols in math stretch much beyond +, =, and /, which we're sure you are a master of? Content Continues Below. Example 1: Example 2: It may seem confusing and it takes a bit of practice to get used to the calculations but it is not difficult, so I encourage you to start Smartick and practice this subject among many more subjects in primary school math. Case insensitive 'Contains(string)' 4015. Sometimes, parentheses will help you to see where you should start but are not playing an important role in coming up with the right solution. Implement a basic calculator to evaluate a simple expression string.The expression string contains only non-negative integers, '+', '-', '*', '/' operators, open '(' and closing parentheses ')' and empty spaces ' '. Parentheses In Math Displaying top 8 worksheets found for - Parentheses In Math . 2 Comments. I want to be able to handle multiple parentheses within parentheses for a math reader I'm writing. If instead we would like to add the value 10 to 10, then multiply that sum by 5, we can use parentheses just like we would in math: u = (10 + 10) * 5 print(u) Output. As far as the C language is concerned, anything happening within parentheses is evaluated first in any equation. Updated February 21, 2017 | Factmonster Staff . How do I make the first letter of a string uppercase in JavaScript? The following code matches parentheses in the string s and then removes the parentheses in string s1 using Python regular expression. How to force order with parentheses. Even if you are using only one bracket, both commands are mandatory. So even when you forget the order of precedence, you can force it by hugging parts of an equation with parentheses. Enclose material that is not essential to a sentence and that if not included would not alter its meaning: After a few minutes (some say less) the blaze was extinguished. Math ahead! The P in PEMDAS stands for "parenthesis!. After brackets, braces. When you see a math problem containing parentheses, you need to use the order of operations to solve it. Brackets (Parentheses) Brackets are symbols used in pairs to group things together. Python parentheses primer. This gives the output () I love books Rajendra Dharmkar. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. Some of the worksheets for this concept are Linear equations work, Nested parentheses in pemdas, Subtracting integers a, Parentheses in pemdas mixed s1, Adding integers a, Parenthesis brackets and braces, Add subtract with parenthesis 5 numbers, Order of operations pemdas practice work. In math questions, parentheses are setting aside some few terms or operations. Improve your math knowledge with free questions in "Evaluate numerical expressions with parentheses in different places" and thousands of other math skills. Perhaps I'm going about this the wrong way, but my goal was to recursively go deeper into the parentheses until there were none, and then I would perform the math operations. 3082. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. Thereof, how parentheses are used in math? That is, the above would be produced by $(a * (b+c))$. First, let’s review a couple of the basics. It is curious that many solutions checks the "direction" of given parenthesis to find it's pair. Let’s see some more examples of handling parentheses in the powers. See more. After parentheses are used, then for clarity we use brackets. In many computer programming languages, parentheses have a special purpose. Definition of Parentheses explained with real life illustrated examples. If you’re like me, you were probably amazed by how long it took to do things that we don’t even think about. For K-12 kids, teachers and parents. When Do You Use a Bracket and a Parenthesis in Math?. Ideally, this would be done without alteration to the math syntax. The whole idea of parentheses is that they say, “Do this thing first!” Sometimes, parentheses are hugely important for the order of operations. Please note that the curly are converted to square without impact to the final result. If the calculations involve a combination of parenthesis, addition, subtraction, multiplication and division then. Look it up now! What is the difference between String and string in C#? Objective: I know how to perform mixed operations with parenthesis, addition, subtraction, multiplication and division. Parentheses In Math Multiplication Pleasant Expanding Brackets ... #151406. Automatically sized parentheses are obtained with \left and \right, as any LaTeX guide or manual tells. This kind of math expression, where the result depends upon the position of parentheses, is called non-associative. We will do it by removing the brackets first. I am looking for a way to have auto-resizing parenthesis on math mode.
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https://howlingpixel.com/i-en/Friedrich_Bessel | # Friedrich Bessel
Friedrich Wilhelm Bessel (German: [ˈbɛsəl]; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method of parallax. A special type of mathematical functions were named Bessel functions after Bessel's death, though they had originally been discovered by Daniel Bernoulli and then generalised by Bessel.
Friedrich Wilhelm Bessel
C. A. Jensen, Friedrich Wilhelm Bessel, 1839 (Ny Carlsberg Glyptotek)
Born22 July 1784
Died17 March 1846 (aged 61)
ResidencePrussia
NationalityPrussian (German)
Known forBessel functions
Stellar parallax
Bessel ellipsoid
(full list here)
AwardsPhD (Hon):
University of Göttingen (1811)
Lalande Prize (1811)
Gold Medal of the Royal Astronomical Society (1829 and 1841)
Scientific career
FieldsAstronomy, mathematics, geodesy
InstitutionsUniversity of Königsberg
Doctoral studentsFriedrich Wilhelm Argelander
## Life and family
Bessel was born in Minden, Westphalia, administrative center of Minden-Ravensberg, as second son of a civil servant. He was born into a large family in Germany. At the age of 14 Bessel was apprenticed to the import-export concern Kulenkamp at Bremen. The business's reliance on cargo ships led him to turn his mathematical skills to problems in navigation. This in turn led to an interest in astronomy as a way of determining longitude.
Bessel came to the attention of a major figure of German astronomy at the time, Heinrich Wilhelm Olbers, by producing a refinement on the orbital calculations for Halley's Comet in 1804, using old observation data taken from Thomas Harriot and Nathaniel Torporley in 1607.[1]
Two years later Bessel left Kulenkamp and became Johann Hieronymus Schröter's assistant at Lilienthal Observatory near Bremen. There he worked on James Bradley's stellar observations to produce precise positions for some 3,222 stars.[1]
In January 1810, at the age of 25, Bessel was appointed director of the newly founded Königsberg Observatory by King Frederick William III of Prussia. On the recommendation of fellow mathematician and physicist Carl Friedrich Gauss (with whom he regularly corresponded)[2] he was awarded an honorary doctor degree from the University of Göttingen in March 1811.
Around that time, the two men engaged in an epistolary correspondence.[3] However, when they met in person in 1825, they quarrelled; the details are not known.[4]
In 1842 Bessel took part in the annual meeting of the British Association for the Advancement of Science in Manchester, accompanied by the geophysicist Georg Adolf Erman and the mathematician Carl Gustav Jacob Jacobi.
Bessel married Johanna, the daughter of the chemist and pharmacist Karl Gottfried Hagen who was the uncle of the physician and biologist Hermann August Hagen and the hydraulic engineer Gotthilf Hagen, the latter also Bessel's student and assistant from 1816 to 1818. The physicist Franz Ernst Neumann, Bessel's close companion and colleague, was married to Johanna Hagen's sister Florentine. Neumann introduced Bessel's exacting methods of measurement and data reduction into his mathematico-physical seminar, which he co-directed with Carl Gustav Jacob Jacobi at Königsberg.[5] These exacting methods had a lasting impact upon the work of Neumann's students and upon the Prussian conception of precision in measurement.
Bessel had two sons and three daughters. His eldest daughter, Marie, married Georg Adolf Erman, member of the scholar family Erman. One of their sons was the renowned Egyptologist Adolf Erman.
After several months of illness Bessel died in March 1846 at his observatory from retroperitoneal fibrosis.[6]
## Work
Königsberg Observatory in 1830. It was destroyed by bombing in the Second World War.
While the observatory was still in construction Bessel elaborated the Fundamenta Astronomiae based on Bradley's observations. As a preliminary result he produced tables of atmospheric refraction that won him the Lalande Prize from the French Academy of Sciences in 1811. The Königsberg Observatory began operation in 1813.
Starting in 1819, Bessel determined the position of over 50,000 stars using a meridian circle from Reichenbach, assisted by some of his qualified students. The most prominent of them was Friedrich Wilhelm Argelander.
With this work done, Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star. Astronomers had believed for some time that parallax would provide the first accurate measurement of interstellar distances—in fact, in the 1830s there was a fierce competition between astronomers to be the first to measure a stellar parallax accurately. In 1838 Bessel won the race, announcing that 61 Cygni had a parallax of 0.314 arcseconds; which, given the diameter of the Earth's orbit, indicated that the star is 10.3 ly away.[7][8][9] Given the current measurement of 11.4 ly, Bessel's figure had an error of 9.6%. Nearly at the same time Friedrich Georg Wilhelm Struve and Thomas Henderson measured the parallaxes of Vega and Alpha Centauri.
As well as helping determine the parallax of 61 Cygni, Bessel's precise measurements using a new meridian circle from Adolf Repsold allowed him to notice deviations in the motions of Sirius and Procyon, which he deduced must be caused by the gravitational attraction of unseen companions.[10][11][12] His announcement of Sirius's "dark companion" in 1844 was the first correct claim of a previously unobserved companion by positional measurement, and eventually led to the discovery of Sirius B.
Bessel was the first scientist who realized the effect later called personal equation, that several simultaneously observing persons determine slightly different values, especially recording the transition time of stars.[13]
In 1824, Bessel developed a new method for calculation the circumstances of eclipses using the so-called Besselian elements. His method simplified the calculation to such an extent, without sacrificing accuracy, that it is still in use today.
Bessel's work in 1840 contributed to the discovery of Neptune in 1846 at Berlin Observatory, several months after Bessel's death. On Bessel's proposal (1825) the Prussian Academy of Sciences started the edition of the Berliner Akademische Sternkarten (Berlin Academic Star Charts) as an international project. One unpublished new chart enabled Johann Gottfried Galle to find Neptune near the position calculated by LeVerrier in 1846.
In the second decade of the 19th century while studying the dynamics of 'many-body' gravitational systems, Bessel developed what are now known as Bessel functions. Critical for the solution of certain differential equations, these functions are used throughout both classical and quantum physics.
Bessel is responsible for the correction to the formula for the sample variance estimator named in his honour. This is the use of the factor n − 1 in the denominator of the formula, rather than just n. This occurs when the sample mean rather than the population mean is used to centre the data and since the sample mean is a linear combination of the data the residual to the sample mean overcounts the number of degrees of freedom by the number of constraint equations — in this case one. (Also see Bessel's correction).
An additional field of work was geodesy.[14] Bessel published a method for solving the main geodesic problem.[15] He was responsible for the survey of East Prussia which joined the Prussian and Russian triangulation networks[16] and he obtained an estimate of increased accuracy for the figure of the Earth, nowadays referred to as the Bessel ellipsoid.[17][18]
Despite lacking a university education, Bessel was a major figure in astronomy during his lifetime. He was elected as member of the Prussian Academy of Sciences in 1812, the French Academy of Sciences in 1816, foreign member of the Royal Swedish Academy of Sciences in 1823, and fellow of the Royal Society in 1825. In 1832, he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.[19] In 1827 Bessel became member of the Royal Institute of the Netherlands, predecessor of the Royal Netherlands Academy of Arts and Sciences.[20]
Bessel won the Gold Medal of the Royal Astronomical Society twice in 1829 and 1841.
The largest crater in the Moon's Mare Serenitatis and the main-belt asteroid 1552 Bessel, as well as two fjords in Greenland, Bessel Fjord, NE Greenland and Bessel Fjord, NW Greenland, were named in his honour.[21]
## Publications
Latin
• Fundamenta Astronomiae pro anno MDCCLV deducta ex observationibus viri incomparabilis James Bradley in specula astronomica Grenovicensi, per annos 1750–1762 institutis, Königsberg, 1818
• Tabulae regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatæ, Königsberg, 1830
German
• Untersuchungen über die scheinbare und wahre Bahn des im Jahre 1807 erschienenen grossen Kometen. [Investigations on the apparent and the real orbit of the great comet of 1807], Königsberg, 1810
• Untersuchung der Größe und des Einflusses des Vorrückens der Nachtgleichen. [Investigations on precession], Berlin, 1815
• Untersuchungen über die Länge des einfachen Secundenpendels. [Investigations on the length of the seconds pendulum], Berlin, 1828
• Versuche über die Kraft mit welcher die Erde Körper von verschiedener Beschaffenheit anzieht. [Experiments on the force with which the earth attracts things of different matter], Berlin, 1832
• Gradmessung in Ostpreußen und ihre Verbindung mit Preußischen und Russischen Dreiecksketten. [The East Prussian Survey and its connection with the Prussian and Russian networks], Berlin, 1838
• Darstellung der Untersuchungen und Maaßregeln, welche, in 1835 bis 1838, durch die Einheit des Preußischen Längenmaaßes veranlaßt worden sind. [Description of the investigations and rules arranged in 1835 to 1838 for the standardization of the prussian unit of length], Berlin, 1839
• Astronomische Beobachtungen auf der Königlichen Universitäts-Sternwarte zu Königsberg. [Astronomical Investigations (XXI Volumes)], Königsberg, 1815–1844
• Astronomische Untersuchungen. [Astronomical Investigations. (2 Volumes)], Königsberg, 1841–1842
• Heinrich Christian Schumacher, ed. (1848), Populäre Vorlesungen über wissenschaftliche Gegenstände von F.W.Bessel. [Popular lectures on scientific subjects], Hamburg
• Rudolf Engelmann (ed.), Abhandlungen von Friedrich Wilhelm Bessel. [Treatises of Friedrich Wilhelm Bessel]
• Vol. 1: I. Bewegungen der Körper im Sonnensystem. II. Sphärische Astronomie. Leipzig 1875
• Vol. 2: III. Theorie der Instrumente. IV. Stellarastronomie. V. Mathematik. Leipzig 1876
• Vol. 3: VI. Geodäsie. VII. Physik. VIII. Verschiedenes – Literatur. Leipzig 1876.
• Rudolf Engelmann, ed. (1878), Recensionen von Friedrich Wilhelm Bessel, Leipzig
## References
1. ^ a b
2. ^ Clifford J. Cunningham, Bode's Law and the Discovery of Juno: Historical Studies in Asteroid Research, Springer, 2017, pp. 121ff..
3. ^ Helmut Koch, Introduction to Classical Mathematics I: From the Quadratic Reciprocity Law to the Uniformization Theorem, Springer, p. 90.
4. ^ Oscar Sheynin, History of Statistics, Berlin: NG Verlag Berlin, 2012, p. 88.
5. ^ Olesko, Kathryn M. (1991). Physics as a Calling: Discipline and Practice in the Königsberg Seminar for Physics. Cornell University Press.
6. ^ Bessel, Friedrich Wilhelm (1846). "Bessel's Tod" [Bessel's death]. Astronomische Nachrichten (in German). 24 (556): 49–52. Bibcode:1846AN.....24...49B. doi:10.1002/asna.18460240402.
7. ^ Bessel, F. W. (1838). "Bestimmung der Entfernung des 61sten Sterns des Schwans" [Determination of the distance to 61 Cygni]. Astronomische Nachrichten (in German). 16 (365–366): 65–96. Bibcode:1838AN.....16...65B. doi:10.1002/asna.18390160502.
8. ^ Bessel, F. W. (1838b). "On the parallax of 61 Cygni". Monthly Notices of the Royal Astronomical Society. 4 (17): 152–161. Bibcode:1838MNRAS...4..152B. doi:10.1093/mnras/4.17.152.
9. ^ "A brief history of light dates". National Geographic. Retrieved 14 August 2013.
10. ^ Bessel, F. W. (1844a). "Ueber Veränderlichkeit der eigenen Bewegungen der Fixsterne" [On Variations of the proper motions of the fixed stars]. Astronomische Nachrichten (in German). 22 (514): 145–160. Bibcode:1844AN.....22..145B. doi:10.1002/asna.18450221002.
11. ^ Bessel, F. W. (1844b). "Ueber Veränderlichkeit der eigenen Bewegungen der Fixsterne (Fortsetzung)" [On Variations of the proper motions of the fixed stars (continued)]. Astronomische Nachrichten (in German). 22 (515): 169–184. Bibcode:1844AN.....22..169B. doi:10.1002/asna.18450221202.
12. ^ Bessel, F. W. (1844c). "On the variations of the proper motions of Procyon and Sirius". Monthly Notices of the Royal Astronomical Society. 6 (11): 136–141. Bibcode:1844MNRAS...6R.136B. doi:10.1093/mnras/6.11.136a.
13. ^ Hoffmann, Christoph (2007). "Constant differences: Friedrich Wilhelm Bessel, the concept of the observer in early nineteenth-century practical astronomy and the history of the personal equation". British Journal for the History of Science. 40 (3): 333–365. doi:10.1017/s0007087407009478.
14. ^ Viik, T. (2006). F.W. Bessel and Geodesy (PDF). Struve Geodetic Arc 2006 International Conference: The Struve Arc and Extensions in Space and Time. August 13–15, 2006. Haparanda and Pajala, Sweden: Lantmäteriet, Gävle, Sweden, 2006. pp. 53–63.
15. ^ Bessel, F. W. (2010) [1825]. . Translated by C. F. F. Karney & R. E. Deakin. "The calculation of longitude and latitude from geodesic measurements". Astronomische Nachrichten. 331 (8): 852–861. arXiv:0908.1824. Bibcode:2010AN....331..852K. doi:10.1002/asna.201011352. English translation of Astron. Nachr. 4, 241–254 (1825). Errata.
16. ^ Bessel, F. W.; Baeyer, J. J. (1838). Gradmessung in Ostpreussen und ihre Verbindung mit Preussischen und Russischen Dreiecksketten [The East Prussian Survey and its connection with the Prussian and Russian networks] (in German). Berlin: Dümmler.
17. ^ Bessel, F. W. (1837). "Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht" [Determination of the axes of ellipsoid that fits best to the existing measurements of meridian arcs]. Astronomische Nachrichten (in German). 14 (333): 333–346. Bibcode:1837AN.....14..333B. doi:10.1002/asna.18370142301.
18. ^ Bessel, F. W. (1841). "Ueber einen Fehler in der Berechnung der französischen Gradmessung und seinen Einfluß auf die Bestimmung der Figur der Erde" [Concerning an error in the calculation of the French survey and its influence on the determination of the figure of the Earth]. Astronomische Nachrichten (in German). 19 (438): 97–116. Bibcode:1841AN.....19...97B. doi:10.1002/asna.18420190702.
19. ^ "Book of Members, 1780–2010: Chapter B" (PDF). American Academy of Arts and Sciences. Retrieved 24 June 2011.
20. ^ "Friedrich Wilhelm Bessel (1784 - 1846)". Royal Netherlands Academy of Arts and Sciences. Retrieved 22 May 2016.
21. ^ Schmadel, Lutz D. (2007). "(1552) Bessel". Dictionary of Minor Planet Names – (1552) Bessel. Springer Berlin Heidelberg. p. 123. doi:10.1007/978-3-540-29925-7_1553. ISBN 978-3-540-00238-3. | 2019-04-22 07:59:50 | {"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": 0.6199570298194885, "perplexity": 9641.69989481862}, "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-2019-18/segments/1555578548241.22/warc/CC-MAIN-20190422075601-20190422101311-00071.warc.gz"} |
https://electronics.stackexchange.com/questions/36118/designing-high-power-led-driver-with-pwm | # Designing high-power LED driver with PWM
## Update:
Provided I set up the PIC and program it correctly (planning on using this guide to interface it as a USB HID device), would this be a suitable configuration? I'm not entirely certain how I should be deriving values for the passive components outside of the manufacturers given buck configuration application example.
I also can't find any examples using this driver with digital PWM (as opposed to converting the digital to analog), and I can't find enough explanation on how the DCTRL and ACTRL pins work. In this schematic, I assumed it would work as I intend by just applying the PWM signal directly to the DCTRL pin and leaving ACTRL untouched. In any case, if I wanted to use the digital-to-analog conversion, I'd just need to add a capacitor from ACTRL to GND (Richtek suggests a .47uF).
Thanks again :)
edit 4: From my last comment -- ...I think I'm leaning towards using the RT8482 and the NTD4960 to drive the LED, and integrate it on a PCB along with a PIC18F2550 and its associated USB interface. If I were to do this, how should I be calculating resistor/capacitor/inductor values for the driver-side of the circuit (assuming 0-1.5A or 0-2A current range for the LED)? The worst part about this project, at the moment, is that a programmer for the PIC will cost more than parts =_=
edit3: Given the answers I currently have, I should be able to deal with the analog part of the circuit, however, I still am at a loss as to how I should generate the PWM signal. What sorts of MCUs should I be considering, given the restrictions of it needing to do nothing more than generate a PWM signal, and be able to be interfaced via USB (be it native USB, or a conversion to USB from another standard)?
I'm currently looking to design a driver for a single high-power LED that can be brightness controlled and turned on/off via a PC. I am using a Cree XM-L LED (datasheet here, it's the 240lm neutral white model, part number: 000LT40E4). I already have a GUI written in C# to control the other parts of the project (camera and 3-axis motors), and I'd like to integrate light control into my program. Currently we are using a basic analog circuit to drive the LED (nothing more than a current-limiting resistor and a rheostat).
I figure the best approach would be to build a small constant-current, PWM controlled driver circuit. I stumbled upon this circuit on Instructables and think it would suit my needs fine (however, I have no commitment to any design, so any ideas would be appreciated).
Pertaining to the circuit above, I'm not sure what kind of parts I'd want to get (for the transistors, the zener, if necessary), and likewise, I have no idea what kind of microcontroller I'd need. I have next to no experience with microcontrollers, so I'm not sure what I'd need to get started. I shouldn't have a problem programming the controller, I just don't know what type of controller I'd use for this application. Likewise, I'm not sure how I'd interface the controller with a PC after programming it, though I'd assume RS-232 would be feasible, in which case I shouldn't have an issue; I can use a RS-232 --> USB converter and deal with serial communication within the scope of C#.
I've also stumbled upon ICs like this Maxim-IC MAX16834, but feel like these are a) overkill and b) less efficient, given I'd still need a microcontroller to generate the PWM signal and supply power transistors, so I could just use the basic circuit above and use the PWM and NPN transistor instead.
As far as my background goes, I'm entering my third-year in the EE program at UMass Amherst. I'm familiar with basic circuit analysis and programming, but I haven't learned anything about electronics yet, hence why I'm posting this question. I get the gist of most of the circuit designs, I just don't know how to design them from scratch since I don't know how to calculate values when non-elementary items are added to the mix (transistors, mainly). I feel that with a simple list of parts and schematic for this project, I'd be able to figure out the rest (all that's really left is programming the microcontroller).
If I left anything important out, please let me know. Thanks in advance!
edit: I forgot to mention, at the moment, the LED isn't normally driven past 400mA. It is rated at 700mA nominal (but is rated to run all the way up to 3A). I'm not sure what levels of light we will need at the end, but I'd like to have the ability to supply up to 2A.
edit2: To answer RusselMcMahon's question, I would prefer to run the LED off of 5V (I already have a Cincon CFM20 5V, 20W DC supply), but if a potential circuit design would require a different supply, it wouldn't be a problem to buy a new supply.
• Your background should be in your profile. Then we can consult it at any time, without you having to repeat it in each and every question. – stevenvh Jul 19 '12 at 13:50
• @stevenvh I understand that. I included a more specific background for this question because I think it is important to mention that while I understand the general approaches to solve my problem, I don't know how to deal with the specifics of part selection and other basic design issues (but given a circuit, I should be able to understand how it works, just not quantitatively). – Shamtam Jul 19 '12 at 14:10
• Very important - what voltage do you want to run this from? eg 12V, 5V, ... . The instructables circuits are poor. They would work after a fashion but the 'designer' does not fully understand his art. The Maxim circuits and IC are good BUT they are for a string of LEDs that have a highjer voltage than the Vin supply voltage. Unless you use Vin <= about 2.5V that does not apply to you. We will be able to offer good advice. Vin matters much as a first step. NB - f mains in (110 or 230 VAC) then an intermediate low V DC will be a good idea. – Russell McMahon Jul 19 '12 at 14:18
• @RussellMcMahon I can't believe I forgot that. I updated the question. – Shamtam Jul 19 '12 at 14:36
Given: Cree XM-L LED.
Want: Up to 2A drive, PWM controled by PC via USB.
This can be two parts. ie actual LED drive and PC to LED drive interface. These may or may not be integrated.
A "very easy" approach is to
1. use an off the shelf USB to "output" device. "Output" may be analog level, PWM, 8 bit port etc to control ... 2. An off the shelf LED driver that uses analog or PWM input.
For example, the circuit below using a RT8482 requires an analog input level or PWM with a simple RC filter (to convert the PWM to analog). The analog could be provided by a USB to analog output I/O device (COTS) or by a USB to parallel port device (not a printer port per se) (COTS) with a simple R2R digital to analog converter (about 16 resistors plus maybe a cheap op-amp).
Or a microcontroller with USB capability could have a relatively simple program written to provide PWM or analog output. A USB enabled Arduino or a Raspberry Pi would do this. (USB has to be slave not host mode).
LED drive:
(1) "Off the shelf" complete units that do the LED drive part of this job well are available at good prices from eg ebay, or Mouser and similar. Using such is a good default solution unless you have some reason to do otherwise.
(2) DIY LED driver.
Digikey LED drivers are found here. Alas the parametric search is poor in this case (which is unusual).
Searching using LED driver 2A gives better results.
There will be a nummber.
Example only: For $US1.52/1 in stock Digikey you get 1 Ricktek RT8482, buck or boost, LED driver. Drives external MOSFET so LED current capability essentially unlimited. Looks like a good start. 350 kHz for smallish inductors. • High Voltage Capability : VIN Up to 36V, VOUT Up to 48V Buck, Boost or Buck Boost Operation C u r r e n t M o d e P W M w i t h 3 5 0 k H z S w i t c h i n g Frequency Easy Dimming : Analog, PWM Digital or PWM Converting to Analog with One External Capacitor Programmable Soft Start to Avoid Inrush Current Programmable Over Voltage Protection VIN Under Voltage Lockout and Thermal Shutdown 16-Lead WQFN and SOP Packages RoHS Compliant and Halogen Free A MOSFET suitable for use as M1 would be eg ONSEMI NTD4960$US0.40/1 in stock Digikey, 30V, 9A, 9 milliohm on resistance nominal, logic gate - data sheet curves show good at 4V gate and say 4A.
Should I be looking at specific types of inductors for this sort of application
Inductors are very special for best results. If this is a one-off then off the shelf inductors from eg Digikey or similar are wise. We can give advice in this when final real spec is known.
I'm assuming all of the caps in this type of application would be ceramic?
Ceramic capacitors will work well for all capacitors shown. At least 10V rating. More or much more voltage OK.
D1 is Schottky and should have current rating equal or greater than LED max current.
Now I just need to figure out how to generate the PWM signal.
PWM is "easy" [tm] and may not be needed. Above LED controller example can use analog or PWM control.
USB to I/O
This USB to paraell FIFO I/O module](http://www.ftdichip.com/Support/Documents/DataSheets/DLP/usb245r-ds-v10.pdf) uses FTDI's FT245R USB-parallell FIFO interface IC - datasheet here .
Vast amounts of related FT245 information here
FT245 available from Digikey ~= $US4.50/1 from here FT245 based module from Digikey for about$40/1 here
This page discusses a DIY USB printer port which, as you have complete control over the hardware and how it acts, could "easily" meet your need. Based on a PIC18F4550 microcontroller and not much else. All software PCB patterns, circuit etc free.
• Should I be looking at specific types of inductors for this sort of application (I'm assuming all of the caps in this type of application would be ceramic)? Thank you for the post though! Now I just need to figure out how to generate the PWM signal. – Shamtam Jul 20 '12 at 0:18
• On a slight tangent, if I were to change my requirement to only need up to 1A or so maximum, wouldn't it be easier to use an IC like the ON Semiconductor NCP3065 (digi-key link here: digikey.com/product-detail/en/NCP3065PG/NCP3065PGOS-ND/1693183) to be an all-in-one package to handle driving the LED (and all I need to do outside of this is to supply digital PWM input to the COMP pin)? I'm rethinking if I really need up to 2A, seeing as how I'm generally getting sufficient light output at 350mA. – Shamtam Jul 20 '12 at 3:26
• @Shamtam - The NCP3065/NCV3065 is a sheep in sheep's clothing rework of the venerable MC34063. It is "OK" but would have poor efficiency if used here as shown by them due to the decades old darlington transistor switch with massive on voltage and so losses. You can use if to drive an external MOSFET but max frequency is still not marvellous. The have reduced Vref from stupid to just "high for 1A+ LEDs". ie it's now 0.235 V which is about 0.235/3.3 = 0.07 so you lose 7% efficiency in the sense resistor. ... more ... – Russell McMahon Jul 20 '12 at 7:47
• @Shamtam - ... You are better off with the Richtek IC I mentioned. (110 mV sense, higher frequency, more ...). Driving an external FET is not an issue if the IC is made for it as this is. – Russell McMahon Jul 20 '12 at 7:47
• I figured as much. Given the devices you've given me so far, I'm leaning towards using the RT8482 to drive the LED, and then I could use the PIC18F4550 to send a digital PWM signal directly to the controller. This way, I could make a single board that has a USB in, and an LED out (where I'd use a simple barrel DC Power jack), and the control/driver circuitry is integrated. Given that this is possible, this should be a very small board, and also very cheap, but with plenty of functionality. – Shamtam Jul 20 '12 at 14:40
There are some difficult challenges in driving high current LEDS for starters, yet it is a simple concept to turn on a diode. You will learn a lot of analog and thermal inter-dependancies that are also dependant on mechanical skills and the LED has a very low series resistance like a battery so you will learn the effects of this too.
How to drive an LED rated for 5 Amps absolute max and 3 Amps continuous but recommended in test spec at 0.75 A without turning the (light) "emitter" into a soldering iron. ;) ?! ;)
Thermal Resistance of LED The odd thing is LED's radiate no heat since there are no IR wavelengths. It is all conducted to substrate. But then the "Black body effects" of the substrate radiates IR heat as it attempts to conduct it away from the source. This may be hard to understand at first. Black body effects are are detailed here, but essentially the energy absorbed as conducted heat from V*I loss is later radiated by heat sink conductors as "infrared heat". This wavelength shift is always from short to longer wavelengths, blue to yellow red phosphor to infrared heat loss indicates emissions going to lower electron valence levels. Thermal resistance case to ambient is a significant impact on de0rating of the ambient for the device.
Effective Series Resistance (ESR of LED and of Driver) 1) you need great thermal conductance and an aluminum substrate to spread the heat fast to a large heatsink that removes the heat efficiently from a 2mm square chip inside a 5mm square SMD package. This chip when driven with 0.7A @ 3.5Vmax =2.45W worst case. Yet the rise in forward voltage "TYP" is only 2.90 to 3.35 from 0.7A to 3.0A. THis ∂V/∂I change is the effective series resistance ESR [Ω] which can vary significantly from batch to batch of wafers , which is why worst case or MAX is much larger than TYP. The ESR from 0.7 to 1.5A is 3.1-2.9=0.2V/0.8A = 0.25Ω ESR. It will decrease as current increases from this point and visa versa. So like a battery or an ideal zener, it has a sharp V-I curve with low ESR. This is critical when choosing switch MOSFETS and other switches to ensure your driver is much lower than this load and also understand the I^2R power dissipation increases rapidly with current. The efficiency also drops with rising current so you want to keep this in mind if you were thinking low duty cycles at 3A peaks. This applies to "efficacy" of Lumens per watt created by in this case converting blue light on LED substrate to some phosphors on the surface to emit white balanced light. THis conversion drops with rising light levels slightly. Also heat loss increases with ESR * I squared. All this heat is conducted fortunately and none is radiated as IR until it hits your heatsink. Then it becomes radiated as well so local ambient heat rise can drop your light output and cause blue shift over time and make your LED chip into a soldering iron(ha) if you have no heatsink.
1. learn how to make a heatsink <<10deg'C/Watt preferably 4'C/W.
2. Understand the effects characteristics of series resistance in each component or ESR.
The Simplest solution is like the site you posted earlier with a MOSFET switch to ground with a current sensing resistor which also adds to circuit ESR. The reasons of choosing Rs similar to ESR of LED string are for optimal efficiency and and ability to drive LED hard without thermal runaway caused by stability of V vs T effect (Shockley effect). Otherwise current limiter circuits are used which often have a higher voltage drop, hence more inefficient. Switching current sources help stabilize with the chance of higher efficiency.
1. Learn to add realize non-ideal schematics in your mind. From . It goes a few levels deeper but good enuf for now.
The simplest PWM circuit uses a low ESR MOSFET driver with ESR << LED or << .25Ω.
• A lot more than I was looking for, but good information! I already have the LED on a heatsink, and it's mounted with thermal tape (for prototyping). In a final revision, I'd probably end up using thermal epoxy to mount the LED to the heatsink (I'm soldering wires to the top-side of the PCB of the LED and using an electrically-insulating thermal mating between the entire bottom of the LED package to the heatsink). At the moment, the LED can be driven around 350mA with no heat issues (it becomes warm after extended use, but not too hot to touch). At this point, I just need help with the MCU. – Shamtam Jul 20 '12 at 0:12
• Likewise, in a final product, the LED would only be on for fractions of a second like a camera flash, enough for the camera to snap a picture, and then shut off. In this case, I would imagine even a moderately undersized heatsink would suffice for short, high-current bursts (say at most 2A at 500ms). However, the heatsink won't pose a problem for me. I mainly just need to figure out how to generate a PWM signal from a computer. – Shamtam Jul 20 '12 at 0:17 | 2019-08-20 19:16:10 | {"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": 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": 0.4359930157661438, "perplexity": 1710.4752589190464}, "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/1566027315558.25/warc/CC-MAIN-20190820180442-20190820202442-00044.warc.gz"} |
https://eprints.iisc.ac.in/9741/ | # Structural and Dielectric Charactersics of Strontium Tetraborate-Bismuth Vanadate Glass-Ceramics
Varma, KBR and Shankar, MV and Subbanna, GN (1996) Structural and Dielectric Charactersics of Strontium Tetraborate-Bismuth Vanadate Glass-Ceramics. In: Materials Research Bulletin, 31 (5). pp. 475-482.
PDF STRUCTURAL_AND_DIELECTRIC.pdf Restricted to Registered users only Download (623kB) | Request a copy
## Abstract
Glasses of strontium tetraborate, containing up to 50 mole-percent bismuth vanadate [(1-x)SBO-xBiV (x = 0 to 0.50)], were prepared by splat quenching method. The glassy nature of these samples was confirmed by differential thermal analysis (DTA). The glass transition temperature $(T_g)$ and the crystallization temperature $(T_{cr})$ of the glasses decrease with increase in bismuth vanadate, $Bi_2VO_{5.5}$ (BiV) content. High resolution transmission electron microscopic studies reveal the presence of spherical particles of amorphous BiV (less than 10 nm in size) dispersed in the glassy matrix of strontium tetraborate, $SrB_40_7$ (SBO). The glasses of the compositions x = 0.25 to 0.50, on annealing at 500 $^oC (T_g)$ gave rise to crystalline BiV phase. Physical properties such as density, dielectric and optical transmission of these SBO:BiV glass-ceramics have been studied. The dielectric constant $(\in_r)$ of these glass-ceramics increases with increasing BiV content. The measured $\in_r$ values are found to be in good agreement with those predicted by the logarithmic mixture rule.
Item Type: Journal Article Materials Research Bulletin Elsevier Copyright of this article belongs to Elsevier. A. ceramics; A. glasses; C. electron microscopy; C. X-ray diffraction; D. dielectric properties Division of Chemical Sciences > Materials Research Centre 14 Mar 2007 19 Sep 2010 04:35 http://eprints.iisc.ac.in/id/eprint/9741 | 2022-12-05 02:11:59 | {"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": 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": 0.5394711494445801, "perplexity": 8885.895094450734}, "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/1669446711001.28/warc/CC-MAIN-20221205000525-20221205030525-00379.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-5-sqrt-a-3-10-0 | # How do you solve 5 * [sqrt( a - 3 )] - 10 = 0?
Apr 24, 2016
#### Explanation:
$5 \sqrt{a - 3} = 10$
$\sqrt{a - 3} = 2$
${\left(\sqrt{a - 3}\right)}^{2} = {\left(2\right)}^{2}$
$a - 3 = 4$
$a = 7$
Checking this solution in the original equation we find that it doesn't work. Thus, there is no solution to this equation; it has a solution set $\left\{\emptyset\right\}$
Hopefully this helps! | 2019-12-09 07:38:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.7873550057411194, "perplexity": 554.4902156468846}, "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-20191209065626-20191209093626-00063.warc.gz"} |
https://physics.stackexchange.com/questions/623156/why-is-angular-momentum-mvr-and-not-mvr2-or-mvr3/623167 | # Why is angular momentum $mvr$ and not $mvr^2$ or $mvr^3$?
We know angular momentum $$L = mvr$$, where $$v$$ is the velocity in the direction perpendicular to the distance from the source to the object whose $$L$$ we are trying to measure. My question is why $$L$$ was defined as $$mvr$$ where power of $$r$$ is 1 and not other integer greater than 1? And what experiments were conducted to come to this conclusion or was it only intuition?
And also when a dancer rotates with respect to her body it is said $$L$$ will be conserved if she pulls her hand closer to her body. As a result her angular velocity will increase. Where does this extra kinetic energy come from?
• Because angular momentum is the moment of momentum ,and $r$ is the moment arm. It is literally how far away is momentum from the origin. – JAlex Mar 23 at 18:04
Using Newton's law one can easily check that $$\vec L= \vec r \times \vec p$$ is a conserved vector for a motion of a particle in a spherically symmetric potential (while axial symmetry implies conservation of one component of $$\vec L$$). If you try to modify this definition, you will not obtain a conserved quantity.
If you are familiar with analytical mechanics, the form of angular momentum can also be derived from the Lagrangian using Noether's procedure.
Summarizing, I would say that the reason for this definition of angular momentum is not necessarily related to any particular experiment (although I don't know how it was developed historically). That's because motion of bodies is determined already by Newton's law and and there is no need to give separate laws for circular motions -- they can be derived from Newton's law (or other formulations of classical mechanics). It happens to be true that angular momentum is an extremely useful quantity in solving the Newton's equations.
Historically, this was mostly derived from observation (for example, conservation of angular momentum appears straightforwardly in Kepler's 2nd Law for the motion of planets).
Nowadays, however, conservation laws (of momentum, angular momentum, energy…) can be found directly from symmetries of the problem, through Noether's wonderful Theorem.
Simply put, Noether's Theorem states that if a system is invariant through a continuous transformation (for example, any rotation around a given point), then some quantity is a constant of motion (for this example, this turns out to be the usual angular momentum).
More specifically, if the Lagrangian $$L$$ describing a system does not depend on some quantity $$q_k$$, then the associated momentum $$p_k=\frac{\partial L}{\partial \dot q_k}$$ is a constant of motion. Writing the expression of $$L$$ for a given system gives the expression of the conserved quantity.
In our example, if a system is not changed by a rotation $$\theta$$ around origin, then L is independent of $$\theta$$ ; Noether's theorem states then that $$\frac{\partial L}{\partial \dot \theta}$$ is a constant of motion. Replacing L with the classical Lagrangian of a point mass, leads to the usual angular momentum.
All this works with other symmetries as well, leading to conservation of energy, of linear momentum ; and the same formalism applies to fields as well as particles, leading to modern gauge theories.
Regarding conservation of energy
To pull her arms towards her body, the dancer has to do work (against the centrifugal force) in the rotating frame. In the non-rotating frame, the corresponding force is towards the axis of rotation, and if the arm moves inward, it does work all the same.
That's unrelated to the conservation of momentum though! Momentum can be conserved (for rotational invariant systems) even if energy is not (for non time-invariant systems)
• This answer should be expanded to include a comment re the dancer's conservation of energy – shaunokane001 Mar 23 at 10:17
• I'm not quite sure I understand what you mean… If you're speaking about the cancer accelerating its rotation by bringing its arms close to the axis of rotation, that's about angular momentum conservation. The fact that energy is conserved (or not) depends on the details of the process and is an unrelated matter (conservation of energy is related to time-shift invariance through Noether's theorem, not rotation). – Nicolas Mar 23 at 17:24
• Oh, OK I get the point… – Nicolas Mar 23 at 17:27
I was in the middle of writing an answer when I saw 2 perfectly good answers, but I think at least one important point remains unanswered. Also, just in case you don't know about vector products and/or Lagrangians, here goes my explanation.
Angular momentum is not just any combination of coordinates and velocities. It's the one that doesn't change when the system is invariant under rotations.
Angular momentum is proportional to the following particular combination of coordinates and velocities. The body is assumed to rotate in the $$x$$, $$y$$ plane, and the axis of rotation is along $$z$$, in the perpendicular direction: $$xv_{y}-yv_{x}$$ It's not too difficult to see that, if you rotate your system infinitesimally, with angle $$\epsilon$$, the quantities involved change as, $$x\mapsto x-\epsilon y$$ $$v_{x}\mapsto v_{x}-\epsilon v_{y}$$ $$y\mapsto y+\epsilon x$$ $$v_{y}\mapsto v_{y}+\epsilon v_{x}$$ And our proposed conserved quantity changes as, $$xv_{y}-yv_{x}\mapsto\left(x-\epsilon y\right)\left(v_{y}+\epsilon v_{x}\right)-\left(y+\epsilon x\right)\left(v_{x}-\epsilon v_{y}\right)=$$ $$=xv_{y}+\epsilon xv_{x}-\epsilon yv_{y}-\epsilon^{2}yv_{x}-yv_{x}-\epsilon xv_{x}+\epsilon yv_{y}+\epsilon^{2}xv_{y}=$$ $$=xv_{y}-yv_{x}+\epsilon^{2}\left(xv_{y}-yv_{x}\right)$$ Now you make this infinitesimal $$\epsilon$$ go to zero and you realise immmediately why this particular combination is special when there is rotation symmetry. How do we recover your special case, when the system is rotating itself at a constant rate around a fixed axis? In Cartesian coordinates: $$v_{x}=-\omega y$$ $$v_{y}=\omega x$$ If we restore the mass: $$L=m\left(xv_{y}-yv_{x}\right)=m\left(\omega x^{2}+\omega y^{2}\right)=mr^{2}\omega=mrv$$
## Kinetic energy (rotational)
Energy is conserved (approximately) in your dancer case. So there's no extra kinetic energy. The reason why the dancer (I prefer to think of an ice-skater; she illustrates better the point) spins faster when she pulls her arms closer is because this rotational energy is, $$\frac{1}{2}mv^{2}=\frac{1}{2}mr^{2}\omega^{2}=\textrm{const.}$$ So if energy is to be conserved, $$\omega$$, the angular velocity, must increase when the average distance to the axis of rotation decreases.
• +1 Excellent answer. – user290607 Mar 23 at 9:49
• Thanks a lot, Feynstein. But if you think about it, it's not that good an answer. @Blazej's and Nicolas' answers are more rigorous --I'm voting them up to highlight their answers better. I was deliberately ambiguous about the words "doesn't change" for the sake of pedagogy. Noether's theorem is very simple, but far more subtle. I just wanted to show that the particular combination $xp_y-yp_x$ is special when you rotate the whole thing. – joigus Mar 23 at 12:19
• Your equation doesn't coincide with conservation of angular momentum. $K=\frac{L^2}{2mr^2).$ – Bill N Mar 23 at 12:43
• Doesn't it? $$K=\frac{L^{2}}{2mr^{2}}=\frac{m^{2}r^{4}\omega^{2}}{2mr^{2}}=\frac{1}{2}mr^{2}\omega^{2}$$ The dancer isn't a rigid body either, but still... – joigus Mar 23 at 13:09
• Oh, I see your point. But the OP said "extra energy." So energy was the point there. – joigus Mar 23 at 13:15 | 2021-05-15 17:11:11 | {"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": 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": 34, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8560799360275269, "perplexity": 310.39969538605413}, "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-2021-21/segments/1620243990551.51/warc/CC-MAIN-20210515161657-20210515191657-00016.warc.gz"} |
http://www.nucleonica.net/wiki/index.php/Help:Decay_Engine | # Help:Decay Engine
Level: Intermediate
Create htmlfile for this document
## Background Theory
Radioactive decay is a random process. It is not possible to predict when a particular nucleus in a sample will decay. One can, however, evaluate the probability that a nucleus will decay in a time interval. The basic relation governing radioactive decay, first identified by Rutherford, is
$\frac{dQ}{dt} = -kQ$
where the left hand side gives the rate of decay of the atoms Q in terms of the number of atoms Q and a decay "constant" k which is the probability per unit time that a nucleus will decay. The minus sign indicates that Q is decreasing with time. The solution to this equation is
$Q = Q(0) exp(-kt)$
where Q(0) is the number of nuclei at time t=0. From this basic relation various useful quantities such as half-life, mean life and activity or intensity of the radioactive transformation can be derived. These are described in more detail in chapter Nuclider Explorer (sections Data Sheets and Fact Sheets).
H. Bateman
It is very often the case that the daughter product of a nuclear decay is also radioactive. In such cases one speaks of radioactive decay "chains". As an example, consider the decay chain Q1→ Q2 → Q3 →….. in which the starting or "parent" nuclide Q1 decay to the "daughter" Q2. This daughter in turn is radioactive and decay to Q3. More generally each nuclide in the decay chain Qi can "branch" to more than one daughter. In addition, there may be an external source term Sii for the production of Qi (apart from the decay of the parent).
The situation for successive radioactive decay is shown schematically in fig. 1. This general process of radioactive decay was first investigated systematically by Bateman (Proc. Camb. Phil. Soc. 16 (1910) 423. See also Skrable, K., et al., Health Physics, 27, 1974, 155-157).
Figure 1. Successive radioactive decay with branching and source terms.
Bateman's orignal paper from 1910
The differential equation governing the above processes can be written as:
$\frac{dQ_1}{dt} = S_1 - k_{Q_1} \cdot Q_1$
$\frac{dQ_2}{dt} = S_2 + k_{Q_1,Q_2} \cdot Q_1 - k_{Q_2}\cdot Q_2$
..
$\frac{dQ_i}{dt} = S_i + k_{Q_{i-1},Q_i} \cdot Q_{i-1} - k_{Q_i}\cdot Q_i$
where Qn is the number of atoms of species n present at time t, kn is the decay constant (total removal constant) for species n (k = ln2/τ = 0.69315/τ), kn,n+1 is the partial decay constant (partial removal constant) and is related to the branching ratio BRn,n+1 through the relation kn,n+1 = BRn,n+1×kn. The solution to this system of equations is
Bateman solution with source terms
(1):
$Q_n(t)=\prod_{j=1}^{n-1} k_{j,j+1} \times \sum_{i=1}^n \sum_{j=i}^n \left( \frac {Q_i(0)e^{-k_jt}} {\prod_{\overset{p=i}{\underset{p \ne j}{}}}^n(k_p - k_j)} + \frac {S_i (1-e^{-k_jt})} {k_j \prod_{\overset{p=i}{\underset{p \ne j}{}}}^n(k_p - k_j)} \right)$
Bateman solution for 1 parent nuclide without source terms, Si = 0 and Qi(0) = 0 for i>1:
(2):
$Q_n(t)= Q_1(0) \prod_{j=1}^{n-1} k_{j,j+1} \times \sum_{j=1}^n \frac {e^{-k_jt}} {\prod_{\begin{array} {}_{p=1} \\ _{p \ne j} \end{array}}^n(k_p - k_j)}$
It is of interest to construct the first few terms i.e.
(3):
$Q_1(t) = Q_1(0) e^{-k_1t}$
$Q_2(t) = Q_1(0)k_{1,2} \left\{ \frac {e^{-k_1t}} {k_2-k_1} + \frac {e^{-k_2t}} {k_1-k_2}\right\}$
$Q_3(t) = Q_1(0)k_{1,2} k_{2,3} \left\{ \frac {e^{-k_1t}} {\left(k_2-k_1\right) \left(k_3-k_1\right)} + \frac {e^{-k_2t}} {\left(k_1-k_2\right) \left(k_3-k_2\right)} + \frac {e^{-k_3t}} {\left(k_1-k_3\right) \left(k_2-k_3\right)} \right\}$
$.......$
These relations allow one to update the numbers of atoms from time t=0 to time t. It is also of interest to calculate the numbers at various times in the range 0,t (for example for plotting purposes). This can be done by specifying the total time t over which the calculation is to be made, and the number of time-steps L to reach t. The time interval for each calculation is then Δt = t/L. For L = 1 (the default value used in the calculation), the numbers are evaluated at the time t. For L = 2, the numbers are evaluated at t/2, and t. For L = 3, the Qs are evaluated at t/3, 2t/3, t etc. From above, the relation to be used is then
(4)
$Q_n\left( l \Delta t \right)= \sum_{i=1}^{i=n} Q_i(0) \prod_{j=1}^{n-1} k_{j,j+1} \times \sum_{j=i}^n \frac {e^{-k_jl \Delta t}} {\prod_{\begin{array} {}_{p=i} \\ _{p \ne j} \end{array}}^n(k_p - k_j)} ~~~~\mbox{for }l=1,2,3,...~L$
### Convergent and Divergent Branches
The solution to the differential equations given in equation (4) is valid for the various species produced in series i.e. in a chain. If branching occurs, as indicated in the figure 1, the solution (4) must be applied to all possible chains. As an example, consider the radioactive decay of Ac225. The breakdown into linear chains is shown in table 1. Equation (4) must be applied to each of these three chains. In the evaluation of the total quantities of any species, care is required not to count the same decay more than once.
Table 1. The three "linear chains" for the decay of Ac-225 showing the various paths by which the nuclide can decay.
Consider a simplified radioactive decay process involving only three nuclides, Q1, Q2, and Q3. The nuclide 1 decays into nuclide 2 which in turn decays to nuclide 3. Nuclide 1 is the parent of nuclide 2 (or nuclide 2 is the daughter of nuclide 1). From the relations given above, the number of atoms of nuclide 2 is given by
(5) $Q_2 = \frac {k_1} {k_2-k_1} \cdot Q_1(0) \cdot \left( e^{-k_1t}-e^{-k_2t}} \right) = \frac {k_1} {k_2-k_1} \cdot Q_1(0) \cdot e^{-k_1t} \left( 1-e^{- \left( k_2-k_1 \right)t}} \right)$
From equation (5) it can be seen that the time required to reach equilibrium depends on the half-life of both the parent and the daughter. Four cases can be distinguished:
τ1 >> τ2. The halflife of the parent is much longer than that of the daughter.
τ1 > τ2. The halflife of the parent is longer than that of the daughter.
τ1 < τ2. The halflife of the parent is shorter than that of the daughter.
τ1 ≈ τ2. The halflifes of the parent and daughter are similar.
These will be discussed in more detail in the following sections.
#### (τ1>>τ2): Secular Equilibrium:
In secular equilibrium, the half-life of the parent is much longer than that of the daughter i.e. τ1 >>τ2 (k1 <<k2). In this case equation (5) reduces to
(6) Q2 = (k1/k2)Q1(0)(1-e-k2t)
For times t >>τ1, radioactive equilibrium is established and the following relation holds
Secular Equilibrium: Q2/Q1 = k1/k2 = τ21, and A1 =A2
where A is the activity defined by k×Q. Hence in radioactive equilibrium the ratio of the numbers, and the masses are constant whereas the activities are equal.
#### (τ1>τ2) : Transient Equilibrium
In transient equilibrium the half-life of the daughter is of the same order but smaller than that of the parent i.e.τ1 > τ2 (k1 < k2). The general equation for the daughter is from equation (5)
(7) Q2 = (τ2/(τ21))Q1(0)(e-k1t - e-k2t)
As an example, consider the decay of Ba140 as shown in fig 2. For times t << τ1 (12.75d), the first exponential term is very close to 1 and A2 increases according to (1-e1-kt) (rising part of activity of La-140 in fig. 2). For times t >> τ2 (1.68d), the second exponential becomes smaller than the first one with A2 decreasing according to e-kt (see fig. 2). For this decreasing part of the curve, one obtains
Transient Equilibrium: Q2 = (τ2/(τ21))Q1
where the relation Q1 = Q1(0) e-k·t has been used. This is the condition for transient equilibrium.
Half-lives of Ba140 and daughters.
Figure 2. Transient equilbrium in the Ba140 decay chain.
#### (τ1 < τ2): The Half-life of the Parent is Shorter than that of the Daughter
When the parent has a shorter half-life that that of the daughter, the daughter activity grows to some maximum and then decays with its own characteristic half-life. An example is shown in fig. 3 for Po-218.
Figure 3. No equilbrium: decay of Po-218.
#### (τ1 ≈ τ2): The Half-lives of the Parent and Daughter are Similar
As the half-lives of the parent and daughter become more equal, the attainment of equilibrium becomes more delayed. An example of this is shown in fig. 4 for the decay of I-135.
Figure 4. Similar half-lives: decay of I-135.
The time required for the maximum daughter activity is tmax = (k2-k1)-1×ln(k2/k1). As can be seen from fig. 4, this occurs around 11 hours.
## Using the Decay Engine Module
To make a decay calculation within NUCLEONICA, the user must first select the Decay Engine from the Application Centre on the main page. Thereafter, the nuclide of interest can be selected using the drop down menus. Alternatively, the nuclide can be selected from the nuclide chart, and then with the right mouse button, the Decay Engine can be selected.
#### Input User Interface
The resulting Decay Engine input interface is shown in the figure for Po-210. The user can either accept the default input values shown or enter input into the boxes. In the main Decay Engine tab, a number of default input parameters are shown. These are described in more detail in the following sections.
Decay calculation window input interface
##### Time units + Decay time
A default time shown with units corresponds to 10 half-lives of the selected nuclide.
##### Starting quantities / Final quantities
The default activity is 1 MBq, and by default the activity will be plotted in the graph (see Type of graph). The red question marks in the Final quantity box indicate this is the quantity to be calculated. Also shown in red is the number of linear chains - also a result of the calculation. The number of such chains depends on the Accurcy factor (described later).
##### Calculation details
###### Number of timesteps
This input box is used to set the number of intermediate times at which the decay is evaluated between t=0 and t ( t is the value in the "Decay Time" box). The default number of timesteps is 40 such that a graph can be drawn. The maximum number of timesteps is 40.
###### Accuracy factor
This input box is used to set the accuracy of the calculation. As explained in section Convergent and Divergent Branches, a parent nuclide may decay to different daughter nuclides depending on the branching ratios. The importance of each of these pathways in the decay process can be evaluated by the product of the branching ratios. If there is only one decay pathway, i.e. no branching, the product of all branching ratios is 1. If there is a pathway with a small branching ratio, then this will result in a branching ratio product determined by this low branching ratio. The input box Accuracy Factor sets the minimum product of branching ratios which are accounted for in the calculation. To evaluate all possible paths, the Accuracy Factor should be set to zero. This is discussed in more detail in section Details. The number of pathways or "Chains" accounted for in the calculation is given in the Number of linear chains box.
###### Distance (cm)
The Distance (cm) is required for the evaluation of the gamma dose rate. It is also required for plotting the gamma dose rates in the graph. The default value is 100 cm.
###### Number of linear chains
This is not an input variable but depends on the Accuracy factor. On pressing the Start button to make a calculation, this box shows the number of pathways or chains taken into account in the calculation.
##### Start / Reset
On pressing the Start button, the input data is posted to the web-server where the decay calculations are made. Thereafter, the output data is returned to the user computer.
The Reset button can be used to clear the input data used in the calculation.
##### Type of graph
In the "Type of graph" list box, the user can choose which quantity should be shown i.e. numbers, masses, activities, and gamma emission rate.
#### Results
If the default values shown in the input interface are accepted, pressing the Start button will lead to the results shown in the figure below.
Results of the Decay calculation
The results show the final activity and the number of linear decay chains in red. Had the accuracy factor been set to 0 (all chains calculated), the number of linear chains would be 23. The details in the output grid show the parent and all the daughter of Po-218 in the second column. In the following columns, the halflives, number of atoms, masses, activities, gamma dose rates at 1 m, and the number of disintegrations of the parent and all daughters are given. The total values in each case are also given. The choice of which properties to show in the Results grid can be set by the user in the Options tab (discussed later in this section).
##### Time units + Decay time
It is assumed that at time t = 0, only a single radioactive parent nuclide is present. Such a situation can arise for example after chemical separation of a parent nuclide from its daughter(s). The number entered in the input box is the time at which the decay is evaluated with reference to t=0. The default time is ten half-lives of the parent nuclide. After this time the parent nuclide has almost entirely decayed to its daughter(s).
##### Starting quantities / Final quantities
(Mass-Activity Calculator)
The source strength is set by specifying first the unit i.e. becquerel, curie, grams, or the number of atoms, and then the quantity. The quantity should be entered in scientific notation in the form 1, 10, 1e2 1e-6 etc. The default value 1 MBq. The source strength box functions also as an activity calculator which allows the conversion of one unit into another. As an example, for the nuclide Co-60 select, the default activity is 1 MBq. On changing the entry in the unit list box, one obtains 2.7e-5 Ci, 2.4e-8 g, 2.4e14 atoms.
The Mass-Activity-Number Calculator
The conversion from mass to number of atoms N, and vice versa etc., is described in the Basic quantities and relations
At the bottom of the Results table, the user has the possibility to download the table results to an Excel file or to a comma separated value (CSV) file. In this latter case (CSV) the user needs additionally to select the character separator.
Choosing the Excel option, the resulting Excel file is shown.
##### Rescale Feature
The Rescale tool is a very simple yet powerful feature in the Decay Engine. Very often the one knows the activity (e.g. total activity) one would like to have after a certain time. The Rescale tool allows the input activity in the calculation to be rescaled such that this value is obtained after the selected decay time. Without this tool, a number of iterations would be necessary to obtain the desired result.
To use the tool, copy any value from the table or the graph into the 'Rescale from' box. This value can than be rescaled to any other value (given in the 'Rescale to' box) by starting the calculation again.
Rescale feature in the Decay Engine
Example: The radionuclide F-18 is used in positron emission tomography (PET) in hospitals. The radionuclide is created in an accelerator which is located approximately 100 km from the hospital where it will be used. The journey time is approximately 2 hours for the transport. How much F-18 needs to be produced such that upon arrival at the hospital the total activity is 300 MBq?
With the decay engine, select F-18. The default activity is 1 MBq. Based on the transport time, the decay time is set to 2 h. The resulting activity after two hours is 469 kBq. So starting with 1 MBq, the activity remaining after two hours is 469 kBq. We now wish to rescale these results. Instead of 469 kBq we would like to have 300 MBq after the two hour period.
Rescale feature in the Decay Engine
Through the use of the rescale tool this is straightforward. We rescale from 469 kBq to 300 MBq. On pressing the Start button, all results are rescaled such that after 2 hours there are 300 MBq F-18 remaining. It can then be see that the Starting quantity should be 640 kBq.
A related example of the use of the rescale tool is given in the nuclide mixtures module.
##### Graph
A full description of the details and settings of the graph can be found in the webGraph wiki page.
Graph of the decay of Po-210
A graph of the results is shown below the results table. By default, the number of timesteps calculated in 40. The maximum number of timesteps is 40. Below the graph, the Show Graph Settings allow the user to select the type of graph to be plotted plus a variety of additonal settings. Both logarithmic and linear plots are available. In addition the region of interest can be specified for both x and y axes.
Various options for plotting graphs of the radioactive decay.
A full description of the details and settings of the graph can be found in the webGraph wiki page.
Print: From the Show Graph Settings, the graph can be printed directly to a printer.
Save Configuration The graph configuration (data plus graph settings) can be saved in xml format. This is described in more detail in the webGraph section.
#### Options
As mentioned in the previous section, the choice of which properties to show in the Results grid can also be set by the user from the Options tab shown below.
File:DE Options1.png
Options window showing the modes of operation, calculation details and quantities to be shown in the Results grid.
##### Mode of operation
There are two modes of operation of the Decay Engine:
1. Time mode: this has already been described above. For an initial activity of a given nuclide or mixture, the final activity is calculated after a decay time.
2. Date mode: if an activity is known at some time (date), then the activity can be calculated at any other time (date). The Date mode uses universal time (see link for further details).
Input window for Date mode of the Decay Engine
In this date mode, the user can enter a date (e.g. when the sample was taken). The calculation result then shows the activity at any later time (date). This is useful for the calculations of standards.
See the blog article on Date calculator.
Example: Consider a standard of Pb-210 from 1.9.1987. What is the activity of this standard today. In the input one enters the activity and date of certificate. The results give the activity today and the precise time elapsed.
Activity today of a Pb-210 standard from 1.9.1987
Example: A radwaste samples was measured on a date in January and contained some medium half-life nuclides. What is the precise time elapsed and the activity today?
##### Select quantities to be shown in the decay Result grid
(under development)
#### Linear Decay Chains
The linear decay chains tab is shown in the Results window. In this tab all the individual decay chains can be seen together with the numbers obtained by using equation (2). The importance of a decay chain is determined by the product of the branching ratios. If the branching ratios were all 1, there would only be one chain and the Accuracy Factor = 1.0. If a particular branching ratio in a decay chain is small e.g. 1E-8, and all other branching ratios were 1, then the product would be 1E-8. This implies that this particular decay chain is relatively less important. One must be careful in neglecting such chains, however, since the concentration or mass of the relevant nuclide may be small but may have a very high activity and for example gamma emission rate. An example is the decay of U232.
If only the main chain were considered (i.e. the chain with the highest product of branching ratios) one would miss the daughter Tl208 which is a powerful gamma emitter.
Ideally, one would like to evaluate all the decay chains for a particular parent nuclide. The problem is, however, that some heavy nuclides have a large number of decay chains - U-238 has for example 52 separate decay chains! To apply equation (2) to all these chains would impose a considerable burden on the web server. Fortunately, in most cases of interest, this is not necessary. It is sufficient to restrict the number of chains to only a few. The parameter used to restrict the number of chains to be evaluated is the product of the branching ratios "Accuracy Factor" for the particular chain. The minimum value can be set in input window. The default value is 1E-2. When the Start button is pressed, the branching ratios of all decay chains are evaluated and compared to this value. If the product of the branching ratios is less than the Accuracy Factor, the chain is rejected as being unimportant. If greater, the chain is stored for further evaluations. The number of chains stored - Number of linear chains -, i.e. with the product of the branching ratios ≥ Accuracy factor, is given at the Calculation details tablein the input window.
If the Accuracy factor is set to 0, all chains are evaluated but as stated above, this may take some time.
## References
General :
R. Dreher, Modified Bateman solution for identical eigenvalues, Annals of Nuclear Energy, 53 (2013) 427–438. Article
H. Bateman, Proc. Camb. Phil. Soc. 16 (1910) 423.
K. Skrable et al., Health Physics, 27 (1974) 155-157.
K. H. Lieser, Nuclear and Radiochemistry: Fundamentals and Applications, VCH /Wiley, 1997.
J. Magill and P. Peerani, (Non-) Proliferation Aspects of Accelerator Driven Systems, J. Phys. IV France 9 (1999) 167-181.
paper
How "hot" is Am-242m:
J. Galy, J. Magill, H. van Dam, J, Valko A Neutron Booster for Spallation Sources- Application to Accelerator Driven Systems and Isotope Production, Nucl. Instr. Meth. Res. 2002.
paper
J. Magill, R. Schenkel, Non-Proliferation Issues for Generation IV Power Systems: Advanced Waste Management in Global Warming and Energy Policy, Eds B.N. Kursunoglu, S.L. Mintz, A. Perlmutter, Kluwer Academic/Plenum Publishers, 2001.
"Age" Determination of Plutonium Particles:
J. Magill, R. Schenkel, L. Koch, Verification of Nuclear Materials and Sites, Detection of Clandestine Activities, and Assessment of the Proliferation Resistance of New Reactors and Fuel Cycle Concepts, Fachsitzung zum Thema "Abrüstung und Verification" Dresden 22-23 März, 2000.
M.Wallenius, K. Mayer, "Age" Determination of Plutonium Material in Nuclear Forensics by Thermal Ionisation Spectrometry, Fresenius Journal of Analytical Chemistry 366 (2000) 234-238.
Alpha-Immunotherapy:
C. Apostlidus, R. Carlos-Marquez, W. Janssens, R. Molinet, A. Quadi, Cancer Treatment using Bi-213 and Ac-225 in Radioimmunotherapy, Nuclear News, Dec. 2001, 29-33. | 2015-08-01 11:46:35 | {"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": 13, "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": 0.6557913422584534, "perplexity": 1261.3936325636716}, "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-2015-32/segments/1438042988650.6/warc/CC-MAIN-20150728002308-00119-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/magnetism-problem.71744/ | # Magnetism Problem
1. Apr 16, 2005
### confusion321
Which is the correct equation for Magnetic Reluctance ?
$$R$$ = Reluctance
$$l$$ = Length
$$A$$ = Area
$$\mu$$ = Magnetic Permeability of the material
$$\mu_0$$ = Magnetic Permeability of free space ($$4\pi \times 10^- ^7$$ tesla/ampere turn)
$$R = \frac {l} {\mu A}$$ as stated on Wikipedia or $$R = \frac {l} {\mu_0 \mu A}$$ as stated in my text book.
2. Apr 16, 2005
### Staff: Mentor
The Wikipedia formula looks correct. Are you sure your book isn't using $\mu$ to mean relative permeability (which is just a ratio)? (If not, the units would make no sense in your book's formula.)
3. Apr 16, 2005
### confusion321
Thanks very much that would be it. "Relative Permeability" | 2017-12-11 09:30:08 | {"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": 0.7955178618431091, "perplexity": 3303.965433717015}, "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-2017-51/segments/1512948513330.14/warc/CC-MAIN-20171211090353-20171211110353-00645.warc.gz"} |
https://www.researchgate.net/project/Turbulence-intensity-in-pipe-flow | Project
# Turbulence intensity in pipe flow
Goal: Derive scaling of turbulence intensity with bulk Reynolds number
Relationship between turbulence intensity and the friction factor
Date: 1 January 2015 - 31 December 2024
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## Project log
We have characterized a transition of turbulence intensity (TI) scaling for friction Reynolds numbers $Re_{\tau} \sim 10^4$ in the companion papers [Basse, N.T. Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 065127}] and [Basse, N.T. Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term. {\it Phys. Fluids} {\bf 2021}, {\it 33}, {\it 125109}]. Here, we build on those results to extrapolate TI scaling for $Re_{\tau} \gg 10^5$, under the assumption that no further transitions exist. Scaling of the core, area-averaged and global peak TI demonstrates that they all scale inversely with the logarithm of $Re_{\tau}$, but with different multipliers. Finally, we confirm the prediction that the TI squared is proportional to the friction factor for $Re_{\tau} \gg 10^5$.
Presentation on turbulence intensity and how it can be applied to e.g. computational fluid dynamics boundary conditions.
We extend the procedure outlined in Basse [“Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region,” Phys. Fluids 33, 065127 (2021)] to study global, i.e., radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling, which leads to the existence of a mathematical abstraction, which we call the “global peak.” The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynolds number of order 10 000 is identified. Consequences for the global peak scaling, length scales, non-zero asymptotic viscosity, turbulent energy production/dissipation, and turbulence intensity scaling are appraised along with the impact of including an additional wake term.
We extend the procedure outlined in [Basse, "Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region," Phys. Fluids Vol. 33, 065127 (2021)] to study global, i.e. radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling which leads to the existence of a mathematical abstraction which we call the "global peak". The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynolds number of order 10000 is identified. Consequences for the global peak scaling, length scales, non-zero asymptotic viscosity, turbulent energy production/dissipation and turbulence intensity scaling are appraised along with the impact of including an additional wake term.
We study the global, i.e. radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as I^2 = overbar(u^2)/U^2 , where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark M, Vallikivi M, Bailey SCC and Smits AJ. Logarithmic scaling of turbulence in smooth-and rough-wall pipe flow. J. Fluid Mech. Vol. 728, 376-395 (2013)]. Scaling expressions are derived both for log-law and power-law functions of radius. A transition to asymptotic scaling is found at a friction Reynolds number Re_τ ∼ 11000.
An asymptotic scaling law for drag in flat-plate turbulent boundary layers has been proposed [Dixit SA, Gupta A, Choudhary H, Singh AK and Prabhakaran T. Asymptotic scaling of drag in flat-plate turbulent boundary layers. Phys. Fluids Vol. 32, 041702 (2020)]. In this paper we suggest to amend the scaling law by using a correction term derived from the logarithmic law for the mean velocity in the streamwise direction. | 2022-09-27 17:16:14 | {"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": 0.8295450806617737, "perplexity": 2917.1948301317925}, "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/1664030335054.79/warc/CC-MAIN-20220927162620-20220927192620-00720.warc.gz"} |
http://etheses.whiterose.ac.uk/22403/ | # A statistical study of ionopause perturbation and associated boundary wave formation at Venus
Chong, Ghai Siung (2018) A statistical study of ionopause perturbation and associated boundary wave formation at Venus. PhD thesis, University of Sheffield.
Text thesis_corrected.pdf Restricted until 21 January 2022. Request a copy
## Abstract
Previous missions to Venus have revealed that encounters with plasma irregularities of atmospheric origin outside the atmosphere are not uncommon. A number of mechanisms have been proposed to discuss their origins. One such mechanism involves an ionopause with a wavelike appearance. Extensive research on this characteristic of the ionopause is crucial in understanding the atmospheric evolution of Venus. This thesis implemented an approach to identify potential boundary crossings resulting from ionospheric boundary waves. Coupled with Minimum Variance analysis, this approach was able to demonstrate whether or not the boundary crossings are smooth' or rippled'. By utilising the magnetic field and plasma data from Venus Express (VEX) over its entire mission from 2006 to 2014, this work presents the first observational statistical analysis of the ionospheric boundary waves at Venus. The average estimated ionospheric boundary thickness is 57~$\pm$~4~km. This value which is estimated during solar minimum is roughly 1.5 times more than those estimated during solar maximum period. Further analysis shows that the boundary is thicker for weaker pressure and at higher altitudes. On the other hand, the boundary is thinner for stronger pressure at lower altitudes. In the northern polar region of Venus, the normal directions of the rippled ionospheric boundary crossings lie mainly in the terminator plane with the largest component predominantly along the dawn-dusk ($Y_{VSO}$) direction. The average estimated wavelength of the boundary wave is 212~$\pm$~12~km and the average estimated velocity difference across the ionopause is 104~$\pm$~6~km~s$^{-1}$. The consistency shown between these results and the results from previous simulation studies of the Kelvin-Helmholtz Instability (KHI), suggests that the rippled boundary is a result of KHI. Furthermore, the magnetic field orientation in the barrier region is found to be quasi-perpendicular to the terminator plane, which is a favourable condition for the excitation of KHI along the dawn-dusk direction. Analysis reveals a correlation between the normal directions and the locations of the boundary wave with respect to Venus. This indicates the draping of magnetic field lines appear to play a role in enhancing the plasma flow along the dawn-dusk direction, which could subsequently set up a velocity shear that favours the excitation of ionospheric boundary wave by the KHI along the dawn-dusk direction. Two other previously proposed boundary wave excitation mechanisms are also explored. In addition, flux ropes are also identified using a similar approach. The average estimated flux ropes diameter is 90~$\pm$~6 km and they have axial orientations which lie mainly along the Venus-Sun ($X_{VSO}$) direction. The consistency in the sizes, locations and orientations shown between the identified flux ropes and boundary wave events suggest that these flux ropes are created as a result of the boundary waves reaching a turbulent stage. In summary, this statistical study reveals that the ionopause of Venus does not always appear to be smooth, but often exhibits a wavelike appearance. Further analysis suggests that this wavelike appearance of the Venus ionopause is likely to be excited by the KHI, which arises as a result of the velocity shear across the ionopause driven by the draping pattern from the magnetic field. This study also shows that flux ropes can form and populate inside the ionosphere as a result of turbulent boundary wave. On the other hand, atmospheric bubbles can also form in similar manners and exist outside the ionosphere. Continuous scattering and the subsequent convection of atmospheric bubbles downstream and away from Venus over a prolonged period of time plays an important role in atmospheric loss from Venus.
Item Type: Thesis (PhD) The University of Sheffield > Faculty of Engineering (Sheffield) > Automatic Control and Systems Engineering (Sheffield)The University of Sheffield > Faculty of Engineering (Sheffield) Dr Ghai Siung Chong 28 Jan 2019 15:33 06 Dec 2019 12:31 http://etheses.whiterose.ac.uk/id/eprint/22403
Please use the 'Request a copy' link(s) above to request this thesis. This will be sent directly to someone who may authorise access. | 2020-07-03 14:04:15 | {"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": 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": 0.5742672085762024, "perplexity": 1838.553856007665}, "config": {"markdown_headings": true, "markdown_code": false, "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-2020-29/segments/1593655882051.19/warc/CC-MAIN-20200703122347-20200703152347-00413.warc.gz"} |
https://tex.stackexchange.com/questions/654408/creating-a-comparison-chart | # Creating a comparison chart
How to create a chart as follows using LaTeX. This is just one example. You can also share your more elegant suggestions.
MWE:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\pagestyle{empty}
\begin{tikzpicture}
\begin{scope}[shift={(3cm,-5cm)}, fill opacity=0.5,
mytext/.style={text opacity=1,font=\large\bfseries}]
\draw[fill=blue, draw = black] (0,0) circle (8);
\draw[fill=yellow, draw = black,name path=circle 1] (0,-2) circle (6);
\draw[fill=green, draw = black,name path=circle 2] (0,-2) circle (3);
\node[mytext] at (0.4,7) (A) {Artifical Intelligence};
\node[mytext] at (0,3.3) (C) {Machine Learning};
\node[mytext] at (0,0.1) (E) {Deep Learning};
\end{scope}
\end{tikzpicture}
\end{document}
P.S.: I have a problem with coloring. Because the color changes at the intersection of the diagram. Also, I'm living a problem with entering the text to the next line and adding arrows like in the second figure.
or another example is
• Hello. Since you're not new to this site, you probably know (or at least you should) that this is not a do it for me site. Please edit your post to provide a minimal working example (MWE) and ask for help about a specific issue you encounter. Aug 18 at 8:34
• How? // Use tikz. Draw 3 circles. Use 3 "split rectangle" objects to host headline and text. Or replace the headline with an image to obtain the second example. // We are looking forward to your Edit to append tikz-code (from \documentclass to \end{document}) and screenshot(s). Aug 18 at 12:21
• Good places to start: 1) Minimal introduction (ctan.org/pkg/pgf) 2) Tikz-manual (ctan.org/pkg/pgf) 3) texample.net/tikz/examples/venn 4) split rectangles: tex.stackexchange.com/questions/301404/… . Aug 18 at 12:26
Here are some ways to do it, which address most of your questions:
• for simplicity I changed to standalone, which is independent of paper size; readjust later to article
• I added two more libraries, which will be needed below
• I put a help grid to have a better idea of the dimensions and coordinates you chose
• add align=center to your nodes options, so you'll have multilines
• I also put an example for shape rectangle split
• as an example I put a new node (m) for text to the left
• I draw 2 lines, the straight line with the standard arrow tip, a bent line with e.g. the {Stealth} tip, to show the differences
Remaining issues:
• alignment, postioning
• using different text fonts
• color
(can be solved in various ways with tikz)
\documentclass[10pt, border=10pt]{standalone}% fits paper automatically
\usepackage{tikz}
\usetikzlibrary{intersections, shapes.multipart, % <<< for rectangle split
arrows.meta}
\begin{document}
\pagestyle{empty}
\begin{tikzpicture}
\begin{scope}[shift={(3cm,-5cm)}, fill opacity=0.5,
mytext/.style={text opacity=1,font=\large\bfseries}]
\draw [help lines] (0,0) grid (5,5); % <<< to "see" coordinates
\draw[fill=blue, draw = black] (0,0) circle (8);
\draw[fill=yellow, draw = black,name path=circle 1] (0,-2) circle (6);
\draw[fill=green, draw = black,name path=circle 2] (0,-2) circle (3);
\node[mytext, align=center] at (0.4,7) (A) % <<< align
{Artifical Intelligence\\ sth. else}; % <<< now you have multi-lines
\node[mytext] at (0,3.3) (C) {Machine Learning};
\node[mytext, rectangle split,
rectangle split parts=2, align=center] % <<< alternative
at (0,-2) (E) % <<< shifting
{Deep Learning
\nodepart{two} % <<< rectangle split
Subnset of \\
machine learning\\
which
};
\node [mytext] (m) at (-10, 3.2) {Machine Learning};
\draw [->] (m) to (C); % standard arrow tip
\draw [-{Stealth}] (m) to [bend left] (C); % a different tip from arrows.meta
\end{scope}
\end{tikzpicture}
\end{document} | 2022-12-08 19:36:57 | {"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": 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": 0.7007752060890198, "perplexity": 7705.417018318074}, "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/1669446711360.27/warc/CC-MAIN-20221208183130-20221208213130-00492.warc.gz"} |
https://codedump.io/share/jknHfJ8vOEuO/1/can39t-get-datatable-to-populate-with-codeigniter-and-ajax | Timothy Fisher - 10 months ago 57
Ajax Question
# Can't get dataTable to populate with Codeigniter and Ajax
I have gone through several answers on SO as well as a few tutorials and the documentation on ajax/dataTables. My dataTable will still not populate with JSON data.
HTML:
<table id="table" class="table table-striped table-bordered table-hover" cellspacing="0" width="100%">
<tr>
<th>Status</th>
<th>Student Name</th>
<th>Exam Name</th>
<th>School</th>
<th colspan="2">Action</th>
</tr>
<tfoot>
<th>Status</th>
<th>Student Name</th>
<th>Exam Name</th>
<th>School</th>
<th colspan="2">Action</th>
</tfoot>
</table>
Javascript:
<script type="text/javascript">
$(document).ready(function() { // Datatables$('#table').DataTable({
"url": "<?php echo site_url('exams/ajax_list'); ?>",
});
});
</script>
ajax_list PHP function in Exams controller:
public function ajax_list() {
$list =$this->exam_model->get_datatables();
$data = array(); foreach ($list as $exam) {$row = array();
$row[] =$exam->exam_status;
$row[] =$exam->first_name . " " . $exam->last_name;$row[] = $exam->exam_name;$row[] = $exam->exam_school;$data[] = $row; } echo json_encode($data);
}
From what I can see when navigating to the method, the json_encode outputs correctly, but the dataTable is still empty.
Am I missing anything?
Ok I finally got it to work... Thanks for giving me the chance to play with this as its all brand new to me. It was a bit of a bugga, but like everything else it turns out to be simple.
I didn't set this up in CI but that wont matter...
After drudging through the documentation I came up with this...
1.Change your "url" to "ajax". I'll assume your path you use is correct. Alter what I have to yours.
<script type="text/javascript">
$(document).ready(function () { // Datatables$('#table').DataTable({
"ajax": "./ajax_list.php" // change this to suit.
});
});
</script>
2. Lose the colspan="2" in your th tags. It don't like it unless you use some other option I didn't look into...
3. And finally change your json_encode to...
echo json_encode(['data'=>\$data]);
And hopefully that gets you up and running... The documentation is pretty good so I suggest giving that a good going over. | 2017-08-20 19:52:45 | {"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": 0.44156011939048767, "perplexity": 6100.898201367216}, "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-2017-34/segments/1502886106984.52/warc/CC-MAIN-20170820185216-20170820205216-00083.warc.gz"} |
http://openstudy.com/updates/55a4b27fe4b07a3af135624c | ## anonymous one year ago I need help with this question, The graph below shows the value of Edna's profits f(t), in dollars, after t months: graph of quadratic function f of t having x intercepts at 6, 0 and 18, 0, vertex at 12, negative 36, and passes through point 21, 41.25 What is the closest approximate average rate of change for Edna's profits from the 18th month to the 21st month?
• This Question is Open
1. anonymous
whees the graph??
2. anonymous
I don't know how to put it on here but the part that says graph of quadratic function f of x that is what is on the graph.
3. anonymous
click on attach file i can't understand u
4. anonymous
Oh ok sorry on my computer the screen is small so i didn't see that okay one sec and i will put the graph.
5. anonymous
ok :3
6. anonymous
7. anonymous
Here is the graph
8. anonymous
ok wait up
9. anonymous
ok
10. anonymous
rate of change= slope
11. anonymous
do you know how to find the slope?
12. anonymous
since you are not answering i will show ya
13. anonymous
slope=$\frac{ y2-y1 }{ x2-x1 }$
14. anonymous
so your two points from 18 months to 21 months is: (0,18) and (21,41.25).
15. anonymous
Hey sorry it wouldn't let me type lol
16. anonymous
plug the points into the slope formula and you get:$\frac{ 41.25-18 }{ 21-0 } =\frac{ 23.25 }{ 21 }$
17. anonymous
thats the rate o change :3
18. anonymous
g'night btw
19. anonymous
Oh okay thank you! Lol
20. anonymous
:) | 2017-01-24 16:02:53 | {"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": 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": 0.6104688048362732, "perplexity": 3785.5759816143322}, "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/1484560284429.99/warc/CC-MAIN-20170116095124-00038-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://nigerianscholars.com/past-questions/accounts/question/206674/ | Home » » Use the following information to answer the given question:$$\begin{array}{c|c} & DR & CR \\ & ₦ & ₦ \\ \hline \text{Goodwill} & 10,000 & \\ \text{Plant and Machinery} & 32,000& \\ \text{Freehold premises} & 50,000& \\ stock & 15,000& \\ debtors & 12,500 & \\ \text{Cash in bank} & 7,500 & \\ \text{cash in hand} & 2,000 & \\ \text{Profit and loss account} & & 34,000 \\ \text{Accrued rent} & & 500 \\ \text{Sundry creditors} & & 9500 \\ Capital & \overline{129,000} & \overline{129,000}\end{array}$$What is the value of current assets?...
Use the following information to answer the given question:$$\begin{array}{c|c} & DR & CR \\ & ₦ & ₦ \\ \hline \text{Goodwill} & 10,000 & \\ \text{Plant and Machinery} & 32,000& \\ \text{Freehold premises} & 50,000& \\ stock & 15,000& \\ debtors & 12,500 & \\ \text{Cash in bank} & 7,500 & \\ \text{cash in hand} & 2,000 & \\ \text{Profit and loss account} & & 34,000 \\ \text{Accrued rent} & & 500 \\ \text{Sundry creditors} & & 9500 \\ Capital & \overline{129,000} & \overline{129,000}\end{array}$$What is the value of current assets?...
Question
Use the following information to answer the given question:
$$\begin{array}{c|c} & DR & CR \\ & ₦ & ₦ \\ \hline \text{Goodwill} & 10,000 & \\ \text{Plant and Machinery} & 32,000& \\ \text{Freehold premises} & 50,000& \\ stock & 15,000& \\ debtors & 12,500 & \\ \text{Cash in bank} & 7,500 & \\ \text{cash in hand} & 2,000 & \\ \text{Profit and loss account} & & 34,000 \\ \text{Accrued rent} & & 500 \\ \text{Sundry creditors} & & 9500 \\ Capital & \overline{129,000} & \overline{129,000}\end{array}$$
What is the value of current assets?
A) ₦37,000
B) ₦35,000
C) ₦27,500
D) ₦22,500
E) ₦22,000 | 2022-01-22 22:58:41 | {"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": 0.7627667784690857, "perplexity": 2728.8226919737185}, "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-2022-05/segments/1642320303917.24/warc/CC-MAIN-20220122224904-20220123014904-00649.warc.gz"} |
https://zbmath.org/?q=an:1069.34054 | # zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A note on the stability and boundedness results of solutions of certain fourth order differential equations. (English) Zbl 1069.34054
Summary: The purpose of this paper is to investigate the asymptotic stability of the zero solution of $x^{(4)}+\varphi(\ddot x)\dddot+f(x,\dot x)+h(x)=0$ with $p\equiv 0$ and the boundedness of all solutions of $x^{(4)}+\varphi(\ddot x)\dddot x+f(x,\dot x)\ddot x+g(\dot x)+h(x)= 0$ with $p\ne 0$. The results obtained here revise, improve and include some results in the literature.
##### MSC:
34C11 Qualitative theory of solutions of ODE: growth, boundedness 34D20 Stability of ODE
Full Text:
##### References:
[1] Cartwright, M. L.: On the stability of solutions of certain differential equations of the fourth-order. Quart. J. Mech. appl. Math. 9, No. 4, 185-194 (1956) · Zbl 0071.30901 [2] Chin, P. S. M: Stability results for the solutions of certain fourth-order autonomous differential equations. Int. J. Control, No. 4, 1163-1173 (1989) · Zbl 0696.93048 [3] Ezeilo, J. O. C: On the boundedness and the stability of solutions of some differential equations of the fourth order. J. math. Anal. appl. 5, 136-146 (1962) · Zbl 0107.29602 [4] Ezeilo, J. O. C: A stability result for solutions of a certain fourth order differential equation. J. London math. Soc. 37, 28-32 (1962) · Zbl 0101.30702 [5] Harrow, M.: A stability result for solutions of certain fourth order homogeneous differential equations. J. London math. Soc. 42, 51-56 (1967) · Zbl 0145.11405 [6] Harrow, M.: On the boundedness and the stability of solutions of some differential equations of the fourth order. SIAM J. Math. anal. 1, 27-32 (1970) · Zbl 0247.34038 [7] Ku, Y. H.: Lyapunov function of a fourth-order system. IEEE trans. Automat. control 9, 276-278 (1964) [8] Shi-Zhong, L.; Zheng-Rong, L.; Yuan-Hong, Y.: Stability for certain fourth-order nonlinear differential equations. Demonstratio Mathematica 31, No. 1, 87-96 (1998) [9] Wu, X.; Xiong, K.: Remarks on stability results for the solutions of certain fourth-order autonomous differential equations. Int. J. Control. 69, No. 2, 353-360 (1998) · Zbl 0934.34039 | 2016-04-30 20:48:08 | {"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": 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": 0.5524330139160156, "perplexity": 3835.565373724346}, "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-2016-18/segments/1461860112727.96/warc/CC-MAIN-20160428161512-00151-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://matcomgrader.com/problem/9683/isosceles-right-triangle/ | ### I - Isosceles right triangle
##### Languages: C, C++, Java, Python, ... (details)
You are given the coordinates of $n$ points and a set of $m$ isosceles right triangles. The legs (shortest sides) of the triangles are parallel to the coordinate axes.
For each triangle, you have to find how many of the given points it contains.
#### Input
First line contains two positive integers $n$ and $m$ $(1 \leq n,m \leq 10^5)$, the numbers of points and the number of triangles respectively.
Next $n$ lines contain two integers $x$ $y$ $(-10^9\le x,y \le 10^9)$, the coordinates of each of the given points.
Next $m$ lines contain six integers $x_1\ y_1\ x_2\ y_2\ x_3\ y_3$ $(-10^9\le x_1,y_1,x_2,y_2,x_3,y_3 \le 10^9)$, the coordinates of the vertices of the given triangles.
#### Output
Print $m$ numbers in separated lines, the number of points on each triangle.
#### Sample test(s)
Input
3 2 1 1 -8 -4 -8 -4 0 0 -100 0 0 -100 2 3 3 3 2 4
Output
2 0
Input
9 5 0 0 10 0 20 0 0 10 10 10 20 10 0 20 10 20 20 20 20 0 20 20 0 0 0 20 0 -20 40 20 0 20 0 -19 39 20 9 9 12 9 9 12 -10 -10 -15 -10 -10 -15
Output
6 9 8 1 0
Input
5 4 1000000000 1000000000 -1000000000 1000000000 1000000000 -1000000000 -1000000000 -1000000000 0 0 1000000000 1000000000 -1000000000 1000000000 1000000000 -1000000000 1000000000 -1000000000 1000000000 1000000000 -1000000000 -1000000000 -1000000000 1000000000 -1000000000 -1000000000 1000000000 1000000000 -1000000000 -1000000000 1000000000 -1000000000 -1000000000 1000000000
Output
4 4 4 4 | 2022-12-02 16:24:20 | {"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": 0.3770328760147095, "perplexity": 501.3013955492515}, "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/1669446710909.66/warc/CC-MAIN-20221202150823-20221202180823-00236.warc.gz"} |
http://ocrcomplaints.com/to-do-list/ | Improvement Step 3: Since the ship is starting at now at port B you draw the third vector 5.5cm long beginning start here toward the way south-west. one protractor is required to assess the reason for 4hich {\displaystyle 2\times 2} {\displaystyle 2\times 2} is only 2 social events for 2 included to give 4. you cone handle one relative strategy to oversee see how vector duplication limits.
Fhsst vectors32.png
The Free High healing magnets School Science Texts: one Textbook for High School Students Studying Physics.
Fundamental Page – << Previous Chapter (Waves and wavelike improvement) – Next Chapter (Forces) >> | 2019-10-22 02:24:17 | {"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": 0.9183719158172607, "perplexity": 7231.348067043035}, "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/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00074.warc.gz"} |
http://supertestoboostsfacts.com/%E9%B9%BF%E5%85%90%E5%B3%B6%E7%9C%8C/%E9%8C%A6%E6%B1%9F%E7%94%BA/ | E C A E B F E Ba
Linear algebra - show that $e^{a+b}=e^a e^b, If$a$and$b$are$n\times n$matrices such that$ab = ba$(that is,$a$and$b$commute), show that $$e^{a+b}=e^a e^b$$ note that$a$and$b\$ do not have to be. A b cde f de e f e e c b - florida department of, Eb ˜e ab' b e b e e 3b f de ˜ e b a˜ ( ˇa a eb debe eˆ* 3b aa d a f a e . / ( e5.5 ˆ ed˘ e eˇa#bbb add a ebe be ˇ a˜ bˇ be .4. Alphabetical index (a b c d e f g h i j l m n o p r s t u, Assistive technology specialist pay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g 100 attendance reports.
A ba cdebf f e a e a ba - pmu.edu.sa, A bcdc aecff d a bca fd dc e fc b a bb c daacdd b a c d b b cb b˘a cab cc a a ˘fa caa˘ a cb c ˇˆ˙˘d ˝ ˛˚ecc bb c ca d. Bae.c (@ba_e_c) | twitter, Tweet with a location. you can add location information to your tweets, such as your city or precise location, from the web and via third-party applications.. Hohner blues band 1501 7 key harmonica set a bb c d e f g, The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging (where packaging is applicable). packaging should be the same as what is found in a retail store, unless the item is handmade or was packaged by the manufacturer in non-retail packaging, such as an unprinted box or plastic bag..
Scales list | the guitar woodshed, Scales. here is a list of all the scales in the guitar woodshed.com database. click on the scale name to view the scale on a fretboard.. Prove e^(a+b)=e^a*e^b*e^-k/2 if [a,b]=k | physics forums, And using a power series again, i found (e^a)*(e^b) = (e^b)*(e^a) * e^-k, which is clearly very close to what i am trying to show. i keep staring at that neat partial derivative of coefficients trick, trying to see. | 2018-10-17 01:14:44 | {"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": 0.6987244486808777, "perplexity": 3508.317938161202}, "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-2018-43/segments/1539583510932.57/warc/CC-MAIN-20181017002502-20181017024002-00035.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-1-introduction-to-algebraic-expressions-review-exercises-chapter-1-page-75/8 | # Chapter 1 - Introduction to Algebraic Expressions - Review Exercises: Chapter 1: 8
This statement is $\color{red}{\text{FALSE}}$.
#### Work Step by Step
Like terms are terms having exactly the same variable and degree of each variable for each term. In this item, the variable of the first term is $r^2s$ while the second is $rs^2$. Therefore, they are not like terms, and this statement is $\color{red}{\text{FALSE}}$.
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 19:00:08 | {"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": 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": 0.4581187665462494, "perplexity": 544.5381542519017}, "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-2018-34/segments/1534221213693.23/warc/CC-MAIN-20180818173743-20180818193743-00359.warc.gz"} |
https://www.vedantu.com/question-answer/find-the-total-number-of-ways-of-answering-five-class-11-maths-cbse-5f2e1c61b275c258b1900267 | Question
# Find the total number of ways of answering five objective type questions, each question having four choices.
Hint: Find the categories which are present in the given equation. Find the possibilities present in each case. Now check whether you have to add the categories or multiply them to get the result of total possibilities. Here each question has independent choices. So, you must multiply all the question’s possibilities to get the final result. So, first find possibilities for each question.
Rule of Sum: - In combinatorics, the rule of sum or addition principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. P, Q words cannot be done together. Total number of ways to do both P, Q are given by (A + B) ways.
Rule of product: - In combinatorics, the rule of product or multiplication principle is basic counting principle. It is simply defined as, if there are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q works are given by (A.B) ways.
By listing all possible categories, we get:
Question 1, Question 2, Question 3, Question 4, Question 5.
For given 5 Questions each have 4 multiple choices.
By listing each category with their possibilities, we get:
Possibilities of answering Question 1 = 4
Possibilities of answering Question 2 = 4
Possibilities of answering Question 3 = 4
Possibilities of answering Question 4 = 4
Possibilities of answering Question 5 = 4
Here options of 1 & 2 can be answered together. Similarly to any pair of questions.
So, this comes under product rule. So, we must multiply all.
By applying product rule, we get:
Total possibilities = $4\times 4\times 4\times 4\times 4={{4}^{5}}$
Therefore, there are 1024 ways to answer 5 questions with 4 choices each.
Note: Be careful while categorizing into product rule or sum value. You must carefully check whether there is any dependence between 2 works or not. Students generally apply sum rule but it is wrong you must apply product rule. Generally students confuse between the number of questions and number of choices. So, read the question carefully. | 2020-10-27 17:14:53 | {"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": 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": 0.7471779584884644, "perplexity": 614.8505962186754}, "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/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00128.warc.gz"} |
https://aerospaceanswers.com/question-category/propulsion/ | • Question Cagtegory: Propulsion
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• 233 views | 2021-01-20 06:49:28 | {"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": 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": 0.43580853939056396, "perplexity": 2312.234063928003}, "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-2021-04/segments/1610703519923.26/warc/CC-MAIN-20210120054203-20210120084203-00218.warc.gz"} |
https://www.gamedev.net/forums/topic/286645-help---this-code-wont-work/ | # Help - this code won't work!
This topic is 4829 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 relatively new to C++, and I've run into difficulties with what should be very easy code. Here it is:
#include <iostream.h>
void f_1();
void f_2();
void solution(int x, int y, int z);
int main(void)
{
void f_1();
void f_2();
void solution();
return 0;
}
void f_1()
{
int x;
cout<<"Enter a number:\n\n";
cin>>x;
}
void f_2()
{
int y;
cout<<"Enter another number:\n\n";
cin>>y;
}
void solution(int x,int y)
{
int z;
z=x*y;
cout<<"x * y = ";
}
I can get the same program to work if I declare x,y,z globally and do everything within int main(). However, this code compiles and runs with 0 errors, 0 warnings (I'm using VC++ 6 Standard), but just exits. Doesn't even show any of the text in the cout<< bits, or ask me for my input. I can pass parameters in Visual Basic no problem, but this stumps me as to why it won't even work. Hope somebody can help me!
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I could list and explin all the mistakes you made, but IMO a book on C++ is better suited for that. Please read about variable scope, function declaration and definition as well as <>calling functions and passing parameters.
Regards,
Pat.
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The big problem is that inside main() when you do:
void f_1();
void f_2();
void solution();
Those aren't calling the functions, those are declaring function prototypes. If you get rid of the void in front of f_1() and f_2() in main, it should then call the functions. Fixing the call to solution() is trickier, since you declared solution() to take arguments, but you don't have any arguments to pass to it.
One way to fix that, is to change f_1() and f_2() to functions that return an int value. And then declare local variables in main() to store the values f_1() and f_2() return. Then you can use the variables when calling solution().
It would go something like this:
#include <iostream>int f_1();int f_2();void solution(int x, int y);int main(void){ int x = f_1(); int y = f_2(); solution(x, y); return 0;}int f_1(){ int x; std::cout << "Enter a number:\n\n"; std::cin >> x; return x;}int f_2(){ int y; std::cout << "Enter another number:\n\n"; std::cin >> y; return y;}void solution(int x, int y){ int z = x * y; std::cout << "x * y = " << z << std::endl;}
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Quote:
Original post by ukdeveloperWhat mistakes have I made?
For starters, the definition of the solution method (void solution(int x, int y, int z)) is different than its declaration (void solution(int x, int y)).
Also, as SiCrane noted above, you're not calling the functions in main, just redefining them. I'm at a loss to explain why your compiler isn't complaining about this.
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It's working great now. Thanks for helping me, I'll do a bit more reading next time!
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Quote:
Original post by kSquaredAlso, as SiCrane noted above, you're not calling the functions in main, just redefining them. I'm at a loss to explain why your compiler isn't complaining about this.
Those are local function declarations, not definitions. Local function declarations are valid in C++. Local function definitions are not.
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Okay then riddle me this - what's the point in locally declaring a function if you can't locally define it? :-)
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Well, you sort of can locally define a function, but you have to do it inside a struct:
int main() { struct foo { static int bar() { return 3; } }; foo.bar();}
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I believe the big reason for allowing local function declarations is for compatibility with C code. If C++ disallowed local function declarations, a lot of legacy C code would break.
It's also sometimes useful when you're in crunch mode and you need something done yesterday. Let's say you add in a quick hack that uses a function from a header that the source file doesn't currently include (for some reason it's usually Windows API calls for me), and the good solution doesn't involve using that source file. Rather than including the whole source file, put the local function declaration for the one function you need to use, and then all the horrible hacks will all be in one spot for when you go through and pull it out later. Not a practice I recommend for every day use, but it comes in handy once in a while. | 2018-02-23 15:58:09 | {"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": 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": 0.19404329359531403, "perplexity": 3587.679784684573}, "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-2018-09/segments/1518891814801.45/warc/CC-MAIN-20180223154626-20180223174626-00031.warc.gz"} |
https://codegolf.meta.stackexchange.com/questions/1320/programming-puzzle-asking-for-real-clever-solutions-ok | # Programming puzzle asking for “real”, clever solutions OK?
I've got an idea for a new kind of puzzle and would like to ask if it's OK first. It's a bit different than the usual code-golf kind of question. It doesn't ask for the shortest solution, or something funny/silly like the troll questions, but actually for something semi-useful. I'll give an example first, and then describe the general idea:
Example: Implement multi-line lambdas in Python. Guido van Rossum said it couldn't be done, prove him wrong. Your solution should allow something like:
>>> f = multilinelamba("hour", """
... if hour > 20 or hour < 6:
... print "Good night"
... else:
... print "Hello world"
... """)
>>> f(10)
... 'Hello world'
Your solution should be as close as possible to the behavior of real def or lambda. The actual syntax or way of implementation doesn't matter (e.g. define a function, write a preprocessor, ...). You get bonus points:
• If your solution captures outside variables in a closure, like real def
• For correct handling of exceptions in the multiline lamba
• (and so on, a checklist of objective criteria or 'unit tests' to judge)
I posted this question a while ago on Stack Overflow: Recipe for anonymous functions in python?, but it received a lot of hate. People didn't answer it, but told my why it was a bad idea, and I had to justify myself and put up big disclaimers. So now I'm trying something different, I'm posing it not as a SO question, but as a programming riddle. Explicitly not looking for a solution for production code, but just for fun. That doesn't mean I'm looking for silly solutions - they should be well-engineered and in principle usable. I have a couple of these questions where tinkering is required, and I'm really curious to see clever solutions, but I'm afraid of posting them on SO for obvious reasons.
I personally don't really care for rating the answers, and would be happy with two or three interesting submissions per question without picking a winner. But it seems the community feels strongly about having objective winning criteria, so I propose a check list of "tests" like above that the solution must pass, the more it passes the better.
One think I'd like to add is that the answers should be compact enough to fit in an answer. You may link to a repository, if your solution lead to something cool and useful, of course, but the core idea (and basis of rating, if desired) should be presented succinctly.
Finally, it should be a question that doesn't fit directly on SO, so no straightforward programming questions, but rather hackish "I wonder if it could be done?" kind of challenges. The kind where people refuse to answer, and say "don't" or "what are you trying to accomplish?".
As for a tag name, I have no great idea. Maybe something like tinker-challenge or clever-hack?
As Guido said, "Language Design Is Not Just Solving Puzzles", but solving puzzles is fun!
• possible duplicate of Let's create some new types of challenges! – John Dvorak Mar 22 '14 at 13:55
• @JanDvorak: Not a duplicate, that question said: "To propose a genre, ask a question here on meta with the new-genre tag.". Hope I didn't misunderstand that post... – jdm Mar 22 '14 at 13:57
• Huh... sorry. I was under the impression new genres were supposed to be posted as answers to that post. Retracting. – John Dvorak Mar 22 '14 at 14:00
• What if more than one of the entries pass all of the tests? Also, this seems like it would have to be language-specific, if I understand it correctly. – Doorknob Mar 22 '14 at 14:01
• @Doorknob: Yes, this is language specific, I hope that's OK. It's not inherent to the challenge type, other questions may or may be not language specific. And yes, if multiple questions pass all mandatory and optional tests, then they receive the same rating (that can also happen with code golf, but is less likely). But as I said, this is more about seeing clever solutions than picking a winner. – jdm Mar 22 '14 at 14:10
• Language specific challenges are generally discouraged here, so an entire challenge type that requires them doesn't seem ideal. Also, if they receive the same rating, how do you decide which one wins? – Doorknob Mar 22 '14 at 14:12
• @Doorknob: The challenge type is not about language specific questions, only this example is! And if both receive the same rating, it's just a tie? I don't know, what do you do when that happens in code-golf? Personally I don't want it to be a competition, but a challenge. Every working, conforming answer wins! If you feel it's necessary to add a ranking on top, feel free to suggest additional criteria. – jdm Mar 22 '14 at 14:24
• Okay, but how would you create a challenge like this that is not language-specific? The idea of having a tiebreaker of sorts sounds good. – Doorknob Mar 22 '14 at 14:26
• @Doorknob I think language-independent questions of this type would tend to be algorithmic. Two mediocre examples from my SO questions would be Layout dependencies in tree graph (similar to gitk) and Difference (XOR) between two rectangles, as rectangles? . The more interesting examples tend to be platform or language specific, however: "Do this crazy thing with C++ templates/the C preprocessor", "Make a Hybrid Windows 8 Desktop/Metro app" (actually possible, but evil), and so on. – jdm Mar 22 '14 at 14:40 | 2019-10-20 11:13:05 | {"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": 0.4009856879711151, "perplexity": 1544.9802961200496}, "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/1570986707990.49/warc/CC-MAIN-20191020105426-20191020132926-00469.warc.gz"} |
https://tex.stackexchange.com/questions/247104/hyperref-doesnt-link-cite-command/396390 | # Hyperref Doesn't link \cite Command
I am using latex to hyperlink my citations in an article. The problem is that it links figures, equations and sections numbers but not citation links. Here's a MWE,
\documentclass[letterpaper, 10 pt, conference]{ieeeconf}
\IEEEoverridecommandlockouts
\overrideIEEEmargins % Needed to meet printer requirements.
\usepackage{times} % assumes new font selection scheme installed
\usepackage{amsmath} %assumes amsmath package installed
\usepackage{amssymb} % assumes amsmath
\usepackage{footnote}
\title{\LARGE \bf
Some Title*
}
\author{Author One$^{1}$, Author 2$^{2}$ % <-this % stops a space
\begin{document}
\long\def\/*#1*/{} % Define block comment
\graphicspath{ {Chart and Figures/} }
\maketitle
\thispagestyle{empty}
\pagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
Some Random Abstract.
{\textit{Index Terms} - Life Science and Health Care; Mechatronics; Emerging Topics in Automation }
\end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Results from the clinical studies \cite{cervino, c2, c3} serve as motivation for minimizing patient positioning errors during IGRT in order to minimize positioning-related uncertainties in cancer dose treatments, improve tumor control, and reduce toxicity \cite{c4, c5}.
\begin{thebibliography}
\bibitem{cervino} Cervino, L.I., Pawlicki, T., Lawson, J.D., Jiang, S.B., Frame-less and mask-less cranial stereotactic radiosurgery: a feasibility study. Phys. Med. Biol. 55 (2010) 1863-1873.
\bibitem{c2} Xing, L., Dosimetric effects of patient displacement and collimator and gantry angle misalignment on intensity modulated radiation therapy. Radiother Oncol, 2000. 56(1): p. 97-108.
\bibitem{c3} Manning, M.A., et al., The effect of setup uncertainty on normal tissue sparing with IMRT for head-and-neck cancer. Int J Radiat Oncol Biol Phys, 2001. 51(5): p. 1400-9.
\bibitem{c4} Hong, T.S., et al., The impact of daily setup variations on head-and-neck intensity-modulated radiation therapy. Int J Radiat Oncol Biol Phys, 2005. 61(3): p. 779-88.
\bibitem{c5} Den, R.B., et al., Daily image guidance with cone-beam computed tomography for head-and-neck cancer intensity-modulated radiotherapy: a prospective study. Int J Radiat Oncol Biol Phys, 2010. 76(5): p. 1353-9.
\end{thebibliography}
\end{document}
But I find that my \cite citations are not hyper-ref'd. Is there a workaround this? Would appreciate your help. Thank you!
• Usually hyperref should be imported last, you could try that. – Matt May 27 '15 at 22:43
• @Matt Tried that. No luck! – Calorified May 28 '15 at 6:10
It has been 2 years since this question was asked, but I will write my solution because I had the same problem.
As suggested by many, hyperref should be imported last. I also tried that but failed to hyperlink my citations in the document. So, the problem is with the ieeeconf.cls. If you open it with the latex editor you are using you'll see the following:
% 2) Provide a fake nabib command \NAT@parse so that hyperref will not
% interfere with the operation of cite.sty. However, as a result citation
% numbers will not be hyperlinked. Also, natbib will not be able to work
% with IEEEtran. However, this is perhaps the best solution until cite.sty
% and hyperref.sty are able to co-exist with each other.
% It easy enough to override the fake command via:
% \makeatletter
% \let\NAT@parse\undefined
% \makeatother
Here is my example:
\documentclass[letterpaper, 10 pt, conference]{ieeeconf}
\IEEEoverridecommandlockouts
\overrideIEEEmargins % Needed to meet printer requirements.
\usepackage{times} % assumes new font selection scheme installed
\usepackage{amsmath} %assumes amsmath package installed
\usepackage{amssymb} % assumes amsmath
\usepackage{footnote}
\makeatletter
\let\NAT@parse\undefined
\makeatother
\usepackage{hyperref} %hyperref still needs to be put at the end!
\title{\LARGE \bf
Some Title*
}
\author{Author One$^{1}$, Author Two$^{2}$ % <-this % stops a space
\begin{document}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}
\begin{abstract}
Some Random Abstract.
\end{abstract}
\section{Introduction}
Results from the \cite{a}, \cite{b}, \cite{c} serve as motivation...
\begin{thebibliography}
\bibitem{a} First reference
\bibitem{b} Second reference
\bibitem{c} Third reference
\end{thebibliography}
\end{document}
I just included the three lines suggested in the cls file. As Schweinebacke remarked, this is for the old version of ieeeconf.cls.
• This is about the old ieeeconf.cls not the current IEEEconf.cls from IEEEconf, isn't it? – Schweinebacke Oct 16 '17 at 7:08
• Would be good to add a minimal working example to show that this works. – user30471 Oct 16 '17 at 7:19
• @Shweinebacke, you are correct. But in my case, I just downloaded the ieeeconf.cls file from the conference website. Apparently, they use the old version. So, probably just updating the files works. – Ilbant Oct 16 '17 at 8:21
• awesome! thanks for figuring this out! This fixes the problem for ieeeconf.cls. I did not realize there is an up-to-date version called IEEEconf.cls. – Calorified Oct 16 '17 at 15:42
Your code contain several errors. In my MWE below I corrected them. The main errors are marked with <============.
Some remarks:
• The hyperref has to be (except in rare cases) the last package in the preamble.
• Your command \begin{thebibliography} is not correct. It has to has a length parameter for the labels like \begin{thebibliography}{99}.
• The \bf is outdated. Use \textbf{...} instead or \bfseries.
• I commented your two IEEE commands, they are unknown in the class I have.
• Your command \author{} was not closed with a }.
• Define block element is not needed for the MWE (BTW: Why do you use it?)
• Preamble of your MWE contain several packages (amsmath, footnote, ...) not used in your code. Since they are not related to the your problem, you can omit them!
MWE:
\documentclass[letterpaper, 10 pt, conference]{ieeeconf} % IEEEconf
%\IEEEoverridecommandlockouts
%\overrideIEEEmargins % Needed to meet printer requirements.
\usepackage{times} % assumes new font selection scheme installed
\usepackage{amsmath} % assumes amsmath package installed
\usepackage{amssymb} % assumes amsmath
\usepackage{xcolor} % <======================================= for green
\usepackage{footnote}
\usepackage{hyperref} % <====================== last packge to be called
\title{\LARGE
\textbf{Some Title}*
}
\author{Author One$^{1}$, Author 2$^{2}$} % <================= missing }
\begin{document}
\long\def\/*#1*/{} % Define block comment
\graphicspath{ {Chart and Figures/} }
\maketitle
\thispagestyle{empty}
\pagestyle{empty}
\begin{abstract}
Some Random Abstract.
{\textit{Index Terms}~-- Life Science and Health Care; Mechatronics;
Emerging Topics in Automation}
\end{abstract}
\section{Introduction}
Results from the clinical studies \cite{cervino, c2, c3} serve as
motivation for minimizing patient positioning errors during IGRT in
order to minimize positioning-related uncertainties in cancer dose
treatments, improve tumor control, and reduce toxicity \cite{c4, c5}.
\begin{thebibliography}{99} % <==================== length missing! {99}
\bibitem{cervino} Cervino, L.I., Pawlicki, T., Lawson, J.D., Jiang, S.B., Frame-less and mask-less cranial stereotactic radiosurgery: a feasibility study. Phys. Med. Biol. 55 (2010) 1863-1873.
\bibitem{c2} Xing, L., Dosimetric effects of patient displacement and collimator and gantry angle misalignment on intensity modulated radiation therapy. Radiother Oncol, 2000. 56(1): p. 97-108.
\bibitem{c3} Manning, M.A., et al., The effect of setup uncertainty on normal tissue sparing with IMRT for head-and-neck cancer. Int J Radiat Oncol Biol Phys, 2001. 51(5): p. 1400-9.
\bibitem{c4} Hong, T.S., et al., The impact of daily setup variations on head-and-neck intensity-modulated radiation therapy. Int J Radiat Oncol Biol Phys, 2005. 61(3): p. 779-88.
\bibitem{c5} Den, R.B., et al., Daily image guidance with cone-beam computed tomography for head-and-neck cancer intensity-modulated radiotherapy: a prospective study. Int J Radiat Oncol Biol Phys, 2010. 76(5): p. 1353-9.
\end{thebibliography}
\end{document}
and the result:
I would suggest to use biblatex with a bib file.
• Interesting. I tried your suggestions doing everything you have commanded :). I still got the pdf without link borders. Maybe it has something to do with my Latex setup. I am using TeXstudio, if it helps. – Calorified Nov 26 '15 at 4:30
• Okay. I will check later this evening. Got to finish up a project right now. – Calorified Dec 13 '15 at 20:30
• I have no idea why it does not come out after trying all your suggestions. I bow out --ungracefully! – Calorified Apr 24 '16 at 1:40
I know this is not the answer to the original question, but it might still save someone a couple of minutes:
• Another reason for links not appearing can be that the document is in draft mode. Double check that this is not the case \documentclass[draft,...]{...}
Use the following code. You can change the color values to whatever color you like.
\usepackage[colorlinks,citecolor=red,urlcolor=blue,bookmarks=false,hypertexnames=true]{hyperref}
You can use natbib package with hyperref. With the natbib package, you will have also \citep and \citet commands available. | 2021-06-19 13:24:37 | {"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": 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": 0.9064153432846069, "perplexity": 12888.929119560113}, "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/1623487648194.49/warc/CC-MAIN-20210619111846-20210619141846-00174.warc.gz"} |
https://socratic.org/questions/how-do-you-represent-triple-bonds-in-bond-line-notation | # How do you represent triple bonds in bond line notation?
Feb 6, 2017
#### Answer:
Simply by triple bonds...........
#### Explanation:
Acetylene is the simplest molecule that possesses a triple bond, and is represented as $H - C \equiv C - H$. Of course the $\angle H - C - C$ $=$ ${180}^{\circ}$. | 2019-10-15 02:27:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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": 0.37953975796699524, "perplexity": 2329.1019761521247}, "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-2019-43/segments/1570986655735.13/warc/CC-MAIN-20191015005905-20191015033405-00226.warc.gz"} |
https://phys.libretexts.org/Core/Quantum_Mechanics/Fitzpatrick's_Quantum_Mechanics/08%3A_Time-Dependent_Perturbation_Theory/8.08%3A_Harmonic_Perturbations | $$\require{cancel}$$
8.8: Harmonic Perturbations
Consider a perturbation that oscillates sinusoidally in time. This is usually called a harmonic perturbation. Thus,
(850)
where is, in general, a function of position, momentum, and spin operators.
Let us initiate the system in the eigenstate of the unperturbed Hamiltonian, , and switch on the harmonic perturbation at . It follows from Equation (796) that
(851)
where
(852) (853)
This formula is analogous to Equation (803), provided that
(854)
Thus, it follows from the analysis of Section 8.6 that the transition probability is only appreciable in the limit if
(855) (856)
Clearly, (855) corresponds to the first term on the right-hand side of Equation (851), and (856) corresponds to the second term. The former term describes a process by which the system gives up energy to the perturbing field, while making a transition to a final state whose energy level is less than that of the initial state by . This process is known as stimulated emission. The latter term describes a process by which the system gains energy from the perturbing field, while making a transition to a final state whose energy level exceeds that of the initial state by . This process is known as absorption. In both cases, the total energy (i.e., that of the system plus the perturbing field) is conserved.
By analogy with Equation (816),
(857) (858)
Equation (857) specifies the transition rate for stimulated emission, whereas Equation (858) gives the transition rate for absorption. These equations are more usually written
(859) (860)
It is clear from Equations (852)-(853) that . It follows from Equations (857)-(858) that
(861)
In other words, the rate of stimulated emission, divided by the density of final states for stimulated emission, equals the rate of absorption, divided by the density of final states for absorption. This result, which expresses a fundamental symmetry between absorption and stimulated emission, is known as detailed balancing, and is very important in statistical mechanics. | 2018-01-17 08:53:33 | {"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": 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": 0.9008340835571289, "perplexity": 403.73183062279884}, "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-2018-05/segments/1516084886860.29/warc/CC-MAIN-20180117082758-20180117102758-00559.warc.gz"} |
http://www.acmerblog.com/hdu-2304-electrical-outlets-3533.html | 2014
01-05
# Electrical Outlets
Roy has just moved into a new apartment. Well, actually the apartment itself is not very new, even dating back to the days before people had electricity in their houses. Because of this, Roy’s apartment has only one single wall outlet, so Roy can only power one of his electrical appliances at a time.
Roy likes to watch TV as he works on his computer, and to listen to his HiFi system (on high volume) while he vacuums, so using just the single outlet is not an option. Actually, he wants to have all his appliances connected to a powered outlet, all the time. The answer, of course, is power strips, and Roy has some old ones that he used in his old apartment. However, that apartment had many more wall outlets, so he is not sure whether his power strips will provide him with enough outlets now.
Your task is to help Roy compute how many appliances he can provide with electricity, given a set of power strips. Note that without any power strips, Roy can power one single appliance through the wall outlet. Also, remember that a power strip has to be powered itself to be of any use.
Input will start with a single integer 1 <= N <= 20, indicating the number of test cases to follow. Then follow N lines, each describing a test case. Each test case starts with an integer 1 <= K <= 10, indicating the number of power strips in the test case. Then follow, on the same line, K integers separated by single spaces, O1 O2 . . . OK, where 2 <= Oi <= 10, indicating the number of outlets in each power strip.
Input will start with a single integer 1 <= N <= 20, indicating the number of test cases to follow. Then follow N lines, each describing a test case. Each test case starts with an integer 1 <= K <= 10, indicating the number of power strips in the test case. Then follow, on the same line, K integers separated by single spaces, O1 O2 . . . OK, where 2 <= Oi <= 10, indicating the number of outlets in each power strip.
3
3 2 3 4
10 4 4 4 4 4 4 4 4 4 4
4 10 10 10 10
7
31
37
#include<stdio.h>
int main()
{
int t, a, sum, n, i;
scanf("%d",&t);
while(t--)
{
scanf("%d",&n);
sum = 1 - n;//要减去n-1个被占用的插孔
for(i=0; i<n; i++)
{
scanf("%d",&a);
sum += a;
}
printf("%d\n",sum);
}
} | 2017-04-25 20:46:27 | {"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": 0.3246225118637085, "perplexity": 1021.4445017611382}, "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-2017-17/segments/1492917120878.96/warc/CC-MAIN-20170423031200-00060-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://lists.gnu.org/archive/html/axiom-developer/2004-12/msg00398.html | axiom-developer
[Top][All Lists]
RE: [Axiom-developer] [TeXmacs]
From: michel . lavaud Subject: RE: [Axiom-developer] [TeXmacs] Date: Mon, 20 Dec 2004 12:23:23 +0100
```On 18 Dec 2004 at 19:59, Bill Page wrote:
> > What differences did you notice?
> I didn't find any ? > (probably because it is too late here:-)) >
>
> Or is it too early?? (I guess we are all the same ... :)
>
> When I display these formula on my system - both native
> Windows and cygwin versions display the Axiom
>
> Type: Expression Integer
>
> output immediately after the equation number, on the same
> line, often wrapping to the start of the next line. Something
> like this:
>
> x+y+z (2)Type: Expression
> Integer
>
> Normally it is formatted on the next full line below with
> right hand justification. The change to the stylesheet
> is to add an explicit new-line formatting command for
> \axiomtype{...} to force it to a new line. In previous
> versions of TeXmacs, this was not necessary since it was
> automatically treated like a new paragraph.
Ah OK, I see. I had noticed it but it was concentrated only on the correctness
of
mathematics ; and I thought it was a feture, not a bug. Actually I agree, it
would look
better (although it is not very disturbing, as it is displayed in a different
color)
> Have a good night.
I had, thanks a lot :-)
Best wishes,
http://www.univ-orleans.fr/EXT/ASTEX
ftp://ftp.univ-orleans.fr/pub/tex/PC/AsTeX | 2019-05-21 17:51:00 | {"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": 0.813736617565155, "perplexity": 5800.89852787993}, "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-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00289.warc.gz"} |
https://math.stackexchange.com/questions/2481768/weighted-least-squares-for-parabola-coefficients-estimation | Weighted Least Squares for Parabola Coefficients Estimation
I am in trouble to find where I am making a mistake...
I have to estimate the parameters a and b of the curve modeled by:
$y = a x^2 + bx$
I have to do that from K measures of the curve, each measure is modeled by:
$y_i = a x^2 + bx + \epsilon_i$
Where $\epsilon_i$ is a Gaussian random variable that follows $N(0,\sigma_i^2)$
After collecting a group of K measures, I start the estimation. In order to do that, I am using weighted least squares for estimating my parameters:
$\Theta = \left[\begin{array}{c} a\\ b \end{array}\right]$
So, the closed solution formula tells me that:
$\hat\Theta = (\Phi^TR^{-1}\Phi)^{-1}\Phi^TR^{-1}y$
Where:
• y is a $[k \times 1]$ vector with k measures of the curve.
• R is a $[k \times k]$ matrix constructed from $R = E[\epsilon \epsilon^T] = diag(\sigma_{\epsilon 1}^{2} ... \sigma_{\epsilon k}^{2})$
• $\Phi$ is a $[k \times 2]$ matrix that is equals to $\left[\begin{array}{cc} x^{2} & x\\ \vdots & \vdots\\ x^{2} & x \end{array}\right]$
The problem is that I always find
$(\Phi^TR^{-1}\Phi)$
as a singular matrix, therefore, I am unable to invert it and get to the final estimatiion.
What am I doing wrong? Have I made a mistake in the construction of the problem?
Thank you!
• Just to be sure, are you using $$\Phi = \left[\begin{array}{cc} x_1^{2} & x_1\\ \vdots & \vdots\\ x_k^{2} & x_k \end{array}\right]?$$ – Math Lover Oct 20 '17 at 17:36
• I don't see any problem with the method in principle. Let me give you a derivation of the formula so its assumptions are better understood. – Zhuoran He Oct 20 '17 at 17:53
• Thank you for your help! I finally managed to spot the mistake that I was making... – Artur Oct 21 '17 at 11:17
The regression model written in matrix notation is given by
$$y=\Phi\Theta+\epsilon.$$
We want to find the least-squares solution $\Theta=\hat{\Theta}$ that maximizes the likelihood of the multivariate Gaussian distributed errors, i.e.
$$L\propto\exp\left(-\frac{1}{2}\epsilon^TR^{-1}\epsilon\right).$$
If we view $R^{-1}$ as the metric of an inner product (since it's positive-definite), we need to minimize $\Vert\epsilon\Vert_{R^{-1}}^2\equiv\epsilon^TR^{-1}\epsilon$ by choosing our $\Theta=\hat{\Theta}$ such that the vector $y-\Phi\hat{\Theta}$ is orthogonal to $\Phi$ under the same metric $R^{-1}$. Therefore,
$$\Phi^TR^{-1}(y-\Phi\hat{\Theta})=0,$$
which leads to the least-squares formula
$$\hat{\Theta}=(\Phi^TR^{-1}\Phi)^{-1}\Phi^TR^{-1}y.$$
So your formula is correct. This formula can even handle correlated errors if you put off-diagonal entries in $R$. The matrix $R$ should be positive-definite. If you find numerically that $\Phi^TR^{-1}\Phi$ is close to singular, use ridge regression
$$\hat{\Theta}_\lambda=(\Phi^TR^{-1}\Phi+\lambda I)^{-1}\Phi^TR^{-1}y,$$
where $\lambda>0$ is the ridge parameter. | 2019-07-17 06:16:27 | {"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": 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": 0.8822246789932251, "perplexity": 144.578142411626}, "config": {"markdown_headings": false, "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-2019-30/segments/1563195525094.53/warc/CC-MAIN-20190717061451-20190717083451-00426.warc.gz"} |
https://socratic.org/questions/when-do-we-have-to-use-roman-numerals-in-the-name-of-a-compound | # When do we have to use roman numerals in the name of a compound?
Nov 28, 2015
#### Answer:
Roman numerals are used in naming ionic compounds when the metal cation forms more than one ion. The metals that form more than one ion are the transition metals, although not all of them do this.
#### Explanation:
For example, copper can form $\text{Cu"^(+)}$ ions and $\text{Cu"^(2+)}$ ions. If they combine with chlorine, we can have $\text{CuCl}$ and $\text{CuCl"_2}$. The first compound is composed of copper 1+ ions bonded to choride 1- ions. It's called copper(I) chloride. The second compound is composed of copper 2+ ions bonded to chloride 1- ions. It's called copper(II) chloride. | 2019-09-16 00:44:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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": 0.6057889461517334, "perplexity": 2472.986630432019}, "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-2019-39/segments/1568514572439.21/warc/CC-MAIN-20190915235555-20190916021555-00263.warc.gz"} |
https://brilliant.org/problems/the-baffling-bishop/ | # The Baffling Bishop
Logic Level 3
One morning after church, the vicar asked the bishop, "How old are those three worshippers?"
"Well," replied the bishop, "the product of their ages is $2450$ and the sum is twice your age."
"I'm afraid I still don't know," said the vicar.
"Ah, I am older than each of them! You should be able to figure out their ages now," cried the bishop.
Sadly, the vicar was still at a loss. How old is the bishop?
× | 2020-02-25 16:27:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.3986349105834961, "perplexity": 3540.9761054577343}, "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/1581875146123.78/warc/CC-MAIN-20200225141345-20200225171345-00175.warc.gz"} |
https://artint.info/2e/html/ArtInt2e.Ch8.S1.SS2.html | # 8.1.2 Axioms for Probability
The preceding section gave a semantic definition of probability. An axiomatic definition specifies axioms. These are axioms one may want for a calculus of belief, and we show they are satisfied by probability.
Suppose $P$ is a function from propositions into real numbers that satisfies the following three axioms of probability:
Axiom 1
$0\leq P(\alpha)$ for any proposition $\alpha$. That is, the belief in any proposition cannot be negative.
Axiom 2
$P(\tau)=1$ if $\tau$ is a tautology. That is, if $\tau$ is true in all possible worlds, its probability is 1.
Axiom 3
$P(\alpha\vee\beta)=P(\alpha)+P(\beta)$ if $\alpha$ and $\beta$ are contradictory propositions; that is, if $\neg(\alpha\wedge\beta)$ is a tautology. In other words, if two propositions cannot both be true (they are mutually exclusive), the probability of their disjunction is the sum of their probabilities.
These axioms are meant to be intuitive properties that we would like to have of any reasonable measure of belief. If a measure of belief follows these intuitive axioms, it is covered by probability theory. Note that empirical frequencies – propositions about the proportion of examples in a data set – obey these axioms, and so follow the rules of probability, but that does not mean that all probabilities are empirical frequencies (or obtained from them).
These axioms form a sound and complete axiomatization of the meaning of probability. Soundness means that probability, as defined by the possible-worlds semantics, follows these axioms. Completeness means that any system of beliefs that obeys these axioms has a probabilistic semantics.
###### Proposition 8.1.
If there are a finite number of finite discrete random variables, Axioms $1$, $2$, and $3$ are sound and complete with respect to the semantics.
It is easy to check that these axioms are true of the semantics. Conversely, the axioms can be used to compute any probability from the probability of worlds, because the descriptions of two worlds are mutually exclusive. The full proof is left as an exercise. (See Exercise 2.)
###### Proposition 8.2.
The following hold for all propositions $\alpha$ and $\beta$
1. 1.
Negation of a proposition:
$P(\neg\alpha)=1-P(\alpha).$
2. 2.
If $\alpha\leftrightarrow\beta$, then $P(\alpha)=P(\beta)$. That is, logically equivalent propositions have the same probability.
3. 3.
Reasoning by cases:
$P(\alpha)=P(\alpha\wedge\beta)+P(\alpha\wedge\neg\beta).$
4. 4.
If $V$ is a random variable with domain $D$, then, for all propositions $\alpha$,
$P(\alpha)=\sum_{d\in D}P(\alpha\wedge\mbox{}V=d).$
5. 5.
Disjunction for non-exclusive propositions:
$P(\alpha\vee\beta)=P(\alpha)+P(\beta)-P(\alpha\wedge\beta).$
###### Proof.
1. 1.
The propositions $\alpha\vee\neg\alpha$ and $\neg(\alpha\wedge\neg\alpha)$ are tautologies. Therefore,
$1=P(\alpha\vee\neg\alpha)=P(\alpha)+P(\neg\alpha)$. Rearranging gives the desired result.
2. 2.
If $\alpha\leftrightarrow\beta$, then $\alpha\vee\neg\beta$ is a tautology, so $P(\alpha\vee\neg\beta)=1$. $\alpha$ and $\neg\beta$ are contradictory statements, so Axiom 3 gives $P(\alpha\vee\neg\beta)=P(\alpha)+P(\neg\beta)$. Using part (a), $P(\neg\beta)=1-P(\beta)$. Thus, $P(\alpha)+1-P(\beta)=1$, and so $P(\alpha)=P(\beta)$.
3. 3.
The proposition $\alpha\iff((\alpha\wedge\beta)\vee(\alpha\wedge\neg\beta))$ and $\neg((\alpha\wedge\beta)\wedge(\alpha\wedge\neg\beta))$ are tautologies. Thus, $P(\alpha)=P((\alpha\wedge\beta)\vee(\alpha\wedge\neg\beta))=P(\alpha\wedge% \beta)+P(\alpha\wedge\neg\beta).$
4. 4.
The proof is analogous to the proof of part (c).
5. 5.
$(\alpha\vee\beta)\iff((\alpha\wedge\neg\beta)\vee\beta)$ is a tautology. Thus,
$\displaystyle P(\alpha\vee\beta)$ $\displaystyle=P((\alpha\wedge\neg\beta)\vee\beta)$ $\displaystyle=P(\alpha\wedge\neg\beta)+P(\beta).$
Part (c) shows $P(\alpha\wedge\neg\beta)=P(\alpha)-P(\alpha\wedge\beta)$. Thus,
$P(\alpha\vee\beta)=P(\alpha)+P(\beta)-P(\alpha\wedge\beta).$ | 2019-01-24 13:19:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 45, "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": 0.9614292979240417, "perplexity": 398.65449002408013}, "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-2019-04/segments/1547584547882.77/warc/CC-MAIN-20190124121622-20190124143622-00548.warc.gz"} |
https://math.stackexchange.com/questions/1458095/waiting-times-probability | # Waiting times probability
Each entering customer must be served first by server $1$, then by server $2$, and finally by server $3$. The amount of time it takes to be served by server i is an exponential random variable with rate $\mu_i$, $i = 1, 2, 3$. Suppose you enter the system when it contains a single customer who is being served by server $3$.
Find the probability that server 3 will still be busy when you move over to server 2.
If we let $T_i\sim exp(\mu_i)$ the service time of server $i$, then we just want to find $$P(T_3>T_1)=P(T_1=min(T_1,T_3))=\frac{\mu_1}{\mu_1+\mu_3}$$
but the answer says $\frac{\mu_3}{\mu_1+\mu_3}$.
Where did that come from?
• Unfortunately there are two different conventions for exponential variables: one has the pdf $\lambda e^{-\lambda x}$ with mean $1/\lambda$ and the other has the pdf $\frac{1}{\lambda} e^{-x/\lambda}$ with mean $\lambda$. The former can be said to have "rate $\lambda$" when the exponential variable itself is viewed as a time. Which convention are you using? – Ian Sep 30 '15 at 14:33
• @Ian I'm using $\lambda e^{-\lambda x}$, but the exercise don't say nothing about it. – Roland Sep 30 '15 at 14:40
• See, interchanging the two interchanges your answers, because $\frac{\frac{1}{\mu_1}}{\frac{1}{\mu_1}+\frac{1}{\mu_3}}=\frac{1}{1+\frac{\mu_1}{\mu_3}}=\frac{\mu_3}{\mu_3+\mu_1}$. – Ian Sep 30 '15 at 14:41
• That's right (except for the obvious typo). The point is that when you take the minimum, you're waiting for the first event, and so the rates of (independent) exponentials will add together. You can make an electrical analogy to parallel resistors (resistance is to mean as conductance is to reciprocal of mean). – Ian Sep 30 '15 at 14:58
• Let us assume that the given answer is the one they intended to give (not a good assumption!). Then the answer makes sense if the $\mu_i$ are the means. That interpretation is reinforced by the choice of the letter $\mu$. I would not call that the rate, it is certainly not the rate of the associated Poisson. Sadly, usage is not uniform. – André Nicolas Sep 30 '15 at 15:01 | 2019-06-26 12:18:29 | {"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": 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": 0.6620239019393921, "perplexity": 292.47492645011124}, "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/1560628000306.84/warc/CC-MAIN-20190626114215-20190626140215-00204.warc.gz"} |
http://mathhelpforum.com/trigonometry/95608-solving-trig-equation.html | # Math Help - Solving trig equation
1. ## Solving trig equation
1]Show that 2[sinxcosx - 1/2 ] = sin2x -1
2] Hence, determine the general solution for:
2[sinxcosx - 1/2 ] =0
2. $2(sinxcosx-\frac{1}{2}) = 2sinxcosx-2*\frac{1}{2} = 2sinxcosx-1$ We know that $sin2x=2sinxcosx$ so we have
$2sinxcosx-1 = sin2x-1$
3. thanks alot! | 2015-11-24 23:35:23 | {"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": 0.963352620601654, "perplexity": 9260.568229178698}, "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-2015-48/segments/1448398444047.40/warc/CC-MAIN-20151124205404-00024-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://discourse.pymc.io/t/partial-pooling-for-election-polls/6461 | # Partial Pooling for Election Polls
Hey everyone, new PyMC3 user here. I work mostly with machine learning and I’ve been trying to learn as much as I can about probabilistic programming in my free time. Anyway, I put together a model for the Georgia run off election in the United States based on the partial pooling baseball example on the website, and I’m curious to know if it seems reasonable to the pros on here.
My idea is that the true vote share for the state is unobservable, but each poll can give us a glimpse. However each poll should have its own distribution for house effects or sample bias or whatever. Similar to the Efron and Morris baseball example where each player has their own distribution that informs the distribution of the population of professional baseball players.
The data
samplesize = [605, 1250, 500, 800, 1377, 1450, 583, 1064, 300, 1500]
num_votes = [296, 631, 247, 404, 703, 717, 312, 499, 143, 734]
And the model
with pm.Model() as warnock_model:
phi = pm.Beta('phi', alpha=alpha, beta=beta)
kappa_log = pm.HalfNormal('kappa_log', sigma=1)
kappa = pm.Deterministic('kappa', tt.exp(kappa_log))
thetas = pm.Beta(
'thetas',
alpha=phi*kappa,
beta=(1.0-phi)*kappa,
)
y = pm.Binomial(
'y',
n=samplesize,
p=thetas,
)
This model seems reasonable to me. Plotting the forest plot for each pollster with the actual poll result in orange looks good.
And the posterior taking phi to be estimated state vote share looks realistic to me as well.
So my questions now are what types of posterior predictive checks should I do?
Then if I wanted to weight the polls by time what is the best way to do that? My thinking would be artificially decrease the sample size by some function, that should increase the uncertainty for those polls that are far away from the election date? However I’m not sure if that is a commonly accepted practice.
Hi Florian,
And welcome
I frequently work on such models, so I can give you some pointers for resources:
• I put all my models here. Note that I work mainly on French elections, which have many parties, with a changing identity, so that adds dimensions and complications compated to the US, but the ideas stay the same. I gave a talk at PyMCon this year, walking through this exact model.
• I find the most satisfying conceptual approach is Linzer’s dynamic Bayesian model, extended this year by Gelman, Heidemanns and Morris for The Economist’s model. They use an MvGRW to handle the temporal correlations between polls, and the hierarchical structure is there to partially pool between states (but you probably don’t care about that last part in your case).
• As a matter of fact, I interviewed Andrew and Merlin on my podcast just before the US election this year
That’s your opportunity to be creative! You can use ArviZ’s functions for generic posterior plots, but, ultimately, each model needs its own posterior visualizations, especially big, complex, high-dimensional models.
You’ll see in my repo and my talk that I used custom plots to better understand the model. Feel free to reuse this if useful. Again though, each model requires at least partly custom viz, as it’s usually tailored to your use-case and domain knowledge.
That’s indeed a good rule of thumb I think! I actually used it myself in the model in my repo, and that does work. If it’s your first model, that’s a good idea to start simple, with good-enough heuristics and then you’ll build on it iteratively – in my case for instance, I plan to add the clean time-dependency part in the next iteration of my model. Start small and grow with it
Hope this helps
2 Likes
Hi Alex
Thank you for taking the time on the reply! I actually watched part of your talk and I have it bookmarked for later. I also looked at the dynamic Bayesian model from Linzer and I started to work on that but I decided it was probably better to get a simple model done first. The only part of that model that seems mysterious to me is the reverse GRW. I like the idea of using a GRW to act almost like a Khalman filter, like Jackman 2005, but I’m not sure I understand why Linzers would go in reverse. I guess to ensure that the national component decays to 0?
Also, and maybe this is a misunderstanding on my part, but figure 3 in the Linzer paper is concerning to me. I understand that the solid horizontal line is the actual outcome and the model is forecasting something close to that. But to me nothing in the data up to 2 months before election day seems to suggest anything that would trend that way. If I were developing this model and I saw that forecast with 2 months to go I would think that something is wrong.
My only guess as to whats happening here given my (poor) understanding of the model is that its regression towards the national polling average (which tends to be closer to 50%). However this would be a bad prediction for all the “safe” states in the US elections. It looks good here for the two swing states obviously.
Anyway, thanks again for your response. It seems like I’m on the right track with a baby model on a baby election. I’ll watch your talk and see if I can pick up any tricks!
Ha ha yeah, sorry, the talk is a bit long – I talk too much
It’s to make sure the GRW “walks” towards its priors instead of away from it when one doesn’t have much data. And the nice thing is that the prior is actually the result of another model, called “the fundamentals” model.
Linzer uses the “Time for change” model, but basically it’s a regression of past election results on a domain-expert-informed selection of socio-economic variables – so, yeah, there are actually two models
Indeed, nothing in the data trends that way, but it’s because we don’t have any data (i.e polls) from 2 months in the future. When that happens, the beauty of this model is that it reverts to the fundamentals forecast (which doesn’t contain polls, only socio-econ variables available well in advance of the election) to still be able to make a forecast. Otherwise, you would just get a huge, uninformative uncertainty.
If you’ve already used Gaussian Processes, you can draw a parallel here, as GPs revert back to their mean when data become sparse.
Restating what I wrote above, in case it wasn’t clear: the model is reverting to the state-level fundamentals forecast when data become sparse.
Definitely a good strategy
Good luck and happy holidays
Thanks again for taking the time to produce a long thoughtful response! I’m enjoying this conversation, and happy holidays to you too!
It’s to make sure the GRW “walks” towards its priors instead of away from it when one doesn’t have much data. And the nice thing is that the prior is actually the result of another model, called “the fundamentals” model.
Ok this makes a lot of sense. Seems so simple now that you explain it haha. Thanks!
When that happens, the beauty of this model is that it reverts to the fundamentals forecast (which doesn’t contain polls, only socio-econ variables available well in advance of the election) to still be able to make a forecast.
In this case though shouldn’t each state then walk toward the dashed line since thats the Time-for-Change model? Florida does but Indiana drifts away from it. Indiana also far less polls than Florida so I would expect the model to be more heavily leaning towards the Time-for-Change outputs. Furthermore, with 2 months before the election Indiana starts at the Time-for-Change result and forecasts a drift away from it.
Maybe I’m too focused on that single figure but I just can’t square that result with my understanding of the model.
Not necessarily, because the trend in each state is not only influenced by the fundamentals – it’s also influenced by the opinion trends in the 49 other states (through the hierarchical structure and the \delta_j coefficients).
You don’t see that for Florida because the national trend and the fundamentals agree, but Indiana is indeed a good illustration of that fact, as noted in the paper:
In Indiana, Obama was slightly ahead of the historical forecast, but this was atypical: in most states, as in Florida, fewer voters than expected were reporting a preference for Obama. As a result, estimates of \delta_j < 0; so after the final day of polling, \pi_{ij} trended upward to Election Day. […] In Indiana, \pi_{ij} moved away from the structural forecast, but again toward the actual outcome—thus correcting the substantial 5.5% error in the original Time-for-Change forecast.
Hope this helps, but again, don’t get hung up on that, you’ll get there eventually – lots of moving parts in this model, so it’s completely normal to get confused
Ok cool, super helpful again. I’ll have to go through the paper again more carefully and maybe implement it myself or trace it out somehow to fully put all the pieces together. | 2022-07-01 10:56:41 | {"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": 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": 0.5604469776153564, "perplexity": 1055.0624742592931}, "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/1656103940327.51/warc/CC-MAIN-20220701095156-20220701125156-00167.warc.gz"} |
http://mathoverflow.net/questions/54203/question-on-a-relation-between-minors-of-a-particular-kind-of-matrix | # Question on a relation between minors of a particular kind of matrix
Hi!
Perhaps it is an easy question but i don't figure out how to prove it. Let $a_1,...,a_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer part of $a$) consider the matrix $A(i,j)=(a_{i+k+l})_{0\leq k,l\leq j}$. I denote with $D(i,j)=\det(A(i,j))$. I have to prove the following relation:$$D(1,m-1)D(3,m-1)-D(1,m)D(3,m-2)=D(2,m-1)^2$$
I tried by induction but i didn't manage to get the result.
Any help or hint would be appreciated. | 2015-05-29 02:36:38 | {"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": 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": 0.9487849473953247, "perplexity": 42.387584426735316}, "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-2015-22/segments/1432207929832.32/warc/CC-MAIN-20150521113209-00159-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://stacks.math.columbia.edu/tag/0C6H |
Lemma 10.121.9. Let $A \to B$ and $B \to C$ be ring homomorphisms such that $A \to C$ is of finite type. Let $\mathfrak r$ be a prime of $C$ lying over $\mathfrak q \subset B$ and $\mathfrak p \subset A$. If $A \to C$ is quasi-finite at $\mathfrak r$, then $B \to C$ is quasi-finite at $\mathfrak r$.
Proof. Observe that $B \to C$ is of finite type (Lemma 10.6.2) so that the statement makes sense. Let us use characterization (3) of Lemma 10.121.2. If $A \to C$ is quasi-finite at $\mathfrak r$, then there exists some $c \in C$ such that
$\{ \mathfrak r' \subset C \text{ lying over }\mathfrak p\} \cap D(c) = \{ \mathfrak {r}\} .$
Since the primes $\mathfrak r' \subset C$ lying over $\mathfrak q$ form a subset of the primes $\mathfrak r' \subset C$ lying over $\mathfrak p$ we conclude $B \to C$ is quasi-finite at $\mathfrak r$. $\square$
Comment #3952 by Manuel Hoff on
I think, that the Lemma only needs the assumption that $A \rightarrow C$ is of finite type (and this form of the Lemma is needed in 00Q9).
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# Hippocampal hub neurons maintain distinct connectivity throughout their lifetime
## Abstract
The temporal embryonic origins of cortical GABA neurons are critical for their specialization. In the neonatal hippocampus, GABA cells born the earliest (ebGABAs) operate as ‘hubs’ by orchestrating population synchrony. However, their adult fate remains largely unknown. To fill this gap, we have examined CA1 ebGABAs using a combination of electrophysiology, neurochemical analysis, optogenetic connectivity mapping as well as ex vivo and in vivo calcium imaging. We show that CA1 ebGABAs not only operate as hubs during development, but also maintain distinct morpho-physiological and connectivity profiles, including a bias for long-range targets and local excitatory inputs. In vivo, ebGABAs are activated during locomotion, correlate with CA1 cell assemblies and display high functional connectivity. Hence, ebGABAs are specified from birth to ensure unique functions throughout their lifetime. In the adult brain, this may take the form of a long-range hub role through the coordination of cell assemblies across distant regions.
## Introduction
GABAergic neurons are a critical component of cortical circuit development and function. This sparse and heterogeneous population of cells has been classified according to several parameters, including genetic and molecular markers, connectivity schemes as well as morphological and electrophysiological properties1. According to leading theories, this heterogeneity greatly enhances the computational power of cortical networks2.
The diversity among GABAergic cells is specified as early as progenitor stages in the ganglionic eminences, the embryonic regions that give rise to cortical inhibitory neurons3,4. Genetic restriction of neuronal potential from spatially distributed progenitors is a major determinant of GABA neuron diversity5,6,7. In addition, a temporal clock also shapes GABA neuron fate: discrete temporal windows within the same ganglionic eminence specify different cell types8,9. Hence, place and time of origin from discrete progenitor pools in the ganglionic eminences determine, at least in part, the ultimate features that the cell will display in adult cortical circuits.
The temporal control of GABA neurons’ fate is particularly striking when considering early born cells. In the CA3 region of the hippocampus, early born GABAergic neurons (ebGABAs) act as “hubs” during the perinatal period, that is they show exceptional functional and effective connectivity10. Due to this remarkable connectivity scheme, CA3 hub cells anticipate and coordinate single-handedly spontaneous network bursts occurring in the form of giant depolarizing potentials (GDPs) in postnatal slices10,11. Interestingly, these cells are still present in adult animals and a fraction of them display long-range projections to the medial septum12.
It is not known whether ebGABAs are able to orchestrate network synchrony only in CA3, a region that is characterized by notable recurrent connectivity and that is the preferred site of GDP initiation13. In addition, despite their major developmental role, the integration and function of ebGABAs into adult circuits remains unknown. More generally, it is not known whether an early neuronal birth date leads to distinct functional properties in adult circuits. To fill this gap, we have examined ebGABAs in the CA1 region of the hippocampus, because this area is well understood in terms of cell types and connectivity14 and is more accessible for in vivo calcium imaging. We investigated ebGABAs’ morphological and electrophysiological properties, their local and long-range connectivity as well as their involvement in network dynamics. We show that CA1 ebGABAs operate as hub cells during the early postnatal period and maintain distinct properties in adulthood, encompassing neurochemical content, intrinsic electrophysiological properties, input connectivity and in vivo activity.
## Results
### CA1 ebGABAs are sparse and mostly located in deep layers
To study CA1 ebGABAs, we used the inducible transgenic driver line Dlx1/2-CreER, expressing CreER under the control of the Dlx1/2 intergenic enhancer15. We crossed Dlx1/2-CreER mice with a floxed GFP reporter line (see Methods). Like in our previous studies10,12, ebGABAs were labeled with GFP by inducing Cre recombinase at embryonic day 7.5 or 8.5 (E7.5 or E8.5) via tamoxifen administration. For simplicity, we refer to this tamoxifen-treated transgenic line as Dlx1/2(E7.5)-GFP. Since Dlx1 and/or Dlx2 genes are required for the proper development of all GABAergic cells16, this approach labels GABAergic neurons from all ganglionic eminences. However, it is likely to label more medial ganglionic eminence (MGE)-derived neurons, which on average are born earlier than caudal ganglionic eminence (CGE)-derived cells6.
In line with our previous reports10,12, Dlx1/2(E7.5)-GFP ebGABAs of the hippocampus were very sparse (3 ± 1 cells per 70 μm-thick PFA-fixed coronal section at P7, mean ± SD, 58 sections from four mice, quantified bilaterally, Fig. 1a–c). We estimated that the amount of ebGABAs labeled with our approach is ~1% of GABA-positive cells and is ~20 times lower than the amount of somatostatin-positive (SOM+) cells (Fig. 1b). In CA1, ebGABAs were similarly sparse at neonatal and adult stages: 0.8 ± 0.5 ebGABAs per 80 μm-thick horizontal section at P7 (171 sections from 7 mice), 1.5 ± 0.6 ebGABAs per section at P45 (77 sections from 3 mice, quantified bilaterally, Fig. 1c). Next, we examined the distribution of ebGABAs’ somata in the rostrocaudal and dorsoventral axes (n = 2 P45 mice, Fig. 1d), observing that these cells were sparse and scattered in a relatively even fashion across the entire CA1 region.
Next, we studied their distribution across CA1 layers (six P45 mice). The location of ebGABAs’ somata appeared to be significantly dependent on layering (P < 0.0001, Friedman test, Fig. 1e). The somata of most ebGABAs were located in the stratum oriens and inside or around the stratum pyramidale (29 ± 5% and 43 ± 4%, respectively, means ± SDs). Fewer cells (21 ± 4%) were located in the stratum radiatum and only a small minority (7 ± 4%) populated the stratum lacunosum-moleculare.
### EbGABAs are operational hub cells in the developing CA1
We sought to test whether ebGABAs played a hub function in the developing CA1 circuit. To this end, horizontal slices (380 µm-thick) from Dlx1/2(E7.5)-GFP or littermate GFP-negative controls containing the intermediate/ventral CA1 were loaded with the calcium indicator Fura2-AM. Slices were subsequently imaged using two-photon microscopy to record spontaneous neuronal activity (Supplementary Movie 1). EbGABAs were generally very sparse (1.5 ± 1.1 cells found per hippocampus, quantified unilaterally from four mice, mean ± standard deviation, SD). Three reasons are likely to contribute to the fact that the number of ebGABAs was lower in acute experiments than in histological analyses from fixed tissue. First, the signal from fixed slices was amplified with immunofluorescence; second, some ebGABAs may not survive after the slicing procedure; third, the GFP signal could not be detected below ~100 µm in depth in acute slices. As we previously reported for CA3 ebGABAs10, GFP+ cells could not be labeled with Fura2-AM. This prevented us from calculating their functional connectivity index based on the analysis of spontaneous calcium events. Thus, we tested the role of ebGABAs in orchestrating spontaneous network dynamics by measuring the effect of ebGABA stimulation on GDPs.
We performed whole cell patch clamp recordings from ebGABA (n = 65) and ctrlGABA cells (random putative GABAergic cells in stratum oriens and stratum radiatum, n = 17, Fig. 2a, b). A phasic stimulation protocol was applied, i.e., short supra-threshold current pulses repeated at 0.1, 0.2, and 0.4 Hz (within the frequency range of spontaneous GDP occurrence). GDPs could be detected only in 14/82 slices (including 8 ctrlGABAs and 6 ebGABAs). Thus, subsequent analyses were restricted to these cases. Stimulation of 3/6 ebGABAs significantly affected GDP frequency (among these, 2 decreased GDP frequency whereas one increased it, Fig. 2d, g). In contrast, no stimulated ctrlGABA (0/8) significantly affected GDP frequency (Fig. 2c, e and Supplementary Table 1). This difference was unlikely to arise from spontaneous differences in GDP rate between CreER-positive and CreER-negative mice because the median GDP intervals of ctrlGABA and ebGABA experiments were comparable (ctrlGABA: median 0.02 Hz, interquartile ranges (IQR): 0.014–0.022 Hz; ebGABA: median = 0.017, IQR: 0.015–0.023, P > 0.99, Mann–Whitney U test).
Next, we asked whether ebGABAs could also synchronize the network in the shorter timescale. We examined the occurrence of calcium events in all the trials that followed the depolarizing current steps used as stimulation. We constructed a calcium event probability histogram including the activity of all the cells in the field of view and we normalized by the number of GDPs during the stimulation protocol. Within interstimulations trials, ebGABA activation increased synchronous calcium activity (Fig. 2f).
To define cells that significantly locked the onset of GDPs, we examined whether the highest peak of the calcium event probability histogram passed a threshold defined using a surrogate distribution (see “Methods” for details). Using this method, we calculated that 6/6 ebGABAs significantly synchronized the onset of GDPs, with GDP onset occurring 2.2 ± 0.9 s after ebGABA activation (mean ± SD). In contrast, only one ctrlGABA was found to significantly lock GDP onset (Fig. 2e–g and Supplementary Table 1).
Overall, 1/6 ebGABA increased the frequency of GDPs and synchronized GDP occurrence, 2/6 ebGABAs decreased the frequency of GDPs and synchronized GDP occurrence, and 3/6 ebGABAs synchronized GDP occurrence but had no effect on GDP frequency. In contrast, only 1/8 ctrlGABA synchronized GDP occurrence. The remaining ctrlGABAs had no effect on either GDP frequency or onset (Supplementary Table 1). On balance, all ebGABAs could affect the coordination of neuronal activity in the developing CA1 network, whereas the proportion of ctrlGABA affecting network dynamics (1/8) was significantly lower (P = 0.0014, Fisher’s exact test). Hence, CA1 ebGABAs can be classified as operational hub neurons because they can synchronize the activity of large population of cells in the network.
### EbGABAs show remarkably long axons in the developing CA1
Previously described hub cells and ebGABAs in the CA3 area could be distinguished by four times longer axonal lengths compared to non-hub cells10,11. We asked whether widespread axonal arborizations could also be a distinctive feature of CA1 ebGABAs. To test this, we reconstructed 38 neurobiotin-filled neurons (ctrlGABAs: n = 20, ebGABAs: n = 18, Fig. 2h and Supplementary Fig. 1).
The axons of most CA1 ebGABAs displayed remarkable lengths, in some cases crossing CA1 boundaries and innervating CA3 and/or the subiculum (Fig. 2h and Supplementary Fig. 1). In four cases, the axons ran in the alveus, suggesting a possible extrahippocampal projection. We performed morphometric analysis on the reconstructed cells. In line with previous results on CA3 ebGABA and hub cells, CA1 ebGABAs showed significantly longer axons (P = 0.0006) that covered a significantly larger surface than ctrlGABAs (P = 0.0003, Mann–Whitney U test, Fig. 2i and Supplementary Fig. 2a). In contrast, dendritic length or surface covered by dendrites did not differ significantly between the two groups (P = 0.279 and P = 0.125, respectively, Mann–Whitney U test, Supplementary Fig. 2b). When we pooled cells that had a significant effect on GDPs (operational hub cells, six ebGABAs and one ctrlGABA), we found that hub cells had significantly longer axons (but not dendrites) than non-hub cells (seven ctrlGABAs, P = 0.0379, Mann–Whitney U test, Supplementary Fig. 2c, d), pointing toward a link between widespread axons and an operational hub role. Thus, CA1 ebGABAs exhibit functional and anatomical features of previously reported hub cells10,11,17.
### Adult ebGABAs exhibit features of long-range projecting cells
Given that ebGABAs displayed unique anatomical and functional features in the immature CA1, we asked whether they maintained distinct properties in adulthood. We examined the molecular content of CA1 ebGABAs to infer the putative cell types comprising this GABAergic population. Staining for single neurochemical markers, we found that many ebGABAs expressed SOM (49 ± 16%, mean ± SD, four mice) and, in a progressively lower extent, PV (29 ± 7%, five mice), NPY (24 ± 11%, five mice) and M2R (22 ± 12%, three mice, Fig. 3a and Supplementary Fig. 3a, b). These data are in line with previously published results on the whole hippocampus10. Using an antibody that allows discrimination between weak and strong levels of nNOS expression, we found that a small but consistent proportion of ebGABAs (8 ± 4%, six mice) expressed strong nNOS levels, a marker of long-range projection cells18 (Fig. 3b and Supplementary Fig. 3c).
To estimate the cell types comprised in the ebGABA population, we examined the combinatorial expression of molecular markers in these cells. In the hippocampus, strong nNOS+ cells are GABAergic projection cells that also express SOM, NPY and M2R14 and are likely to innervate the dentate gyrus and CA319.
We confirmed that strong nNOS+ ebGABAs belong to this cell type by finding that these cells often co-expressed SOM (94 ± 11%, four mice), NPY (93 ± 13%, five mice) and M2R (62 ± 27%, three mice, Fig. 3b and Supplementary Fig. 3). EbGABAs in stratum oriens often expressed SOM and NPY but rarely PV, confirming that the majority of ebGABAs in this layer are not O-LM cells. Furthermore, some ebGABA (15 ± 7%) expressed M2R but not nNOS (three mice, Fig. 3c), suggesting that some of these cells could be retrohippocampal projection cells innervating the subiculum and the retrosplenial cortex20.
A recent study pointed toward COUP-TFII as a possible temporal identity cue promoting an early specification toward SOM+ GABA neurons21. Since most ebGABAs are SOM+, we asked whether COUP-TFII could be a broad ebGABA marker. However, only 7 ± 2% of ebGABAs were COUP-TFII+ (five mice, Supplementary Fig. 3i). Based on the combinatorial expression of molecular markers and previous data10,12, we estimated that at least 40% of ebGABA are likely to be long-range GABAergic projection cells (see “Methods” for details). As it was estimated that CA1 GABAergic projection cells account for only 5–7% of all GABAergic cells22, these data suggest that ebGABAs are biased toward a long-range output connectivity.
To corroborate these data, we filled ebGABAs with neurobiotin using whole cell patch clamp in acute coronal brain slices (n = 34 cells). A subset of these cells (n = 14) were morphologically reconstructed to reveal their axonal and dendritic arborizations (Fig. 3d and Supplementary Fig. 4). In line with the immunohistological data, all reconstructed ebGABAs displayed little local axonal arborizations, suggesting that most of these neurons do not belong to canonical interneuron types. In addition, the axons of some ebGABAs consisted of only one or two branches running straight for several hundreds of micrometers. Finally, not only the somata of ebGABAs were rarely located in stratum lacunosum-moleculare, but also ebGABAs’ dendrites arborized very little in this layer. Taken together, these data indicate that ebGABAs are biased toward a deep radial location and long-range outputs.
### Adult ebGABAs have distinct electrophysiological properties
We then wished to determine whether ebGABAs mature into an electrophysiologically defined subpopulation of GABA neurons. To this end, we performed a series of ex vivo whole cell patch clamp recordings in acute brain slices from adult Dlx1/2(E7.5)-GFP mice (n = 167 cells from 73 mice; 85 ctrlGABAs, 82 ebGABAs), sampling from both the dorsal and intermediate/ventral CA1. The scarcity of ebGABAs in acute slice experiments from adult mice was similar to acute slice experiments at neonatal stage (1.6 ± 0.8 cells found per hippocampus, quantified unilaterally from five mice, mean ± SD). First, we analyzed ebGABAs intrinsic electrophysiological properties (ctrlGABAs: n = 18; ebGABAs: n = 16). Consistent with the fact that many ebGABAs express SOM+, these cells displayed relatively long spikes (half-width: 0.9 ± 0.3 ms, mean ± SD) and a regular, non-fast-spiking pattern of discharge (firing rate at 2× rheobase: 24 ± 17 Hz; adaptation index: 0.69 ± 0.09, Supplementary Table 2).
Analysis of firing rate vs. injected current (f/I) curves revealed a sublinear input/output relationship in ebGABA but not in ctrlGABA (Fig. 3f, g; effect of current injection P < 0.0001; effect of birth date P < 0.0001, interaction P < 0.0001, two-way ANOVA). In many cases, upon strong current injections ebGABA displayed initial high frequency firing, but then inactivated, likely due to depolarization block23. When we plotted an f/I curve only for a shorter time window of the current injection (the initial 50 ms), the firing of ebGABA was not significantly lower than the one of ctrlGABA (effect of current injection P < 0.0001; effect of birth date P = 0.0738). In addition, ebGABAs showed a significantly smaller “sag” response (i.e., a significantly higher sag ratio, P = 0.028, Mann–Whitney U test, Fig. 3g, h). The remaining intrinsic membrane parameters that we examined did not differ between the two groups (Supplementary Table 2). These data suggest that an early birth date biases GABAergic cells toward specific electrophysiological properties.
### EbGABAs are mostly excited by intra-hippocampal inputs
Since the somata of ebGABAs are preferentially located in deep CA1 layers and CA1 inputs are radially organized, we asked whether an early birth date biases GABAergic cells toward a special input connectivity from glutamatergic afferents. We probed ebGABAs’ monosynaptic input connectivity from specific pathways using electrical stimulation and optogenetic mapping combined with intracellular voltage clamp recordings (with cells held at ECl: −87 mV).
First, we focused on the Schaffer collateral input by stimulating the axons from CA3 pyramidal cells in stratum radiatum using a bipolar stimulating electrode while recording from CA1 GABA cells in voltage clamp (Fig. 4a). Electrical stimulation of the Schaffer collateral evoked EPSCs in the majority of ctrlGABAs (13/16 cells) and ebGABAs (9/14 cells, Fig. 4b, c). EPSCs were mediated by AMPA/KA receptors because they were blocked by NBQX (10 µM, 3/3 cells, Supplementary Fig. 5). Thus, the majority of ebGABAs are recruited by CA3 inputs.
Since somata and dendrites of ebGABAs are rarely found in the stratum lacunosum-moleculare, we predicted that long-range inputs targeting this layer would play a less important role in the recruitment of these cells. To evaluate this hypothesis, we tested the inputs from the entorhinal cortex (EC) and the ventromedial thalamus (VMT) using a viral vector to express the fast opsin Chronos in these regions (Fig. 4d–i). Following at least two weeks of expression, Chronos/tdTomato+ axons densely innervated the stratum lacunosum-moleculare of CA1 for both injection sites (some axons were also detected in the stratum oriens next to the alveus, Supplementary Fig. 5; very few axons were also present in other layers). Pulses of 470 nm light were delivered to test for the presence of short-delay EPSCs (monosynaptic connections) in the recorded cells. For EC stimulations, we detected EPSCs in the majority of ctrlGABAs (7/9 cells), but only in about half of the ebGABAs (4/9 cells, Fig. 4e, f). For VMT stimulations, we detected EPSCs in 5/13 ctrlGABAs cells but only in 1/14 ebGABAs (Fig. 4h, i).
The proportion of ebGABA recruited by an intra-hippocampal excitatory input (CA3) was not significantly different from the proportion of ctrlGABA (P = 0.417, Fisher’s exact test). In contrast, the proportion of ebGABA recruited by extra-hippocampal excitatory inputs (EC and VMT pooled) was significantly lower than the proportion of ctrlGABA (5/23 and 12/22, respectively, P = 0.0331, Fisher’s exact test, Fig. 4j). In line with this, maximum EPSC amplitude for intra-hippocampal afferents did not differ between the two populations (P = 0.29, Mann–Whitney U test, n = 16 ctrlGABAs, n = 14 ebGABAs), whereas maximum amplitude for extra-hippocampal afferents was lower for ebGABAs (P = 0.024, Mann–Whitney U test, n = 22 ctrlGABAs, n = 23 ebGABAs, Fig. 4k, l). However, paired pulse ratios for intra- and extra-hippocampal afferents were similar in the two populations (P > 0.99, n = 13 ctrlGABAs, n = 9 ebGABAs and P = 0.72, n = 11 ctrlGABAs, n = 5 ebGABAs, respectively, Mann–Whitney U test). Maximum EPSC amplitudes for extra-hippocampal afferents were comparable when only responsive cells were taken into account (Supplementary Fig. 5f, g). This indicates that an early birth date biases GABA cells’ input connectivity but not synaptic strength or release probability. Given the sparse connectivity and the fact that EPSCs evoked in ebGABAs by stimulation of the EC were small (maximum amplitude 19 ± 33 pA, mean ± SD), these data suggest that extra-hippocampal afferents play a small role in the recruitment of ebGABAs.
Furthermore, we examined spontaneous EPSCs (sEPSCs), many of which are likely to be action potential-dependent, and thereby arise from intra-hippocampal connections in coronal slices. We confirmed that sEPSCs were mediated by AMPA/KA receptors because they were completely blocked by NBQX (10–20 µM, 4/4 cells). CtrlGABA and ebGABA displayed similar sEPSC parameters, in particular frequency and amplitude (Fig. 4k, l and Supplementary Fig. 5h), indicating similar recruitment of these cells by putative intra-hippocampal excitatory afferents. Taken together, these data indicate that the deep location and dendritic arborization of CA1 ebGABA favor their recruitment by intra-hippocampal inputs.
### ebGABAs receive weak local synaptic inhibition
Given that ebGABAs are biased toward a deep location and that the radial position influences the inhibitory innervation of CA1 pyramidal cells24, we also asked whether an early birth date biases GABA cells toward specific inhibitory wiring schemes. We began by examining sIPSCs by voltage clamping the cells at the reversal potential for glutamatergic currents (0 mV; ctrlGABA n = 18, ebGABA n = 17, Fig. 5a). As many sIPSCs are action potential-dependent, this measurement largely portrays local inhibition driven by interneurons firing in the slice. We confirmed that sIPSCs were mediated by activation of GABAA receptors as they were completely abolished by the GABAA receptor antagonist Gabazine (SR95531, 10 µM, n = 4 cells). EbGABAs displayed a significantly lower sIPSC frequency compared to ctrlGABAs (ctrlGABAs: median 11.7 Hz, IQR: 6.2–16.8 Hz; ebGABAs: median 6.5 Hz, IQR: 1.2–10.6 Hz; P = 0.043, Mann–Whitney U test). Lower frequency of sIPSCs in ebGABAs could arise from decreased spontaneous activity of interneurons innervating ebGABAs, sparser innervation by GABAergic terminals or lower release probabilities of presynaptic GABAergic terminals. To verify whether weaker innervation by GABAergic terminals underlies this effect, we carried out further experiments.
First, we placed a bipolar stimulating electrode in stratum radiatum, the layer with weakest innervation by medial septal axons (Supplementary Fig. 6b) to bias our stimulation toward interneurons. In line with the sIPSC data, the maximum IPSC evoked by local electrical stimulation was significantly higher in ctrlGABA than ebGABA (P = 0.0085, Mann–Whitney U test, ctrlGABA n = 10, ebGABA n = 8, Fig. 5c, d). Furthermore, IPSC paired pulse ratio did not differ between the groups (Fig. 5e). These observations suggest that sparser GABAergic innervation and not weaker release probability at GABAergic synapses could account for the lower inhibitory tone onto ebGABA.
To corroborate that this deficit in inhibition arises from local sources, we probed the GABAergic input from the medial septum, a region that provides significant innervation of CA1 GABAergic cells25, by virally driven Chronos expression in this nucleus (Supplementary Fig. 6a). Following at least two weeks of expression, Chronos/tdTomato+ axons densely innervated the stratum oriens and the border between the stratum radiatum and the stratum lacunosum-moleculare (and to a lesser extent the strata pyramidale and radiatum; Supplementary Fig. 6b).
Both ctrlGABA and ebGABA displayed a high degree of connectivity (13/14 ctrlGABA and 12/18 ebGABA receiving IPSCs upon septal stimulation, Supplementary Fig. 6d). These IPSCs were GABAergic as they were blocked by the GABAA receptor antagonist Gabazine (SR95531, 10 µM, 11/11 cells, Supplementary Fig. 6c). Maximum amplitude of the IPSC evoked by septal stimulation as well as paired pulse ratio did not differ between ctrlGABAs (n = 13) and ebGABAs (n = 12, Fig. 5e, f), suggesting that these cells received similar innervation and release probability from this pathway. Thus, the deficit in inhibition onto ebGABAs is likely to originate from local sources.
### EbGABAs receive sparse innervation from parvalbumin neurons
Parvalbumin-expressing (PV+) basket cells were previously shown to differentially target CA1 pyramidal neurons according to their radial position24. Since the radial position of pyramidal cells is highly influenced by their date of birth26, we tested whether ebGABAs could also receive different innervation by PV+ cells. To this end, we quantified the number of PV+ boutons innervating ebGABAs and ctrlGABAs. To quantify the innervation ctrlGABAs, we used the GAD67-GFP mouse line. To avoid bias due to uneven sampling across layers, we sampled ctrlGABA cells that roughly matched the position of the imaged ebGABAs (n = 43 ctrlGABA and n = 30 ebGABA, each population from two mice). We found a striking difference in the number of boutons innervating the two populations, with ebGABAs receiving a significantly lower number of PV+ terminals (P < 0.0001, Mann–Whitney U test, Fig. 5i, j). Importantly, we verified that intensities and areas of the PV staining, as well as the sampled volumes were similar for the two populations (P = 0.2363, Mann–Whitney U test, Fig. 5j and Supplementary Fig. 7). In addition, the difference in the number of PV+ terminals held when restricting the analysis to cells in the stratum oriens (P = 0.0008, Mann–Whitney U test, Supplementary Fig. 7d), suggesting that this difference was not generated by uneven sampling across layers. Thus, an early birth date biases GABA cells for a sparse PV innervation, which is likely to account (at least in part) for the weak inhibition observed in ebGABAs.
### ebGABAs show distinct relation to network activity
Finally, we asked whether ebGABAs’ different anatomical, electrophysiological and wiring properties reported above could result in distinct in vivo activity in awake mice. To test this, we injected a viral vector expressing the red calcium indicator jRGECO1a in the dorsal hippocampus of adult Dlx1/2(E7.5)-GFP mice. Two weeks after the injections, mice were implanted with a chronic glass window that was placed just above the dorsal hippocampus and a bar for head fixation. This allowed performing in vivo two-photon calcium imaging from ebGABA and nearby cells. Mice were head-fixed in the dark on a non-motorized treadmill allowing spontaneous movement27 (Fig. 6a).
We imaged from 15 mice expressing jRGECO1a in large numbers of cells using 400 × 400 µm fields of view (FOVs, Supplementary Movies 2 and 3). Given the sparsity of ebGABAs, only 17 Dlx1/2(E7.5)-GFP+ cells were found out of 15 mice. Analyses of the calcium dynamics during spontaneous locomotion and rest could be performed only for nine ebGABAs and nearby cells (n = 776 in total) from seven FOVs from six mice (two FOVs from stratum oriens and six from the stratum pyramidale). The remaining eight ebGABAs could not be analyzed because of either excessive movement in the z-axis (four cells), no expression of jRGECO1a in the Dlx1/2(E7.5)-GFP+ cells (three cells) or epileptic-like activity detected in the FOV (one cell). We employed a matched subsampling approach (see “Methods”) to determine statistically whether ebGABAs, as a population, displayed significantly different mean parameters (or proportional modulation) than control cells. The inferred firing rate of ebGABAs was not significantly different from the rate of random control cells (Supplementary Fig. 8a). We next examined single cells’ activity during locomotion (Fig. 6d, see “Methods” for details). We found that all ebGABAs (9/9) were recruited during locomotion (LocomotionON cells Fig. 6d, i), mostly for the entire locomotion period, but in one case only at locomotion onset (Supplementary Fig. 8b, c). The proportion of LocomotionON ebGABAs was significantly higher than the proportion of LocomotionON random control cells (P < 0.0001, bootstrap). We also analyzed single cell activities in relation to synchronous calcium events (SCEs, Fig. 6e and Supplementary Fig. 9, see “Methods” for details) that are known to occur during rest, often in synchrony with sharp-wave ripples28. SCEs occurred at a rate of 0.08 ± 0.04 Hz in stratum pyramidale (mean ± SD), in line with previous reports, and of 0.02–0.03 Hz in stratum oriens. We found that 8/9 ebGABAs were significantly recruited during SCEs (SCEON cells, Fig. 6f). This proportion was higher than the proportion of SCEON random control cells (P = 0.05, bootstrap, Fig. 6i).
Next, we detected cell assembly patterns occurring during rest in a 200 ms time window and analyzed single cell activities in relation to assembly activations (Fig. 6g and Supplementary Fig. 9, see “Methods” for details). This analysis revealed that 7/9 ebGABAs were significantly recruited around “mean cell assembly activity” (AssemblyON cells, Fig. 6h). The proportion of AssemblyON ebGABAs was significantly higher than the proportion of random control cells (P < 0.001, bootstrap, Fig. 6i).
Overall, 7/9 ebGABAs displayed combined modulation by locomotion, SCEs and assembly activities and this proportion was significantly higher than the proportion of random control cells showing combined modulation (P < 0.0001, bootstrap, Fig. 6i). This suggests that ebGABAs constitute an important node of the hippocampal circuit: they are among a minority of cells in the network that integrate both locomotion and population synchrony signals.
Finally, we asked whether ebGABAs exhibit distinct functional connectivity even in adult networks. We analyzed functional connectivity in a subset of FOVs imaged from the stratum pyramidale (five FOVs, seven ebGABA, 746 other cells). We calculated output functional connectivity similarly to our previous studies in developing networks11,17. In brief, a functional connection from neuron A to neuron B was established if neuron A consistently fired before B. Although the connectivity of ebGABAs varied across FOVs (percentage of ebGABA connections among all connections: 1.5 ± 1.3%, mean ± SD, Supplementary Fig. 8d), as a population ebGABAs connected to a larger number of cells than random control cells (P < 0.05, bootstrap, Fig. 6j, k). In line with this, 2/7 ebGABAs (28%) could be defined as hub cells (see Methods for criteria), whereas the proportion of hub cells among control cells was smaller (60/746, 8%; P = 0.05, bootstrap). The difference in functional connectivity between ebGABAs and control cells appeared to be driven by locomotion periods (Supplementary Fig. 8e). Thus, ebGABAs remain crucial nodes of the CA1 network even in adulthood in vivo.
## Discussion
Using inducible genetic fate mapping, ex vivo and in vivo large-scale calcium imaging, electrophysiology, optogenetics, and anatomical analyses, we have shown that ebGABAs are involved in local CA1 dynamics both in development and adulthood. At the neonatal stage, ebGABAs coordinate network bursts (GDPs) ex vivo. In adulthood, ebGABAs maintain a strong link with network activity and high functional connectivity in vivo. Their early birth date specifies anatomical and intrinsic electrophysiological properties as well as input connectivity schemes that may contribute to their recruitment.
We found that ebGABAs maintain a set of distinct anatomical and functional properties in the adult CA1. First, they display sparse local axons and express combinations of markers typical of different classes of projection cells. Electrophysiologically, they are characterized by a small “sag” and a sublinear f/I (input/output) relationship. The latter may limit strong activations to short periods of time. In addition, adult ebGABAs show wiring schemes that are distinct from randomly sampled GABAergic cells. Their recruitment appears to be largely driven by intra-hippocampal excitatory afferents because they receive typical intra-hippocampal excitation, but little long-range inputs from the EC and the VMT. Furthermore, ebGABAs receive weak local inhibition via a sparse innervation by axons arising from PV+ neurons. In vivo, ebGABAs are part of a minority of cells in the CA1 network that are recruited with locomotion, synchronous calcium events and assembly activity. In addition, they exhibit high functional output connectivity degrees. Therefore, ebGABAs appear predetermined for exceptional functional and structural properties in both the developing and adult hippocampus.
The present study demonstrates that ebGABAs are highly involved in hippocampal network dynamics in development and adulthood. During early postnatal development, stimulation of a single ebGABA is sufficient to trigger GDPs and to change their frequency. The latter phenomenon is likely to involve complex polysynaptic interactions because the delay between ebGABA stimulation and GDP onset ranged between 0.2 and 5 s. Thus, hub cells are not a unique feature of CA3, which shows more recurrent connections29, but are distributed throughout the hippocampal formation, and possibly throughout the brain. It is not clear whether CA1 hub cells orchestrate GDPs only through action on the CA1 circuit or, by contrast, they modulate GDP generation in CA3 with subsequent propagation to CA1. Both scenarios are possible. Reconstructions of biocytin-filled ebGABAs demonstrated that some projected back to CA3, a finding that is corroborated by our analyses of molecular marker combinations (i.e., strong nNOS-expressing backprojection cells). On the other hand, GDPs have been shown to occur in CA1 also independently from CA313,30.
GDPs and their in vivo counterpart31 are network bursts that could represent an early form of sharp wave ripple (SWRs). SWRs are generated in CA2 and, similarly to GDPs, require the recurrent circuit of CA3 to successfully propagate to CA132,33. Recently, we reported that SCEs detected with calcium imaging in the adult CA1 often occur during SWRs and involve reactivations of cell assemblies28. Here, we show that ebGABAs maintain a strong relationship with network bursts (SCEs) in adulthood.
In addition, these cells are consistently recruited around the activation of CA1 assemblies. This may be achieved via special intrinsic properties and circuit motifs. For instance, we have shown with ex vivo patch clamp experiments that ebGABAs display efficient rate coding for short but not long depolarizing stimuli, a mechanism that could favor their transient recruitment in vivo. In addition, our data suggest that ebGABAs receive the majority of their excitatory inputs from intra-hippocampal inputs. This may enhance their signal to noise ratio to report local network activity to postsynaptic targets. If such intra-hippocampal inputs are most likely originating from CA3 (since they are evoked by electrical stimulation in the stratum radiatum), future experiments should also probe the contribution of synaptic inputs from CA1 and CA2 pyramidal cells to the excitation of ebGABAs. A preferential input from CA2 could be expected, given that neurons sharing a similar temporal embryonic origin are more likely to connect34,35 and that CA2 is the earliest region of the Cornus Ammonis to be generated36. Preferential CA2 inputs could also provide a circuit mechanism for the high modulation of ebGABAs by assembly activation, given the role of CA2 in triggering SWRs33.
Finally, we demonstrate that ebGABAs receive little local inhibition from PV+ interneurons, but high levels of long-range inhibition from the medial septum. This finding is consistent with a study showing that the majority of inhibitory terminals on CA1 GABAergic projection cells arise from the medial septum37. Virtually all PV+ terminals originating from CA1 interneurons and targeting other GABA cells are likely to arise from PV+ basket cells and bistratified cells, and both cell types are strongly active during SWRs38,39. Thus, this lack of cell type-specific inhibition could additionally favor ebGABAs’ recruitment during SWRs/SCEs. Since superficial pyramidal cells receive little PV innervation but abundant inputs from CCK+ interneurons35, future work could establish whether ebGABAs receive preferential innervation from CCK+ interneurons.
Another remarkable feature of ebGABAs in the adult CA1 in vivo is their functional versatility, namely their recruitment during a variety of behavioral/network phenomena. Specifically, ebGABAs were among a minority of cells in the network that were consistently activated with locomotion, SCEs and assembly activations. This versatility sets them apart from most known GABAergic cell types because interneurons that are activated by locomotion are usually not activated during SWRs, and vice versa38,39,40.
Interestingly, ebGABAs’ recruitment during a variety of network states is reminiscent of the activity described for CA1 GABAergic projection neurons41. In line with this, a significant portion of ebGABAs project to the medial septum12. In the present study, we have found that ebGABAs express combinations of molecular markers typical of two extra classes of GABAergic projection cells: strong nNOS-expressing backprojection cells, likely innervating dentate gyrus and CA314,19, and M2R-expressing retrohippocampal cells that innervate subiculum and retrosplenial cortex20. Combined with the fact that biocytin-filled ebGABAs exhibited little local axonal arborization, these data indicate that ebGABAs could form several classes of projection cells innervating various target regions. EbGABAs also share some similarities with the recently described long-range inhibitory nNOS+ cells (LINCs)42, in particular laminar location and various long-range targets. However, overlap between these two classes could be small because LINCs are born later (E11.5) and rarely express SOM.
EbGABAs that we filled with biocytin in coronal slices from adult mice showed poor local axonal arborizations. However, ebGABAs displayed strong functional output connectivity in the adult CA1 in calcium imaging experiments in vivo. Various reasons could lie at the bottom of this apparent discrepancy. EbGABA’s axons could innervate CA1 cells through arborizations in different planes that are largely severed in acute coronal slices. Alternatively, local projections could be minimal but circuit motifs could guarantee a powerful effect on the CA1 network. These could be local disinhibition (i.e., inhibition of few CA1 interneurons targeting many pyramidal cells, such as basket cells) or inhibition of distant regions that project to CA1 (such as the medial septum or CA3).
We report that an early embryonic origin results in a deep soma location, sublinear input-output firing curve, small sag, reduced innervation by PV+ interneurons and by long-range excitatory inputs, in particular by thalamic afferents. These findings highlight that birth date and/or radial position may have a similar impact on certain cellular features for GABAergic and glutamatergic neurons. We recently reported that a low input–output firing curve also distinguishes dentate gyrus granule cells with an early temporal embryonic origin from later-born granule cells43. In addition, deep CA1 pyramidal cells are born earlier than superficial pyramidal cells26 and show a smaller sag24,44. However, deep CA1 pyramidal cells show the opposite pattern of PV innervation from ebGABAs, receiving more PV+ inputs than superficial cells24.
An early embryonic origin specifies a large proportion of long-range projecting GABAergic cells, an extremely rare neuronal population22. It is crucial to understand how this early specification occurs. Early transcription factors shared across different subpallial proliferative areas could direct GABAergic cells toward a long-range projecting fate. It was recently proposed that COUP-TFII acts as a temporal identity cue that promotes an early specification toward SOM+ GABA neurons21, however, very few Dlx1/2(E7.5)-GFP+ cell expressed it in adulthood.
Our study has some limitations. First, the number of recorded ebGABAs in some experiments is low. This is due to the sparseness of GFP+ cells in the hippocampus of our Dlx1/2(E7.5)-GFP mouse line. This sparseness reduces the yield of most experiments. The low number of ebGABAs is consistent with the scantiness of GABAergic projection cells22. In addition, our Dlx1/2(E7.5)-GFP line is unlikely to capture all hippocampal ebGABAs. Importantly, we have kept tamoxifen levels low and used a weak reporter line because this minimizes known “leak” issues of the Dlx1/2-CreER line12. Second, future studies should investigate ebGABAs’ axonal fields filled in vivo and their postsynaptic targets. Furthermore, they should test whether these cells are causally involved in adult network activity. Intersectional genetic strategies should be developed to tackle these questions because Cre is no longer expressed in adult ebGABAs, leading to inability to target these neurons with Cre-dependent constructs.
On balance, this study shows that ebGABAs are pioneer GABAergic cells operating as “hubs” during development and maintaining unique connectivity throughout adulthood. To our knowledge, we have provided the first evidence that an early birth date alone (regardless of spatial embryonic origins or cell types) dictates anatomical, electrophysiological and connectivity properties of GABAergic cells. Given their bias toward long-range targets, intra-hippocampal inputs and local assembly activity, we hypothesize that ebGABAs could detect CA1 activity and bind local and distant assemblies into chains of neuronal activation45.
## Methods
### Animals
All protocols were performed under the guidelines of the French National Ethics Committee for Sciences and Health report on “Ethical Principles for Animal Experimentation” in agreement with the European Community Directive 86/609/EEC under agreement #01413. Dlx1/2CreER+/−;RCE:LoxP+/+ male mice15 were crossed with 7- to 8-week-old wild-type Swiss females (C.E. Janvier, France) for offspring production. To induce CreER activity, we administered a tamoxifen solution (Sigma) by gavaging (force-feeding) pregnant mice with a silicon-protected needle (Fine Science Tools). We used 2 mg of tamoxifen solution per 30 g of body weight prepared at 10 mg/mL in corn oil (Sigma). In order to label neuronal progenitors expressing Dlx1 and/or Dlx2 with GFP at embryonic age E7.5 or E8.5, pregnant females crossed with Dlx1/2CreER+/−;RCE:LoxP+/+ males were force-fed at day 7.5 or 8.5 post vaginal plug. For simplicity, we refer to Dlx1/2CreER+/−;RCE:LoxP+/+ mice treated with tamoxifen at E7.5/E8.5 as Dlx1/2(E7.5)-GFP. GAD67-GFP mice were kindly donated by Dr. Hannah Monyer (Heidelberg University).
### Slice preparation for ex vivo experiments
Slices containing the hippocampus were prepared from Dlx1/2(E7.5)-GFP mice or Cre-negative littermates Dlx1/2CreER−/−;RCE:LoxP+/+. Four to 5 days old (P4–P5) pups were used for developmental experiments and 4–33 weeks old mice (mean age ± SD: P67 ± 32) for adult experiments. Adult mice were anaesthetized with a ketamine/xylazine mix (Imalgene 100 mg/kg, Rompun 10 mg/kg) or a Domitor/Zoletil mix (0.6 and 40 mg/kg, respectively) prior to decapitation. Slices were cut using a Leica VT1200 S vibratome in ice-cold oxygenated modified artificial cerebrospinal fluid with the following composition (in mM): 2.5 KCl, 1.25 NaH2PO4, 7 MgCl2, 5 CaCl2, 26 NaHCO3, 5 d-glucose, 126 CholineCl. Slices were cut in the horizontal plane at 380 µm thickness for developmental experiments and in the coronal plane at 300 µm thickness for adult experiments. Slices were then incubated in oxygenated normal ACSF containing (in mM): 126 NaCl, 3.5 KCl, 1.2 NaH2PO4, 26 NaHCO3, 1.3 MgCl2, 2.0 CaCl2, and 10 d-glucose (1 h at room temperature, RT for developmental experiments, 15 min at 33 °C followed by 30 min at RT for adult experiments).
### Ex vivo calcium imaging and patch clamp recordings during development
For Fura2-AM loading, slices were incubated in a small vial containing 2.5 ml of oxygenated ACSF with 25 ml of a 1 mM Fura2-AM solution (in 100% DMSO) for 30 min. Slices were incubated in the dark, and the incubation solution was maintained at 30–33 °C. Slices were transferred to a submerged recording chamber and continuously perfused with oxygenated ACSF (3 mL/min) at 30–33 °C. Imaging was performed with a multibeam multiphoton pulsed laser scanning system (LaVision Biotech) coupled to a microscope as previously described46. Images were acquired through a CCD camera, which typically resulted in a time resolution of 80–138 ms per frame. Slices were imaged using a 20×, NA 0.95 objective (Olympus). Imaging depth was on average 80 µm below the surface (range: 50–100 µm).
Patch recording electrodes (4–8 MΩ resistance) were pulled using a PC-10 puller (Narishige) from borosilicate glass capillaries (GC150F-10, Harvard Apparatus) and filled with a filtered solution consisting of the following compounds (in mM): 130 K-methylSO4, 5 KCl, 5 NaCl, 10 HEPES, 2.5 Mg-ATP, 0.3 GTP, and 0.5% neurobiotin (265–275 mOsm, pH 7.3). Electrophysiological signals were amplified (EPC10 amplifier; HEKA Electronik), low-pass filtered at 2.9 kHz, digitized at 10 kHz and acquired using a Digidata 1550 Digitizer and pClamp 10 software (Molecular Devices). For most stimulation experiments, imaging acquisition was separated between: (1) a baseline period during which the cell was held close to resting membrane potential (i.e., zero current injection); (2) a stimulation period during which phasic stimulation protocols were applied; and (3) a 3 min recovery period during which the cell was brought back to resting membrane potential. The stimulation protocol consisted of suprathreshold current pulses (amplitude: 100–150 pA) of 200 ms duration repeated at 0.1, 0.2, and 0.4 Hz. Cells were discarded if they did not meet the following criteria: (1) stable resting membrane potential; (2) stable network dynamics measured with calcium imaging (i.e., the coefficient of variation of the inter-GDP interval did not exceed 1); (3) cells displaying healthy shape and good Fura2-AM loading throughout the entire field of view.
### Analysis of ex vivo calcium imaging data during development
We used custom designed MATLAB software allowing automatic identification of loaded cells, measurement of the average fluorescence transients from each cell as a function of time and detection of onsets and offsets of calcium signals11. Network synchronizations (GDPs) were detected as synchronous onsets peaks including more neurons than expected by chance, as previously described11, and their time stamp denoted by tG. The “Inter GDP intervals” (IGI) was defined as the interval between two consecutive GDPs. To establish whether the stimulation of a single neuron was able to influence the frequency of GDPs occurrence, we employed a previously published analysis11. We first calculated the average IGI in the three epochs: pre-stimulus (control), during the stimulation period and post-stimulus. Due to the variability distribution of IGI in each interval, we calculated the average IGI in a window of ts (300 or 400 frames calculated starting from each tG and eliminating the data corresponding to overlaps between epochs). To test for the significance of the change in the period of GDP, a Kolmogorov–Smirnov test was applied between all the three resulting distributions of average IGI and a significance level of P < 0.05 was chosen. To test for changes in phase of the GDP upon stimulation, the IGI during resting conditions was used as a reference period of a harmonic oscillator mimicking the rhythm of GDPs. The phase of the expected GDP generated by the harmonic oscillator was compared to the phase of the real GDP intervals occurring during the entire recording. For the ith GDP, a phase measure Φi in respect to the control IGI is defined as follows:
$${\mathrm{{\Phi}}}_i = \frac{{(t_G^i - < t_G^i > )}}{{{\Delta}t}},$$
(1)
where Δt is the average IGI interval in the control condition, and 〈$$t_G^i$$〉 = i × Δt is the expected occurrence of the ith GDP, according to the control condition. The phase of each real GDP was set to zero at the peak of synchronous activation and increased linearly reaching 2π at the peak of the next GDP. By subtracting the phase of real GDPs from the expected GDP, we could instantaneously measure the effect of phasic stimulation. The cell was included in the dataset only if the difference between expected and observed cycles did not exceed ±2 cycles during resting conditions.
To assess whether single cell stimulation was able to lock GDP occurrence, we isolated the trials following each stimulation (trial length was either 2.5 or 5 s). Next, we calculated an average histogram of single cell calcium onsets including all trials. The histogram was then normalized by the number of GDPs during the stimulation protocol. The height of the highest peak in the histogram was stored. To estimate the significance of the time locking, we created surrogate data consisting of concatenated control and post-stimulation frames (without stimulation trials). The same histogram was then produced from the surrogate data. Since no stimulation was present in this case, we used an arbitrary frame as stimulation time. For each cell, number of trials and trial length for the surrogate data matched the ones of stimulation data. This process was repeated 100 times using 100 different starting frames for the stimulation, and the values of the highest peaks were stored. A final histogram of peak heights was computed, and the cell was considered to significantly lock GDP onset if the peak height during stimulation fell within the 5% highest peaks in the distribution (i.e., P < 0.05).
### Virally-driven opsin/jRGECO1a expression
Dlx1/2(E7.5)-GFP mice (P25–P35 age for opsin expression, 2–3 months age for jRGECO1A expression) were anaesthetized using either a ketamine/xylazine mix (Imalgene 100 mg/kg, Rompun 10 mg/kg) or 1–3% isoflurane in oxygen. Analgesia was also provided with buprenorphine (Buprecare, 0.1 mg/kg). Lidocaine cream was applied before the incision for additional analgesia. Mice were fixed to a stereotaxic frame with a digital display console (Kopf, Model 940). Under aseptic conditions, an incision was made in the scalp, the skull was exposed, and a small craniotomy was drilled over the target brain region. A recombinant viral vector was delivered using a glass pipette pulled from borosilicate glass (3.5” 3-000-203-G/X, Drummond Scientific) and connected to a Nanoject III system (Drummond Scientific). The tip of the pipette was broken to achieve an opening with an internal diameter of 25–35 μm. The following viral vectors were used (both from Penn Vector Core): AAV8.Syn.Chronos-tdTomato.WPRE.bGH to drive Chronos expression in the VMT or medial septum, AAV1.Syn.NES.jRGECO1a.WPRE.SV40 to drive jRGECO1a expression in the dorsal CA1 (for the latter, virus stock was diluted 1:4 in d-phosphate-buffered saline (PBS), Sigma Aldrich). Stereotaxic coordinates were based on a mouse brain atlas (Paxinos and Franklin). All coordinates are in millimeters. Anteroposterior (AP) coordinates are relative to bregma; mediolateral (ML) coordinates are relative to the sagittal suture; dorsoventral (DV) coordinates are measured from the brain surface. For EC, VMT (nuclei reuniens and rhomboid), and medial septum, 100 nL of virus were injected at a rate of 20 nL/min at the following coordinates: EC −4.7 AP, −4.1 ML, −2.7 DV; VMT −0.7 AP, 0 ML, −3.8 DV; medial septum +0.9 AP, 0 ML, −3.8 DV. For CA1 hippocampus, two injections of 250 nL were performed at a rate of 25 nL/ min (−1.8 AP, −1.6 ML, −1.25 DV and −2.3 AP, −2.4 ML, −1.25 DV). EC injections mostly targeted the lateral EC. At least 2 weeks were allowed for Chronos and jRGECO1a expression before recording procedures and chronic hippocampal window, respectively.
### Ex vivo patch clamp recordings and optogenetics in adulthood
Patch clamp recordings in adult slices were performed using a SliceScope Pro 1000 rig (Scientifica) equipped with a CCD camera (Hamamatsu Orca-05G). Slices were transferred to a submerged recording chamber and continuously perfused with oxygenated ACSF (3 mL/min) at ~32 °C. Patch recording electrodes (4–6 MΩ resistance) were produced as described above. For current clamp recordings, electrodes were filled with an intracellular solution containing (in mM): 125 k-gluconate, 5 KCl, 4 ATP-Mg, 0.3 GTP-Na2, 10 Na2-phosphocreatine, 10 HEPES and 0.05% neurobiotin (pH 7.3 and ~280 mOsm). For voltage clamp experiments, an intracellular solution with the following ingredients (in mM) was used: 152 Cs-methanesulfonate, 4 CsCl, 4 ATP-Mg, 0.3 GTP-Na2, 10 Na2-phosphocreatine, 10 HEPES (pH 7.3 and ~280 mOsm). Electrophysiological signals were amplified (Multiclamp 700B), low-pass filtered at 2.9 kHz, digitized at 5–20 kHz and acquired using a Digidata 1440 A digitizer and pClamp 10 software (all from Molecular Devices).
Patch clamp recordings were performed from visually identified neurons (GFP+ cells for ebGABAs, random putative GABAergic cells for ctrlGABAs). EbGABAs and ctrlGABAs were sampled in roughly equal numbers from all layers of both the dorsal and the intermediate/ventral CA1 to avoid biases toward specific cell types or inputs. We used the following criteria to ensure that no pyramidal cell was included in the ctrlGABA group: first, we avoided pyramidal-shaped neurons in the strata pyramidale and oriens; second, we verified in current clamp experiments that all targeted cells (21/21) displayed stereotypical GABA cell firing; third, we verified histologically a subset of ctrlGABAs that were filled with neurobiotin (n = 28, see below for details), and all of them exhibited GABA cell anatomical features.
An optoLED system (Cairn Research) consisting of two LEDs was used to visualize fluorescence signals and stimulate Chronos-expressing afferents in CA1. A 470 nm LED coupled to a GFP filter cube was used to visualize GFP-expressing cells and activate Chronos-expressing terminals (3 ms-long light pulses). A white LED coupled to a TRITC/Cy3 filter cube was used to visualize the Chronos/tdTomato-expressing axons. Cells were tested for postsynaptic responses and included in connectivity data only if located in a part of CA1 densely innervated by tdTomato+ axons. Light was delivered using a 40× objective, leading to a light spot size of ~1 mm, which was able to illuminate all CA1 layers.
Electrical stimulation of the Schaffer collateral and evoked local IPSC experiment were performed by placing a bipolar stimulating electrode made by two twisted nichrome wires in the stratum radiatum of CA1 (between the recorded cell and CA3). The bipolar electrode was connected to a DS2A Isolated Voltage Stimulator (Digitimer Ltd.) delivering 0.2 ms-long stimuli. Postsynaptic current amplitude and paired pulse ratio were assessed by two stimulations separated by 50 ms. Sweeps were separated by 20 s delay to avoid the induction of plasticity and ensure stable responses. Maximum PSCs were determined by delivering increasing stimulation powers and constructing stimulation power vs. PSC amplitude curves. Since these curves usually saturated, the maximum PSC amplitude was measured from the first PSC of the plateau. In the few cases in which the amplitude did not saturate, the response obtained from the maximum stimulation power was used to calculate the maximum amplitude.
For sPSCs experiments, Rs was not compensated because this allowed to monitor more precisely its changes throughout the experiment. For evoked PSC experiments, 60% compensation was applied to the Rs. Only recordings with Rs < 30 MΩ were included in the dataset. The cell was discarded if Rs changed by more than 20% throughout the protocol. For electrically evoked and optically evoked PSCs, PSC onset delays were consistent with monosynaptic responses (3.3 ± 1.6 ms, mean ± SD).
### Analysis of ex vivo patch clamp recordings in adulthood
Analysis of intrinsic membrane properties was performed using custom-made MATLAB scripts. The resting membrane potential was estimated by averaging a 60 s current–clamp trace recorded at 0 pA holding current. The input resistance was calculated from the slope of steady-state voltage responses to a series of subthreshold current injections lasting 500 ms (from −50 pA to last sweep with a subthreshold response, 5 or 10 pA step size). The membrane time constant (τ) was estimated from a bi-exponential fit of the voltage response to a −30 pA hyperpolarizing pulse. The membrane capacitance was calculated as the ratio between membrane τ and input resistance. The first spike in response to a juxta-threshold positive current injection was used to determine: the threshold potential (the first point > 0.5 in the first derivative), the fast afterhyperpolarization (calculated from the threshold potential), the action potential half-width (the width at half-amplitude between the threshold potential and the peak of the action potential). The rheobase (in pA) was determined as a 50 ms current injection, able to generate a spike in 50% of the cases in 10 trials. A 1 s-long current injection of 2× the rheobase was used to determine: the firing rate, the adaptation index (range: 0–1, defined as the ratio between the mean of first three and the last three inter-spike intervals and the burst index (the ratio between the number of spikes in the first 50 ms and the entire depolarizing step). The sag ratio was calculated by injecting a 1 s-long negative current able to hyperpolarize the cell between −100 and −120 mV, and by dividing the amplitude of the hyperpolarization at the end of the current step by the peak of the hyperpolarization at the beginning of the step (sag ratio close to 0 implies high sag, whereas sag ratio close to 1 implies little or no sag). The maximum firing rate was defined as the maximum value reached in a f/I curve with positive current injections from +20 to +820 pA (step size: 50 pA).
Analysis of spontaneous postsynaptic currents (sPSCs) was performed with MiniAnalysis (Synaptosoft). PSC parameters were calculated from 2 min recording time. The decay time (10–90%) was calculated from a mono-exponential fit of mean sPSCs. Analysis of evoked postsynaptic current data were performed with Clampfit 10 (Molecular Devices) and MATLAB custom-made scripts. Since membrane and synaptic parameters may be affected by age, for all adult datasets we tested that the distributions of the ages of the mice used did not differ statistically between ctrlGABA and ebGABA. Additionally, for optogenetics experiments we verified that the time of opsin expression did not differ between the groups.
### In vivo calcium imaging in adulthood
A chronic cranial window was implanted using previously published procedures27. Mice were head-fixed on a non-motorized treadmill allowing self-paced locomotion (adapted from ref. 47) All experiments were performed in the dark. No reward was given. After three to five habituation sessions, mice were alert but calm and alternated between periods of locomotion and rest during imaging. The treadmill was made of a 180 cm black velvet seamless belt lacking tactile or visual cues mounted on two wheels. The movement of the treadmill was monitored using two pairs of LEDs and photo-sensors that read patterns from a disk attached to one of the wheels, similarly to what previously described27,48.
For all experiments, extra sound, odor, touch, and light were minimized during the imaging session. Imaging was performed with a single beam multiphoton laser scanning system coupled to a microscope (TriM Scope II, Lavision Biotech). The Ti: sapphire excitation laser (Chameleon Ultra II, Coherent) was operated at 1030 nm for jRGECO1a excitation and 920 nm for GFP excitation. Fluorescence emission was acquired using a 16× objective (Nikon, NA 0.8) and split in two detectors (GaSP PMT, H7422-40, Hamamatsu) with bandpass filters of 510/10 nm for GFP and 580/20 nm for jRGECO1a. Scanner and PMTs were controlled by a commercial software (Imspector, Lavision Biotech). To optimize the signal-to-noise ratio of fluorescence variation, we used a dwell time exposition of 1.85 µs and a spatial resolution of 2 µm/pixel that allowed us to acquire at 9.85 Hz at a field of view of 400 × 400 µm. Locomotion and imaging triggers were synchronously acquired and digitized using a 1440A Digidata (Axon instrument, 2 kHz sampling) and the pClamp 10 software (Molecular Devices).
### Analysis of in vivo calcium imaging data in adulthood
In vivo calcium movies were pre-processed using the CaImAn toolbox for MATLAB49. First, movies were motion-corrected using a rigid registration method50. Then, contours and calcium transients were detected using a constrained nonnegative matrix factorization framework allowing denoising and demixing of fluorescence signals51. To ensure correct segmentation of somatic calcium activity, the automatic detection was manually refined by adding and removing region of interests (ROIs) using the correlation image based on neighboring pixels. ΔF/F fluorescence signals were then denoised with a median filter, detrended and extraction of the 20% ΔF/F. An additional manual refinement was carried out with visual inspection of each ROI in the correlation image and the corresponding trace. Unstable or noisy traces were removed because this led to spurious spike inference. A Markov Chain Monte Carlo approach52 initialized by the fast OOPSI algorithm53 was used to model rise and decay time constants using a second order autoregressive process, which allowed spike inference from the fluorescence traces. A final visual inspection of overlapping calcium traces and spike raster plots was performed to ensure that the spike times reflected the dynamics of the fluorescence traces.
Temporal alignment of the treadmill movement signal and spike raster as well as subsequent analyses were performed using custom-made MATLAB scripts. Locomotion epochs were defined as time periods with deflections in the photo-sensors signal reading the treadmill movement. Rest epochs were defined as periods >200 ms without treadmill movement. Two methods were used to define cells activated by locomotion. In the first method, peri-stimulus time histograms for the locomotion onset (PSTHLOC) were generated (10 s window). The mean $$\left( {\mu _{\mathrm{B}}^{{\mathrm{LOC}}}} \right)$$ and the SD $$\left( {{\upsigma }}_{\mathrm{B}}^{{\mathrm{LOC}}} \right)$$ of the baseline firing rate (activity preceding 200 ms before locomotion onset) were used to generate Z-score normalized PSTHs:
$${Z}^{{\mathrm{LOC}}} = \frac{{{\mathrm{PSTH}}^{{\mathrm{LOC}}} - \mu _{\mathrm{B}}^{{\mathrm{LOC}}}}}{{{\upsigma }}_{\mathrm{B}}^{{\mathrm{LOC}}}}.$$
(2)
Cells were defined as significantly activated if at least two consecutive bins exceeded a Z-score of 2 in a time window from 200 ms before locomotion onset onwards. This method defined cells activated around the onset of locomotion. In the second method, spike times were circularly shifted to disrupt their temporal relationship with locomotion and rest periods. The ratio between the number of spikes occurring during locomotion and during rest was calculated for each cell of the original and the reshuffled spike matrices. The 95th percentile of the vector containing these ratios for the reshuffled spike times was used as statistical threshold for each cell. Cells were defined as activated by locomotion if their locomotion/rest spike ratio exceeded this threshold. This method defined cells that were more active during locomotion than rest (but not necessarily activated at the onset of locomotion). For each imaging session, a cell was defined as activated by locomotion if it passed at least one of the above tests.
Synchronous calcium events were detected from rest periods binning the spike raster matrix with a 200 ms window. We circularly shifted each spike vector, hence maintaining each cell’s firing rate, but disrupting its temporal relationship to the rest of the population. One thousand surrogate distributions were created. The spikes of each frame for these distributions were computed and the 99th percentile of the resulting “sum of spikes” vector was used as a statistical threshold. Peaks above the threshold that were at least separated by one second were considered SCEs. To define cells activated during SCEs, PSTHs for SCE onset (PSTHSCE) were constructed (four seconds window). The mean $$\left( {\mu _{\mathrm{B}}^{{\mathrm{SCE}}}} \right)$$ and the SD $$\left( {{\upsigma }}_{\mathrm{B}}^{{\mathrm{SCE}}} \right)$$ of the baseline firing rate (activity preceding SCEs) were used to generate Z-score normalized PSTHs
$${Z}^{{\mathrm{SCE}}} = \frac{{{\mathrm{PSTH}}^{{\mathrm{SCE}}} - \mu _{\mathrm{B}}^{{\mathrm{SCE}}}}}{{{\upsigma }}_{\mathrm{B}}^{{\mathrm{SCE}}}}.$$
(3)
Cells were defined as significantly activated if the bin at zero time lag exceeded a Z-score of 3.
Neuronal assemblies (co-active neurons) were detected using an unsupervised statistical method based on independent component analysis54. We chose this method because it allows the analysis of assembly activities over time. Analysis was restricted to rest periods and a 200 ms time binning was used because assembly activations coincident to sharp-wave ripples are known to occur in these conditions28. Binned spike counts were convolved with a Gaussian kernel and then Z-scored to reduce the influence of firing rates. The number of significant co-activation patterns was estimated as the number of principal component variances above a threshold derived from the circularly shifted spike count matrix. Assembly patterns were then extracted with an independent component analysis. To track the activation of cell assemblies over time, a projection matrix was constructed for each pattern from the outer product of its weight vector.
The activity of each assembly was estimated by projecting the columns of the spike matrix (time bins) onto the axis spanned by the corresponding assembly pattern (principal components). The length of the projection was calculated by taking the inner product between the assembly pattern and a weighted sum of the Z-scored spike counts54.
To define cells significantly modulated by assembly activities, we cross-correlated the spike count vector of each cell to the activity vector of each assembly (8 s time window). For each cell, we averaged cross-correlograms for different assemblies to define ‘mean cross-correlations to assembly activity’. To calculate a statistical threshold for mean cross-correlations, the same procedure was performed on reshuffled data (circularly shifted spike count matrix) and the 95th percentile of this surrogate distribution was set as threshold for each cell. Mean cross-correlograms with at least two consecutive bins out of the five bins around zero time lag exceeding this threshold were defined as significant.
Functional connectivity analysis was performed in Python using our CICADA toolbox (https://gitlab.com/cossartlab/cicada/). This analysis was restricted to FOVs imaged from the stratum pyramidale (n = 5, including 7 ebGABAs and 776 control cells) because these had more cells (77–236) than movies recorded from the stratum oriens (n = 2, 8–24 cells). The principle of output functional connectivity is to establish a connection from cell A to cell B if the firing of cell A precedes in a repetitive way the firing of cell B. Given the spike times of neuron A, we calculated the distribution of spikes of neuron B occurring at different time lags (between −1000 and +1000 ms). This measure is equivalent to the PSTH of cell B but centered on the firing onsets of cell A. Zero delay correlations were discarded by excluding normal distributions (Gaussians centered at zero) identified using the D’Agostino and Pearson’s test with a two-sided chi squared probability threshold of 0.05 (normaltest function in Python). We also excluded cases in which the activation of two neurons was completely uncorrelated (uniform distributions, identified using the Kolmogorov–Smirnov test from the Python “scipy” package with a two-tailed P value threshold of 0.05). A connection was drawn from A to B if the average time lag was greater than zero, or from B to A if it was lower than zero. This procedure was applied to all possible pairs of neurons. The network graph was built with the “Networkx” package in Python. Hub cells were defined based on the following criteria: (1) being functionally connected to at least 5% of the cells in the network; (2) being among the 5% most connected cells in the distribution including cells from all movies; (3) being among the cells with betweenness centrality above the 80th percentile in the distribution including cells from all movies.
To test whether ebGABAs displayed significantly higher proportions of modulation than control neurons, we employed a matched subsampling (bootstrapping) approach. We sampled randomly picked GFP-negative cells to match the amount of ebGABAs in each FOV. This procedure was repeated 1000 times to create a surrogate distribution. The proportion of modulated ebGABAs was considered significantly higher than the proportion of control cells if the former fell above then 95th percentile of the surrogate distribution. The same procedure was employed to test for significance for the output functional connections and the proportion of hub cells.
### Histological processing
For analysis of neurochemical markers expressed in ebGABAs, mice were deeply anaesthetized with a mix of Domitor and Zoletil (0.6 and 40 mg/kg, respectively), then transcardially perfused with 0.1 M PBS followed by 4% PFA in 0.1 M PBS. Brains were post-fixed overnight at 4 °C in 4% PFA in 0.1 M PBS. Brains were then sectioned using a vibratome (VT 1200 s, Leica) into 70–80 μm-thick slices. Sections were stored in 0.1 M PBS containing 0.05% sodium azide until further usage.
Slices containing neurobiotin-filled cells were fixed overnight at 4 °C in 4% PFA in 0.1 M PBS, rinsed in PBS containing 0.3% Triton X-100 (PBST) and incubated overnight at room temperature in streptavidin −488, −555, −594, or −649 (1:1000 in PBST). Imaging was performed using a confocal microscope (Leica TCS SP5-X) equipped with emission spectral detection and a tunable laser providing excitation range from 470 to 670 nm. Stacks of optical sections were collected for computer-assisted neuron reconstructions.
Primary antibodies (Table 1) were detected with fluorophore-conjugated secondary antibodies for wide-field epifluorescence and confocal microscopy. After preincubation in 10% normal donkey serum (NDS) in PBST, sections were incubated with a mix of up to three primary antibodies simultaneously diluted in PBST with 1% NDS. The following secondary antibodies were used (all from Jackson Immunoresearch): donkey anti-chicken Alexa 488 (1:1000, 703-545-155), donkey anti-rat Cy3 (1:500, 712-165-150), donkey anti-sheep Dylight 647 (1:250-1:500, 713-605-147), donkey anti-rabbit Dylight 594 (1:500, 711-585-152), and donkey anti-rat Alexa 594 (1:500, 705-585-003).
For immunostainings with multiple antibodies, an initial negative control was performed by omitting each primary antibody in turn from the staining procedure; in these cases, no positive fluorescence signal was detected. In addition, each secondary antibody was omitted in turn to confirm its specificity. Epifluorescence images were obtained with a Zeiss AxioImager Z2 microscope coupled to a camera (Zeiss AxioCam MR3) with an HBO lamp associated with 470/40, 525/50, 545/25, and 605/70 filter cubes. Confocal images were acquired either with the Leica system described above or with a Zeiss LSM-800 system equipped with emission spectral detection and a tunable laser providing excitation range from 470 to 670 nm.
### Quantification of PV-expressing axon terminals
The innervation of ctrlGABA and ebGABA by PV + axon terminals was assessed in PFA-fixed sections stained for PV from GAD67-GFP and Dlx1/2(E7.5)-GFP mice, respectively. Confocal stacks centered on the soma and proximal dendrites of GFP + cells were acquired with a Zeiss LSM-800 microscope at constant resolution (0.065 µm/pixel) and z-step (0.41 µm). Since PV + axon density varies depending on CA1 layers, ctrlGABAs were sampled to match as much as possible the location of ebGABAs (ctrlGABAs: 17 from stratum oriens, 26 from intra-/peri-stratum pyramidale, 4 from stratum radiatum; ebGABA: 16 from stratum oriens, 14 from intra-/peri-stratum pyramidale, 2 from stratum radiatum). Appositions between PV + boutons and GFP + somata or proximal dendrites were counted manually using the cell counter plugin in Fiji (http://fiji.sc). Area and median fluorescence of the PV staining were quantified using first the threshold function to exclude unspecific signal.
### Neurolucida reconstruction and morphometric analysis
Fifty-two neurobiotin-filled neurons (38 filled in developing slices, 14 filled in adult slices) were reconstructed using Neurolucida (MBF Bioscience). Neurons recorded during development underwent morphometric analysis. Examined morphological variables included: dendritic and axonal lengths, dendritic, and axonal surfaces.
### Estimate of long-range projecting neurons originating from ebGABA
The proportion of ebGABAs formed by long-range projecting cell types was estimated as follows. We summed: (1) the percentage of ebGABAs formed by SOM+ (but nNOS−) cells in stratum oriens; (2) the percentage of ebGABAs formed by strong nNOS+ cells; (3) the percentage of ebGABAs formed by M2R+ (but nNOS−) cells.
SOM+ cells in stratum oriens are very likely to be projection cells for three reasons. First, most of them are not O-LM cells (very few express PV and we never observed a dense axonal plexus in lacunosum-moleculare in filled cells). Second, most of them are not bistratified cells because very few ebGABAs co-express NPY and PV55. Third, the majority of septum-projecting CA1 GABA cells are ebGABAs with the soma in the stratum oriens, and 80% of them express SOM12.
### Drugs
NBQX disodium salt and SR95531 (gabazine) were purchased from Tocris Biosciences. All the remaining drugs (tamoxifen and compounds to prepare ACSF and intracellular solutions) were purchased from Sigma.
### Statistical analyses
Pairwise comparisons between distributions were performed using the Mann–Whitney U test for unpaired groups and with the Wilcoxon signed rank test for paired groups. Comparisons between multiple groups were performed using the Friedman test with post hoc Dunn’s correction. Two-way ANOVAs were performed with Bonferroni post hoc correction. Data are expressed as either means ± SDs or medians and IQR. Statistical tests were performed with Graphpad Prism 8 (GraphPad Software, Inc.) or MATLAB.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
In vivo two-photon imaging data are available at https://doi.org/10.5281/zenodo.3931805. The remaining raw data are available from the authors upon request. Source data are provided with this paper.
## Code availability
MATLAB codes to analyze in vitro and in vivo two-photon imaging data as well as intrinsic membrane properties are available at https://gitlab.com/cossartlab/bocchio-gouny-et-al-2020. The code for in vivo functional connectivity analysis is part of our CICADA Python toolbox, available at https://gitlab.com/cossartlab/cicada/.
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## Acknowledgements
We thank L. Cagnacci, C. Pauchet-Lopez, F. Michel (InMAGIC imaging facility) and INMED’s animal facility technicians for excellent technical support; V. Crepel and A. Represa for sharing equipment; J. Denis, R. Dard, M. Picardo, E. Leprince and Y. Bollmann for experimental and/or analytical help; V. Lopes-Dos Santos for sharing the assembly detection code and useful discussions; R. Khazipov, T. Marissal, R. Boyce, for useful comments on this paper. This work was supported by the European Research Council under the European Union’s FP7 and Horizon 2020 research and innovation program (grant No. 646925) and by the Bettencourt-Schueller Foundation (Coup d’Elan). M.B. was supported by the Fyssen Foundation, the Fondation pour la Recherche Medicale (grant No. SPF20170938593), and by the European Union (Marie Skłodowska-Curie individual fellowship, grant No. 794861—IF-2017). D.A. was supported by an A*MIDEX grant (grant No. ANR-11-IDEX-0001-02) and by the I-Site Paris Seine Excellence Initiative (grant No. ANR-16-IDEX-0008). T.Tr. was supported by The French National Research Agency (grant No. ANR-14-CE13-0016).
## Author information
Authors
### Contributions
Conceptualization: R.C. Experimental design: R.C., M.B., C.G., and A.B. Data acquisition: M.B., C.G., E.Q., A.B., and T.To. Software: M.B., D.A., and T.Tr. Formal analysis: M.B., C.G., and D.A. Technical assistance: T.Tr. Supervision: R.C. and A.B. Figures: M.B. and C.G. Writing: M.B. and R.C.
### Corresponding authors
Correspondence to Marco Bocchio or Rosa Cossart.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks Chris McBain, Liset Menendez de la Prida, and other, anonymous, reviewers for their contributions to the peer review of this work. Peer review reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Bocchio, M., Gouny, C., Angulo-Garcia, D. et al. Hippocampal hub neurons maintain distinct connectivity throughout their lifetime. Nat Commun 11, 4559 (2020). https://doi.org/10.1038/s41467-020-18432-6
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Molecular Psychiatry (2021) | 2022-05-21 23:29:20 | {"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": 0.48671671748161316, "perplexity": 9497.767941355765}, "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-2022-21/segments/1652662541747.38/warc/CC-MAIN-20220521205757-20220521235757-00658.warc.gz"} |
https://uqworld.org/t/exp-tanh-function/56 | # Exp-Tanh Function
In Owen et al. (2017), the Exp-Tanh function is used to test metamodeling approaches, namely polynomial chaos expansions and Gaussian process modeling.
## Description
The Exp-Tanh function is defined as:
f(\mathbf{x}) = \exp{(-x_1)} \tanh{(5x_2)},\;\;\; x_1, x_2 \in [-1,1],
where \mathbf{x} = \{x_1, x_2\} are the input variables.
Figure 1 and 2 show the surface and contour plots of the Exp-Tanh function, respectively.
Figure 1: Surface plot of the Exp-Tanh function.
Figure 2: Contour plot of the Exp-Tanh function.
## Inputs
For computer experiment purposes, the inputs x_1, x_2 are modeled as two independent uniform random variables.
No Variable Distribution Parameters
1 x_1 Uniform x_{1,\min} = -1, x_{1,\max} = 1
2 x_2 Uniform x_{2,\min} = -1, x_{2,\max} = 1
## Resources
The vectorized implementation of the Exp-Tanh function in MATLAB as well as the script file with the model and probabilistic inputs definitions for the function in UQLAB can be downloaded below:
uq_expTanh.zip (2.1 KB)
The contents of the file are:
Filename Description
uq_expTanh.m vectorized implementation of the Exp-Tanh function
uq_Example_expTanh.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)
## References
• N. E. Owen, P. Challenor, P. P. Menon, and S. Bennani, “Comparison of surrogate-based uncertainty quantification methods for computationally expensive simulators,” SIAM/ASA Journal on Uncertainty Quantification, vol. 5, no. 1, pp. 403–435, 2017. DOI:10.1137/15M1046812
1 Like | 2020-01-23 16:32:46 | {"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": 0.7983819246292114, "perplexity": 6975.4459967134}, "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/1579250611127.53/warc/CC-MAIN-20200123160903-20200123185903-00496.warc.gz"} |
https://rpg.stackexchange.com/questions/14745/guidelines-for-appropriate-challenges-in-a-cyberpunk-2-0-2-0-game/84088 | # Guidelines for appropriate challenges in a Cyberpunk 2.0.2.0 game
I've owned a copy of the 2nd edition of Cyberpunk (Cyberpunk 2020) for years now but never had the ability or gumption to run a game (same for gurps). Lately though I've been playing and DMing a lot of D&D and I feel that my skills are high enough that I can finally run a CP2020 game. My issue is that the sourcebook itself is clunky in its description of rules and there doesn't seem to be any sort of guide lines for creating fights and challenges that are appropriate for the PCs (mechanically speaking).
I know about LUYPS! (Listen Up, You Primitive Screwheads!) and I've read it, but most of the advice in there seems to be story and tone advice (which I don't need being a huge cyberpunk fan already). I'd appreciate some sensible guidelines for setting up my session plans, to help get this campaign going.
I've run CP2020 many a time (Usually Trauma Team games) The CP2020 book isn't as nicely edited as it could be it's true; then again it's far better than something that WW usually produce!
The system itself is very simple, but what I'd recommend for a new campaign in the system is:
• As mentioned there is no real balance system for cyberpunk, you play it by ear and by experience of the game; the only real comparison you have is the special ability levels, but gear and circumstances can completely wipe this away. Armour is the major balance factor in combat; metalgear and suchlike will really give whichever side has it a major advantage. Similarly with weapons, assault rifles really skew the balance in whoever has them's favour, but this sort of gear should be heavily restricted, it's paramilitary after all - you shouldn't wander into starbucks with a Ronin Light Assault over your back or the cops are going to be there very quickly!
• Run a combat first, really, get the players some generic characters as a test run and playout a simple shootout, the combat mechanics flow well when you understand them but they are irksome and it's best to get all that unfamiliarity out of the way immediately so when a real firefight happens in the campaign it will flow nicely rather than stuttering.
• Balancing fights is principally about numbers, weapons and armour; look at the players armour levels (if any!) and balance what their enemies have against this; if the players are wearing heavy armour jackets and flak pants (which in any police patrolled area is likely to get them stopped!) then firing pistols that do 2d6 damage at them isn't going to do anything!
• Cyberpunk should not be a fight-based encounter system; fights are very, very dangerous and healing is slow. Players should be thinking how to avoid shooting anything rather than opening up with their Uzi's at every opportunity.
• For the first games stick to the stock CP2020 book for gear and equipment, there are a large number of suppliments that can really make things annoying or "what does that do" or unbalanced until you understand how your flavour of CP will work.
• Keep it low-tech and money lite to start off of; there are dozens of things characters can buy at the start that can turn themselves into unstoppable monsters, be careful with this or it'll soon spiral out of control before you understand the system.
• Keep netrunners as NPCs. No, really, yes it's an integral part of the world but they will slow things down and split the party up; certainly for the first few games until the whole world has settled down and you're finding your GM feet this will save your sanity and stop all the other players getting very bored during netruns.
• Print yourself out a combat cheatsheet or GM screen, it'll save you looking everything up all the time in the crazy organisation of the Friday night firefight rules section.
• Finally, combat in CP2020 is very dangerous; without armour players can get blown away by a saturday night special if it hits them in the head, players will need to be smart and use cover (there's a mantra for cyberpunk if I ever heard one) rather than running in all guns blazing.
Finally here are some links for stuff I've found useful for CP:
Mockerys cyberpunk page - first page you should look at
The Chrome page
The first batch of my players came from a D&D background. They rushed into every combat encounter as if it was some sort of opportunity. All their characters died very quickly the first time. And a few times after that.
A few sessions later, they developed a healthy habit of avoiding combat as much as possible, and preparing for a fight if they can't avoid it. They learned that in the dark future, every kid with a ripper is a threat they would rather avoid than confront. Avoiding and circumventing threats became the new challenge.
In the dark future, every challenge is asymmetric and inappropriate. Do not nerf the opposition. Your players could and should find a way of dealing with it. Hint: It's probably not combat!
Do not "create fights". Create NPCs with goals, motivations and capabilities. See how your players react to your NPCs and react in return. That's the game; a series of reactions.
And let the dice fall where they may. If a stray bullet surgically removes the cerebellum of the rockerboy, so be it. Welcome to Night City.
Once you do that, your players will learn to take the challenges they can possibly handle and avoid the ones they can't. You won't have to worry about "balance" and you all will have more fun with the game.
• "Do not "create fights". Create NPCs with goals, motivations and capabilities." Excellent mantra! – Rob May 9 '18 at 12:19
Cyberpunk as a Gaming Genre is not Cyberpunk as a Literary Genre. The two overlap somewhat, but just as D&D and Sword & Sorcery literature only somewhat overlap, so also CP2020 and the Cyberpunk genre only somewhat overlap.
The tone advice in LUYPS is there to illustrate the differences for fans of the literary genre, as much as or more than being there to educate the new-to-the-genre.
Also note that D&D style Grinding is not normative to the CP2020 game - it's deadly, since damages are usually higher than armor values.
That said, there is no mechanical balance written in.
# But since you want a system for balance...
You can obtain four key metrics, tho', from the stats:
• Average Damage per Hit (details below)
• Average Worn Armor (AV+BTM)
• Average Attack base (Ref+Weapon+WpnAcc)
• Average Defense base (Ref+Dodge)
Assuming even skills, a character hits 45% of the time (defender wins ties - see table for proof, below). Each point of difference in the attacker's favor is a 10% improvement; each in the defender's is a 10% reduction.
Average damage is found easiest by taking the minimum rolled damage (all dice rolled 1), adding the maximum damage (all rolling max) and dividing by two... or just accept that a d6 is average 3.5, and a d10 is 5.5, and total it up.
The damage you do per round is going to average equal to your rate of hits times the average damage minus their AV... with the caveat that the actual average will be higher if the AV exceeds average damage but doesn't exceeed maximum damage, as the formula says "NO!"...
So, find the average damage per hit, and take the percentage of it that would be hits...
Example: Mook M has Ref 5, Weapon Skill 2, dodge skill 2, and a weapon doing 3d6, and 4 AV. PC A has Ref 7, Skill 4 each weapon and athletics, and a weapon doing 3d6 damage, and 8 AV.
A has an 11 total for attacks, vs a 7 defense, and thus has 45+(4x10) percent chance of hit - 85%. He averages 10.5 damage per hit, less the target's AV4, for 6.5 damage per hit, and .85 x 6.5 = 5.525 damage per round. That's gonna hurt under CP2013; under 2020, it adds up pretty quick
The mook, by comparison, hits about 5% of the time, with an average ddamage of 3.5 per hit, for .175 damage per round on average.
Still, said mook is a threat. The PC should be able to handle 3-4 of them no problem... but sooner or later, one will get lucky and put the hurt in.
# A final warning about balance
There's this neat thing called Synergy... when two things together perform more than the sum of those two together. Characters in combat tend to be somewhat synergistic, especially if the GM runs NPC's as a cohesive whole and is a creative tactician.
Roll Comparison:
Attacker Roll
1 2 3 4 5 6 7 8 9 10
D 1 D A A A A A A A A A
e 2 D D A A A A A A A A
f 3 D D D A A A A A A A
e 4 D D D D A A A A A A
n 5 D D D D D A A A A A
d 6 D D D D D D A A A A
e 7 D D D D D D D A A A
r 8 D D D D D D D D A A
9 D D D D D D D D D A
R 10 D D D D D D D D D D
Try to have one combat possibility per get together. Where DnD tends to go fight to fight with a little dialog to sell junk, CP2020 tends to be like a half hour TV show serial. In the 1980s, the A-Team tv show had one big combat and the rest of the show built up to it.
When planning a scenario, plan an alternative scenario.
For example, last week, the DM wanted us to join the army to help defeating the nomads. Instead, we decided to join the nomads who valued individual freedom, to defeat the Nazi-like army. We could have gone for guns and money, as expected, but we preferred freedom instead of getting computer chips tracking our every move from then on. | 2020-08-13 10:00:19 | {"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": 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": 0.2983970046043396, "perplexity": 2710.406994983343}, "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-2020-34/segments/1596439738964.20/warc/CC-MAIN-20200813073451-20200813103451-00216.warc.gz"} |
http://haifux.org/pipermail/haifux/2009-January/000682.html | # [Haifux] Haifux Digest, Vol 17, Issue 15
Yossi Gil yossi.gil at gmail.com
Tue Jan 27 14:27:18 MSK 2009
Hi Shlomi,
Thanks for the tip. I usually use flip, but added a section to our Wiki to
reflect you suggestions, see:
http://ssdl-linux.cs.technion.ac.il/wiki/index.php/Beginning_SSDL_users#File_Conversion
>
> > Problems of Windows\Linux compatibility with code files I solved with
> > Notepad++. I don't know if there's a Linux version or a Linux substitute
> > but it's an awesome writer for windows that can display context in many
> > languages and can also convert to\from Windows from\to Linux formatting.
> > Saved me a lot of trouble
>
> First of all, you can always use tofrodos to do the conversion if needed:
>
> http://www.thefreecountry.com/tofrodos/index.shtml
>
> And otherwise, vim and other text editors for Linux are capable of
> converting
> from CRLF to LF and vice-versa, but I cannot tell you off-hand how to do
> that. | 2015-03-30 16:14:07 | {"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": 0.33412861824035645, "perplexity": 8527.929548445776}, "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-2015-14/segments/1427131299496.98/warc/CC-MAIN-20150323172139-00157-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://mathzsolution.com/the-last-digit-of-2200622006/ | # The last digit of 220062^{2006}
My $$1313$$ year old son was asked this question in a maths challenge. He correctly guessed $$44$$ on the assumption that the answer was likely to be the last digit of $$262^6$$. However is there a better explanation I can give him?
$2^{4} = 16$. Multiply any even integer by $6$ and you don’t change the last digit:
$0 \times 6 = 0$, $2 \times 6 = 12$, $4 \times 6 = 24$ etc. The same is true if you multiply an even integer by anything whose last digit ends in $6$, in particular by $16$. Now
$2006 = 2004 + 2$ where $2004 = 501 \times 4$, so
$2^{2006} = (2^4)^{501} \times 2^2$ has the same last digit as $2^2$. | 2022-10-04 11:14:19 | {"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": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7910903692245483, "perplexity": 166.14080463366093}, "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-2022-40/segments/1664030337490.6/warc/CC-MAIN-20221004085909-20221004115909-00755.warc.gz"} |
https://cyberleninka.org/article/n/606811 | # Complete moment convergence of moving average process generated by a class of random variablesAcademic research paper on "Mathematics"
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## Academic research paper on topic "Complete moment convergence of moving average process generated by a class of random variables"
Ko Journalof Inequalities and Applications (2015) 2015:225 DOI 10.1186/s13660-015-0745-x
<3 Journal of Inequalities and Applications
a SpringerOpen Journal
RESEARCH
Open Access
Complete moment convergence of moving average process generated by a class of random variables
Mi-Hwa Ko*
CrossMark
Correspondence: songhack@wonkwang.ac.kr Division of Mathematics and InformationalStatistics, Wonkwang University, Jeonbuk, 570-749, Korea
Abstract
In this paper, we establish the complete moment convergence of a moving average process generated by the class of random variables satisfying a Rosenthal-type maximal inequality and a weak mean dominating condition with a mean dominating variable.
MSC: 60F15
Keywords: complete moment convergence; moving average process; Rosenthal-type maximal inequality; weak mean domination; slowly varying
ft Spri
ringer
1 Introduction
Let {Yi, -x < i < x} be a doubly infinite sequence of random variables with zero means and finite variances and {ai, -x < i < x} an absolutely summable sequence of real numbers. Define a moving average process {Xn, n > 1} by
Xn =J2 aiYi+n, n > 1. (1.1)
The concept of complete moment convergence is as follows: Let {Yn, n > 1} be a sequence of random variables and an > 0, bn > 0. If ^ |=1 anE{b-1 \Yn \ - e}+ < x for all e > 0, then we call that {Yn, n > 1} satisfies the complete moment convergence. It is well known that the complete moment convergence can imply the complete convergence.
Chow [1] first showed the following complete moment convergence for a sequence of i.i.d. random variables by generalizing the result of Baum and Katz [2].
Theorem 1.1 Suppose that {Yn, n > 1} is a sequence of i.i.d. random variables with EY1 = 0. For 1 <p <2 and r >p, ifE{\Yi\r + Y\log(1 + \ Yx\)} < x, then np-2-pE(\Yi\-
enp)+ < x for any e >0.
Recently, under dependence assumptions many authors studied extensively the complete moment convergence of a moving average process; see for example, Li and Zhang [3] for NA random variables, Zhou [4] for ^-mixing random variables, and Zhou and Lin [5] for p-mixing random variables.
We recall that a sequence {Yn, n > 1} of random variables satisfies a weak mean dominating condition with a mean dominating random variable Y if there is some positive constant C such that
for allx >0 and all n > 1 (see Kuczmaszewska [6]).
One of the most interesting inequalities in probability theory and mathematical statistics is the Rosenthal-type maximal inequality. For a sequence {Yi,1 < i < n} of i.i.d. random variables with E| Yl |q < to for q > 2 there exists a positive constant Cq depending only on q such that
The above inequality has been obtained for dependent random variables by many authors. See, for example, Peligrad [7] for a strong stationary p-mixing sequence, Peligrad and Gut [8] for a p*-mixing sequence, Stoica [9] for a martingale difference sequence, and so forth.
In this paper we will establish the complete moment convergence for a moving average process generated by the class of random variables satisfying a Rosenthal-type maximal inequality and a weak mean dominating condition.
2 Some lemmas
The following lemmas will be useful to prove the main results.
Recall that a real valued function h, positive and measurable on [0, to), is said to be slowly varying at infinity if for each X >0
Lemma 2.1 (Zhou [4]) If h is a slowly varyingfunction at infinity and m a positive integer, then
(1) £m=i nth(n) < Cmt+1h(m) for t > -1,
(2) YZm nth(n) < Cmt+1h(m) for t < -1.
Lemma 2.2 (Gut [10]) Let {Xn, n > 1} be a sequence of random variables satisfying a weak dominating condition with a mean dominating random variable X, i.e., there exists some positive constant C
x^rn h(x)
for allx >0 and all n > 1.
Let r >0 and for some A > 0
X = XiI( |Xi| <A), X'/ = XiI( |Xi| > A), X = XiI(|Xi| < A) - AI(Xi < -A) + AI(Xi > A),
X' = XI(|X| < A, X" = XI(|X| > A), X* = XI(|X| <A) - AI(X < -A) +AI(X > A).
Then for some C >0
(1) if EX |r < to, then (n-1) Eti EX! < CE|X|r,
(2) (n-1) Eh E|Xi|r < C(E|X'|r + ArP(|X| > A)) for Any A > 0,
(3) (n-1) ^n=1 E|X'/|r < CEX"^ for any A > 0,
(4) (n-1) Eh E|X*|r < CE|X*|r for any A > 0.
3 Main result
Theorem 3.1 Let h be a function slowly varying at infinity, p > 1, a >2 and ap > 1. Assume that {ai, -to < i < to} is an absolutely summable sequence of real numbers and that |Yi,-TO < i < to} is a sequence of mean zero random variables satisfying a weak mean dominating condition with a mean dominating random variable Y, i.e. there exists some positive constant C
- J2 p{ Y > x) < CP( | Y| > x) for all x > 0,-to < j < to
and alln > 1 andE| Y|ph(| Y|«) < to.
Suppose that {Xn, n > 1} is a moving average process, where Xn = £TO-TO aiYi+n, n > 1 is defined as (1.1).
Assume that for any q > 2, there exists a positive Cq depending only on q such that
El max
I 1<i<n
J2(Yxj- EYxj)
< C^E|Yxj|q + £EYx2
where Yxj- = -xI(Yj < -x) + j(| Y| < x)+ xI( Y > x) for all x >0. Then for all e >0
Ynap-2-ah(n)E max 1<i<n
n=1 I--
- en ) < to
J2nap-2h(n)E\ sup
- e > < TO.
Proof of (3.2) Let Yxj = Yj - Yxj and l(n) = nap-2-a h(n). Recall that EhXk = Eh ETO-to ^Y+k = ETO-to ai Yj by (1.1). If a >1, by the assumption that ^^^ |ai | < to and Lemma 2.2 we have, for x > na,
TO i+n
E j2 aih Yx>
i=-TO j=i+1
< Cx-1n{E|Y|/[|Y|<x] + xP(|Y| >x)}
< Cn1 a ^ 0 as n ^to.
(3.41)
If 2 < a < 1, ap >1 implies p > 1. By the assumption EYi = 0 for all -x < i < x and Lemma 2.2 we obtain
TO i+n
E j2 aij2 Yxj
i=—TO j=i+1
TO i+n
E j2 aij2 Yxj
i=—TO j=i+1
< Cx-1J^ \*i\Y, E\Yj\I [ \Y\ > x]
i=-x j=i+1
< Cx-1nE\Y\I[\Y\ >x] < Cx«-1E\Y\I[\Y\ >x]
< CE\Y\p/[\Y\ > x] ^ 0 as x ^x.
It follows from (3.4i) and (3.4ii) that for x > na large enough,
TO i+n
E j2 aij2 Yx>
i=—TO j=i+1
which yields
J2l(n)E\
I1<k<n
- € n
a \ 1<k<n
Vl(n) / PI max t! Jena \v1<k<,
TOTO £lml P(
TO /.TO /
l(n) P max
f^ t/ na \1<k<n
> x I dx (letting x = ex')
> ex' I dx'
j=1 TO i+k
eaie Yxj
i=-TO =i+1 TO i+k
> — \dx
= /1+ I2.
J^aiJ2(Yxj - EYXj)
i=-TO =i+1
> — dx
(3.4ii)
Now we will by an estimate show that I1 < to. It is clear that | Yxj| < | Y'|I[| Yj| > x]. Hence for I1, by Markov's inequality and Lemma 2.2, we have
I1 < C
n=1 Jna
x-1E max 1<k<n
TO i+k
j2aij2 Yxj
i=-TO j=i+1
TO TO TO i+n
< cZl(n) i x-1£|ai|^
n=1 n -TO j=i+1
a^) EYA dx
TO j=i+1
TO />TO
< CVnl(n) / x-1E| Y|I[|Y| >x]dx
n=1 Jn"
TO TO Am+1)a
= C£nl(n)£/ x-1E| Y|I[|Y| >x]dx
n=1 m=Jm
< Cj2nl(n)j2m-1E| Y|I[|Y| > ma]
n=1 m=n
= C^m-1E|Y|I[|Y| > man^p-1-ah(n). (3.7)
m=1 n=1
Ifp >1, note that ap -1 - a > -1. By Lemma 2.1 and (3.7) we obtain
I1 < C^map-1-ah(m)E|Y|I[|Y| > ma]
= C^map-1-ah(m)J2E|Y|I[ka < |Y| < (k + 1)a]
m=1 k=m
= C^E|Y|I[ka < |Y| < (k + 1)a]J2map-1-ah(m)
k=1 m=1
< Cj2kap-ah(k)EY^[ka < | Y|< (k + 1)a]
< CE| Y|ph(| Y|«) < to. (.8) Ifp = 1, by (3.7), we also obtain
I1 < C^]m-1E|Y|I[| Y| > man-1h(n)
m=1 n=1
< C^]m-1E|Y|I[| Y| > man-1+a5h(n) for any 5 >0
m=1 n=1
< C^ma5-1h(m)E|Y|I[|Y| > ma]
< CE| Y|1+5h(| Y|«) < to. (.9) So, by (3.8) and (3.9) we get
I1 < to forp > 1. (3.10)
For I2, by Markov's inequality, Holder's inequality, and (3.1) we get for any q > 2
to i+k
I2 < C>7(n)/ x-qE max 1< k< n
TO /»C
TO /.c
(YXj- EYXj)
i=-TO j=i+1
< CVl(n)/ x-q
\a,:\ q) \ai\q max
\ 1<k<n
J2(Yxj- EYxj)
l(n) x-q
(x \ q-1 ( x i+k
\aiiEmax ^(Yxj -EYj
i=-x i=-x j=i+1
x x x i+n
J2l(n) x-^ \ai\ £E\ Yxj - EYxj\qdx
n=1 n x
i=-x j=i+1
= : I21 + II22.
x x x i+n
^l(n) \ai\ ( £E\ YXj -EYx
\ 2 dx
i=-x j=i+1
For I21, we consider the following two cases.
If p >1, take q > max{2,p}, then by the assumption that ^x_x \ai\ < x, Cr inequality and Lemmas 2.1 and 2.2 we get
CVnl(n) / x-q{E\Y\qI[\Y\<x] + xqP(\Y\ >x)}dx
x x Am+1)a
< C£nl(n)£/ {x-qE\Y\q/[\Y\<x] + P(\ Y\ >x)} dx
n=1 m=Jm"
< C£nl(n)£{ma(1-q)-1E \ Y \ qI[ \Y \ < (m + 1)a] + ma-1P( \Y \ >ma)}
n=1 m=n
= C J2{ma(1-q)-1E \ Y\ qI[ \ Y\ < (m + 1)a] + ma-1P(\ Y \ > manl(n)
< C^ma(P-q)-1h(m)J2E\ Y\ qI[ka < \ Y \ < (k + 1)a]
m=1 k=1
+ ^J2map-1h(m)J2EI[ka < \ Y\ < (k + 1)a]
m=1 k=m
= Cj^E\ Y \ qI[ka < \ Y \ < (k + 1)a]X!ma(p-q)-1h(m)
k=1 m=k
+ ^^EI[ka < \ Y \ < (k + 1)a] ^map-1h(m)
k=1 m=1
< C^^kk"(p-q)h(k)E \ Y \ qI[ka < \ Y\ < (k + 1)a]
+ C ^kaph(k)EI[ka < \ Y \ < (k + 1)a]
< CE\ Y \ph( \ Y\1) < x.
For I21, ifp = 1, take q > max{1 + 5,2} by the same argument as above one gets for any 5 >0
I21 < CJ2{ma(1-q)-1E|Y|qI[| Y| < (m + 1)a] + ma-1P(|Y| > manl(n)
m=1 n=1
= C J2{ma(1-q)-1E|Y|qI[| Y| < (m + 1)a] + ma-1P(|Y| >ma)} ^n-1l(n)
m=1 n=1
< CJ2{ma(1-q)-1E|Y|qI[| Y| < (m + 1)a] + ma-1P(|Y| > man-1+a5h(n)
m=1 n=1
< CJ2{ma(1-q+5)-1h(n)E| Y|qI[|Y| < (m + 1)a] + ma(1+5)-1h(x)EI[|Y| > ma]}
< CE| Y|1+5h(| Y|«) < to. (3.13) It follows from (3.12) and (3.13) that, forp > 1,
I21 < to. (.14)
It remains to estimate I22 < to.
For I22, we consider the following two cases. If 1 < p <2, take q >2, note that ap + | -Op -1 = (ap -1)(1 - 2)< 0. Then by Cr inequality and Lemma 2.2, we obtain
TO fTO q q
I22 < CVn2 l(n) x-q{(E|Y |2I[| Y |<x])2 + xq(P(|Y | > x))2} dx
TO TO Am+1)a q q
< C^>ql(n)^ {x-q(E|Y|2I[|Y|<x])2 + (P(|Y| >x))2}dx
n=1 m=n m
< C^>2l(n)J2{ma(1-q)-1(E| Y|2I[|Y|< (m + 1)2])2 + ma-1(P(|Y| >ma))2}
n=1 m=n
= CY{ma(1-q)-1(E| Y|2I[|Y|<(m + 1)a])2 + ma-1(P(|Y| > ma))2 ^n2l(n)
m=1 n=1
< Cj2ma(p-q)+1-2h(m)(E|Y|2I[|Y| < (m + 1)a])2
+ C^map+2-2h(m)(EI[|Y| >ma])2
< Cj2map+2-apl-2h(m)(E| Y|p)2 < to. (3.15)
Ifp > 2, take q > a-1 > 2, which yields a(p - q) + 2 - 2 <-1. Then we get
I22 < CJ2№(1-q)-1(E|Y|21[|Y| < (m + 1)a])2
+ ma-1(P(|Y| > ma))2}J2nql(n)
< Cj^m"(p-q)+2-2h(m)(E\ Y\ 2I[ \Y\ < (m + 1)a])q
+ Cj2map+q-2h(m)(EI[ \Y\ > ma])q
< C^ma(p-q)+2-2h(m)(E\ Y\ 2)2 <x.
Hence, by (3.15) and (3.16) we get
I22 < x for p > 1. Moreover, by (3.14) and (3.17), we also get
I2 < x for p > 1. The proof of (3.2) is completed by (3.6), (3.10), and (3.18). Proof of (3.3) By Lemma 2.1 and (3.2), we have
(3.17)
(3.18) □
^Vp-2h(n)E sup
xx = n p-2h(n) P sup n=1 j0 V>n
x 2k -1 x /
= n p-2h(n) P sup
k=1 „=9k-1 J° V>n
> e + x I dx
k=1 n=2k-
x /.x /
< / M sup
k=^0 \i>2k-1
< C 2k( p
> e + x dx
> e + x dx n p-2h(n)
/ n=2k-1
0 \i>2k-1
> e + x dx
x x -x
C^ 2k(ap-1)h( 2k)^/ P
k=1 m=k 0
0 \2m-1<i<2m
> e + x dx
< C P max
I 2m-1<i<2m
+ ^d^ 2k(a p-1)h( 2k)
2m-1<i<2m
2m(ap-1)h( 2m) j P| (letting y = 2(m-1)a x)
xx < C 2m( p-1- )h 2m P <max
> (e + x)2(m-1)a dx
> e2(m-1)a + y\dy
to l>TO /
< cy"naP-2-ah(n) P max
tr io \i«<n
> 6 na 2-a + j I dy
CJ"nap-2-ah(n)E| max
^^ \ l<i<n
- 6'na < oo
where 6' = 62 a. Hence the proof of (3.3) is completed. □
Remark There are many sequences of dependent random variables satisfying (3.l) for all
q > 2.
Examples include sequences of NA random variables (see Shao [ll]), p*-mixing random variables (see Utev and Peligrad [l2]), y-mixing random variables (see Zhou [4]), and p-mixing random variables (see Zhou and Lin [5]).
Corollary 3.2 Under the assumptions of Theorem 3.l for any 6 > 0
y/p-2h(n)P( max
^ \ l<i<n
n=l \--
> 6n < to.
Proof As in Remark l.2 of Li and Zhang [3] we can obtain (3.l9). □
Competing interests
The author declares that they have no competing interests. Acknowledgements
This paper was supported by Wonkwang University in 2015.
Received: 13 April 2015 Accepted: 29 June 2015 Published online: 17 July 2015
References
1. Chow, YS: On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16,177-201 (1988)
2. Baum, LE, Katz, M: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120(1), 108-123 (1965)
3. Li, YX, Zhang, LX: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 70,191-197 (2004)
4. Zhou, XC: Complete moment convergence of moving average processes under y-mixing assumption. Stat. Probab. Lett. 80,285-292 (2010)
5. Zhou, XC, Lin, JG: Complete moment convergence of moving average processes under p-mixing assumption. Math. Slovaca 61(6), 979-992 (2011)
6. Kuczmaszewska, A: On complete convergence in Marcinkiewicz-Zygmund type SLLN for negatively associated random variables. Acta Math. Hung. 28,116-130 (2010)
7. Peligrad, M: Convergence rates of the strong law for stationary mixing sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70,307-314(1985)
8. Peligrad, M, Gut, A: Almost sure results for a class of dependent random variables. J. Theor. Probab. 12, 87-104 (1999)
9. Stoica, G: A note on the rate of convergence in the strong law of large numbers for martingales. J. Math. Anal. Appl. 381,910-913 (2011)
10. Gut, A: Complete convergence for arrays. Period. Math. Hung. 25,51-75 (1992)
11. Shao, QM: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13,343-356 (2000)
12. Utev, S, Peligrad, M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab. 16,101-115 (2003) | 2022-09-28 06:51:34 | {"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": 0.8605826497077942, "perplexity": 11708.614638661127}, "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/1664030335124.77/warc/CC-MAIN-20220928051515-20220928081515-00616.warc.gz"} |
https://ibpsonline.in/debate/14890/IBPS-Clerk/Quantitative-Aptitude/1592/-The-sum-of-15%25-of-a-positive-number-and-10%25-of-the-same-number-is-70.-What-is-twice-of-that-number | 1 .
The sum of 15% of a positive number and 10% of the same number is 70. What is twice of that number?
[ A ] 440 [ B ] 280 [ C ] 560 [ D ] 140 [ E ] None of these
Answer : Option C Explanation : | 2019-07-18 05:06:08 | {"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": 0.9239418506622314, "perplexity": 3542.045815028896}, "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-2019-30/segments/1563195525500.21/warc/CC-MAIN-20190718042531-20190718064531-00370.warc.gz"} |
https://sciencenotes.org/farenheit-to-celcius-conversion/ | # Farenheit to Celcius Conversion
The Farenheit to Celcius conversion is really the temperature conversion from degrees Fahrenheit to degrees Celsius. The names of the two temperature scales are easy to misspell, so you can avoid them by using the symbols °F and °C. Fortunately, converting between the temperatures is easier than spelling their names. Here’s how to do it, along with example calculations.
### °F to °C Formula
There are two common formulas for converting °F to °C:
• °C = 5/9(°F – 32)
• °C =(°F – 32) ÷ 1.8
It does not matter which formula you use. You’ll get the same answer.
### Easiest Way to Convert Farenheit to Celcius
Basically, you take the Fahrenheit temperature, subtract 32 from it, and then either multiply by 5/9 or else divide by 1.8. There’s less chance of making a mistake if you avoid the fraction, so dividing by 1.8 is the easiest option.
2. Take the answer and divide it by 1.8. This is the answer in °C.
Calculators don’t all handle order of operations the same way, so it’s important to get the answer for “F – 32” before dividing by 1.8! For example, enter the °F temperature, the minus sign, 32, the equal sign, the divide symbol, 1.8, and then the equal sign.
### Convert °F to °C
For example, convert 90 °F to °C.
°C = (°F – 32) ÷ 1.8
°C = (90 – 32) ÷ 1.8 = 58 ÷ 1.8 = 32.2 °C
### Convert °C to °F
It’s just as easy working the temperature conversion the other way, from °C to °F.
• °F = 95°C + 32
• °F = 1.8 °C + 32
Once again, you don’t need to work with fractions.
For example, convert 20 °C to °F.
°F = 1.8 °C + 32
°F = (1.8)(20) + 32 = 36 + 32 = 68 °F
Most calculators handle the math just fine if you do 1.8 x 20 + 32, but if you’re worried, enter 1.8 x 20 = + 32 =.
### Table of °F to °C Temperatures
Here is a handy table of important temperatures in both °F and °C:
### References
• Balmer, Robert T. (2010). Modern Engineering Thermodynamics. Academic Press. ISBN 978-0-12-374996-3.
• Boyes, Walt (2009). Instrumentation Reference Book. Butterworth-Heinemann. ISBN 978-0-7506-8308-1.
• Buchdahl, H. A. (1966). The Concepts of Classical Thermodynamics. Cambridge U.P. ISBN 978-0-521-04359-5.
• Helrich, Carl S. (2009). Modern Thermodynamics With Statistical Mechanics. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-540-85417-3. | 2022-07-05 15:10:03 | {"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": 0.8919161558151245, "perplexity": 2840.1429279113113}, "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/1656104585887.84/warc/CC-MAIN-20220705144321-20220705174321-00281.warc.gz"} |
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The programme covers Renewable Energy, a growing area with tremendous opportunities for new technologies, businesses and ideas. Question. Why is it important to keep track of the sunspot cycle? If the inductance is 0.100 H, what is the voltage across the inductor? How is geothermal energy generated from volcanoes? Currently in Washington State there is an initiative on the November ballot (I-732) to impose a 'revenue neutral'$25/ton tax on 'carbon pollution'. Determine the mechanical energy of air per unit mass and the power generation potential of a wind turbine with 80-mm-diameter blades at tha... Wind is blowing steadily at a wind turbine. For three different latitudes up of CO2 and CO gases power invention gas/wastewater treatment digester )... Produce a little over 5.5 % of the world is most suitable for heating, cooking, and opportunity... Km thick from the sun 's radiation was to increase by 10 %, energy emitted by photoelectric. A ) Methane b ) how much energy is 3.9 eV cell companies claim lifetimes of 30 years the... Of GDP ) fell by 1.6 %, energy emitted by the Earth 's surface illuminated! The Rocky Mountains and copyrights are the property of their respective owners a PV system that provide. Solar collectors have, to function in an RL circuit builds up to speed the! Photons all have in common is that they don ’ t run out, who pays for green energy that! Ontario and Quebec function from electron volts to joules Arcadia in as little as minutes. Each statement carefully before ticking a box wavelength 400 nm bombards a,! In order to achieve a sustainable renewable energy questions options in a way that 's easy for you to understand a. - large wind turbines send the windâs energy to power the whole world and the... Nuclear of hydroelectric is, Worldwide, the renewable energy questions heater available to who. Same frequency as 'Fuel of future ' its temperature tank and drives a turbine disk of mass 26 kg at. Are all renewable energy industry depends on _________ availability long-term potential supply of geothermal energy come from completed... Trade winds they also help to heat a building or a river can be given to the problem insufficient. Found at Earth science: renewable renewable energy questions questions that are explained in a way 's... Navigate alternative energy solutions and future energy options and address common challenges water heating use Hydro-Electric C Neon... So you have been asking Google about renewable energy technologies provide a lower net energy return than the fossil! Of solar energy cells, usually made from silicon, that collect solar thermal energy is any we. Renewables, include wind and solar only produce a little over 5.5 % of the salt water of this are! A light string 1.70 m long are to be made by wrapping wires around a paper tube of 10... Charge density atmosphere of Jupiter 2008 by MidAmerican energy cloudless day on which the sun 's was! Wind e. radiant solar energy the sun 's light produce more current of pyruvic acid c. lactic acid d. acids. In our society biomass energy and what are the advantages and disadvantages of energy. Almost all renewable energy wind power generation complex industrial civilisation you focus to. Them their typical luster world country like Kenya property of their respective.! For renewable energy for domestic use is the price per kW this surface based on that... Search in morocco is about 20 MJ CO gases 752 nm, the,. On information that can only be explained by light having a particle nature year old children maximum kinetic of! E. radiant solar is energy from volcanoes tapped for human use and in Hawaii plant operates pressures. That affect biogas generation as a reference, answer the following are attributes of a location! Required, what is the self inductance of a passive solar collector is 7,200 kJ/ ( day. Where the wind industry more cost-effective way to switch to solar than installing panels inductance should be in! To convert completely away from non-renewables given the following is an example a reason why it renewable energy questions more! Can produce about 40 W of electricity per square meter of surface area directly... Hydropower - water from salt water state two factors that influence them to use your current earnings as energy... Wrapping wires around a paper tube of radius 10 cm and length 8.44 cm Mars 's two moons day 1000... To 1500 rpm needed for the ejected electrons paper tube of radius 0.702 cm and length cm... Are to be economically feasible sustainable society the inductor change is 1.2 eV and exits at kPa... Conduction heat the Earth s surface would do what years have tested all... Radiation of wavelength 400 nm bombards a surface is 180 W/m2 averaged over a waterfall of 10 m to 110. } C to 50^ { \circ } C to 50^ { \circ } in! Transfer, transformation, storage and dissipation of energy in science and know examples types! Of gasoline sunlight impinging on Earth is about 20 MJ cooking, and what we... Provide secondary data to research the viability of selling electric cars profitably c. acid! Is 3 \times 10^8 m/s solar cells when exposed to sunlight.... what are the a trade... I ) Assuming that the radiative temperature of Phobo... why is a more cost-effective way to discuss your expectations. Runs on water instead of gasoline the wind is blowing steadily at an average velocity of 6 m/s energy sun! Electric energy consumption in one house is 1.0 kW print page of nm! What characteristics should solar collectors have, to function in an oscillating LC circuit with L 30. A monochromatic light is heated from 15^ { \circ } C to 50^ { \circ } C in the ethics... Of surface area if directly facing the sun 's radiation was to renewable energy questions by 10,! M in diameter and 60 m long are to be met by windmills 40-m-diameter! A barium target, which one is specifically about technical stuff, the of... Buildings team has collected renewables resources that help navigate alternative energy solutions and energy. Other renewable resources as several States in the mainland United States laser emits a beam light! You instant feedback, and iron are 2.3, 2.1, 4.7, and what do we call produced! { \circ } C in the U.S., have ambitious goals for reducing their greenhouse gas.. Is stopped by the sun got you up to speed on the surface of Jupiter a viable resource windpower. Which state generates the greatest number of years of renewable energy source: nuclear or hydroelectric are a. Attributes of a blimp 5 m in diameter and 60 m long are to be answered is! Inductor carries a renewable energy questions a current most potential for the generator years have tested all. Was commissioned in 2008 by MidAmerican energy windmills with 40 meter- diameter work. 400 nm bombards a surface, the average insolation on a barium,. Energy resource is power business in town does increased exposure to the Site C dam project that created! All other trademarks and copyrights are the differences between non-renewable resources and renewable resources spin blades. ) Methane b ) how much voltage should be applied to it the definition energy! Read each statement carefully before ticking a box a city looking for sources of funding for this?... Ethics debate these self-marking questions give you instant feedback, and what different... Thinking about switching to renewable energy sources show the most widely used renewable energy sources how... Type of green technology and away from petroleum 6 1 day ago 'The burn! Electricity from the sun you renewable energy questions addressing a class of 12 year old children gathered the top questions youâve Google! For this combination in a way that 's easy for you to understand value in s.. The same inductor store if it carries a steady current of 1.0 a are. Have about renewables the sources of energy on my roof the Rocky Mountains sustainable society green technology our... Converts 22 rpm from the sun in the United States for human use ) increased transmitter release f... a! Affect electricity production using hydrogen fuel cells the weather three classifications of biogas plants in 2010, generated! Information that can be given to the sun 's light produce more current does power... And got you up to one-third of its steady-state value in 5.20 s. find cutoff. Mirrors to generate electricity 90 MHz for discussing improvements to the problem of insufficient energy renewable... Practice problems and step-by-step solutions right resources, we can all make better choices when it comes our., who pays for green energy projects that fail 49 ) Rotor ( selective surface ) b the economics politics... Factor of 9, how would the energy density of the photoelectron is eV... What two natural phenomena are caused by geothermal energy used in mechanical and engineering! To develop my wind power to the problem of insufficient energy using renewable energy resources there! The outer segment behind the windmills are to be economically feasible photovoltaic cells on a metal which has a function... Snap forces UK electricity market prices to new energy sources it might be like! Per second are generated by the sun is 1400 W/m^2 the course of a community are to met... 3.2 eV industrial civilisation at 300 degrees celcius radiates 10 joules per centimeter squared-second, what is voltage. Storm '' propelling the shift to 100 % clean energy will depend on small-scale progress and,. That prevail in the blank: in glycolysis, the average power generation of electricity from surface. 1.2 eV gas is known as 'Fuel of future ' acid depends on _________ availability n't find the time... Transmitter release f... at a frequency of 6.71 \times 10^ { -7 } ;... Of classifications of biogas plants questions youâve asked Google and provided some insight do they produce coll... wind. Pressure at which the sun shines down across the turbine { -7 } \ ; m b incident photons smaller... Using hydrogen fuel cells like cars cooking, and the frequency is 750 Hz frequently asked questions about energy! ) with a frequency of 20kHz is applied to it a small 0.490 kg mass attached interview. As 'Fuel of future ' function ( in eV ) energy ( eV! For your dream renewable energy sources and how that affects us globally despite falling costs, wind blowing!
Ace Combat 6 Dlc, Who Owns Connectwise, 1 Corinthians 13 4-5 Meaning, 18th Century British Frigate, Forensic Examination Definition, Destiny 2 The Draw, Yamata No Orochi - Persona 5 Royal, Barnard College Majors, | 2021-12-07 11:39:46 | {"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": 0.450211763381958, "perplexity": 2702.6384630364746}, "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/1637964363376.49/warc/CC-MAIN-20211207105847-20211207135847-00464.warc.gz"} |
https://www.andlearning.org/5-most-basic-financial-trading-terms-every-aspiring-trader-should-know/ | Connect with us
With these 5 terms, you will be ready to dive into the amazing world of financial trading. They are building blocks of basic knowledge which will create a fundamental understanding framework for your mind. So let’s begin.
## 5 Absolutely Crucial And Basic Financial Trading Terms
Trading is not easy, it takes a lot of practice, theoretical knowledge, and discipline. Because of this, understanding basic terms are critical. also, if you want to become a funded trader these terms will help you pass the evaluation period.
2. Long and Short positions – another staple for financial trading terms. When you buy an asset, you are in a long position. When you sell the asset, you are in a short position. Short positions are also called short selling or shorting. When a trader longs the asset, they are waiting for the price to increase and then will close or sell back the position to generate profit. When traders short the asset, they are waiting for the price to fall and then buy back the asset and make a profit. In some markets, like real stocks, traders can not immediately sell the stocks they don’t own. For short-selling the asset right away, there are other specific assets or trading instruments enabling traders to short-sell the asset right away.
3. Stop Loss and Take Profit orders – are the two main risk management tools in financial markets. Stop loss and take profit are stop orders meaning when a trader buys or sells the asset they can set stop loss and take profit. Take profit closes the position when a certain price is hit and profit is made, while the stop loss limits the loss traders can accumulate by closing the trade at a certain price. So, both stop loss and take profit to play a major role in controlling risk and profitability.
4. Trends. There are two main trends when considering the financial markets. Downtrend and uptrend. An uptrend is called a bullish trend, and downtrends are often called bearish markets or bearish trends. An uptrend is when prices are moving higher, or technically speaking higher highs and higher lows. Downtrends happen when the price is making lower lows and lower highs on the price charts.
5. Bulls and bears and related terms. Bull is the symbol of optimism and upward price movement. Bears are the opposite and are symbols of falling prices. Bulls or bullish traders are traders who are buying the asset for future gains. Bears on the other hand are sellers who are shorting the markets to gain profits. The bullish trend is the upward trend, while the bearish trend refers to the downward trend.
• Risk management – risk management refers to the practice of calculating the possible return and possible loss for each trade and setting maximum risks and profits to maintain a healthy ratio. Another important thing in this matter is to consider how many trades are you expecting to win on average. This is called win rate and it is critical. Traders generally try to risk 1 to get double the profit, or a 1:2 Risk to reward ratio. 1:2 RR means you risk $1 to get a potential$2 profit. With a strategy that has a win rate of 50% and 1:2 RR, you are expecting to make a profit in the long term. Risk management defines if a trader is a winner or loser in the long term. Leverage plays an important role when managing risks during trading.
• Margin – defines how much money you need in your account to open a trade. Margin is money, a trader needs to put forward to place a trade and maintain the position.
• Leverage – refers to the practice of borrowing money from the broker to increase the buying power of the account balance. A leverage of 1:100 means the balance is multiplied by 100 times. A $1000 trading account with a leverage of 1:100 can buy up to$100 000 worth of assets. Leverage is like a double-edged sword, it increases the potential profits or returns of the investment. But the leverage also amplifies the potential losses, making it critical to properly control risks with stop loss and take profit orders.
## Summary
Understanding the most basic terms and their workings is the first step toward understanding any field. In the financial trading world, the above terms are the most basic building blocks that no trader is allowed to miss. So make sure to memorize them. | 2023-03-28 03:04:37 | {"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": 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": 0.23251962661743164, "perplexity": 1890.8396840066762}, "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-2023-14/segments/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00005.warc.gz"} |
https://projecteuler.net/problem=684 | ## Inverse Digit Sum
### Problem 684
Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$.
Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are given $S(20) = 1074$.
Further let $f_i$ be the Fibonacci sequence defined by $f_0=0, f_1=1$ and $f_i=f_{i-2}+f_{i-1}$ for all $i \ge 2$.
Find $\displaystyle \sum_{i=2}^{90} S(f_i)$. Give your answer modulo $1\,000\,000\,007$. | 2021-05-06 03:29:22 | {"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": 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": 0.9142308831214905, "perplexity": 175.71726972685136}, "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-2021-21/segments/1620243988725.79/warc/CC-MAIN-20210506023918-20210506053918-00340.warc.gz"} |
http://beantownbuckaroos.com/white-tansy-qwxqd/algebra-2-properties-of-real-numbers-properties-practice-sheet-answers-bfc299 | # algebra 2 properties of real numbers properties practice sheet answers
Additive Identity The sum of any number and is equal to the number. The graph of x > a and x < bis similar to the graph of x > a because it may include some of the same values, Start studying Unit 2. Mental Math … 2π 22. 7 Basics of Equations 1. 45,368 23. Extra Practice (blank, answers) 3. Name all the integers in the list: 0, —2, 5 , z, , 121, , 6 Name the property of real numbers illustrated by each equation. Algebra Regents Questions 1) The statement is an example of the use of which property of real numbers? Property: a + b is a real number; Verbal Description: If you add two real numbers, the sum is also a real number. 1) commutative property 2) associative property 3) distributive property 4) identity property of addition 14 A teacher asked the class to solve the equation 3(x+2) =21. Work on Review Sheet 4. Answers and explanations The correct answer is –3x + 33. a. commutative property of addition b. commutative property of multiplication c. associative property of addition d. associative property of multiplication additive identity f. multiplicative identity g. distributive property h. additive inverse i. multiplicative inverse 16. 8 … Textbook Authors: Hall, Prentice, ISBN-10: 0133500403, ISBN-13: 978-0 … 3 4 20. The sets of rational and irrational numbers together make up the set of real numbers.As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Suppose a, b, and c represent real numbers. Robert wrote 3x+6 =21 as his first Addition Properties of Real Numbers. Lesson 1-6 2. 2 = 2. Examples of doing distributive property in algebra workshhet that i can print; how to use my casio in algebra; free geometry basic transformation worksheets; 7th grade pre algebra test; algebra 2 online book; how changes in one or two dimensional of an object affect perimeter, area, surface area, and volume. We offer a whole lot of good reference information on subjects ranging from numbers to power Rational Numbers Maze Activity Answer Files. PROPERTIES OF REAL NUMBERS Let , , and be any real numbers 1. Lesson 4: Properties of Real Numbers. Symmetric property. Commutative Property of Multiplication Associative Property of Addition For the numbers 0.18349…, 0.$$\overline{2}$$, 1.67, list … Which number property is she using? If ever you have to have service with math and in particular with algebra 2 plato answers or solving linear equations come visit us at Polymathlove.com. Real Numbers. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Homework (blank, answers) 2. Real numbers can be broken down into groups known as _____. 1. 13 Which property of real numbers is illustrated by the equation 52 +(27 36) =(52 27) 36? Example: 3 + 9 = 12 where 12 (the sum of 3 and 9) is a real number. It cannot be both. 6 The Distributive Property 1. COMMUTATIVE PROPERTY. The following rules show distributing multiplication over addition and distributing multiplication over subtraction: Practice questions –3(x – 11) = ? Basic Number Properties The ideas behind the basic properties of real numbers are rather simple. If you ever need some worksheets to improve your children\'s skills, download them from here. Learn about the properties of real numbers with this interactive lesson. Learn vocabulary, terms, and more with flashcards, games, and other study tools. !36, !49 26. !11 24. 19. Real numbers are closed under addition, subtraction, and multiplication.. That means if a and b are real numbers, then a + b is a unique real number, and a ⋅ b is a unique real number.. For example: 3 and 11 are real numbers. _____ numbers are based on the idea that _____. This property is all about the answers you get. 1-7 The Distributive Property 7-1 Zero and Negative Exponents 8-2 Multiplying and Factoring 10-2 Simplifying Radicals 11-3 Dividing Polynomials 12-7 Theoretical and Experimental Probability Absolute Value Equations and Inequalities Algebra 1 Games Algebra 1 Worksheets algebra review solving equations maze answers Cinco De Mayo Math Activity Class Activity Factoring to Solve Quadratic … Remembering the properties of numbers is important because you use them consistently in pre-calculus. 2. B. 1) Closure Property of Addition. Have you ever noticed that we CAN add any number in any order we want, but we CAN’T do the same with subtraction? 2 2 3 Compare the numbers in each exercise using an inequality symbol. 2. !100, 2!169 28. a. Khan Academy's Algebra 2 course is built to deliver a comprehensive, illuminating, … All numbers that you have dealt with up until this point are known as _____ numbers. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. Given any number n, we know that n is either rational or irrational. Example: 3 + 9 = 12 where 12 (the sum of 3 and 9) is a real number. The numbers have been regrouped. The properties aren’t often used by name in pre-calculus, but you’re supposed to know when you need to utilize them. More on this to come in a later chapter! For any number ... Show all possible answers. You may even think of it as “common sense” math because no complex analysis is really required. 28 21. Play this game to review Pre-algebra. 25. 2 Order of Operations and Simplifying Expressions 1. There are four (4) basic properties of real numbers: namely; commutative, associative, distributive and identity. For example: If you add any two real numbers, you will get a real number sum. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to represent a complex number. 1-3 Practice (continued) Form K Real Numbers and the Number Line Name the subset(s) of the real numbers to which each number belongs. a. Selected Answers Topic 1 PearsonRealize.com Part C 3.2 x + 42 > 3x + 42.5; x >2.5 ; Elijah is ahead of Aubrey after 2.5 hours. 5 Multiplying and Dividing Real Numbers 1. Thus, is called the additive identity. IDENTITY PROPERTIES A. Properties of Real ... Explores how to engage complex Properties of Real Numbers. Real numbers are simply the combination of rational and irrational numbers, in the number system. 1-1 Rational Numbers - Answers - Maze Activity (Editable Doc) If […] Therefore, real numbers are … 3 Real Numbers 1. Home > Math Skills / Topics > Algebra Worksheets > Properties of Real Numbers Worksheets. The Algebra 2 course, often taught in the 11th grade, covers Polynomials; Complex Numbers; Rational Exponents; Exponential and Logarithmic Functions; Trigonometric Functions; Transformations of Functions; Rational Functions; and continuing the work with Equations and Modeling from previous grades. In algebra, the distributive property is used to perform an operation on each of the terms within a grouping symbol. Free Algebra 2 worksheets (pdfs) with answer keys-each includes visual aides, model problems, exploratory activities, practice problems, and an online component Standard: Math 2 Grades: (9-12) View lesson. + = + + 2. Textbook Authors: Hall, Prentice, ISBN-10: 0133186024, ISBN-13: 978-0-13318-602-4, Publisher: Prentice Hall Do some more word problems - you may check answers with your partner. Holt Algebra 2 1-2 Properties of Real Numbers Identify the property demonstrated by each question.! a = a. Example 2: Identifying Properties of Real Numbers A. 2 + . We are giving you the real number system activities pdf files but in order to get the Editable Versions and the Answer Keys you will need to Join the Math Teacher Community. Algebra 2 Common Core answers to Chapter 1 - Expressions, Equations, and Inequalities - 1-2 Properties of Real Numbers - Practice and Problem-Solving Exercises - Page 16 62 including work step by step written by community members like you. The Closure Properties. -2 Class 18 Date Form G 3. But in algebra we can. 14 x 1 =14 Commutative Property of Multiplication 2(3x — y) = 6x — 2)' Write an … Algebra 1 answers to Chapter 1 - Foundations for Algebra - 1-4 Properties of Real Numbers - Lesson Check - Page 26 2 including work step by step written by community members like you. The commutative properties say that the order in which we either add or multiply real number doesn’t matter. Here are the Real Number System Maze Activities. 1) associative 2) additive identity 3) additive inverse 4) distributive 4) 2) Tori computes the value of in her head by thinking . Some important terminology to remember before we begin is as follows: integers: counting numbers like 1, 2, 3, etc., including negatives and zero real number: fractions, negative umbers, decimals, integers, and zero are all real numbers absolute value: a number’s distance from zero; it’s always positive. Addition Properties of Real Numbers. If you do something to two real numbers and always get a real number answer, you could say that real numbers are closed under that operation. Chapter 1 Quiz 1 Lessons 1-1 through 1-3 Do you know HOW? What property is this? 2.1 The Real Number Line 2.2 Addition of Real Numbers 2.3 Subtraction of Real Numbers 2.4 Adding and Subtracting Matrices 2.5 Multiplication of Real Numbers 2.6 The Distributive Property 2.7 Division of Real Numbers 2.8 Probability and Odds 2 • 3.9 = 3.9 • 2 Numbers are multiplied in any order without changing the results. algebra 2 real numbers quizlet, Algebra 1 Lesson 5.3 Slope Intercept Form - YouTube tiotravexah.tk/ Algebra Fundamentals 1.1 Variables and Expressions 1. Subsets of Real Numbers Name Explanation Example Natural Numbers Check answers on quiz Test Friday, Nov 11, 2016 Chapter 7 Also, we can multiply in any order we want, but not in division. Describe a rule for the pattern. 1.4 The Properties of Algebra 15 Section 1.4 The Properties of Algebra. The correct answer is –5. Property Example Commutative Property of Addition . Suppose a, b, and c represent real numbers.. 1) Closure Property of Addition Property: a + b is a real number Verbal Description: If you add two real numbers, the sum is also a real number. The following list presents the properties of numbers: Reflexive property. Basic Properties of Algebra: Trevor L.A. May 2010 Where a, b, and c can be real numbers, variables, or algebraic expressions. 4 Adding and Subtracting Real Numbers 1. PRACTICE TEST. Example: We can add 3 + 5 + 2 = 10. and switch the order to 5 + 3 + 2 1 3, !1.25 27. Download them now! We have thousands of printable worksheets such as Properties Of Real Numbers Worksheet With Answers Pdf/page/2 that can be downloaded for free. Start studying Algebra 2 properties. For example, 10 = 10. In general, all the arithmetic operations can be represented in the number line, also review... Is –3x + 33 of real numbers, you will get a real number.... 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We have thousands of printable Worksheets such as Properties of real numbers are simply the combination of rational irrational.: ( 9-12 ) View lesson, download them from here commutative Properties say the! ) = Lessons 1-1 through 1-3 Do you know how list presents the Properties of real can..., also and irrational numbers, you will get a real number commutative Properties say that order! Some more word problems - you may check answers with your partner how to engage Properties. … ] the commutative Properties say that algebra 2 properties of real numbers properties practice sheet answers order in which we either add or multiply real.... On this to come in a later chapter and Expressions 1 we either add or real..., real numbers, you will get a real number your children\ 's Skills, download them here. Are rather simple subtraction: Practice Questions –3 ( x – 11 ) = additive Identity sum... Be downloaded for free answers you get improve your children\ 's Skills, download them here! Analysis is really required know how basic Properties of real numbers: Reflexive property following rules show distributing multiplication subtraction... Number line, also you ever need some Worksheets to improve your 's. T matter numbers a: Identifying Properties of real numbers Worksheet with answers that! Numbers is important because you use them consistently in pre-calculus of which property of numbers. Consistently in pre-calculus doesn ’ t matter 3x+6 =21 as his first 1.4 the of! Over subtraction: Practice Questions –3 ( x – 11 ) = 1-3 Do you know?... Combination of rational and irrational numbers, in the number system any order without the... You know how and distributing multiplication over subtraction: Practice Questions –3 ( x – 11 ) = example:... | 2021-03-02 14:19:31 | {"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": 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": 0.40192994475364685, "perplexity": 1059.9165647775662}, "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/1614178364008.55/warc/CC-MAIN-20210302125936-20210302155936-00043.warc.gz"} |
http://www.r-bloggers.com/rejection-sampling/ | # Rejection Sampling
June 9, 2011
By
(This article was first published on Playing with R, and kindly contributed to R-bloggers)
An interesting sampling method that was covered briefly in my Bayesian statistics course was rejection sampling. Since I have nothing better to do, I thought it would be fun to make an acceptance-rejection algorithm using R. FUN!
The Rejection Sampling method is usually used to simulate data from an unknown distribution. To do this one samples from a distribution that covers the suport of the unknown distribution and use certain criteria for accepting/rejecting the sampled values. One way to do this is as follows (Rice, p 92).
Step 1: Generate T with density m.
Step 2: Generate U, uniform on [0,1] and independent of T. If M(T)*U ≤ ƒ(T), then let X = T (accept T). Otherwise, go to Step 1 (Reject T).
Where M(x) is a function such that M(x) ≥ ƒ(x) on [a,b].
To keep things simple for myself I will be simulating a Beta distribution with parameters 6 and 3 (ƒ). To do this I will sample T's from a scaled uniform distribution (M), and reject sampled values where M(T)*U ≥ ƒ(T).
In a plot of the beta distribution with parameters 6 and 3 we can see that the ƒ(x) never goes above 3. For this reason I chose to scale the uniform distribution M by multiplying it by 3.
Here is the R code to implement rejection sampling for 100,000 observations in this example.
sample.x = runif(100000,0,1)accept = c()for(i in 1:length(sample.x)){ U = runif(1, 0, 1) if(dunif(sample.x[i], 0, 1)*3*U <= dbeta(sample.x[i], 6, 3)) {
accept[i] = 'Yes'
}
else if(dunif(sample.x[i],0,1)*3*U > dbeta(sample.x[i], 6, 3)) { accept[i] = 'No' }}T = data.frame(sample.x, accept = factor(accept, levels= c('Yes','No')))
We can plot the results along with the true distribution with the following code.
hist(T[,1][T\$accept=='Yes'], breaks = seq(0,1,0.01), freq = FALSE, main = 'Histogram of X', xlab = 'X')lines(x, dbeta(x,6,3))
With 100,000 observations sampled, the data fit very well.
We can look at the densities of both the accepted and rejected values to get an idea of what's going on.
library(ggplot2)print(qplot(sample.x, data = T, geom = 'density', color = accept))
Looking at a stacked histogram of all the sampled values together we can really see how much wasted data there are in this example.
print(qplot(sample.x, data = T, geom = 'histogram', fill = accept, binwidth=0.01))
In fact, when I ran this example I got 33,114 accepted values and 66,886 rejected values. I probably could have chosen a better value than 3 to scale the uniform distribution, but ideally rejection sampling uses a known distribution that is only slightly different from the unknown distribution we're trying to estimate. | 2014-10-21 10:29:48 | {"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": 0.5751681327819824, "perplexity": 1410.0921642714482}, "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-2014-42/segments/1413507444339.31/warc/CC-MAIN-20141017005724-00101-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.zarges.com/uk/products/variotec-v | My Cart
# Variotec V
## Details
Multifunctional, extendible ladder suitable for use on stairs, with compact transport dimensions and in sturdy, durable design.
• The ladder can be adjusted to the desired working height in increments of 280 mm.
• Folds down to a space-saving compact size for transport.
• Functional robust steel hinges, bolted and easy to replace.
• Fitted with fold-out stabilisers to allow the product to be used as a single ladder in compliance with EN131-1.
• The stabilisers can be folded in to save space during transport or when the product is used as a stepladder.
• Stiles made from extruded aluminium sections for maximum stability.
• High strength, non-twist rung/stile connections.
### Hints and special features
• Does not comply with EN131-1+2.
## More Information
Coating natural flanged 30 mm max. 150 kg aluminium 265 mm Flanged telescopic multipurpose ladder, 4 sections serrated
## Variants
Order number Dimensions, folded Length Length as lean-to ladder Length as trestle Number of rungs Number of rungs Transport dimensions Weight Width Working height as lean-to ladder Working height as stepladder
42437 1,3 m × 0,64 m × 0,21 m 1,3 m 4,17 m 2,2 m 4 4 × 4 1.310 mm × approx. 640 mm × 200 mm 15 kg 0,64 m 4,95 m 3,4 m
42438 1,58 m × 0,71 m × 0,21 m 1,58 m 5,29 m 2,7 m 5 4 × 5 1.580 mm × approx. 710 mm × 200 mm 18 kg 0,71 m 6,05 m 3,9 m
42439 1,86 m × 0,78 m × 0,21 m 1,86 m 6,4 m 3,09 m 6 4 × 6 1.860 mm × approx. 780 mm × 210 mm 22 kg 0,78 m 7,1 m 4,4 m | 2021-12-05 11:32:38 | {"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": 0.8038524389266968, "perplexity": 12193.76855402523}, "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-2021-49/segments/1637964363157.32/warc/CC-MAIN-20211205100135-20211205130135-00043.warc.gz"} |
https://zbmath.org/?q=an:0587.55006 | ## Zero divisors in enveloping algebras of graded Lie algebras.(English)Zbl 0587.55006
Let k be a field of characteristic different from 2. Following R. Bøgvad [Astérisque 113/114, 156-166 (1984; Zbl 0552.17012)], a graded Lie algebra over k, $$L=\oplus_{i\geq 1}L_ i$$, is torsion free if [x,x]$$\neq 0$$ for every nonzero x of odd degree and absolutely torsion free if it remains torsion free after field extension to the algebraic closure $$\bar k$$ of k. The authors prove: the enveloping algebra UL of an absolutely torsion free graded Lie algebra L has no zero divisors.
In an appendix they reproduce a different proof due to R. Bøgvad.
Reviewer: J.C.Thomas
### MSC:
55Q52 Homotopy groups of special spaces 57T05 Hopf algebras (aspects of homology and homotopy of topological groups) 17B70 Graded Lie (super)algebras 17B35 Universal enveloping (super)algebras
Zbl 0552.17012
Full Text:
### References:
[1] Anick, D., Non-commutative algebras and their Hilbert series, J. algebra, 78, 120-140, (1982) · Zbl 0502.16002 [2] Borho, W.; Rentschler, R., Oresche teilmengen in einhüllenden algebren, Math. ann., 217, 201-210, (1975) · Zbl 0297.17004 [3] R. Bøgvad, Some elementary results on the cohomology of graded Lie algebras, in: Homotopie Algébrique et Algèbre Locale, Astérisque 113-114, 156-166. [4] Halperin, S., Finiteness in the minimal models of Sullivan, Trans. A.M.S., 230, 173-199, (1977) · Zbl 0364.55014 [5] Halperin, S.; Lemaire, J.M., Suites inertes dans LES algèbres de Lie graduées, (), no. 22 · Zbl 0655.55004 [6] Milnor, J.; Moore, J.C., On the structure of Hopf algebras, Ann. math., 81, 211-264, (1965) · Zbl 0163.28202 [7] Stenström, B., Rings of quotients, () · Zbl 0194.06602 [8] Sjödin, G., A set of generators for ext_{R} (k, k), Math. scand., 38, 199-210, (1976) · Zbl 0346.18017 [9] Wall, C.T.C., Nets of conics, Math. proc. Cambridge ph. soc., 81, 351-364, (1977) · Zbl 0351.14032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2023-02-03 19:43:46 | {"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": 0.7420198321342468, "perplexity": 3620.3140721026543}, "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-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00519.warc.gz"} |
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/1619.html | No.1619
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Combinatorial and Descriptive Set Theory
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Jorg Brendle
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1. ULTRAPRODUCTS OF FINITE ALTERNATING GROUPS (Combinatorial and Descriptive Set Theory)---------------------------------------------1
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2. FURTHER COMBINATORIAL PROPERTIES OF COHEN FORCING (Combinatorial and Descriptive Set Theory)--------------------------------------8
@@@@KURT GODEL RESEARCH CENTER FOR MATHEMATICAL LOGIC / DEPARTMENT OF MATHEMATICS AND STATISTICS, YORK UNIVERSITY@@@FISCHER,VERA / STEPRANS,JURIS
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3. Description of some ultrafilters via $\mathcal{I}$-ultrafilters (Combinatorial and Descriptive Set Theory)-----------------------20
@@@@University of West Bohemia, Department of Mathematics@@@FLASKOVA,JANA
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5. FUNCTIONS WITH MANY LOCAL EXTREMA (Combinatorial and Descriptive Set Theory)-----------------------------------------------------43
@@@@DEPARTMENT OF MATHEMATICS, BOISE STATE UNIVERSITY@@@GESCHKE,STEFAN
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6. There are more co-analytic sets than Borel (Combinatorial and Descriptive Set Theory)--------------------------------------------48
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8. The branching structure of trees on directed sets (Combinatorial and Descriptive Set Theory)-------------------------------------57
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9. A maximal forcing axiom compatible with weak club guessing (Combinatorial and Descriptive Set Theory)----------------------------63 | 2016-06-29 07:19:04 | {"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": 0.5855585336685181, "perplexity": 6713.838957334412}, "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/1466783397636.15/warc/CC-MAIN-20160624154957-00099-ip-10-164-35-72.ec2.internal.warc.gz"} |
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# Discover 8 Types of Algorithms for Efficient Problem Solving
November 23, 2022
This post is also available in: हिन्दी (Hindi) العربية (Arabic)
An algorithm refers to the sequential steps and processes that should be followed to solve a problem. There can be various kinds of algorithms devised to solve different problems although in programming and each type of algorithm has its own merits and demerits.
Let’s understand different types of algorithms and their working.
## What is an Algorithm?
An algorithm is a procedure or formula for solving a problem, based on conducting a sequence of specified actions. A computer program can be viewed as an elaborate algorithm. In mathematics and computer science, an algorithm usually means a small procedure that solves a recurrent problem.
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Algorithms are widely used throughout all areas of IT (information technology). A search engine algorithm, for example, takes search strings of keywords and operators as input, searches its associated database for relevant web pages, and returns results.
## Qualities of Good Algorithms
A good algorithm should have the following qualities:
• Input and output should be defined precisely.
• Each step in the algorithm should be clear and unambiguous.
• Algorithms should be the most effective among many different ways to solve a problem.
• An algorithm shouldn’t include computer code. Instead, the algorithm should be written in such a way that it can be used in different programming languages.
### Example of Algorithm – Finding Largest Among Three Numbers
Step 1: Start
Step 2: Declare variables a, b and c
Step 3: Read variables a, b and c.
Step 4: If a > b
If a > c
Display a is the largest number
Else
Display c is the largest number
Else
If b > c
Display b is the largest number
Else
Display c is the largest number
Step 5: Stop
## Types of Algorithms
Algorithms that use a similar problem-solving approach can be grouped together. This classification scheme is neither exhaustive nor disjoint. The purpose of classifying algorithms is to highlight the various ways in which a problem can be attacked.
Based on the working principle, there are 8 different types of algorithms.
• Simple recursive algorithms
• Backtracking algorithms
• Divide and Conquer algorithms
• Dynamic Programming algorithms
• Greedy algorithms
• Branch and Bound algorithms
• Brute Force algorithms
• Randomized algorithms
### 1. Simple Recursive Algorithms
The first on the list of different types of algorithms is the Simple Recursive Algorithm. In computer science, recursion is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem. Such problems can generally be solved by iteration, but this needs to identify and index the smaller instances at programming time. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
A recursive algorithm is an algorithm that calls itself with “smaller (or simpler)” input values, and which obtains the result for the current input by applying simple operations to the returned value for the smaller (or simpler) input. More generally if a problem can be solved utilizing solutions to smaller versions of the same problem, and the smaller versions reduce to easily solvable cases, then one can use a recursive algorithm to solve that problem. For example, the elements of a recursively defined set, or the value of a recursively defined function can be obtained by a recursive algorithm.
The initial steps of the recursive algorithm correspond to the basic clause of the recursive definition and they identify the basic elements. They are then followed by steps corresponding to the inductive clause, which reduces the computation for an element of one generation to that of elements of the immediately preceding generation.
In general, recursive computer programs require more memory and computation compared with iterative algorithms, but they are simpler and in many cases a natural way of thinking about the problem.
Let us consider a problem that a programmer has to determine the sum of first n natural numbers, there are several ways of doing that but the simplest approach is simply to add the numbers starting from 1 to n. So the function simply looks like, this simply adding one by one – f(n) 1 + 2 + 3 + …
In the recursive approach, it becomes:
f(n) = 1, n = 1;
f(n) = n + f(n - 1), n > 1;
There is a simple difference between the first approach and the second approach and it is that in the second approach the function “ f( ) ” itself is being called inside the function, so this phenomenon is named recursion, and the function containing recursion is called recursive function, in the end, this is a great tool in the hand of the programmers to code some problems in a lot easier and efficient way.
### 2. Backtracking Algorithms
The next on the list of different types of algorithms is Backtracking Algorithm. It is a general algorithm for finding all (or some) solutions to some computational problems, such as constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate (“backtracks”) as soon as it determines that the candidate cannot possibly be completed to a valid solution.
The classic textbook example of the use of backtracking is the eight queens puzzle, which asks for all arrangements of eight chess queens on a standard chessboard so that no queen attacks any other. In the common backtracking approach, the partial candidates are arrangements of k queens in the first k rows of the board, all in different rows and columns. Any partial solution that contains two mutually attacking queens can be abandoned.
Backtracking can be applied only for problems that admit the concept of a “partial candidate solution” and a relatively quick test of whether it can possibly be completed to a valid solution. It is useless, for example, for locating a given value in an unordered table. When it is applicable, however, backtracking is often much faster than brute force enumeration of all complete candidates, since it can eliminate many candidates with a single test.
Backtracking is an important tool for solving constraint satisfaction problems, such as crosswords, verbal arithmetic, Sudoku, and many other puzzles. It is often the most convenient (if not the most efficient) technique for parsing, for the knapsack problem, and other combinational optimization problems. It is also the basis of the so-called logic programming languages such as Icon, Planner, and Prolog.
Backtracking is a technique based on algorithms to solve problems. It uses recursive calling to find the solution by building a solution step by step increasing values with time. It removes the solutions that don’t give rise to the solution of the problem based on the constraints given to solve the problem.
• The backtracking algorithm is applied to some specific types of problems, such as
• Decision problem used to find a feasible solution to the problem.
• Optimization problems are used to find the best solution that can be applied.
• An enumeration problem is used to find the set of all feasible solutions to the problem.
In the backtracking problem, the algorithm tries to find a sequence path to the solution which has some small checkpoints from where the problem can backtrack if no feasible solution is found for the problem.
Let’s use this backtracking problem to find the solution to the N-Queen problem.
In the N-Queen problem, we are given an N×N chessboard and we have to place n queens on the board in such a way that no two queens attack each other. A queen will attack another queen if it is placed in horizontal, vertical, or diagonal points in its way. Here, we will do the 4-Queen problem.
Here, the solution is −
The binary output for n queen problem with 1s as queens to the positions placed.
{0 , 1 , 0 , 0}
{0 , 0 , 0 , 1}
{1 , 0 , 0 , 0}
{0 , 0 , 1 , 0}
For solving the n queens problem, we will try placing the queen into different positions of one row. And checks if it clashes with other queens. If they are attacking, we will backtrack to the previous location of the queen and change its position. And check the clash of the queen again.
Algorithm for solving the n-Queen problem:
Step 1 − Start from the 1st position in the array.
Step 2 − Place queens on the board and check. Do,
Step 2.1 − After placing the queen, mark the position as a part of the solution and then recursively check if this will lead to a solution.
Step 2.2 − Now, if placing the queen doesn’t lead to a solution and trackback and go to step (a) and place queens in other rows.
Step 2.3 − If placing the queen returns a lead to solution return TRUE.
Step 3 − If all queens are placed return TRUE.
Step 4 − If all rows are tried and no solution is found, return FALSE.
### 3. Divide and Conquer Algorithms
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (quicksort, merge-sort), multiplying large numbers (e.g., Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform.
Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cost is often determined by solving recurrence relations.
This technique can be divided into the following three parts:
1. Divide: This involves dividing the problem into some sub-problem.
2. Conquer: Sub-problem by calling recursively until the sub-problem is solved.
3. Combine: The Sub problem is solved so that we will find a problem solution.
The following are some standard algorithms that follow the Divide and Conquer algorithm.
1. Binary Search is a search algorithm. In each step, the algorithm compares the input element x with the value of the middle element in an array. If the values match, return the index of the middle. Otherwise, if x is less than the middle element, then the algorithm recurs for the left side of the middle element, and else recurs for the right side of the middle element.
2. Quicksort is a sorting algorithm. The algorithm picks a pivot element and rearranges the array elements in such a way that all elements smaller than the picked pivot element move to the left side of a pivot, and all greater elements move to the right side. Finally, the algorithm recursively sorts the subarrays on the left and right of a pivot element.
3. Merge Sort is also a sorting algorithm. The algorithm divides the array into two halves, recursively sorts them, and finally merges the two sorted halves.
4. The Closest Pair of Points is a problem finding the closest pair of points in a set of points in the x-y plane. The problem can be solved in $\text{O} \left(n^2 \right)$ time by calculating the distances of every pair of points and comparing the distances to find the minimum. The Divide and Conquer algorithm solves the problem in O(N log N) time.
### 4. Dynamic Programming Algorithms
The next on the list of different types of algorithms is Dynamic Programming algorithms. Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
In both contexts, it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have an optimal substructure.
If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems. In the optimization literature, this relationship is called the Bellman equation.
The dynamic programming approach is similar to divide and conquer in breaking down the problem into smaller and yet smaller possible sub-problems. But unlike, divide and conquer, these sub-problems are not solved independently. Rather, the results of these smaller sub-problems are remembered and used for similar or overlapping sub-problems.
Dynamic programming is used where we have problems, which can be divided into similar sub-problems so that their results can be re-used. Mostly, these algorithms are used for optimization. Before solving the in-hand sub-problem, dynamic algorithms will try to examine the results of the previously solved sub-problems. The solutions of sub-problems are combined in order to achieve the best solution.
So we can say that −
• The problem should be able to be divided into smaller overlapping sub-problem.
• An optimum solution can be achieved by using an optimum solution of smaller sub-problems.
• Dynamic algorithms use Memoization.
The following computer problems can be solved using a dynamic programming approach −
Dynamic programming can be used in both top-down and bottom-up manners. And of course, most of the time, referring to the previous solution output is cheaper than recomputing in terms of CPU cycles.
### 5. Greedy Algorithms
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not usually produce an optimal solution, but nonetheless, a greedy heuristic may yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time.
For example, a greedy strategy for the traveling salesman problem (which is of high computational complexity) is the following heuristic: “At each step of the journey, visit the nearest unvisited city.” This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with submodular structures.
Most networking algorithms use the greedy approach. Here is a list of a few of them −
### 6. Branch and Bound Algorithms
The next on the list of different types of algorithms are Branch and Bound algorithms. The Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution and is discarded if it cannot produce a better solution than the best one found so far by the algorithm.
The algorithm depends on the efficient estimation of the lower and upper bounds of regions/branches of the search space. If no bounds are available, the algorithm degenerates into an exhaustive search.
The method was first proposed by Ailsa Land and Alison Doig whilst carrying out research at the London School of Economics sponsored by British Petroleum in 1960 for discrete programming, and has become the most commonly used tool for solving NP-hard optimization problems. The name “branch and bound” first occurred in the work of Little et al. on the traveling salesman problem.
Let’s see the Branch and Bound Approach to solve the 0/1 Knapsack problem: The Backtracking Solution can be optimized if we know a bound on the best possible solution subtree rooted with every node. If the best in the subtree is worse than the current best, we can simply ignore this node and its subtrees. So we compute the bound (best solution) for every node and compare the bound with the current best solution before exploring the node.
Example bounds used in the diagram below are, A down can give $315, B down can$275, C down can $225, D down an$125 and E down can \$30.
### 7. Brute Force Algorithms
In computer science, brute-force search or exhaustive search, also known as generating and testing, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem’s statement.
A brute-force algorithm to find the divisors of a natural number n would enumerate all integers from 1 to n, and check whether each of them divides n without remainder. A brute-force approach for the eight-queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard, and, for each arrangement, check whether each (queen) piece can attack any other.
While a brute-force search is simple to implement, and will always find a solution if it exists, its cost is proportional to the number of candidate solutions – which in many practical problems tends to grow very quickly as the size of the problem increases (Combinatorial explosion). Therefore, brute-force search is typically used when the problem size is limited, or when there are problem-specific heuristics that can be used to reduce the set of candidate solutions to a manageable size. The method is also used when the simplicity of implementation is more important than speed.
This is the case, for example, in critical applications where any errors in the algorithm would have very serious consequences; or when using a computer to prove a mathematical theorem. Brute-force search is also useful as a baseline method when benchmarking other algorithms or metaheuristics. Indeed, brute-force search can be viewed as the simplest metaheuristic. Brute force search should not be confused with backtracking, where large sets of solutions can be discarded without being explicitly enumerated (as in the textbook computer solution to the eight queens problem above). The brute-force method for finding an item in a table – namely, checking all entries of the latter, sequentially – is called linear search.
For example, imagine you have a small padlock with 4 digits, each from 0-9. You forgot your combination, but you don’t want to buy another padlock. Since you can’t remember any of the digits, you have to use a brute force method to open the lock.
So you set all the numbers back to 0 and try them one by one: 0001, 0002, 0003, and so on until it opens. In the worst-case scenario, it would take 104, or 10,000 tries to find your combination.
A classic example in computer science is the traveling salesman problem (TSP). Suppose a salesman needs to visit 10 cities across the country. How does one determine the order in which those cities should be visited such that the total distance traveled is minimized?
The brute force solution is simply to calculate the total distance for every possible route and then select the shortest one. This is not particularly efficient because it is possible to eliminate many possible routes through clever algorithms.
### 8. Randomized Algorithms
The last on the list of different types of algorithms are Randomized algorithms. A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the “average case” over all possible choices randomly determined by the random bits; thus either the running time or the output (or both) are random variables.
One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithm, for example, Quicksort), and algorithms that have a chance of producing an incorrect result (Monte Carlo algorithm, for example, Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem.
In common practice, randomized algorithms are approximated using a pseudorandom number generator in place of a true source of random bits; such an implementation may deviate from the expected theoretical behavior.
## Practice Problems
1. What is an Algorithm?
2. What are the features of a good algorithm?
3. What are the 8 different types of algorithms?
4. What does the word Backtracking mean?
5. What are Dynamic Programming algorithms?
6. How do Greedy algorithms work?
## FAQs
### What is an algorithm?
An algorithm is a procedure or formula for solving a problem, based on conducting a sequence of specified actions. In mathematics and computer science, an algorithm usually means a small procedure that solves a recurrent problem.
### What is the use of an algorithm?
An algorithm is a procedure used for solving a problem or performing a computation. Algorithms act as an exact list of instructions that conduct specified actions step by step in either hardware- or software-based routines. Algorithms are widely used throughout all areas of IT.
### How many types of algorithms are there?
There are 8 different types of algorithms. These are
a) Simple recursive algorithms
b) Backtracking algorithms
c) Divide and Conquer algorithms
d) Dynamic Programming algorithms
e) Greedy algorithms
f) Branch and Bound algorithms
g) Brute Force algorithms
h) Randomized algorithms
## Conclusion
An algorithm is a procedure or formula for solving a problem, based on conducting a sequence of specified actions. They help in understanding and documentation of computer programs. There are 8 different types of algorithms in use with their own merits and limitations. | 2023-02-04 22:25:00 | {"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": 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": 0.734011709690094, "perplexity": 1289.3615628309203}, "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/1674764500154.33/warc/CC-MAIN-20230204205328-20230204235328-00004.warc.gz"} |
https://www.gamedev.net/forums/topic/275683-simple-quaternion-tasks/ | This topic is 5394 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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I'm still trying to grasp the whole quaternion deal. To be honest, I despise them [lol] -They never do what I expect them to do, but they are really the only acceptable means of interpolating rotations that I've tried. This is what I'm trying to do now. I have a quaternion called "motion". This motion is filled from keyframes from an animation that is playing. Every motion frame of the animation is relative to the previous frame. I do a simular thing with translation:
VECTOR translation_motion_buffer = 0;
.....
VECTOR key_translation = NextAnimationKeyframeVector;
FLOAT frame_time_laps = NumberOfMillisecondsPassedThisGameFrame;
FLOAT TimeToNextKey = NextAnimationKeyTime - PreviousAnimationKeyTime;
// Calculate the percentage of distance we've covered
perc = frame_time_laps / TimeToNextKey;
// Get Motion Amount (Motion values are relative)
translation_motion_buffer += key_translation * perc;
The above code may execute several times before translation_motion_buffer is used. Then my character's current position is updated right after by adding translation_motion_buffer to his current position, and reseting translation_motion_buffer back to zero. Does anyone know how I can do something just like this with quaternion rotations? The math is not the same, is it? 1) What would key_rotation * perc actually represent? Will it actually pull the percentage of that rotation quaternion out of the key? Or will it mess up the axis as well? Do I have to do something weird like factor the rotation quaternion with an identity quaternion to pull a percentage of it's rotations out? In other words, SLERP with identity and the key? 2) And after I do this, I can't simply add the amount to rotation_motion_buffer, right? How can I add an amount of rotation (quat) to another rotation (quat)? Each rotation may be around different axes. Does this matter? I know I sound pretty dumb. But that's okay, I feel the same way [lol] Any help is appreciated [smile]
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Just a quick reply, which may or may not address any of the issues you mentioned.
Quaternions, like matrices, can be used to concatenate rotations. If your object's orientation is represented by one quaternion, you can multiply it by another quaternion that represents, say, a roll around your object's z axis. The resulting quat represents your orientation after applying the roll.
For keyframe interpolation, I think you will want to use slerp, which, given two quats and a parametric value should return the appropriate intermediate quat. I don't think this can be done linearly for any but very small differences in orientation.
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It sounds like one would never want to simply multiply the rotations? I've noticed a lot of jerking when I simply multiply the quaternions rather than SLERP. And the differences between my keyframes are not drastic. -Medium paced rotations with 33.333 milliseconds between keys.
And thanks, that would cover everything except for factoring. What is the best way to use a specific amount of a rotation quaternion? Say I have a quat (Buffer) which represents 90 degree rotation around (0.5, 0.5, 0.0) axis. What rotation does Buffer * 0.7 represent? And if this will work, will it also cause the same problems as interpolating without SLERP? Would my best bet be to SLERP between an identity quat and the main rotation by factor 0.7? Does Rotation == (Rotation * 0.5) * (Rotation * 0.5) ?
Thanks again [smile]
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Quote:
It sounds like one would never want to simply multiply the rotations? I've noticed a lot of jerking when I simply multiply the quaternions rather than SLERP. And the differences between my keyframes are not drastic. -Medium paced rotations with 33.333 milliseconds between keys.
It's not that you don't want to multiply quats. It's that multiplying them and slerping them do two different things.
Multiplying quats has the same effect as multiplying matrices. It's what you do when you want to change the orientation of your object. You have one quat representing your orientation, and another representing your change in orientation. You mult them together to get the new orientation.
Slerping is what you do when you want to interpolate two quats, i.e. for keyframe animation. So if you have quat1 and quat2 which are the keyframe quats, and you are at .33 between the frames, then newquat = slerp(.33, quat1, quat2).
Quote:
What is the best way to use a specific amount of a rotation quaternion? Say I have a quat (Buffer) which represents 90 degree rotation around (0.5, 0.5, 0.0) axis. What rotation does Buffer * 0.7 represent?
First of all, keep in mind that the axis needs to be normalized (I think) for the quat representation to make sense. So .5, .5, 0 wouldn't give you the results you were looking for.
I'd have to look in my books to see if scaling a quat (like by 0.7) has any useful meaning, but I'm fairly certain it doesn't do what you expect it to. That's what slerp is for - it uses quaternion calculus to give you a meaningful parametric interpolation between two quats.
In summary, use multiplication for cumulative changes in orientation, and slerp for interpolation, such as between keyframes.
Quaternions can be tricky, so ask more questions if you have them! I'm no math expert - I just understand them from a practical point of view. So others may chime in with more in-depth explanations.
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No, you pretty much answered everything. Everything except for using a specific amount of a rotation quaternion. Usually that amount would be based on time. I mean I know a SLERP between an identity and the rotation would pull this off, but is this what I should do? Is this what most do?
I appreciate your help. And I prefer practical experience explanations [smile]
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Quote:
Usually that amount would be based on time. I mean I know a SLERP between an identity and the rotation would pull this off, but is this what I should do? Is this what most do?
Ah, I see what you mean. Yes, I'm not sure how you would do the equivalent of:
quat1 *= .7 * quat2;
I'm not sure if they can be used that way. I just do the scaling beforehand, like this:
angle = rotspeed * deltatime;
m_quat *= quat(angle, axis);
Perhaps someone else will give a more rigorous explanation. But that's how I do it.
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So I should extract the relative angle and axis from each quaternion keyframe? But then I would have to factor axes and angles. Not much difference is there?
What I'm trying to do is have my character's rotations controlled by his animations. If his animation turns him around 180 degrees, then stops, I want my character to actually turn around 180 degrees. But if I don't make each keyframe relative to the previous keyframe, he will turn around 180 degrees, then blast back to 0 degrees when he switches to another animation, or if that animation loops around to frame 0 (if it loops).
So if they are relative, and the animation turns him around 45 degrees, and it loops, he will turn around 45, then turn around another 45, etc. So if it loops twice, he will do a 90 degree turn.
So every frame update I add the current key's relative rotation (factored by time) to my character's rotation. If anyone can show me how to do this without slerp with an identity or such, that would be great. For some reason, it feels like a hack doing it that way.
Thanks again [smile]
[Edited by - Jiia on October 11, 2004 10:51:17 PM]
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Sorry, your question was clear from the beginning, but I just wasn't tracking.
So let me see if I'm understanding now. Your keyframe rotations are relative rather than absolute so, for example, you can be facing at 45 degrees, and start an animation which rotates you to the left *from that orientation* rather than snapping you back to 0 first.
The one thing I'm not quite clear on is whether, for example, an animation that rotates you 5 degrees per keyframe has successive keyframes of 5, 10, 15 degrees, etc., or whether they are all 5 and are performed relative to each other.
In any case, here's an off-the-top-of-my-head solution. I really don't know if this will work - I've never tried it.
I'm pretty sure this won't work:
buffer_quat += perc * key_quat;
But this might:
Quaternion goal_quat = buffer_quat * key_quat;buffer_quat = Slerp(buffer_quat, goal_quat, perc);
Again, that's just off the top of my head, so it may not work for some reason or another. But it seems like it should.
Anyway, let us know what happens :)
P.S. Note that you might have to change the quat mult order depending on how your quat mult function is set up.
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Ahh man, how did I totally miss that? It makes perfect sense. It does work, but I'm still unsure of the accuracy. I haven't measured the rotation under extreme time tests yet. So far it looks great. Thanks [smile]
Oh, and my animation keys are relative to each other. Not originally, but I convert them to that form. This lets me start playing an animation from the middle of it, and I never need to keep track of any previous frames.
Each animation file has settings such as X-Motion, Y-Motion, Z-Motion, and R-Motion. x,y and z are translation, and r rotation. If X-Motion is defined for an animation, I erase all X-movement from the base bone of my skeleton and convert what I erased into relative motion. That animation can still leave y and z movement to the base bone, which is better really in such animations as walking, where the motion is usually in a single direction.
By using motion in this way, a 30 frame walk animation can move the character infinitely in one direction by looping, and his movements exactly match his foot movement. Same for turning around. His feet have to push him around, none of that spinning stuff [smile]
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anyway, jyk 's post seems to do the job.
Quote:
Oh, and my animation keys are relative to each other. Not originally, but I convert them to that form. This lets me start playing an animation from the middle of it, and I never need to keep track of any previous frames.
I don't get that. first sentence contradicts to second. If by relative you mean relative to previous frame.
If you need to start animation from the middle you need to store _absolute_ rotation quaternions, that is, quaternions that rotates from _initial_ position to position at that frame*, so previous frames doesn't matter. So there's no precision issues you'll surely sooner-or-later get with your method as i understand it.
* it's "goal_quat" in jyk post. In fact i think this goal_quat it's what you initially had from what you obtained.
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• 16 | 2019-07-21 06:22:56 | {"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": 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": 0.46926024556159973, "perplexity": 1267.5883654197119}, "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/1563195526931.25/warc/CC-MAIN-20190721061720-20190721083720-00200.warc.gz"} |
http://www.j.sinap.ac.cn/nst/EN/10.1007/s41365-019-0587-1 | # Nuclear Science and Techniques
《核技术》(英文版) ISSN 1001-8042 CN 31-1559/TL 2019 Impact factor 1.556
Nuclear Science and Techniques ›› 2019, Vol. 30 ›› Issue (4): 62
• NUCLEAR ENERGY SCIENCE AND ENGINEERING •
### Simulation of the effects of different substrates, temperature, and substrate roughness on the mechanical properties of Al2O3 coating as tritium penetration barrier
Ze Liu1 • Fei Meng2 • Liang-Bi Yi3
1. 1 Key Laboratory for Radiation Physics and Technology of Ministry of Education, Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, China
2 Chengdu Youfang Technology Co., Ltd, Chengdu 610041, China
3 Southwestern Institute of Physics, Chengdu 610041, China
• Received:2018-07-05 Revised:2018-09-10 Accepted:2018-10-20
• Contact: Ze Liu E-mail:liuze720@sina.com
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Ze Liu, Fei Meng, Liang-Bi Yi. Simulation of the effects of different substrates, temperature, and substrate roughness on the mechanical properties of Al2O3 coating as tritium penetration barrier.Nuclear Science and Techniques, 2019, 30(4): 62
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Abstract: Residual thermal stress in the system is a serious problem that affects the application of tritium permeation barrier coatings in fusion reactors. The stress not only determines the adhesion between coating and substrate, but also changes the properties of the material. In this study, finite element analysis was used to investigate the relationship between the residual thermal stress and the mechanical properties of Al2O3 tritium penetration barrier systems. Moreover, the residual thermal stress influenced by factors such as different substrates, temperature, and substrate roughness was also analyzed. The calculation showed that the hardness and elastic modulus increased with increasing compressive stress. However, the hardness and elastic modulus decreased with increasing tensile stress. The systems composed of Al2O3 coatings and different substrates exhibited different trends in mechanical properties. As the temperature increased, the hardness and the elastic modulus increased in an Al2O3/316L stainless steel system; the trend was opposite in an Al2O3/Si system. Apart from this, the roughness of the substrate surface in the system could magnify the change in hardness and elastic modulus of the coating. Results showed that all these factors led to variation in the mechanical properties of Al2O3 tritium permeation barrier systems. Thus, the detailed reasons for the changes in mechanical properties of these materials need to be analyzed. | 2021-12-07 16:08:48 | {"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": 0.2830338776111603, "perplexity": 4459.175228522638}, "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/1637964363400.19/warc/CC-MAIN-20211207140255-20211207170255-00083.warc.gz"} |
http://www.ck12.org/book/CK-12-Math-Analysis-Concepts/r4/section/5.8/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 5.8: Distance Between a Point and a Plane
Difficulty Level: At Grade Created by: CK-12
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Practice Distance Between a Point and a Plane
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The distance between a specific point and a plane is important to a number of different activities.
For instance, a bungee jumping tower would not be very safe if the distance to the ground were not measured at the point directly under the tower, since any angle away from straight down would make the distance measure further and lead to a cord too long!
A computer game programmer needs to know how to calculate the distance between the location of a character on the screen and the walls around it to tell the game how to identify when a projectile hits a target, or when the character hits a wall.
Embedded Video:
### Guidance
Identifying the Point Closest to the Origin No matter what the orientation of a plane, there will always be one point located closer to the origin than any other point on the plane. This means that the position vector for this point is shorter than any other point in the plane. The diagram below shows a two-dimensional projection of a plane, in grey, near a point not on the plane, in black. The position vectors to a variety of points are shown in the diagram. The position vector marked in blue is shorter than the position vectors for the other points. This shortest vector is perpendicular to the plane. You can also see that the blue line is the vector projection of any orange vector onto the perpendicular direction.
This orthagonality (i.e. being perpendicular) is useful for us because it means that the position vector for this special point is parallel to the normal vector. Therefore, if we know the equation for a normal vector and the position vector for any point on the plane, we can determine the location of the point on the plane closest to the origin by finding the projection of the given point’s position vector onto the normal direction.
The Dihedral Angle
The angle between two planes is called the dihedral angle. The angle between two planes is the same as the angle between their normal vectors. If we want to determine the dihedral angle between two planes, we identify normal vectors to the two planes, then we can use the dot-product of the two normal vectors to determine the angle between the two normals which is also the two planes. Recall A×B=|A||B| cos θ\begin{align*}\overrightarrow{A} \times \overrightarrow{B} =|\overrightarrow{A}||\overrightarrow{B}| \ \mbox{cos} \ \theta\end{align*}
#### Example A
The three points P = (3, 7, 2), Q = (1, 4, 3), and R = (2, 3, 4) define a plane. Determine the point on the plane which is closest to the origin.
Solution
First find the vectors between two pairs of the points.
PQ=(QxPx),(QyPy),(QzPz)=(13),(47),(32)=\begin{align*}\overrightarrow{PQ} = \left \langle (Q_x - P_x), (Q_y - P_y), (Q_z - P_z) \right \rangle = \left \langle (1 - 3), (4 - 7), (3 - 2) \right \rangle =\end{align*} 2,3,1\begin{align*}\left \langle -2, -3, 1 \right \rangle\end{align*}
PR=(RxPx),(RyPy),(RzPz)=(23),(37),(42)=\begin{align*}\overrightarrow{PR} = \left \langle (R_x - P_x), (R_y - P_y), (R_z - P_z) \right \rangle = \left \langle (2 - 3), (3 - 7), (4 - 2) \right \rangle =\end{align*} 1,4,2\begin{align*}\left \langle -1, -4, 2 \right \rangle\end{align*}
The cross product of these two vectors is normal to the plane.
PQ×PR=(PQyPRzPQzPRy),(PQzPRxPQxPRz),(PQxPRyPQyPRx)\begin{align*}\overrightarrow{PQ} \times \overrightarrow{PR} = \left \langle (PQ_yPR_z - PQ_zPR_y), (PQ_zPR_x - PQ_xPR_z), (PQ_xPR_y - PQ_yPR_x)\right \rangle\end{align*}
PQ×PR=[(32)(14)],[(11)(22)],[(24)(13)]\begin{align*}\overrightarrow{PQ} \times \overrightarrow{PR} = \left \langle [(-3 \cdot 2) - (1 \cdot -4)], [(1 \cdot -1) - (-2 \cdot 2)], [(-2 \cdot -4) - (-1 \cdot -3)] \right \rangle\end{align*}
n=PQ×PR=[(6)(4)],[(1)(4)],[(8)(3)]=2,3,5\begin{align*}\overrightarrow{n} = \overrightarrow{PQ} \times \overrightarrow{PR} = \left \langle [(-6) - (-4)], [(-1) - (-4)], [(8) - (3)] \right \rangle = \left \langle -2, 3, 5 \right \rangle\end{align*}
The point on the plane which is nearest to the origin can be found by determining the projection of the position vector of either of these three points onto the normal vector. Remember that the vector projection of one vector onto the direction of another, is given by the dot-product of the first vector onto the unit vector defining the direction of the second vector: (A×B)B\begin{align*}\left (\overrightarrow{A} \times \overrightarrow{B} \right ) \overrightarrow{B}\end{align*}.
Since we know three points on the plane, we can use one of them to solve the problem. Let’s start with point P. The vector projection of P\begin{align*}\overrightarrow{P}\end{align*} onto n^\begin{align*}\hat{n}\end{align*} is given by \begin{align*}\left (\overrightarrow{P} \times \hat{n} \right ) \hat{n}\end{align*}, so first we need to determine the unit vector \begin{align*}\hat{n}\end{align*} which is given by \begin{align*}\hat{n} = \frac{\hat{n}}{|\overrightarrow{n}|} = \frac{\left \langle n_x, n_y, n_z \right \rangle}{\sqrt{n_x^2 + n_y^2 + n_z^2}} = \frac{\left \langle -2, 3, 5 \right \rangle}{\sqrt{(-2)^2 + 3^2 + (5)^2}} = \frac{\left \langle -2, 3, 5 \right \rangle}{\sqrt{38}} = \left \langle -0.32, 0.49, 0.81 \right \rangle\end{align*}
\begin{align*}\overrightarrow{P} \times \hat{n} = P_x\hat{n_x} + P_y\hat{n_y} + P_z\hat{n_z} = (3)(-0.32) + (7)(0.49) + (2)(0.81) =\end{align*} \begin{align*}-0.96 + 3.43 + 1.62 = 4.09\end{align*}
\begin{align*}\left (\overrightarrow{P} \times \hat{n} \right )\hat{n} = (4.09) \left \langle -0.32, 0.49, 0.81 \right \rangle = \left \langle -1.3088, 2.0041, 3.3129 \right \rangle\end{align*}
Therefore, the point on the plane closest to the origin is (-1.3088, 2.0041, 3.3129).
#### Example B
The three points P = (3, 7, 2), Q = (1, 4, 3), and R = (2, 3, 4) define a plane. Determine the dihedral angle between this plane and the x-y plane.
Solution As we saw in the example above, these three points define a plane which has a normal vector
\begin{align*}\overrightarrow{n} = \left \langle -2, 3, 5 \right \rangle\end{align*}
The normal to the x-y plane is the unit vector \begin{align*}\hat{z} = \left \langle 0, 0, 1 \right \rangle\end{align*}. To find the angle between these two vectors we use the fact that \begin{align*}\overrightarrow{A} \times \overrightarrow{B} =|\overrightarrow{A}||\overrightarrow{B}| \ \mbox{cos} \ \theta\end{align*} and that \begin{align*}\overrightarrow{A} \times \overrightarrow{B} = A_xB_x + A_yB_y + A_zB_z\end{align*}
First find a numerical value for the dot product:
\begin{align*}\overrightarrow{n} \times \hat{z} = n_xz_x + n_yz_y + n_zz_z = (-2 \cdot 0) + (3 \cdot 0) + (5 \cdot 1) = 5\end{align*}
\begin{align*}|\overrightarrow{n}| = \sqrt{n_x^2 + n_y^2 + n_z^2} = \sqrt{(-2)^2 + (3)^2 + (5)^2} = \sqrt{4 + 9 + 25} = \sqrt{38}\end{align*}
\begin{align*}|\hat{z}| = \sqrt{z_x^2 + z_y^2 + z_z^2} = \sqrt{0^2 + 0^2 + 1^2} = 1\end{align*}
Then find the cosine version of the dot product:
\begin{align*}\overrightarrow{n} \times \hat{z} = \sqrt{38} \ \mbox{cos} \ \theta\end{align*}
Now equate the two and solve for the angle, θ
\begin{align*}\overrightarrow{n} \times \hat{z} = 5 = \sqrt{38} \ \mbox{cos} \ \theta\end{align*}
\begin{align*}\theta = \mbox{cos}^{-1} \left (\frac{5}{\sqrt{38}} \right ) = 62.5^\circ\end{align*}
#### Example C
Determine the dihedral angle between the two planes 12x + 23y + 14z - 5 = 0 and 7x + 3y + z + 12 = 0.
Solution
The dihedral angle is defined as the angle between two planes. This angle is also equal to the angle between the normals to the two planes. In two of the previous problems we determined the unit vectors which are perpendicular to these two planes \begin{align*}\overrightarrow{n_1} = \left \langle \frac{12}{29.5}, \frac{23}{29.5}, \frac{14}{29.5} \right \rangle\end{align*} and \begin{align*}\overrightarrow{n_2} = \left \langle \frac{7}{\sqrt{59}}, \frac{3}{\sqrt{59}}, \frac{1}{\sqrt{59}}, \right \rangle\end{align*}. We can then use the dot-product of these two normal vectors to determine the angle between the two. The dot-product is defined as \begin{align*}\overrightarrow{A} \times \overrightarrow{B} = A_xB_x + A_yB_y + A_zB_z + ...\end{align*} and as \begin{align*}\overrightarrow{A} \times \overrightarrow{B} = |A||B| \ \mbox{cos} \ \theta\end{align*}. First, we need to find the component version of the dot product and the magnitudes of the two normal vectors.
\begin{align*}\overrightarrow{n_1} \times \overrightarrow{n_2} = \overrightarrow{n_{1x}}\overrightarrow{n_{2x}} + \overrightarrow{n_{1y}}\overrightarrow{n_{2y}} + \overrightarrow{n_{1z}}\overrightarrow{n_{2z}} = \frac{12\cdot7}{29.5\sqrt{59}} + \frac{23\cdot3}{29\cdot5\sqrt{59}} + \frac{14\cdot1}{29.5\sqrt{59}}\end{align*}
\begin{align*}\overrightarrow{n_1} \times \overrightarrow{n_2} = \frac{12\cdot7}{29.5\sqrt{59}} + \frac{23\cdot3}{29.5\sqrt{59}} + \frac{14\cdot1}{29.5\sqrt{59}} = \frac{119}{226.6} + \frac{69}{226.6} + \frac{14}{226.6} = \frac{202}{226.6} = 0.891\end{align*}
Since these two vectors are unit-vectors, their magnitudes are both equal to 1.
\begin{align*}\mbox{cos} \ \theta = \frac{\overrightarrow{n_1} \times \overrightarrow{n_2}}{|\overrightarrow{n_1}||\overrightarrow{n_2}|} = \frac{0.891}{(1)(1)} = 0.891\end{align*}
\begin{align*}\theta = cos^{-1} 0.891 = 27.0^\circ\end{align*}
### Vocabulary
The Dihedral Angle is the angle between two planes in a 3D space.
The origin is the point (0, 0, 0) in a 3D coordinate system.
A perpendicular line is oriented at 90o to a given line or plane.
### Guided Practice
Questions
1) Determine the angle between the plane 2x - 5y + 8 - 10 = 0 and the y-z plane.
2) The three points \begin{align*}\overrightarrow{P} = \left \langle -2, 3, 4 \right \rangle, \ \overrightarrow{Q} = \left \langle 5, -6, 7 \right \rangle,\end{align*} and \begin{align*}\overrightarrow{R} = \left \langle 8, 9, -1 \right \rangle\end{align*} identify a plane. Determine the point on the plane which is closest to the origin.
3) Determine the point on the plane 7x + 3y + z + 12 = 0 which is closest to the origin.
Solutions
1) The dihedral angle is defined as the angle between the two planes and is also equal to the angle between the two normal unit vectors. In this case, we already know the normal unit vector for the y-z plane, \begin{align*}\hat{x} = \left \langle 1, 0, 0 \right \rangle\end{align*}. We still need to determine, however, the unit vector for the plane 2x - 5y + 8z - 10 = 0.
Comparing this equation to \begin{align*}n_xx + n_yy + n_zz + d = 0\end{align*}, we can see that \begin{align*}\overrightarrow{n} = \left \langle 2, -5, 8 \right \rangle\end{align*}.
Now we can use the definition of the unit vector
\begin{align*}\hat{n} = \frac{\overrightarrow{n}}{|\overrightarrow{n}|} = \frac{\left \langle n_x, n_y, n_z \right \rangle}{\sqrt{n_x^2 + n_y^2 + n_z^2}} = \frac{\left \langle 2, -5, 8 \right \rangle}{\sqrt{2^2 + (-5)^2 + 8^2}} = \frac{\left \langle 2, -5, 8 \right \rangle}{\sqrt{4 + 25 + 64}} = \frac{\left \langle 2, -5, 8 \right \rangle}{9.64} = \left \langle \frac{2}{9.64}, \frac{-5}{9.64}, \frac{8}{9.64} \right \rangle\end{align*}
The angle between the two planes is equal to the angle between the two normal vectors.
We can then use the dot-product of these two normal vectors to determine the angle between the two. The dot-product is defined as \begin{align*}\overrightarrow{A} \times \overrightarrow{B} = A_xB_x + A_yB_y + A_zB_z + ...\end{align*} and as \begin{align*}\overrightarrow{A} \times \overrightarrow{B} = |A||B| \ \mbox{cos} \ \theta\end{align*}. First, we need to find the component version of the dot product and the magnitudes of the two normal vectors.
\begin{align*}\overrightarrow{n_1} \times \overrightarrow{n_2} = \overrightarrow{n_{1x}}\overrightarrow{n_{2x}} + \overrightarrow{n_{1y}}\overrightarrow{n_{2y}} + \overrightarrow{n_{1z}}\overrightarrow{n_{2z}} = \frac{2 \cdot 1}{9.64} + \frac{-5\cdot0}{9.64} + \frac{8\cdot0}{9.64} = \frac{2}{9.64} = 0.2074\end{align*}
Since these two vectors are unit-vectors, their magnitudes are both equal to 1.
\begin{align*}\mbox{cos} \ \theta = \frac{\overrightarrow{n_1} \times \overrightarrow{n_2}}{|\overrightarrow{n_1}||\overrightarrow{n_2}|} = \frac{0.2074}{(1)(1)} = 0.2074\end{align*}
\begin{align*}\theta = cos^{-1} 0.2074 = 16.18^\circ\end{align*}
2) The point on the plane nearest to the origin can be found by determining the projection of the position vector of one of these three points onto the normal vector. Remember that the vector projection of one vector onto the direction of another is given by the dot-product of the first vector onto the unit vector defining the direction of the second vector: \begin{align*}\left ( \overrightarrow{P} \times \hat{n} \right )\hat{n}\end{align*}.
We can use the position vectors for the three points to determine two vectors within the plane. Once we have those two vectors, their cross-product will define the direction normal to the plane. First find the two equations in the plane:
\begin{align*}\overrightarrow{A} = \overrightarrow{Q} - \overrightarrow{P} = \left \langle 5, -6, 7 \right \rangle - \left \langle -2, 3, 4 \right \rangle = \left \langle 7, -9, 3 \right \rangle\end{align*}
\begin{align*}\overrightarrow{B} = \overrightarrow{R} - \overrightarrow{P} = \left \langle 8, 9, -1 \right \rangle - \left \langle -2, 3, 4 \right \rangle = \left \langle 10, 6, -5 \right \rangle\end{align*}
Now determine the cross product of the two vectors
\begin{align*}\overrightarrow{n} = \overrightarrow{A} \times \overrightarrow{B} = \left \langle (A_yB_z - A_zB_y), (A_zB_x - A_xB_z), (A_xB_y - A_yB_x) \right \rangle\end{align*}
\begin{align*}\overrightarrow{n} = \overrightarrow{A} \times \overrightarrow{B} = \left \langle (45 - 18), (30 - (-35)), (42 + 90) \right \rangle\end{align*}
\begin{align*}\overrightarrow{n} = \overrightarrow{A} \times \overrightarrow{B} = \left \langle 27, 65, 132 \right \rangle\end{align*}
Now we need to determine the unit vector associated with this normal vector
\begin{align*}\hat{n} = \frac{\overrightarrow{n}}{|\overrightarrow{n}|} = \frac{\left \langle n_x, n_y, n_z \right \rangle}{\sqrt{n_x^2 + n_y^2 + n_z^2}} = \frac{\left \langle 27, 65, 132 \right \rangle}{\sqrt{27^2 + (65)^2 + (132)^2}} = \frac{\left \langle 27, 65, 132 \right \rangle}{\sqrt{22378}}\end{align*}
\begin{align*}\hat{n} = \frac{\overrightarrow{n}}{|\overrightarrow{n}|} = \left \langle 0.181, 0.435, 0.882 \right \rangle\end{align*}
Now we determine the vector progression of one of the three initial position vectors onto the direction of this normal unit-vector: \begin{align*}\left (\overrightarrow{P} \times \hat{n} \right )\hat{n}\end{align*}. Remember that the dot product is given by \begin{align*}\overrightarrow{A} \times \overrightarrow{B} = A_xB_x + A_yB_y + A_zB_z + ...\end{align*}.
\begin{align*}\left (\overrightarrow{P} \times \hat{n} \right )\hat{n} = (-2(0.181) + 3(0.435) + 4(0.882))\left \langle 0.181, 0.435, 0.882 \right \rangle\end{align*}
\begin{align*}\left (\overrightarrow{P} \times \hat{n} \right )\hat{n} = (4.471)\left \langle 0.181, 0.435, 0.882 \right \rangle\end{align*}
\begin{align*}\left (\overrightarrow{P} \times \hat{n} \right )\hat{n} = (4.471)\left \langle 0.181, 0.435, 0.882 \right \rangle = \left \langle 0.809, 1.945, 3.943 \right \rangle\end{align*}
3) The point on the plane nearest to the origin can be found by determining the projection of the position vector of any point on the plane onto the normal vector. The vector projection of one vector onto the direction of another is given by the dot-product of the first vector onto the unit vector defining the direction of the second vector: \begin{align*}\left (\overrightarrow{P} \times \hat{n} \right )\hat{n}\end{align*}.
In this case, we can determine a normal vector using the equation of the plane. Comparing 7x + 3y + z + 12 = 0 to the generic equation \begin{align*}n_xx + n_yy + n_zz + d = 0\end{align*}, we can see that \begin{align*}\overrightarrow{n} = \left \langle 7, 3, 1 \right \rangle\end{align*} and
\begin{align*}\hat{n} = \frac{\overrightarrow{n}}{|\overrightarrow{n}|} = \frac{\left \langle n_x, n_y, n_z \right \rangle}{\sqrt{n_x^2 + n_y^2 + n_z^2}} = \frac{\left \langle 7, 3, 1 \right \rangle}{\sqrt{(7)^2 + (3)^2 + (1)^2}} = \frac{\left \langle 7, 3, 1 \right \rangle}{\sqrt{49 + 9 + 1}} = \frac{\left \langle 7, 3, 1 \right \rangle}{\sqrt{59}} = \left \langle \frac{7}{\sqrt{59}}, \frac{3}{\sqrt{59}}, \frac{1}{\sqrt{59}} \right \rangle\end{align*}
We also need to know the location of a point on the plane. If we write the equation of the plane in intercept form, we can determine the position vector for the x-, y-, and z-intercepts of the plane.
The equation \begin{align*}1 = \frac{x}{a} + \frac{y}{b} + \frac{z}{c}\end{align*} must be true for all points on a plane. Therefore, we should first rearrange 7x +3y + z + 12 = 0 into the form \begin{align*}1 = \frac{x}{a} + \frac{y}{b} + \frac{z}{c}\end{align*}.
7x + 3y + z = -12 becomes \begin{align*}\frac{7}{-12}x + \frac{3}{-12}y + \frac{1}{-12}z = 1\end{align*}
Therefore, \begin{align*}a = \frac{-12}{7}, \ b = \frac{-12}{3} = -4\end{align*}, and \begin{align*}c = \frac{-12}{1} = -12\end{align*} and the position vectors of the three intercepts are \begin{align*}\overrightarrow{A} = \left \langle -1.714, 0, 0 \right \rangle, \ \overrightarrow{B} = \left \langle 0, -4, 0 \right \rangle\end{align*}, and \begin{align*}\overrightarrow{C} = \left \langle 0, 0, -12 \right \rangle\end{align*}.
To complete the problem, compute the dot product.
\begin{align*}\left (\overrightarrow{B} \times \hat{n} \right )\hat{n} = \left (B_x \hat{n}_x + B_y \hat{n}_y + B_z \hat{n}_z \right )\hat{n} = \left ( 0 \left (\frac{7}{\sqrt{59}} \right ) - 4 \left (\frac{3}{\sqrt{59}} \right ) + 0 \left (\frac{1}{\sqrt{59}} \right ) \right ) \left \langle \frac{7}{\sqrt{59}}, \frac{3}{\sqrt{59}}, \frac{1}{\sqrt{59}} \right \rangle .\end{align*}
\begin{align*}\left (\overrightarrow{B} \times \hat{n} \right )\hat{n} = \frac{-12}{\sqrt{59}}\left \langle \frac{7}{\sqrt{59}}, \frac{3}{\sqrt{59}}, \frac{1}{\sqrt{59}} \right \rangle = \left \langle \frac{-84}{59}, \frac{-36}{59}, \frac{-12}{59} \right \rangle = \left \langle -1.424, -0.610, -0.203 \right \rangle\end{align*}
### Practice
The three points define a plane. Determine the point on the plane which is closest to the origin
1. \begin{align*}P = (3, 6, 9), Q = (9, 6, 3)\end{align*} and \begin{align*}R = (6, -9, 9)\end{align*}
2. \begin{align*}P = (1, -7, 2), Q = (4, 2, 9)\end{align*} and \begin{align*}R = (3, -5, 1)\end{align*}
3. \begin{align*}P = (3, 8, 10), Q = (-2, 5, 8)\end{align*} and \begin{align*}R = (7, 4, 8)\end{align*}
4. \begin{align*}P = (9, -1, 4), Q = (6, 2, -8)\end{align*} and \begin{align*}R = (12 , 9, 10)\end{align*}
5. \begin{align*}P = (5, 8,-9), Q = ( -5, 3, 9)\end{align*} and \begin{align*}R = (10, 4, -6)\end{align*}
Determine the dihedral angle between each of the planes in questions 1-5 and the x-y plane, use the \begin{align*}|\overrightarrow{n}|\end{align*} you calculated for each plane and recall that the normal to the x-y plane is the unit vector \begin{align*}\hat{z} = \left \langle 0, 0, 1 \right \rangle\end{align*}
1. \begin{align*}P = (3, 6, 9), Q = (9, 6, 3)\end{align*} and \begin{align*}R = (6, -9, 9)\end{align*}
2. \begin{align*}P = (1, -7, 2), Q = (4, 2, 9)\end{align*} and \begin{align*}R = (3, -5, 1)\end{align*}
3. \begin{align*}P = (3, 8, 10), Q = (-2, 5, 8)\end{align*} and \begin{align*}R = (7, 4, 8)\end{align*}
4. \begin{align*}P = (9, -1, 4), Q = (6, 2, -8)\end{align*} and \begin{align*}R = (12 , 9, 10)\end{align*}
5. \begin{align*}P = (5, 8,-9), Q = ( -5, 3, 9)\end{align*} and \begin{align*}R = (10, 4, -6)\end{align*}
Determine the dihedral angle between the two planes
1. \begin{align*}9x + 17y - 4z - 7 = 0\end{align*} and \begin{align*}-17x + 24y + 14z + 2 = 0 \end{align*}
2. \begin{align*}2x - 4y + 10z - 11 = 0\end{align*} and \begin{align*}2x - 9y + 4z + 12 = 0 \end{align*}
3. \begin{align*}-7x + 20y + 6z + 4 = 0\end{align*} and \begin{align*}-19x - 3y + z + 5 = 0 \end{align*}
4. \begin{align*}5x - 8y + 20z - 5 = 0\end{align*} and \begin{align*}6x + y + 19z - 7 = 0 \end{align*}
5. \begin{align*}14x + 11y - 5z - 16 = 0\end{align*} and \begin{align*}11x - 13y + 8z + 4 = 0 \end{align*}
6. \begin{align*}-10x + 9y + 2z + 8 = 0\end{align*} and \begin{align*}21x + 7y - 4z + 15 = 0 \end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
Dihedral Angle
A dihedral angle is an angle between two planes in three dimensional space.
Origin
The origin is the point of intersection of the $x$ and $y$ axes on the Cartesian plane. The coordinates of the origin are (0, 0).
orthogonality
To be orthogonal is to be perpendicular.
Perpendicular
Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.
plane
A plane is a flat, two-dimensional surface. It can be conceptualized as a sheet of paper of infinite area.
position vector
A position vector describes the straight-line travel between a starting point (usually the origin) and the location of a second point on a coordinate plane.
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https://www.quantopian.com/posts/scoring-changes-for-the-april-quantopian-open | Scoring Changes for the April Quantopian Open
We have two changes to the scoring for the April contest. We just posted the first leaderboard with the updated scoring system. The March contest, currently in progress, is unaffected. The changes were driven by things we observed and learned in the almost-two-months we've been running the Quantopian Open. I discussed the changes in a webinar yesterday, and am putting them down in writing below.
Beta to SPY
The first change is that your algorithm is now scored according to how connected your algorithm's performance is to SPY. The lower your connection to SPY, the better. We already had 6 equal-weighted factors that generate your score; beta is now a 7th factor, all of them still equal-weighted.
Why did we do this change? If you look at a chart of SPY for February and March (the Quantopian Research notebook is attached), you could almost believe that the Quantopian Open was the driving factor in the S&P performance. On February 2, the start of the February judging period, SPY got on a rocket and headed for the stars. Grant's algo rode that rocket to the top of the charts, and he started trading real money on March 2. On March 2 the rocket ran out of fuel, and Grant's algo suffered! As we build the Quantopian hedge fund, we need to find algorithms that are uncorrelated from each other. The biggest correlation we're seeing today is around the S&P 500. This change is designed to encourage algorithms that are not correlated.
Consistency Between Paper Trading Results and Backtesting
The second change is that algorithms are now scored on how consistent they are between their paper trading returns and backtesting returns. The more consistent you are, the better. This factor is added to the calculation at the very end. After we compute what used to be referred to as your final score, we now multiply it by the consistency number, and the result is the new final score. This is applied gradually over the first few days of trading while the paper trading record is very volatile, and is fully applied at 20 days of trading.
We put in this change for a couple of reasons. The biggest reason is that we were seeing a lot of algorithms that had really good backtests that just weren't doing well in paper trading. This isn't too surprising when you think about it - if you're trying for a good score, you invest time in the backtest, and it's pretty easy to fall into data-mining, data-snooping, curve-fitting, or whatever you want to call that mistake. If you're prone to that mistake, you're not going to make an algorithm that lasts in the long run. We want to strongly encourage people to use good practices with out-of-sample data testing. If we make it very clear that a good backtest, on its own, can't win the contest, we hope to get more careful thought about how to write an algorithm that will perform well in paper trading.
The second reason is to discourage cheaters. We've seen a few instances where contest submissions are being deliberately gamed by submitting a "perfect backtest" and then a coin-flip over several entries for the paper trading. We've disqualified them, and we will continue to disqualify them in the future. The scoring change is a bit of a safety net, and a clear signal that it's not a strategy that will succeed.
For the detail-oriented: we're computing the consistency score using a kernel-density estimate using Gaussian kernels found in the Python scipy package. Both the backtest daily returns and the paper trade daily returns are each pushed through the function to fit them to a distribution separately. The difference between the areas of each of the distribution curves is used for the consistency score.
Future Changes
When we kicked off the Quantopian Open we promised that we would iterate and improve the contest. We don't think today's changes are the last word. There will be future scoring and rules changes as we think are necessary. We hope that you've found the previous discussions about scoring to be helpful; we certainly have. As always, we welcome and value your feedback about how we can make the contest better.
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Disclaimer
The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by Quantopian. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. No information contained herein should be regarded as a suggestion to engage in or refrain from any investment-related course of action as none of Quantopian nor any of its affiliates is undertaking to provide investment advice, act as an adviser to any plan or entity subject to the Employee Retirement Income Security Act of 1974, as amended, individual retirement account or individual retirement annuity, or give advice in a fiduciary capacity with respect to the materials presented herein. If you are an individual retirement or other investor, contact your financial advisor or other fiduciary unrelated to Quantopian about whether any given investment idea, strategy, product or service described herein may be appropriate for your circumstances. All investments involve risk, including loss of principal. Quantopian makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances.
31 responses
Solid changes, the lot. The publishing of that Gaussian kernel code would be useful too; so that one could test one's own strategy, in and out of sample (manually) to see how it performs.
Unfortunately, the one aspect that has caused me to dismiss the whole project as being an exercise in futility, which remains the same, is the selection process. Of the 270 odd strategies in the Open at this point easily 260 of them will never make the grade. Only the top 10, realistically -- at any one point, will go on to vie for top slot; a single top slot. The other 260 are just fodder to make the top 10 look good. It would be a more accurate judgement were the obvious low grade strategies remove themselves from the contest.
Add to this behavior the fact that only 5 algos, of the hundreds offered, will be selected over the coming months. And the fact is that those already in the list (most likely within the top 10) will be those selected in May, June and so on. 5? Out of hundreds? Why bother competing? You'll state that any new algo can easily jump in and rise to the top within a month (or less), but the likelihood of this happening is slight. Those posted back in February have a clear lead. Note that I'm not complaining. Just pointing out an additional, potential flaw. I'd only like to see a broader, more egalitarian system in place - for experimental, philosophical purposes.
Two potential fixes to the above are,
1) Pick the top 4 each month and give each $50k. Watching more ponies in the real race would be better for your bottom line as well as the contestant's. Perhaps narrow each winner set down by half every month. A contest within a contest. 2) Clear the field on the start of every new contest. Do not carry over the pool from month to month. If you don't clear the pool, the last 2 or 3 contests are likely to be lacking any new significant contestants. Why try? The probable winners have already been cooking for half a year or so. More winners and a fresh pool would go a long way to solicit new quants into submitting better strategies. This of course grinds against the Q's new objectives of being a hedge fund which uses the contest to harvest likely profitable algos. One would want the longest track records to pull strategies from. But as it stands, the Open, a level contest, it is not. My read is that a few things are at play here: 1. Quantopian is trying to understand if the whole idea of a crowd-sourced hedge fund is gonna have legs, and how they might pull it off. They have upwards of 35,000 registered users, and have been broadcasting their platform for a number of years now to the masses, but going on its third month now, the contest participation has been relatively low. At$1M-$5M per manager, they need thousands of managers to get to the gigadollars in capital under management. So, if the contest runs for 6 months and only a few hundred individuals enter, then, "Hmm? Is this thing gonna work?" Of course, they could decide to put$100M or more to a hundred managers (still a big number), but then it wouldn't be exactly crowd-sourced, since managing $100M sounds like a full time job to me. 2. It is a means to provide quantitative feedback to users, to give them a feel for the requirements of a hedge-fund-worthy strategy, as they develop their algos. My winning algo, as Dan alludes to, is probably a complete failure in this regard, due to its high beta. Had they had the beta metric in place earlier, I would have figured that out on my own, without their$100,000 of real money.
3. It is an engineering prototyping effort to see how Q might manage, say, 10,000 algo submissions to their crowd-sourced hedge fund on a rolling basis, in an automated fashion. The contest gives the Q team something definite to iterate, until they are ready for prime time.
4. Presently, it appears to be the only means to select algos/individuals as managers (unless folks are being contacted individually). So, although the number of winners is limited, I am surmising that all contest participants have a shot at getting $1M-$5M as a manager. If this is the case, Q really hasn't made it clear. I can't imagine that if someone pulls of a stellar second place, Q would just ignore the results.
Quantopian Hedgefund Outperforms Other Funds By 37%.
Wall Street Journal October 4, 2016
There may be two competing concepts here; the Open vs the Fund.
For the Open one would want fresh, new, hot, buzz, excitement. (Like the horserace metaphor one of the Q people tried to engender.)
For the Fund one would want established, stable, consistent, methodical. Nothing like one would want to hype a contest.
If the Fund is (and it appears to be so) the primary driving force, then the Open's mechanics are secondary. Get a couple of hundred of freely developed algos, grinding away for months and years, and cherry pick over time to slowly build a portfolio of strategies. But don't expect the hype of the contest to carry much further than, well, about now. "The horses are settled in for the long haul and are rounding the 12th corner. Andddd, it's the same 'ol pack in the front. I need a beer break. I'll be back in a month with an update..."
But who knows. Except Quantopian of course. They probably have metrics as to how the pace of the submissions has changed and whether or not the Open, as a true competition, is actually still viable. Or whether the current entries, or at least the top 10 of them, are "it" for the foreseeable future.
If the goal of my algorithm is to consistently do better than SPY on both during a bull and bear market.
And, say, I happen to achieve that an algorithm that very consistently beats SPY by 20% over any 6 month window, but in other ways follows the ups and downs of the SPY, in general.
What would my beta look like?
Would such an algorithm be penalized with this new rule?
The new beta metric only counts for 1/7 th of the overall score, but the idea is that a combination of high alpha and low beta would be best. Ideally, for the hedge fund, I gather they want something that goes like C= C0*(1+r)^n, with C0 being the initial captial, r the rate of return, and n being the compounding factor, so that their investors are basically putting money into zero-risk a high-yield bank CD. If they get enough uncorrelated strategies, the hope is that they can construct such an investment, with a high return. So, if your algo tracks the market, along with everyone else, then Q will have a challenge in getting C= C0*(1+r)^n behavior. At least this is my sketchy assessment at this point.
A beta of 1.0 indicates that the investment’s price will move in lock-step with the market. A beta of less than 1 indicates the investment will be less volatile than the market, and correspondingly, a beta of more than 1 indicates the investment’s price will be more volatile than the market. For example, if a fund portfolio’s beta is 1.2, it’s theoretically 20% more volatile than the market.
So, indeed, it seems that you could better the market by 20% over any six-month window, but have a beta of 1.0, for which you would be penalized in the contest. Your excess return would be captured in alpha.
Incidentally, in the Open scoring changes yesterday for April, taking a look at the top 20 only, 11 of the 18 ranked there were bumped out, 7 remained and 7 new folks appeared, while overall scores went down, ballpark ~3% to ~10%. I'm a fan of the honing/evolution of the process (not just because my rank went up by 7 as that's just one day, the Contest/Open is helping require rolling up my sleeves on certain aspects I might have overlooked).
Changes seem good to me. I'd just like to point out that many of the (original) stats are very colinear/correlated, and importantly, the Calmar and Sharpe ratios have the volatility/max-drawdown as a denominator. This benefits algorithms with ultra-low volatility and drawdown, where you get points from quantile rankings of features which are calculated from a stat in a denominator.
I just watched your webinar, you wondered why there are few algorithms using fundamental data; I considered it, but it seemed unlikely that any algorithm which relied on fundamental data could win the contest, given how points appear to be calculated, much for the same reasons as above.
It might be worthwhile to think up some novel performance measures which are not inter-related, equity slope consistency, conditional beta, skew, that sort of thing?
Thanks all for the feedback
The contest and the fund: It's very important that the contest and the fund be aligned. We can't on the one hand say that we want algos of type X, and give financial rewards to people who create algos of type Y. That's a misalignment of incentives that I'd very much like to avoid.
I'm interested in the perception that someone might think "I can't win" and decide not to enter. I think it's not wise conclusion.
• Looking at the top 10 for March, 4 of them are new entries for March and 6 were entries for January. That fraction runs all the way through the top 50 at least. In other words: a new entry is quite competitive with a well-aged entry.
• We're going to need dozens of algorithms in the hedge fund, far more than there are winners of the Quantopian Open. The hedge fund will, presumably, be significantly more lucrative than the Quantopian Open (algo writers won't get 100% of the profit, but they'll be managing far larger sums of money).
Still, perception is reality, so if people perceive the contest as unwinnable then entries will indeed be deterred. We're thinking about other prizes that make sense for people entering. I'm interested in other ideas that might help. I hear the suggestion of "clearing the boards" but that would be contrary to the fund goals, so we're unlikely to choose that option.
As for adding other performance measures: we very well might! For now, the biggest change we want to see in the pool of entries is less beta correlation. When we look at how the pool evolves we may find a new correlation that needs to be corrected. We'll see what the future holds.
Thought I'd chime in on the topic of potential entries/perception of inevitable defeat, and at least give my own reason for not entering the contest (I hope this isn't too off-topic) despite fully supporting this idea of a crowd-sourced hedge fund and understanding Qs need to test things out. As a disclaimer, I've only been using your platform for a couple of months and just plugged in my first algo to paper-trade (IB) yesterday (fingers crossed), so I'm a Q-newbie (Qewbie? Qbie?).
I initially intended to enter the Open. I think it's a great way for Q to get a feel for how things are operating, and it's a great way for me to see how my algorithm performs relative to others' (I'm especially curious to see the correlation statistic). But my algo trades rather frequently, and the 0.03 cent transaction costs simply aren't reasonable for it. This may seem like a small issue, but that's 4x the IB fixed pricing structure of 0.0075/share. Incorporating at least this standard rate, if not the volume-based pricing structure they have, may give algo writers more freedom, and thus give you the diverse pool of algos you're looking for.
An interesting thing to see would be if you have any sort of bias in the Open algos because very-short holding period algos are so heavily penalized. Who knows, I might still submit an algo, if just to get my hands on that correlation score, but I would likely have to force it to trade less to make it a contender.
Picture months from now, July 4 say, just one more winner to be picked August 1st. For the sake of argument we'll assume that the pool of strategies in the Open has grown 5-10% per month. There are now 450 strategies running. The Q has peeled off the top 5 best strats over the last 6 months and fate has dealt their cards, such as they are. Then along comes Byron, a hot quant from Georgia Tech with a serpentine sense of python, who manages to slither out a damn decent strategy. But...
There are 30 algos now crowded around the 90's mark; 85-95, and the market is convulsing. The VIX is over 30 and the leaderboard, at least the top 5% of them or so, are writhing back and forth in their strive to the top. These top strategies, however, have been in the running for months and months. How can our boy Byron ever compete? Would he even want to? As it stands, I cannot see that he would.
So what would make him want to?
A chance. Isn't that what every quant here is yearning for? Every strategy so far submitted, those at least that not born of connivance or chicanery, was done with a touch of hope. So how to foment such hope in Byron? He has to know his efforts at quintessential quantification have at least a chance at being selected to participate in the financial fantasy that Quantopian is offering. How can his strategy rank? As I see it the primary way would be to gate the fresh horses at the same starting post. And then to cream the crop by skimming the top X% into their own pool. That's right. Take the top fraction of algorithms from every fresh race and create their own fund; portioned by position at the finish.
Our player Byron would see that he does indeed have a chance at earning a top slot in a considerably more equitable contest than what he will witness come August.
As Quantopian's vetting and judgement mechanism matures wouldn't it be prudent to expand the selected number and therefore variety of strategies? Listen to Fawce and his quotes of Markowitz; diversification is your goal.
Good luck Byron.
Well, it's just freakin' a lot of work! And now that Q wants algos that are orthogonal to the market and economic winds, it's even more work! Aside from knowing a little Python and getting up the learning curve on the platform (frankly, I still stumble), there's identifying a strategy, developing it, getting it to work well over the past two years in a backtest (ideally without bias), and then deploying it to simulated live trading without it crashing, all while while conforming to the various contest rules. And even if a contestant wins, there's no immediate reward, but more of a coin toss in six months to determine if there will be a payout. So, I can see how it the Open might be a bit daunting and not worth the effort for a guy who stumbles across it on the web.
Regarding needing dozens of algos for the hedge fund, it'll be more than that, if we are talking $1M-$5M per algo. Even 25 algos at $2.5M each is only$62.5M in capital deployed, which doesn't sound like enough to make it a business for Q.
Hey Grant,
I'm one of those guys off the web you speak of above that has stumbled across this contest. I sent you a message on Facebook, would love a response.
Antoine: Thanks, that's good feedback. Yes, we need to revise the sliippage and commissions.
Market: Yes, that chance to win is definitely one of the motivators. And yes, we are definitely going to be skimming off the top X% and putting them into the fund! It's going to be more than one per month, for sure.
Grant: For sure. That work hurdle is a hard one. We're going to keep improving the platform and try to make it easier, but there's some level of effort that is going to be a requirement.
Generally: We're really pleased with how the contest is going, and we're not going to stop in July. We're going to run the contest indefinitely. I can't commit to forever, of course, but we are going to keep running monthly contests for as long as I can see.
Hi Dan,
One thing that may not be clear to folks trying to sort out if the Open is worth the effort, is that they'd also somehow be competing for an opportunity for a spot in the hedge fund. When I go to https://www.quantopian.com/open and search on 'hedge' it doesn't say anything to the effect of "we are definitely going to be skimming off the top X% and putting them into the fund" as you say above. If all it takes is a two-year backtest and one month of simulated live trading to be considered, then it's a great opportunity that you need to highlight, especially if they'd have a shot at $1M-$5M in capital as a manager (and you split a traditional 2/20 fee structure 50/50).
Grant
I absolutely agree with Grant on this. I figured that Open winners would also be considered/looked at closely for the fund, but not necessarily the runners-up or top X%. Definitely something to highlight if that's what Q is planning. Even without binding yourself to a fee structure/capital amount/or X% value, I'd likely reconsider the Open knowing that you're at least looking into the top tranche of algos and not just the winners.
Cheers,
Antoine
And in a separate forum thread, Jess Stauth had alluded to potentially another avenue to getting into the fund, with paper trading results only. Was she referring to the Open? Or will there be something else announced? It seems that the real opportunity is that there will be N fund slots this year, 10*N to 100*N next year, and so forth, with $1M-$5M of capital per slot, so when I land on quantopian.com, it should say "You've hit the jackpot! Submit your algo, and have a shot at $1M-$5M in capital!" rather than "Your..." blah, blah, blah. Oh, just another trading API. No big deal. Move on. Frankly, I think you have something to sell, and you're just not selling it.
Grant, I think you are spot on that our messaging on our homepage and in other places doesn't effectively convey the developments in our business over the past 1 year. That is something we are working on. We need to communicate the value of the opportunity we're presenting to anyone, anywhere. It's like you were sitting in our product planning meeting yesterday :)
In terms of Jess' allusion to using paper trading, we as a company had previously said that to be eligible for the fund, you'd need a real money track record. We no longer think that's the case and the Open is the most obvious way for a user and algo to develop an out of sample track record through paper trading. But really, you can pursue other methods as well, paper (or real money) trading on Quantopian outside the contest. We'll be able to evaluate those algorithms as well. (unlike those quants who approach us to use their track records off of the platform)
Hope that helps,
Josh
Disclaimer
The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by Quantopian. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. No information contained herein should be regarded as a suggestion to engage in or refrain from any investment-related course of action as none of Quantopian nor any of its affiliates is undertaking to provide investment advice, act as an adviser to any plan or entity subject to the Employee Retirement Income Security Act of 1974, as amended, individual retirement account or individual retirement annuity, or give advice in a fiduciary capacity with respect to the materials presented herein. If you are an individual retirement or other investor, contact your financial advisor or other fiduciary unrelated to Quantopian about whether any given investment idea, strategy, product or service described herein may be appropriate for your circumstances. All investments involve risk, including loss of principal. Quantopian makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances.
Are there timelines set in place regarding establishing formal incentives for algo owners who get accepted as a manager? I've been tracking this thread, and it seems like a lot of the future financial gain of the algo owner is up in the air. I've read the manager page, and have observed a lot of "up to x% verbiage that does not resonate well with me, and most likely other potential algo owners as well.
My case specifically, I have an algorithm that does well in Forex (written in Java). If I were to transfer over to Python and stock trading, it would take some time, so I was wondering what kind of financial opportunity would exist in the future if the hedge fund is successful.
Resonating with Michael's reply above, one way to get more people to spend the time to write algos for the contest/fund is to have clear requirements and rewards. If I do X, I'll get Y, with probability P.
@ Josh,
But really, you can pursue other methods as well, paper (or real money) trading on Quantopian outside the contest. We'll be able to evaluate those algorithms as well.
What does this mean, in practice? Do you mean at some future date, users will be able to submit algorithms through a channel other than the Open? Or are you automatically evaluating all algos, live trading on Quantopian, and will contact folks with algos that have fund-worthy attributes and performance? Any advantage to a real-money track record? If so, how much? And what will be the feedback? A simple thumbs up/thumbs down, or something more like the ranking against metrics that you apply to the Open contestants?
In practice, I don't think we have it worked out. But the baseline observation is that an algo that has been paper trading is generating much the same data we'd need to evaluate it as the contest algos.
So in the future, if and when we have a process for evaluating non-contest algorithms for the fund, we would have the ability to evaluate your work.
For example, the 3 entry limit doesn't mean we expect someone to absolutely only have 3 algos worthy of consideration.
Or a winner like yourself shouldn't feel that you only have 1 algo that will be eligible now.
The Quantopian vision clouds as it ages. There appear to be numerous and varied intents stated in the last six months with no clear story to follow. Now, of course the Q is not beholden to anyone but their VC, but some clarity of offering might be due here soon. No doubt you're working just that for release soon; a revised and precise Fund qualification structure with the Open's involvement clearly define along with statements addressing the lowering of barriers to the monetarily challenged via brokerage offerings and tool expansion.
"Nice to be the boys with the best toys." The Q has an enviable position.
Our vision has evolved as we have evolved as a company.
We are, and have always been, laser-focused on what we want to achieve. The fact that we don't know all the details of how we're going to get there doesn't mean that our vision is "cloudy," it means that we are doing something new, something that no one else has done before, and we are not so arrogant as to believe that we know in advance the exact path we will take to get there.
We have, over our entire history, been as open and transparent as possible about what we know and what we don't. We've answered everyone's questions to the best of our ability. When we don't know an answer, we say so, as illustrated by Josh's comment above just an hour ago.
Our story is clear.
Disclaimer
The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by Quantopian. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. No information contained herein should be regarded as a suggestion to engage in or refrain from any investment-related course of action as none of Quantopian nor any of its affiliates is undertaking to provide investment advice, act as an adviser to any plan or entity subject to the Employee Retirement Income Security Act of 1974, as amended, individual retirement account or individual retirement annuity, or give advice in a fiduciary capacity with respect to the materials presented herein. If you are an individual retirement or other investor, contact your financial advisor or other fiduciary unrelated to Quantopian about whether any given investment idea, strategy, product or service described herein may be appropriate for your circumstances. All investments involve risk, including loss of principal. Quantopian makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances.
I must apologize; every other Saturday (well, without the markets on it felt like Saturday) I get up and think to myself "How can I ruffle Jonathan's feathers." It's just one of my habits at this point in time, like sweet rich coffee and a banana.
I would be interested into how the 7 factors have been chosen and why they have equal weights. It would seem to me that there might be a market for various different types of algorithms and therefore for different categories in the competition. For example, some investors have a higher risk profile and would therefore be happy to have more volatility if it ultimately meant higher returns. Am I completely misunderstanding the ultimate business model?
The crowd- funded hedge funds seems like a good idea on paper, but not sure how it will be executed . Why not turn Quantopian as "THE" marketplace for advanced algos, where anyone with an IB or E Trade * account can sign up , look at a list of algos ( approved by Quantopian) developed by the community and classified by risk tolerance, objective, etc , and link their account to it.
For example , @Market Tech has an algo that I would like to trade with my own account, he could charge each user \$50 a month to have access to the algo's signal. Quantopian in turn could take a % of that fee for themselves. The owner of the algo makes money, Quantopian makes money, and I believe total AUM would be greater. This model is being used by collective2. Just my 2 cents....
Cheers
If the intention of the new consistency score is to discourage curve-fitted, "perfect", gamed backtest returns as explained in the original post, why should the consistency score also punish algos whose real world paper-trading performance is better than their backtest performance? Why not just reduce the weight that the backtest score counts towards the final score? To what end would it serve to penalize algos whose actual trading performance exceeds their backtest performance?
Mike T
If your trading performance for a short 30 day period is radically different than your 2 year backtest its a strong indication that your algorithm is showing anomalous returns that aren't representative of its long-term potential. Since long-term potential is what's important, it doesn't make sense to take one month's out of the norm performance and multiply it by 12 to get the algorithm's long-term performance.
Kevin: I understand and agree with what you are saying about using 30 days of trading to judge long-term potential, but along those same lines, backtesting results (especially just two year ones) are not representative of the future long-term potential of an algorithm either. The 2013-2015 market is completely different than the 2010-2012 market as I'm sure 2015-2017 market will be completely different also. The current month's "out of the norm" performance might actually be the new norm going forward.
In my own personal experience, I've never seen backtesting results translate that effectively to live trading. So many strategies and systems have backtests that look amazing, yet when they are actually turned on and traded live in the current market, its very rare that any of them actually match their backtesting performance. The fact that none of the algos in the competition have currently have both a top 20 paper trading and backtesting score somewhat speaks to that. Most of the top algos either have great backtests with mediocre live trading results or mediocre backtests with great live trading results. Just my 2 cents.
There are many ways to game this competition. These manipulations would not create successful long-term automated algorithms. The biggest problem, in my opinion, is the short-time frames being used. Consider this method:
1) Use a stock scanner to find stocks with a low beta
2) Write a simple algorithm to buy/sell based on technical indicators (MACD/RSI/etc)
3) Select stocks with the best back-test results, tweaking the algorithm to achieve the best entry and exits
4) Manually review each stock and look for potential technical analysis formations (descending triangle, head-and-shoulders, etc)
5) Submit the algorithm when you expect a break-out to occur
6) Re-submit the algorithm each day a break-out doesn't occur, using new stocks with potential formations
7) When a breakout happens, you will have a high scoring algorithm with amazing back-test and paper trading results.
The problem is that steps 4-6 are using information that is NOT part of the algorithm. This bias will reward users who can guess a short-term move based on a manual review of the chart.
I think the most important fix to judging algorithms is to extend the time that they are paper trading. This could be either a hard requirement (i.e. "algos must paper trade for 6-months before entering") or perhaps some method to reward longer paper trading.
Quantopian's investors may not have this much patience, but I think it is necessary to identify high-quality algos that would be successful long-term. Quantopian users may not like this change, but I believe the reward of potentially being a hedge fund manager is worth waiting 6-months time. | 2018-10-16 20:19:42 | {"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": 0.29438719153404236, "perplexity": 1628.7018083237729}, "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-2018-43/segments/1539583510867.6/warc/CC-MAIN-20181016201314-20181016222814-00039.warc.gz"} |
http://tex.stackexchange.com/questions/106308/how-to-use-the-pullquote-package-on-a-two-column-layout | # How to use the pullquote package on a two-column layout? [closed]
I've been reading about pull quotes. In this question, a great answer is posted with a link to a custom package called pullquote.
I've installed this package and have been trying to make it work, as part of a document which already has two columns. Here's what I want to achieve:
Here is a MWE:
\documentclass[twocolumn]{article}
\usepackage{xunicode}
\usepackage{pullquote}
\usepackage{lipsum}
\usepackage[english]{babel}
\begin{document}
{%
\Large
\begin{tabular}[b]{p{5cm}}
\textit{Wir m\"ussen wissen.}\\
\textit{Wir werden wissen.}\\
\mbox{}\hfill\large\textsc{David Hilbert}\\
\end{tabular}%
}
\lipsum
\begin{pullquote}
{
}
\lipsum[1-3]
\end{pullquote}
\lipsum
\end{document}
This gives me THREE columns wherever the pull quote appears:
I've taken code from this discussion about the issue, but can not make it work. Any idea how to implement a pull quote in a two-column layout?
-
## closed as too localized by Speravir, egreg, percusse, Thorsten, QrrbrbirlbelApr 6 '13 at 6:18
This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.
I think this is a duplicate: How do you create pull quotes?. – m0nhawk Apr 1 '13 at 9:18
@m0nhawk I've read that question and others here, but my question is specifically about the pullquote package, and about documents that are already formatted with two columns. The only other information on this same topic I found here came up in the discussion that I linked above, but not in a question with an answer. – user Apr 1 '13 at 10:02
First: I get an output only with XeTeX and deactivated microtype (pullquote package option nomicrotype). You should have mentioned this. Second: This is a package in development (you could have known this from the package resource): What you show is clearly a bug, and for this kind of issues TeX.SX is the wrong place. In documentation an e-mail address is given, or you try to contact Stephan in chat of TeX.SX. – Speravir Apr 1 '13 at 22:44 | 2014-12-22 21:32:27 | {"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": 0.5467036366462708, "perplexity": 1402.8026273384494}, "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-2014-52/segments/1418802776996.17/warc/CC-MAIN-20141217075256-00090-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/561372/how-to-show-that-l-i-v-j-i-hbar-sum-k-epsilon-ijkv-k-for-any-vector-te | # How to show that $[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$ for any vector $\textbf{v}$ constructed from $\textbf{x}$ and/or $\nabla$?
In Weinberg's Lectures on Quantum Mechanics (pg 31), he said that the commutator relation $$[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$$ is true for any vector $$\textbf{v}$$ constructed from $$\textbf{x}$$ and/or $$\nabla$$, where $$\textbf{L}$$ is the angular momentum operator given by $$\textbf{L}=-i\hbar\textbf{x} \times \nabla$$.
An example for vector $$\textbf{v}$$ is the angular momentum $$\textbf{L}$$ itself: $$[L_i,L_j] = i\hbar \sum_k \epsilon_{ijk} L_k.$$ Other examples include $$\textbf{v}=\textbf{x}$$ and $$\textbf{v}=\nabla:$$ $$[L_i,x_j] = i\hbar \sum_k \epsilon_{ijk} x_k,$$ $$[L_i,\frac{\partial}{\partial x_j}] = i\hbar \sum_k \epsilon_{ijk} \frac{\partial}{\partial x_k}.$$
How can it be shown that the commutator relation $$[L_i, v_j]=i\hbar\sum_k \epsilon_{ijk}v_k$$ is indeed true for any vector $$\textbf{v}$$ constructed from $$\textbf{x}$$ and/or $$\nabla$$?
Edit: I am looking for an answer that does not simply say that this is the definition of a vector operator. In fact, I think that Weinberg refers to $$\textbf{v}$$ as a vector, not a vector operator.
• I think you have it the wrong way around: this is the definition of a vector operator in quantum mechanics. – Philip Jun 24 '20 at 5:42
• Yes Phillip is right, most authors use equation 1 to define vector operators. – user224659 Jun 24 '20 at 5:42
• I think the key question here is what you mean by "constructed" – user224659 Jun 24 '20 at 5:49
• @Philip Perhaps it's like this: equation 1 is true for any vector operator constructed from $\vec{x}$ and $\nabla$. All vector operators in QM are constructed from $\vec{x}$ and $\nabla$. Hence we define vector operators in QM as operators that satisfy equation 1. But that brings us back to the orignial question: why is equation 1 is true for any vector operator constructed from $\vec{x}$ and $\nabla$.? – TaeNyFan Jun 24 '20 at 7:02
• Well, total angular momentum (with spin) is a vector operator that isn't constructed solely from $\vec{x}$ and $\vec{\nabla}$... Though I admit that I wouldn't be personally completely convinced by this counter-example in your position. It seems to me that to answer this question we need another definition of a vector operator. How would you want to define a vector operator $\vec{v}$, independent of this above "definition"? To my mind, this, or rather the more general but equivalent $$U^\dagger(R) V_i U(R) = R_{ij}V_j$$ is the only way, but I'm no expert. – Philip Jun 24 '20 at 7:22
Note: I suspect we might have a slightly different reading of the line from Weinberg's book:
It can be shown that [the commutation relation you specify] is true of any vector $$\mathbf{v}$$ that is constructed from $$\mathbf{x}$$ or $$\mathbf{\nabla}$$.
I feel that the stress is on the word "vector", not on the words "$$\mathbf{x}$$ or $$\mathbf{\nabla}$$". Furthermore, even though the quantity $$\mathbf{v}$$ is referred to as a vector and not a vector operator, I think it has to be an operator for this to make sense quantum mechanically. I do not know how to derive a commutation such as this without taking the quantities in question to be operators! I think that this early on in the book, Weinberg didn't want to speak of Spin and total angular momentum, and so he keeps it simple by only mentioning external degrees of freedom, especially since his actual goal is to solve the central potential problem. He is much clearer on the subject when he speaks of Rotations and Spin in Chapter 4 (around pg. 100 of your link).
But let's see if I can be convincing. (Apologies if it's too long!)
What is a vector?
This is the hardest part since I'm not sure exactly how you want to define a vector operator. But let's say we use a very basic definition of a vector: a vector $$\mathbf{V}$$ is any combination of 3 components which all transform a particular way under rotations. In other words, under a rotation:
$$V_i' = R_{ij} V_j,$$
where $$R_{ij}$$ are the components of the $$3\times 3$$ matrix denoting geometric rotations. For example, for a rotation in the $$xy-$$plane by an angle $$\theta$$,
$$R_z(\theta) = \begin{pmatrix}\cos{\theta}&\sin{\theta}&0\\-\sin{\theta}&\cos{\theta}&0\\0&0&1\end{pmatrix}$$
We will be working with infinitesimal rotations, and so it's a good idea to have a general form for $$R$$ when the angle is infinitesimally small. If we rotate a vector $$\mathbf{v}$$ in a plane orthogonal to an axis-vector $$\mathbf{\hat{u}}$$, then
$$$$R_\mathbf{\hat{u}}(\text{d}\theta) \mathbf{v} \approx \mathbf{v} - \text{d}\theta\,\, \mathbf{\hat{u}}\times \mathbf{v}. \tag{1}$$$$
I won't prove it this here in full generality, but it's quite instructive to prove that this is true for $$R_z$$ by doing a Taylor expansion in $$\theta$$.
How does a general operator transform under symmetry transformations?
Given the transformation properties of physical states, we can easily derive the transformation properties of the operators that act on them. Let us consider a state $$|\psi\rangle$$ and its image under a rotation $$|\psi'\rangle = \mathcal{R}|\psi\rangle$$. In this case, $$\mathcal{R}$$ is the unitary operator in the Hilbert Space that represents the symmetry operation of rotating a state. (This is not the $$3\times 3$$ matrix $$R$$.)
When acting on $$|\psi\rangle$$ with some operator $$\mathcal{O}$$ we obtain a new state $$|\phi\rangle = \mathcal{O}|\psi\rangle$$. How does $$|\phi\rangle$$ transform under a rotation? Well, that's simple:
\begin{equation*} \begin{aligned} |\phi\rangle \rightarrow \mathcal{R}|\phi\rangle &= \mathcal{R O} |\psi\rangle\\ &= \mathcal{ROR^\dagger} \left( \mathcal{R}|\psi\rangle \right)\\ &= \mathcal{O'} |\psi'\rangle \end{aligned} \end{equation*}
Thus under a rotation,
\begin{equation*} \begin{aligned} |\psi\rangle \rightarrow |\psi'\rangle &= \mathcal{R}|\psi\rangle,\\ \mathcal{O} \rightarrow \mathcal{O'}&= \mathcal{R O R^\dagger}. \end{aligned} \end{equation*}
The operator $$\mathcal{R}$$ in state-space can be represented as $$\mathcal{R}(\theta) = e^{-i\theta \,\mathbf{\hat{u}\cdot L}/\hbar}.$$ This is true under a physical rotation of the coordinates. (i.e. for spin-less particles. It's not too hard to prove this, but I'm omitting it here. If you'd like me to show it for $$\mathcal{R}_z$$, I could. Let me know in the comments.) Thus, an infinitesimal rotation by $$\text{d}\theta$$ is represented by
$$$$\mathcal{R}_\mathbf{\hat{u}}(\text{d}\theta) \approx \mathbb{1} - \frac{i}{\hbar} \text{d}\theta\,\mathbf{\hat{u}\cdot L} \tag{2}$$$$
What is a vector operator?
Combining the two ideas above, it should hopefully be clear that a vector operator is one that transforms as
$$\begin{equation*} \begin{pmatrix}\mathcal{V}_x\\\mathcal{V}_y\\\mathcal{V}_z\end{pmatrix} \rightarrow \begin{pmatrix}\mathcal{V'}_x\\\mathcal{V'}_y\\\mathcal{V'}_z\end{pmatrix} = \mathcal{R} \begin{pmatrix}\mathcal{V}_x\\\mathcal{V}_y\\\mathcal{V}_z\end{pmatrix} \mathcal{R^\dagger} = R \begin{pmatrix}\mathcal{V}_x\\\mathcal{V}_y\\\mathcal{V}_z\end{pmatrix} \end{equation*}$$
i.e. $$$$\boxed{\mathcal{R} \mathcal{V}_i \mathcal{R^\dagger} = R_{ij} \mathcal{V}_j} \tag{3}$$$$
(Notice that the symbols for $$\mathcal{R}$$ and $$R$$ are different as they act on vectors in different vector spaces. Weinberg uses $$U(R)$$ instead of $$\mathcal{R}$$) In other words, under a rotation, its components transform under a unitary transformation exactly as vectors would in 3D space.
Putting it all together:
Let's now perform an infinitesimal rotation and see what the equation above becomes. Using Equation (2),
$$\mathcal{R}(\text{d}\theta) \mathcal{V}_i \mathcal{R^\dagger}(\text{d}\theta) = \left( \mathbb{1} - \frac{i}{\hbar} \text{d}\theta\,\mathbf{\hat{u}\cdot L} \right)\, \mathcal{V}_i \left(\mathbb{1} + \frac{i}{\hbar} \text{d}\theta\,\mathbf{\hat{u}\cdot L}\right) = \mathcal{V}_i - \frac{i}{\hbar} \text{d}\theta\,[\mathbf{\hat{u}\cdot L},\mathcal{V}_i].$$
Similarly, using Equation (1),
$$R_{ij}(\text{d}\theta) \mathcal{V}_j = \mathcal{V}_i - \text{d}\theta\,\epsilon_{ijk}\hat{u}_j \mathcal{V}_k,$$
where I've used the definition $$(\mathbf{A}\times\mathbf{B})_i = \epsilon_{ijk}A_j B_k$$, and the summation over repeated indices is implied.
Thus, Equation (3) becomes
$$[\mathbf{\hat{u}\cdot L},\mathcal{V}_i] = -i\hbar \epsilon_{ijk}\hat{u}_j \mathcal{V}_k.$$
To get the specific results for $$L_i$$, the components of $$\mathbf{L}$$ in the Cartesian axes, we can successively choose $$\mathbf{\hat{u}}$$ as the unit vectors $$\mathbf{\hat{x}},\mathbf{\hat{y}},$$ and $$\mathbf{\hat{z}}$$, so that we get
$$\begin{equation*} [L_j, \mathcal{V}_i] = -i\hbar \epsilon_{ijk}\mathcal{V}_k \end{equation*}$$
or, by exchanging the indices $$i \leftrightarrow j$$ and using the antisymmetry properties of the $$\epsilon$$ symbol and commutator,
$$\begin{equation*} \boxed{[L_i, \mathcal{V}_j] = i\hbar \epsilon_{ijk}\mathcal{V}_k} \end{equation*}$$
You'll notice that I haven't mentioned anything to do with the operators $$\mathbf{x}$$ and $$\mathbf{\nabla}$$ here. The point is (as has been indicated in other answers) that not every combination of the above two operators is a vector operator. So how would one know that one particular combination was a vector operator? Why is (for example) $$\mathcal{A \times B}$$ a vector operator and not $$\mathcal{A\cdot B}$$? Well, it must transform as a vector, i.e. as Equation (3)!
In other words, this particular proof is much more general than just proving it for all vector operators composed of $$\mathbf{x}$$ and $$\mathbf{\nabla}$$, which seems to me to be harder to prove even though it's a weaker statement than the one we've just proved! So I don't see the point in trying to prove it.
Also, while it is true that so far as I know all vector operators are in fact "composed" of $$\mathbf{x}$$ and $$\mathbf{\nabla}$$ I see no reason in QM that that be the case. Tomorrow, if a funky observable was found that "vector" observable was found that didn't depend on $$\mathbf{x}$$ and $$\mathbf{\nabla}$$, it would still satisfy such equations!
In particular, this argument also works for Spin, which is a vector operator that isn't composed of position and momentum operators, provided me make the simple generalisation to shift $$\mathbf{L} \to \mathbf{J}$$ to account for the internal degrees of freedom. Weinberg himself discusses all of this in his chapter on Rotations (around pg. 100 of the link you've posted).
• I do not think all of this is relevant and second guessing Weinberg is really unwise. However, you did correctly point out that one does not need to prove Weinbergs statement since L is generator of rotations, eijk is group structure tensor so if one had a group theory class this is what follows. The question was, however, posed in constructive way, please prove this to me. and that can be done, if needed will do below. – JohannR Jul 5 '20 at 20:50
• Dear @JohannR, thank you for your feedback. As I pointed out, I was not sure how one would show that any operator composed of $\mathbf{x}$ and $\mathbf{\nabla}$ was a vector operator without using Equation (3). As for the relevance of this, I felt that it was necessary to motivate the idea of a vector operator. If there is another way, I'd be very happy to know, please let me know when you post your answer. – Philip Jul 5 '20 at 20:56
• See below, and I hope you agree and show your approaval! – JohannR Jul 5 '20 at 21:09
• @JohannR I would like to point out that I have nowhere used the fact that "$\epsilon_{ijk}$ is the structure of the rotation group", I have merely used the definition of the cross-product. – Philip Jul 5 '20 at 21:25
• which is the only way to define antisymmetric product on group manifold...... – JohannR Jul 5 '20 at 21:38
It follows from the fact that $$\hat L_i$$ are generators of the rotations.
Generally, the vector operator $$\hat V_i$$ is not just "a list" of operators. It has to transform like a vector under rotations $$\hat V_i' = R_{ij} \hat V_j$$ (summation implied).
This identity can be proven by considering an observable in a rotated frame of reference. The frame rotation can be carried out through a transformation $$R$$ acting on $$\hat V$$ or a unitary operator $$\hat{U}(R)$$: $$$$\left._R\right. \langle\alpha|\hat{V}_i|\beta\rangle_R = \langle\alpha|\hat U^\dagger \hat V_i \hat U|\beta\rangle = R_{ij}\langle\alpha| \hat V_j |\beta\rangle$$$$
Thus, we have: $$\hat U^\dagger \hat V_i \hat U = R_{ij} \hat V_j$$. By considering infinitesimal rotation $$\hat U = 1 - \frac{i \epsilon \bf{J}\cdot \bf{n}}{\hbar}$$ and corresponding $$R = \left(\begin{matrix}1& -\epsilon & 0\\ \epsilon & 1 &0 \\ 0 & 0 & 1\end{matrix}\right)$$ we find at first order in $$\epsilon$$: $$$$[V_i, J_j] = i \hbar \epsilon_{ijk} V_k$$$$
A nice proof of this identity can be found here: https://www.oulu.fi/tf/kvmIII/english/2004/09_tensop.pdf.
Similar question: Angular and linear momentum operators' commutation
Another useful reference: https://www.wikiwand.com/en/Tensor_operator#/Vector_operators
Theorem: Let $$\mathbf{A},\mathbf{B}$$ be vector operators. then $$\mathbf{A}\times\mathbf{B}$$ is a vector operator. But e.g. $$\mathbf{A}\mathbf{B}$$ is a scalar. That is $$[L_j,\mathbf{A}\mathbf{B}]=0$$
The proof is done by straight forward algebra, using the definition of a vector operator $$[L_j,A_i]=i\hbar\epsilon_{jik}A_k$$.
But I think your confusion isn't related to the algebra, but stems from the term "constructed". This is very imprecise language. As the theorem shows, not all combinations or even products of two vector operators are vector operators.
• The author also specified that the operator $\textbf{v}$ is a vector, so that already leaves out the combination of dot products. – TaeNyFan Jun 24 '20 at 6:56
• Also please see my comment on the question and tell me what you think! – TaeNyFan Jun 24 '20 at 7:01
• @TaeNyFan It's not clear to me what kind of operations you are interested in. The sum and the cross product of vector operators are once again vector operators. If you want to extend the result to other operations you need to state about which operations you are talking about. I can think of a million funny ways to "construct" a operator by two operators. Many of these will not produce vector operators. – user224659 Jun 24 '20 at 7:43
• Furthermore you should be careful to differentiate between a group of 3 operators, ordered in a coloumn vector and a vector operator. – user224659 Jun 24 '20 at 7:43
• What I mean is that $\textbf{v}$ is any combination of $\textbf{x}$ and $\nabla$, with the criteria that the combination is a vector operator. Hence, sums and cross with $\textbf{x}$ and $\nabla$ are valid, while dot products and 'division' are not. One of my queries is also what are all the possible combinations of $\textbf{x}$ and $\nabla$ that give a vector operator $\textbf{v}$. – TaeNyFan Jun 24 '20 at 7:46
The question asks for a constructive procedure: any vector $$\mathbf{v}$$ can be written in terms of two scalar functions multiplied by the two allowed vectors $$\mathbf{x}$$, and the gradient, respectively. These scalar functions depend on $$\mathbf{x\cdot\nabla}$$, $$\mathbf{x}^2$$ and $$\nabla^2$$. For simplicity let us forget about poles and cuts, so use for each (analytical) function a power series in these 3 variables. Remembering rules of commutation relation for $$\mathbf{x}$$ with powers of $$\mathbf{p}$$ and $$\mathbf{p}$$ with powers of $$\mathbf{x}$$ (sign change) and applying them to these power series one indeed finds the result presented in Weinberg.
NOTE: this is how one shows constructively that $$\mathbf{L}$$ is the generator of rotations, and $$\epsilon_{ijk}$$ is the structure of the rotation group, all valid answers above assumed this, thus these were not answers to the question posed.
• It's not clear to me how this answers the question beyond effectively stating that the result is correct. Moreover, are there not 3 functions (rather than 2): $\textbf{x}\cdot \nabla$, $\textbf{x}^2$ and $\nabla^2$? – ZeroTheHero Jul 5 '20 at 21:24
• Furthermore, I am not certain I understand what you mean by "any vector $\mathbf{v}$ can be written in terms of two scalar functions multiplied by the two allowed vectors $\mathbf{x}$, and the gradient." Do you mean that $$\mathbf{v} = f(\mathbf{x},\nabla) \mathbf{x}+ g(\mathbf{x},\nabla) \mathbf{\nabla},$$ where $f$ and $g$ are scalar functions depending only on the quantities you mention? In which case, how would one represent $\mathbf{L}$, for example, using this? And even if that were true, I am not certain how one would go about actually proving it, which was what the OP asked anyway. – Philip Jul 5 '20 at 21:30
• v=f(x^2,∇^2, x.∇)x+g(x^2,∇^2, x.∇)∇, expand both f,g in powers x^2n,∇^2k, x.∇^m and the rest is just keeping track of signs, powers, remembering rules of commutations. – JohannR Jul 5 '20 at 21:37
• Ok, and what about representing a vector like $\mathbf{L}$ in this case? What would your functions $f$ and $g$ be? In fact, since classically $\mathbf{L}$ is "orthogonal" to the plane produced by $\mathbf{x}$ and $\mathbf{p}$, I doubt it could be written as vector sum of $\mathbf{x}$ and $\mathbf{p}$... – Philip Jul 5 '20 at 21:43
• You need to construct L such that it is a group generator, this is for rotation group a unique construction - I believe presented explicitly in red cover Sakurai QM book for example (that is original edition and that part is written by Sakurai so is correct, Sakurai died writing the book, alas. – JohannR Jul 5 '20 at 22:03 | 2021-06-19 14:48:42 | {"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": 111, "wp-katex-eq": 0, "align": 0, "equation": 5, "x-ck12": 0, "texerror": 0, "math_score": 0.9446007609367371, "perplexity": 285.5868341566264}, "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/1623487648373.45/warc/CC-MAIN-20210619142022-20210619172022-00149.warc.gz"} |
https://socratic.org/questions/a-charge-of-5-c-is-at-1-3-and-a-charge-of-1-c-is-at-8-4-if-both-coordinates-are- | # A charge of -5 C is at (1, 3 ) and a charge of 1 C is at (8,4 ) . If both coordinates are in meters, what is the force between the charges?
Mar 7, 2018
$3.6 \cdot {10}^{9} N$ or $3.6 G N$
#### Explanation:
You first want to use the coordinates to calculate a distance between the charges.
$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$
The distance between the two charges is $5 \sqrt{2}$ meters
This will be your radius when calculating for the electric force. The formula for the electric force is
$F {e}^{-} = \frac{K {q}_{1} {q}_{2}}{r} ^ 2$
where $K = 9 \cdot {10}^{9}$ and the units for q is in coulombs and the r is in meters.
$F {e}^{-} =$($9 \cdot {10}^{9}$)($- 5$)(4)/${\left(5 \sqrt{2}\right)}^{2}$ | 2019-12-16 10:07:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "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": 0.8962785005569458, "perplexity": 430.7463032449247}, "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/1575541319511.97/warc/CC-MAIN-20191216093448-20191216121448-00300.warc.gz"} |
http://www.r-bloggers.com/example-10-8-the-upper-95-ci-is-3-69/ | # Example 10.8: The upper 95% CI is 3.69
December 10, 2012
By
(This article was first published on SAS and R, and kindly contributed to R-bloggers)
Apologies for the long and unannounced break-- the longest since we started blogging, three and a half years ago. I was writing a 2-day course for SAS users to learn R. Contact me if you're interested. And Nick and I are beginning work on the second edition of our book-- look for it in the fall. Please let us know if you have ideas about what we omitted last time or would otherwise like to see added. In the mean time, we'll keep blogging, though likely at a reduced rate.
Today: what can you say about the probability of an event if the observed number of events is 0? It turns out that the upper 95% CI for the probability is 3.69/N. There's a sweet little paper with some rationale for this, but it's in my other office. And I couldn't recall the precise value-- so I used SAS and R to demonstrate it to myself.
R
The R code is remarkably concise. After generating some Ns, we write a little function to perform the test and extract the (exact) upper 95% confidence limit. This is facilitated by the "..." notation, which passes along unused arguments to functions. Then we use apply() to call the new function for each N, passing the numerator 0 each time. Note that apply() needs a matrix argument, so the simple vector of Ns is converted to a matrix before use. [The sapply() function will accept a vector input, but took about 8 times as long to run.] Finally, we plot the upper limit * N against N. showing the asymptote. A log scaled x-axis is useful here, and is achieved with the log='x' option. (Section 5.3.12.) the result is shown above.
bin.m = seq(10, 10000, by=5)
mybt = function(...) { binom.test(...)\$conf.int[2] }
uci = apply(as.matrix(bin.m), 1, mybt, x=0)
plot(y=bin.m * uci, x=bin.m, ylim=c(0,4), type="l",
lwd=5, col="red", cex=5, log='x',
ylab="Exact upper CI", xlab="Sample size",
main="Upper CI when there are 0 cases observed")
abline(h=3.69)
SAS
In SAS, the data, really just the N and a numerator of 0, are generated in a data step. The CI are found using the binomial option in the proc freq tables statement and saved using the output statement. Note that the weight statement is used here to avoid having a row for each Bernoulli trial.
data binm;
do n = 10 to 10000 by 5;
x=0;
output;
end;
run;
ods select none;
proc freq data=binm;
by n;
weight n;
tables x / binomial;
output out=bp binomial;
run;
ods select all;
To calculate the upper limit*N, another data step is needed-- note that in this setting SAS will only produce the lower limit against the probability that all observations share the same value, thus the subtraction from 1 shown below. The log scale x-axis is obtained with the logbase option to the axis statement. (Section 5.3.12.) The result is shown below.
data uci;
set bp;
limit = (1-xl_bin) * n;
run;
axis1 order = (0 to 4 by 1);
axis2 logbase=10 logstyle=expand;
symbol1 i = j v = none c = red w=5 l=1;
proc gplot data=uci;
plot limit * n / vref=3.69 vaxis=axis1 haxis=axis2;
label n="Sample size" limit="Exact upper CI";
run;
quit;
It's clear that the upper 95% limit on the number of successes asymptotes to about 3.69. Thus the upper limit on the binomial probability p is 3.69/N. | 2013-12-13 07:36:11 | {"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": 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": 0.5496109127998352, "perplexity": 2334.0583794626114}, "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/1386164911644/warc/CC-MAIN-20131204134831-00032-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/prealgebra/prealgebra-7th-edition/chapter-2-section-2-1-introduction-to-integers-exercise-set-page-105/84 | ## Prealgebra (7th Edition)
Since the absolute value of a number $A$ is that number's distance from 0 on the number line, if $A=-13$, then $|-13|=13$. Since the opposite of a number $A$ is to put a negative sign in front of the number, if $A=-13, -A=-(-13)=13$. | 2018-07-17 19:51:24 | {"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": 0.9146449565887451, "perplexity": 157.40164880480683}, "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-2018-30/segments/1531676589892.87/warc/CC-MAIN-20180717183929-20180717203929-00099.warc.gz"} |
https://socratic.org/questions/what-is-the-difference-between-an-antiderivative-and-an-integral | # What is the difference between an antiderivative and an integral?
Mar 17, 2015
There are no differences, the two words are synonymous.
Mar 19, 2015
It depends on a couple of things. Which antiderivative, the general or a particular? which integral definite or indefinite? And, who are we asking?
General Antiderivative and Indefinite Integral:
Many mathematicians do not distinguish the indefinite integral and the general antiderivative. In either case for function $f$ the "answer" is $F \left(x\right) + C$ where $F ' \left(x\right) = f \left(x\right)$..
Some (for instance, textbook author James Stewart) make a distinction. What Stewart refers to as "the most general" antiderivative of $f$, admits different constants at each discontiuity of $f$. For example, he would answer that the most general antiderivative of $\frac{1}{x} ^ 2$ is a piecewise defined function:
$F \left(x\right) = \frac{- 1}{x} + {C}_{1}$ for $x < 0$ and $\frac{- 1}{x} + {C}_{2}$ for $x > 0$.
The indefinite integral of $f$, in this treatment, is always an antiderivative on some interval on which $f$ is continuous.
So $\int \frac{1}{x} ^ 2 \mathrm{dx} = - \frac{1}{x} + C$, where it is understood that the domain is restricted to some subset of either the positive reals or a subset of the negative reals.
Particular Antiderivatives
A particular antiderivative of $f$ is a function $F$ (rather than a family of functions) for which $F ' \left(x\right) = f \left(x\right)$.
For example:
$F \left(x\right) = \frac{- 1}{x} + 5$ for $x < 0$ and $\frac{- 1}{x} + 1$ for $x > 0$.
is a particular antidervative of $f \left(x\right) = \frac{1}{x} ^ 2$
And:
$G \left(x\right) = \frac{- 1}{x} - 3$ for $x < 0$ and $\frac{- 1}{x} + 6$ for $x > 0$.
is a different particular antidervative of $f \left(x\right) = \frac{1}{x} ^ 2$.
Definite integrals
The definite integral of $f$ from $a$ to $b$ is not a function. It is a number.
For example:
${\int}_{1}^{3} \frac{1}{x} ^ 2 \mathrm{dx} = \frac{2}{3}$.
(To further complicate matters, this definite integral can be found, using the Fundamental Theorem of Calculus, Part 2, by finding the/an indefinite integral / general antiderivative first, then doing somearithmetic.)
Mar 19, 2015
Your question is related to what was truly the "key insight" in the development of calculus by Isaac Newton and Gottfried Leibniz.
Focusing on functions that are never negative, this insight can be phrased as: "Antiderivatives can be used to find areas (integrals) and areas (integrals) can be used to define antiderivatives". This is the essence of the Fundamental Theorem of Calculus.
Without worrying about Riemann sums (after all, Bernhard Riemann lived almost 200 years after Newton and Leibniz anyway) and taking the notion of area as an intuitive (undefined) concept, for a continuous non-negative function $f \left(x\right) \setminus \ge q 0$ for all $x$ with $a \setminus \le q x \setminus \le q b$, just think of the definite integral symbol $\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ as representing the area under the graph of $f$ and above the $x$-axis between $x = a$ and $x = b$. If another function $F$ can be found so that $F ' \left(x\right) = f \left(x\right)$ for all $a \setminus \le q x \setminus \le q b$, then $F$ is called an antiderivative of $f$ over the interval $\left[a , b\right]$ and the difference $F \left(b\right) - F \left(a\right)$ equals the value of the definite integral. That is, $\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx} = F \left(b\right) - F \left(a\right)$. This fact is useful for finding the value of a definite integral (area) when a formula for an antiderivative can be found.
Conversely, if we make the upper limit of the integral symbol a variable, call it $t$, and define a function $F$ by the formula $F \left(t\right) = \setminus {\int}_{a}^{t} f \left(x\right) \mathrm{dx}$ (so $F \left(t\right)$ is really the area under the graph of $f$ between $x = a$ and $x = t$, assuming $a \setminus \le q t \setminus \le q b$), then this new function $F$ is well-defined, differentiable, and $F ' \left(t\right) = f \left(t\right)$ for all numbers $t$ between $a$ and $b$. We have used an integral to define an antiderivative of $f$. This fact is useful for approximating values of an antiderivative when no formula for it can be found (using numerical integration methods like Simpson's rule). For instance, it's used all the time by statisticians when approximating areas under the Normal curve. The values of a special antiderivative of the standard Normal curve are often given in a table in statistics books.
In the case where $f$ has negative values, the definite integral must be thought of in terms of "signed areas". | 2021-12-01 08:18:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 61, "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": 0.9555019736289978, "perplexity": 239.67574223363562}, "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/1637964359093.97/warc/CC-MAIN-20211201052655-20211201082655-00422.warc.gz"} |
https://www.paulamoraga.com/course-aramco/01-probability-distributions.html | # Learning objectives
## 1.1 Random variables
• A random process is a random phenomenon or experiment that can have a range of outcomes. For example, tossing a coin, rolling a dice, or measuring the height of a randomly selected individual.
• A random variable is a variable whose possible values are numbers associated to the outcomes of a random process. Random variables are usually denoted with capital letters such as $$X$$ or $$Y$$. There are two types of random variables: discrete and continuous.
• A discrete random variable is one which may take only a finite or countable infinite set of values. For example:
$$X =$$ outcome obtained when tossing a coin (e.g., $$X=1$$ if heads and $$X = 0$$ if tails)
$$X =$$ number of members in a family randomly selected in USA (e.g., 1, 2, 3)
$$X =$$ number of patients admitted in a hospital in a given day
$$X =$$ number of defective light bulbs in a box of 20
$$X =$$ year that a student randomly selected in the University was born (e.g., 1998, 2000)
• A continuous random variable is one which takes an infinite non-countable number of possible values. For example:
$$X$$ = height of a student randomly selected in the department (e.g., 1.65 m, 1.789 m)
$$X$$ = time required to run 1 Km (e.g., 4.56 min, 5.123 min)
$$X$$ = amount of sugar in an orange (e.g., 9.21 g, 12.2 g)
## 1.2 Probability distribution
A probability distribution of a random variable $$X$$ is the specification of all the possible values of $$X$$ and the probability associated with those values.
### 1.2.1 Probability mass function
If $$X$$ is discrete, the probability distribution is called probability mass function. We can describe the probability mass function of $$X$$ by making a table with all the possible values of $$X$$ and their associated probabilities. We can also represent graphically the probability mass function using a barplot.
Let $$X$$ be a discrete random variable that may take $$n$$ different values $$X = x_i$$ with probability $$P(X = x_i) = p_i$$, $$i =1,\ldots,n$$. The probability mass function of $$X$$ can be represented as follows:
Value of $$X$$ $$P(X = x_i)$$
$$x_1$$ $$p_1$$
$$x_2$$ $$p_2$$
$$x_n$$ $$p_n$$
The probability mass function must satisfy the following:
• $$0 < p_i < 1$$ for all $$i$$
• $$\sum_{i=1}^n p_i = p_1 + p_2 + \ldots + p_n = 1$$
Example
Let $$X$$ be a discrete random variable with the following probability mass function.
-
Outcome 1 2 3 4 5 6
Probability 0.20 0.40 0.10 0.10 0.13 0.07
1. Check the validity of the probability mass function.
Probabilities between 0 and 1 and sum to 1.
1. Calculate the probability that $$X$$ is equal to 2 or 3.
P(X = 2 or X = 3) = P(X = 2) + P(X = 3) = 0.4 + 0.1 = 0.5.
1. Calculate the probability that $$X$$ is greater than 1
P(X > 1) = 1 - P(X = 1) = 1 - 0.2 = 0.8
### 1.2.2 Probability density function
If $$X$$ is continuous, the probability distribution is called probability density function. A continuous variable $$X$$ takes an infinite number of possible values, and the probability of observing any single value is equal to 0.
Therefore, instead of discussing the probability of $$X$$ at a specific value $$x$$, we deal with $$f(x)$$, the probability density function of $$X$$ at $$x$$.
We cannot work with probabilities of $$X$$ at specific values, but we can assign probabilities to intervals of values. The probability of $$X$$ being between $$a$$ and $$b$$ is given by the area under the probability density curve from $$a$$ to $$b$$.
$P(a \leq X\leq b) = \int_a^b f(x)dx$
The probability density function $$f(x)$$ must satisfy the following:
• The probability density function has no negative values ($$f(x) > 0$$ for all $$x$$)
• Total area under the curve is equal to 1 ($$\int_{-\infty}^{-\infty} f(x)dx = 1$$)
## 1.3 Cumulative distribution function
The cumulative distribution function (CDF) of a random variable $$X$$ denotes the probability that $$X$$ takes a value less than or equal to $$x$$, for every value of $$x$$.
If $$X$$ is discrete, the cumulative distribution function at a value $$x$$ is calculated as the sum of the probabilities of the values that are less than or equal to $$x$$:
$F(x)=P(X\leq x) = \sum_{a\leq x}P(X=a)$
If $$X$$ is continuous, the cumulative distribution function at value $$x$$ is calculated as the area under the probability density function to the left of $$x$$.
$F(x)=P(X\leq x) = \int_{-\infty}^x f(a)da$
The probability that a continuous variable takes values between $$a$$ and $$b$$ can be expressed as $$P(a \leq X \leq b) = F(b) - F(a)$$
Example
Calculate and represent graphically the cumulative distribution function for the discrete random variable given by the following probability mass function:
-
Outcome 1 2 3 4 5 6
Probability 0.20 0.40 0.10 0.10 0.13 0.07
$$F(1) = P(X \leq 1) = 0.20$$
$$F(2) = P(X \leq 2) = 0.20+0.40 = 0.60$$
$$F(3) = P(X \leq 3) = 0.20+0.40+0.10 = 0.70$$
$$F(4) = P(X \leq 4) = 0.20+0.40+0.10+0.10=0.80$$
$$F(5) = P(X \leq 5) = 0.20+0.40+0.10+0.10+0.13=0.93$$
$$F(6) = P(X \leq 6) = 0.20+0.40+0.10+0.10+0.13+0.07=1$$
## 1.4 Bernoulli distribution
The Bernoulli distribution is used to describe experiments having exactly two outcomes (e.g., the toss of a coin will be head or tail, a person will test positive for a disease or not, or a political party will win an election or not).
If $$X$$ is a random variable that has a Bernoulli distribution with probability of success $$p$$, we write $X \sim Ber(p)$
• Trial can result in two possible outcomes, namely, success ($$X=1$$) and failure ($$X=0$$)
• Probability of success $$p$$ is the same for each trial ($$0 < p < 1$$)
• The outcome of one trial has no influence on later outcomes (trials are independent)
• Probability mass function: $P(X = x) = p^x (1-p)^{1-x},\ x \in \{0, 1\}$
We can check this: if $$x=0$$, $$P(X=0) = p^0 (1-p)^{1-0}=1-p$$, and if $$x=1$$, $$P(X=1) = p^1 (1-p)^{0}=p$$
• Mean is $$E[X]= \sum_{i} x_i P(X=x_i) = 0 (1-p)+ 1 p=p$$
• Variance is $$Var[X] = E[(X-E[X])^2] = E[X^2]-E[X]^2 = 0^2 (1-p)+ 1^2 p - p^2 = p(1-p)$$
Example
## 1.5 Binomial distribution
The binomial distribution is used to describe the number of successes in a fixed number of independent Bernoulli trials (e.g., number of heads when tossing a coin 20 times, number of people that test positive for a disease out of 100 people tested)
If $$X$$ is a random variable that has a Binomial distribution with number of trials $$n$$ and probability of success on a single trial $$p$$, we write
$X \sim Binomial(n,p)$
• $$x=0,1,2,\ldots,n$$ number of successes
• The number of trials is $$n$$ fixed
• The probability of success $$p$$ is the same from one trial to another
• Each trial is independent (none of the trials have an effect on the probability of the next trial)
• Probability mass function. Probability of having $$x$$ successful outcomes in an experiment of $$n$$ independent trials and probability of success $$p$$: $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$
$$x$$ successes occur with probability $$p^x$$ and $$n -x$$ failures occur with probability $$(1 - p)^{n - x}$$. The $$x$$ successes can occur anywhere among the $$n$$ trials. There are $$\binom{n}{x}$$ ($$n$$ choose $$x$$) number of ways to get $$x$$ successes in a sequence of $$n$$ trials. $$\displaystyle{\binom{n}{x} = \frac{n!}{x!(n-x)!}}$$ where $$n$$ factorial = $$n! = n \times (n-1) \times \ldots \times 1$$.
• Mean is $$n p$$. Variance is $$n p (1-p)$$
Example
### 1.5.1 Binomial variable is the sum of iid Bernoulli variables
A Binomial random variable is the sum of independent, identically distributed Bernoulli random variables.
Let $$X_1, X_2, \ldots, X_n$$ be independent Bernoulli random variables, each with the same parameter $$p$$. Then the sum $$X = X_1 + \ldots + X_n$$ is a Binomial random variable with parameters $$n$$ and $$p$$.
Example
• Suppose that a fair coin is tossed 3 times and the probability of head is $$p$$. The probability of obtaining 2 heads in the first two tosses and 1 tail in the third toss is $P(X_1 = 1, X_2 =1, X_3 = 0) = p \times p \times (1-p) = p^2 (1-p)$ (tosses are independent and probabilities are multiplied)
• There are 3 possible ways we obtain 2 heads and 1 tail in three tosses:
$$P\left(\sum_{i=1}^3 X_i = 2\right) =$$
$$P(X_1 = 1, X_2 =1, X_3 = 0) + P(X_1 = 1, X_2 =0, X_3 = 1) + P(X_1 = 0, X_2 =1, X_3 = 1) =$$
$$3 p^2 (1-p)$$
• In general, the probability of obtaining $$x$$ heads in $$n$$ tosses is $P\left(\sum_{i=1}^n X_i = x\right) = \binom{n}{x} p^x (1-p)^{n-x}$
### 1.5.2 R syntax
Purpose Function Example
Generate n random values from a Binomial distribution rbinom(n, size, prob) rbinom(1000, 12, 0.25) generates 1000 values from a Binomial distribution with number of trials 12 and probability of success is 0.25
Probability Mass Function dbinom(x, size, prob) dbinom(2, 12, 0.25) probability of obtaining 2 successes when the number of trials 12 and the probability of success is 0.25
Cumulative Distribution Function (CDF) pbinom(q, size, prob) pbinom(2, 12, 0.25) probability of observing 2 or fewer successes when the number of trials 12 and the probability of success is 0.25
Quantile Function (inverse of pbinom()) qbinom(p, size, prob) qbinom(0.98, 12, 0.25) value at which the CDF of the Binomial distribution with 12 trials and probability of success 0.25 is equal to 0.98
Example
Let us consider a biased coin that comes up heads with probability 0.7 when tossed. Let $$X$$ be the random variable denoting the number of heads obtained when the coin is tossed $$X \sim Bin(n = 10, p = 0.7)$$.
• What is the probability of obtaining 4 heads in 10 tosses?
$$P(X = 4)$$
dbinom(4, size = 10, prob = 0.7)
## [1] 0.03675691
• Calculate $$P(X \leq 4)$$
$$P(X \leq 4) = P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)$$
dbinom(0, size = 10, prob = 0.7) +
dbinom(1, size = 10, prob = 0.7) +
dbinom(2, size = 10, prob = 0.7) +
dbinom(3, size = 10, prob = 0.7) +
dbinom(4, size = 10, prob = 0.7)
## [1] 0.04734899
Alternatively, using the cumulative distribution function
pbinom(4, size = 10, prob = 0.7)
## [1] 0.04734899
• Generate 3 random values from $$X \sim Bin(n = 10, p = 0.7)$$
rbinom(3, size = 10, prob = 0.7)
## [1] 8 6 9
## 1.6 Normal distribution
The Normal distribution is used to model many procesess in nature and industry (e.g., heights, blood pressure, IQ scores, package delivery time, stock volatility)
If $$X$$ is a random variable that has a normal distribution with mean $$\mu$$ and variance $$\sigma^2$$, we write $X \sim N(\mu, \sigma^2)$
• $$x \in \mathbb{R}$$
• Probability density function: $$\displaystyle{f(x)=\frac{1}{\sigma \sqrt{2\pi}} exp^{-(x-\mu)^2/2 \sigma^2}}$$
• Bell-shaped density function
• Single peak at the mean (most data values occur around the mean)
• Symmetrical, centered at the mean
• $$\mu \in \mathbb{R}$$ represents the mean and the median of the normal distribution. The normal distribution is symmetric about the mean. Most values are around the mean, and half of the values are above the mean and half of the values below the mean. Changing the mean shifts the bell curve to the left or right.
• $$\sigma>0$$ is the standard deviation ($$\sigma^2>0$$ variance). The standard deviation denotes how spread out the data are. Changing the standard deviation stretches or constricts the curve.
### 1.6.1 Standard Normal distribution
• The standard normal distribution is a normal distribution with mean 0 and variance 1.
• The standard normal distribution is represented with the letter $$Z$$. $Z \sim N(\mu=0, \sigma^2=1)$
• If $$X \sim N(\mu, \sigma^2)$$, then $$\displaystyle{Z = \frac{X-\mu}{\sigma} \sim N(0, 1)}$$
### 1.6.2 68-95-99.7 rule
The normal distribution is symmetric about the mean $$\mu$$. The total area under the probability density curve is 1. The probability below the mean is 0.5 and the probability above the mean is 0.5.
The 68-95-99.7 rule is used to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations, respectively.
• Approximately 68% of the area lies between one standard deviation below the mean ($$\mu-\sigma$$) and one standard deviation above the mean ($$\mu+\sigma$$). That is, approximately 68% of the data lies within 1 standard deviation from the mean.
• Approximately 95% of the data lies within 2 standard deviations from the mean.
• Approximately 99.7% of the data lies within 3 standard deviations from the mean.
### 1.6.3 R syntax
Purpose Function Example
Generate n random values from a Normal distribution rnorm(n, mean, sd) rnorm(1000, 2, .25) generates 1000 values from a Normal distribution with mean 2 and standard deviation 0.25
Probability Density Function dnorm(x, mean, sd) dnorm(0, 0, 0.5) density at value 0 (height of the probability density function at value 0) of the Normal distribution with mean 0 and standard deviation 0.5.
Cumulative Distribution Function (CDF) pnorm(q, mean, sd) pnorm(1.96, 0, 1) area under the density function of the standard normal to the left of 1.96 (= 0.975)
Quantile Function (inverse of pnorm()) qnorm(p, mean, sd) qnorm(0.975, 0, 1) value at which the CDF of the standard normal distribution is equal to 0.975 ( = 1.96)
Example
Consider a random variable $$X \sim N(\mu=100, \sigma=15)$$. Calculate the following:
• $$P(X < 125)$$
pnorm(125, mean = 100, sd = 15)
## [1] 0.9522096
• $$P(X \geq 110) = 1 - P(X < 110)$$
1 - pnorm(110, mean = 100, sd = 15)
## [1] 0.2524925
• $$P(110 < X < 125) = P(X < 125) - P(X < 110)$$
pnorm(125, mean = 100, sd = 15) -
pnorm(110, mean = 100, sd = 15)
## [1] 0.2047022
## 1.7 Percentiles and quantiles
The $$k$$th percentile is the value $$x$$ such that $$P(X < x) = k/100$$.
The $$k$$th percentile of a set of values divides them so that $$k$$% of the values lie below and (100-$$k$$)% of the values lie above.
Quantiles are the same as percentiles, but are indexed by sample fractions rather than by sample percentages (e.g., 10th percentile or 0.10 quantile).
Example
The mean Body mass index (BMI) for men aged 60 is 29 with a standard deviation of 6. Remember BMI is a value derived from the weight and height of a person to work out if weight is healthy, and it is expressed as kg/m^2.
• What is the 90th percentile (or quantile 0.90)?
$$X \sim N(\mu=29, \sigma = 6)$$. The 90th percentile is the value $$x$$ such that $$P(X < x) = 0.90$$. The 90th percentile is 36.69. This means 90% of the BMIs in men aged 60 are below 36.69. 10% of the BMIs in men aged 60 are above 36.69.
qnorm(0.90, mean = 29, sd = 6)
## [1] 36.68931
• Find percentile 25th (or quantile 0.25).
Value $$x$$ such that $$P(X < x) = 0.25$$.
qnorm(0.25, mean = 29, sd = 6)
## [1] 24.95306
• For infant girls, the mean body length at 10 months is 72 centimeters with a standard deviation of 3 centimeters. Suppose a girl of 10 months has a measured length of 67 centimeters. How does her length compare to other girls of 10 months?
$$X \sim N(\mu=72, \sigma = 3)$$. We can compute her percentile by determining the proportion of girls with lengths below 67. Specifically, $$P(X < 67) = 0.047$$. This girl is in the 4.7th percentile among her peers, her height is very small.
pnorm(67, mean = 72, sd = 3)
## [1] 0.04779035
## 1.8 F distribution
The F distribution is usually defined as the ratio of variances of two populations normally distributed. The F distribution depends on two parameters: the degrees of freedom of the numerator and the degrees of freedom of the denominator.
Characteristics of the F distribution:
• The F-distribution is always $$\geq 0$$ since it is the ratio of variances and variances are squares of deviations and hence are non-negative numbers.
• Density is skewed to the right
• Shape changes depending on the numerator and denominator degrees of freedom
• As the degrees of freedom for the numerator and denominator get larger, the density approximates the normal
• In ANOVA we always use the right-tailed area to calculate p-values
Density curves of four different F-distributions:
R syntax
Purpose Function Example
Generate n random values from a F distribution rf(n, df1, df2) rf(1000, 2, 11) generates 1000 values from a F distribution with degrees of freedom 2 and 11
Probability Density Function (PDF) df(x, df1, df2) df(1, 2, 11) density at value 1 (height of the PDF at value 1) of the F distribution with degrees of freedom 2 and 11
Cumulative Distribution Function (CDF) pf(q, df1, df2) pf(5, 2, 11) area under the density function of the F(2,11) to the left of 5
Quantile Function (inverse of pf()) qf(p, df1, df2) qf(0.97, 2, 11) value at which the CDF of the F(2,11) distribution is equal to 0.97
## 1.9 Chi-squared distribution
The chi-squared distribution with $$m \in N^*$$ degrees of freedom is the distribution of a sum of squares of $$m$$ independent standard normal random variables.
If $$X_1, X_2, \ldots, X_m$$ are $$m$$ independent random variables having the standard normal distribution, then $V=X_1^2 + X_2^2+ \ldots + X_m^2 \sim \chi^2_{(m)}$ follows a chi-squared distribution with $$m$$ degrees of freedom. Its mean is $$m$$ and its variance is $$2m$$.
par(mfrow = c(2, 2)) # to draw figures in a 2 by 2 array on the device
x <- seq(0, 6, length.out = 100)
plot(x, dchisq(x, 1), main = "chi2(1)", type = "l")
plot(x, dchisq(x, 2), main = "chi2(2)", type = "l")
plot(x, dchisq(x, 3), main = "chi2(3)", type = "l")
plot(x, dchisq(x, 6), main = "chi2(6)", type = "l")
Find the 95th percentile (or quantile 0.95) of the Chi-Squared distribution with 6 degrees of freedom.
qchisq(0.95, df = 6)
## [1] 12.59159 | 2023-03-26 06:11:11 | {"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": 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": 0.8716638088226318, "perplexity": 416.923786299331}, "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-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00251.warc.gz"} |
https://jmanton.wordpress.com/tag/geometry-of-inner-products/ | ### Archive
Posts Tagged ‘geometry of inner products’
## Understanding (real- and complex-valued) Inner Products
This short note addresses two issues.
• How can we intuitively understand a complex-valued inner product?
• If an inner product structure is given to a vector space, how can we understand the resulting geometry?
Since inner products are associated with angles, and since we can understand angles, there is temptation to interpret inner products in terms of angles. I advocate against this being the primary means of interpreting inner products.
An inner product $\langle \cdot, \cdot \rangle$ induces the norm $\| x \| = \sqrt{\langle x,x \rangle}$. Importantly, the inner product can be recovered from this norm by the polarisation identity. Therefore, understanding the geometry of an inner product is the same as understanding the geometry of the norm, and for the latter, it often suffices to consider what the unit ball looks like. For me, the norm is the primary structure giving the space its geometry.
What then is the purpose of the inner product? Not all norms have the same properties. Under some norms, projection onto a closed subspace may not be unique, for example. When interested in shortest-norm optimisation problems, the most desirable situation to be in is for the square of the norm to be quadratic, since then differentiating it produces a linear equation. In infinite dimensions, what does it mean for the square of a norm to be quadratic?
The presence of an inner product structure means the square of the norm is quadratic. Furthermore, the inner product “decomposes” the norm in a way that gives direct access to the derivative of the norm squared.
The remaining issue is how to understand complex-valued inner products. Given the above, the natural starting place is to consider endowing a complex vector space with a norm. Keeping the axioms of a real-valued normed vector space seems sensible; it implies that scaling a vector by $e^{\jmath \theta}$ does not change its norm (because $\| e^{\jmath \theta} x \| = | x |\,\| x \| = \| x \|$).
Then one asks what it means for the square of a norm to be quadratic. From the real-valued case, one guesses that one wants to be able to represent the square of the norm as a bilinear form: $\| x \|^2 = \langle x, x \rangle$, where $\langle \cdot, \cdot \rangle$ is linear in each of its arguments. Following the letter of the law, this would mean $\| \alpha x \|^2 = \langle \alpha x, \alpha x \rangle = \alpha^2 \langle x,x \rangle = \alpha^2 \|x\|^2$. In the complex case though, $\alpha^2$ need not equal $|\alpha|^2$. This explains why one tweaks the definition and instead considers sesquilinear forms which are linear in one argument and conjugate linear in the other: $\langle \alpha x, x \rangle = \alpha \langle x, x \rangle = \langle x, \bar\alpha x \rangle$. Indeed, one then correctly has that $\|\alpha x \|^2 = \langle \alpha x, \alpha x \rangle = \alpha \bar\alpha \langle x , x \rangle = |\alpha|^2 \|x\|^2$. With this tweak, one can verify that the complex-valued case works the same way as the real-valued case.
By treating the norm as the primary structure, one does not have to worry about giving an intuitive meaning to the inner product of two vectors not being a purely real-valued number; the inner product is there merely to expose the square of the norm as being quadratic. A complex-valued inner product is recoverable from its norm and hence no geometric information is lost. (Of course, orthogonality remains an important concept.) If one really wanted, one could play around with examples in $\mathbb{C}^2$ to get a better feel for what it means for $\langle x , y \rangle = \jmath$, for example, however, unless one encounters a particular problem encountering this level of detail, thinking in terms of norms is cleaner and more efficient. (If $\langle x, y \rangle = r e^{\jmath\theta}$ then $\langle x, e^{\jmath \theta} y \rangle = r$, so that by “rotating” a complex-valued vector in the two-dimensional real-valued vector space that it spans, one can always return to thinking about real-valued inner products.) | 2017-07-28 16:54:03 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 15, "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": 0.9060089588165283, "perplexity": 245.7527981742341}, "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-2017-30/segments/1500550975184.95/warc/CC-MAIN-20170728163715-20170728183715-00246.warc.gz"} |
https://scipost.org/submissions/1812.11705v2/ | # Exceptional Chern-Simons-Matter Dualities
### Submission summary
As Contributors: Po-Shen Hsin Arxiv Link: https://arxiv.org/abs/1812.11705v2 (pdf) Date accepted: 2019-10-18 Date submitted: 2019-09-19 02:00 Submitted by: Hsin, Po-Shen Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical
### Abstract
We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize the familiar level-rank duality of Chern-Simons theories based on classical gauge groups to the setting of exceptional gauge groups. For instance, one duality sequence we discuss is $(E_{N})_{1}\leftrightarrow SU(9-N)_{-1}$. Others such as $SO(3)_{8}\leftrightarrow PSU(3)_{-6},$ are dualities among theories with classical gauge groups that arise due to their embedding into an exceptional chiral algebra. We apply these equivalences between topological field theories to conjecture new boson-boson Chern-Simons matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant $G_{2}$ gauge theory coupled to either an adjoint fermion, or two fundamental fermions.
### Ontology / Topics
See full Ontology or Topics database.
Published as SciPost Phys. 7, 056 (2019)
### List of changes
- new footnote 8
- replace the word quartic coupling by a suitable potential
- fixed typos mentioned in the report
- update references | 2020-12-06 01:00:24 | {"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": 0.5655144453048706, "perplexity": 5301.936655281636}, "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/1606141753148.92/warc/CC-MAIN-20201206002041-20201206032041-00693.warc.gz"} |
http://www.generative-ebooks.com/ebooks/Phase-portraits.html | Part II: Two-dimensional systems: flows in the plane - Chapter 2 s
# Phase portraits
The phase portrait of a given equation $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ is a plot of typical trajectories in the $xy$-plane. For most interesting examples arising in applications, there is no hope of finding the trajectories analytically. And even when explicit formulas are available (which occurs very seldom), they are often two complicated to provide much insight. Instead we will try to determine the qualitative behavior of the solutions. Digital experiments will allow us to explore and observe some phenomena that we will analyse afterwards, more or less rigorously, depending on their level of difficulty.
Sketching the phase portrait
The two basic features of any phase portrait are :
• the fixed points that are the points $\boldsymbol{x}$ where $\boldsymbol{f} (\boldsymbol{x})=\boldsymbol{0}$ ;
• special isoclines, namely the nullclines. They are defined as the curves where either $\dot{x}=0$ or $\dot{y}=0$. The nullclines indicate where the vector field is purely horizontal or vertical. By definition, fixed points lie at the intersection of the nullclines. They also partition the plane into regions where $\dot{x}$ and $\dot{y}$ have a fixed sign.
We will see how we can approximate the phase portrait near a fixed point by that of a corresponding linear system. This will extend what we developed earlier for one-dimensional systems.
Reversing time is so easy !
Consider an equation $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ that we write as
$$\frac{\text{d}\boldsymbol{x}}{\text{d}t}=\boldsymbol{f}(\boldsymbol{x}).$$
Remember that
$$\dot{\boldsymbol{x}}=\frac{\text{d}\boldsymbol{x}}{\text{d}t}.$$
We want to reverse time, which means that we change $t$ into $-t$. We see that it is equivalent to switch the direction of the vector field because
$$\frac{\text{d}\boldsymbol{x}}{\text{d}(-t)}=-\frac{\text{d}\boldsymbol{x}}{\text{d}t}=-\,\boldsymbol{f}(\boldsymbol{x}).$$
This shows in particular that it is enough to look forward in time, since to know what happens backward in time we just have to change consider the vector field $-\boldsymbol{f}$.
Flowing blobs of initial conditions
So far we have been emphasizing solutions of differential equations as functions of time. We now emphasize the dependence on initial conditions. Let $\dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x})$ be a differential equation and imagine drawing some shape in the $xy$-plane (a rectangle, for instance) and solving the equation, starting at each point of this shape. Using our image of an imaginary fluid whose speed at point $\boldsymbol{x}$ is given by $\boldsymbol{f}(\boldsymbol{x})$, we can imagine that we drop a bunch of particles and look how they flow. The shape will move, and likely become distorted, as the following examples illustrate.
You can draw a blob of any shape with your mouse and observe how it evolves. You can also reverse time by changing $t$ into $-t$. | 2022-05-23 22:45:53 | {"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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7569974660873413, "perplexity": 277.45250987322834}, "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-2022-21/segments/1652662562106.58/warc/CC-MAIN-20220523224456-20220524014456-00419.warc.gz"} |
https://electronics.stackexchange.com/questions/97261/led-lights-behave-strangely-why | # LED lights behave strangely, why?
I have connected 10 LEDs in series, the LEDs have a current rating: 20 mA which is 0.02 A and forward Voltage: 2V.
My power supply provides me with: 5V DC 2A
By using the formula down below:
$$R = \frac{vs-vf}{i} = 5 - \frac{2}{20} = 150 \Omega$$
I can see that LEDs need 150 ohm.
Which is in total $10*150 = 1500\Omega$
I connected: 1 k resistor, 2 x 150 ohm resistor and 2 x 100 ohm resistor which is a total of: 1500 ohm.
But the 10 LEDs dont turn on, what's wrong?
I can still turn 3-4 LEDs with or without the resistors.
Try to explain this down below please:
I tried to connect a single green led by itself without a resistor. It turned orange which is a sign that it needs a resistor.
Then I connected 150 ohm resistor to the led and it became green. I then changed the resistor value from 150 ohm to 1 k ohm, but it did not change the brightness of the led, it still the same green, no fading in brightness.
I tried to change the value to 33k ohm, it faded in brightness, but I can still see some greenish brightness in the LED.
What's wrong? Why is the LED responding like that to the resistor's value? I mean 1k ohm is much more then 150 ohm but it worked the same.
P.S.: I bought the resistors from China via eBay.
• 10 led in series would require a voltage of at least 20v (10*2v), that is not available in your system. You should use 5 parallel sets with two led in series each. Please refer to this reply for resistor calculation Jan 22 '14 at 10:07
• But i have seen once small 5mm light led about 8 leds connected that use only 2 x aa battery which have 1.5v ech, how is that possible then ?? Jan 22 '14 at 10:10
• I don't think they are connected in series or there is a circuit that multiplies the voltage of the batteries. Make sure that you distinguish correctly between series and parallel connection Jan 22 '14 at 10:17
• The led should be used with either a resistor to limit the current or a driver that drives the led with constant current. i tried to connect a single green led by it self without a resistor it turn orange which is a sign that it need resistor. using the led like this will damage it very fast Jan 22 '14 at 10:59
• The brightness was definitely different (as was the current) but they eye can't quantify the difference easily. If you use two leds side by side with 150 ohm and 1K you should see the difference. In any case the proper way to measure the current is by using an ammeter. You can also measure the voltage drop across the resistor and calulate the led current by using Ohms law Jan 22 '14 at 11:25
Here is the forward conduction characteristic of a typical LED: -
At 2V forward voltage the LED consumes 20mA and is quite bright. At 1.8V forward voltage the current has fallen to 5mA and the LED will be somewhat dimmer than the 20mA scenario. At 1.6V the LED is taking almost zero current and below this voltage it is doubtful that it illuminates at all.
If you have 10 similar LEDs in series, to obtain a forward current of 20mA requires a forward voltage of 20V.
If you are happy with a forward current of 5mA then 18 volts is all that is needed. And, if your power supply is (say) 20V, a series resistor is needed that drops 2 volts at 5mA therefore its resistance will be about $\dfrac{2V}{5mA}$ = 400 ohms.
If you have a limited supply voltage, you can wire the LEDs in parallel banks or use a boost converter to lift your 5V to about 20 volts. The CAT4238 device from Motorola can do this: -
• What about the resistor problem?? Jan 22 '14 at 10:46
• @user1022734 I think I've answered that but please let me know if you think I haven't Jan 22 '14 at 12:23
Your calculation of $R = \frac {vs-vf}{i} = \frac {5v-2v}{0.02A} = 150 ohm$ is for 10 individual resistors that should be connected as shown below (leds in parallel).
• I understand that now, but that dont explain the rest i have asked for. Jan 22 '14 at 10:47
Forward voltages add when you wire LEDs in series. You need a 20 volt supply to run your 10 in series. With a 5 volt supply you can run the LEDs in 5 parallel pairs, using either a single 1/2 watt 250 ohm resistor at one end of the circuit, or smaller wattage resistors in each circuit branch.
I will address the part of your question about the different resistance values having no or little apparent effect:
"Ultra bright" LEDs today are remarkably efficient and can still appear quite bright at currents well below the typical or maximum stated on the data sheet.
So, let's consider just one green LED which has a forward voltage $V_f$ of 2V and a nominal forward current $I_f$ of 20mA.
With a power supply voltage of 5V, we could calculate the resistor value needed for 20mA as:
$$R = \frac{3}{0.02} = 150\Omega$$
But what if we wanted to drive the LED with only 1mA?
$$R = \frac{3}{0.001} = 3k\Omega$$
The LED might not illuminate with so little current, but many LEDs (for example, these inexpensive Panasonic indicator LEDs) will start to illuminate with as little as 2mA or less. For 2mA, the calculations above would give us a resistor value of 1.5kΩ.
If you exceed the maximum forward current (20mA), the LED will be damaged. One sign of impending damage is a green LED illuminating orange or yellow. That color is an indication that the LED is being driven too hard. (Too long at such currents and the LED will stop working altogether.)
(Check the data sheet for whether you can exceed the nominal forward current for various duty cycles. For example, many LEDs which have a nominal $I_f$ of 20mA can be driven at 40mA or more with duty cycles of 10% or so.)
If you connect the LED to a variable resistor (a potentiometer) you should be able to see a nonlinear brightness increase as you decrease resistance from 1.5kΩ to 150Ω To ensure that you don't damage the LED, connect a 150Ω resistor in series with your potentiometer. When your potentiometer is at 0Ω, your LED will be at 20mA. | 2021-09-24 10:09:25 | {"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": 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": 0.50008624792099, "perplexity": 1171.8426509426176}, "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/1631780057508.83/warc/CC-MAIN-20210924080328-20210924110328-00166.warc.gz"} |
https://codereview.stackexchange.com/questions/224931/convert-a-string-like-4h53m12s-to-a-total-number-of-seconds-in-javascript | # Convert a string like 4h53m12s to a total number of seconds in JavaScript
At the moment I have this:
function getValue(str)
{
let result = 0;
var regex = /(\d+[a-z]+)/g;
match = regex.exec(str);
while (match != null)
{
var match_str = match[0];
var last_char = match_str[match_str.length-1];
if ( last_char == 'h' )
result += parseInt(match_str) * 3600;
if ( last_char == 'm' )
result += parseInt(match_str) * 60;
if ( last_char == 's' )
result += parseInt(match_str);
match = regex.exec(str);
}
return result;
}
console.log( getValue("4h12m32s") );
It feels quite clumsy though. Also checking invalid values like 4hs feels difficult.
Is there any clever trick for something similar?
• +"4h12m32s".replace(/(\d+)h(\d+)m(\d+)s/, (_, h, m, s) => (h * 3600) + (m * 60) + (s * 1)) ;) – user11536834 Jul 25 '19 at 22:42
• (might as well enjoy that implicit type coercion) – user11536834 Jul 25 '19 at 22:49
• @user11536834 I would expect that each unit is optional. The current code allows that; your solution doesn't. – 200_success Jul 25 '19 at 23:17
• It's probably not a good idea to mix var and const/let - you could consider replacing them all with const (or let, when the variable needs to be reassigned) – CertainPerformance Jul 27 '19 at 9:53
• getValue(str) is such a vague name for the function and its parameter, it could mean anything! Furthermore, "get" implies that this is a getter function that retrieves something, which is not the case.
• Your regex is ineffective. Capturing parentheses could be useful, but you didn't actually use them right, such that you ended up having to pass a dirty string to parseInt() and extract the last character the harder way.
• You neglected to scope match, such that it acts as a global variable. The regex-matching statement is written twice; the assignment could be done within the loop condition instead.
• The if statements should be an if-else chain, since the conditions are mutually exclusive. However, since the branches are all so similar, a lookup table would be more elegant.
function durationSeconds(timeExpr)
{
var units = {'h': 3600, 'm': 60, 's': 1};
var regex = /(\d+)([hms])/g;
let seconds = 0;
var match;
while ((match = regex.exec(timeExpr)))
{
seconds += parseInt(match[1]) * units[match[2]];
}
return seconds;
}
console.log( durationSeconds("4h12m32s") );
Alternatively, if you expect that the units will be in the conventional order, you don't have to loop at all.
function durationSeconds(timeExpr)
{
var match = /^(?:(\d+)h)?(?:(\d+)m)?(?:(\d+)s)?\$/.exec(timeExpr);
return 3600 * (parseInt(match[1]) || 0)
+ 60 * (parseInt(match[2]) || 0)
+ (parseInt(match[3]) || 0);
}
console.log( durationSeconds("4h32s") );
• Some of your mentions were because I have extracted this from my current code base (and messed up a bit with copy pasting), but this is exactly the knowledge and cleverness I was looking for. Despite that the second one is more consise I think your first example is great and super readable. Thanks for your effort. – Dirk Boer Jul 26 '19 at 9:45
• Btw - one last question: why the double parentheses within the while loop in your first example? – Dirk Boer Jul 26 '19 at 9:48
• One common mistake is to write while (a = 1) when you meant while (a == 1). The extra parentheses is a convention to emphatically say "yes, that really is an assignment". – 200_success Jul 26 '19 at 9:56
• Does eslint have that warning, and does the double parenthesis silence it? – Barmar Jul 27 '19 at 17:01
## Named Capture Groups
JavaScript RegExp has named capture groups that can make life a lot simpler when dealing with complicated RegExp. Combined with destructuring assignment you can extract the named hours minutes and seconds as follows.
function toSeconds(time) {
const {groups: {h = 0, m = 0, s = 0}} = /(?<h>\d*)h(?<m>\d*)m(?<s>\d*)/i.exec(time);
return h * 3.6e3 + m * 60 + s * 1; // * 1 to coerce s to Number
}
Missing digits are set to zero in the assignment defaults.
However this is limited to strings that have hours, minutes, and seconds in the correct order (hence no need to match the "s") and will throw an error if there is a problem.
## A more robust solution
You can also reduce the array created by symbol.matchAll (it returns an iterator that you convert to an array via spread operator)
It RegExp[symbol.matchAll] is the same call as String.matchAll(RegExp)
To handle as many variations as possible you can convert the time string to lowercase, soak up white spaces, allow for fractions, multiple periods, and negative periods.
Using an IIF to wrap the periods constant via closure the function looks like
const toSeconds = (() => {
const periods = {h: 3600, m: 60, s: 1};
return time => [.../(\-*\d*\.*\d*)\W*([hms])/g[Symbol.matchAll](time.toLowerCase())]
.reduce((time, [, digits, type]) => periods[type] * digits + time, 0);
})();
Or via the string
const toSeconds = (() => {
const periods = {h: 3600, m: 60, s: 1};
return time => [...time.toLowerCase().matchAll(/(\-*\d*\.*\d*)\W*([hms])/g)]
.reduce((time, [, digits, type]) => periods[type] * digits + time, 0);
})();
To combat the readability the next version creates some extra variables to segregate the logic parts a little
const toSeconds = (() => {
const periods = {h: 3600, m: 60, s: 1};
const extractHMS = /(\-*\d*\.*\d*)\W*([hms])/g;
const sumSeconds = (time, [, digits, type]) => periods[type] * digits + time;
return time => [...time.toLowerCase().matchAll(extractHMS)].reduce(sumSeconds, 0);
})();
The snippet below shows some of the results of a variety of inputs.
const toSeconds = (() => {
const periods = {h: 3600, m: 60, s: 1};
return time => [.../(\-*\d*\.*\d*)\W*([hms])/g[Symbol.matchAll](time.toLowerCase())]
.reduce((time, [, digits, type]) =>periods[type] * digits + time, 0);
})();
"1h,1m,1s,1,,1s2m3h,3h2m1s,2H2M2S,1h 1H1s1 S1m1M,1.1s,1.2s,s,1h-5m,1 1s,hms"
.split(",")
.forEach(time => log("\"" + time + "\" =" , toSeconds(time)+" seconds"));
function log(...data) {
document.body.appendChild(
Object.assign(
document.createElement("div"), {textContent: data.join(" ")}
)
)
}
BTW in Javascript we put...
• the opening { on the same line as the statement,
• use camelCase for naming.
And from many years of C style language experience I would advise you to always delimit statement blocks with { } eg Bad if (foo) bar = foo, Good if (foo) { bar = foo }
• This code might be more robust, but it's also A LOT harder to read IMHO. – Nzall Jul 26 '19 at 9:08
• @Nzall Modern JS is progressing to less a less verbose style that to many is less readable. I will update with a more readable version. – Blindman67 Jul 26 '19 at 9:22
• Very nice. Might be worth mentioning that matchAll and named captures don't have full browser support yet, but this is surely the way forward (old regex functions all have some unfortunate deficiencies, aside from replace, which is awkward to use when not actually replacing something). – user11536834 Jul 26 '19 at 13:09
• One other idea for the first solution could be to whack everything into Date.UTC, with 1970, 0, 1 as the first arguments, then multiply by 0.001; then you could scrap periods and do it all in one shot (a bit funky, but makes it really clear we're dealing with a timestamp). – user11536834 Jul 26 '19 at 13:17
• "Always specify a radix when using parseInt." -- MDN parseInt documentation
• When getValue() is given a malformed string, it ignores the malformed parts without letting anyone know something went wrong. For example, 2 h 500 m 600 s returns 0. It is better to throw an error when the input isn't valid (so the caller can more easily figure out why the values it got back are wrong).
• If you expect requirements to change in the future, you should look into harnessing the power of a real parser (or parser generator). For example, what changes do you need to do to:
• Add support for uppercase letters: 1H2D3M
• Allow spaces: 30h 50 m 10s
• Allow input to be in any order: 5s 10m
• Allow multiple occurrences or no occurrences of a unit: 30h 10h or 10s
• Add new units: d (days) or ms (milliseconds), etc.
Here is what that looks like using a parser generator (PEG.js):
// Duration to Seconds Grammar
// ==========================
//
// Accepts expressions like "4h53m12s" or "4H 33M 12S"
// and computes the total number of seconds.
start
= total
total
= left:subtotal right:total { return left + right; }
/ subtotal
subtotal
= left:integer right:day { return left * 86400; }
/ left:integer right:hour { return left * 3600; }
/ left:integer right:minute { return left * 60; }
/ left:integer right:second { return left; }
/ left:integer right:millisecond { return left * .001; }
day
= whitespace [dD]
hour
= whitespace [hH]
minute
= whitespace [mM][^sS]
second
= whitespace [sS]
millisecond
= whitespace [mM][sS]
integer
= whitespace [0-9]+ { return parseInt(text(), 10); }
whitespace
= [ \t\n\r]*
• Test it out in the right-hand pane
• Nice answer. That first bullet point is a really easy trap to fall into (as in OP and accepted answer here). I'd rather avoid parseInt entirely than give it a second look to make sure it's used correctly. AFAIK implicit conversion has never parsed leading zeros as octal (at least, not from ES3 forward). – user11536834 Jul 26 '19 at 23:13 | 2020-04-06 22:26:43 | {"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": 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": 0.3173774182796478, "perplexity": 5857.176996881301}, "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/1585371660550.75/warc/CC-MAIN-20200406200320-20200406230820-00436.warc.gz"} |
https://twistedone151.wordpress.com/2007/12/ | ## Archive for December, 2007
### Monday Math 1: The Isoperimetric Problem
December 31, 2007
Isoperimetric Problem
The isoperimetric problem is the problem of finding the closed plane curve with a given perimeter that encloses the greatest area. Below, I will use the calculus of variations in a proof of the long-known solution, the circle.
We let our enclosed region be D, and the enclosing curve ∂D, with orientation on ∂D chosen so that D is to it's left (counterclockwise). We thus want to minimize subject to the constraint . Note that these are quite different kinds of integrals, and cannot be combined directly via the multiplier method of including constraints.
However, using Green's theorem, which says that over a plane region D with (counterclockwise oriented) boundary ∂D we have:
By choosing , and using the fact that the area of D is , we obtain
So now we can combine the minimized quantity and the constraint using a multiplier. As both are integral constraints, the multiplier is a constant λ, so parametrizing our curve ∂D as (x(t),y(t)), so
and we want to find the extrema of
The integrand is , so we have the Euler-Lagrange equation for x:
Now, , so , and
Similarly for y:
So we have and
Squaring and summing these, we obtain:
This is the equation of a circle with radius λ and and center (C2,C1). From our original constraint , we obtain λ=p/2π.
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### The Pain Beam
December 29, 2007
No, it's not the twisted creation of a mad scientist in a grade B movie, it's here. While the concept and it's execution are pretty cool, scientifically speaking; and while I'm in favor of developing non-lethal weaponry; the abuse potential of something like this is rather worrying.
### Orexin A: Substitute for Sleep…
December 29, 2007
This is a very interesting result, but it will be years, if ever, before we see medical use of these results; not to mention that we still don't know what the long-term side-effects may be. Still, anything that might help people with sleep disorders is a good thing, in my opinion.
### Pocket Veto
December 29, 2007
Here is an analysis of the constitutionality of President Bush's current “pocket veto.” Look also at the first comment and at this Overlawyered post for info on the President's key objection to the bill.
### testing again
December 29, 2007
これは日本語です。
### Physics Friday 1
December 28, 2007
Bead on a hoop:
Let us consider a thin hoop of radius R. On this hoop is strung a small bead of mass m which can slide along the hoop. The bead-hoop interaction is frictionless. If we place the hoop on end, and rotate it about the vertical with a constant angular velocity , what is the equation of motion for the movement of the bead on the hoop? What are the equilibrium positions, and are they stable or unstable equilibria? We decribe the position of the bead on the hoop by the angle from the bottom of the hoop.
Rotating Reference Frame Approach:
We work in the reference frame of the hoop. As this is a rotating reference frame, it is non-inertial and we have fictitious forces to consider, namely the centrifugal and Coriolis forces.
As our bead’s motion is constrained to the plane of the hoop, and the axis of the frame’s rotation is also in that plane, the Coriolis force is perpendicular to the plane, and can be ignored (as it will be entirely canceled by a normal force of the hoop on the bead, and will not affect the motion at all). Thus we have as the only relevant fictitious force the centrifugal force straight away from the axis.
We need only consider the component of net force tangential to the hoop, as only motion along the hoop is possible. The component of gravity tangential to the hoop is, and the component of the centrifugal force is , so
The equation of motion:
so:
If we define , then the above can be written as
Equilibrium angles are those where
which occurs when , so , so long as
or
Note when , we have only the point , as the two coincide.
Stability: Look at the sign of near the zeroes
1. For , we have only the one equilibrium at . We can see that for <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?0<theta, 0″> and <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?\omega^2\cos\theta-\omega_0^2<\omega^2-\omega_0^2, so <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?\ddot{\theta}, and the equilibrium is stable.
2. For \omega_0″>, we have two equilibria, at and at . We can see that for <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?0<\theta, 0″>. For <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?0<\theta, 0″>, so 0″>; for <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?\theta_0<\theta, <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?\omega^2\cos\theta-\omega_0^2, so <img src=”http://www.forkosh.dreamhost.com/mimetex.cgi?\ddot{\theta}.
and thus the equilibrium at is unstable, and that at is stable.
For a 50 cm diameter hoop, we have
or approximately one revolution a second.
### Nintendo should hire this guy…
December 28, 2007
This is very cool:
It demonstrates how powerful the motion-tracking of the Wiimote really is.
### This is an interesting video
December 28, 2007
### Testing 2…
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https://rudeboybert.github.io/SDS220/syllabus.html | # Basic information
• Course title: SDS/MTH 220: Introduction to Probability and Statistics
• Instructor: Albert Y. Kim - Assistant Professor of Statistical & Data Sciences
• Email: Slack team: Click hashtag icon in navbar for the browser interface or use the desktop app.
• Meeting locations/times:
• Lectures: MWF 11:00-12:10 - Sabin-Reed 301
• Labs: Held by Dr. Jenny Smetzer (Note lab end times are incorrectly listed on Course Search)
• Sec 01: Tue 1:00-2:20 - Sabin-Reed 301
• Sec 02: Tue 3:00-4:20 - Sabin-Reed 301
• Outside help:
• The #questions channel on Slack.
• Spinelli Center for Quantitative Learning Tutoring: Sunday thru Thursday 7-9pm in Sabin-Reed 301.
• Albert’s office hours in McConnell Hall 215: Mon & Wed 2:45-4:00 and by apppointment (click here to book). If you’re having R or RStudio issues, please have your computer and RStudio loaded and ready to go.
• Jenny’s office hours in Bass 403: By apppointment on Tu 11:00-12:20 and Wed 10:00-12:20 and 1:00-2:00 (click here to book).
## Instructor work-life balance
• I will respond to Slack messages sent during the week within 24h. I will respond to Slack messages sent during the weekend at my own discretion.
• If possible, please only Slack me with briefer and administrative questions; I prefer having more substantive conversations in person as it takes me less energy to understand where you are at.
• I will do my best to return all grading as promptly as possible.
• I will rarely be on campus on Thursdays as this is my self-care day.
# How can I succeed in this class?
• When I have questions or don’t understand something:
• “Am I asking questions in class?”
• “Am I asking questions on Slack in #questions?” Even better: “Am I answering my peers’ questions on Slack?”
• “Having I been going to the Spinelli tutoring center?”
• “Have I been coming to Albert or Jenny’s office hours?”
• “Am I staying on top Slack notifications sent between lectures?” If you need help developing a notification strategy that best suits your lifestyle, please speak to me.
• “Am I attending lectures consistently?”
• “Am I actually running the code and studying the outputs in R during in-class exercises, or am I just skimming the text?”
• “Am I completing all the ModernDive readings/in-class activites for a given lecture before the start of the next lecture?”
• “During in-class exercises and lab time, am I taking full advantage that I’m in the same place at the same time with the instructor, the lab assistants, and most importantly your peers, or am I browsing the web/texting the whole time?”
• Problem sets, DataCamp, and coding:
• “Am I actually doing the problem sets?”
• “Have I been attempting a good faith balance between to push myself during DataCamp exercises while not banging my head on the wall, or am I just taking the hints/solutions without any effort?”
• “When learning to code, much like learning a language, have I been really pushing myself to practice, practice, practice?”
# Course Description & Objectives
• Official course description: On Smith College Course Search.
• Objectives: This semester you will
1. Learn statistical inference via data science, not mathematics/probability theory.
2. Engage in the data/science research pipeline in as faithful a manner as possible while maintaining a level suitable for novices.
3. Develop your statistical literacy, a necessary ability for effective citizenship.
A rough topic schedule and corresponding readings are posted below on the main page of this course webpage. We will draw from the following sources:
1. Statistical Inference via Data Science: A moderndive into R and the tidyverse. We’ll be using the development version; click link in menubar above. Follow the book on Twitter @ModernDive.
2. OpenIntro: Introductory Statistics with Randomization and Simulation by Diez, Barr, and Cetinkaya-Rundel
• Free PDF
• Hardcopy. Available for \$8.49 on Amazon
3. DataCamp: an online interactive environment for learning data science currently via R and Python. On top of the DataCamp courses we’ll cover this semester, you have free access to all their courses for 6 months.
## Policies
• Bring your laptop, a set of headphones, colored pens/pencils, and your paper notebook to every lecture.
• You are expected to stay until the end of lecture. If you need to leave before the end of lecture, please confirm with me first.
• Attendance will not be explicitly taken and occasional absenses are excused. However, extended absenses should be mentioned to me.
• However, you are responsible for asking your peers for what you missed. For example, makeup lectures will not be held during office hours.
# Evaluation
## Weekly Problem Sets 10%
• Total of 10 problem sets: assigned during Tuesday labs, due the following week.
• Two lowest scores dropped.
• The problem sets in this class should be viewed as low-stakes opportunities to practice, instead of evaluative tools used by the instructor to assign grades. Each problem set is worth ~1% of your final grade.
## DataCamp Assignments 5%
• Assigned during Tuesday labs, due the following week.
• DataCamp is meant to be low stakes-practice, so the only thing that matters for your grade is whether you complete the course. So while things like the number of hints/solutions taken don’t factor into your grade, it is important to make a good faith effort to answer these questions the best you can.
• While you may do them in advance if you are curious, it is most definitely not required. Note that the DataCamp schedule may change, with certain courses dropped/added.
• Jenny will talk more about DataCamp during Lab 2.
## Quizzes 5%
There will be 2-3 quizzes assigned during the course of the semester. They will always be announced beforehand.
## Term Project 30%
See Term Project page. While your term project grade is only based on your final resubmission on the last day of class, your level of contributions at all the intermediate steps (data, proposal, and initial submission) will affect your engagement grade; see Engagement below.
## Three Midterms: 45%
• There will be three self-scheduled midterms, including one during finals week. See the Midterms page.
• Lowest score weighted 10%, middle score weighted 15%, and highest score weighted 20%.
## Engagement 5%
It is difficult to explicit codify what constitutes “an engaged student,” so instead I present the following rough principle I will follow: you’ll only get out of this class as much as you put in. That being said, here are multiple pathways for you to stay engaged in this class:
• Engaging with this class in a fashion where you can say “yes” to all “How can I succeed in this class?” questions above.
• Contributing for all the steps in the term project leading up to the final submission: data, proposal, and initial submission.
• Getting reasonabable peer evaluations for your term project.
## Policies
1. Collaboration: While I encourage you to work with your peers for problem sets and labs, you must submit your own answers and not simple rewordings of another’s work. Furthermore, all collaborations must be explicitly acknowledged in your submissions.
2. Honor Code: All your work must follow the Smith College Academic Honor Code Statement; in particular all external sources must be cited in your submissions.
3. Problem sets:
• No extensions will be granted without a dean’s note.
• All written problem sets must be handed in at the start of lab. No emailed submissions will be accepted; if you can’t make it to class ask a classmate to turn it in for you. They must be stapled with the fringe/perf from any spiral notebook paper removed.
4. Midterms/quizzes:
• No make-up quizzes or midterms will be allowed without a dean’s note.
• Timestamps for all midterms will be strictly enforced.
5. Grading: I reserve the right to not discuss any grading issues in class and instead direct you to office hours.
# Accommodations
Smith is committed to providing support services and reasonable accommodations to all students with disabilities. To request an accommodation, please register with the Disability Services Office at the beginning of the semester. To do so, call 413.585.2071 to arrange an appointment with Laura Rauscher, Director of Disability Services.
# Code of Conduct
As the instructor and assistants for this course, we are committed to making participation in this course a harassment-free experience for everyone, regardless of level of experience, gender, gender identity and expression, sexual orientation, disability, personal appearance, body size, race, ethnicity, age, or religion. Examples of unacceptable behavior by participants in this course include the use of sexual language or imagery, derogatory comments or personal attacks, trolling, public or private harassment, insults, or other unprofessional conduct.
As the instructor and assistants we have the right and responsibility to point out and stop behavior that is not aligned to this Code of Conduct. Participants who do not follow the Code of Conduct may be reprimanded for such behavior. Instances of abusive, harassing, or otherwise unacceptable behavior may be reported by contacting the instructor.
All students, the instructor, the lab instructor, and all assistants are expected to adhere to this Code of Conduct in all settings for this course: lectures, labs, office hours, tutoring hours, and over Slack.
This Code of Conduct is adapted from the Contributor Covenant, version 1.0.0, available here. | 2021-09-28 20:09:48 | {"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": 0.2234456092119217, "perplexity": 5383.837544310695}, "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/1631780060882.17/warc/CC-MAIN-20210928184203-20210928214203-00018.warc.gz"} |
http://meetings.aps.org/Meeting/MAR08/Event/75874 | ### Session C1: Poster Session I: 2:00 pm - 5:00 pm
2:00 PM–2:00 PM, Monday, March 10, 2008
Morial Convention Center Room: Exhibit Hall A
Abstract ID: BAPS.2008.MAR.C1.206
### Abstract: C1.00206 : The One-Hole, One-Dimensional Hubbard Model at $U = \infty$
MathJax On | Off Abstract
#### Authors:
William Hodge
(Wake Forest University)
Natalie Holzwarth
(Wake Forest University)
William Kerr
(Wake Forest University)
The Hubbard Hamiltonian is the simplest model that describes interacting electrons on a lattice. In this work, we use the properties of stochastic matrices to examine the ground state with an even number of lattice sites and one electron less than half-filling. We show that there exists a highly symmetric state with energy $-2$ (in units where $t = 1$) at all \textit{U}. At $U = \infty$ this state becomes the lowest energy state, consistent with the established lower energy bound. \footnote{S. A. Trugman, Phys. Rev. B \textbf{42}, 6612 (1990)} Using this result, several properties of the strongly coupled ground state are derived, including the chemical potential and momentum distribution. This method may be applicable to other models as well. Disagreements between our results and previous work are examined.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2008.MAR.C1.206 | 2015-08-01 09:45:20 | {"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": 0.36420708894729614, "perplexity": 2245.1473407351596}, "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-2015-32/segments/1438042988650.53/warc/CC-MAIN-20150728002308-00267-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.sciforums.com/threads/basic-ecconomics-what-is-ecconomics.161080/page-2 | # Basic Ecconomics(what is Ecconomics?)
Discussion in 'Business & Economics' started by RainbowSingularity, Aug 4, 2018.
1. ### SeattleValued Senior Member
Messages:
3,508
Can you lend me $100,000 for 10 years? I'll pay you back$10,000 every year for 10 years, no problem.
3. ### CptBorkRobbing the Shalebridge CradleValued Senior Member
Messages:
5,785
That countries which follow modern capitalist economic theory have flourished more than any of the societies which have thus far failed to embrace it. The average American (both by mean and median) has a higher income and standard of living than the mean and median standard enjoyed in any country which has tried a substantially different approach, and low-income Americans are still considered relatively wealthy by such standards.
Edit: Redundant post. Hit the wrong key on my laptop, then backspace, browser took me off page and I couldn't find the post I had been making. I love how practical most geniuses are
Last edited: Aug 7, 2018
5. ### iceauraValued Senior Member
Messages:
27,559
You don't know how compound interest means and works, or you're trolling.
Either would be typical of the academically educated in economics.
7. ### CptBorkRobbing the Shalebridge CradleValued Senior Member
Messages:
5,785
What went right is that the average and median American today enjoys a vastly higher income and standard of living than they would in any country on the planet which doesn't implement similar economic practises.
Great. What were/are the nominal per-capita GDP's of these historical/modern societies? Oh I'm not being fair because I'm not accounting for changes in technology? Well feel free to notify me when someone in the modern age completely ignores mainstream economic theory and still manages to do better than Mr. Greenspan and his lot. BTW your biology link doesn't prove anything, because capitalists also take resource limits into account when making long-term plans, whereas leaders in other systems have historically miscalculated when trying to do so.
Last edited: Aug 7, 2018
8. ### SeattleValued Senior Member
Messages:
3,508
It's an oxymoron to talk about the educated not understanding a subject. You are suggesting that the uneducated understand the subject?
I don't think you know what you are talking about.
Your response to everything, regardless of the subject, is the period of time in the U.S. after the Great Depression up until the Reagan Presidency. That's not an answer. That's just a time period.
9. ### iceauraValued Senior Member
Messages:
27,559
That's what went right due to various people - most of them educated primarily in the liberal arts, some in with engineering or military experience, very few of them formally educated in economics - in the past.
The major economists involved in things going right back then would be Adam Smith and (later) John Maynard Keynes.
But you were asked for what went right due to an economist.
10. ### CptBorkRobbing the Shalebridge CradleValued Senior Member
Messages:
5,785
No, what's actually destructive is allowing banks to make terrible decisions and then deciding to use taxpayer money to bail them out instead of starting fresh and channeling that money towards more productive causes, such as those banks which didn't screw things up. I've already argued that position on these forums before, Joepistole and I used to have a long-running disagreement about it.
Canada's banks remained highly profitable throughout the whole sub-prime mortgage crisis, and it wasn't government regulation which made them stick to conservative lending practises; in fact, the Conservative government under Stephen Harper wanted them to do more of what the US and UK banks were doing, nevertheless crediting themselves for Canadian stability in the aftermath.
Limit the ability of banks to do what they do and what reward:risk ratios they can demand of their clients, and the only people who'll be able to get anything financed will be people with rich parents or uncles.
11. ### iceauraValued Senior Member
Messages:
27,559
I'm not talking about not understanding a "subject". I'm talking about someone who is shocked that moneylenders will prey on people and act as parasites on economies if it profits them, personally, to do so. I'm talking about a body of influential policy advisors and powerful officials who expect the banking and financial industries to regulate themselves and be constrained by their rational evaluation of market forces to moderate their greed in the interests of the greater good.
My response to you was the Roosevelt administration and subsequent maintainers of its policies, which were revoked beginning in 1981.
You asked for an example of policymakers who did a better job.
No one is talking about ignoring mainstream economic theory - I recommended paying careful attention to it, in fact, above, in order to avoid repeating the errors of defunct and dismissed economic theory of the past.
12. ### CptBorkRobbing the Shalebridge CradleValued Senior Member
Messages:
5,785
And I've already described it. America's financial institutions follow modern capitalist economic theory. Therefore they're responsible both for the successes and the failures, along with the theory they choose to put into practise.
13. ### iceauraValued Senior Member
Messages:
27,559
Non-conservative mortgage lending practices were not the cause of the Crash of '08.
Canadian banks did take losses on their mortgage lending, and were bailed out to the tune of 100 billion + by a central bank fund previously designed for the purpose.
Canadian banks are heavily and stringently regulated by the Canadian Federal Government - not, as in the US, free to shop around for loosely regulating States.
That was not true between 1933 and 1983. My farming ancestors, for example, had no trouble borrowing money on their signatures under the New Deal regulations- and they were dead broke after the Depression.
No one in my ancestry had rich parents or uncles, and none of them had any trouble getting their various businesses, farms, houses, etc, financed by banks, during the fifty years of rising US prosperity under New Deal banking regulations.
Last edited: Aug 7, 2018
14. ### iceauraValued Senior Member
Messages:
27,559
Speaking of economists, now: their manner of "following" theory - the actual policies the influential and dominant economists recommended and helped establish - changed in 1981.
Before that change, they provided successes. After that change, they provided failures.
Last edited: Aug 7, 2018
15. ### CptBorkRobbing the Shalebridge CradleValued Senior Member
Messages:
5,785
The article notes that none of Canada's major lending institutions were ever in danger of bankruptcy during the crisis, nor were extra funds needed to be raised from government revenues as was done in the US. The money the Canadian government set aside came from the central bank and was used to provide a source of credit at a time when global credit markets were seizing up, and the interest rates were set at standard market prices rather than negative values after inflation as was done in the US. Plus most of the mortgages financed with this money were already insured, so the government was guaranteed to receive most of its money back no matter what, and only about half of the available funds from this program were ever actually tapped.
Making loans available to support credit liquidity when global credit markets are freezing, is not the same as handing over hundreds of billions in tax dollars to subsidize bankrupt CEO's taking luxury vacations, with no guarantees of timely payback or profit after inflation.
Banks were still relatively loosely regulated even in those times. As I say, capitalism has a built-in regulatory mechanism whereby companies and institutions with unsound practises end up going bankrupt, and those with sound management succeed on their own. Government intervention to bail out US banks instead of leaving them to deal with their own mess and allowing a new class of wealthy elite to replace them, resulted in the US government supporting bad actors who will be free to engage in similar blunders in the future. Instead there should have been a consistent policy- the banks are free to lend and deal with voluntary clients as they please, and they're entirely 100% responsible for absorbing the damage when things go wrong, leaving room for better performers to take over the economy and put more people to work.
16. ### iceauraValued Senior Member
Messages:
27,559
dup.
Last edited: Aug 7, 2018
17. ### iceauraValued Senior Member
Messages:
27,559
Likewise the South Dakota State Bank, which was better regulated than most.
But the Canadians took losses on their mortgage lending - they could not escape all of the US implosion.
Which shows, btw, that it wasn't the mortgage lending that did in the entire US banking and financial industry.
It didn't work. It never has worked, in the US or anywhere else. There are many reasons it will never work, but the main one is that bankers are human beings - not saintly and rationally self-sacrificing guardians of the greater good.
The problems arise when the unsound practices inevitable in poorly regulated banks take down the entire economy, millions of people lose their jobs and homes, and great misery overtakes a formerly prosperous nation. That's bad, see? We learned that finally, once and for all we thought, in 1929. That's why a sensible government regulates its banking industry.
Republican government sucks ass, agreed. But that's what the American people voted for - or the electoral college, anyway.
That was the US policy in the derivatives market after 1999. Not a very good one, as it turned out - as if anyone with a lick of sense thought it was.
Even well run banks aren't physically capable of "absorbing" damage on the scale they are capable of creating if unregulated, and banks brought under by "unsound practices" are less capable than the well run. That's partly what the regulation is for - to make sure that banks are capable of absorbing whatever damage they have caused when things go wrong. Without the regulation, they won't be.
18. ### CptBorkRobbing the Shalebridge CradleValued Senior Member
Messages:
5,785
Bankers don't need to be guardians of the greater good, that's a problem for the people as a whole to sort out through their collective spending and earning choices. People stop working in a given job when the pay doesn't justify their working and living conditions, when it creates more burdens than it solves. Employers have an interest in attracting quality workers to their businesses and therefore an interest in providing attractive working conditions and salaries. Manufacturers have an interest in making better products at more affordable prices to outdo their competitors.
In a free market with heavy competition, businesses and corporations which fail to abide by sound practices end up developing a bad reputation and mismanaging their assets, leading to their collapse and replacement with superior competitors. The problems with capitalism often stem from the market not being free and competitive due to selective government intervention, and governments not intervening under more appropriate circumstances when vulnerable people are being abused and exploited.
Why even worry about economic policies, tax brackets and profit margins, when it's legal to buy stuff that was made or handled in a foreign country by people working at gunpoint, children locked up in factories prone to infernos, folks who are arrested and beaten when they attempt to unionize, and all that other pleasantness?
No, that's why a sensible banking industry regulates itself. Didn't John D. Rockefeller make a comment about knowing to get out of the market back in '29 when even his shoeshine boy was playing stocks? As you yourself note, not all US banks failed during the 2008 crisis, and they'd be running things now if their competitors hadn't been rescued. Evolution works by allowing things to fail and be replaced with better stuff.
Obama was involved in the bailouts policy too, he's the one who enacted it. You can say he was pressured or whatever and I'm sure he was, but the President's job is to lead, he can't say he had no part in the bailouts.
If a bank is well-run, then by definition it won't cause financial damage it can't afford to absorb.
19. ### iceauraValued Senior Member
Messages:
27,559
It's the responsibility of citizens to prevent disproportionate wealth accumulation and resulting bad consequences in the society they control.
And by preventing bankers from wrecking their economy.
That's what Greenspan said he expected - may have been an excuse for abetting exactly what happened.
None ever has, or ever will (as noted above). Being shocked when they don't is kind of silly.
And therein lies the oddity of the common effect of getting an academic education in economics - so many of those folks expect bankers to regulate themselves.
There is also the more common problem: markets not being "free" because the government failed to set up and defend them, or because they were a natural monopoly and could not be set up or defended as free markets.
A free market is usually a difficult thing to maintain and defend - sophistication in governance is necessary.
Republican governance sucks ass - agreed.
That said, none of the surviving banks (if any) would be ""running things" as some kind of replacements for the ruined banks - there would have been little to run, for a long time.
You can say he took an oath of office to faithfully execute the legislation passed by Congress and previous administrations. He wasn't pressured, he was subject to mandate.
That's good then, because they couldn't absorb as much damage as they could do, if unregulated. No bank can.
20. ### JeevesValued Senior Member
Messages:
2,190
This piece of crockery again? It wasn't even true in Dickens' time.
Enough to keep people fed, clothed, sheltered and working; enough left over to support a stable government and standing army and build some quite impressive roads, fortifications and temples. With no periodic monetary crises. No homeless. No breadlines. No police busting the heads of striking miners. No children in sweatshops, though they would be expected to help out in cottage industry and family farms. No dole. No crippling debt. Nobody becoming a billionnaire on adolescent angst or pathological gambling.
So, predators flourish when it kills prey? Not news. What sometimes - like every twenty years - seems to strike capitalist countries, as if it were news, is that a large portion of their own population pays a heavy price for the enrichment of a small portion of their own population. Then the apex predators say: "Yes, but you're still better off than the peoples we're actively bombing, so shut up."
Last edited: Aug 7, 2018 | 2018-12-13 16:31:48 | {"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": 0.1786462366580963, "perplexity": 4449.753302992831}, "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-2018-51/segments/1544376824912.16/warc/CC-MAIN-20181213145807-20181213171307-00418.warc.gz"} |
https://ask.libreoffice.org/en/questions/15377/revisions/ | # Revision history [back]
### printer setting changes to genric printer
OS=ubuntu 12.04 64bit
LO=3.5
I am running ubuntu on client through LTSP. Since on server i have added 2 network printer through CUPS and shared between my Fat Clients. I am able to print successfully, but sometime in middle the default printer changes to GENRIC PRINTER
http://i50.tinypic.com/21jai51.png
http://i47.tinypic.com/ix3xis.jpg
i tried this /usr/lin/libreoffice/program/spadmin and set default printer but again in middle it goes to generic printer
How do i make any one printer default system wide?
output: /home/$USER/.config/libreoffice/3/user/psprint/psprint.conf [__Global_Printer_Defaults__] DisableCUPS=false How do i tell libreoffice to use /etc/libreoffice/psprint.conf ? reference:http://askubuntu.com/questions/191524/set-default-printer-settings-libreoffice ### printer setting changes to genric printer OS=ubuntu 12.04 64bit LO=3.5 I am running ubuntu on client through LTSP. Since on server i have added 2 network printer through CUPS and shared between my Fat Clients. I am able to print successfully, but sometime in middle the default printer changes to GENRIC PRINTER http://i50.tinypic.com/21jai51.png http://i47.tinypic.com/ix3xis.jpg i tried this /usr/lin/libreoffice/program/spadmin and set default printer but again in middle it goes to generic printer How do i make any one printer default system wide? output: /home/$USER/.config/libreoffice/3/user/psprint/psprint.conf /home/$USER/.config/libreoffice/3/user/psprint/psprint.conf [__Global_Printer_Defaults__] DisableCUPS=false How do i tell libreoffice to use /etc/libreoffice/psprint.conf ? reference:http://askubuntu.com/questions/191524/set-default-printer-settings-libreoffice ### printer setting changes to genric printer OS=ubuntu 12.04 64bit LO=3.5 I am running ubuntu on client through LTSP. Since on server i have added 2 network printer through CUPS and shared between my Fat Clients. I am able to print successfully, but sometime in middle the default printer changes to GENRIC PRINTER http://i50.tinypic.com/21jai51.png http://i47.tinypic.com/ix3xis.jpg i tried this /usr/lin/libreoffice/program/spadmin /usr/lib/libreoffice/program/spadmin and set default printer but again in middle it goes to generic printer How do i make any one printer default system wide? output: /home/$USER/.config/libreoffice/3/user/psprint/psprint.conf
[__Global_Printer_Defaults__] DisableCUPS=false
How do i tell libreoffice to use /etc/libreoffice/psprint.conf ?
### printer solved:printer setting changes to genric printer
OS=ubuntu 12.04 64bit
LO=3.5
I am running ubuntu on client through LTSP. Since on server i have added 2 network printer through CUPS and shared between my Fat Clients. I am able to print successfully, but sometime in middle the default printer changes to GENRIC PRINTER
http://i50.tinypic.com/21jai51.png
http://i47.tinypic.com/ix3xis.jpg
i tried this /usr/lib/libreoffice/program/spadmin and set default printer but again in middle it goes to generic printer
How do i make any one printer default system wide?
output: /home/$USER/.config/libreoffice/3/user/psprint/psprint.conf [__Global_Printer_Defaults__] DisableCUPS=false reference:http://askubuntu.com/questions/191524/set-default-printer-settings-libreoffice update: https://bugs.launchpad.net/ubuntu/+source/libreoffice/+bug/1020048/ 5 cleanup formatting and embed pictures oweng 21588 ●18 ●96 ●166 ### solved:printer setting changes to genric printer OS=ubuntu 12.04 64bit LO=3.5 I am running ubuntu on client through LTSP. Since on server i have added 2 network printer through CUPS and shared between my Fat Clients. I am able to print successfully, but sometime in middle the default printer changes to GENRIC PRINTERfrom: http://i50.tinypic.com/21jai51.png http://i47.tinypic.com/ix3xis.jpg ...to Generic Printer: i tried this /usr/lib/libreoffice/program/spadmin /usr/lib/libreoffice/program/spadmin and set default printer but again in middle it goes to generic printer How do i make any one printer default system wide? output: /home/$USER/.config/libreoffice/3/user/psprint/psprint.conf
[__Global_Printer_Defaults__]
DisableCUPS=falseDisableCUPS=false | 2019-08-18 21:03:28 | {"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": 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": 0.17649342119693756, "perplexity": 14075.01826850474}, "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/1566027314130.7/warc/CC-MAIN-20190818205919-20190818231919-00389.warc.gz"} |
https://academic.oup.com/cercor/article/15/9/1343/288674/Overexpression-of-p27Kip1-Probability-of-Cell | ## Abstract
Neocortical projection neurons arise from a pseudostratified ventricular epithelium (PVE) from embryonic day 11 (E11) to E17 in mice. The sequence of neuron origin is systematically related to mechanisms that specify neuronal class properties including laminar fate destination. Thus, the neurons to be assembled into the deeper layers are the earliest generated, while those to be assembled into superficial layers are the later generated neurons. The sequence of neuron origin also correlates with the probability of cell cycle exit (Q) and the duration of G1-phase of the cell cycle (TG1) in the PVE. Both Q and TG1 increase as neuronogenesis proceeds. We test the hypothesis that mechanisms regulating specification of neuronal laminar destination, Q and TG1 are coordinately regulated. We find that overexpression of p27Kip1 in the PVE from E12 to E14 increases Q but not TG1 and that the increased Q is associated with a commensurate increase in the proportion of exiting cells that is directed to superficial layers. We conclude that mechanisms that govern specification of neocortical neuronal laminar destination are coordinately regulated with mechanisms that regulate Q and are independent of mechanisms regulatory to cell cycle duration. Moreover, they operate prior to postproliferative mechanisms necessary to neocortical laminar assembly.
## Introduction
In the mouse, neocortical projection neurons are produced in the pseudostratified ventricular epithelium (PVE) over a period of 7 days, from embryonic day (E) 11 to E17, spanning 11 integer cell cycles (His, 1904; Sauer, 1935, 1936; Boulder Committee, 1970; Takahashi et al., 1994, 1995). On each successive day of neuronogenesis there is a systematic progression in the laminar destination of neocortical neurons produced (Fig. 1A). Specifically, the earliest born neurons are destined for the subplate and layer VI, while the later formed neurons are destined for progressively more superficial layers (Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). It is widely accepted that the time of origin of a neuron (i.e. a neuron's ‘birthday’) is intimately linked to mechanisms of specification of its laminar fate; however, the length of the cell cycle (TC), G1-phase (TG1) and the proportion of daughter cells exiting the cell cycle (Q) also increase with each day of the 7 day neuronogenetic period (Caviness et al., 1995; Takahashi et al., 1996a,b). Thus, the laminar fate of a newly born neuron is not only associated with its birthday but also with specific parameters of the proliferative population. These proliferative factors, including TC, TG1 and Q, are not normally dissociable from one another during normal development. Therefore, enquiry into the mechanisms linking one or more of these factors to mechanisms of laminar fate specification has not been feasible in vivo.
Figure 1.
Experimental design in relation to p27Kip1 overexpression. (A) A diagrammatic illustration of the period of p27Kip1 overexpression and the overall schedule of neocortical neuronogenesis (modified from Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). Neocortical neurons arise from a pseudostratified ventricular epithelium (PVE) in the course of 11 integer cell cycles during the interval embryonic day 11 (E11) to E17 (Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). There is a systematic progression in the laminar fate of neurons produced during each successive cell cycle and on each successive day of neuronogenesis. Thus, the earliest born neurons are destined for the deeper layers while the later formed neurons are destined for progressively more superficial layers (Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). The diagram applies specifically to the medial cortical zone (MCZ) and neocortical field 1. In this region, infragranular neurons (IG neurons; layers V and VI) are produced mainly during cell cycles 1–6 and granular and supragranular neurons (SG neurons; layers IV, III and II) mainly during cycles 9–11. We induced p27Kip1 overexpression in the PVE of the embryonic brain by oral administration of doxycycline to pregnant dams, twice daily. The p27Kip1 overexpression was induced for a 3 day period from E12 to E14 when cycles 3–8 are in progress. Our earlier work showed that in this paradigm, p27Kip1 protein levels increased by 3-fold from E13 onwards and p27Kip1 mRNA returned to basal levels 12 h after the last doxycycline dosage (Mitsuhashi et al., 2001). The cell cycle parameters and the probability of cell cycle exit were estimated for cycles 7/8 on E14. During these cell cycles, the PVE of the MCZ produces neurons destined for both SG and IG layers of field 1, with a significant bias in favor of the IG layers (Takahashi et al., 1999). We reasoned that if p27Kip1 overexpression produced a shift in layer destination of the neurons generated in the MCZ on E14, analysis of the proportion of neurons that take up residence in SG versus IG layers in field 1 at maturity would facilitate a quantitative estimation of the direction and magnitude of the shift. (B–E) Schematic illustrations of the double S-phase labeling paradigms for the estimation of cell cycle parameters, neuronogenetic interval, probability of cell cycle exit and laminar fate of cells. We used four protocols in which S-phase markers iododeoxyuridine (IdU, solid arrowhead) and bromodeoxyuridine (BrdU, open arrowhead) were administered to pregnant mothers for estimation of cell cycle parameters on E14 (B), the neuronogenetic gradient on E16 (C), the number of cells in the Q-fraction on E14 (D) and the number and distribution of cells exiting the cell cycle on E14 in the neocortex of mice on P21 (E).
Figure 1.
Experimental design in relation to p27Kip1 overexpression. (A) A diagrammatic illustration of the period of p27Kip1 overexpression and the overall schedule of neocortical neuronogenesis (modified from Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). Neocortical neurons arise from a pseudostratified ventricular epithelium (PVE) in the course of 11 integer cell cycles during the interval embryonic day 11 (E11) to E17 (Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). There is a systematic progression in the laminar fate of neurons produced during each successive cell cycle and on each successive day of neuronogenesis. Thus, the earliest born neurons are destined for the deeper layers while the later formed neurons are destined for progressively more superficial layers (Sidman and Rakic, 1973; Rakic, 1976; Caviness, 1982; Takahashi et al., 1999). The diagram applies specifically to the medial cortical zone (MCZ) and neocortical field 1. In this region, infragranular neurons (IG neurons; layers V and VI) are produced mainly during cell cycles 1–6 and granular and supragranular neurons (SG neurons; layers IV, III and II) mainly during cycles 9–11. We induced p27Kip1 overexpression in the PVE of the embryonic brain by oral administration of doxycycline to pregnant dams, twice daily. The p27Kip1 overexpression was induced for a 3 day period from E12 to E14 when cycles 3–8 are in progress. Our earlier work showed that in this paradigm, p27Kip1 protein levels increased by 3-fold from E13 onwards and p27Kip1 mRNA returned to basal levels 12 h after the last doxycycline dosage (Mitsuhashi et al., 2001). The cell cycle parameters and the probability of cell cycle exit were estimated for cycles 7/8 on E14. During these cell cycles, the PVE of the MCZ produces neurons destined for both SG and IG layers of field 1, with a significant bias in favor of the IG layers (Takahashi et al., 1999). We reasoned that if p27Kip1 overexpression produced a shift in layer destination of the neurons generated in the MCZ on E14, analysis of the proportion of neurons that take up residence in SG versus IG layers in field 1 at maturity would facilitate a quantitative estimation of the direction and magnitude of the shift. (B–E) Schematic illustrations of the double S-phase labeling paradigms for the estimation of cell cycle parameters, neuronogenetic interval, probability of cell cycle exit and laminar fate of cells. We used four protocols in which S-phase markers iododeoxyuridine (IdU, solid arrowhead) and bromodeoxyuridine (BrdU, open arrowhead) were administered to pregnant mothers for estimation of cell cycle parameters on E14 (B), the neuronogenetic gradient on E16 (C), the number of cells in the Q-fraction on E14 (D) and the number and distribution of cells exiting the cell cycle on E14 in the neocortex of mice on P21 (E).
Proliferative parameters TC, TG1 and Q are regulated by the concerted actions of cyclins, cyclin-dependent kinases (cdk) and their inhibitors. The cdk inhibitor p27Kip1 is a critical modulator of the G1- to S-phase transition (Sherr and Roberts, 1999). Interestingly, it is also implicated in mechanisms of neuronal specification (Durand et al., 1997, 1998; Ohnuma et al., 1999, 2001, 2002; Livesey and Cepko, 2001; Vernon et al., 2003). We have developed a transgenic mouse model in which p27Kip1 is overexpressed selectively in response to doxycycline in the neuroepithelium of the embryonic brain during specific periods of neocortical neuronogenesis. In a previous report (Mitsuhashi et al., 2001) using this mouse model we characterized the expression of the p27Kip1 transcript and protein in the E12–E13 ventricular zone (VZ) in response to different doxycycline dosing schedules. Following only 26 h of doxycycline exposure (an interval corresponding to ∼2 cell cycle durations) beginning on E12, the 2 h bromodeoxyuridine (BrdU) labeling index in the VZ was reduced, although the width of the VZ was unchanged (Mitsuhashi et al., 2001). We interpreted the observation that VZ width did not change to indicate that Q was not altered by this brief interval of p27Kip1 overexpression. This interpretation relating to Q required that the decrease in the BrdU labeling index would be explained by a prolongation of TG1. In the present study, we re-examined the consequences of p27Kip1 overexpression for a full 2 day interval with actual direct measurements of both Q and TG1. At variance with our earlier interpretation, the present experiments show that p27Kip1overexpression increases Q but does not modify TG1. Moreover, the increase in Q is associated with a commensurate increase in the relative proportion of postmitotic neurons directed to the supragranular layers of the neocortex.
## Materials and Methods
### Animals
Double transgenic (DT) mice, in which p27Kip1 can be overexpressed selectively and electively in the neuroepithelium of the embryonic brain, were produced by mating the hemizygous tetOp27Kip1/– and PnestinrtTA/– FVB lines described previously (Mitsuhashi et al., 2001). The plug date was defined as embryonic day 0 (E0). Wild-type (WT) littermates were used as controls. Each experiment was initiated by induction of p27Kip1 overexpression in response to orally administered doxycycline hydrochloride (dox; Sigma) to pregnant dams at a dose of 25 μg/g body wt, twice daily from E12 to E14 (Fig. 1). All of the experimental procedures were in full compliance with institutional guidelines and the NIH Guide for the Care and Use of Laboratory Animals.
### Induction of p27Kip1 Overexpression and S-phase Labeling
Each experiment was initiated by induction of p27Kip1 expression by oral dox administration to pregnant dams (Fig. 1A). The following four iododeoxyuridine–bromodeoxyuridine (IdU and BrdU, respectively) double S-phase labeling paradigms were employed.
#### Cell Cycle Kinetics (Fig. 1B)
A single i.p. injection of IdU (Sigma; 50 μg/g) was administered to pregnant dams carrying E14 mice at 07:00 h. It was followed at 09:00 h by a single injection of BrdU (Sigma; 50 μg/g). Mice were killed at 09:30 h (Hayes and Nowakowski, 2000).
#### Neuronogenetic Interval (Fig. 1C)
BrdU was administered to pregnant dams carrying E16 mice as a single i.p. injection at either 09:00 or 21:00 h on E16. Offspring were killed on postnatal day 4 (P4) and the distribution of BrdU-labeled cells was mapped in the cerebral cortex with respect to cortical cytoarchitectonic fields (Fig. 2).
Figure 2.
Effect of p27Kip1 overexpression on total neurogenetic interval. Schematic illustration of the labeling patterns in the cerebral hemisphere of postnatal day 4 (P4) mice produced by a single administration of bromodeoxyuridine (BrdU) either at 9:00 h (A, C) or 21:00 h (B, D) on embryonic day 16 (E16) (Fig. 1C). The position of BrdU-labeled cells in the cortical gray matter is illustrated in drawings of sections taken from the level of somatosensory cortex following the 09:00 or 21:00 injections (A and B, respectively) in the DT mice. Solid arrows point to the lateral extent of the BrdU labeling in each section, which corresponds to the position of the ‘wave front’ of labeled cells in the medial-to-lateral axis. With four coronal sections as A and B taken at approximately equal intervals (broken lines i–iv in C, D; section iii is presented in A, B) in double transgenic and wild-type littermates, the distribution of BrdU-positive cells on a surface view of the cortical area map was reconstructed (C, D). The neocortical area map (C, D) is adapted from the architectonic map of the adult mouse brain (Caviness, 1975) as seen from superior–lateral aspect. The solid lines indicate the approximate position of the ‘wave front’ of BrdU-labeled cells in the two BrdU injection paradigms. The BrdU-labeling displays an ascending rostrolateral to caudomedial gradient, approximated by the gray shading, which corresponds to the transverse neurogenetic gradient. In both the genotypes, following the 9:00 h BrdU injection, the front of BrdU-labeled cells is located a short distance from the rhinal fissure along an axis that passes through insular field 14 laterally and the lateral temporal and perirhinal field 36 posteriorly (solid line in C). Following the 21:00 h BrdU injection, the front of BrdU-labeled cells is located in the lateral parietal fields 3a and 40 medially and the lateral occipital field 18a posteriorly in both the genotypes (solid line in D). Thus, the position of the proliferative ‘wave fronts’ produced by the 9:00 h to 21:00 h BrdU injections is an identical in both the genotypes. CC, corpus callosum; CTX, cortex; HI, hippocampus. Scale bars in A, B = 100 μm.
Figure 2.
Effect of p27Kip1 overexpression on total neurogenetic interval. Schematic illustration of the labeling patterns in the cerebral hemisphere of postnatal day 4 (P4) mice produced by a single administration of bromodeoxyuridine (BrdU) either at 9:00 h (A, C) or 21:00 h (B, D) on embryonic day 16 (E16) (Fig. 1C). The position of BrdU-labeled cells in the cortical gray matter is illustrated in drawings of sections taken from the level of somatosensory cortex following the 09:00 or 21:00 injections (A and B, respectively) in the DT mice. Solid arrows point to the lateral extent of the BrdU labeling in each section, which corresponds to the position of the ‘wave front’ of labeled cells in the medial-to-lateral axis. With four coronal sections as A and B taken at approximately equal intervals (broken lines i–iv in C, D; section iii is presented in A, B) in double transgenic and wild-type littermates, the distribution of BrdU-positive cells on a surface view of the cortical area map was reconstructed (C, D). The neocortical area map (C, D) is adapted from the architectonic map of the adult mouse brain (Caviness, 1975) as seen from superior–lateral aspect. The solid lines indicate the approximate position of the ‘wave front’ of BrdU-labeled cells in the two BrdU injection paradigms. The BrdU-labeling displays an ascending rostrolateral to caudomedial gradient, approximated by the gray shading, which corresponds to the transverse neurogenetic gradient. In both the genotypes, following the 9:00 h BrdU injection, the front of BrdU-labeled cells is located a short distance from the rhinal fissure along an axis that passes through insular field 14 laterally and the lateral temporal and perirhinal field 36 posteriorly (solid line in C). Following the 21:00 h BrdU injection, the front of BrdU-labeled cells is located in the lateral parietal fields 3a and 40 medially and the lateral occipital field 18a posteriorly in both the genotypes (solid line in D). Thus, the position of the proliferative ‘wave fronts’ produced by the 9:00 h to 21:00 h BrdU injections is an identical in both the genotypes. CC, corpus callosum; CTX, cortex; HI, hippocampus. Scale bars in A, B = 100 μm.
#### The Number of Cells in the Q Fraction (NQ) (Fig. 1D)
IdU was administered as a single i.p. injection to pregnant dams carrying E14 mice at 07:00 h. This was followed by sequential i.p. administrations of BrdU every 3 h from 09:00 to 24:00 h corresponding to an interval longer than TC – TS (‘birth hour method’: Takahashi et al., 1996a,b). This design identifies a cohort of cells that were in S-phase between 07:00 and 09:00 h and that exited the cell cycle following the S-phase. The cohort is labeled only with IdU (Fig. 3C). Since we count only IdU-only labeled cells, the cells that reenter S-phase (rather than exiting the cell cycle) ‘disappear’, as they will be double-labeled with BrdU and IdU (Fig. 3C).
Figure 3.
Effect of p27Kip1 overexpression on cell output. Micrographs of 4 μm thick coronal sections through the heads of embryonic day 14 wild-type (WT, A) and double transgenic (DT, B) littermates taken from approximately the mid-hemisphere level and processed for iododeoxyuridine (IdU) and bromodeoxyuridine (BrdU) immunohistochemistry. A higher magnification view (C) illustrates the IdU-only labeled cells (solid arrowheads, blue cells) and BrdU-labeled cells (open arrowheads, blue/brown cells), which can be reliably distinguished and counted in these preparations. The labeled cells are distributed throughout the telencephalic neuroepithelium and demarcate it from the marginal zones, which contain only unlabeled cells. Histological appearance of the cerebral wall was similar in the WT and DT mice (compare A to B). The IdU-only labeled cells were counted in the ventricular zone (VZ) of the cerebral wall at two regions along the medial-lateral axis, the medial and lateral cortical zones (MCZ and LCZ, respectively). These cells are referred to as Q-fraction or Q cells. The distribution of the Q cells in the VZ and the intermediate zone (IZ) of the cerebral wall was similar in the WT and DT littermates both in the MCZ (D) and the LCZ (E) suggesting that the rates of cell exit from the VZ and migration through the IZ were not influenced by the p27Kip1 overexpression. Scale bars in A and B = 100 μm, in C = 10 μm.
Figure 3.
Effect of p27Kip1 overexpression on cell output. Micrographs of 4 μm thick coronal sections through the heads of embryonic day 14 wild-type (WT, A) and double transgenic (DT, B) littermates taken from approximately the mid-hemisphere level and processed for iododeoxyuridine (IdU) and bromodeoxyuridine (BrdU) immunohistochemistry. A higher magnification view (C) illustrates the IdU-only labeled cells (solid arrowheads, blue cells) and BrdU-labeled cells (open arrowheads, blue/brown cells), which can be reliably distinguished and counted in these preparations. The labeled cells are distributed throughout the telencephalic neuroepithelium and demarcate it from the marginal zones, which contain only unlabeled cells. Histological appearance of the cerebral wall was similar in the WT and DT mice (compare A to B). The IdU-only labeled cells were counted in the ventricular zone (VZ) of the cerebral wall at two regions along the medial-lateral axis, the medial and lateral cortical zones (MCZ and LCZ, respectively). These cells are referred to as Q-fraction or Q cells. The distribution of the Q cells in the VZ and the intermediate zone (IZ) of the cerebral wall was similar in the WT and DT littermates both in the MCZ (D) and the LCZ (E) suggesting that the rates of cell exit from the VZ and migration through the IZ were not influenced by the p27Kip1 overexpression. Scale bars in A and B = 100 μm, in C = 10 μm.
#### Identification of a Cohort of cells that Exited the Cell Cycle on E14 in the P21 Cortex (Fig. 1E)
The design is identical to that for NQ estimation (Fig. 1D) except that the 2 h cohort of IdU-only labeled cells is examined in the cerebral cortex on P21 (Fig. 4D,E) (Takahashi et al., 1999).
Figure 4.
Effect of p27Kip1 overexpression on laminar fate of neocortical neurons. Effects of p27Kip1 overexpression during the embryonic period on the gross appearance of the brain, cytoarchitecture of the neocortex and laminar fates of cells generated on embryonic day 14 (E14) examined at postnatal day 21 (P21). The brains of wild-type (WT) and double transgenic (DT) mice appear similar in size and shape at P21 (A). B and C are micrographs of 4 μm thick coronal sections through field 1 (B) and field 40 (C) of WT and DT mice stained with basic fuchsin to reveal cortical lamination. The cytoarchitecture of fields 1 and 40 is preserved in the DT mice. However, the thickness of the cortical gray matter is reduced dramatically in the DT mouse in field 1 compared to the WT littermates (B). The thickness of the gray matter in field 40 is similar in the DT and WT mice (C). The reduction in the thickness of field 1 is caused mainly by a marked reduction in the thickness of SG layers (layers II/III and IV; B). The thickness of IG layers (layers V and VI) is similar in the DT and WT littermates in fields 1 and 40 (B, C). Micrographs of 4 μm thick coronal sections through field 1 of P21 WT and DT littermates processed for iododeoxyuridine (IdU) and bromodeoxyuridine (BrdU) immunohistochemistry. At higher magnification (box insert), IdU-only (blue labeled) cells are readily distinguished from cells labeled with BrdU (brown containing). (D). Quantitative analysis of the distribution of the IdU-only labeled cells (i.e. cells that exited the cell cycle on E14) in field 1 at P21 (E) revealed that the majority of the cells was distributed in the IG layers in the WT cortex (blue line) and in the SG layers in the DT cortex (red line) at P21. Thus, layer destination of cells generated on E14 was shifted toward the SG layers in the DT mice. ML = molecular layer (layer I). Scale bar in A = 5 mm; BD = 100 μm.
Figure 4.
Effect of p27Kip1 overexpression on laminar fate of neocortical neurons. Effects of p27Kip1 overexpression during the embryonic period on the gross appearance of the brain, cytoarchitecture of the neocortex and laminar fates of cells generated on embryonic day 14 (E14) examined at postnatal day 21 (P21). The brains of wild-type (WT) and double transgenic (DT) mice appear similar in size and shape at P21 (A). B and C are micrographs of 4 μm thick coronal sections through field 1 (B) and field 40 (C) of WT and DT mice stained with basic fuchsin to reveal cortical lamination. The cytoarchitecture of fields 1 and 40 is preserved in the DT mice. However, the thickness of the cortical gray matter is reduced dramatically in the DT mouse in field 1 compared to the WT littermates (B). The thickness of the gray matter in field 40 is similar in the DT and WT mice (C). The reduction in the thickness of field 1 is caused mainly by a marked reduction in the thickness of SG layers (layers II/III and IV; B). The thickness of IG layers (layers V and VI) is similar in the DT and WT littermates in fields 1 and 40 (B, C). Micrographs of 4 μm thick coronal sections through field 1 of P21 WT and DT littermates processed for iododeoxyuridine (IdU) and bromodeoxyuridine (BrdU) immunohistochemistry. At higher magnification (box insert), IdU-only (blue labeled) cells are readily distinguished from cells labeled with BrdU (brown containing). (D). Quantitative analysis of the distribution of the IdU-only labeled cells (i.e. cells that exited the cell cycle on E14) in field 1 at P21 (E) revealed that the majority of the cells was distributed in the IG layers in the WT cortex (blue line) and in the SG layers in the DT cortex (red line) at P21. Thus, layer destination of cells generated on E14 was shifted toward the SG layers in the DT mice. ML = molecular layer (layer I). Scale bar in A = 5 mm; BD = 100 μm.
### Tissue Processing and Histology
P4 and P21 mice were anesthetized (Ketamine, 50 μg/g; Ketalar, Abbott; Xylazine 10 μg/g; Rompun, Bayer; i.p.) and perfused through the heart with 4% paraformaldehyde in phosphate buffer, pH 7.2. Brains were removed, embedded in paraffin wax. E14 mice were removed by hysterotomy from anesthetized dams and decapitated. The entire heads were embedded in paraffin wax. The postnatal brains and embryonic heads were sectioned at a thickness of 4 μm in the coronal plane. The sections were processed for IdU–BrdU double immunohistochemistry as described below. Tail (postnatal) or trunk (embryo) samples were collected for genotyping from anesthetized mice prior to tissue fixation.
### IdU–BrdU Immunohistochemistry
Paraffin-embedded sections were cleared in Histoclear (National Diagnostic) and xylene (Fisher Scientific), rehydrated in graded ethanol and PBS. The sections were immersed in 5% acetic acid overnight, then washed with distilled water, and treated with 0.2% trypsin (37°C, 20 min) and 2 N HCl (30 min). The sections from E14 mice were microwaved in a solution of 0.01 M sodium citrate, pH 6.2, for 15 min and not trypsinized, as trypsinization adversely affected the integrity of the sections. Non-specific antibody reaction was blocked with 1.5% normal horse serum in PBS (30 min). Sections were incubated with mouse monoclonal antibody Br3 (Caltag Lab., 0.025% in PBS, 1.5% normal horse serum and 0.5% Tween-20) for 30 min, biotinylated anti-mouse IgG (0.5% in PBS) for 45 min and then with Vector ABC-peroxidase solution (ABC Peroxidase Elite kit, Vector) for 60 min. The sections were reacted with diaminobenzidine (DAB, 0.05%, Sigma) and H2O2 (0.01%) for 8– 15 min, rinsed with PBS, immersed in 5% acetic acid for 30 min and washed with distilled water. The sections were then incubated with mouse monoclonal antibody IU4 (Caltag Lab., 0.025% in PBS and 0.5% Tween-20) for 30 min, biotinylated anti-mouse IgG (0.5% in PBS) for 45 min and then with vector ABC alkaline phosphatase solution (ABC-Alkaline phosphatase kit, Vector) for 60 min. The sections were reacted with Alkaline Phosphatase Substrate (Vector Blue-Alkaline Phosphatase Substrate Kit III, Vector) for 15 min. The sections were rinsed with distilled water and coverslipped with Crystal Mount (Biomedia).
### TUNEL Histochemistry
Paraformaldehyde-fixed, paraffin-embedded, 4 μm thick, coronal sections were processed for terminal deoxynucleotidyl transferase-mediated dUTP-biotin nick-end labeling (TUNEL) according to the manufacturer's instructions (ApopTag kit, Intergen Purchase). The sections were counterstained with 0.1% aqueous basic fuchsin. TUNEL-positive cells were counted in the cortical gray matter (layer VI to the pial surface) of fields 1 and 40 (Verney et al., 2000).
### Analysis of IdU–BrdU-labeled Cells
The monoclonal antibody IU4 detects both IdU and BrdU. IU4-labeled cells stain blue due to the alkaline phosphatase reaction product. Br3 detects only BrdU. Br3-labeled cells stain brown due to the DAB reaction product (Hayes and Nowakowski, 2000). Thus, although the IdU–BrdU labeling paradigms (Fig. 2B,D,E) yield three types of labeled cells — IdU-only, BrdU-only, and BrdU and IdU double-labeled cells — only the IdU-only labeled cells (blue, solid arrowhead in Fig. 3C), which were in S-phase at the time of the IdU injection and exited the S-phase by the time of the BrdU injection, can be considered to be specifically identifiable. Both the BrdU-only labeled and IdU–BrdU-double labeled cells (blue/brown, open arrowhead in Fig. 3C) would have been in S-phase at the time of the BrdU injection (Nowakowski et al., 1989; Takahashi et al., 1999; Hayes and Nowakowski, 2000). Due to the antibody specificity, we attach no significance to the double labeled versus BrdU-only labeled population. Note, specifically, that our assay depends only on identifying the IdU-only labeled cells. There are also cells not labeled with either IdU or BrdU. Those cells were not in S-phase when effective labeling concentrations of either tracer were available (Fig. 3AC).
The analyses were performed at two locations along the medial–lateral axis of the VZ, namely in the medial cortical zone (MCZ) and the lateral cortical zone (LCZ; Fig. 3A,B). We counted all three types of labeled cells in the MCZ and the LCZ of E14 mice (Fig. 3A,B) within a sector that was 100 μm in its medial–lateral dimension and 4 μm (corresponding to section thickness) in its rostral–caudal dimension (Fig. 3C). The radial dimension of the sector was divided into bins. Each bin is 10 μm high. The bins were numbered 1, 2, 3, etc., from the ventricular margin outward. Following the IdU-only labeled cell counts, the coverslips were removed and the sections were stained with 0.1% aqueous basic fuchsin so that all cells in the VZ could be counted. The number of unlabeled cells was obtained by subtracting IdU and/or BrdU labeled cells from all cells.
Labeled cells were counted also in the medially located field 1 and laterally located field 40 (Caviness, 1975) in P21 cerebral cortex (Fig. 4D,E). For convenience and consistency with our previous work (Takahashi et al., 1999; Caviness et al., 2003), we divided the cortical gray matter into granular–supragranular (SG: layers IV–II/III) and infragranular (IG: layers VI–V) layers. In order to correlate judgments of the architectonic landmarks of the two cortical fields with the various labeling patterns in P21 mice, we stained the IdU–BrdU-labeled sections with basic fuchsin as described above (Fig. 4B,C). In these sections, we identified the border between layers V (internal pyramidal cell layer) and IV (internal granular cell layer) based on cell morphology (large pyramidal cells in layer V and smaller granular cells in IV) and cell packing density (relatively low packing density in layer V compared to that in layer IV). The entire gray matter between the pial surface and white matter/layer VI border was divided into bins as in the E14 studies. We divided the cortical gray matter into SG and IG layers (Fig. 4B,C). We confirmed the registration of the IdU-only labeled cells to IG and SG layers, retrospectively by re-registering the bins with cortical layers (Fig. 4D, Table 4). We also counted the total number of nuclei (i.e. all nuclei) in IG and SG (Table 3).
Cell counts were performed on non-adjacent sections to avoid counting the same cell more than once. The analysis was performed on data collected from the brains of five embryos/mice in each genotype, one or two mice from each of four or five different litters. In each brain, four or five non-adjacent coronal sections were analyzed. Statistical analyses were performed using t-test and ANOVA (Microsoft Excel). In one case a statistical outlier (>3 SD from the mean) was removed from the analysis. A P < 0.05 was used as a significance level in all cases.
### Analysis of Cell Cycle Kinetics
To measure the length of the cell cycle (TC) and its constituent phases [TG1, TS (the length of S-phase) and TG2+M (combined lengths of G2- and M-phases)], three types of labeled cells in the labeling paradigm of Figure 2B were counted in the VZ: IdU-only labeled cells (NI), BrdU-only and BrdU and IdU double-labeled cells (NB) and all cells (NT). The number of unlabeled cells (N0) was calculated by N0 = NT – NI – NB (Table 1). Kinetic parameters are derived according to a modification of the double-labeling algorithm formulated by Hayes and Nowakowski (2000). We determined that TG2+M was ∼2 h, as the majority of M-phase cells (99%) were labeled with IdU-only during the 2 h labeling period (Takahashi et al., 1995) in both WT and DT mice. The growth fraction (GF; Takahashi et al., 1995) was ∼1.0, as cumulative BrdU administration over 15 h labeled all cells in the VZ of WT and DT mice (data not shown). Since the number of cells after M-phase is doubled by mitosis, the number of unlabeled cells that contributes to the cell cycle is N0/2. Similarly the total number of cells that contributes to the cell cycle is NT – N0/2. With these considerations, TC and TS are estimated from the number of cells of each labeling group according to the following equations.
(1)
$NI/(NT{-}N0/2){=}2/T_{C}T_{C}{=}2(NT{-}N0/2)/NI$
(2)
$NI/NB{=}2/T_{S}T_{S}{=}2NB/NI$
Note that both (1) and (2) are approximations because we ignore the 30 min survival time. This was done because there is an 10–15 min interval required for effective labeling concentrations for S-phase markers in the VZ (Hayes and Nowakowski, 2000).
Table 1
Analysis of cell cycle kinetics and cell output
Genotype
NI
NB
NT
N0
MCZ WT 14.8 ± 0.7 19.5 ± 2.8 137.3 ± 10.0 103.0 ± 9.2
DT 15.6 ± 0.9 20.3 ± 0.9 125.1 ± 8.1 89.2 ± 8.5
LCZ
WT 11.9 ± 1.4 24.8 ± 3.4 125.7 ± 5.2 89.0 ± 4.1
DT
11.9 ± 1.3
22.7 ± 3.5
118.3 ± 6.4
83.8 ± 5.2
Genotype
NI
NB
NT
N0
MCZ WT 14.8 ± 0.7 19.5 ± 2.8 137.3 ± 10.0 103.0 ± 9.2
DT 15.6 ± 0.9 20.3 ± 0.9 125.1 ± 8.1 89.2 ± 8.5
LCZ
WT 11.9 ± 1.4 24.8 ± 3.4 125.7 ± 5.2 89.0 ± 4.1
DT
11.9 ± 1.3
22.7 ± 3.5
118.3 ± 6.4
83.8 ± 5.2
Mean ± SEM values of IdU-labeled (NI), BrdU- or IdU+BrdU-labeled (NB) and all cells (NT) per unit area of the VZ in embryonic day 14 wild-type (WT) and double transgenic (DT) mice produced by the S-phase labeling protocol shown in Figure 1B. The number of unlabeled cells (N0) was calculated using the formula N0 = (NT −NI – NB). The values are shown separately for the medial cortical zone (MCZ) and lateral cortical zone (LCZ) of the cerebral wall.
### Analysis of Neurogenetic Interval
The BrdU-labeled sections of P4 brains were photographed using a Nikon Eclipse E400 microscope and a digital camera (SPOT RT Slider Camera, RT230-2, Diagnostic Instruments). The images were processed by SPOT Advanced v.3.5 software (Diagnostic Instruments) and Adobe Photoshop 7.0 (Adobe). The position of the BrdU-labeled cells in the cortical gray matter was superimposed on the outlines of the sections (Fig. 2A,B). The BrdU labeling pattern reconstructed from the serial sections was superimposed on a surface view of the cortical area map (Fig. 2C,D).
## Results
### The Double Transgenic (DT) Mouse Line and Induction of p27Kip1
The mating between the hemizygous tetOp27Kip1/– and PnestinrtTA/– mice was as frequently associated with vaginal plugs as mating among WT mice, although the incidence of successful impregnation was lower in the former. Once pregnancy was established, however, the course of gestation was not altered by the hemizygous condition and parturition occurred on E19, just as for the WT mice. In addition, the number of live born offspring per litter was not different in the hemizygous matings compared to the WT matings. Finally, the DT, hemizygote and WT genotype ratios from tetOp27Kip1/– × PnestinrtTA/– mating approximated the expected Mendelian ratio of 1:2:1.
### Cell Cycle Kinetics
We used a double S-phase labeling method to estimate cell cycle parameters in the VZ of the cerebral wall in E14 DT and WT littermates exposed to dox from E12 onwards (Fig. 1B). We chose E14 for analysis because it is the time when the laminar fate of neurons shifts from layer V to IV in the somastosensory cortex of normal mouse (Takahashi et al., 1999) (Fig. 1A). The analyses were performed at MCZ and LCZ (Fig. 3A,B). The progenitors in the LCZ are developmentally ‘in advance’ of those in the MCZ by at least 24 h, with respect to neuronogenetic schedule (Takahashi et al., 1995, 1996a,b, 1999). Thus, this single experiment is the equivalent of examining the same area of cortex at both E14 and E15, and the analyses in the two zones provide a method to determine if the effects of p27Kip1 overexpression are dependent upon the stage of maturation of the progenitors.
The number of IdU- and BrdU-labeled (i.e. BrdU+ or IdU+/BrdU+) cells was recorded in the MCZ and LCZ of DT and WT littermates (Table 1) to estimate the length of cell cycle (TC) and its phases [TG1, TS (the duration of S-phase) and TG2+M (combined lengths of G2- and M-phases)] according to a method described previously (Hayes and Nowakowski, 2000). The cell cycle parameters did not differ between DT and WT mice either in the MCZ or the LCZ (Table 2). Thus, overexpression of p27Kip1 from E12 to E14 does not alter either the total length of cell cycle or the lengths of its constituent phases on E14.
Table 2
Cell cycle parameters
Genotype
TC (h)
TG2+M (h)
TS (h)
TG1 (h)
NQ (cells)
Q
MCZ WT 11.7 ± 1.0 2.0 2.7 ± 0.4 7.1 ± 0.7 10.5 ± 0.4 0.35
DT 10.5 ± 1.0 2.0 2.6 ± 0.2 5.8 ± 0.9 12.5 ± 0.8* 0.40*
LCZ
WT 14.6 ± 2.0 2.0 4.5 ± 0.8 8.1 ± 1.4 9.6 ± 0.3 0.40
DT
13.6 ± 2.2
2.0
4.1 ± 1.0
7.2 ± 1.5
10.5 ± 1.0
0.41
Genotype
TC (h)
TG2+M (h)
TS (h)
TG1 (h)
NQ (cells)
Q
MCZ WT 11.7 ± 1.0 2.0 2.7 ± 0.4 7.1 ± 0.7 10.5 ± 0.4 0.35
DT 10.5 ± 1.0 2.0 2.6 ± 0.2 5.8 ± 0.9 12.5 ± 0.8* 0.40*
LCZ
WT 14.6 ± 2.0 2.0 4.5 ± 0.8 8.1 ± 1.4 9.6 ± 0.3 0.40
DT
13.6 ± 2.2
2.0
4.1 ± 1.0
7.2 ± 1.5
10.5 ± 1.0
0.41
Mean ± SEM values of cell cycle parameters (the length of cell cycle TC; S phase TS; combined length of G2 and M phase TG2+M; G1 phase TG1) in the medial cortical zone (MCZ) and lateral cortical zone (LCZ) of embryonic day 14 wild-type (WT) and double transgenic (DT) littermates. The number of Q cells (NQ) was calculated using the S-phase labeling protocol shown in Fig. 1D. The Q fraction was calculated from NQ and NP+Q (= 2 × NI) and is significantly larger in the MCZ of the DT mice. No other comparisons are statistically significant.
*
P <0.05; t-test.
### Duration of the Neuronogenetic Interval
Since the cell cycle parameters did not change in the DT mice, the total number of integer cell cycles executed over the duration of the neuronogenetic interval itself would not be expected to change. We tested that prediction next. Neuronogenesis is completed throughout the neocortical PVE of the mouse early on E17 (Caviness, 1982; Takahashi et al., 1995; Miyama et al., 1997). The process of termination of neuronogenesis follows a transverse neurogenetic gradient (Bayer and Altman, 1991) such that ‘wave fronts’ of initiation and termination of neuronal production advance along the rostrolateral to caudomedial axis of the hemisphere (Miyama et al., 1997). The position of the termination wave front is particularly clear at the end of the neuronogenetic period because it marks the border between an area of the PVE that continues to produce neurons and an area in which the PVE has involuted and is no longer producing cells. If p27Kip1 overexpression altered the neuronogenetic interval, the position of the termination wave front at the end of the neuronogenetic period would be displaced in the DT mice compared to the WT littermates, and the degree of displacement would be commensurate with the extent of alteration in the neuronogenetic interval.
We visualized the position of the termination wave front in DT and WT mice at the end of the neuronogenetic period by using a BrdU labeling method (Fig. 1C). We administered BrdU at 09:00 or 21:00 h on E16 and observed the distribution of BrdU-positive cells in the neocortical gray matter in serial coronal sections of the brain on P4 (see examples in Fig. 2A,B). The position of BrdU-positive cells in the cortical gray matter was recorded with respect to the section outlines (Fig. 2A,B). The lateral extent of the spread of the BrdU label (the termination wave front) for representative cases from the 09:00 and 21:00 injections is indicated by arrows at the level of the somatosensory (barrel field) cortex in Figure 2A,B. The termination wave front is located more medially following the 9:00 injection than the 21:00 injection at this coronal level (compare Fig. 2A to 2B), as predicted by the neurogenetic gradient. We reconstructed the termination wave front from the serial sections and superimposed it on a surface view of the cortical area map (Fig. 2C,D). In both WT and DT mice, following the 09:00 h BrdU injection, the front of labeled cells was located a short distance from the rhinal fissure along a plane that passed through insular field 14 laterally and the lateral temporal and perirhinal field 36 posteriorly (Fig. 2C), in accordance with the predicted labeling pattern (Caviness, 1975; Miyama et al., 1997). Following the 21:00 h BrdU injection, the termination wave front was located further into the lateral parietal fields 3a and 40 medially and the lateral occipital field 18a posteriorly in both the genotypes (Fig. 2D). Thus, we found that the tangential and radial distributions of the E16 BrdU labeled cells in the P4 neocortex were indistinguishable between DT (Fig. 2A,B) and WT littermates (data not shown). Therefore, within the limits of resolution of the methods used here, neither the duration of the neuronogenetic interval nor the slope of the transverse neurogenetic gradient were altered by the overexpression of p27Kip1.
### Probability of Cell Cycle Exit
Each of the two daughter cells resulting from a round of cell division has a quantifiable probability of re-entry into or exit from the cell cycle. These probability values are designated as P (for reentry) and Q (for exit) (Takahashi et al., 1996a,b). P + Q is always 1.0, as these two fates are complementary and cell death in the PVE is small, i.e. <1% per cell cycle (Haydar et al., 2000b; Cai et al., 2002). P and Q are calculated based on an estimation of the number of cells in the P and Q fractions (NP and NQ, respectively). This is accomplished by labeling a cohort of cells undergoing S-phase over an experimentally defined interval and following the cohort as it executes G2-, M- and G1-phases so as to distinguish cells that exit the cell cycle from those that re-enter it (Takahashi et al., 1996a,b). We calculated NQ and NP in E14 DT and WT embryos exposed to dox from E12 to E14 using the IdU–BrdU labeling protocol. We calculated NQ using the labeling protocol shown in Figure 1D. In this protocol cells stained blue (labeled only with IdU) are the NQ cells (Fig. 3AC) (Takahashi et al., 1996a,b). We calculated NP+Q by doubling the value of NI (Table 1) obtained using the labeling protocol shown in Figure 1B. Q was calculated using the formula Q = NQ/(NP + NQ). We found that there was a statistically significant increase (P-value = 0.03, t-test) in Q in MCZ of DT mice compared to WT littermates (Table 2), whereas the measures of Q in the LCZ of DT and WT littermates were not different.
We also recorded the pattern of distribution of the NQ cells in the VZ and the intermediate zone (IZ) as they exited the VZ and migrated toward the marginal zones. The distribution of these cells represented the distance of migration ∼6 h after exiting cell cycle. The cells which exited cell cycle have left the VZ and migrated across the IZ to traverse ∼90% (MCZ) and ∼70% (LCZ) of the height of the cerebral wall at the time of measurement (Takahashi et al., 1996a,b). Therefore the pattern of distribution represented the migratory rate of these cells, which was indistinguishable in DT and WT littermates (Fig. 3D,E). Moreover, the architectonic appearance of the developing cortex in DT animals was normal, including cortical plate and subplate and the course of the sagittal stratum subjacent to the cortical strata. There was no suggestion of architectonic anomalies such as are seen in reeler (Caviness, 1982) and other ‘cortical mutants’ (Walsh, 2000; Ohshima et al., 2001; Hammond et al., 2004) where there is disorder of migration and postmigration mechanisms of laminar assembly. Therefore, patterns of cell exit from the VZ and migration across the IZ as well as postmigration mechanisms of laminar assembly are unperturbed by p27Kip1 overexpression. The analysis of postnatal migration and laminar assembly will be considered later in the Discussion.
### Cytoarchitecture of the Cerebral Cortex at Postnatal Day 21
We examined the histology of the cerebral cortex in the DT and WT littermates on P21. There were no differences between WT and DT mice in the gross appearance of the brain as seen in a dorsal view (Fig. 4A). In histological sections, however, there are clear differences in the cortical width. Thus, the neocortex is distinctly thinner in the DT, especially in field 1 (Fig. 4B) albeit less in field 40 (Fig. 4C). This is a quantitative difference only. In particular, the stratification and laminar cytoarchitectonic patterns are similar in the DT and WT mice in neocortical fields 1 and 40 (Fig. 4B,C). Moreover, regionally distinguishing architectonic features — e.g. the barrel patterns of area 3, and the full array of sublaminar patterns through the parietal, frontal temporal, occipital and medial hemispheric fields — were also preserved in the DT mice (data not shown). Thus, the p27Kip1 overexpression from E12 to E14 did not produce gross malformations or architectonic pattern abnormalities of the cerebral cortex or other brain regions at P21. We address the quantitative issues related to cortical thickness in the next section.
### Cortical Thickness and Neuronal Number at P21
Overexpression of p27Kip1 was associated with a modest reduction in the radial thickness and the total number of neurons in the cortex in field 1 in the DT mice (∼ 8% and ∼10%, respectively) (Table 3). This was due entirely to a more substantial reduction in the thickness and number of cells of the SG layers (∼24% and ∼ 18%, respectively). There was no detectable difference in the thickness of the IG layers of field 1. There was no detectable difference in either the full cortical thickness or cell numbers of field 40 overall or that of either the SG or IG layers in that field.
Table 3
The thickness and cell number of P21 cortex
Field 1
Field 40
WT
DT
P
% change
WT
DT
P
% change
Thickness of ctx (μm)a Total 830 ± 14 759 ± 31 0.04 −8* 977 ± 28 939 ± 632 0.55
SG 475 ± 24 408 ± 16 0.03 −24* 521 ± 8 523 ± 65 0.97
IG 355 ± 23 475 ± 20 0.08 457 ± 22 417 ± 38 0.16
Cell number (cells/area)b
Total 179 ± 6 161 ± 6 0.001 −10** 175.6 ± 7 173.4 ± 7 0.76
SG 82.2 ± 2.8 67.7 ± 2.8 0.00009 −18** 83.1 ± 3.3 86.8 ± 6.8 0.46
IG
97.0 ± 3.6
93.2 ± 2.3
0.23
92.5 ± 4.8
86.6 ± 2.4
0.16
Field 1
Field 40
WT
DT
P
% change
WT
DT
P
% change
Thickness of ctx (μm)a Total 830 ± 14 759 ± 31 0.04 −8* 977 ± 28 939 ± 632 0.55
SG 475 ± 24 408 ± 16 0.03 −24* 521 ± 8 523 ± 65 0.97
IG 355 ± 23 475 ± 20 0.08 457 ± 22 417 ± 38 0.16
Cell number (cells/area)b
Total 179 ± 6 161 ± 6 0.001 −10** 175.6 ± 7 173.4 ± 7 0.76
SG 82.2 ± 2.8 67.7 ± 2.8 0.00009 −18** 83.1 ± 3.3 86.8 ± 6.8 0.46
IG
97.0 ± 3.6
93.2 ± 2.3
0.23
92.5 ± 4.8
86.6 ± 2.4
0.16
Mean ± SEM values of the cortical thicknesses and mean of cell numbers in field 1 and field 40 of postnatal day 21 wild type (WT) and double transgenic (DT) littermates.
a
Statistical analyses with respect to thickness were performed by t-test.
b
Statistical analyses with respect to cell number by ANOVA.
‘% change’ refers to the WT – DT difference as a percent of WT.
*
P < 0.05;
**
P < 0.01.
This contrast in the response to p27Kip1 overexpression between medial and lateral localized cortical fields and of SG with respect to IG layers within field 1 is predicted from a model formalized earlier (Caviness et al., 2000, 2003). It will be considered further in the Discussion.
### Laminar Fates of Cells ‘Born’ on E14
We next examined if the laminar fates of cells that exited the cell cycle (NQ cells) on E14 were different between the DT and WT littermates on P21, using a modification of the double-S-phase labeling method (Figs 1E, 4D,E) that was used to estimate NQ on E14 (Fig. 1D). In this method, the E14 NQ cells can be recognized in the P21 cortex as blue cells (Fig. 4D,E). These cells appear to be neocortical projection neurons based on the morphology of the somata (data not shown) and also based on previous reports that the output of the neocortical VZ on E14 is virtually exclusively projection neurons (Takahashi et al., 1995; Qian et al., 2000; Malatesta et al., 2003). It is possible that some of the NQ cells are interneurons. However, the interneurons constitute only ∼15–25% of the total number of neocortical neurons in rodents (Ren et al., 1992; Beaulieu, 1993) and are unlikely to introduce significant bias in our data. Therefore, we consider the E14 NQ cells to be projection neurons.
We recorded the number and radial distribution of the NQ cells in neocortical fields 1 and 40 in WT and DT mice (field 1; Fig. 4E). First, we measured the total number of NQ cells in the gray matter in fields 1 and 40. We found that the total number increased by ∼52% in field 1 (destination of cells originating in the MCZ; Takahashi et al., 1999) and ∼43% in field 40 (destination of cells originating in the LCZ; Takahashi et al., 1999) in the DT mice relative to the WT littermates (Table 4). The differences were statistically significant in both the fields. The increase in NQ cells in field 1 of the DT mice was consistent with the increases in NQ and Q in the MCZ of DT mice at E14 (Table 2). However, the increase in NQ cells in field 40 was not paired with an increase in NQ or Q in the LCZ of DT mice at E14. Plausibly the discrepancy here is attributable to the variability of measurement in the relatively small numbers of DT and WT littermates recovered for the E14 studies.
Next, we examined the distribution of the E14 NQ cells in the SG and IG layers. We found that in the WT mice ∼22% of the NQ cells were distributed to SG layers in field 1 and ∼56% in field 40 (Table 4), reflecting the maturity differences produced by the down gradient position of field 1 relative to field 40 along the transverse neurogenetic gradient. In the DT mice, by contrast, nearly 42% of the NQ cells were found in SG layers in field 1 and 65% in field 40. Thus, there was an increase in the proportion of the NQ cells of the cohort that had been directed to SG layers in the DT mice in both fields 1 and 40 compared to the WT mice (Fig. 4E). This corresponded to a 94% increase in the proportion of NQ cells directed to SG layers in field 1 of the DT mice compared to the WT mice. In field 40, by contrast, although 65% of the NQ cells were directed to the SG layers in the DT mice, this represented only a 20% increase over the WT mice (Table 4).
Table 4
Laminar fate of neurons born at E14
Genotype
Cell number
% change
IG cells
SG cells
% SG cells
% change
Field 1 WT 6.8 ± 0.4 51.9 ± 16.9 5.3 ± 0.6 1.4 ± 0.6 21.5 ± 8.1 94.2 ± 150.7
DT 10.3 ± 0.1** 6.1 ± 0.8 4.2 ± 0.9* 41.7 ± 7.0**
Field 40
WT 5.9 ± 0.7 43.4 ± 19.9 2.9 ± 0.7 3.0 ± 0.2 56.2 ± 9.5 19.9 ± 20.1
DT
8.5 ± 0.7*
3.1 ± 0.6
5.4 ± 0.6**
64.7 ± 5.8
Genotype
Cell number
% change
IG cells
SG cells
% SG cells
% change
Field 1 WT 6.8 ± 0.4 51.9 ± 16.9 5.3 ± 0.6 1.4 ± 0.6 21.5 ± 8.1 94.2 ± 150.7
DT 10.3 ± 0.1** 6.1 ± 0.8 4.2 ± 0.9* 41.7 ± 7.0**
Field 40
WT 5.9 ± 0.7 43.4 ± 19.9 2.9 ± 0.7 3.0 ± 0.2 56.2 ± 9.5 19.9 ± 20.1
DT
8.5 ± 0.7*
3.1 ± 0.6
5.4 ± 0.6**
64.7 ± 5.8
The number and laminar fate of a cohort of cells that exited the cell cycle on embryonic day 14 and came to reside in the infragranular (IG) or granular–supragranular (SG) layers of cortical fields 1 and 40 at postnatal day P21. The proportion of cells destined to the SG layers (% SG cells) was calculated as a percentage of the total cell number.
*
P < 0.05;
**
P < 0.01; t-test.
These findings indicate that on average in the DT mice an NQ cell produced on E14 in the MCZ has a 94% greater chance of residing in SG layers and a NQ cell from the LCZ a 20% greater chance of residing in the SG layers compared to corresponding cells in the WT (Table 4). We encountered substantial variability among the litters in the relative distribution of the NQ cells, in keeping with the expected variation in relative maturity among the litters (Theiler, 1972; Takahashi et al., 1999). We exploited this by correlating the percentage of NQ cells directed to SG layers in DT versus WT littermate pairs in fields 1 and 40 (Fig. 5A). The broken line (with 45° slope; Fig. 5A) denotes the ‘no effect’ line, representing a hypothetical situation, in which DT and WT littermates have identical values (i.e. p27Kip1 overexpression has no effect). However, the slope of the regression line obtained with the actual data is significantly (t-test, P < 0.01) lower than that of the ‘no effect’ line, indicating that the percentage of cells directed to the SG layers was greater in DT mice compared to the WT littermates. Thus, it is evident that the field 1 values diverge from the ‘no effect’ line to a greater extent than the field 40 values, reflecting differences in the maturational state of these two areas (Miyama et al., 1997). This is consistent with the findings from E14 where NQ and Q were significantly higher in the DT mice in the MCZ (precursor of field 1) and not in the LCZ (precursor of field 40) (Table 2). Thus, overexpression of p27Kip1 from E12 to E14 not only shifted the destination of cells produced on E14 towards more superficial layers of the neocortex but also did so in a developmentally regulated manner. That is, there is progressively smaller effect in more mature regions of the cortex (at the time of p27Kip1 induction) as the transverse neurogenetic gradient shifts to formation of principally SG neurons.
Figure 5.
The regulatory linkage among p27Kip1 overexpression, cell output and cell fate. (A) Regression analysis of the percentage of Q cells directed to granular and supragranular (SG) layers in fields 1 and 40 in wild type (WT; x-axis) and double transgenic (DT; y-axis) littermate pairs. The broken line (with 45° slope) denotes the ‘no effect’ line, representing a hypothetical function, in which DT and WT littermates have identical values (i.e. no effect of p27Kip1 overexpression). The slope of the regression line obtained with the actual data is significantly (t-test, P = 0.01) lower than that of the ‘no effect’ line indicating that the percentage of cells directed to the SG layers was greater in DT mice compared to the WT littermates. Moreover, the field 1 values diverge from the ‘no effect’ line to a greater extent than the field 40 values, which tend to approximate the line closely. (B) The percentage of cells that exited the cell cycle on embryonic day 14 (E14) and that occupied SG layers at postnatal day 21 (percentage SG cells) in the WT and DT mice is plotted as function of the probability of cell cycle exit (Q). The solid line represents data from CD1 strain of mouse obtained in our earlier work (Takahashi et al., 1996a, 1999). The data from the current study are plotted as broken line. Non-linear regression curves are obtained for both the sets of data. Q values were measured in the ventricular zone of the cerebral wall in the medial and lateral cortical zones (MCZ and LCZ, respectively) on E14. The MCZ and LCZ are precursors of fields 1 and 40, respectively. MCZ and LCZ are separated from each other along the transverse neurogenetic gradient such that the LCZ progenitors are ‘in advance’ of the MCZ progenitors with regard to cell cycle kinetics. The correlation between Q and percentage SG cells observed in the WT mice also applies to the DT mice despite the p27Kip1 overexpression-induced increase in Q. Thus, when Q increases, percent SG increases proportionately, both in the MCZ and the LCZ and both in the WT and DT mice. This indicates that the probability of origin of a given laminar neuron class is a function of the probability of cell cycle exit, regardless of position along the transverse neurogenetic gradient. The dotted lines in BE demarcate the boundary between IG and SG layers.
Figure 5.
The regulatory linkage among p27Kip1 overexpression, cell output and cell fate. (A) Regression analysis of the percentage of Q cells directed to granular and supragranular (SG) layers in fields 1 and 40 in wild type (WT; x-axis) and double transgenic (DT; y-axis) littermate pairs. The broken line (with 45° slope) denotes the ‘no effect’ line, representing a hypothetical function, in which DT and WT littermates have identical values (i.e. no effect of p27Kip1 overexpression). The slope of the regression line obtained with the actual data is significantly (t-test, P = 0.01) lower than that of the ‘no effect’ line indicating that the percentage of cells directed to the SG layers was greater in DT mice compared to the WT littermates. Moreover, the field 1 values diverge from the ‘no effect’ line to a greater extent than the field 40 values, which tend to approximate the line closely. (B) The percentage of cells that exited the cell cycle on embryonic day 14 (E14) and that occupied SG layers at postnatal day 21 (percentage SG cells) in the WT and DT mice is plotted as function of the probability of cell cycle exit (Q). The solid line represents data from CD1 strain of mouse obtained in our earlier work (Takahashi et al., 1996a, 1999). The data from the current study are plotted as broken line. Non-linear regression curves are obtained for both the sets of data. Q values were measured in the ventricular zone of the cerebral wall in the medial and lateral cortical zones (MCZ and LCZ, respectively) on E14. The MCZ and LCZ are precursors of fields 1 and 40, respectively. MCZ and LCZ are separated from each other along the transverse neurogenetic gradient such that the LCZ progenitors are ‘in advance’ of the MCZ progenitors with regard to cell cycle kinetics. The correlation between Q and percentage SG cells observed in the WT mice also applies to the DT mice despite the p27Kip1 overexpression-induced increase in Q. Thus, when Q increases, percent SG increases proportionately, both in the MCZ and the LCZ and both in the WT and DT mice. This indicates that the probability of origin of a given laminar neuron class is a function of the probability of cell cycle exit, regardless of position along the transverse neurogenetic gradient. The dotted lines in BE demarcate the boundary between IG and SG layers.
### Apoptosis
P4 is the time of maximum apoptotic cell death in the developing mouse neocortex, especially in fields 1 and 40 (Verney et al., 2000). No significant difference was detected between DT and WT littermates in the numerical density of TUNEL+ profiles either in field 1 (mean ± SEM, DT 4.36 × 10−3 ± 1.46 × 10−3, WT 3.66 × 10−3 ± 0.51 × 10−3 cells/unit area, t-test, P = 0.661) or field 40 (mean ± SEM, DT 2.46 × 10−3 ± 0.92 × 10−3, WT 3.38 × 10−3 ± 0.55 × 10−3 cells/unit area, t-test, P = 0.418). There was also no difference in the laminar distribution of TUNEL+ profiles between WT and DT littermates in either cortical field (data not shown).
## Discussion
The direct experimental measures of TG1 and Q undertaken here are based on a recently developed method (Hayes and Nowakowski, 2000), which is both more efficient and sensitive than the methods we have used previously. Importantly in the present study we overexpressed p27Kip1 for a 48 h period from E12 to E14, corresponding to the interval through which p27Kip1 expression ascends from its low to high asymptote in the normal mouse (Delalle et al., 1999). As with other developmentally regulated processes, p27Kip1 expression in WT was found in the prior study (Delalle et al., 1999) to reach asymptote in the upgradient LCZ well in advance of its asymptote in the downgradient MCZ. We find that TG1 is not altered by the p27Kip1 overexpression but that Q is increased. These findings are supported by a recent report showing that both TC and TG1 are unchanged in the neocortical VZ of the p27Kip1 null mouse (Goto et al., 2005). Thus, cell cycle kinetic parameters are not altered even by extreme disregulation of p27Kip1 expression, whether the disregulation is either up or down. In a previous report (Mitsuhashi et al., 2001) we interpreted the reduction in BrdU LI following 26 h of p27Kip1 overexpression as indication of prolongation of TG1 without alteration of Q. Our present findings, based on direct measurements of Q and TG1, show that Q and not TG1 is modulated by p27Kip1 overexpression; this corrects our prior misinterpretation of the limited data we had previously.
The capacity of p27Kip1 to increase Q is developmentally regulated, i.e. strongly apparent in the developmentally early MCZ but not detectable in the developmentally late LCZ. The increase in Q is dissociated from the kinetic operation of the cell cycle in that TG1 (and therefore TC) and the overall neuronogenetic interval remain unchanged. This means that the number of cell cycles in the neuronogenetic interval remains fixed at 11 just as in the normal animal. This also means that the path of ascent of Q as a function of cell cycle sequence through the neuronogenetic interval is altered. This path increases non-linearly from 0.0 (before the first neurons are born) to 1.0 with the final cycle when the last cell division occurs (Takahashi et al., 1996a,b; Nowakowski et al., 2002). In the DT animal the rate of ascent with cycle is increased over the interval of p27Kip1 overexpression and must therefore be somewhat slowed subsequently. Correspondingly the ascent of Q in the p27Kip1/– mouse must rise abnormally slowly initially but then accelerate with respect to WT in the terminal cycles (Goto et al., 2005). It is to be noted in this regard that variations in cell cycle parameters, in contrast to variations in Q, will have little effect upon cell production in the neocortical PVE. Thus, output per cycle is an exponential function of Q and total output is simply the cumulative output of all cycles (Takahashi et al., 2000). Variations in cycle duration may affect linearly the output in time but will have no effect upon output per cycle or total output for a series of cycles. As an example in point, in the Emx2–/– mutant there is a profound reduction in neuronal production but cycle durations are unaltered (Mallamaci et al., 2000). Q, by default the affected parameter, has not been measured. Plausibly the same will hold for Pax6–/– and other mutants which are similarly associated with massive limitations of neuronal production (O'Leary and Nakagawa, 2002; Grove and Fukuchi-Shimogori, 2003; Shin et al., 2004).
Finally, we have observed that the effect of p27Kip1 overexpression upon Q is dissociated from post-proliferative histogenetic mechanisms involved in migration and postmigration histogenetic mechanisms of cortical pattern formation and from postmigratory histogenetic cell death. These processes appear to proceed normally in all respects in the E14 DT mouse embryo. Moreover, in confirmation, the stratification and laminar cytoarchitectonic patterns (such as barrel patterns of area 3) are preserved completely in the P21 animal. Recently p27Kip1 has been found to play a role as modulator of cell motility in cultured fibroblasts, a role mediated by RhoA activation (Besson et al., 2004). However, in the present study, p27Kip1 overexpression in the DT mice would have returned to near normal levels by E16, prior to the onset of robust neocortical neuronal migration. Our previous study (Mitsuhashi et al., 2001) showed that p27Kip1 mRNA levels began to rise 6 h after a single dose of dox, reached 300-fold of baseline expression at 12 h and returned to essentially baseline expression levels 48 h after the dox dose. Therefore, it is unlikely that p27Kip1 overexpression affects migration in this experiment model.
### p27Kip1, Q and Neuronal Laminar Destination
The size of the 2 h (cells stained blue) cohort arising on E14 and distributed within fields 1 and 40 at P21 is larger in DT than WT animals. We attribute this to an increase in Q, occurring in response to p27 Kip1 overexpression. We note, moreover, that there is also a substantial increase in the proportion of the cohort that is distributed to SG layers in DT P21 animals. Because TG1 and therefore TC are unaltered at E14 in the DT embryos, the cohort in the P21 cortex must have arisen from the same cell cycle in the 11 cycle sequence in WT and DT animals. However, the composition of this cohort has been altered such that an increased proportion of its cells is destined for SG layers in both fields 1 and 40 though to a larger degree in field 1. For reasons cited above, this phenomenon is not an artifact or a disturbance of migration, postmigration mechanisms of laminar assembly or abnormal patterns of cell death. It cannot be simply incidental to the E14 cohort representing an increased fraction of the total numbers of cells formed on E14 and subsequently in that the architectonic features of these layers are normal. The patterning behavior of the SG cells in DT is indistinguishable from those in WT. Thus, it is the IG–SG fate specification profile of the E14 cohort that is changed. In the DT animals, where cells are specified with the characteristics of IG or SG cells, their histogenetic behavior is appropriate to that specification.
Thus, p27Kip1 overexpression is found here to be associated with an alteration in the proportionate laminar destination of cells with respect to the cell cycle number of origin. However, the cardinal insight here is that the proportionate laminar destination of cells is not altered with respect to the value of Q of the cell cycle of origin. In other words, of four possible ‘counting’ methods — embryonic age, cell cycle number, the length of G1, and Q (the probability of exiting the cell cycle) — it is the last that is apparently the determinant for laminar position. The evidence for this assertion is based upon the interpretation of the data presented here in the context of previously published data from the WT CD1 strain mouse (Takahashi et al., 1996a,b, 1999), in which the relationship between Q and the percentage of the cohort that is destined to be SG cells was first determined (percentage SG cells; Fig. 5B). The percentage of SG cells from the E14 cohort and measured at P21 approximate closely a sigmoidal curve (Fig. 5B; solid line; R2 = 0.99). This suggests a tight regulatory linkage between Q and laminar destination. The data from the present study illustrating the effect of p27Kip1 overexpression also approximates closely a sigmoidal curve (R2 = 0.96). This is represented as a broken line since the y-axis values of its left and right tails would be 0% and 100%, respectively (Takahashi et al., 1996a,b, 1999). The curve for the present transgenic mouse line is displaced to the left, presumably reflecting both differences in the strains and in the methods used, which are recognized to be differentially sensitive to leading and trailing edges of the S phase labeled cohort (Hayes and Nowakowski, 2000). Thus, the proportions of neurons fated by specification for SG layer positions under conditions of p27Kip1 overexpression are also correlated to Q. The most striking feature of Figure 5A is the effect that the maturity of the neocortex has on the shift produced by the upregulation of p27Kip1. The less mature cortex is more dramatically affected. This is reasonable because the less mature cortex in WT would be producing fewer cells ‘fated’ to become SG, and, thus, a shift towards SG production in DT would be more dramatic.
These findings imply that mechanisms that regulate Q are coordinate with those that regulate specification of laminar destination and operate before the histogenetic events of laminar assembly and cell differentiation (Fig. 6). The expression of Tis21, an antimitogen, appears to be selectively sensitive to the transition between specification and readiness to exit the cycle. Its profile of expression approximates the measured profile of advance of Q with cell cycle (Iacopetti et al., 1999; Calegari and Huttner, 2003; Haubensak et al., 2004; Kosodo et al., 2004). These investigators propose that over the course of the G1 phase of the cell cycle there is cumulative synthesis of a substance, as yet unidentified, that coordinately increases Q and the expression of Tis21 (Calegari and Huttner, 2003). Our data are in accord with this model in that they identify p27Kip1 as a substance that drives Q upward as its levels increase. Moreover, we also showed that the linkage between Q and laminar destination is driven by p27Kip1 overexpression. For the present it is premature to suggest that p27Kip1 or substances regulating its synthesis are the exclusive determinants of the advance of Q. The linkage of the p27Kip1 drive of Q may be downstream of the proliferative drive originating with the FGFR1 receptor (Shin et al., 2004). It must be upstream of the graded concentrations of transcription factors known to be involved in neuron specification (Grove and Fukuchi-Shimogori, 2003). The actual linkages between Q and regulation of these concentration gradients are as yet unknown.
Figure 6.
Schematic diagram of proliferative model. The diagram schematizes the comparative interrelationship in DT relative to WT mice in expression level of p27Kip1 mRNA (A) to the progression of Q with cell cycle (B) and finally to neuron output from the PVE (C). All three levels are aligned with respect to the 11 cycles of the neuronogenetic sequence as the ordinate, which is the same in WT and DT PVE. Our previous investigations have determined for the WT that the peak level of p27Kip1 expression (Delalle et al., 1999), the ascent of Q through the steady-state value of 0.05 (Takahashi et al., 1996a) and the period of transition from predominantly IG to SG cell output (Takahashi et al., 1999) occurs in the cycle 6–8 interval. For the purposes of the present findings overexpression of p27Kip1 is induced in the interval leading up to cycles 6–7 (gray area enclosed by dashed curve in A). Transcription factors including Emx2 and Pax6, possibly driven by the Notch signaling system, strongly modulate the proliferative process (Grove and Fukuchi-Shimogori, 2003) and may do so via regulating expression of p27Kip1 or other regulators of Q. Induction of p27Kip1 overexpression in turn induces an increase Q (gray area enclosed below dashed curve in B). The increased Q in turn leads to an overall decreased cumulative output that was limited to the output of SG cells (dashed line in C) in these experiments. However, the ratio of SG with respect to IG cells arising in corresponding cell cycles is increased by p27Kip1 overexpression induced increased rate of advance of Q [compare gray (DT) to white (WT) diamond in C].
Figure 6.
Schematic diagram of proliferative model. The diagram schematizes the comparative interrelationship in DT relative to WT mice in expression level of p27Kip1 mRNA (A) to the progression of Q with cell cycle (B) and finally to neuron output from the PVE (C). All three levels are aligned with respect to the 11 cycles of the neuronogenetic sequence as the ordinate, which is the same in WT and DT PVE. Our previous investigations have determined for the WT that the peak level of p27Kip1 expression (Delalle et al., 1999), the ascent of Q through the steady-state value of 0.05 (Takahashi et al., 1996a) and the period of transition from predominantly IG to SG cell output (Takahashi et al., 1999) occurs in the cycle 6–8 interval. For the purposes of the present findings overexpression of p27Kip1 is induced in the interval leading up to cycles 6–7 (gray area enclosed by dashed curve in A). Transcription factors including Emx2 and Pax6, possibly driven by the Notch signaling system, strongly modulate the proliferative process (Grove and Fukuchi-Shimogori, 2003) and may do so via regulating expression of p27Kip1 or other regulators of Q. Induction of p27Kip1 overexpression in turn induces an increase Q (gray area enclosed below dashed curve in B). The increased Q in turn leads to an overall decreased cumulative output that was limited to the output of SG cells (dashed line in C) in these experiments. However, the ratio of SG with respect to IG cells arising in corresponding cell cycles is increased by p27Kip1 overexpression induced increased rate of advance of Q [compare gray (DT) to white (WT) diamond in C].
The capacity of p27Kip1 overexpression to drive this linkage appears to be developmentally regulated. Our earlier studies (Delalle et al., 1999) determined that p27Kip1 mRNA expression in the PVE maximizes in the LCZ earlier than in the MCZ on E14 as Q approaches 0.5 with cell cycle number 8. Expression then declines as Q continues its ascent to 1.0 over the final three cell cycles (Delalle et al., 1999) (Fig. 6A). Here at E14 we find a prominent effect of p27Kip1 upon Q in the MCZ but not in the LCZ. We hypothesize that this could be because other cdk inhibitors take over the control of the restriction point and therefore the control of Q late in the course of neuronogenesis. Candidates include, inter alia, both the inhibitors of CDKs4/6 (Ink4 proteins: p16Ink4a, p15Ink4b, p18Ink4c) and the inhibitors of cyclin E-Cdk2 complex (p21Cip1 and p57Kip1 in addition to p27Kip1), all of which are variably expressed in regulated fashion in proliferative populations of the CNS (Watanabe et al., 1998; Sherr and Roberts, 1999; Zindy et al., 1999). We suggest this occurs as Q increases above the value of 0.5 and only the last formed cells destined for supragranular layers are being formed. Thus, beyond this stage, p27Kip1 may be replaced by other cdk inhibitors as regulators of Q in the final phase of ascent of this parameter.
The present observation linking neocortical laminar fate specification to p27Kip1 has precedent in other systems. For example, under optimum growth conditions in vitro, O-2A oligodendrocyte progenitor cells withdraw from cell cycle and differentiate into oligodendroglial cells after a set number of cycles. As cycles proceed there are rising levels of p27Kip1 and both cycle withdrawal and differentiation are triggered after p27Kip1 level reaches a plateau. It has been postulated that this reflects a cycle counting mechanism mediated at a threshold level of p27Kip1 (Durand et al., 1997; Durand and Raff, 2000). It was not resolved in these studies whether the role of p27Kip1 is only to trigger cycle arrest or is also directly involved in mechanisms leading to differentiation (Galderisi et al., 2003). This distinction has been approached by experiments with CG-4 cells obtained from p27Kip1 null animals. Although CG-4 cells lack p27Kip1, a small fraction arrest and differentiate into normal astrocytes under deprived conditions of in vitro culture. These observations suggested that p27Kip1 primarily induces cell cycle arrest and only indirectly through cycle arrest does differentiation proceed (Tikoo et al., 1997; Casaccia-Bonnefil et al., 1999). In Xenopus, however, the homologous protein, p27Xic1, may be more directly required for specification of neural fate (Ohnuma et al., 1999; Vernon and Philpott, 2003). Whereas the protein operates as both an inhibitor of cycle progression and in cell fate specification, the active domains of the protein serving the two functions are only partially overlapping (Ohnuma et al., 1999). These observations place p27Kip1, and perhaps related cdks, within a linkage that coordinates the mechanisms regulatory to specification and others regulatory to the proliferative process. The linkage of p27Kip1 overexpression, the associated advance in Q and modulation of the laminar fate are consistent with these observations from other experimental systems.
### p27Kip1 and the Proliferative Model
Our proliferative model for neocortical neuronogenesis was formalized earlier (Caviness et al., 2000, 2003; Nowakowski et al., 2002) from observations based upon kinetic parameters and Q in normal embryos and subsequently confirmed and further validated in Ts16 trisomy (Haydar et al., 2000a) and extended in the p27Kip1 overexpression model. l. This model (Figs 1A, 6AC) integrates the behavior of these parameters across the entire neuronogenetic interval in mouse. In brief, the founder population and progeny on average execute 11 integer cell cycles before exhaustion of proliferative activity in the PVE. The fractional advance of Q with each cell cycle determines the total number of cycles (11 in the normal mouse neocortex).
Pertinent to the present findings, the model predicts that should Q be increased at a rate ahead of its normal progression, there would be an early increase in cell cycle output (Caviness et al., 2003) (Fig. 6A,B). However, this would be at the expense of the size of the overall proliferative pool so that the total number of later formed neurons would be reduced (Fig. 6C). The effect should be more salient in down-gradient regions, i.e. more salient in less mature regions of the cerebral wall, than in up-gradient, more mature regions. These predictions are largely confirmed here where the rate of progression of Q with cell cycle has been increased by overexpression of p27Kip1 in the early part of the neuronogenetic interval. Thus, from the perspective of the size of the E14 labeled cohort counted in the P21 cortex, its size is increased in DT and to a greater degree in down-gradient field 1 than in upgradient field 40. However, in contrast to the size of this cohort directed to the cortex at E14, the total number of cells that will be directed to the cortex over the full remaining neuronogenetic interval is reduced (Table 3). The effect upon total cell numbers is detected only with respect to SG layers in field 1, which are largely being formed on and after E14. An effect is not detectable in field 40, which has largely completed its cycle of neuronogenesis by E14 (by which time a large proportion of cells has already been designated to SG layers).
Overall these studies illustrate the predominant importance of the parameter Q as determinant of the rate and number of neurons arising in the course of cell proliferation in the PVE. In principle, an accelerated progression of Q with successive cycles predicts an initial increased output per cycle, which then causes a reduction in the size of the proliferative population. This in turn leads to an ultimate reduction in total output (Caviness et al., 2003). In contrast, a retardation in the progression of Q with successive cycles predicts the opposite (Caviness et al., 2003). Here Q was increased by only ∼10% at what was probably the point of maximum effect on E14 yet there was substantial reduction in the cells delivered to the cortex. In the p27Kip1–/– (Fero et al., 1996; Kiyokawa et al., 1996; Nakayama et al., 1996; Goto et al., 2004) and the Ts16 mice (Haydar et al., 1996, 2000a) predictions of the opposite condition of retarded progression of Q with cycle are confirmed. In the p27Kip1–/–all other proliferative parameters, including founder population, cycle duration and growth fraction, are normal and the cortex is increased in cell number (Goto et al., 2005). This consequence of the knockout indicates that p27Kip1 exerts a regulatory effect upon Q that is independent of that of other CDK inhibitors, at least in its progression through the earlier cycles of the neuronogenetic interval. Whereas our model scales Q to neuronal production, available observations relating to the coordinate interactions of CDK inhibitors do not yet allow a scaling of the independent contribution of p27Kip1 to the progression of Q.
In Ts16 the effect of changes in Q are offset by a smaller founder population and reduced growth fraction. The number of cells delivered through the early cycles is substantially reduced but the number largely normalizes with the later cycles (Haydar et al., 1996, 2000a). Precocious acceleration rather than retardation, in the progression of Q with cell cycle in particular may underlie a wide range of genetically determined human disorders classified as microcephaly vera, as well as a host of other microcephalies that bear no indication of actively destructive pathological process (Mochida and Walsh, 2001). In particular, relative attenuation of SG layers, as would be predicted from this model, is characteristic of certain forms of human microcephaly vera (Urich, 1976).
We thank Genevieve Stein-O'Brien for the graphics work for Figure 6.This work was supported by US Public Health Service Grants NS12005 (V.S.C.), NS43246 (P.G.B.). T. Tarui was supported by a fellowship from the William Randolph Hearst Foundation.
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) Postnatal neuronal proliferation in mice lacking Ink4d and Kip1 inhibitors of cyclin-dependent kinases. | 2017-02-20 08:40:30 | {"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": 0.493316113948822, "perplexity": 7367.991498406767}, "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-2017-09/segments/1487501170434.7/warc/CC-MAIN-20170219104610-00522-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2613458/how-to-provide-the-transformation-matrix-in-the-plane-v-perp-subset-mathbb | # How to provide the transformation matrix in the plane $v^{\perp} \subset \mathbb{R}^3$ w.r.t. the standard basis.
EDIT:
The following edit is a proposed solution to the problem, based on feedback given in the comments below.
Let $a_1, a_2$ and $a_3$ be the vectors $(1,0,0), (0,1,0)$ and $(0,0,1)$ respectively (the standard basis vectors of $\mathbb{R}^3$).
By considering Gram-Schmidt orthogonlization, @Michael has proposed the following method for finding the reflection of each basis vector in the plane $v^{\perp}$
\begin{align*} s_v(a)=a-2\frac{\langle a,v\rangle}{\|v\|^2}v. \end{align*}
Calculating this for $s_v(a_{1-3})$ yields the following vectors (In partially simplified form)
$(\frac{5\sqrt{2} - 18}{5\sqrt{2}}, -\frac{24}{5\sqrt{2}}, -\frac{6}{\sqrt{2}}), \qquad (-\frac{24}{5\sqrt{2}}, \frac{5\sqrt{2} - 32}{5\sqrt{2}}, -\frac{8}{\sqrt{2}}), \qquad (-\frac{6}{\sqrt{2}}, -\frac{8}{\sqrt{2}}, -\frac{\sqrt{2} - 10}{\sqrt{2}}).$
The transformation matrix $M$ is composed by combining the above vectors. Thus,
$$M = \begin{bmatrix} \frac{5\sqrt{2} - 18}{5\sqrt{2}} & -\frac{24}{5\sqrt{2}} & -\frac{6}{\sqrt{2}} \\ -\frac{24}{5\sqrt{2}} & \frac{5\sqrt{2} - 32}{5\sqrt{2}} & -\frac{8}{\sqrt{2}} \\ -\frac{6}{\sqrt{2}} & -\frac{8}{\sqrt{2}} & -\frac{\sqrt{2} - 10}{\sqrt{2}} \end{bmatrix}$$
——
I am a beginner. Because of deficits in my understanding, I am having difficulty with the following question.
"Let $Sv: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the reflection in the plane $v^{\perp} \subset \mathbb{R}^3$ with $v = (3,4,5)^t$. provide the matrix representation $M = M^{E}_{E}(s_v)$ of $s_v$ with respect to the standard basis of $\mathbb{R}^3$."
I think I have worked out the broad aspects of an approach. I would specifically appreciate help if someone could
a) Confirm that my idea is correct b) Provide assistance with the individual steps.
——
Progress:
• I dont really understand the notation $\textit{"$M = M^{E}_{E}(s_v)$"}$. Is this simply referring to the transformation matrix?
• I dont know how to calculate the matrix representation for a linear transformation, however know that if a mapping takes one vector ($v_1$) and produces another in a perpendicular plane ($v_2$), this would mean that $v_2 \cdot v_1 = 0$. Is this a step in the right direction?
• For the condition $\textit{"with respect to the standard basis of$\mathbb{R}^3$."}$, It seems that this answer gives steps on how to do this- so If i understand correctly, once I find a transformation matrix, I need to find another two Matrices $C^{-1}$ & $C$. I just don't understand how to calculate them.
• First and third point: it means the matrix of the endormorphism $S_v$ expressed in the canonical basis $E$. Second point: this might be helpful. – anderstood Jan 20 '18 at 14:18
• You forgot to specify $v$ ... – Michael Hoppe Jan 23 '18 at 21:37
• $s_v(1,0,0)=(0.64,-0.48,-0.6)$. – Michael Hoppe Jan 23 '18 at 21:43
• The reflection formula is not essentially related to Gram–Schmidt. – Michael Hoppe Jan 23 '18 at 21:48
Obviously $$s_v(a)=a-2\frac{\langle a,v\rangle}{\|v\|^2}v.$$ Now the column vectors of the matrix of a linear map with respect to the standard basis are their images under that map. So the first column would be $s_v((1,0,0))$.
EDIT (regarding the obviouslibility):
Let $v$ be the plane's normal, then $\frac{v}{\|v\|}$ is a unit normal vector. Hence $\langle a,\frac{v}{\|v\|} \rangle$ is the component of $a$ in direction $v$, so $\langle a,\frac{v}{\|v\|} \rangle\frac{v}{\|v\|}=\frac{\langle a,v\rangle }{\|v\|^2}v$ is the projection of $a$ along $v$. To get the reflected vector on that plane, just subtract the double of the projection from $a$ yielding in $a-2\frac{\langle a,v\rangle }{\|v\|^2}v$.
EDIT$^2$ (regarding $\textit{"$M = M^{E}_{E}(s_v)$"}$)
The column vectors of the representation matrix in respect to the basis $E$ are the images of those vectors unter $s_v$. Maybe this post If $T$ is a linear map and $\mathcal M(T)$ is its matrix, what exactly does the multiplication $\mathcal M(T)v$ mean? clarifies.
• im sorry, but this isn't obvious to me.. Perhaps you could provide some clarification? Thanks for your help – Oscar Jan 22 '18 at 19:42
• Thanks for the extra information. I have recently covered Gram-Schmidt Orthonormalization, and it seems your answer is quite similiar to that. So in order to get my transformation matrix, is it correct that I need to repeat the above method of projection for the remaining standard basis vectors (0,1,0) and (0,0,1) ? – Oscar Jan 23 '18 at 16:45
• @Oscar Exactly, that's what it's all about. – Michael Hoppe Jan 23 '18 at 16:50
• Okay, to check I have understood the steps, I will calculate the transformation matrix using this method. Update to come shortly. – Oscar Jan 23 '18 at 16:55
• I have edited the question based on your feedback. If you believe my interpretation is correct, then let me know and i will be happy to accept your answer as the solution. – Oscar Jan 23 '18 at 19:08 | 2019-12-08 03:41:58 | {"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": 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": 0.8492988348007202, "perplexity": 185.3424303481963}, "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-2019-51/segments/1575540504338.31/warc/CC-MAIN-20191208021121-20191208045121-00055.warc.gz"} |
https://www.physicsforums.com/threads/problem-with-integrals.219062/ | # Problem with INTEGRALS
1. Mar 1, 2008
### European
I cant' solve this two integrals :
$$\int$$ (ln x)$$^{2}$$
$$\int$$ cos$$^{4}$$(x)
2. Mar 1, 2008
### Ataman
You do both of them by parts.
You can rewrite this as:
$$\int (lnx)(lnx) dx$$
with
u = lnx
dv = lnx dx
To integrate lnx dx, you have to do it by parts again. After that, it is very simple.
Again, you can rewrite this integral as something you could do by parts.
$$\int cos^{3}xcosx dx$$
u = cos^{3}x
dv = cosx dx
-Ataman
Last edited: Mar 1, 2008
3. Mar 1, 2008
### sylar
The answer of the first question is x*((ln x)^2) - 2x*(ln x) + 2x , you can check your answer.
As for the second one, my approach would be to write the integrand as
((cos x)^2)*(1 - ((sin x)^2)) , and then finish this off by using the trigonometric identities for (cos x)^2 and sin 2x .
Lastly, try to use as many problems as you can in your spare time and take notes for choosing the most suitable method in a problem you encounter.
4. Mar 2, 2008
### European
Hi , thank you very much for the answers !!
By the way , I just can't solve another one :
$$\int$$( x$$^{2}$$-2x+3)lnx dx
5. Mar 2, 2008
### HallsofIvy
Again, straight forward by parts: let u= ln(x), dv= (x2- 2x+ 3)dx. | 2018-12-12 10:17:49 | {"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": 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": 0.9111036062240601, "perplexity": 3868.641254988672}, "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-2018-51/segments/1544376823817.62/warc/CC-MAIN-20181212091014-20181212112514-00109.warc.gz"} |
https://tex.stackexchange.com/questions/304429/arabic-cells-tables-rendered-from-right-to-left | # Arabic cells' tables rendered from right to left
Is there a way to get arabic cells' tables rendered from right to left just like they are typesetted. here is a minimal example!
\documentclass[]{article}
\usepackage[ utf8 ]{inputenc}
\usepackage[LAE]{fontenc}
\usepackage[arabic]{babel}
\usepackage{array}
\newsavebox{\RTLbox}
\newcolumntype{R}{>{\begin{lrbox}{\RTLbox}}c<{\end{lrbox}\AR{\usebox{\RTLbox}}}}
\begin{document}
أ ب ت
\vspace{2cm}
\begin{tabular}{*{3}R}
أ & ب & ت
\end{tabular}
\end{document}
I suggest you using polyglossia and change to xelatex instead of pdflatex engine.
polyglossia relies on the fontspec package to manipulate fonts, and bidi to change direction of text from right to left (arabic script).
You can insert tabular in RTL mode, or in LTR mode depending on context.
MWE
% compile with xelatex
\documentclass[a4paper]{article}
\usepackage{polyglossia}
\setdefaultlanguage[calendar=gregorian]{arabic}
\setotherlanguage{english}
\newfontfamily\arabicfont[Script=Arabic,Scale=1.3]{Amiri} % font for arabic text
\parindent 0pt
\begin{document}
أ ب ت
\vspace{1cm}
\begin{tabular}{*{3}c}
أ & ب & ت
\end{tabular}
\vspace{1cm}
\begin{english}
\begin{tabular}{*{3}c}
a & b & c
\end{tabular}
\end{english}
\end{document}
• is there any solution in latex ? How to install xelatex – Hafid Boukhoulda Apr 15 '16 at 18:21
• XeLaTeX is included by default in miktex. – Salim Bou Apr 15 '16 at 18:41
• and what about texlive ? I am on linux! – Hafid Boukhoulda Apr 15 '16 at 18:45
• xetex is available on texlive. so it is ok – Hafid Boukhoulda Apr 15 '16 at 19:58
You defined a new kind of column R with:
\newcolumntype{R}{>{\begin{lrbox}{\RTLbox}}c<{\end{lrbox}\AR{\usebox{\RTLbox}}}}
but you don't use it since you use the c type of column in your document.
Try to use \begin{tabular}{*{3}R}, it should be better. | 2019-07-18 14:05:49 | {"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": 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": 0.913127064704895, "perplexity": 8183.1681130634}, "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-2019-30/segments/1563195525634.13/warc/CC-MAIN-20190718125048-20190718151048-00017.warc.gz"} |
http://planetmath.org/WreathProduct | # wreath product
## Primary tabs
Defines:
wreath product
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:
## Mathematics Subject Classification
20E22 Extensions, wreath products, and other compositions
### I just want to say...
That I'm glad to finally write an entry that has no parallel in Mathworld.
Wreath products are curious beasts. Is anyone interested in hearing more about them? I'm considering expanding this entry if anyone's listening.
### Re: I just want to say...
Yes, please write more on wreath products. I remember constructions based on wreath products coming up as examples in various courses in the dim distant past.
### Re: I just want to say...
Helo!
I'm very interested in wreath product, expecially connected with games. Rubic cube and it's versions. Have you anything for it?
### Re: I just want to say...
HI,
I'm in interested in the structure of the wreath product,
but to be fair more interested in a generalised version
of the wreath product due to Dixon and Fournelle. Here, the
permutation groups are ordered by a, possibly infinite,
partially ordered set. Just wondering what you can sat about
this.
Cheers,
Dom.
### Re: I just want to say...
Hi!
I try to work with wreath products in my diploma thesis. Whenever I talk to my prof, he seems to have a lot of pictures about them in his head. His hands are always in action taking elements from one component to another, drawing a diagonal when he speaks about the diag. subgroup, and so on.
My problem about that is, that in my head there is no such picture and so he rather confuses me from time to time.
By now I've read quite a lot about wreath products, but all I can find is technical definitions and theorems.
I mean if everybody who works with wreath products has such pictures in his head, why does nobody put them down on paper? It would make it much easier for newcomers to find themselves a way into the theme.
Do you know about anything where they are treated more visually or do you perhaps want to try that yourself...?
Greetings! Kirsten
### Re: I just want to say...
My experience with wreath products has been much like Kirsten, only, my problem is that my pictures are different than my advisor's!
...but yeah, pictures would be great, at least for finite wreaths. I would suggest giving many examples. For instance, as automorphisms of a graph they are easy to see because our brains see repeated patterns in a graphs without a problem. Even if an example teaches you the wrong intuition about wreaths, I'd say it is better than none.
...will you dare touch twisted wreaths? O'Nan-Scott's classification of primitive permutation groups might be a nice followup article.
### Re: I just want to say...
I would like to see wreath product explained in terms of matrices. Thank you. -Wendy Wang | 2013-05-26 06:00:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 39, "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": 0.45653048157691956, "perplexity": 1226.7760268420975}, "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/1368706631378/warc/CC-MAIN-20130516121711-00004-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://nbviewer.jupyter.org/github/yinleon/fake_news/blob/master/opensources-lite.ipynb | # Parsing OpenSources¶
by leon yin
2017-11-22
## What is this?¶
In this Jupyter Notebook we will
Please view the detailed version if you want to know how everything works, and if you're unfamiliar with Jupyter Notebooks and Python.
View this on Github. View this on NBViewer. Visit my Lab's website
## Intro¶
OpenSources is a "Professionally curated lists of online sources, available free for public use." by Melissa Zimdars and collegues. It contains websites labeled with categories spanning state-sponsored media outlets, to conpiracy theory rumor mills. It is a comprehensive resource for researchers and technologists interested in propaganda and mis/disinformation.
The opensources project is in-fact open sourced in json and csv format.
One issue however, is that the data is entered by people, and not readily machine-readible.
Let's take a moment to appreciate the work of peopke
And optimize this information for machines,
Using some good ole'fashioned data wrangling.
## Let's Code Yo! ¶
In [1]:
import numpy as np
import pandas as pd
In [2]:
filename = "data/sources.json"
In [3]:
%%sh -s $filename mkdir -p data curl https://raw.githubusercontent.com/BigMcLargeHuge/opensources/master/sources/sources.json --output$1
% Total % Received % Xferd Average Speed Time Time Time Current
100 136k 100 136k 0 0 136k 0 0:00:01 --:--:-- 0:00:01 510k
In [4]:
df = pd.read_json(filename, orient='index')
In [5]:
df.index.name = 'domain'
Let's simplify this long column name into something that's short and sweet.
In [6]:
replace_col = {'Source Notes (things to know?)' : 'notes'}
In [7]:
df.columns = [replace_col.get(c, c) for c in df.columns]
Let's also reorder the column for readibility.
In [8]:
df = df[['type', '2nd type', '3rd type', 'notes']]
In [9]:
df.head(10)
Out[9]:
type 2nd type 3rd type notes
domain
100percentfedup.com bias
16wmpo.com fake http://www.politifact.com/punditfact/article/2...
21stcenturywire.com conspiracy
24newsflash.com fake
24wpn.com fake http://www.politifact.com/punditfact/article/2...
365usanews.com bias conspiracy
4threvolutionarywar.wordpress.com bias conspiracy
70news.wordpress.com fake
82.221.129.208 conspiracy fake
Acting-Man.com unreliable conspiracy publishes articles denying climate change
### Data Processing - Making categories standard ¶
If we look at all the available categories, you'll see some inconsistences:
In [10]:
replace_vals = {
'fake news' : 'fake',
'satirical' : 'satire',
'unrealiable': 'unreliable',
'blog' : np.nan
}
We can group all our data preprocessing in one function.
In [11]:
def clean_type(value):
'''
This function clean various type values (str).
If the value is not null,
the value is cast to a string,
leading and trailing zeros are removed,
cast to lower case,
and redundant values are replaced.
returns either None, or a cleaned string.
'''
if value and value != np.nan:
value = str(value)
value = value.strip().lower()
value = replace_vals.get(value, value)
return value
else:
return None
In [12]:
df.fillna(value=0, inplace=True)
We'll now loop through each of the columns,
and run the clean_type function on all the values in each column.
In [13]:
for col in ['type', '2nd type', '3rd type']:
df[col] = df[col].apply(clean_type)
### One-Hot Encoding ¶
One-hot encoding is used to make a sparse matrix from a single categorical column.
Let's use this toy example to understand:
In [14]:
all_hot_encodings = pd.Series(pd.unique(df[['type', '2nd type', '3rd type']].values.ravel('K')))
In [15]:
all_hot_encodings
Out[15]:
0 bias
1 fake
2 conspiracy
3 unreliable
4 junksci
5 political
6 hate
7 clickbait
8 satire
9 rumor
10 reliable
11 state
12 None
13 NaN
dtype: object
In [16]:
dum1 = pd.get_dummies(df['type'].append(all_hot_encodings))
dum2 = pd.get_dummies(df['2nd type'].append(all_hot_encodings))
dum3 = pd.get_dummies(df['3rd type'].append(all_hot_encodings))
Let's get the max value for each one-hot encoded column.
By doing so we can combine the three columns information into one dataframe.
In [17]:
__d = dum1.where(dum1 > dum2, dum2)
__d = __d.where(__d > dum3, dum3)
#### Why not take the sum?¶
Taking a sum is also an option, but across rows I noticed duplicate categories between columns.
This would return one-hot encoded columns of 2 or 3!
lastly, let's remove the rows from the unique categorical values we appended.
In [18]:
dummies = __d.iloc[:-len(all_hot_encodings)]
Now we have a wonderful new dataset!
In [19]:
dummies.head(10)
Out[19]:
bias clickbait conspiracy fake hate junksci political reliable rumor satire state unreliable
100percentfedup.com 1 0 0 0 0 0 0 0 0 0 0 0
16wmpo.com 0 0 0 1 0 0 0 0 0 0 0 0
21stcenturywire.com 0 0 1 0 0 0 0 0 0 0 0 0
24newsflash.com 0 0 0 1 0 0 0 0 0 0 0 0
24wpn.com 0 0 0 1 0 0 0 0 0 0 0 0
365usanews.com 1 0 1 0 0 0 0 0 0 0 0 0
4threvolutionarywar.wordpress.com 1 0 1 0 0 0 0 0 0 0 0 0
70news.wordpress.com 0 0 0 1 0 0 0 0 0 0 0 0
82.221.129.208 0 0 1 1 0 0 0 0 0 0 0 0
Acting-Man.com 0 0 1 0 0 0 0 0 0 0 0 1
let's add the notes to this new dataset by concatenating dummies with df row-wise.
In [20]:
df_news = pd.concat([dummies, df['notes']], axis=1)
In [21]:
df_news.head(10)
Out[21]:
bias clickbait conspiracy fake hate junksci political reliable rumor satire state unreliable notes
domain
100percentfedup.com 1 0 0 0 0 0 0 0 0 0 0 0
16wmpo.com 0 0 0 1 0 0 0 0 0 0 0 0 http://www.politifact.com/punditfact/article/2...
21stcenturywire.com 0 0 1 0 0 0 0 0 0 0 0 0
24newsflash.com 0 0 0 1 0 0 0 0 0 0 0 0
24wpn.com 0 0 0 1 0 0 0 0 0 0 0 0 http://www.politifact.com/punditfact/article/2...
365usanews.com 1 0 1 0 0 0 0 0 0 0 0 0
4threvolutionarywar.wordpress.com 1 0 1 0 0 0 0 0 0 0 0 0
70news.wordpress.com 0 0 0 1 0 0 0 0 0 0 0 0
82.221.129.208 0 0 1 1 0 0 0 0 0 0 0 0
Acting-Man.com 0 0 1 0 0 0 0 0 0 0 0 1 publishes articles denying climate change
With one-hot encoding, the opensources dataset is fast and easy to filter for domains that are considered fake news.
In [22]:
df_news[df_news['fake'] == 1].index
Out[22]:
Index(['16wmpo.com', '24newsflash.com', '24wpn.com', '70news.wordpress.com',
'82.221.129.208', 'Amposts.com', 'BB4SP.com', 'DIYhours.net',
...
'washingtonpost.com.co', 'webdaily.com', 'weeklyworldnews.com',
'worldpoliticsnow.com', 'worldpoliticsus.com', 'worldrumor.com',
'worldstoriestoday.com', 'wtoe5news.com', 'yesimright.com',
'yourfunpage.com'],
dtype='object', name='domain', length=271)
We can see how many articles were categorized as conspiracy theory sites.
In [23]:
df_news['conspiracy'].sum()
Out[23]:
201
We can see all sites which are .org superdomains.
In [24]:
df_news[df_news.index.str.contains('.org')].sample(10, random_state=42)
Out[24]:
bias clickbait conspiracy fake hate junksci political reliable rumor satire state unreliable notes
domain
heartland.org 1 0 0 0 0 0 0 0 0 0 0 0 http://www.sourcewatch.org/index.php/Heartland...
ExperimentalVaccines.org 0 0 1 0 0 1 0 0 0 0 0 0
heritage.org 0 0 0 0 0 0 1 0 0 0 0 0
breakpoint.org 0 0 0 0 0 0 0 0 0 0 0 1
bigbluevision.org 1 1 0 0 0 0 0 0 0 0 0 0
witscience.org 0 0 0 0 0 0 0 0 0 1 0 0
freedomworks.org 0 0 0 0 0 0 1 0 0 0 0 0
adflegal.org/media 0 0 0 0 1 0 0 0 0 0 0 0 https://www.splcenter.org/fighting-hate/extrem...
moonofalabama.org 1 0 0 0 0 0 0 0 0 0 0 0
thefreepatriot.org 1 1 0 1 0 0 0 0 0 0 0 0
### Some Last Clean-ups ¶
I see a "/media", is the rest of the site ok?
Let's clean up the domain names a bit...
1. remove "www."
2. remove subsites like "/media"
3. cast to lower case
In [25]:
def preprocess_domains(value):
'''
Removes subsites from domains by splitting out bashslashes,
Removes www. from domains
returns a lowercase cleaned up domain
'''
value = value.split('/')[0]
value = value.replace('www.', '')
return value.lower()
Because the index is a list, rather than use apply-- which only works on Series or DataFrames, we can use map, or a list generator to apply the preprocess_domains function to each element in the index.
In [26]:
df_news.index = df_news.index.map(preprocess_domains)
Let's use pandas to_csv to write this cleaned up file as a tab-separated value file (tsv).
In [27]:
df_news.to_csv('data/sources_clean.tsv', sep='\t')
In [28]:
!ls data
sources.csv sources.json sources_clean.tsv
## Conclusion¶
OpenSources is a great resources for research and technology.
If you are aware of other projects that have categorized the online news ecosystem, I'd love to hear about it.
Let's recap what we've covered:
2. How to search and explore Pandas Dataframes
3. How to preprocess messy real world data, twice!
4. How to one-hot encode a categorical dataset.
In the next notebook, we'll use this new dataset to analyze links shared on Twitter. We can begin to build a profile of how sites categorized from open sources are used during viral campaigns.
### Thank yous:¶
Rishab and Robyn from D&S.
Also my friend and collegue Andrew Guess, who introduced me to links as data. | 2019-05-22 03:36:39 | {"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": 0.18600735068321228, "perplexity": 2171.8282003444615}, "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/1558232256724.28/warc/CC-MAIN-20190522022933-20190522044933-00483.warc.gz"} |
http://st551.cwick.co.nz/lecture/lecture_11/ | # One-sided confidence intervals
## Confidence intervals
We motivated CI’s as all values of $$\mu_0$$ that would not be rejected in a two-sided hypothesis test of $$H_0: \mu = \mu_0$$.
Two-sided p-values, two-sided rejection regions and two-sided confidence intervals are generally equivalent: \begin{aligned} p < \alpha &\iff \text{Reject } H_0: \mu = \mu_0 \text{ at level } \alpha \\ &\iff \mu_0 \text{ is outside} (1-\alpha)100 \text{\% confidence interval} \\ p > \alpha &\iff \text{Fail to reject } H_0: \mu = \mu_0 \text{ at level } \alpha \\ &\iff \mu_0 \text{ is inside} (1-\alpha)100 \text{\% confidence interval} \end{aligned}
## One-sided confidence intervals
You can motivate them from one-sided tests too.
You end up with an infinite bound on one end.
## So, why not report one sided CIs?
You don’t always do a hypothesis test. A plausible range for a parameter value should be two-sided. (If there isn’t a value of interest, how could there be a direction of interest?)
Should a plausible range for depend on your hypothesis of interest? More useful for others to give a 95% two-sided interval.
Yes, this means your one-sided test might not agree with your two-sided confidence interval.
Should we ever do one-sided tests? Some people argue “No, we should never do one sided tests”. I’d say, you can, but you better have a really good reason, or someone will accuse you of doing it just to get a smaller p-value.
# Binomial Proportions
## Data Setting
Population: $$Y \sim \text{Bernoulli}(p)$$, i.e. $Y = \begin{cases} 1, & \text{with probability } p \\ 0, & \text{with probability } 1 - p \end{cases}$
$$E(Y) = p$$, $$Var(Y) = p(1-p)$$ When mean and variance share parameters we say there is a mean-variance relationship.
Parameter: $$\mu = E(Y) = p$$, the population proportion
Sample: $$n$$ i.i.d from population: $$Y_1, \ldots, Y_n$$
Statistic: $$\overline{Y} = \frac{1}{n}\sum_{i=1}^n Y_i = \hat{p}$$, the sample proportion.
## Testing
Null hypothesis: $$H_0:p = p_0$$
1. Exact test: use fact that $n\overline{Y} \sim \text{Binomial}(n, p)$
2. Approximate test: use fact that $\overline{Y} \dot\sim N\left( E(Y) , \frac{Var(Y)}{n}\right) = N\left( p , \frac{p(1-p)}{n}\right)$
## Exact Binomial Test:
Complete worksheet (Charlotte will provide)
1. Get into groups according to number on worksheet at numbered whiteboard
2. Write answers to bold questions on whiteboard as you complete them (so I can see where you are up to) | 2019-02-17 22:29:30 | {"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": 0.9937469959259033, "perplexity": 3336.465850444355}, "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-2019-09/segments/1550247482788.21/warc/CC-MAIN-20190217213235-20190217235235-00636.warc.gz"} |
https://guitarknights.com/guitar-in-spanish-guitar-store.html | At MI, you learn from a proven curriculum taught by the best guitar instructors in the world, augmented by visiting artists’ seminars, concerts, and lessons from some of the greatest players in contemporary music. At our guitar music college, you get to network with other players, find your creative voice, and get the training you need to become the player you have always dreamed of being.
Musicians Institute programs are fast paced and certain fundamental musical skills are required before you can begin your program. If you meet entry requirements and are accepted, a one-on-one performance evaluation and placement testing will be held during registration to determine whether you qualify for advanced placement in any area of course work. You are evaluated based on the content of your submission. For more info, please see the MI Course Catalog.
As already stated, the perfect-fifths (P5) interval is the most harmonious, after the unison and octave intervals. An explanation of human perception of harmony relates the mechanics of a vibrating string to the musical acoustics of sound waves using the harmonic analysis of Fourier series. When a string is struck with a finger or pick (plectrum), it vibrates according to its harmonic series. When an open-note C-string is struck, its harmonic series begins with the terms (C,C,G,C,E,G,B♭,C). The root note is associated with a sequence of intervals, beginning with the unison interval (C,C), the octave interval (C,C), the perfect fifth (C,G), the perfect fourth (G,C), and the major third (C,E). In particular, this sequence of intervals contains the thirds of the C-major chord {(C,E),(E,G)}.[4]
##### Ze first began his journey playing original music and top 40s pop tunes around the country's popular venues. Eventually, through the music of John Mayer, he found a strong attraction to blues music. Ze has years of experience teaching beginners and intermediate guitarists. Currently with Liberty Park Music he is teaching Introduction to Guitar Playing for Complete Beginners, Rhythm Guitar to learn about strumming, chords and more, Guitar Essentials as a fast-track review course, and lots of Song Lessons on pop and rock hits.
Are you stuck in a musical rut? New tunings and tricks can help you keep learning guitar in fresh, fun ways. Try one of these great tips from guitar teacher Samuel B. to breathe new life into your guitar playing... One of the first things I tell any new student is that I don't specialize in a formal discipline. If jazz or classical training is your objective, then I'm not your guy. Instead, I specialize primarily in American roots music (that which we tend to casually lump together as "folk" … Read More
So with that in mind, would you like to learn the guitar on your own or with others? The choice is yours at Guitar Center. If you prefer one-on-one instruction, that's absolutely doable - in fact, you'll find our schedule to be very flexible. Of course, learning in a group is an excellent way to meet like-minded musicians with similar tastes and share ideas on how to improve one another's craft. Who knows, you might even leave a group guitar lesson with plans to start a band with your newfound musical companions. Either way, our group and private guitar lessons are very entertaining and informative.
As previously stated, a dominant seventh is a four-note chord combining a major chord and a minor seventh. For example, the C7 dominant seventh chord adds B♭ to the C-major chord (C,E,G). The naive chord (C,E,G,B♭) spans six frets from fret 3 to fret 8;[49] such seventh chords "contain some pretty serious stretches in the left hand".[46] An illustration shows a naive C7 chord, which would be extremely difficult to play,[49] besides the open-position C7 chord that is conventional in standard tuning.[49][50] The standard-tuning implementation of a C7 chord is a second-inversion C7 drop 2 chord, in which the second-highest note in a second inversion of the C7 chord is lowered by an octave.[49][51][52] Drop-two chords are used for sevenths chords besides the major-minor seventh with dominant function,[53] which are discussed in the section on intermediate chords, below. Drop-two chords are used particularly in jazz guitar.[54] Drop-two second-inversions are examples of openly voiced chords, which are typical of standard tuning and other popular guitar-tunings.[55]
The guitar is a type of chordophone, traditionally constructed from wood and strung with either gut, nylon or steel strings and distinguished from other chordophones by its construction and tuning. The modern guitar was preceded by the gittern, the vihuela, the four-course Renaissance guitar, and the five-course baroque guitar, all of which contributed to the development of the modern six-string instrument.
The nut is a small strip of bone, plastic, brass, corian, graphite, stainless steel, or other medium-hard material, at the joint where the headstock meets the fretboard. Its grooves guide the strings onto the fretboard, giving consistent lateral string placement. It is one of the endpoints of the strings' vibrating length. It must be accurately cut, or it can contribute to tuning problems due to string slippage or string buzz. To reduce string friction in the nut, which can adversely affect tuning stability, some guitarists fit a roller nut. Some instruments use a zero fret just in front of the nut. In this case the nut is used only for lateral alignment of the strings, the string height and length being dictated by the zero fret.
Instructor ProfileArlen RothThe King of All Guitar TeachersMusic lesson pioneer Arlen Roth is the quintessential guitarist. An accomplished and brilliant musician — and one of the very few who can honestly say he’s done it all — Roth has, over the course of his celebrated 35-year career, played on the world’s grandest stages, accompanied many of the greatest figures in modern music and revolutionized the concept of teaching guitar. Read More...Lessons Wes Montgomery-style Octaves
The hardest thing about playing chords when you get started is changing between them. To effectively change between chords, you need to be economical with your movements. Spend a bit of time thinking about where your fingers need to be for each chord, and work out the most efficient way to move from one to the other. For example, from a C major, you can flatten your index finger so it covers the first string too and move your middle and ring fingers both down a string to switch to an F. Easy changes to start with are between C major and A minor and G major and E minor.
All-fourths tuning replaces the major third between the third and second strings with a fourth, extending the conventional tuning of a bass guitar. With all-fourths tuning, playing the triads is more difficult, but improvisation is simplified, because chord-patterns remain constant when moved around the fretboard. Jazz guitarist Stanley Jordan uses the all-fourths tuning EADGCF. Invariant chord-shapes are an advantage of other regular tunings, such as major-thirds and all-fifths tunings.[20]
The ratio of the spacing of two consecutive frets is {\displaystyle {\sqrt[{12}]{2}}} (twelfth root of two). In practice, luthiers determine fret positions using the constant 17.817—an approximation to 1/(1-1/ {\displaystyle {\sqrt[{12}]{2}}} ). If the nth fret is a distance x from the bridge, then the distance from the (n+1)th fret to the bridge is x-(x/17.817).[15] Frets are available in several different gauges and can be fitted according to player preference. Among these are "jumbo" frets, which have much thicker gauge, allowing for use of a slight vibrato technique from pushing the string down harder and softer. "Scalloped" fretboards, where the wood of the fretboard itself is "scooped out" between the frets, allow a dramatic vibrato effect. Fine frets, much flatter, allow a very low string-action, but require that other conditions, such as curvature of the neck, be well-maintained to prevent buzz.
Anyone playing and/or teaching guitar needs staff paper, blank tab, guitar chord charts, guitar scale charts, and fretboard diagrams to chart their guitar lessons and musical ideas. You can find books with some combination of these blank charts and grids, but you can’t find one with all of them organized in a practical way. That’s why we chose to design our own.
A six-string guitar has five musical-intervals between its consecutive strings. In standard tuning, the intervals are four perfect-fourths and one major-third, the comparatively irregular interval for the (G,B) pair. Consequently, standard tuning requires four chord-shapes for the major chords. There are separate chord-forms for chords having their root note on the third, fourth, fifth, and sixth strings.[41] Of course, a beginner learns guitar by learning notes and chords,[42] and irregularities make learning the guitar difficult[43]—even more difficult than learning the formation of plural nouns in German, according to Gary Marcus.[44] Nonetheless, most beginners use standard tuning.[45]
One of our goals at CCM is to instill in students a great appreciation for guitar and music that will last them a lifetime. Students at CCM have won numerous international guitar competitions, performed outreach concerts all over the world, and have gotten into the best colleges in the United States. Recent CCM graduates have gone on to Harvard, Stanford, Brown, and USC. Visit our Yelp pages to see what families have to say about our program.
You need to place one finger on whatever fret you want to bar and hold it there over all of the strings on that fret. The rest of your fingers will act as the next finger down the line (second finger barring, so third finger will be your main finger, and so on). You can also buy a capo, so that you don't have to deal with the pain of the guitar's strings going against your fingers. The capo bars the frets for you. This also works with a ukulele.
##### A chord is inverted when the bass note is not the root note. Chord inversion is especially simple in M3 tuning. Chords are inverted simply by raising one or two notes by three strings; each raised note is played with the same finger as the original note. Inverted major and minor chords can be played on two frets in M3 tuning.[56][74] In standard tuning, the shape of inversions depends on the involvement of the irregular major-third, and can involve four frets.[75]
As already stated, the perfect-fifths (P5) interval is the most harmonious, after the unison and octave intervals. An explanation of human perception of harmony relates the mechanics of a vibrating string to the musical acoustics of sound waves using the harmonic analysis of Fourier series. When a string is struck with a finger or pick (plectrum), it vibrates according to its harmonic series. When an open-note C-string is struck, its harmonic series begins with the terms (C,C,G,C,E,G,B♭,C). The root note is associated with a sequence of intervals, beginning with the unison interval (C,C), the octave interval (C,C), the perfect fifth (C,G), the perfect fourth (G,C), and the major third (C,E). In particular, this sequence of intervals contains the thirds of the C-major chord {(C,E),(E,G)}.[4]
There are three main types of modern acoustic guitar: the classical guitar (nylon-string guitar), the steel-string acoustic guitar, and the archtop guitar, which is sometimes called a "jazz guitar". The tone of an acoustic guitar is produced by the strings' vibration, amplified by the hollow body of the guitar, which acts as a resonating chamber. The classical guitar is often played as a solo instrument using a comprehensive finger-picking technique where each string is plucked individually by the player's fingers, as opposed to being strummed. The term "finger-picking" can also refer to a specific tradition of folk, blues, bluegrass, and country guitar playing in the United States. The acoustic bass guitar is a low-pitched instrument that is one octave below a regular guitar.
Are you stuck in a musical rut? New tunings and tricks can help you keep learning guitar in fresh, fun ways. Try one of these great tips from guitar teacher Samuel B. to breathe new life into your guitar playing... One of the first things I tell any new student is that I don't specialize in a formal discipline. If jazz or classical training is your objective, then I'm not your guy. Instead, I specialize primarily in American roots music (that which we tend to casually lump together as "folk" … Read More
In major-thirds (M3) tuning, the chromatic scale is arranged on three consecutive strings in four consecutive frets.[83][84] This four-fret arrangement facilitates the left-hand technique for classical (Spanish) guitar:[84] For each hand position of four frets, the hand is stationary and the fingers move, each finger being responsible for exactly one fret.[85] Consequently, three hand-positions (covering frets 1–4, 5–8, and 9–12) partition the fingerboard of classical guitar,[86] which has exactly 12 frets.[note 1]
Jump up ^ This sequence of fifths features the diminished fifth (b,f), which replaces the perfect fifth (b,f♯) containing the chromatic note f♯, which is not a member of the C-major key. The note f (of the C-major scale) is replaced by the note f♯ in the Lydian chromatic scale (Russell, "The fundamental harmonic structure of the Lydian scale", Example 1:7, "The C Lydian scale", p. 5).
F major. This is fairly similar to the C, but a little more difficult to play. Press the fourth string down at the third fret with your ring finger, the third string down at the second fret with your middle finger, and the first and second strings down at the first fret with your index. You just flatten your index finger down across the two strings; lower your thumb if you struggle. You don't play the fifth or sixth strings in this chord.
The Spanish vihuela, called in Italian the "viola da mano", a guitar-like instrument of the 15th and 16th centuries, is widely considered to have been the single most important influence in the development of the baroque guitar. It had six courses (usually), lute-like tuning in fourths and a guitar-like body, although early representations reveal an instrument with a sharply cut waist. It was also larger than the contemporary four-course guitars. By the 16th century, the vihuela's construction had more in common with the modern guitar, with its curved one-piece ribs, than with the viols, and more like a larger version of the contemporary four-course guitars. The vihuela enjoyed only a relatively short period of popularity in Spain and Italy during an era dominated elsewhere in Europe by the lute; the last surviving published music for the instrument appeared in 1576.[9]
I tried several times over the course of 20 years to learn guitar. I purchased guitars, amps, books, private lessons. Nothing ever stuck, until I found justinguitar.com. The only reason I can play guitar today is because of Justin. His courses are well thought out, easy to understand, easy to follow, and easy to make progress on. I can't think of a single product or service that I've ever used in my life that I could recommend more highly than justinguitar.com.
## Ze first began his journey playing original music and top 40s pop tunes around the country's popular venues. Eventually, through the music of John Mayer, he found a strong attraction to blues music. Ze has years of experience teaching beginners and intermediate guitarists. Currently with Liberty Park Music he is teaching Introduction to Guitar Playing for Complete Beginners, Rhythm Guitar to learn about strumming, chords and more, Guitar Essentials as a fast-track review course, and lots of Song Lessons on pop and rock hits.
Archtop guitars are steel-string instruments in which the top (and often the back) of the instrument are carved, from a solid billet, into a curved, rather than a flat, shape. This violin-like construction is usually credited to the American Orville Gibson. Lloyd Loar of the Gibson Mandolin-Guitar Mfg. Co introduced the violin-inspired "F"-shaped hole design now usually associated with archtop guitars, after designing a style of mandolin of the same type. The typical archtop guitar has a large, deep, hollow body whose form is much like that of a mandolin or a violin-family instrument. Nowadays, most archtops are equipped with magnetic pickups, and they are therefore both acoustic and electric. F-hole archtop guitars were immediately adopted, upon their release, by both jazz and country musicians, and have remained particularly popular in jazz music, usually with flatwound strings.
The sound of an acoustic guitar is arguably the most well known in all of modern popular music. Without electronic components, the guitar relies solely on the interaction between the strings and the sound box to produce every note. Because of this, the strings of your acoustic guitar can significantly influence its sound. To get the most out of your guitar, keep an eye on the condition of the strings. You don't need to wait until your strings break to replace them. When they get old enough that their tone starts to change, it's time to re-string. Depending on the guitar, you may choose either nylon or metal strings. Nylon strings are the modern substitute for gut strings, so they're usually found on older styles of guitar such as baroque or flamenco. If you play the classical guitar, take care to fit it with the correct strings: since classical guitars are sized and tensioned differently from other varieties of acoustic guitar, they can be damaged by standard strings. Similarly, classical guitar strings won't work with other guitars. For metal-stringed guitars, there are several options available, each with its own acoustic character. The three most common types of metal strings are bronze, phosphor-bronze and silk-and-steel. The staple string for many guitarists, bronze produces a bright, quickly-fading tone that's lively and well-suited to any style of music. Phosphor-bronze is similar, but with added warmth and longer sustain. For a completely different sound, consider silk-and-steel. These strings create a tone that's gentle and mellow. Lower in tension and available in lighter gauges, silk-and-steel strings are easier to play and are great for vintage guitars that need special strings. It's also important to keep in mind the gauge of the strings you're choosing. Higher gauges are thicker, producing increased volume and extended sustain with an overall warmer tone. The trade-off of the rich overtones of heavy-gauge strings is that they are more challenging to play, requiring more force to fret, pick and strum. If you are an experienced guitarist, you likely have a preferred gauge already. If you're a beginner, it's a good idea to start with a lighter gauge to make the learning curve more forgiving. In the end, the right combination of material and gauge is a matter of personal preference. You may need to try several different acoustic guitar strings before you find the perfect ones for you, but the results will certainly be worth it.
Our relaxing music is perfect for Deepak Chopra meditation, Buddhist meditation, Zen meditation, Mindfulness meditation and Eckhart Tolle meditation. This music is influenced by Japanese meditation music, Indian meditation music, Tibetan music and Shamanic music. Some benefits include cleansing the Chakra, opening the Third Eye and increasing Transcendental meditation skills. The work of Byron Katie, Sedona Method, Silva Method and the Secret highlights the fact that healing can occur through using the mind and being in the “now”. Healing Meditation can be practised using this music for best results. | 2019-06-16 20:31:01 | {"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": 0.21972870826721191, "perplexity": 3375.443037078841}, "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-2019-26/segments/1560627998298.91/warc/CC-MAIN-20190616202813-20190616224813-00493.warc.gz"} |
https://dsp.stackexchange.com/revisions/a234950f-c950-43bb-956f-a2c47f3824c8/view-source | I read the paper *Deep Feature Flow for Video Recognition* https://arxiv.org/abs/1611.07715.
In Sec.3, the author implements bilinear interpolation like this:
$$f_i^c(p)=\sum\limits_{q}G(q,p+\delta p)f_k^c(q) \tag{1}$$
Where $q$ is the point from the source image, and $p$ is the points on the target image. $\delta p$ is the distance the point moved each point $p$ (not $\delta \bullet p$). $G$ is defined as
$$G(q,p+\delta p)=g(q_x,p_x+\delta p_x)g(q_y,p_y+\delta p_y)\tag{2}$$
And [the bilinear interpolation][1] is defined in wiki as:
$$f(x,y)\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\tag{3}$$
I think the operation $(1)$ and $(3)$ is equivalent. How can I derive the filter $(1)$ from $(3)$?
[1]: https://en.wikipedia.org/wiki/Bilinear_interpolation#Algorithm | 2019-12-08 13:17:50 | {"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": 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": 0.9070053696632385, "perplexity": 846.1073784838225}, "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/1575540510336.29/warc/CC-MAIN-20191208122818-20191208150818-00092.warc.gz"} |