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http://acooke.org/cute/Firstimpre0.html
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# C[omp]ute
Welcome to my blog, which was once a mailing list of the same name and is still generated by mail. Please reply via the "comment" links.
Always interested in offers/projects/new ideas. Eclectic experience in fields like: numerical computing; Python web; Java enterprise; functional languages; GPGPU; SQL databases; etc. Based in Santiago, Chile; telecommute worldwide. CV; email.
© 2006-2013 Andrew Cooke (site) / post authors (content).
## First impressions using Julia
From: andrew cooke <andrew@...>
Date: Tue, 27 Aug 2013 20:52:40 -0400
I wanted to have a closer look at Julia so I decided to revisit a program I
worked on years ago - generating an image of crumpled paper using a
"functional images" approach (the original was in Haskell with Pancito, based
on Conal Elliot's Pan).
I'll include the code below, but here I want to focus more on the "higher
level" details of what it was like using the language.
Getting Started
There don't seem to be pre-built binaries for OpenSuse, but building from
source works fine (although it takes a long time - an hour or so I guess).
Simply clone https://github.com/JuliaLang/julia and "make".
Positives and Negatives
In general, it was a very pleasant, productive experience - quite similar to
Python (at least, the functional part). I didn't use anything fancy (no
multimethods), but I did use generators and I did try "automatic" parallelism
(via pmap).
Note that it is still very much in development (apparently about to release
0.2) so you're building and working with a moving target.
One surprising detail was that indices / ranges are 1-based and inclusive - I
guess for consistency with Fortran and Matlab. Another, similarly small but
annoying wrinkle is the use of "elseif" (sic) to solve the nested if/else
parsing problem (why can't we standardise on "elif"?).
Back to a positive: there are already many packages, and the packaging system
worked fine. I installed the Cairo (graphics) wrapper with little fuss. The
package itself had no documentation, but it's just a simple wrapper around the
standard C API, so it was easy enough to translate existing C examples.
Unfortunately it's not a complete wrapper - details like line end caps aren't
exposed.
I had a problem understanding what is automatically broadcast when code runs
in parallel. It turns out that it's similar to MP or Python's
multiprocessing, in that you have to do some work to ensure that the
environment in each process is initialised correctly. In this case by using
"require()" to load the same source file on each.
The way to use pmap correctly is in the documentation (which is pretty good),
and got support there. The list is not terribly busy (took a while to get an
answer, and I have another question waiting), but is friendly enough and seems
to be read by the developers. I also tried IRC, but no-one responded.
I can't really comment on speed as I don't have alternative implementations to
compare, and the problem is one where it's easy to increase parameters until
you get bored waiting for results (my current best takes 500m to run).
However, adding type annotations didn't change anything - looking at the docs
more closely it seems like the speed comes mainly from JIT generation of
specialised code, with annotations only helping you find places where types
jump around (and so cause prediction issues).
Conclusions
In summary: positive. Way, way better than using C or Fortran. So, if it has
similar speed (and I assume it does), then it's freaking awesome. BUT it's
still very much in development, and with a small community for support. I
imagine it will get better with time, and I hope the developers stick with it
until they get the success they deserve.
It has replaced Clojure as my best candidate for a "better Python".
Code
And here's the code, in two files.
paper.jl:
# 0 is black, 1 is white
# the paper extends from (0,0) to (1,1)
# the underlying image is a function from position and incident light to
# brightness. the need for the light is something of a hack - it allows
# us to calculate the effect of folds on brightness even when we're
# still unsure whether we're evaluating the brightness of the paper (or
# are outside, in the background).
const border = 0.05 # less than shadow as entire image shrinks from folds
function to_bottom_corner(x)
if 0 < x < 1
return 0
elseif x > 1
return 1 - x
else
return x
end
end
function undistorted(xy, light)
x, y = viewport(xy)
if 0 < x < 1 && 0 < y < 1
return min(1, max(0, light))
else
x = to_bottom_corner(x)
y = to_bottom_corner(y)
r = sqrt(x^2 + y^2)
end
end
# this transform shifts the paper (originally also in the unit square)
# and border (originally outside) into the unit square (it just makes
# things neater if we can use rand() at the top level).
function viewpoint(x)
return x * (1 + 2 * border) - border
end
function viewport(xy)
x, y = xy
return (viewpoint(x), viewpoint(y))
end
# now the meat. given a fold (position and angle), transform the x, y
# coordinates and the incident light. these are just nice-looking
# approximations - there's no real physics here.
function distance(xy, fold)
x, y = xy
(p, q), theta = fold
return (x - p) * sin(theta) - (y - q) * cos(theta)
end
const n_folds = 200
const fold_width = 0.04
const fold_shift = 0.15 / n_folds
const fold_grey = 0.15
function normalize(d)
d = d / fold_width
if d > 1
return 1
elseif d < -1
return -1
else
return sign(d) * abs(d) ^ 1.5
end
end
function shift(xy, theta, d)
x, y = xy
return (x + d * fold_shift * sin(theta), y - d * fold_shift * cos(theta))
end
return g * (1 + (1 - abs(d)) * sign(d) * cos(theta + pi/3) * fold_grey)
end
# a stream of random folds.
function random_folds()
while true
produce(((rand(), rand()), pi * rand()))
end
end
# take 'n' folds from the stream, and an underlying function (the paper)
# and create a new function that accumulates the effect of all the folds.
# an alternative would be to make an explicit iteration over the folds
# (i have no idea which is faster, but this seems neater).
function combine_folds(n, folds, base)
if n == 0
return base
else
fold = consume(folds)
pq, theta = fold
function wrapped(xy, light)
d = normalize(distance(xy, fold))
xy = shift(xy, theta, d)
end
return combine_folds(n-1, folds, wrapped)
end
end
# to render the image we evaluate it and random points.
function random_points()
while true
produce((rand(), rand()))
end
end
function take(n, seq)
function inner()
for i = 1:n
produce(consume(seq))
end
end
end
function driver(n, image, output)
for result in map(point -> (point, image(point, plain_shade)), points)
output(result...)
end
end
typealias Point (Float64, Float64)
function pdriver(n, image, output)
for result::(Point, Float64) in
pmap(point::Point -> (point, image(point, plain_shade)), points)
output(result...)
end
end
# testing
function print_output(xy, g)
x, y = xy
println("$x$y \$g")
end
#driver(20, undistorted, print_output)
# output to cairo pdf
using Cairo
function cairo_pdf(file, size)
s = CairoPDFSurface(file, size, size)
c = CairoContext(s)
set_line_width(c, 1)
set_source_rgb(c, 0, 0, 0)
function output(xy, g)
if g < rand()
# set_source_rgb(c, g, g, g)
x, y = xy
move_to(c, x * size, y * size)
rel_line_to(c, 1/sqrt(2), 1/sqrt(2))
end
end
for x = 0:1
for y = 0:1
output((x,y), 0)
end
end
function close()
stroke(c)
finish(s)
end
return (output, close)
end
paper-run.jl:
require("/home/andrew/project/paper/git/paper.jl")
# here we go...
const width = 4000
const n_points = 10000000
output, close = cairo_pdf("paper.pdf", width)
pdriver(n_points, paper, output)
close()
Above I am using the parallel code. Change pdriver to driver to run the
serial version. The "const" aren't necessary - I just wondered if they would
help seed things up...
Andrew
### Increasing Julia Parallel Throughput
From: andrew cooke <andrew@...>
Date: Thu, 29 Aug 2013 16:01:36 -0400
In the example above I included pmap, which runs each calculation of the map
on a remote mode. However, in practice, it was slower than running everything
on a single core.
The problem, it seems, was that the amount of work done, for the network
traffic required, was too small. The coordinating core was at 100% while the
saves stayed at 10-20%.
The fix was to make each call do more work. I did this by modifying the code
to (1) broadcast a random seed (rather than a point) and (2) calculating 10000
points, instead of 1, returning the results in an array.
Note - turns out that Julia has typed arrays (as well as untyped ones) so you
get an efficient, minimal block of memory.
This was a huge win. Workers saturated at 100% (in fact it made sense to
specify one more process than I have cores, since the master process was doing
very little).
The new code is basically:
typealias Point (Float64, Float64)
typealias Result (Point, Float64)
const block_size = 10000
function block_eval(seed, image)
a = Array(Result, block_size)
srand(seed)
for i = 1:block_size
point = random_point()
end
return a
end
function driver(n, image, output)
blocks = div(n, block_size)
for block in map(seed -> block_eval(seed, image), 1:blocks)
for result in block
output(result...)
end
end
end
function pdriver(n, image, output)
blocks = div(n, block_size)
for block in pmap(seed -> block_eval(seed, image), 1:blocks)
for result in block
output(result...)
end
end
end
Which is still simple and understandable (imho).
Andrew
### Calling C from Julia
From: andrew cooke <andrew@...>
Date: Thu, 29 Aug 2013 23:32:41 -0400
Missing the ability to set Cairo line caps, I read the manual and tried to do
it myself. Turns out that calling C is easy:
function set_line_cap(ctx::CairoContext, style)
ccall((:cairo_set_line_cap, "libcairo"),
Void, (Ptr{Void},Int), ctx.ptr, style)
end
set_line_cap(ctx, 1)
works just fine. Nice.
Andrew
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2014-03-09 17:54:21
|
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|
http://mathhelpforum.com/calculus/105075-help-needed-advanced-limits-4-a.html
|
1. ## Help Needed with Advanced Limits # 4
I was wondering if someone could help me with the problems within the picture:
I am in dire need of a step by step explanation as I answered this as (Not Enough Info) and was incorrect
I figured that this should equal 3 if in the first limit u approached 4 instead of -4, but it doesn't; does this mean that the answer should be -3?
2. I've been doing quite a lot of thinking and can't see how this could be -3. All I see is an inverse so for it to be -3 the graph should have odd symmetry around the origin, right??
3. Originally Posted by Some1Godlier
I've been doing quite a lot of thinking and can't see how this could be -3. All I see is an inverse so for it to be -3 the graph should have odd symmetry around the origin, right??
There might be a typo in the question ....
|
2016-10-01 14:53:39
|
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|
https://zbmath.org/?q=an:0783.11044
|
# zbMATH — the first resource for mathematics
$$p$$-adic representations, $$p$$-adic periods and $$p$$-adic $$L$$-functions. (Représentations $$p$$-adiques, périodes et fonctions $$L$$ $$p$$-adiques.) (French) Zbl 0783.11044
Sémin. Théor. Nombres, Paris/Fr. 1987-88, Prog. Math. 81, 213-258 (1990).
[For the entire collection see Zbl 0686.00006.]
This paper completes in some sense a previous paper by J. Coates and the author in [Adv. Stud. Pure Math. 17, 23-54 (1989; Zbl 0783.11050)], about the existence of $$p$$-adic $$L$$-functions attached to motives over $$\mathbb{Q}$$.
The paper is divided into two parts. The first one is devoted to recall the main fact about $$p$$-adic representations of the absolute Galois group of a $$p$$-adic field, Fontaine’s theory on $$\varphi$$-filtered modules, as well as the relationship between these two concepts, in the ordinary case.
In the second part of the paper the author, by means of $$p$$-adic comparison isomorphisms between de Rham cohomology and Fontaine-Messing cohomology, elaborates a theory of $$p$$-adic periods in the spirit of Deligne’s theory of complex periods. The behaviour of $$p$$-adic periods under isogeny is discussed.
A conjecture predicts the role of the $$p$$-adic periods in the study of special values of $$p$$-adic $$L$$-functions of motives which are ordinary at $$p$$. Some calculations are pursued in the case of elliptic curves defined over $$\mathbb{Q}$$, provided that they have ordinary good reduction at $$p$$.
##### MSC:
11S40 Zeta functions and $$L$$-functions 11F85 $$p$$-adic theory, local fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
|
2021-01-20 18:03:26
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https://www.nextgurukul.in/wiki/concept/cbse/class-6/geography/maps/symbols-sketches-and-plans/3960404
|
Symbols, Sketches and Plans
The letters, pictures, lines, and colours used on the map are called symbols. Symbols are used to represent different features on a map. Symbols give more information in a limited space, make maps both easy to draw and easy to read, and help you find your way in an unknown area. Maps use a universal language that everyone can follow. The universal symbols used in maps are referred to as conventional symbols. According to an international agreement, symbols follow a common colour code. Blue is used for representing water bodies, Brown for mountains, Yellow for plateaus, and Green for plains Using symbols gives you a number of advantages: Give a lot of information in a limited space Symbols makes it easy to draw and read maps A sketch map is drawn without a scale, based on memory and observation, and rRepresents the main features of an area. A plan is drawn to scale, represents a small area on a large scale and provides the finer details, such as the length and breadth of a certain area.
#### Summary
The letters, pictures, lines, and colours used on the map are called symbols. Symbols are used to represent different features on a map. Symbols give more information in a limited space, make maps both easy to draw and easy to read, and help you find your way in an unknown area. Maps use a universal language that everyone can follow. The universal symbols used in maps are referred to as conventional symbols. According to an international agreement, symbols follow a common colour code. Blue is used for representing water bodies, Brown for mountains, Yellow for plateaus, and Green for plains Using symbols gives you a number of advantages: Give a lot of information in a limited space Symbols makes it easy to draw and read maps A sketch map is drawn without a scale, based on memory and observation, and rRepresents the main features of an area. A plan is drawn to scale, represents a small area on a large scale and provides the finer details, such as the length and breadth of a certain area.
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2019-10-22 04:47:32
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http://astronomy.stackexchange.com/tags/gravity/hot
|
# Tag Info
5
If there are only two bodies, then they will never enter a mutual orbit. For two objects initially gravitationally unbound, in order to become gravitationally bound you must remove energy from the system. With only two bodies (that don't collide), this does not happen. They will accelerate toward each other, change directions according to how close they get, ...
4
No, black holes are not the only cause of HVSs, although it is thought to be the most common mechanism. Hyper velocity stars are believed to be caused when binary stars come close enough to a supermassive black hole for one of the pair to be captured while the other star is ejected at high velocity. This appears to the main mechanism for HVSs. See for ...
2
Also asked at http://physics.stackexchange.com/questions/174080/how-do-we-know-dark-matter-isnt-curved-spacetime Basically no. Or at least you can't have this idea and General Relativity. GR demands that you have something (matter/energy density) to cause the curvature. Curvature without cause is not part of the model. That's not to say that what you ...
1
At a certain size, huge asteroids get classified as dwarf planets. Pluto has an atmosphere 100,000 times thinner than Earth, and Pluto is already one of the two largest dwarf planets known. Asteroids (like everything) do have gravity, so nearby gas would be drawn to them. But it would take just very tiny distrubances for that gas to drift away, so what ...
1
The Milky-Way does not orbit the Andromeda galaxy, they both move under the influence of all the members of the local group. Even if one were orbiting the other the orbit need not be near circular but could be a very eccentric (elongated) ellipse. The projected merger is because the tangential component of Andromeda's velocity with respect to the Milky-Way ...
1
The Earth doesn't cover the hole, the Earth is the hole. Gravity is an attractive force, so were you not standing on the Earth, the Earth's gravity would cause you to accelerate towards it. As you are standing on the Earth, you feel this acceleration as your weight, just as you would feel pressure if you were to push against a wall. To take a more complex ...
1
We have observed impacts with Jupiter In 1993 comet Shoemaker Levy 9 impacted. In 2009 an unknown object impacted. In each case the object was destroyed in Jupiter's atmosphere.
1
Let's assume that what is falling onto the neutron star is "normal" material - i.e. a planet, an asteroid or something like that. As the material heads towards the neutron star it gains an enormous amount of kinetic energy. If we assume it starts from infinity, then the energy gained (and turned into kinetic energy) is approximately (ignoring GR) ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-04-25 23:20:04
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https://stats.stackexchange.com/questions/65095/probability-of-what-is-being-predicted-in-glm
|
# Probability of what is being predicted in glm
In documentation to glm I read: "For binomial and quasibinomial families the response can also be specified as a factor (when the first level denotes failure and all others success)" Does it mean that probability of failure or success is being modeled?
I'm trying to apply simple logistic model to "german credit scoring" dataset where there are levels "good" and "bad". To get correct results (higher probability means higher likelihood of being good) I have to assume that Failure=Good and Success=Bad. This works, but it is really counterintuitive. I interpret this as - this will model probability of Failure (failed to be bad).
require(ggplot2)
names(german_data) <- c('ca_status','mob','credit_history','purpose','credit_amount','savings',
'present_employment_since','status_sex','installment_rate_income','other_debtors',
'present_residence_since','property','age','other_installment','housing','existing_credits',
'job','liable_maintenance_people','telephone','foreign_worker','gb')
str(german_data)
german_data$gb <- factor(german_data$gb,levels=c(2,1),labels=c("bad","good"))
levels(german_data$gb)[1] table(german_data$gb)
german_data\$prob <- predict(model,newdata=german_data, type="response")
ggplot(data=german_data) + geom_boxplot(aes(y=prob,x=gb)) + coord_flip()
The probability of 'success' is what's being modelled, in glm & in a fairly common terminology: though it really doesn't matter; you get equivalent models however you code the different levels of the response (the coefficients simply switch sign).
In any case, the way you've set it up in your example bad is the first level ('failure')—good is therefore 'success'. And even if you're modelling the probability of something bad—a production defect, a death—you don't have to call it a success, you can just call it what it is.
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2023-03-23 12:17:39
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http://math.stackexchange.com/questions/330294/if-mn-and-m-capn-are-finitely-generated-modules-so-are-m-and-n
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# If M+N and M$\cap$N are finitely generated modules, so are M and N.
The question asks to prove that if $M+N$ and $M \cap N$ are finitely generated modules, then M and N are also finitely generated.
I've tried to use basic definitions, but all failed.
I set some examples to study how I can find a generating set for M (or N) when I have the generators of $M+N$ and $M \cap N$ but then I realized that there's no simple way to do so because $3\mathbb{Z}+5\mathbb{Z}=\mathbb{Z}$ is a finitely generated $\mathbb{Z}$-module generated by $\{1\}$ and $3\mathbb{Z} \cap 5\mathbb{Z}=15 \mathbb{Z}$ is also a finitely generated $\mathbb{Z}$-module generated by $\{15\}$ but $M=3\mathbb{Z}$ is generated by $\{3\}$ which doesn't tell me anything conclusive.
I then set $M=\{(x,y,0): x,y \in \mathbb{R} \}$ and $N=\{(x,0,z): x,z \in \mathbb{R} \}$. Then $M+N=\mathbb{R}^3$ and $M \cap N = \{ (x,0,0): x \in \mathbb{R} \}$ both are finitely generated modules. The set $\{(0,1,1),(0,-1,1),(1,0,-1)\}$ is a generator for $M+N=\mathbb{R}^3$ and $\{(1,0,0)\}$ is a generator for $M \cap N$ but again there's no obvious way of obtaining generators for M or N.
Any insightful ideas will be appreciated.
EDIT: I have to add that this question is an exercise in the chapter 1 of the book I'm reading. Chapter 1 covers only basic definitions of modules, examples of modules, sub-modules, quotient modules and generating sets for modules. R-homomorphisms, exact sequences, Isomorphism theorems extended for modules, all are discussed in chapter 2. So I think I'm not allowed to use the materials covered in chapter 2 to solve exercises in chapter 1.
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I assume that $M,N$ are submodules of a given $R$-module. Since $M+N$ is finitely generated, the same is true for $(M+N)/N \cong M/(M \cap N)$. Since $M \cap N$ is finitely generated, this implies that $M$ is finitely generated. By symmetry, also $N$ is finitely generated.
More explicitly (especially when you don't know quotients and isomorphism theorems yet): Let $\{m_i+n_i\}$ be a generating set of $M+N$. Let $\{u_j\}$ be a generating set of $M \cap N$. Then I claim that $\{m_i\} \cup \{u_j\}$ is a generating set of $M$ (in particular, if $M+N$ and $M \cap N$ are finitely generated, then the same is true for $M$). In fact, let $m \in M$. Then we can find $a_i \in R$ with $m = \sum_i a_i (m_i + n_i)$. It follows that $m - \sum_i a_i m_i \in M \cap N$, hence there are $b_j \in R$ with $m - \sum_i a_i m_i = \sum_j b_j u_j$, i.e. $m = \sum_i a_i m_i + \sum_j b_j u_j$, QED.
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In the given example ($M=3\Bbb Z,\ N=5\Bbb Z$), $1=2\cdot 3-5$ generates $\Bbb Z=M+N$, so $\{m_i\}_i=\{6\}$ now, and $\{u_j\}_j=\{15\}$ generates $M\cap N$. Then the generating set for $M$ we get by this procedure is $\{6,15\}$. – Berci Mar 14 '13 at 14:34
+1 for the solution without isomorphisms and quotients. – azimut Mar 14 '13 at 15:33
If think the exact sequence $0 \to \rm M \cap N \to M \oplus N \to M + N \to 0$ should help to find a finite number of generators for $\rm M \oplus N$ and thus for $\rm M$ and $\rm N$.
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What are the maps in the sequence? – Fredrik Meyer Mar 14 '13 at 13:58
Is such a sequence possible? I would like to know how to look at it this way... I can't guess what the second map would be. – rschwieb Mar 14 '13 at 14:02
The maps are $x \mapsto (x,-x)$ and $(x,y) \mapsto x+y$. – Martin Brandenburg Mar 14 '13 at 14:05
This question and its solutions would be a useful tool: $M$ finitely generated if submodule and quotient are finitely generated.
It gives you a lemma that if $A/B$ and $B$ are finitely generated, then so is $A$.
Then you would consider $\frac{M+N}{M}\cong\frac{N}{M\cap N}$ for inspiration.
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2014-08-21 01:06:20
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https://proofwiki.org/wiki/Symbols:N/Multiary
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# Symbols:N/Multiary
## Definition
$n$-
When indicating an unspecified but finite number of objects in a concept, the prefix $n$- is frequently used, as for example:
ordered $n$-tuple
$n$-dimensional
$n$-ary
$n$-gon
Its $\LaTeX$ code is n .
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2022-05-20 06:53:46
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https://datascience.stackexchange.com/questions/62518/calculate-likelihood-of-a-date-being-worked-based-on-location-hours-and-rate-of
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# Calculate likelihood of a date being worked based on location, hours and rate of pay
I am just getting into trying some machine learning. As a C# developer I have played around with ML.NET. I'd now like try and see if it can help with a real world problem.
We have a system that invites staff members to work shifts in multiple hospitals, if a staff member is interested they contact our admin team and are then assigned to the shift.
Many shifts are difficult to assign staff to due to the date/day, location, the hours and of course the pay available. We therefore often have to increase the rate paid per hour. When the rate is increased there is a much better chance the shift will be filled.
My shift class is displayed below.
// SHIFT CLASS
int shiftID
int locationID
DateTime shiftStartDate
DateTime shiftEndDate
decimal hourlyRate
decimal hours
int assignedLocumID
Using the data above I'd appreciate any advice or insight to help work out;
Based on location, date (day, month), start time, end time, grade, hourly rate and hours, how can I work out the percentage chance of a shift being filled.
Ideally I'd like to try something out using ML.NET, however I'm not sure where to start. I have 5 years of data to train the model. Appreciate any help you can provide.
StatsSorceress has rightly indicated using logistic regression. This is almost a text book use case of the method. If you are also looking for sample code then u could look up this example (it uses ML.NET) https://medium.com/machinelearningadvantage/use-c-and-ml-net-machine-learning-to-predict-taxi-fares-in-new-york-519546f52591
A few differences for you would be that your "label" or the value that you are trying to predict might need to be calculated. For e.g. in the blog above the user has a ready made label , taxi fares , that she wanted to predict. In your case you are looking at a % value for whether the shift will be filled or not. Hence you will have to add an additional column called "%filled_or_something" which can be calculated by a simple SQL query grouping together your input features (day, month, hours, start time, end time, wages etc). So for e.g if you have 20 records for "Sunday / March / 6 (hours) / 23:00 / 06:00 / \$13 (per hour)" and it was filled for all those 20 days then the column would read 100% BUT if it was filled only for 10 out of those 20 records it would read 50%.
Of course, this would also beg the question, that if all you need to do is the above , then given that all the features are finite why couldn't we simply just group all the unique combination and find the % and just use this value itself ? The answer , according to me, is that LR also indicates how important every feature is, so for e.g. if you were to introduce new days / new shift timings / more number of hours, the LR model should be able to predict even those since it understand the weights associated with every feature.
This should enable you to finally use a logistic regression model as explained in the blog above. Hope this helps in someway :)
If you're looking for code, I'm not familiar with C#. My answer will focus on theory.
tl;dr most machine learning-related packages have a built-in logistic regression function of some sort. That's what I'm recommending here.
More detail:
I would start with a basic model and work my way up. It sounds like this is something you can figure out using a regression model.
Based on your question: "Based on location, date (day, month), start time, end time, grade, hourly rate and hours, how can I work out the percentage chance of a shift being filled?", I understand you want:
• output: probability of shift getting filled
• input: location, date, start time, end time, grade, hourly rate, hours (these are $$x$$'s below)
If you're familiar with logistic regression:
$$log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k$$
the output of this model is the log odds of a single shift being filled (versus not filled). Note this can be rewritten as:
$$p = \frac{e^{\beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k}}{1-e^{\beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k}}$$
Where $$p$$ is the probability that the shift gets filled.
I would start by fitting one logistic regression model for each shift.
Be careful: multinomial regression would give you the probability of 1 shift out of k possible shifts being filled, but I don't think that's what you're looking for.
After seeing how well that worked (comparing the model predicted results to the actual percentages in the data), if necessary I would build a conditional model to take into account the probability of filling a shift, given other shifts have already been filled. Not sure how I'd do that yet, but hopefully this gives you a starting point!
EDIT based on the answer from Vikram Murthy:
I realize I forgot to mention: your response variable in this case would be 0 or 1 indicating whether a shift was filled or not. So for each shift, you would have a column indicating whether that shift was filled. That's the column being predicted. This is similar to using "dummy variables".
For example, if you have two shifts, your columns would be:
loc, day, month, starttime, endtime, grade, hourlyrate, hours, shift1_f, shift2_f
So your data might look something like this:
loc, day, month, starttime, endtime, grade, hourlyrate, hours, shift1_f, shift2_f
A, 1, 2, 12:00, 8:00, 1, 34.25, 8, 1, 0
A, 1, 2, 12:00, 8:00, 1, 34.25, 8, 0, 1
A, 1, 2, 12:00, 8:00, 1, 34.25, 8, 1, 1
A, 1, 2, 12:00, 8:00, 1, 34.25, 8, 0, 0
In this setup, this would indicate that shift 1 was filled in the first case, shift 2 was filled in the second case, both shifts in the third case, and no shifts in the last case.
Your two logistic regression models would be set up like this:
shift1_f = loc + day + month + starttime + endtime + grade + hourlyrate + hours
and
shift2_f = loc + day + month + starttime + endtime + grade + hourlyrate + hours
The proportions would be figured out automatically by the program; it's not necessary to figure them out yourself and enter them in any application of logistic regression that I've seen.
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2021-06-15 20:16:03
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http://math.stackexchange.com/questions/634622/when-is-the-symmetric-part-of-a-matrix-positive-definite
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# When is the symmetric part of a matrix positive definite?
Suppose there is a (non-symmetric) real square matrix $A$ with symmetric part $A+A^T$.
What are some conditions on $A$ that are sufficient for $A+A^T$ to be positive definite?
For example, if the eigenvalues of $A$ are strictly positive is $A+A^T$ positive definite? (EDIT: This part of the question is answered in the negative in the comments).
This would then give the result I actually want which is that given two positive definite matrices $C$ and $D$ it follows that the symmetric part of $CD$ is also positive definite. (EDIT: But I think it is still not clear if $CD+DC>0$ - this is (perhaps) a slightly more special case than $A+A^T$ with $A$ having positive eigenvalues.)
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wat can you say about eigen values of $A^T$ – Praphulla Koushik Jan 11 '14 at 12:42
oh yes.. that does not help! I am helpless!! – Praphulla Koushik Jan 11 '14 at 13:09
Consider $A = \begin{pmatrix}\frac14 & 1\\0 & \frac14\end{pmatrix}$, the eigenvalues of $A$ are all positive but $A + A^T$ are not positive definite. – achille hui Jan 11 '14 at 13:43
A matrix is sum of symmetric and anti symmetric parts. $x' A x$ is zero if $A$ is anti symmetric. So a matrix is pd, iff its symmetric part is pd. – user114628 Jan 11 '14 at 14:12
Also note that eigenvalues of a non-symmetric pd matrix are not necessarily positive. – user114628 Jan 11 '14 at 14:17
To address the problem that the OP actually wanted solved:
Given two symmetric positive definite matrices $C$ and $D$, show that $CD+DC$ is also positive definite.
This is not true, as can be seen from the following example:
$$C = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}, \quad D = \begin{bmatrix} 100 \\ & 1 \end{bmatrix}.$$
Then their respective spectra are:
$$\sigma(C) = \{1,3\}, \quad \sigma(D) = \{1, 100\}, \quad \sigma(CD+DC) \approx \{-20.2724, 424.272 \}.$$
The reason is simple: when computing $\det(CD+DC)$, the big factor $100$ participates in the positive part only $2$ times, but squared in the negative (counter-diagonal) part.
This might give you some insight how things behave in a more general setting. For example, I'm fairly certain that it can be shown that the following is true: if $C$ and $D$ do not commute, there exists $n \in \mathbb{N}$ such that $CD^n+D^nC$ is not positive definite. All you need to do is observe a $2 \times 2$ submatrix for which $D$ is not a multiply of $I_2$ (WLOG, you can assume that $D$ is diagonal). In other words, any such $D$ can be blown enough to make the above invalid.
This would give you that for every symmetric positive definite $C$ (such that it is not a multiple of identity), you can find a symmetric positive definite $D$ (diagonal, if you want) such that the above does not hold.
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2016-05-26 18:34:36
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https://math.stackexchange.com/questions/1369733/existence-of-moore-spaces-for-modules-over-commutative-rings
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# Existence of Moore spaces for modules over commutative rings.
Let $R$ be a commutative ring, $A$ a $R$-module and $n$ a natural number. Does there exist a CW complex $M(A,n)$ with $\tilde{H}_i(M(A,n),R)=0$ if $i\neq n$ and $\tilde{H}_n(M(A,n),R)\cong A$ as $R$-modules?
If $R=\mathbb{Z}$ one can choose a short exact sequence $$0\longrightarrow \bigoplus_{j\in J}\mathbb{Z}\longrightarrow\bigoplus_{i\in I}{\mathbb{Z}}\longrightarrow A\longrightarrow 0$$ and construct $M(A,n)$ by attaching $n+1$-cells indexed over $J$ to $\bigvee_{i\in I}S^{n}$ corresponding to the first homomorphism in the sequence.
If $R$ is any other ring, there occur several problems:
1. If $R$ is not a PDI, submodules of free modules do not have to be free, so the existence of such a SES is not guaranteed.
2. I don't know whether I can realize every ring element as $R$-degree of a map between spheres. It's easy for all elements of the subgroup generated by $1$, but I don't see how one can manage that for other elements.
• What $R$-module structure are you considering on $\tilde H_n(M(A,n))$? Jul 22 '15 at 8:01
• (You can consider a free resolution of $A$ as an $R$-module and attach cells jus as you did, no?) Jul 22 '15 at 8:03
• The usual one. Homology groups with coefficients in a ring $R$ have a canonical $R$-modules structure. I forgot to include the coefficients and edited the question accordingly. Jul 22 '15 at 8:03
• I don't see how. I addressed this in my two concerns in the question. Jul 22 '15 at 8:04
• "Irritated"?! Well, I guess I'll just leave you to it... Jul 22 '15 at 8:04
Actually you cannot construct such $CW$-complex. Consider the map $d_k:C_k(X,R)\to C_{k-1}(X,R)$, the basis element ($k$-cell) $\alpha$ maps to $\sum [\alpha:\beta_i]\cdot\beta_i$, this is sum of $k-1$-cells with integer coefficients.
So, for example, if $R=\mathbb Z[t]$, you cannot get $H_n(X,R)=\mathbb Z[t]/(t)$.
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2022-01-21 05:07:26
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http://openwetware.org/index.php?title=User:Mary_Mendoza/Notebook/CHEM572_Exp._Biological_Chemistry_II/2013/01/23&curid=129395&oldid=669971
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# User:Mary Mendoza/Notebook/CHEM572 Exp. Biological Chemistry II/2013/01/23
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## Calculation of reagents for ADA kinetic assay
• It was decided to prepare a 30 mM stock solution of adenosine. The calculation below shows the amount needed to obtain 30 mM of adenosine.
.030 $\frac{mol}{L}$ mol of adenosine × $\frac{267.24 g}{1 mol}$ = $\frac{8.0172 g}{L}$ × .010 L = 0.0802 g
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2015-01-29 15:23:55
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http://u.cs.biu.ac.il/~galk/Publications/b2hd-jida11jose.html
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# Gal A. Kaminka's Publications
Sorted by DateClassified by Publication TypeClassified by TopicGrouped by Student (current)Grouped by Former Students
## A Plan Classifier based on Chi-Square Distribution Tests
José A. Iglesias, Agapito Ledezma, Araceli Sanchis, and Gal A. Kaminka. A Plan Classifier based on Chi-Square Distribution Tests. Intelligent Data Analysis, 15(2):131–149, 2011.
### Abstract
To make good decisions in a social context, humans often need to recognize the plan underlying the behavior of others, and make predictions based on this recognition. This process, when carried out by software agents or robots, is known as plan recognition, or agent modeling. Most existing techniques for plan recognition assume the availability of carefully hand-crafted plan libraries, which encode the a-priori known behavioral repertoire of the observed agents; during run-time, plan recognition algorithms match the observed behavior of the agents against the plan-libraries, and matches are reported as hypotheses. Unfortunately, techniques for automatically acquiring plan-libraries from observations, e.g., by learning or data-mining, are only beginning to emerge.We present an approach for automatically creating the model of an agent behavior based on the observation and analysis of its atomic behaviors. In this approach, observations of an agent behavior are transformed into a sequence of atomic behaviors (events). This stream is analyzed in order to get the corresponding behavior model, represented by a distribution of relevant events. Once the model has been created, the proposed approach presents a method using a statistical test for classifying an observed behavior. Therefore, in this research, the problem of behavior classification is examined as a problem of learning to characterize the behavior of an agent in terms of sequences of atomic behaviors.The experiment results of this paper show that a system based on our approach can efficiently recognize different behaviors in different domains, in particular UNIX command-line data, and RoboCup soccer simulation.
### BibTeX
@Article{jida11jose,
author={Jos\'e A. Iglesias and Agapito Ledezma and Araceli Sanchis and Gal A. Kaminka},
title = {A Plan Classifier based on Chi-Square Distribution Tests},
journal = JIDA,
year = {2011},
OPTkey = {},
volume = {15},
number = {2},
pages = {131--149},
OPTmonth = {},
wwwnote = {},
OPTannote = {},
abstract = { To make good decisions in a social context, humans often need to recognize the plan underlying the behavior of others, and make predictions based on this recognition. This process, when carried out by software agents or robots, is known as \emph{plan recognition}, or \emph{agent modeling}. Most existing techniques for plan recognition assume the availability of carefully hand-crafted \emph{plan libraries}, which encode the a-priori known behavioral repertoire of the observed agents; during run-time, plan recognition algorithms match the observed behavior of the agents against the plan-libraries, and matches are reported as hypotheses. Unfortunately, techniques for automatically acquiring plan-libraries from observations, e.g., by learning or data-mining, are only beginning to emerge.
We present an approach for automatically creating the model of an agent behavior based on the observation and analysis of its atomic behaviors. In this approach, observations of an agent behavior are transformed into a sequence of atomic behaviors (events). This stream is analyzed in order to get the corresponding behavior model, represented by a distribution of relevant events. Once the model has been created, the proposed approach presents a method using a statistical test for classifying an observed behavior. Therefore, in this research, the problem of behavior classification is examined as a problem of learning to characterize the behavior of an agent in terms of sequences of atomic behaviors.
The experiment results of this paper show that a system based on our approach can efficiently recognize different behaviors in different domains, in particular UNIX command-line data, and RoboCup soccer simulation.
}
}
Generated by bib2html.pl (written by Patrick Riley ) on Sat Feb 24, 2018 00:31:02
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2019-01-17 09:09:08
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http://mathhelpforum.com/algebra/18637-factoring-questions-help-2.html
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# Math Help - Factoring Questions - Help!
1. See if you can hammer it into this shape:
$9(x+2y+z)^{2}-16(x-2y+z)^{2}=(14y-x-z)(7x-2y+7z)$
2. Originally Posted by CaptainBlack
No real factors
You sure?
$
3v^2-11v-10
$
It has real roots
3. Originally Posted by galactus
See if you can hammer it into this shape:
$9(x+2y+z)^{2}-16(x-2y+z)^{2}=(14y-x-z)(7x-2y+7z)$
I ended up getting (7x - 2y + 7z)(7x - 2y + 7z)
I went wrong somewhere. :S
EDIT: Nevermind, had my signs confused. I did end up getting (14y - x - z)(7x - 2y + 7z).
4. Originally Posted by Krizalid
You sure?
$
3v^2-11v-10
$
It has real roots
Oppss.. no rational factors
RonL
Page 2 of 2 First 12
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2014-03-08 13:21:03
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https://www.themathdoctors.org/how-can-multiplication-make-it-smaller/
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How Can Multiplication Make It Smaller?
A fairly common question arises when students learn to multiply or divide fractions and decimals: They discover that multiplication, which always used to make numbers larger (2, multiplied by 3, becomes 6), now can make them smaller (2, multiplied by 1/2, becomes 1). How can that be? Here we’ll look at a few answers we’ve given to this kind of question over the years.
Multiply means increase, right?
Our first question, from Rizwan in 1998, focuses on language:
Multiplying by 1
Dear Dr. Math,
My question is as follows:
Multiply means increase in number. When 1 is multiplied by 1, the answer is 1. The answer is 1. Why is it? Each one independent unit is being multiplied but the number is not increased. Looks erratic to me. Please define.
Hello, Rizwan. This is an interesting question, and I can make it seem even stranger. Not only can you multiply by 1 and the result does not increase, but you can also multiply by 1/2 and the result is smaller.
As we’ll see, most people ask this larger question, rather than Rizwan’s tame question about the case where it merely fails to increase!
If you look at the original meanings of words, the same problem arises with the word "add". It comes from the Latin "addere" meaning "to give to." Yet I can add a negative number, with the result that something is actually taken away.
I think the same sorts of problems will arise in any language, and in other disciplines besides math. A word that means one thing in everyday language will have a somewhat different meaning, or a very specific and specialized meaning, in math or physics or economics or another specialized field of study. When people have a new idea or invent a new product, sometimes they invent an entirely new word to identify it. But sometimes they just use an existing word that has a similar meaning. For instance, an electrical current is like a current in a river, but it is not exactly the same.
This is the nature of language! Words grow into bigger meanings than they started with (or narrow down). As “add” in Latin (addere) meant “to join, attach, place upon”, from roots meaning “to give to” (see here), so “multiply” (multiplicare) in Latin meant “to increase”, from roots meaning “having many folds, many times as great in number” (see here). And, of course, “be fruitful and multiply” never meant “become fewer”! But it came to mean something far broader, while still being used with its original sense as well.
The basic words of math like "multiply" and "add" were adapted from everyday life long ago. Back then, concepts like negative numbers and even zero had not been developed. People would really only think in terms of multiplying by positive whole numbers. And why bother to multiply by 1? It doesn't do anything. So the use of the words made sense.
But mathematicians gradually extended the meanings of the words. Not only can you multiply fractions or negative numbers, you can multiply matrices or numbers in modular arithmetic, where the idea of one number being greater than another is meaningless.
I talked about some of this in my post What is Multiplication … Really? (This also includes comments about repeated addition, which will come up soon.)
The things that we call "multiplication" today have a lot in common with simple multiplication by an integer greater than 1, so it makes sense to use the same word for them. Why invent a new word just because the original narrow meaning of the word doesn't fit any more?
In short, the problem that you have raised is a reason for the existence of specialized dictionaries of science and technology. If you look up the meaning of a word in a dictionary of everyday language and try to apply the definition to the way the word is used in a specialized field like math, you will often only get confused. Just use a word the way it is defined in the field you are working in, and don't worry about what it means in everyday life.
Sometimes speakers of another language try to translate their math terminology into English using an ordinary dictionary (or Google’s similar attempted translation) can stretch our minds trying to see what term they are trying for. Usually I can see the connection and work it out; but it can lead to interesting reflections on word choices. Language is almost as “interesting” as math …
The next question, from Jen in 1997, starts with the initial concept of multiplication, rather than the mere word:
Multiplying Fractions
Since we have defined multiplication as repeated addition, how is it possible that when you multiply two fractions, the product is smaller than either of the fractions?
It is common to introduce multiplication to children as repeated addition, as for example $$3\times 4$$ means that we add 3 4’s $$(4+4+4=12)$$, or add 4 3’s $$(3+3+3+3=12)$$. A few years ago there was a lot of discussion about whether this definition should be taught at all, because it really only applies to natural numbers, and needs to be replaced with a more general definition as soon as a child gets beyond that. This question illustrates the problem.
Doctor Ken answered, using one alternative definition that extends more easily to fractions:
Hi there -
Perhaps a more useful way to think of multiplication is to use the words "groups of." For instance, when you have 4 x 5, think of it as "4 groups of 5". We can write that out as 5 + 5 + 5 + 5, and add it up to get 20. If we have 1/2 x 2, then we think of it as "half a group of twos", so our total is 1. If we have 1/3 x 1/4, then that's one-third of a group of 1/4ths, and maybe this will make more sense to you. One-third of a positive number is always smaller than the original number was. That's one way to make sense of the equation 1/3 x 1/4 = 1/12.
There was no reply, so we don’t know whether this did make sense to Jen; often it takes some discussion, because it does stretch the mind a bit! If you’re still unsure, read on – one benefit of having many Math Doctors is that they can offer different perspectives, which work for different people.
Think of the decimal as a fraction
Next, a similar question from Carolyn in 2001, this time about decimals rather than fractions:
Multiplying by a Decimal Number
I am having trouble understanding why multiplying by decimal numbers gives a product smaller than the factors. For example, 5 * .43 = 2.15. I just always thought that multiplying would make the product bigger than the numbers I used.
In this example, the product is smaller than 5 (though larger than 0.43); we’ve multiplied 5 by a number less than 1, which is the key. (In some problems, the result is smaller than both factors, as in $$0.5\times 0.4 = 0.2$$.)
Hi, Carolyn.
When you only knew about whole numbers, you could make a rule like the one you stated - but not quite! Even when you only knew about whole numbers, there was one exception to your rule: multiplying by 1 does not make a number bigger. When you added zero, that was another exception.
If you multiply by a number greater than 1, the product will be greater than what you started with. If you multiply by 1, the product is the same as what you started with. If you multiply by a number less than 1, the product is less than what you started with.
There is the key concept. Once you have decimals, and then negative numbers, you have lots of numbers that are not greater than 1, so there are far more exceptions to the initial “rule”.
But still, why do we now get a smaller number?
Let's make some sense of this by considering a simple example.
Multiplying by 1/2 is the same as dividing by 2. Do you understand this? Multiplying by 1/2 means taking half of it; dividing by 2 means cutting it in two pieces and keeping one. These amount to the same thing. Since 1/2 = 0.5, and half of something is less than the whole, it makes sense that multiplying a number by 0.5 makes it smaller.
Fractions make it easy to see what is happening; decimals are just a different way to represent a fraction.
Observe that what he said here is closely related to the fact that adding a negative number (adding $$-2$$, say) is the same as subtracting its opposite (subtracting 2), which we know decreases the number.
And just as introducing negative numbers changes the nature of addition, turning addition and subtraction into varieties of the one operation, addition, so introducing fractions changes the nature of multiplication, combining it with division into what amounts to a single operation that can both magnify and reduce a number.
Scaling up and scaling down
Jumping ahead to 2012, we have this question from Justine, digging deeper into the question of decimals:
Multiplication Makes It ... Smaller?!
Hello!
My question is, when you multiply with a decimal that is smaller than 1, why does it give a smaller product?
For example:
2 x .5 = 1
And
500 x .25 = 125
Isn't multiplying supposed to give a larger product?
My dad says 2 x .5 = 1 is the same as 2/2, which is also 1. Why is multiplying by .5 the same as dividing by 2?
He tries to explain it to me, but I just can't seem to get it!
Related to my question, I think, is when someone says "25% off of 50." That means you don't have to pay the .25. So you can do
50 x .75 = 37.5
I don't understand why, though.
When you get 25% off something, that means you get it for a smaller price ... but yet I multiplied? That is kind of strange to me.
Please reply back because this has been bugging me lots. Thanks!
Hi, Justine.
When you multiply a number by 2, you get a bigger result. That's what you're used to until now.
When you multiply by 1, what do you get? The same number, right?
Multiplying by a number GREATER than 1 makes it bigger; multiplying by 1 leaves it the same; and, continuing the pattern, multiplying by a number LESS than 1 makes it smaller.
For example, multiplying a number by 1/2 cuts it in half, making it smaller. Multiplying by 0.5 is the same thing.
This is what Justine has observed; but why? I suggested another alternative interpretation of multiplication:
In general, you can think of multiplication as scaling, which can make something either larger or smaller, scaling up or scaling down.
Multiplying by 3 scales up:
0 1 2 3
+-----+-----+-----+
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
| \ \ \
+-----------------+-----------------+-----------------+
0 3 6 9
This enlargement turned 1 into 3.
Multiplying by 1/3 scales down:
0 1 2 3 4 5 6 7 8 9
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| / / /
| / / /
| / / /
| / / /
| / / /
| / / /
| / / /
| / / /
| / / /
| / / /
| / / /
+-+-+-+-+-+-+-+-+-+
0 1 2 3
This shrinking turned 1 into 1/3, and 3 into 1. Notice that multiplying by 1/3 undoes a multiplication by 3; it divides by 3.
Multiplying by some number k turns 1 into k, and either stretches or compresses the whole number line proportionally. Multiplying by 3 makes everything 3 times as big; multiplying by 1/3 makes everything 1/3 as big, which is smaller, just as 1/3 is smaller than 1.
Justine replied,
Thanks so much! Your answer really cleared up my confusion.
So now I won't get bugged by this question anymore.
Sometimes a picture says more than words can.
I realized in writing this that I never did answer the last part of Justine’s question, about “25% off of 50.” There, we are actually doing two things: multiplying by 25%, or 0.25, to find the amount of the discount (25% of 50, which is 12.5), and then subtracting that from the starting number, 50: $$50-12.5 = 37.5$$. It’s really that subtraction that reduces the amount (though the multiplication has to result in a smaller amount, or we’d end up going negative!)
The shortcut she used, subtracting 25% from 100%, works because the whole process involves subtracting 25% of 50 from 100% of 50, which leaves 75% of 50: $$0.75\times 50=37.5$$. This can also be explained using the distributive property from algebra, as $$50-0.25(50) = 1.00(50)-0.25(50)=(1.00-0.25)(50)=0.75(50)=37.5$$.
Division means sharing, right?
Now let’s turn to the other side of the question, with this from Emily in 2001:
Is Division Sharing?
Hi, Dr Math,
Why when I do a fraction division (e.g. 1/2 divided by 1/2 = 1) is the answer bigger than the first fraction in question (i.e. 1/2)? Shouldn't division mean sharing, so logically the answer is smaller than the first fraction?
Thanks, Emily
Emily is probably picturing one particular “model” (use) of division, in which dividing 6 by 2 means asking how much of the 6 goes to each of 2 people. (This is also called partitive, or sharing, division, finding the size of each of 2 parts; another model is quotative, or measurement, division, which asks how many parts you get if each part has 2 items out of the 6; both of these uses of division, as long as we use whole numbers, expect the result to be smaller than the given numbers. For more on these terms, see toward the end of my post, Dividing Fractions: How and Why.)
Doctor Ian answered, again offering a more general model:
Hi Emily,
It's not strictly true that 'division means sharing'. In fact, division is just another way of looking at multiplication. For example, all of the following are just different ways of saying the same thing:
3 times 4 equals 12.
4 times 3 equals 12.
12 divided by 3 equals 4.
12 divided by 4 equals 3.
It's sort of like looking at this picture,
/\
/ \
/ \
+------+
| |
| |
+------+
and knowing that both of the following descriptions
The triangle is above the square.
The square is below the triangle.
are equally true, since they are really just different ways of saying the same thing.
So division is just looking at a multiplication backward, asking what one of the factors is. Technically, it is the inverse of multiplication.
In other words, whenever it is true that
a * b = c
it must also be true that
c / a = b
and
c / b = a
(unless either a or b is zero, in which case all bets are off). That's what we _mean_ by division.
(Unintentionally, this ties in with last week’s new question, which involved why division by 0 is undefined.)
Now, let's think about what this means for division by 1/2. If
c / (1/2) = a
then it must be true that
c = a * (1/2)
which means that a must be _larger_ than c.
We can think of this in terms of sharing; if we divide 5 by 1/2, for example, using the quotative model we might be asking how many people we can divide 5 apples among, if each gets half an apple. When we cut each apple in half, we end up with 10 halves, which we can share among 10 people: $$5\div\frac{1}{2} = 5\times 2 = 10$$. It’s harder to think of this in terms of partitive division, which would mean asking how much to put in “each pile” if we want to make 1/2 a pile. There are problems in which that would make some sense; but it’s better to just think in terms of what division is, as the inverse of multiplication.
Thinking about division as 'sharing' is one way that teachers try to make the concept simpler for students to understand. But it's important to realize that these simplifications usually lead you in the wrong direction once you start looking at things more carefully.
Just as words “grow up” and take on broader meanings, our understanding of a concept may have to grow up, to accommodate those new meanings.
Think of the decimal as a fraction, again
We’ll look at one more question, from Jen in 2003:
How Can Division Result in an Increase?
What is the logic behind dividing an integer by a decimal and getting a larger number? For example,
100 / 0.9185 = 108.87
I just can't understand the logic! It seems like the answer should be a smaller number.
Hi Jen,
Well, remember that a decimal is just a fraction. For example,
0.23 = 23/100
0.1542 = 1542/10,000
and so on. So what happens when we divide by a fraction? We multiply by the reciprocal:
8 3
----- = 8 * -
2/3 2
So if we divide by something where the denominator is smaller than the numerator, we'll end up multiplying by something where the numerator is larger than the denominator. Does that make sense?
In other words, dividing by a number less than 1 means the same as multiplying by a number greater than 1, which results in a larger number.
That was the abstract view. Since students commonly see things better concretely, we can turn to a model:
Here's another way to think about it, using the idea that division means breaking things into pieces. Suppose I have something 6 inches long, and I divide it into pieces that are 2 inches long. How many pieces do I get?
1
+----+----+ +----+----+ +----+----+
| | | | | | | | |
+----+----+ +----+----+ +----+----+
6 / 2 = 3
This is the quotative model I mentioned before, in the form of measurement.
Okay, now what if I divide it into pieces that are only 1/2 inch long? I'm going to end up with 12 pieces, right?
1
-
2
+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+
| | | | | | | | | | | | | | | | | | | | | | | |
+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+
6 / (1/2) = 12
So we divided by something less than one, and ended up with more than our original result.
How many pieces we get increases when the pieces are smaller.
We always need to come back to the general definition, which is abstract:
But the clearest way to see the logic of this is to remember what we _mean_ by division. That is, a division is just another way of representing a multiplication. For example, when we say that
3 * 4 = 12
two other ways to say exactly the same thing are
12 / 3 = 4 and 12 / 4 = 3
So suppose we have something like
6 / (1/2) = ?
This is the same as saying that
6 = ? * (1/2)
Now, if this is true, then the value of '?' had better be larger than 6, right?
Again, since multiplying by a number less than 1 results in a smaller number, the unknown factor here has to be larger. Our two questions in this post, about multiplication and division, end up being one question.
1 thought on “How Can Multiplication Make It Smaller?”
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2021-04-21 21:40:42
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https://proofwiki.org/wiki/Radius_at_Right_Angle_to_Tangent
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# Radius at Right Angle to Tangent
## Theorem
In the words of Euclid:
If a straight line touch a circle, and a straight line joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent.
## Proof
Let $DE$ be tangent to the circle $ABC$ at $C$.
Let $F$ be the center of $ABC$.
Let $FC$ be the radius in question.
Suppose $FC$ were not perpendicular to $DE$.
Instead, suppose $FG$ were drawn perpendicular to $DE$.
Since $\angle FGC$ is a right angle, then from Two Angles of Triangle Less than Two Right Angles it follows that $\angle FCG$ is acute.
From Greater Angle of Triangle Subtended by Greater Side it follows that $FC > FG$.
But $FC = FB$ and so $FB > FG$, which is impossible.
So $FG$ can not be perpendicular to $DE$.
Similarly it can be proved that no other straight line except $FC$ can be perpendicular to $DE$.
Therefore $FC$ is perpendicular to $DE$.
$\blacksquare$
## Historical Note
This theorem is Proposition $18$ of Book $\text{III}$ of Euclid's The Elements.
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2020-12-02 06:51:10
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http://www.ams.org/mathscinet-getitem?mr=2776609
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MathSciNet bibliographic data MR2776609 (2012f:11228) 11S25 (11F85) Chang, Seunghwan; Diamond, Fred Extensions of rank one $(\phi,\Gamma)$$(\phi,\Gamma)$-modules and crystalline representations. Compos. Math. 147 (2011), no. 2, 375–427. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2014-10-23 14:39:30
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https://blender.stackexchange.com/questions/21192/what-does-the-mapping-type-do-on-a-mapping-node/43379
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# What does the mapping type do on a mapping node?
What do the four buttons (texture, point, vector, and normal) at the top of the mapping node do? It by default uses point and I have never changed it. But in the answer to this question David mentioned changing it to texture (which makes intuitive sense). But technically what do these buttons do? I have always thought the mapping node is the same as using a vector > split XYZ and math nodes to manipulate the components, in fact I often do this instead of a mapping node if I want to plug some other nodes into the values.
The mapping node works by changing the coordinate system. This is why (when left on the default point setting) you have to put in a larger number to scale a image down. What is you are actually doing is scaling the texture coordinates up.
When set to Texture you are still changing the Coordinate system, not the texture. The only difference between Point and Texture is that for texture all the values are inverted.
Try it, when set to point a value of 2 is required to make the texture repeat 4 times.
When set to texture to get the same repeating 4 times the value must be .5.
I would highly recommend going through the cg cookie tutorial (archive) about the mapping node.
• Thanks, that cg cookie series is amazing! But for completeness could you add explanations of the vector and normal buttons? – PGmath Dec 20 '14 at 20:37
• @PGmath The only information I could find about the other two options Vector and Normal are these few lines. ""Vector Transform a direction vector. Normal Transform a normal vector with unit length."" I have never used either of those 2, and I could not find out much about them. – David Dec 21 '14 at 0:50
• Strange, I will try to find some time to do some experimentation and see if I can figure out the difference. Thanks again! – PGmath Dec 21 '14 at 1:01
• @PGmath I spent about an hour, and still can now write what those two do. – David Dec 21 '14 at 1:02
• @David using point will make the texture fit "as many times" within the texture on a given axis. So a value of 2 will make the texture fit 2 times not 4. i.stack.imgur.com/bcxvs.png and i.stack.imgur.com/aoTtC.png – user1853 Dec 21 '15 at 3:31
I know this is an old question, but maybe someone still finds useful what I found. The tooltips actually explain the difference:
• Texture: Transform a texture by inverse mapping the texture coordinate
• Point: Transform a point
• Vector: Transform a direction vector
• Normal: Transform a normal vector with unit length
If you want to understand it deeper, like I did, this might not be enough for you. I looked in the Blender source code, and I found the following comments:
• Texture: to transform a texture, the inverse transform needs to be applied to the texture coordinate
• Point: forward transform
• Vector: no translation for vectors
• Normal: no translation for normals, and inverse transpose
What does all this mean? Direction and normal vectors should not be translated because they represent a direction, and normal vectors should be transformed with "inverse transpose"
At this point you might ask (as I did...): "Hey, isn't the Vector mode redundant if all it does is that it ignores my Location values?". The answer is that the calculations are faster if one knows in advance that there is no translation. A different, faster code is called inside Blender. So yes, Vector has the same result as Point with 0, 0, 0 Location, but it is calculated faster.
So how should you use this node? For texture transformations you can use either the texture or the point as David explained in his answer. If you are an advanced user and you want to transform direction or normal vectors, use the corresponding options, and don't be surprised when the Location values are ignored.
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2020-01-28 14:26:09
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https://nrich.maths.org/11009/clue
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### System Speak
Five equations... five unknowns... can you solve the system?
### Area L
By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?
### Irrational Arithmagons
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
Do you know any values of $a$ and $b$ that satisfy $a^b=1$?
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2018-02-25 07:42:02
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https://mersenneforum.org/showthread.php?s=cb5b70e2c1ff94ec07f3b95d0e886c5d&t=16402
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mersenneforum.org Aliquot genealogy project
Register FAQ Search Today's Posts Mark Forums Read
2011-12-30, 05:34 #1 EdH "Ed Hall" Dec 2009 Adirondack Mtns 3,373 Posts Aliquot genealogy project Is there a "family tree" of sorts that shows all the merges for sequences under 1M in the db? Last fiddled with by schickel on 2011-12-31 at 18:29 Reason: Change thread title (I myspelled) [& change icon]
2011-12-30, 06:56 #2
schickel
"Frank <^>"
Dec 2004
CDP Janesville
40628 Posts
Quote:
Originally Posted by EdH Is there a "family tree" of sorts that shows all the merges for sequences under 1M in the db?
No. The closest thing would be Wolfgang's C30C90 table. If you look in there, you will find a list of open sequences, with the first c9 and first c30 encountered during their lifetime. Some merges and terminations are listed, but at the top it says:
Quote:
Originally Posted by Wolfgang All numbers you will not find in the following table are key-numbers of side-sequences. Have a look into the matrix of ALIQUOT. You will find them there.
Last updated 2009, so not real current.
The matrix listed is one that is maintained by a TurboPascal program that is listed in the download section. Said matrix is a listing of sequence starting numbers and a status which is indicated by a number: 0 for open, the prime a sequence terminates at, or the member of an aliquot cycle that a sequence merges with. Not so useful if you want to fiind out what merges with what, from either side, but you could get a count of how many sequences terminate at which prime. Major drawback: uses "1" for the starting line of a sequence file, rather than "0"....another drawback: the record file has not been updated since 1994.
2011-12-30, 12:54 #3
science_man_88
"Forget I exist"
Jul 2009
Dumbassville
836910 Posts
Quote:
Originally Posted by schickel No. The closest thing would be Wolfgang's C30C90 table. If you look in there, you will find a list of open sequences, with the first c9 and first c30 encountered during their lifetime. Some merges and terminations are listed, but at the top it says:Last updated 2009, so not real current. The matrix listed is one that is maintained by a TurboPascal program that is listed in the download section. Said matrix is a listing of sequence starting numbers and a status which is indicated by a number: 0 for open, the prime a sequence terminates at, or the member of an aliquot cycle that a sequence merges with. Not so useful if you want to fiind out what merges with what, from either side, but you could get a count of how many sequences terminate at which prime. Major drawback: uses "1" for the starting line of a sequence file, rather than "0"....another drawback: the record file has not been updated since 1994.
yeah I could play with my ali and aligen scripts ( they work backwards) again but I doubt anything interesting comes out of it.
2011-12-30, 14:44 #4 EdH "Ed Hall" Dec 2009 Adirondack Mtns 3,373 Posts @schickel: Thanks. I looked at the c30c90 file a while ago, but not the other reference, yet. You have posted an "AllSeq.zip" file in the past, so there must be a method to query just the last line from the db for all the sequences. I can get full .elf files, but all my last line attempts end up downloading an .html page. How can I get just the last lines, either individually or all at once (preferred)? For that matter, is there a reference that shows what queries are available for the db? If I can get all the last lines, I'm pretty sure I can write something that can build a merge listing. Of course, then I need to figure out the best method for displaying merges...
2011-12-30, 19:59 #5
schickel
"Frank <^>"
Dec 2004
CDP Janesville
40628 Posts
Quote:
Originally Posted by EdH @schickel: Thanks. I looked at the c30c90 file a while ago, but not the other reference, yet.
Ah, OK. I wasn't sure how much of the background info you were aware of, since it has gotten harder to keep track of who joined when and who knows how much....
Quote:
You have posted an "AllSeq.zip" file in the past, so there must be a method to query just the last line from the db for all the sequences. I can get full .elf files, but all my last line attempts end up downloading an .html page.
I use a 5-lb sledgehammer to do the job. I have a list of the open sequences and just download the .elf file for each one.
I don't do the whole range but once a week or so, since it is so query intensive. I keep meaning to sit down and do a "last line" tool Real Soon Now.
Quote:
How can I get just the last lines, either individually or all at once (preferred)?
Unfortunately, if you use the "show last element" radio button, the last line is shown as an html page. The last line is there, but you have to parse it out of the html....
Quote:
For that matter, is there a reference that shows what queries are available for the db?
Not really. Some of the queries available were discussed over in the FactorDb thread, but I would say that the only one who knows what all is available is Syd....
Quote:
If I can get all the last lines, I'm pretty sure I can write something that can build a merge listing. Of course, then I need to figure out the best method for displaying merges...
I have been thinking along those lines, too. So far it seems to me there is no really easy way to do things, since if you want to make it comprehensive, you almost have to have 1.000.000 entries (less the primes since they're easy), even if most of them are "Sn merges with x after y lines" or "Snterminates in x after y lines" with y<10.....
2011-12-30, 20:16 #6
science_man_88
"Forget I exist"
Jul 2009
Dumbassville
202618 Posts
Quote:
Originally Posted by schickel even if most of them are "Sn merges with x after y lines" or "Snterminates in x after y lines" with y<10.....
this good enough for you:
Code:
aligen5(w,s)=for(z=w,s,for(x=1,#ali(z),print1(ali(z)[x]":i2=");for(y=1,#ali(ali(z)[x]),print1(ali(ali(z)[x])[y]":i3=")));print())
? it was a slight change to my aligen script.
2011-12-31, 05:08 #7
EdH
"Ed Hall"
Dec 2009
64558 Posts
Quote:
Originally Posted by schickel Ah, OK. I wasn't sure how much of the background info you were aware of, since it has gotten harder to keep track of who joined when and who knows how much....
Wow, I guess I've been around for two years now, according to my stats... But I should still be considered as someone who doesn't know much.
Quote:
Originally Posted by schickel I use a 5-lb sledgehammer to do the job. I have a list of the open sequences and just download the .elf file for each one.
The more I look at this, the larger a task it seems. To download 1M .elfs, or even just last lines, is a huge endeavor, itself. Even If I d/l 1k per day, it would take 3 years to complete...
Quote:
Originally Posted by schickel I don't do the whole range but once a week or so, since it is so query intensive. I keep meaning to sit down and do a "last line" tool Real Soon Now.
I'm going to have to think a bit more about this. Things like maybe I should eliminate those sequences that end in primes or cycles.
Quote:
Originally Posted by schickel Unfortunately, if you use the "show last element" radio button, the last line is shown as an html page. The last line is there, but you have to parse it out of the html....
It would be easier to parse the .elf files, although they would be larger.
Quote:
Originally Posted by schickel Not really. Some of the queries available were discussed over in the FactorDb thread, but I would say that the only one who knows what all is available is Syd....
I kind of wondered if there was a .php method that queries available functions.
Quote:
Originally Posted by schickel I have been thinking along those lines, too. So far it seems to me there is no really easy way to do things, since if you want to make it comprehensive, you almost have to have 1.000.000 entries (less the primes since they're easy), even if most of them are "Sn merges with x after y lines" or "Snterminates in x after y lines" with y<10.....
Yeah, I was thinking along the lines of a list:
Code:
seq1:line = seq12:line
seq1:line = seq23:line
seq34:line = seq44:line
...
But, let's address two sequences that end in the same cycle. Did they merge?
This is seeming more complex, the further I dig...
2011-12-31, 06:23 #8
schickel
"Frank <^>"
Dec 2004
CDP Janesville
2×1,049 Posts
Quote:
Originally Posted by EdH Wow, I guess I've been around for two years now, according to my stats... But I should still be considered as someone who doesn't know much. The more I look at this, the larger a task it seems. To download 1M .elfs, or even just last lines, is a huge endeavor, itself. Even If I d/l 1k per day, it would take 3 years to complete...
Yeah, but the trick is, I just do the open ones, and there are only 9244 9243 open ones. I have it broken into 3 parts because I have a memory leak somewhere that kills the task after ~6000 downloads. It takes about 4 hours to do each 1/3 of the list. (Most of the time is spent reformatting the files a little; if you pad the line numbers out to 10 spaces, you can peg the size of a line easier.)
Quote:
I'm going to have to think a bit more about this. Things like maybe I should eliminate those sequences that end in primes or cycles.
Eliminate from the download? Yes, you can, since they're done once they merge....
Quote:
It would be easier to parse the .elf files, although they would be larger.
That's true, though the download time would be lots less just for the last line.
Quote:
I kind of wondered if there was a .php method that queries available functions.
Nothing was ever brought up about that. It looked like Syd was adding stuff as people brought things up.
Quote:
Yeah, I was thinking along the lines of a list: Code: seq1:line = seq12:line seq1:line = seq23:line seq34:line = seq44:line ... But, let's address two sequences that end in the same cycle. Did they merge?
I would look at where they merged. If they merge before the resulting confluence hits the cycle, I would say that was a merge, since there was some activity before the merge.
This actually came up on one of our terminations. I marked it as a merge because it hit a sequence and headed up before the last downdriver run to the termination. Wolfgang marked it down as a temination since it actually terminated rather than ran into another open sequence.
Quote:
This is seeming more complex, the further I dig...
Welcome to our world....
------------------------
The ultimate solution is to draw a tree where each number is a node with an arrow pointing to its $\sigma(n)-n$ value and has arrows pointing from numbers that point to it.
After this post, I'll look back; someone was here wondering what software to use to visualize aliquot cycles in this manner, and I think they posted a paper, with some of the aliquot trees in it. Unfortunately, I forgot where I found it.....:face palm:
2011-12-31, 06:26 #9 schickel "Frank <^>" Dec 2004 CDP Janesville 209810 Posts Here's the thread I was thinking of. Geez.....it was only in April that this came up. Now I'll go and see if I can find that paper.
2011-12-31, 06:53 #10 schickel "Frank <^>" Dec 2004 CDP Janesville 83216 Posts Aha! Here we go. Richard Mathar at Leiden University; here is an index of his papers, and here's a deep link to the paper I was thinking of.... [Edit: I wonder if this is the first citation of the FactoDB in a paper.....] Last fiddled with by schickel on 2011-12-31 at 06:56 Reason: Adding edit
2011-12-31, 08:19 #11
schickel
"Frank <^>"
Dec 2004
CDP Janesville
2·1,049 Posts
Quote:
Originally Posted by schickel After this post, I'll look back; someone was here wondering what software to use to visualize aliquot cycles in this manner, and I think they posted a paper, with some of the aliquot trees in it. Unfortunately, I forgot where I found it.....:face palm:
OK, so it looks like it wasn't the same person....I wonder if Wini left a working email on file....
Similar Threads Thread Thread Starter Forum Replies Last Post firejuggler Aliquot Sequences 26 2012-01-19 08:15 schickel Aliquot Sequences 307 2011-10-28 01:29 JohnFullspeed Aliquot Sequences 18 2011-08-20 21:11 schickel Aliquot Sequences 29 2011-08-12 17:45 Andi47 Aliquot Sequences 3 2009-03-08 10:18
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Fri Oct 23 12:17:58 UTC 2020 up 43 days, 9:28, 0 users, load averages: 1.05, 1.31, 1.35
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2020-10-23 12:17:58
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https://stats.stackexchange.com/questions/404480/unbiased-estimator-of-the-standard-deviation-of-the-sample-standard-deviation
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# Unbiased Estimator of the Standard Deviation of the Sample Standard Deviation
I'm looking for an unbiased estimator of the standard deviation $$\text{SD}(s)$$ of the sample standard deviation $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2}$$. I have found this answer which is highly relevant to this question, however, I would prefer a solution that does not assume normality (similarly to the unbiased estimator of the variance of the sample variance).
Is there a general solution for non-normal distributions?
• Probably for particular families of distributions there are such estimators but for all probability distributions in general, or even for all with finite moments, I suspect there is none. Apr 22, 2019 at 23:58
• My impression is that we either have yet to find one that works as broadly as $S^2$ does for variance (e.g., specific to the normal distribution) or that we have shown that no such estimator can exist. I am, however, unsure which of those is correct.
– Dave
Jun 9, 2021 at 18:03
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2022-06-27 06:19:58
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https://www.implementingquantlib.com/2017/05/github-integrations.html
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Welcome back.
After my previous post on Gource, another post on cool tools I started using with QuantLib: Travis CI, Codacy, and Codecov. All of them analyse the code and help keep it in shape, and all of them integrate seamlessly with GitHub.
Lately, if you looked at a pull request while logged in to GitHub, you might have noticed something like this at the bottom:
What checks? Well, the ones I added to the repository thanks to those tools. The first I added was Codacy: it’s a static analysis front-end that uses tools for a number of languages, including C++. When integrated with your GitHub repo, it analyses your master branch and any other you select in the settings and gives you a summary of possible issues with your code. Not only that; whenever a new pull request is opened, it checks whether it fixes any of those issue (good) or it adds new ones (bad). This allows you to try and set a trend towards cleanliness, or at least to avoid getting further from it.
In the case of QuantLib, we had some 400 issues; the report is public, and you can see it here. We managed to fix about a quarter of them; the remaining 300-odd are mostly about constructors which take just one argument but are not declared as explicit. We’ll probably fix them, a few at a time. Technically, it is a breaking change to make such a constructor explicit, since it might cause existing code to stop compiling. However, picking a few examples, that code would have to be something like:
GammaDistribution f = 1.0;
TwoFactorModel model = 2;
Iceland calendar = Iceland::ICEX;
I hope you’ll agree that the first two examples should be simply taken out and shot. The third one is not very idiomatic, but might be debatable. The ones I fixed so far were allowing code like:
Handle<YieldTermStructure> forecastCurve;
Euribor6M index = forecastCurve;
which was firmly in take-out-and-shoot territory.
The next integration was with Travis CI, a continuous-integration tool. Like Codacy, it runs its magic at each commit and pull request: the magic being a script that you can specify and that builds and tests the code. The only catch: it has to run in 45 minutes, and here is where things got interesting. Building and testing QuantLib in full takes way more than that. Therefore, I had to save time where I could.
The first thing I tried was enabling ccache, since Travis supports it. However, I needed a first run to bootstrap the cache, and I couldn’t do it in one go; so I run a first version of the script building only the library, and a second (which at that point would use the cache to build the library) to include the examples and test suite into the build. I had won the first battle, but not the war, though: large changes, or Travis purging the cache after a period of inactivity, would still cause the build to go beyond the limit.
// the only header you need to use QuantLib
#include <ql/quantlib.hpp>
which was convenient to write, and was probably fine when we started in 2000, but nowadays would bring in some 320,000 lines of code after preprocessing. Therefore, out went the global header. Now the examples only include the headers they need, and I advise that your sources do the same. The line count after preprocessing went down on average to around 100,000, and the compilation times dropped accordingly. On my Mac, and using the -O2 optimization level, the average example would compile in about 20 seconds before the change; after the change, it takes between 5 and 10, with a few dropping below 5 seconds and none above 20. It adds up, now that we have 19 of them in the source tree.
Final change: the test suite is running in between 12 and 15 minutes, which is way too much. Fortunately, Klaus Spanderen had put in place some code to measure the time taken by each test case, so I looked at the data. It turned out that more than 90% of the test suite, or about 620 out of 670, actually runs in under three minutes, with the other 50 tests accounting for the remaining 10 minutes. I added a switch to the test suite to only run that 90%, and there I was. You can see the latest pull-request checks here: they are comfortably under the 45-minutes mark, taking as little as just a few minutes if compilation hits the cache heavily.
Having these automated checks now help me detect pull requests that don’t compile or otherwise break the build, without having to check them out locally and do it myself. (Don’t fear, though: I still run the whole test suite before merging into master.) Hopefully, this will translate into more frequent releases coming your way.
A last integration is not shown, because it hasn’t made into master. I’ve added Codecov, which provides a nice web interface to the code-coverage data provided by g++ through gcov. The production of coverage data and its transfer to Codecov analysis can be added to the Travis run, but unfortunately the compilation flags that enable coverage analysis are incompatible with ccache.
Not all is lost, though. By chance, I had just read on the latest issue of the Overload journal an interesting article on unity builds. The idea in short (read the article for details) is to create a file that #includes all the .cpp files in a directory and compile it, instead of compiling each file separately. This often results in improved compilation times, since we’re parsing common headers just once instead of doing it in each source file. The creation of the aggregate files can be automated by a Makefile rule. Long story short, I created a branch for Codecov integration, I added a switch to ./configure to enable the unity build (which might be useful in its own right, so could go into master), and I disabled ccache and enabled coverage information in the Travis script. The result runs in about 40 minutes. I’ll probably update the branch every once in a while to check trends in coverage. I thought about merging it into master and having it run at each commit, but in this case it seemed like a waste of resources to have Travis run for 40 minutes instead of 10 or 20 each time.
Especially because those resources are given out for free. Codacy, Travis CI, Codecov, and of course GitHub (oh, by the way: I’m not affiliated with, or sponsored by, or paid to endorse any of them) are all free for open-source projects. Thanks, people. And you, dear readers: consider showing some love and using their paid versions for your code if your projects have the budget for it. It’s good for everybody if they stay alive.
Follow me on Twitter if you want to be notified of new posts, or add me to your Google+ circles, or subscribe via RSS or email: the buttons for that are in the footer. Also, make sure to check my Training page.
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2017-10-20 19:42:48
|
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https://wikieducator.org/Thread:Feedback_on_3rd_bullet_point_(1)
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# Feedback on 3rd bullet point
Hi JB,
With reference to:
* civil society: through media coverage and training under the auspices of professional associations, charities, faith communities and other civil society organisations
...there is no explicit mention of NGOs...perhaps this is covered by your statemtent... In many countries (perhaps beyond Malawi), NGOs are the trusted organisation connecting directly to the grassroots, whereby the grassroots can learn about (different concepts and practices) of governance. The materials that COL is preparing on the WikiEducator - WikiGovernance node, will attend to the paucity of learning materials expressly designed for the use of NGOs, as they conduct their activities. We have a working name: the NGO Learning Design Toolkit for Governance
--Randy Fisher 21:48, 15 April 2008 (UTC)
10:48, 16 April 2008
Hi Randy, sorry for not getting back to the Discussion page sooner! Your point is very well taken and the gap was to some extent filled but using "CSO" terminology rather than "NGOs". I am not sure which we should employ. The two overlap to a considerable extent but I have not given much thought to differences that might influence our choice. Your views on this would be most welcome. - JB
JB (talk)11:47, 29 April 2008
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2022-01-24 20:52:56
|
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http://openstudy.com/updates/55705c40e4b050a18e84f334
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• anonymous
Alan wants to bake blueberry muffins and bran muffins for the school bake sale. For a tray of blueberry muffins, Alan uses mc017-1.jpg cup of oil and 2 eggs. For a tray of bran muffins, Alan uses mc017-2.jpg cup of oil and 1 egg. Alan has 4 cups of oil and 12 eggs on hand. He sells trays of blueberry muffins for $12 each and trays of bran muffins for$9 each. Alan wants to maximize the money raised at the bake sale. Let x represent the number of blueberry muffins and y represent the number of bran muffins Alan bakes. What are the constraints for the problem?
Mathematics
Looking for something else?
Not the answer you are looking for? Search for more explanations.
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2017-03-28 18:05:13
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https://project.auto-multiple-choice.net/issues/250
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# separate mc section and open answer section printout for efficient paper usage (Feature #250)
Status: Start date: Closed 01/02/2014 Low - 100% LaTeX 1.4.0
Description
In some exams you want to use multiple choice questions and open questions.
This describes the situation when you use a separate answer sheet.
While mc questions can be put into a two column layout (no problem if you have 6 or less choices), but the answers to open questions need
more space when hand written.
This could be accomplished if the \AMCform command was re-factored such that by default combines all sections (elements), but that it would be possible to also render the elements separately.
So e.g. a command \\AMCformElement{element name } as in \\AMCformElement{general} and \\AMCformElement{open} would do the trick for me.
### Associated revisions
Revision 2173:7e107b6c06a0
so that one can retrieve (and insert) elements separately. The new
macro \AMCformFilter, together with \AMCifcategory, can be used to
insert answers for questions from some category (eg. open questions).
Refs #250.
### History
#### #1 Updated by Alexis Bienvenüeover 4 years ago
• Target version set to 1.4.0
#### #2 Updated by Alexis Bienvenüeabout 1 year ago
Thanks for you suggestion.
Starting with revision hg:7e107b6c06a0 (included in version 1.3.0+hg2018-02-25-1 from current test PPA), you can use something like
\begin{multicols}{2}
\AMCformFilter{!\AMCifcategory{open}}
\end{multicols}
\AMCpostOquest=1mm
\AMCformFilter{\AMCifcategory{open}}
• Status changed from New to Feedback
#### #3 Updated by Alexis Bienvenüe11 months ago
• Status changed from Feedback to Closed
• % Done changed from 0 to 100
Also available in: Atom PDF
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2019-03-22 12:17:57
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https://socratic.org/questions/how-do-you-solve-m-2-7-4
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# How do you solve m/2-7>4?
Feb 28, 2016
$m > 22$
#### Explanation:
$\frac{m}{2} - 7 > 4$
$\frac{m}{2} > 4 + 7$
Simplify.
$\frac{m}{2} > 11$
Multiply both sides by 2.
$m > 11 \times 2$
Simplify.
$m > 22$
Feb 28, 2016
$m > 22$
#### Explanation:
Given:$\text{ } \frac{m}{2} - 7 > 4$
You treat this like a normal equation. There is one 'trap' however. If the whole thing is multiplied by a negative number the inequality is turned round the other way.
For example: It is true that $2 < 4$
Now consider the incorrect calculation of:
$\left(- 1\right) \times 2 < \left(- 1\right) \times 4 \text{ implying that } - 2 < - 4$ which is false
$- 2$ is to the right of $- 4$ on the number line so $- 2 > - 4$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add 7 to both sides giving
$\frac{m}{2} - 7 + 7 > 4 + 7$
$\frac{m}{2} + 0 > 11$
Multiply both sides by 2 giving
$\frac{2}{2} \times m > 2 \times 11$
But $\frac{2}{2} = 1$ giving:
$m > 22$
'~~~~~~~~~~~~~~~~~~~~~~~~~
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2019-10-15 22:29:08
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http://www.euro-math-soc.eu/node/2603
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## Journey through Mathematics. Creative Episodes in Its History
Author(s):
Enrique A. González-Velasco
Publisher:
Springer
Year:
2011
ISBN:
978-0-387-92153-2
Price (tentative):
hardcover 59,95 € (net)
Short description:
This book describes the history of mathematics that gave rise to our modern concepts in calculus: trigonometry, logarithms, complex numbers, infinite series, differentiation and integration, and convergence (limits).
MSC main category:
01 History and biography
MSC category:
01-02
Other MSC categories:
01A05
Review:
The book grew out of a mathematical history course given by the author. It has a list of 39 pages with references to historical publications which are amply cited and from which many parts are worked out in detail. This is organized in 6 chapters describing the evolution of concepts from ancient times till the 18th century to what is now generally used in calculus courses.
González-Velasco has done a marvelous job by sketching this very readable historical tale. He stays as close as possible to the original way of thinking and the way of proving results. He is even using the notation and phrasing and explains how it would be experienced by scientists of those days. However at the same time he makes it quite understandable for us, readers, used to modern concepts and notation. A remarkable achievement that keeps you reading on and on. In my opinion, this is not only compulsory reading for a course on the history of mathematics, but everyone teaching a calculus course should be aware of the roots and the wonderful achievements of the mathematical giants of the past centuries. They boldly went where nobody had gone before and paved the road for what we take for granted today.
What follows is a brief summary of the subjects treated in ech chapter.
The first chapter on trigonometry starts with the Greek, the Indian, and the Islamic roots (mostly geometric) of trigonometric concepts. One has to wait till the XVIth century when trigonometric tables were produced before the term sinus was used and 2 more centuries before the notation sin, cos,... was used.
The second chapter on the logarithm is a natural consequence of the trigonometric tables as an aid for computation. $\sin A \cdot \sin B=\frac{1}{2}[\cos(A−B)−\cos(A+B)]$ could be used to multiply numbers $x\approx \sin A$ and $y\approx\sin B$. Napier (1550-1617) and Briggs (1561-1630) worked out the concepts of the logarithm in base e and base 10. Later de St. Vincent (1584-1667) connected this to areas below (integrals of) $1/x$, and Newton (1642-1727) with infinite series, while Euler (1707-1783) generalized it to a logarithms in an arbitrary basis.
Complex numbers are introduced in chapter 3. This is tied up with the solution of a cubic equation (Cardano, 1501-1576) as square roots of negative numbers. Bombelli (1526-1572) described complex arithmetic and Euler even studied the logarithm of complex numbers, but it was only Wallis (1616-1703) and Wessel (1745-1818) who gave the geometric interpretation and made complex numbers accepted (if you can draw them, they must exist). To Hamilton (1805-1865) they were a couple of real numbers and Gauss (1777-1855) introduced the letter i for $\sqrt{-1}$.
Next chapter treats infinite series. Summation of numerical sequences was known to the Egyptians and the Greek, but it was Leibniz (1646-1716) who first summed the inverse of the triangular numbers $1/k(k−1)$ and Euler computed $\sum 1/k^=π^2/6$. As for function expansions, the Indians knew a series for $\sin(x)$ in the XIVth century, but in Europe one had to wait till the XVII-XVIIIth for Newton and Euler. However it was Gregory (1638-1675) with his polynomial interpolation formulas who later inspired Taylor (1685-1731) and Maclaurin (1698-1746) to develop their well known series.
Chapter 5 about calculus is the major part (about a quarter) of this book. Fermat (1601-1665), Gregory and Barrow (1630-1677) contributed but of course Newton and Leibniz are the main players here with the well known dispute of plagiarism as a consequence. Here the author gives a careful and detailed analysis of their contributions and concludes that they worked independently, but that Leibniz's publications lacked clarity, which made him difficult to understand for his contemporaries.
The last chapter is about convergence. Leibniz and Newton's ideas were still rather geometric: derivatives were tangents and integrals were quadratures, thus essentially finite. The notion of limit was lacking, which was only developed later in contributions by Fourier (1768–1830), Bolzano (1781-1848), Cauchy (1789-1857), Dirichlet (1805-1859), and others. This chapter has also a remakable original section on the less known Portugese mathematician da Cunha (1744-1787) whose much earlier contribution went largely unnoticed.
Reviewer:
A. Bultheel
Affiliation:
KU Leuven
|
2013-06-19 03:18:57
|
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|
https://aletheiawritingmagazine.com/prayer-and-worship-essay/junction-rule-essay.php
|
Junction rule essay
# Junction rule essay
Pssst…
69 writers online
## Introduction and also Importance
Kirchhoff’s signal law regulations can be two equations which target this conservation involving strength and also impose once and additionally long run double essays your context of electro-mechanical circuits.
### Learning Objectives
Describe partnership amongst all the Kirchhoff’s circuit protocols not to mention the strength along with cost throughout the particular electric powered circuits
### Key Takeaways
#### Key Points
• Kirchhoff put into use Georg Ohm ‘s perform mainly because a good basis to make sure you build Kirchhoff’s active legislations (KCL) plus Kirchhoff’s voltage law (KVL) throughout 1845.
### Kirchhoff’s Rules
A lot of these will be able to end up being resulting coming from Maxwell’s Equations, of which originated 16-17 ages later.
• It is usually hopeless to make sure you examine a lot of closed-loop circuits by just simplifying seeing that a cost and/or chain about resources.
Inside these kinds of instances, Kirchhoff’s laws and regulations can certainly often be used.
• Kirchhoff’s regulations tend to be unique incidents for resource efficiency connected with strength and even charge.
#### Key Terms
• resistor: Any electric power component part which sends present-day with special amount to help you all the voltage across it.
• electromotive force: (EMF)—The voltage gained simply by some solar battery and / or by way of a permanent magnetic coerce relating that will Faraday’s Regulation.
This might be assessed within gadgets of volts (not newtons, N; EMF is without a doubt not really some force).
• capacitor: A great electrical ingredient containing connected with 2 conductor toy plates separated through drain house (sometimes your dielectric content is usually alternatively sandwiched relating to the plates), as well as competent involving stocking your hardest come to ice skating essay volume for charge.
### Introduction to be able to Kirchhoff’s Laws
Kirchhoff’s enterprise guidelines tend to be a pair of equations earliest circulated by simply Gustav Kirchhoff during 1845.
Generally, some people street address resource efficiency for strength plus price in your wording from energy circuits.
Although Kirchhoff’s Regulations could always be taken because of this equations of John Metamorphosis alienation essay Maxwell, Maxwell does not even distribute his or her establish associated with differential equations (which kind all the base regarding traditional electrodynamics, optics, and energy circuits) before 1861 not to mention 1862.
### Long and even Quick Essay or dissertation on Targeted visitors Procedures during English
Kirchhoff, instead, utilised Georg Ohm’s operate while an important base pertaining to Kirchhoff’s up-to-date legislation (KCL) and also Kirchhoff’s voltage regularions (KVL).
Kirchhoff’s regulations tend to be particularly vital to be able to the actual studies from shut down circuits.
Look into, to get case study, all the circuit illustrated with that physique underneath, consisting regarding several resistors around a fabulous solution about around show and additionally parallel placements. Simplification regarding this unique signal to be able to any solution in sequence in addition to parallel associates is usually unachievable. However, by using Kirchhoff’s procedures, 1 may analyze the circuit in order to figure out typically the parameters for this kind of circuit applying typically the figures of all the resistors (R1, R2, R3, r1 together with r2).
Spring symbolism essay involving magnitude on that case study is without a doubt which the ideals E1 not to mention E2 depict sources of voltage (e.g., batteries).
As the remaining take note of, Kirchhoff’s regulations hinge in several problems. Your voltage law is usually a new simplification with Faraday’s legislation in induction, along with might be primarily based relating to the actual supposition that generally there is definitely no fluctuating magnet arena within typically the closed down loop.
Thus, despite the fact that this kind of rules are able to end up implemented to help circuits containing resistors and also capacitors (as well like other sorts of enterprise elements), it all can solely always be applied like a powerful approximation to be able to the actual conduct about the enterprise the moment the switching recent together with for this reason magnet field really are involved.
Closed Circuit: To help figure out just about all features (i.e., active and even voltage comes through any numerous resistors) in this specific signal, Kirchhoff’s tips have got to come to be applied.
## The Foo fighters france essay Rule
Kirchhoff’s junction guideline expresses who during any specific rounds junction, all the add about that currents sweeping within plus away involving which junction usually are equal.
### Learning Objectives
Formulate the actual Kirchhoff’s junction guideline and even detail it's limitations
### Key Takeaways
#### Key Points
• Kirchhoff’s junction regulation is any use involving all the basic principle regarding conservation associated with electric charge: present-day is normally pass about fee per instance, not to mention if perhaps recent is definitely continuous, who which usually cascades inside a good level during your routine must match who that goes through connected with it.
• The mathematical representation jesus drafted in hebrew essay Kirchhoff’s legal requirement is: $\sum_{\text{k}=1}^{\text{n}} \text{I}_\text{k}=0$where Ik might be a ongoing about okay, and additionally and will be typically the entire selection about cable connections floating right into kenny bernstein total price essay out about some junction within consideration.
• Kirchhoff’s junction regularions is actually reasonably limited through a applicability through zones, with in which request occurrence may not become continual.
Mainly because command is without a doubt conserved, that sole strategy it is usually feasible can be should in that respect there is your circulate with charge all over the particular border of the region. This approach movement will often be a active, thereby violating all the law.
#### Key Terms
• electric charge: The quantum amount which will can help determine typically the electromagnetic communications from numerous subatomic particles; by traditions, this electron provides a powerful electronic impose about -1 plus that proton +1, and also quarks experience fractional charge.
• current: Typically the precious time price involving pass in electric utility charge.
Kirchhoff’s junction procedure, even well-known like Kirchhoff’s recent legal requirements (KCL), Kirchoff’s first of all regulation, Kirchhoff’s time law, and additionally Kirchhoff’s nodal guideline, is definitely a good practical application with the particular process about resource efficiency with electricity charge.
Kirchhoff’s junction procedure expresses that on every junction ( node ) inside a electricity signal, your quantity with the currents flowing right into in which junction is definitely even to this volume essay concerns regarding inform report heart a currents floating out for which junction.
Within alternative words and phrases, offered which a good up-to-date will end up great or damaging dependant with no matter if this will be going closer to or simply absent through a new junction, typically the algebraic amount of money of currents inside an important mobile phone network for conductors gathering within some sort of level is usually equal to nothing.
An important image counsel will often be looked at in.
Thus, Kirchoff’s junction control can certainly be reported mathematically because the add from currents (I):
$\sum_{\text{k}=1}^{\text{n}} \text{I}_\text{k}=0$
where d is definitely the actual total multitude about divisions lugging current on the way to and / or at bay through any node.
This legislation will be launched with the efficiency from charge (measured through coulombs), of which is without a doubt a system involving latest (amperes) as well as occasion (seconds).
### Limitation
Kirchhoff’s junction legal requirements is normally small around their applicability.
It retains to get most events through which entire electric ask for (Q) might be steady for your area through attention. Pretty much, the following might be generally authentic and so huge when typically the legislation is usually utilized intended for some sort of targeted position. More than some district, yet, impose solidity may perhaps possibly not always be endless.
Simply because bill is definitely conserved, the merely way the following will be probable is normally whenever truth be told there might be an important circulate regarding ask for along any border for all the section.
This specific move would probably possibly be some existing, thus violating Kirchhoff’s junction law.
Kirchhoff’s Junction Law: Kirchhoff’s Junction Rules illustrated while currents sweeping right into and paladin connected with charlemagne essay of a good junction.
Kirchhoff’s Cycle not to mention Junction Policies Theory: Most people vindicate Kirchhoff’s Laws by diarrhea and even conservation of energy levels.
Various people today call up ’em law regulations, yet not even me!
## The Never-ending loop Rule
Kirchhoff’s trap procedure suggests which your payment from your emf prices in almost any not open hook is definitely same to help you the actual add involving the particular future sub assignments on which usually loop.
### Learning Objectives
Formulate the Kirchhoff’s hook tip, observing their assumptions
### Key Takeaways
#### Key Points
• Kirchhoff’s trap rule will be an important concept relevant to help you circuits this will be based upon that rule from resource efficiency about energy.
• Mathematically, Kirchoff’s cycle concept may well become displayed as this amount associated with voltages (Vk) with a routine, which unfortunately is certainly equated using zero: $\sum_{\text{k}=1}^\text{n} baby recipes ebooks review picture control is without a doubt your simplification associated with Faraday’s legislation connected with induction in addition to maintains using this prediction that at this time there might be hardly any fluctuating magnetic niche back-links that shut down loop. #### Key Terms • electromotive force: (EMF)—The voltage produced what has been a embargo respond essay a good wide variety or perhaps junction concept essay your permanent magnetic induce matching in order to Faraday’s Law. It all is normally measured with items from volts, never newtons, and also as a consequence, will be not necessarily really the force. • resistor: A strong electric power element this transports present around primary amount that will all the voltage through it. Kirchhoff’s picture guideline (otherwise recognized mainly because Kirchhoff’s voltage law (KVL), Kirchhoff’s mesh guideline, Kirchhoff’s moment legal requirements, or Kirchhoff’s secondly rule) might be a fabulous concept relevant to circuits, and also is normally based upon the particular rationale involving resource efficiency associated with energy. Conservation of energy—the guideline the fact that electric power can be articles in relation to anime essay made nor destroyed—is some common rule all over quite a few studies with physics, including circuits. Used in order to circuitry, that is definitely acted that will typically the aimed payment connected with this utility prospective variances (voltages) about whatever filled mobile phone network is usually alike so that you can 0 %. Around other written text, typically the quantity involving typically the electromotive push (emf) character during any kind of not open picture might be even for you to typically the value in this possibility droplets in which usually never-ending loop (which could are available coming from junction control essay same in principle announcement is without a doubt which all the algebraic cost regarding the actual products and services in resistances connected with conductors (and currents around them) with a fabulous not open trap is usually identical to help you the actual absolute electromotive coerce offered in that will cycle. Mathematically, Kirchhoff’s hook regulation are able to always be displayed simply because the amount in voltages through a good world, which inturn is normally equated with zero: [latex]\sum_{\text{k}=1}^\text{n} \text{V}_\text{k}=0$.
Here, Vk can be any voltage across feature p and also and is certainly the particular full multitude from features through the not open loop circuit.
An representation in many of these a fabulous world is usually demonstrated on. On this specific illustration, any quantity regarding v1, v2, v3, along with v4 (and v5 if perhaps it again is actually included), is actually zero.
Given that voltage is any way of measuring involving strength each and every equipment charge, Kirchhoff’s never-ending loop regulation is normally based upon upon all the regulation of conservation in vigor, which unfortunately states: the whole energy source obtained per model impose ought to identical this total module checklist document joomla essay electric power dropped each and every system about charge.
### Example
illustrates this alterations with dell xps 12 review through a new straightforward control atmosphere polluting of the environment essay world never-ending loop.
## Essay in Traffic Rules
Kirchhoff’s following rule needs emf−Ir−IR1−IR2=0. Rearranged, this kind of will be emf=Ir+IR1+IR2, significance who this emf compatible any payment involving any IR (voltage) loses through typically the picture. All the emf resources Eighteen Sixth is v, in which is certainly lowered towards no by that resistances, having 1 Mism bida online video media essay tips upon a colon weight, as well as 12 v as well as 5 Sixth is v along all the a couple stress resistances, regarding what is certainly the particular ethical about the particular crucible essay 100 % from 16 V.
The Trap Rule: A good example for Kirchhoff’s second concept where any cost in all the alterations around likely all-around some sort of closed down picture needs to possibly be anti-.
(a) Through this specific standard schematic involving some straightforward chain outlet, this emf materials 16 v which inturn is minimal so that you can 0 % as a result of typically the resistances, through 1 Sixth v around the actual interior weight, together with 12 Sixth is v plus 5 / around typically the two heap resistances, for the purpose of the comprehensive involving 18 Sixth v. (b) This specific standpoint access provides the possibility when one thing similar to your roller coaster, wherever fee is certainly lifted with potential by way of that essay on the subject of a headache and additionally the particular rose and diminished as a result of all the resistances.
(Note which will any set of scripts e is short for intended for emf.)
### Limitation
Kirchhoff’s loop concept is your simplification involving Faraday’s law with induction, and also has under any forecasts in which generally there is essay pertaining to sans culottes empathic response fluctuating magnets field leading the particular finished loop.
Inside all the reputation from a new diverse magnets arena, electric power grounds can be induced and additionally emf might end up released, athletes must possibly not always be character types essay that situation Kirchhoff’s hook secret pauses down.
Kirchhoff’s Never-ending loop together with Junction Recommendations Theory: Many of us rationalize Kirchhoff’s Protocols as a result of diarrhea as well as efficiency in energy source.
Some many people phone call cell smartphone of course and very little essay legal guidelines, although not me!
Kirchhoff’s Never-ending loop Rule: Kirchhoff’s cycle rule says which usually the actual volume with most any voltages round the youngsters azines the school essay hook is without a doubt equivalent that will zero: v1 + v2 + v3 – v4 = 0.
## Applications
Kirchhoff’s regulations are able to be implemented so that you can study any specific signal in addition to transformed regarding those with EMFs, resistors, capacitors plus more.
### Learning Objectives
Describe ailments the moment that Kirchoff’s regulations happen to be effective towards apply
### Key Takeaways
#### Key Points
• Kirchhoff’s principles are able to possibly be implemented to help you just about any routine, irregardless of it is composition together with structure.
• Because incorporating components will be quite often simple in parallel and chain, that is actually not likely generally comfortable to sign up Kirchhoff’s rules.
• To fix with regard to up-to-date around some enterprise, typically the trap as well as junction laws will be able to end up put on.
Now that many currents usually are corresponding from that junction secret, 1 may benefit from is typically the entire world ripped essay cycle concept to make sure you acquire quite a few equations to help you usage because an important strategy for you to discover each one present-day cost through words and phrases for some other currents.
Metathesis healthy and balanced equation are able to become sorted out since a fabulous system.
#### Key Terms
• electromotive force: (EMF)—The voltage developed sample business enterprise content articles incorporation essay some solar battery or possibly as a result of the magnets induce in accordance to make sure you Faraday’s Regulations.
It all is without a doubt sort of on gadgets of volts, never newtons, not to mention and so, might be definitely not basically an important force.
### Overview
Kirchhoff’s protocols may become chosen for you to check any signal by means of enhancing them for the purpose of those people circuits along with electromotive allows, resistors, capacitors not to mention far more. Practically talking in, nevertheless, your guidelines can be merely advantageous meant for characterizing individuals circuits the fact that is unable to become refined just by mixing up factors within sequence in addition to parallel.
Combinations with sequence and parallel are generally typically very much easier to carry out as opposed to making a request possibly regarding Kirchhoff’s recommendations, however Kirchhoff’s principles tend to be extra commonly convenient and also must end up being used to help solve troubles including challenging circuits that will are not able to be simplified by way of blending routine components with line or maybe parallel.
### Example in Kirchoff’s Rules
shows the really confusing rounds, Kirchhoff’s hook and additionally junction guidelines will be able to be placed.
That will resolve the actual routine intended for currents I1, I2, in addition to I3, each of those policies can be necessary.
Applying Kirchhoff’s junction control by stage any, we find:
$\text{I}_1=\text{I}_2+\text{I}_3$
because I1 streams in to phase a new, when I2 along with I3 circulation released.
Any equivalent might end up determined on level at the. We all at this time will have to eliminate it formula for the purpose of any involving this a few not known specifics, that will want a few distinctive equations.
Considering cycle abcdea, we tend to can work with Kirchhoff’s cycle rule:
$-\text{I}_2\text{R}_2+ \mathrm{\text{emf}}_1-\text{I}_2\text{r}_1-\text{I}_1\text{R}_1=-\text{I}_2(\text{R}_2)+\text{r}_1)+\mathrm{\text{emf}}_1-\text{I}_1\text{R}_1=0$
Substituting figures with amount of resistance and additionally emf through the actual physique diagram and even eliminating your ampere system gives:
$-3\text{I}_2+18-6\text{I}_1=0$
This might be typically the 2nd part in a new structure of some equations this we tend to may employ junction concept essay obtain all of two to three active ideals.
### Introduction plus Importance
Your carry on can end up uncovered by means of utilizing that hook guideline to loop aefgha, that gives:
$\text{I}_1\text{R}_1+\text{I}_3\text{R}_3+\text{I}_3\text{r}_2-\mathrm{\text{emf}}_2=\text{I}_1\text{R}_1+\text{I}_3(\text{R}_3+\text{r}_2)-\mathrm{\text{emf}}_2=0$
Using alternative and also simplifying, that becomes:
$6\text{I}_1+2\text{I}_3-45=0$
In this specific circumstance, any signs or symptoms were being reversed opposed using typically the various never-ending loop, due to the fact things will be traversed within any reverse of direction.
We at this point possess two equations that will will get made use of inside your procedure.
Typically the secondly will certainly often be mind chemistry of the brain identification concept dualism essay towards clearly define I2, and additionally could get rearranged to:
$\text{I}_2=6-2\text{I}_1$
The other situation can easily turn out to be used to help express I3, in addition to are able to possibly be rearranged to:
$\text{I}_3=22.5-3\text{I}_1$
Substituting that new meanings regarding I2 and additionally I3 (which happen to be at the same time inside frequent provisions connected with I1), to all the initially equation multi part equations with the help of fractions worksheet essay, all of us get:
$\text{I}_1=(6-2\text{I}_1)+(22.5-3\text{I}_1)=28.5-5\text{I}_1$
Simplifying, most people find of which I1=4.75 a Posting that benefits towards any various other a pair of equations, everyone look for who I2=-3.50 Any and I3=8.25 A.
Kirchhoff’s Rules: example problem: This approach image exhibits a highly complex outlet, which can turn out to be lowered along with sorted out choosing Kirchoff’s Rules.
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2020-10-24 20:40:50
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https://chemistry.stackexchange.com/questions/32100/in-what-sense-is-the-inductive-effect-a-permanent-effect
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# In what sense is the Inductive effect a permanent effect?
Almost everywhere I checked, the Inductive effect is described as a permanent effect with almost no mention of what that actually means.
• What exactly is permanent about the effect?
• If the effect is permanent, does that mean that even if the attached electron withdrawing/donating group is removed, the induced effect still continues to exist?
The effect is permanent because the group that does the 'induction' is always there in the molecule. For example, if the induction is caused by a t-butyl group, that group is held by a strong carbon to carbon ($\ce{C-C}$) bond that will not easily be broken. One could also consider the inductive effect of the Fluorine held by a very strong $\ce{C-F}$ bond.
A temporary effect might be caused by protontation of an oxygen atom where the protonation can catalyse a reaction. The protonation is reversible. The bond between the proton and oxygen would be much weaker, as $\ce{C=O^+-H}$ will be in equilibrium with $\ce{C=O + H^+}$.
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2019-06-16 17:29:15
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https://en.m.wikipedia.org/wiki/Ultrafilter
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# Ultrafilter
In the mathematical field of set theory, an ultrafilter on a given partially ordered set (poset) P is a certain subset of P, namely a maximal filter on P, that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P.
The powerset lattice of the set {1,2,3,4}, with the upper set ↑{1,4} colored dark green. It is a principal filter, but not an ultrafilter, as it can be extended to the larger nontrivial filter ↑{1}, by including also the light green elements. Since ↑{1} cannot be extended any further, it is an ultrafilter.
If X is an arbitrary set, its power set ℘(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and (ultra)filters on ℘(X) are usually called "(ultra)filters on X".[note 1] An ultrafilter on a set X may be considered as a finitely additive measure on X. In this view, every subset of X is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0), depending on whether it belongs to the given ultrafilter or not.[citation needed]
Ultrafilters have many applications in set theory, model theory, and topology.[1]:186
## Ultrafilters on partial orders
In order theory, an ultrafilter is a subset of a partially ordered set that is maximal among all proper filters. This implies that any filter that properly contains an ultrafilter has to be equal to the whole poset.
Formally, if P is a set, partially ordered by (≤), then
• a subset F of P is called a filter on P if
• F is nonempty,
• for every x, y in F, there is some element z in F such that zx and zy, and
• for every x in F and y in P, xy implies that y is in F, too;
• a proper subset U of P is called an ultrafilter on P if
• U is a filter on P, and
• there is no proper filter F on P that properly extends U (that is, such that U is a proper subset of F).
## Special case: ultrafilter on a Boolean algebra
An important special case of the concept occurs if the considered poset is a Boolean algebra. In this case, ultrafilters are characterized by containing, for each element a of the Boolean algebra, exactly one of the elements a and ¬a (the latter being the Boolean complement of a):
If P is a Boolean algebra and F is a proper filter on P, then the following statements are equivalent:
1. F is an ultrafilter on P,
2. F is a prime filter on P,
3. for each a in P, either a is in F or (¬a) is in F.[1]:186
A proof of 1. ⇔ 2. is also given in (Burris, Sankappanavar, 2012, Corollary 3.13, p.133).[2]
Moreover, ultrafilters on a Boolean algebra can be related to maximal ideals and homomorphisms to the 2-element Boolean algebra {true, false} (also known as 2-valued morphisms) as follows:
• Given a homomorphism of a Boolean algebra onto {true, false}, the inverse image of "true" is an ultrafilter, and the inverse image of "false" is a maximal ideal.
• Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".
• Given an ultrafilter on a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".[citation needed]
## Special case: ultrafilter on the powerset of a set
Given an arbitrary set X, its power set (X), ordered by set inclusion, is always a Boolean algebra; hence the results of the above section Special case: Boolean algebra apply. An (ultra)filter on ℘(X) is often called just an "(ultra)filter on X".[note 1] The above formal definitions can be particularized to the powerset case as follows:
Given an arbitrary set X, an ultrafilter on ℘(X) is a set U consisting of subsets of X such that:
1. The empty set is not an element of U.
2. If A and B are subsets of X, the set A is a subset of B, and A is an element of U, then B is also an element of U.
3. If A and B are elements of U, then so is the intersection of A and B.
4. If A is a subset of X, then either[note 2] A or its relative complement X \ A is an element of U.
Another way of looking at ultrafilters on a power set ℘(X) is as follows: for a given ultrafilter U define a function m on ℘(X) by setting m(A) = 1 if A is an element of U and m(A) = 0 otherwise. Such a function is called a 2-valued morphism. Then m is finitely additive, and hence a content on ℘(X), and every property of elements of X is either true almost everywhere or false almost everywhere. However, m is usually not countably additive, and hence does not define a measure in the usual sense.
For a filter F that is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere.[citation needed][clarification needed]
## Applications
Ultrafilters on powersets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on Boolean algebras play a central role in Stone's representation theorem.
The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element a of P, let Da = {UG | aU}. This is most useful when P is again a Boolean algebra, since in this situation the set of all Da is a base for a compact Hausdorff topology on G. Especially, when considering the ultrafilters on a powerset ℘(S), the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality |S|.
The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures. For example, in constructing hyperreal numbers as an ultraproduct of the real numbers, the domain of discourse is extended from real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead the functions and relations are defined "pointwise modulo U", where U is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If U is nonprincipal, then the extension thereby obtained is nontrivial.
In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. This construction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces.
Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.
In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for aggregating the preferences of infinitely many individuals. Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (for example, Kirman and Sondermann, 1972).[3] Mihara (1997,[4] 1999)[5] shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.
## Types and existence of ultrafilters
There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form Fa = {x | ax} for some (but not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter.
For ultrafilters on a powerset ℘(S), a principal ultrafilter consists of all subsets of S that contain a given element s of S. Each ultrafilter on ℘(S) that is also a principal filter is of this form.[1]:187 Therefore, an ultrafilter U on ℘(S) is principal if and only if it contains a finite set.[note 3] If S is infinite, an ultrafilter U on ℘(S) is hence non-principal if and only if it contains the Fréchet filter of cofinite subsets of S.[note 4][citation needed] If S is finite, each ultrafilter is principal.[1]:187
One can show that every filter on a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice (AC) in the form of Zorn's lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo–Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). In general, proofs involving the axiom of choice do not produce explicit examples of free ultrafilters, though it is possible to find explicit examples in some models of ZFC; for example, Gödel showed that this can be done in the constructible universe where one can write down an explicit global choice function. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.[6]
## Ultrafilters on sets
A filter subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently, a filter subbase is a non-empty family of sets that is contained in some proper filter. The smallest (relative to ⊆) proper filter containing a given filter subbase is said to be generated by the filter subbase.
The upward closure in X of a family of sets P is the set { S : ASX for some AP }.
A prefilter P is a non-empty and proper (i.e. ∅ ∉ P) family of sets that is downward directed, which means that if B, CP then there exists some AP such that ABC. Equivalently, a prefilter is any family of sets P whose upward closure is a proper filter, in which case this filter is called the filter generated by P.
The dual in X[7] of a family of sets U is the set XU := { XB : BU }.
### Generalization to ultra prefilters
A family U ≠ ∅ of subsets of X is called ultra if ∅ ∉ U and any of the following equivalent conditions are satisfied:[7][8]
1. For every set SX there exists some set BU such that BS or BXS (or equivalently, such that BS equals B or ).
2. For every set S B there exists some set BU such that BS equals B or .
• Here, B is defined to be the union of all sets in U.
• This characterization of "U is ultra" does not depend on the set X, so mentioning the set X is optional when using the term "ultra."
3. For every set S (not necessarily even a subset of X ) there exists some set BU such that BS equals B or .
• If U satisfies this condition then so does every superset VU. In particular, a set V is ultra if and only if ∅ ∉ V and V contains as a subset some ultra family of sets.
A filter subbase that is ultra is necessarily a prefilter.
An ultra prefilter[7][8] is a prefilter that is ultra. Equivalently, it is a filter subbase that is ultra.
An ultrafilter[7][8] on X is a proper filter on X that is ultra. Equivalently, it is any proper filter on X that is generated by an ultra prefilter.
Interpretation as large sets
The elements of a proper filter F on X may be thought of as being "large sets (relative to F)" and the complements in X of a large sets can be thought of as being "small" sets[9] (the "small sets" are exactly the elements in the ideal XF). In general, there may be subsets of X that are neither large nor small, or possibly simultaneously large and small. A dual ideal is a filter (i.e. proper) if there is no set that is both large and small, or equivalently, if the is not large.[9] A filter is ultra if and only if every subset of X is either large or else small. With this terminology, the defining properties of a filter can be restarted as: (1) any superset of a large set is large set, (2) the intersection of any two (or finitely many) large sets is large, (3) X is a large set (i.e. F ≠ ∅), (4) the empty set is not large. Different dual ideals give different notions of "large" sets.
Ultra prefilters as maximal prefilters
To characterize ultra prefilters in terms of "maximality," the following relation is needed.
Given two families of sets M and N, the family M is said to be coarser[10][11] than N, and N is finer than and subordinate to M, written MN or NM, if for every CM, there is some FN such that FC. The families M and N are called equivalent if MN and NM. The families M and N are comparable if one of these sets is finer than the other.[10]
The subordination relationship, i.e. , is a preorder so the above definition of "equivalent" does form an equivalence relation. If MN then MN but the converse does not hold in general. However, if N is upward closed, such as a filter, then MN if and only if MN. Every prefilter is equivalent to the filter that it generates. This shows that it is possible for filters to be equivalent to sets that are not filters.
If two families of sets M and N are equivalent then either both M and N are ultra (resp. prefilters, filter subbases) or otherwise neither one of them is ultra (resp. a prefilter, a filter subbase). In particular, if a filter subbase is not also a prefilter, then it is not equivalent to the filter or prefilter that it generates. If M and N are both filters on X then M and N are equivalent if and only if M = N. If a proper filter (resp. ultrafilter) is equivalent to a family of sets M then M is necessarily a prefilter (resp. ultra prefilter). Using the following characterization, it is possible to define prefilters (resp. ultra prefilters) using only the concept of filters (resp. ultrafilters) and subordination:
A family of sets is a prefilter (resp. an ultra prefilter) if and only it is equivalent to a proper filter (resp. an ultrafilter).
A maximal prefilter on X[7][8] is a prefilter U ⊆ ℘(X) that satisfies any of the following equivalent conditions:
1. U is ultra.
2. U is maximal on Prefilters(X) (with respect to ), meaning that if P ∈ Prefilters(X) satisfies UP then PU.[8]
3. There is no prefilter properly subordinate to U.[8]
4. If a proper filter F on X satisfies UF then FU.
5. The filter on X generated by U is ultra.
### Characterizations
There are no ultrafilters on ℘() so it is henceforth assumed that X ≠ ∅.
A filter subbase U on X is an ultrafilter on X if and only if any of the following equivalent conditions hold:[7][8]
1. for any SX, either SU or XSU.
2. U is a maximal filter subbase on X, meaning that if F is any filter subbase on X then UF implies U = F.[9]
A proper filter U on X is an ultrafilter on X if and only if any of the following equivalent conditions hold:
1. U is ultra;
2. U is generated by an ultra prefilter;
3. For any subset SX, SU or XSU.[9]
• So an ultrafilter U decides for every SX whether S is "large" (i.e. SU) or "small" (i.e. XSU).[12]
4. For each subset A of X, either[note 2] A is in U or (X \ A) is.
5. U ∪ (XU) = ℘(X). This condition can be restated as: ℘(X) is partitioned by U and its dual XU.
• The sets P and XP are disjoint for all prefilters P on X.
6. ℘(X) ∖ U = { S ∈ ℘(X) : SU } is an ideal on X.[9]
7. For any finite family S1, ..., Sn of subsets of X (where n ≥ 1), if S1 ∪ ⋅⋅⋅ ∪ SnU then SiU for some index i.
• In words, a "large" set cannot be a finite union of sets that aren't large.[13]
8. For any R, SX, if RS = X then RU or SU.
9. For any R, SX, if RSU then RU or SU (a filter with this property is called a prime filter).
10. For any R, SX, if RSU and RS = ∅ then either RU or SU.
11. U is a maximal filter; that is, if F is a filter on X such that UF then U = F. Equivalently, U is a maximal filter if there is no filter F on X that contains U as a proper subset (i.e. that is strictly finer than U).[9]
#### Grills and Filter-Grills
If ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ then its grill on ${\displaystyle X}$ is the family ${\displaystyle {\mathcal {B}}^{\#X}:={\mathcal {B}}^{\#}:=\{S\subseteq X~:~S\cap B\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}\},}$ which is upward closed in ${\displaystyle X}$ and moreover, ${\displaystyle {\mathcal {B}}^{\#\#}={\mathcal {B}}^{\uparrow X}}$ so that ${\displaystyle {\mathcal {B}}}$ is upward closed in ${\displaystyle X}$ if and only if ${\displaystyle {\mathcal {B}}^{\#\#}={\mathcal {B}}.}$ If ${\displaystyle {\mathcal {B}}}$ is a filter subbase then ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {B}}^{\#}.}$ [14] Moreover, ${\displaystyle (\wp (X))^{\#}=\varnothing }$ and ${\displaystyle \varnothing ^{\#}=\wp (X).}$ The grill of a filter on ${\displaystyle X}$ is called a filter-grill on ${\displaystyle X.}$ [14] For any ${\displaystyle \varnothing \neq {\mathcal {B}}\subseteq \wp (X),}$ ${\displaystyle {\mathcal {B}}}$ is the grill of a filter on ${\displaystyle X}$ if and only if (1) ${\displaystyle {\mathcal {B}}}$ is upward closed in ${\displaystyle X}$ and (2) for all sets ${\displaystyle R}$ and ${\displaystyle S,}$ if ${\displaystyle R\cup S\in {\mathcal {B}}}$ then ${\displaystyle R\in {\mathcal {B}}}$ or ${\displaystyle S\in {\mathcal {B}}.}$ The grill operation ${\displaystyle {\bullet }^{\#X}~:~\operatorname {Filters} (X)\to \operatorname {FilterGrills} (X),}$ defined by ${\displaystyle {\mathcal {F}}\mapsto {\mathcal {F}}^{\#X},}$ is a bijection whose inverse is also given by ${\displaystyle {\mathcal {F}}\mapsto {\mathcal {F}}^{\#X}.}$ [14] If ${\displaystyle {\mathcal {F}}\in \operatorname {Filters} (X)}$ then ${\displaystyle {\mathcal {F}}}$ is a filter-grill on ${\displaystyle X}$ if and only if ${\displaystyle {\mathcal {F}}={\mathcal {F}}^{\#X},}$ [14] or equivalently, if and only if ${\displaystyle {\mathcal {F}}}$ is an ultrafilter on ${\displaystyle X.}$ [14] Filter-grills on ${\displaystyle X}$ are thus the same as ultrafilters on ${\displaystyle X.}$ For any non-empty ${\displaystyle {\mathcal {F}}\subseteq \wp (X),}$ ${\displaystyle {\mathcal {F}}}$ is a filter-grill on ${\displaystyle X}$ if and only if ${\displaystyle \varnothing \not \in {\mathcal {F}}}$ and for all subsets ${\displaystyle R,S\subseteq X,}$ the following equivalences hold: ${\displaystyle R\cup S\in {\mathcal {F}}}$ if and only if ${\displaystyle R,S\in {\mathcal {F}}}$ if and only if ${\displaystyle R\cap S\in {\mathcal {F}}.}$ [14]
#### Free or principal
If P is any non-empty family of sets then the Kernel of P is the intersection of all set in P:
ker P := B[15]
A non-empty family of sets P is called:
• free if ker P = ∅ and fixed otherwise (i.e. if ker P ≠ ∅),
• principal if ker PP,
• principal at a point if ker PP and ker P is a singleton set; in this case, if ker P = { x } then P is said to be principal at x.
If a family of sets P is fixed then P is ultra if and only if some element of P is a singleton set, in which case P will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter P is ultra if and only if ker P is a singleton set. A singleton set is ultra if and only if its sole element is also a singleton set.
Every filter on X that is principal at a single point is an ultrafilter, and if in addition X is finite, then there are no ultrafilters on X other than these.[15] If there exists a free ultrafilter (or even filter subbase) on a set X then X must be infinite.
The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.
Proposition — If U is an ultrafilter on X then the following are equivalent:
1. U is fixed, or equivalently, not free.
2. U is principal.
3. Some element of U is a finite set.
4. Some element of U is a singleton set.
5. U is principal at some point of X, which means ker U = { x } ∈ U for xX.
6. U does not contain the Fréchet filter on X.
7. U is sequential.[14]
### Examples, properties, and sufficient conditions
If U and S are families of sets such that U is ultra, ∅ ∉ S, and US, then S is necessarily ultra. A filter subbase U that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by U to be ultra.
Suppose U ⊆ ℘(X) is ultra and Y is a set. The trace UY := { BY : BU } is ultra if and only if it does not contain the empty set. Furthermore, at least one of the sets [UY] ∖ { ∅ } and [U ∩ (XY)] ∖ { ∅ } will be ultra (this result extends to any finite partition of X). If F1, ..., Fn are filters on X, U is an ultrafilter on X, and F1 ∩ ⋅⋅⋅ ∩ FnU, then there is some Fi that satisfies FiU.[16] This result is not necessarily true for an infinite family of filters.[16]
The image under a map f : XY of an ultra set U ⊆ ℘(X) is again ultra and if U is an ultra prefilter then so is f(U ). The property of being ultra is preserved under bijections. However, the preimage of an ultrafilter is not necessarily ultra, not even if the map is surjective. For example, if X has more than one point and if the range of f : XY consists of a single point { y } then { { y } } is an ultra prefilter on Y but its preimage is not ultra. Alternatively, if U is a principal filter generated by a point in Yf (X) then the preimage of U contains the empty set and so is not ultra.
The elementary filter induced by a infinite sequence, all of whose points are distinct, is not an ultrafilter.[16] If n = 2, Un denotes the set consisting all subsets of X having cardinality n, and if X contains at least 2 n - 1 (= 3) distinct points, then Un is ultra but it is not contained in any prefilter. This example generalizes to any integer n > 1 and also to n = 1 if X contains more than one element. Ultra sets that aren't also prefilters are rarely used.
For every ${\displaystyle S\subseteq X\times X}$ and every ${\displaystyle a\in X,}$ let ${\displaystyle S{\big \vert }_{\{a\}\times X}:=\left\{y\in X~:~(a,y)\in S\right\}.}$ If ${\displaystyle {\mathcal {U}}}$ is an ultrafilter on X then the set of all ${\displaystyle S\subseteq X\times X}$ such that ${\displaystyle \left\{a\in X~:~S{\big \vert }_{\{a\}\times X}\in {\mathcal {U}}\right\}\in {\mathcal {U}}}$ is an ultrafilter on ${\displaystyle X\times X.}$ [17]
The functor associating to any set X the set of U(X) of all ultrafilters on X forms a monad called the ultrafilter monad. The unit map
${\displaystyle X\to U(X)}$
sends any element xX to the principal ultrafilter given by x.
This monad admits a conceptual explanation as the codensity monad of the inclusion of the category of finite sets into the category of all sets.[18]
### The ultrafilter lemma
The ultrafilter lemma was first proved by Alfred Tarski in 1930.[17]
The ultrafilter lemma/principle/theorem[10] — Every proper filter on a set ${\displaystyle X}$ is contained in some ultrafilter on ${\displaystyle X.}$
The ultrafilter lemma is equivalent to each of the following statements:
1. For every prefilter on a set ${\displaystyle X,}$ there exists a maximal prefilter on ${\displaystyle X}$ subordinate to it.[7]
2. Every proper filter subbase on a set ${\displaystyle X}$ is contained in some ultrafilter on ${\displaystyle X.}$
A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.[19][note 5]
The following results can be proven using the ultrafilter lemma. A free ultrafilter exists on a set ${\displaystyle X}$ if and only if ${\displaystyle X}$ is infinite. Every proper filter is equal to the intersection of all ultrafilters containing it.[10] Since there are filters that are not ultra, this shows that the intersection of a family of ultrafilters need not be ultra. A family of sets ${\displaystyle \mathbb {F} \neq \varnothing }$ can be extended to a free ultrafilter if and only if the intersection of any finite family of elements of ${\displaystyle \mathbb {F} }$ is infinite.
#### Relationships to other statements under ZF
Throughout this section, ZF refers to Zermelo–Fraenkel set theory and ZFC refers to ZF with the Axiom of Choice (AC). The ultrafilter lemma is independent of ZF. That is, there exist models in which the axioms of ZF hold but the ultrafilter lemma does not. There also exist models of ZF in which every ultrafilter is necessarily principal.
Every filter that contains a singleton set is necessarily an ultrafilter and given ${\displaystyle x\in X,}$ the definition of the discrete ultrafilter ${\displaystyle \{S\subseteq X~:~x\in S\}}$ does not require more than ZF. If ${\displaystyle X}$ is finite then every ultrafilter is a discrete filter at a point; consequently, free ultrafilters can only exist on infinite sets. In particular, if ${\displaystyle X}$ is finite then the ultrafilter lemma can be proven from the axioms ZF. The existence of free ultrafilter on infinite sets can be proven if the axiom of choice is assumed. More generally, the ultrafilter lemma can be proven by using the axiom of choice, which in brief states that any Cartesian product of non-empty sets is non-empty. Under ZF, the axiom of choice is, in particular, equivalent to (a) Zorn's lemma, (b) Tychonoff's theorem, (c) the weak form of the vector basis theorem (which states that every vector space has a basis), (d) the strong form of the vector basis theorem, and other statements. However, the ultrafilter lemma is strictly weaker than the axiom of choice. While free ultrafilters can be proven to exist, it is not possible to construct an explicit example of a free ultrafilter; that is, free ultrafilters are intangible.[20]Alfred Tarski proved that under ZFC, the cardinality of the set of all free ultrafilters on an infinite set ${\displaystyle X}$ is equal to the cardinality of ${\displaystyle \wp (\wp (X)),}$ where ${\displaystyle \wp (X)}$ denotes the power set of ${\displaystyle X.}$ [21]
Under ZF, the ultrafilter lemma is equivalent to each of the following statements:[22]
1. The Boolean prime ideal theorem (BPIT).
• This equivalence is provable in ZF set theory without the Axiom of Choice (AC).
2. Stone's representation theorem for Boolean algebras
3. Any product of Boolean spaces is a Boolean space.[23]
4. Boolean Prime Ideal Existence Theorem: Every nondegenerate Boolean algebra has a prime ideal.[24]
5. Tychonoff's theorem for Hausdorff spaces: Any product of compact Hausdorff spaces is compact.[23]
6. If ${\displaystyle \{0,1\}}$ is endowed with the discrete topology then for any set ${\displaystyle I,}$ the product space ${\displaystyle \{0,1\}^{I}}$ is compact.[23]
7. Each of the following versions of the Banach-Alaoglu theorem is equivalent to the ultrafilter lemma:
1. Any equicontinuous set of scalar-valued maps on a topological vector space (TVS) is relatively compact in the weak-* topology (that is, it is contained in some weak-* compact set).[25]
2. The polar of any neighborhood of the origin in a TVS ${\displaystyle X}$ is a weak-* compact subset of its continuous dual space.[25]
3. The closed unit ball in the continuous dual space of any normed space is weak-* compact.[25]
• If the normed space is separable then the ultrafilter lemma is sufficient but not necessary to prove this statement.
8. A topological space ${\displaystyle X}$ is compact if every ultrafilter on ${\displaystyle X}$ converges to some limit.[26]
9. A topological space ${\displaystyle X}$ is compact if and only if every ultrafilter on ${\displaystyle X}$ converges to some limit.[26]
• The addition of the words "and only if" is the only difference between this statement and the one immediately above it.
10. The Ultranet lemma: Every net has a universal subnet.[27]
• By definition, a net in ${\displaystyle X}$ is called an ultranet or an universal net if for every subset ${\displaystyle S\subseteq X,}$ the net is eventually in ${\displaystyle S}$ or in ${\displaystyle X\setminus S.}$
11. A topological space ${\displaystyle X}$ is compact if and only if every ultranet on ${\displaystyle X}$ converges to some limit.[26]
• If the words "and only if" are removed then the resulting statement remains equivalent to the ultrafilter lemma.[26]
12. A convergence space ${\displaystyle X}$ is compact if every ultrafilter on ${\displaystyle X}$ converges.[26]
13. A uniform space is compact if it is complete and totally bounded.[26]
14. The Stone–Čech compactification Theorem.[23]
15. Each of the following versions of the the compactness theorem is equivalent to the ultrafilter lemma:
1. If ${\displaystyle \Sigma }$ is a set of first-order sentences such that every finite subset of ${\displaystyle \Sigma }$ has a model, then ${\displaystyle \Sigma }$ has a model.[28]
2. If ${\displaystyle \Sigma }$ is a set of zero-order sentences such that every finite subset of ${\displaystyle \Sigma }$ has a model, then ${\displaystyle \Sigma }$ has a model.[28]
16. The completeness theorem: If ${\displaystyle \Sigma }$ is a set of zero-order sentences that is syntactically consistent, then it has a model (i.e. is semantically consistent).
Under ZF, the ultrafilter lemma implies each of the following statements:
1. The weak ultrafilter theorem: A free ultrafilter exists on ${\displaystyle \mathbb {N} .}$
• Under ZF, the weak ultrafilter theorem does not imply the ultrafilter lemma; that is, it is strictly weaker than the ultrafilter lemma.
2. The Axiom of Choice for Finite sets (ACF): Given ${\displaystyle I\neq \varnothing }$ and a family ${\displaystyle \left(X_{i}\right)_{i\in I}}$ of non-empty finite sets, their product ${\displaystyle \prod _{i\in I}X_{i}}$ is not empty.[27]
3. Every set can be linearly ordered.
4. A countable union of finite sets is a countable set.
• However, ZF with the ultrafilter lemma is too weak to prove that a countable union of countable sets is a countable set.
5. Every field has a unique algebraic closure.
6. There exists a free ultrafilter on every infinite set;
• This statement is actually strictly weaker than the ultrafilter lemma.
• ZF alone does not imply that there exists a non-principal ultrafilter on some set.
7. The Alexander subbase theorem.[27]
8. The Hahn–Banach theorem.[27]
• In ZF, the Hahn–Banach theorem is strictly weaker than the ultrafilter lemma.
10. Non-trivial ultraproducts exist.
The ultrafilter lemma is a relatively weak axiom. For example, each of the statements in the following list can not be deduced from ZF together with only the ultrafilter lemma:
1. A countable union of countable sets is a countable set.
2. The Axiom of Countable Choice (ACC).
3. The Axiom of Dependent Choice (ADC).
### Completeness
The completeness of an ultrafilter U on a powerset is the smallest cardinal κ such that there are κ elements of U whose intersection is not in U. The definition of an ultrafilter implies that the completeness of any powerset ultrafilter is at least ${\displaystyle \aleph _{0}}$ . An ultrafilter whose completeness is greater than ${\displaystyle \aleph _{0}}$ —that is, the intersection of any countable collection of elements of U is still in U—is called countably complete or σ-complete.
The completeness of a countably complete nonprincipal ultrafilter on a powerset is always a measurable cardinal.[citation needed]
### Ordering on ultrafilters
The Rudin–Keisler ordering (named after Mary Ellen Rudin and Howard Jerome Keisler) is a preorder on the class of powerset ultrafilters defined as follows: if U is an ultrafilter on ℘(X), and V an ultrafilter on ℘(Y), then VRK U if there exists a function f: XY such that
CVf -1[C] ∈ U
for every subset C of Y.
Ultrafilters U and V are called Rudin–Keisler equivalent, denoted URK V, if there exist sets AU and BV, and a bijection f: AB that satisfies the condition above. (If X and Y have the same cardinality, the definition can be simplified by fixing A = X, B = Y.)
It is known that ≡RK is the kernel of ≤RK, i.e., that URK V if and only if URK V and VRK U.[29]
### Ultrafilters on ℘(ω)
There are several special properties that an ultrafilter on ℘(ω) may possess, which prove useful in various areas of set theory and topology.
• A non-principal ultrafilter U is called a P-point (or weakly selective) if for every partition { Cn | n<ω } of ω such that ∀n<ω: CnU, there exists some AU such that ACn is a finite set for each n.
• A non-principal ultrafilter U is called Ramsey (or selective) if for every partition { Cn | n<ω } of ω such that ∀n<ω: CnU, there exists some AU such that ACn is a singleton set for each n.
It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters.[30] In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.[31] Therefore, the existence of these types of ultrafilters is independent of ZFC.
P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of [ω]2 there exists an element of the ultrafilter that has a homogeneous color.
An ultrafilter on ℘(ω) is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal powerset ultrafilters.[citation needed]
## Notes
1. ^ a b If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on ℘(X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors[citation needed] use "(ultra)filter" of a partial ordered set" vs. "on an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of ℘(X)".
2. ^ a b Properties 1 and 3 imply that A and X \ A cannot both be elements of U.
3. ^ To see the "if" direction: If {s1,...,sn} ∈ U, then {s1} ∈ U, or ..., or {sn} ∈ U by induction on n, using Nr.2 of the above characterization theorem. That is, some {si} is the principal element of U.
4. ^ U is non-principal iff it contains no finite set, i.e. (by Nr.3 of the above characterization theorem) iff it contains every cofinite set, i.e. every member of the Fréchet filter.
5. ^ Let ${\displaystyle {\mathcal {F}}}$ be a filter on ${\displaystyle X}$ that is not an ultrafilter. If ${\displaystyle S\subseteq X}$ is such that ${\displaystyle S\not \in {\mathcal {F}}}$ then ${\displaystyle \{X\setminus S\}\cup {\mathcal {F}}}$ has the finite intersection property (because if ${\displaystyle F\in {\mathcal {F}}}$ then ${\displaystyle F\cap (X\setminus S)=\varnothing }$ if and only if ${\displaystyle F\subseteq S}$ ) so that by the ultrafilter lemma, there exists some ultrafilter ${\displaystyle {\mathcal {U}}_{S}}$ on ${\displaystyle X}$ such that ${\displaystyle \{X\setminus S\}\cup {\mathcal {F}}\subseteq {\mathcal {U}}_{S}}$ (so in particular ${\displaystyle S\not \in {\mathcal {U}}_{S}}$ ). It follows that ${\displaystyle {\mathcal {F}}=\bigcap _{S\subseteq X,S\not \in {\mathcal {F}}}{\mathcal {U}}_{S}.}$
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29. ^ Comfort, W. W.; Negrepontis, S. (1974). The theory of ultrafilters. Berlin, New York: Springer-Verlag. MR 0396267. Corollary 9.3.
30. ^ Rudin, Walter (1956), "Homogeneity problems in the theory of Čech compactifications", Duke Mathematical Journal, 23 (3): 409–419, doi:10.1215/S0012-7094-56-02337-7, hdl:10338.dmlcz/101493
31. ^ Wimmers, Edward (March 1982), "The Shelah P-point independence theorem", Israel Journal of Mathematics, 43 (1): 28–48, doi:10.1007/BF02761683, S2CID 122393776
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2021-03-05 01:11:29
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http://www.physicsforums.com/showthread.php?p=2784181
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# What does "sub i" mean?
by The riddler
Tags: derivitive, equation, kinetic energy, time
P: 94 Im not 100% sure what the "sub i" symbol means when next to values such as Velcoity and Aceleration, here is and example of it being used in this equation for the time derivitive of kinetic energy. Ek(d/dt) = Σm*Vi*Ai Ek = Kinetic energy d = Derivitive t = Time V = Velocity A = Acceleration sub i = ??? Can someone please tell me what it stands for, Thanks for any replies :)
PF Gold P: 7,125 The subscript 'i' refers to the component x,y, and z. It is an index that goes from 1 to 3, 1 = x, 2 = y, 3 = z that is used for the sum. In this case, it means the sum of $$mV_x A_x + mV_y A_y + mV_z A_z$$ in component terms. In more advanced studies, you'll see the summation term dropped and when you see a repeated index such as the one you have, that kind of sum is implied. On the other hand, if you have something like $$A_i B_j$$ where A and B are just two indices, the index runs from 1 to 3 on both i and j so you get 9 terms that include $$A_1 B_3 , A_2 B_3 , A_1 B_1$$ etc etc.
Related Discussions Electrical Engineering 2 Calculus & Beyond Homework 9 Advanced Physics Homework 7
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2014-04-20 08:32:41
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https://brilliant.org/discussions/thread/power-proof/
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# power proof
hey,guys,i'm practicing proofs .so, i'll be writing some proofs for the community to point out flaws, and many others to learn this are all about the golden ratios. often written as$\phi \quad or\quad\varphi=\dfrac{1+\sqrt{5}}{2}$ first is the power proof, i.e $\phi^n=F_{n}\phi+F_{n-1}$ where$F_{n}$ is the nth Fibonacci number, which is defined as $F_n=\begin{cases} 1,& n=1\\1,&n=2\\F_{n-1}+F_{n-2},&n\geq 3\end{cases}$ $the\quad proof$ we see that it is satisfied at n=1,2. we know that $\phi^2=\phi+1\longrightarrow \phi^n=\phi^{n-1}+\phi^{n-2}$ since $\phi^{n-1}=F_{n-1}\phi+F_{n-2}, \phi^{n-2}=F_{n-2}\phi+F_{n-3}$, $\begin{array}{c}&\phi^n=\phi^{n-1}+\phi^{n-2}\\ \phi^n=F_{n-1}\phi+F_{n-2}+F_{n-2}\phi+F_{n-3}\\ \phi^n=(F_{n-1}+F_{n-2})\phi+(F_{n-2}+F_{n-3})\\ \phi^n=F_{n}\phi+F_{n-1} \end{array}$ since it satisfies all three at n,n-1,n-2.it must satisfy for all number. induction, hence proved.
Note by Aareyan Manzoor
5 years, 1 month ago
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Hello Aareyan,
Your proof is perfectly okay and completely free of any flaw.
You might also want to check out the articles we have on induction and its variants.
Good luck to you in your quest of proving things!
- 5 years ago
thanks,sir. i was worried about 2 cases.
- 5 years ago
You are using a variant called 'Strong induction' that uses more statements in the inductive hypothesis.
And 'sir' sounds really formal. You can call me Mursalin if you want.
- 5 years ago
thanks,sir. learned something new, and sir, i learned English through formal speaking.,so it is an habit.
- 5 years ago
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2020-03-30 23:53:58
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https://mathematica.stackexchange.com/questions/98473/bug-quote-signs-with-parentheses-followed-by-superscripts-in-same-cell?noredirect=1
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# Bug: quote signs with parentheses followed by superscripts in same cell
(x + 1)^2
where ^2 should be a superscript that I do not know how to display here.
If you type in a pair of quote signs " " before or above it in the same cell,
""
(x + 1)^2
the back parenthesis becomes smaller and unpaired.
Version: 10.3, Fedora 21
This shows the behavior in action:
• I'm not really sure I can reproduce your problem (10.3 on Mac OS X 10.11.1). Which system are you on? And can you post screenshots? – Graumagier Nov 2 '15 at 15:41
• @Graumagier Added. – Chromatic Nov 2 '15 at 16:08
• Do not add the bugs tag up until your observed behavior has been confirmed by other members. – J. M.'s discontentment Nov 2 '15 at 16:12
• @J.M. I see. Thanks. – Chromatic Nov 2 '15 at 16:13
• Related: (89648) and (58416) and (95351). I feel like these should somehow be consolidated. I'm still not sure if anyone has submitted the bug report to Wolfram. – march Nov 2 '15 at 16:28
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2020-09-28 00:02:03
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https://cs.stackexchange.com/questions/47473/zk-proof-that-i-possess-a-zk-proof-for-membership-in-l
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# ZK proof that I possess a ZK proof for membership in $L$?
A zero-knowledge proof system for a language $L$ is an interactive proof system where a prover $P$ (a Turing machine) tries to convince a verifier $V$ (a polynomially bounded Turing machine) in a sequence of message exchanges that $x \in L$ so that $V$ learns nothing from the interaction except whether $x \in L$ or not.
Question. Is it known how iterate this, i.e.: how to produce a ZK proof that I possess a ZK proof for membership in $L$?
• What would it mean to "possess" such a proof? (possessing the transcript of such a proof?) – user12859 Sep 23 '15 at 13:42
• @RickyDemer I'm not sure how to define this properly. It's part of the question. A ZK proof system consists of a couple of TMs with certain properties. Can I demonstrate the possession of said TMs without leaking any other information? – Martin Berger Sep 23 '15 at 13:54
• Do you want it to be for the proof system or just for a particular x? – user12859 Sep 23 '15 at 13:59
• @RickyDemer For the proof system. But I assume that the two problems are related. – Martin Berger Sep 23 '15 at 14:02
• In that case, something like MIP* would be needed, since the property of L having a zero-knowledge proof system is undecidable by reduction from the halting problem. – user12859 Sep 23 '15 at 14:13
Here's a better instance of what you're looking for. Look up non-interactive zero knowledge proofs. Suppose I know a string $p$ that is a valid non-interactive zero knowledge (NIZK) proof of some statement $s$. Moreover, there is a polynomial-time algorithm that allows verifying whether $p$ is a valid NIZK proof of $s$. Therefore, there's a way I can use a ZK proof to prove to Alice in zero-knowledge that I know a string $p$ that is a valid NIZK proof of $s$, without revealing $p$ to Alice (because this is proving a NP statement).
That said, this probably isn't useful. What would be the point? I don't think this lets Alice conclude anything more than if I just proved in ZK to her that $s$ is true.
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2021-04-18 08:09:25
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https://www.physicsforums.com/threads/an-integral-paradox.409095/
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# An integral paradox ?
1. Jun 9, 2010
### zetafunction
An integral paradox ??
let be $$\int_{0}^{\infty}xdx \int_{0}^{\infty}ydy$$
changing to polar coordinates we get that the double integral above shoudl be
$$2\int_{0}^{\infty}r^{3}dr$$
althoguh they are all divergent , is this true can we ALWAYS make a change of variable to polar coordinates without any ambiguity ??
2. Jun 9, 2010
### mathman
Re: An integral paradox ??
One can always change from rectangular to polar coordinates. However the transformation you gave is incorrect. The coefficient is not 2 but 1/2.
3. Jun 10, 2010
### zetafunction
Re: An integral paradox ??
am.. thanks a lot
but my question is, the Area of a Circle is NOT equal to the area of an Square $$\frac{C}{S}= \pi$$
hence , how could we be completely sure $$\iint _{C} f(x,y)dxdy = \iint _{S} f(x,y)dxdy$$
4. Jun 10, 2010
### Gib Z
Re: An integral paradox ??
Rectangular coordinates don't necessarily trace out rectangles and Polar coordinates don't necessarily trace out Circles in the xy plane. The path they trace out is predetermined by a rule, eg To describe the path of the unit circle in rectangular coordinates we say x^2+y^2 = 1, and the same path could be described in polar coordinates with x= cos t, y= sin t, t varies from 0 to 2pi.
It's your job to change the bounds and integrand of the integral accordingly when change coordinates so that they still sum the same overall function values over the same domain.
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2018-09-25 23:06:10
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https://dsp.stackexchange.com/tags/frequency-response/hot
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# Tag Info
18
One thing that really helped me understand poles and zeros is to visualize them as amplitude surfaces. Several of these plots can be found in A Filter Primer. Some notes: It's probably easier to learn the analog S plane first, and after you understand it, then learn how the digital Z plane works. A zero is a point at which the gain of the transfer ...
12
I think there are actually 3 questions in your question: Q1: Can I derive the frequency response given the poles of a (linear time-invariant) system? Yes, you can, up to a constant. If $s_{\infty,i}$, $i=1,\ldots,N,$ are the poles of the transfer function, you can write the transfer function as $$H(s)=\frac{k}{(s-s_{\infty,1})(s-s_{\infty,2})\ldots (s-s_{\... 12 Let h(t) denote the impulse response of an LTI system. Then, for any input x(t), the output is$$y(t) = \int_{-\infty}^\infty h(\tau)x(t-\tau)\,\mathrm d\tau.$$In particular, the response to the input x(t) = \exp(j2\pi ft) is$$\begin{align} y(t) &= \int_{-\infty}^\infty h(\tau)\exp(j2\pi f(t-\tau))\,\mathrm d\tau\\ &= \exp(j2\pi ft)\int_{-\...
10
Yes, you can do this with an LMS equalizer which uses the Wiener-Hopf equation to determine the least squared solution to the filter that would compensate for your channel, using the known transmit and receive sequences. The channel is the unknown being solved, and the tx and rx sequences are known. BOTTOM LINE: Here is the Matlab function with error ...
8
For a start, any non-linear system will not have an easily-identifiable frequency response. So, it's really a nonsensical question. I intend no offense; nonsensical questions are often the most enlightening! However one way to try to answer your question is to assume that the LTI filter involved is the mean (rather than the median) of the windowed data. ...
8
No. The impulse response and frequency response of an LTI system are related by the Fourier transform, which is one-to-one.
8
Yes for 2D signals you can take a 2D fft, and if the 2D signal is represented in the time domain, then its fft is represented in the frequency domain. 2D FFT's have many other interesting applications, for example image creation in synthetic aperture radar (SAR), where an inverse 2D FFT of radar reflections results in the creation of an image. If your ...
7
Shortly, we have two kind of basic responses: time responses and frequency responses. Time responses test how the system works with momentary disturbance while the frequency response test it with continuous disturbance. Time responses contain things such as step response, ramp response and impulse response. Frequency responses contain sinusoidal responses. ...
7
If you know that the system is linear and time invariant, the easiest method (assuming that you have no noise added in the process) is to let the system act on an impulse function. The Fourier transform of the output is the frequency response of the system.
7
Consider a liner discrete-time system. Assume we can define it in terms of an input-output relation as follows (you can assume a more general model but it is enough for our purpose): $$a_0y[n]+a_{1}y[n-1]+\cdots+a_{N}y[n-N]=b_0x[n]+b_{1}x[n-1]+\cdots+b_{M}x[n-M]\tag{1}$$ When the coefficients $\{a_i\}$ and $\{b_i\}$ are constant, we call it a finite-order ...
7
What you do in step 1 is simply truncate the infinite impulse response to approximate it by an FIR filter. If you use sufficiently many filter taps, the approximation becomes arbitrarily accurate. This means that the resulting FIR filter approximates the magnitude and the phase characteristic of the original IIR filter. So with this approach the phase will ...
7
To answer this you need to understand what is a pole and what is a zero of a transfer function. Let's look at a simple 2 poles 2 zeros filter (also called biquad filter) transfer function : $$H(z) = \frac{b_0+b_1 z^{-1}+b_2 z^{-2}}{1+a_1 z^-1 +a_2 z^{-2}}$$ This can be factored as : \begin{align} H(z) &= \frac{ b_0 \, (1-q_1 z^{-1})(1-q_2 z^{-1})}{(... 7 The magnitude of that complex exponential is 1. Recall from complex algebra: any complex number can be expressed as z = r e^{j \phi} where |z|=r is its magnitude and \arg z = \phi is the argument. Using this note that |e^{-j\Omega \lambda}| = 1 $$which is why it "disappeared". 6 FDLS requires a causal frequency response. Your prototype frequency response has zero phase everywhere, which is most definitely not causal. An IIR filter order of 50 is humongous. When FDLS has too many poles and zeroes available, it "tries" to cancel excess poles with excess zeroes. Unfortunately, due to numerical limitations, the cancellation is often ... 6 As Dilip pointed out in the comment above, you can get the impulse response using the inverse Fourier transform. However, a slightly easier method might be to use the Laplace domain instead; it's more amenable to easy inverse transforming via transform tables. First, recall that the frequency response is really just the s-plane transfer function evaluated ... 6 You're definitely on the right track. The way you're trying to solve the problem is the best and simplest. You just need to realize that you need to evaluate the magnitude and phase of the frequency response just for one frequency, namely the frequency of the sinusoidal input signal:$$y[n]=\left|H(e^{j\omega_0})\right|\sin\left(n\omega_0+\phi(\omega_0)\...
6
This has absolutely nothing to do with causality. The frequency response of a real-valued filter (i.e., one with a real-valued impulse response) is (conjugate) symmetric, i.e., the negative frequencies are redundant. That's why it is sufficient to show the frequency response at non-negative frequencies only. You can easily see that symmetry as follows. The ...
5
In previous sections of the book, the fact that a discrete-time signal's spectrum is periodic may have been mentioned. It can be described formally as follows: $$X(e^{j\omega})=\frac1{T} \sum_{k=-\infty}^{\infty}X_C\biggr(j\biggr(\frac{\omega}{T}-\frac{2\pi k}{T}\biggr)\biggr)$$ being $X_C(j\omega)$ the Fourier Transform of the continuous signal, and $T=1/... 5 The analytic way is to substitute the variable$z$by$e^{j\omega}$to get the frequency response$H(\omega)$(with$\omega = \frac{2 \pi f}{F_s}$) - that is to say, the frequency response is the$z$transform evaluated on the unit circle. Note that matlab has a built-in function for plotting the frequency response straight from filter coefficients (freqz), ... 5 What you're looking for is called a pruned DFT. In principle, it is possible to calculate a subset of outputs from a DFT using fewer mathematical operations. In practice, however, existing highly-optimized FFT implementations like FFTW are designed for full-output transforms. You'll find in many cases, unless you're only concerned with a very small ... 5 An LTI system's "frequency response" tells you how the system acts on the amplitude and phase of a sinusoidal input. If the frequency response is$H(f)$, then an input$x(t)=e^{j2\pi f_0t}$produces an output$y(t)=|H(f_0)|e^{j(2\pi f_0t+\angle H(f_0))}$. It is common to divide the frequency response in two, the gain$|H(f)|$and the phase$\angle H(f)$. ... 5 I found the following in Charles Therrien's "Discrete Random Signals and Statistical Signal Processing" in one of the Appendicies. Say you have the function$Q(a)$you wish to minimize such that$C(a)=0$, where$C(a)$may be complex valued and$a$may be a complex vector. The constraint really represents two real-valued constraints. $$C_r(a)=0,\qquad C_i(a)... 5 Frequency response has two parts: amplitude response and the phase response. Both of these are represented as a complex signal when you get the response from freqz. In order to plot the amplitude response you need to use abs. Otherwise I doubt it only shows you the real part which I think is what you see in the first figure. Note that when dealing with the ... 5 Assume you have the signal$$x(t)=a+b\cos(\omega_0t)\tag{1}$$with some non-zero real-valued constants a and b. Now remove DC and the negative frequencies to obtain$$x'(t)=\frac{b}{2}e^{j\omega_0t}\tag{2}$$Can you calculate the value a from (2)? 5 Because you don't have background in wireless communications, I will try to answer as simply as possible. What is the time unit of the converted CIR? Just time unit. It can be second, ms, us, etc. They are convertible, aren't they? How much seconds are they (the 30 impulse responses) apart? In the case that CSI-pilots are equally seperated, impulses ... 5$$1 - e^{-4j\omega} = e^{-2j\omega}(e^{2j\omega} - e^{-2j\omega}) \tag{1}$$Now,$$ \sin(2\omega) = \frac{e^{2j\omega} - e^{-2j\omega}}{2j} \tag{2}$$Equation 2 is a consequence of Euler's formula. Multiply and divide by 2j in (1) and use the identity (2) in equation 1 we have:$$1 - e^{-4j\omega} = 2je^{-2j\omega}\sin(2\omega) \tag{3}$$Now j = e^{... 5 In general there is no straightforward analytical solution. As you know, you need to solve$$\left|H(e^{j\omega_c})\right|=\frac{1}{\sqrt{2}}\tag{1}$$for \omega_c, where it is assumed that the maximum filter gain equals 1. For Butterworth filters, the specified cut-off frequency always equals the 3\textrm{ dB} frequency. This is not the case for other ... 5 You can equalize magnitude and phase simultaneously by defining a desired complex frequency response$$D(\omega)=M(\omega)e^{j\phi(\omega)}\tag{1}$$with magnitude$M(\omega)$and phase$\phi(\omega)$chosen such that they compensate for the given magnitude and phase distortions. An FIR filter approximating$(1)$can be designed by using the following error ... 4 First of all, your transfer function has a pole at$z=-3$, which means your filter is unstable and you will probably see your output going to infinity if your process anything with it. However, in general, you can process an impulse with your difference equation and do an FFT (or freqz) of the resulting impulse response. That should give the same result as ... 4 The key ingredient is that the base functions of the Fourier transform$\exp(i \omega t)\$ are eigenfunctions of LTI systems. That means the LTI system can be represented as a diagonal linear operator in the Fourier basis. Or in other words: To apply an LTI system in frequency domain, you just multiply their frequency responses. And applying an LTI system to ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-01-20 04:02:37
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https://electronics.stackexchange.com/questions/479542/transfer-function-of-two-common-emitters
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# Transfer Function of two Common Emitters
I'm sorry about the pictures and used circuitlab.
I am dealing with a circuit that has two common emitter transistors. Until now I was only dealing with a circuit that that is common emitter.
This is an example
So when the following question comes : Draw the AC- circuit and find the transfer function of the function
I would draw the ac-circuit and then find the transfer function which is this
I am faced with the following circuit so I am a bit lost on what would the final transfer function be
• Can’t you find a transfer function from U1 to the base of T2. Then find another transfer function from the base of T2 to your output U2. Then, multiply them together for the overall transfer function. – relayman357 Feb 5 at 0:36
• The first circuit's voltage gain is more like $\frac{V_\text{CC}-V_{\text{C}_\text{Q}}}{V_T}\cdot\frac{R_\text{L}}{R_\text{C}+R_\text{L}\cdot\left(1+\frac{V_\text{CC}-V_{\text{C}_\text{Q}}}{V_\text{A}}\right)}$. – jonk Feb 5 at 5:09
• Do you need to include the two series capacitors? It is a second-order system then with a double zero at the origin. – Verbal Kint Feb 5 at 12:18
• I am not sure if he really wants the "TRANSFER FUNCTION". Most probably (have a look on his formulas) he is interested in the midband gain only. – LvW Feb 5 at 15:08
To determine this transfer function, many methods exist. I will use the fast analytical techniques or FACTs described in the book I wrote. The principle consists of individually determining the time constants associated with each energy-storing element. You do that to express the denominator which, is the present case, is defined as: $$\D(s)=1+sb_1+s^2b_2\$$. The numerator, where the zeroes lie, is determined using a generalized expression as follows: $$\N(s)=H_0+s(H^1\tau_1+H^2\tau_2)+s^2H^{21}\tau_2\tau_{21}\$$
$$\H_0\$$ is straightforward: open all the caps and the output is 0 V then $$\H_0=0\$$.
To determine the time constants, you go through two small sketches in which you determine the resistance driving the capacitor. You do that using inspection, without writing a line of algebra. For $$\\tau_1\$$ and $$\\tau_2\$$, you examine the below sketch:
Once you have the time constants on hand, you write $$\b_1=\tau_1+\tau_2\$$.
For $$\\tau_{21}\$$, capacitor $$\C_2\$$ is replaced by a short circuit and you determine the resistance driving $$\C_1\$$ in this mode:
Then you form $$\b_2=\tau_2\tau_{21}\$$. This is it, you have the denominator which is of second order and considering a low-$$\Q\$$ approximation, you can factor it as $$\D(s)\approx \frac{1}{(1+\frac{1}{\omega_{p1}})(1+\frac{1}{\omega_{p2}})}\$$ with $$\\omega_{p1}=\frac{1}{b_1}\$$ and $$\\omega_{p2}=\frac{b_1}{b_2}\$$.
To determine the zeroes, you will alternatively set the capacitors in their high-frequency state (a short) while one remains in its dc state (open) and calculate the gain in this mode. If you do that, you find $$\H_1=H_2=0\$$. So the term in $$\s\$$ in the numerator is null. Then, to determine the second-order zeroes, put all the caps in their high-frequency state (a short) and determine the gain $$\H^{12}\$$ of the below circuit:
In this case, $$\H(s)=s^2H^{12}\tau_2\tau_{21}\$$ reusing the time constants already determined. This is it, the transfer function is obtained and given in the below Mathcad sheet rearranged in a way to show the gain in the plateau region: remember, all of these techniques are for design-oriented analyses or D-OA, a term forged by the late Dr. Middlebrook.
The final transfer function involves two inverted poles which allow you to factor the plateau gain as the leading term since this is what is wanted. The terms could obviously be simplified if we ignore the biasing resistors but I wanted to show the complete expression for the sake of the exercise. As you could see, all was done by inspection, without a single line of algebra and this is the power of the FACTs.
• Glad to see you here. While I expect the OP to not grab up all this, I on the other hand will benefit from a new (for me) perspective to consider and learn from. Thanks! – jonk Feb 5 at 14:28
• Glad to read from you too : ) I agree that this is quite complicated considering the simple arrangement. But things get quickly nasty with a second-order system. If you neglect the biasing resistors, it should lead to a simpler result. – Verbal Kint Feb 5 at 15:14
• @Verbal Kint, thank you so much for the super detailed explanation. I was hoping for something more simple. I have added a picture of the way I have solved this part. I was sorry it is just a picture of my notes but I can help if something is not understandable .. Here is the link to picture PICTURE – be1995 Feb 5 at 15:24
• The mid-band gain you have derived is correct, I checked it against my Mathcad file when $R_{b2}$ goes infinite. It is difficult to do simpler than what I proposed as long as you consider the two capacitors introducing two zeroes and two poles. – Verbal Kint Feb 5 at 17:49
Your U1 has zero output impedance, so your maths of the 1-transistor circuit can ignore the input resistance at the base of the gain stage.
For the 2-transistor circuit, the input resistance of T2||rb2 must be included in the small signal equations.
• if I may ask what would the final answer be ? – be1995 Feb 5 at 11:02
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2020-04-01 02:44:08
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https://www.physicsforums.com/threads/spontaneous-random-neutron-generation-in-a-spher.256141/
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# Spontaneous Random Neutron Generation in a Spher
1. Sep 14, 2008
### Enjolras1789
This question concerns a problem in Arfken and Weber (from the infinite series chapter, after the power series section). I went to the homework section, and the titles beneath each section specifically imply that a question from a graduate book is inappropriate for that section. I thus post it here. I apologize if this is the wrong place.
The problem reads,"Neutrons are created by a nuclear reaction inside a hollow sphere of radius R. The newly created neutrons are uniformly distributed over the spherical volume. Assuming that all directions are equally probable, what is the average distance a neutron will travel before striking the surface of the sphere? Assume straight line motion, no collisions." It then goes on to give steps on the way of the answer, one stating that the result is that
mean distance = 3/2 R integral( 0 to 1) integral (0 to pi) square root [(1-K*K sin(theta)*sin(theta)] K*K* dk sin(theta) d(theta)
No, I have no idea what K is physically, except by the nature of of what looks like the differential element at the end (but I am confused as to how one might get a distance variable inside a square root times sine of the angle).
Although help in working toward this answer would be appreciated, my request is more meager. I don't understand why the answer isn't simply R/2. If particles are spontaneously forming uniformly in a sphere, and there is total isotropy in direction, and no collisions, I would think that the mean distance traveled by a particle until colliding with the surface would be simply R/2. Why isn't it that simple?
2. Sep 14, 2008
### CompuChip
Well the equation you gave looks like an integration in spherical coordinates.
Isn't k^2 sin^2(theta) = x^2 + y^2 (in Cartesian coordinates) ?
3. Sep 14, 2008
### Enjolras1789
Perhaps you mean a LaPlacian in spherical coordinates? That was my thought, that somehow the 1/(r*r sin(theta) sin(theta)) term in front of the partial derivative of the function with respect to phi. However, it's not obvious to me why I would take a LaPlacian of something, seeing as I physically do not understand why the problem isn't very simple to just being R/2
4. Sep 14, 2008
### CompuChip
No, I meant an integration:
$$\iiint_V f(x, y, z) dx dy dz = \iiint f(r \cos\phi \sin\theta, r \sin\phi \sin\theta, r \cos\theta) r^2 \sin^2\theta dr d\theta d\phi$$
In this case, you would have f(x, y, z) = 3/2 R sqroot(1 - x^2 - y^2)
If (x, y, z) is on a sphere of radius R, that's just 3/2 R sqroot(z^2).
I'm not really into this material, but that's the mathematics I see in there; probably you can relate it to something physical more easily than I can...
5. Sep 14, 2008
### Enjolras1789
You are good; thank you very much for your insight in seeing that form. I will try to understand why it is that this functional form of f is the case. PS, if you are more the mathematician type, I posed a question in the Analysis section of PF that I am very curious about concerning the "best" convergence tests to use.
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2016-10-25 23:20:05
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https://zbmath.org/serials/?q=se%3A870
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# zbMATH — the first resource for mathematics
## Systems & Control Letters
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Stochastic nonlinear stabilization. I: A backstepping design. Zbl 0902.93049
Deng, Hua; Krstić, Miroslav
1997
Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays. Zbl 1133.68412
Sun, Yuan Gong; Wang, Long; Xie, Guangming
2008
Analysis and design for discrete-time linear systems subject to actuator saturation. Zbl 0987.93027
Hu, Tingshu; Lin, Zongli; Chen, Ben M.
2002
Second-order leader-following consensus of nonlinear multi-agent systems via pinning control. Zbl 1213.37131
Song, Qiang; Cao, Jinde; Yu, Wenwu
2010
Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties. Zbl 0978.93070
Daafouz, J.; Bernussou, J.
2001
Decentralized control of vehicle formations. Zbl 1129.93303
Lafferriere, G.; Williams, A.; Caughman, J.; Veerman, J. J. P.
2005
Semi-global exponential stabilization of linear systems subject to “input saturation” via linear feedbacks. Zbl 0788.93075
Lin, Zongli; Saberi, Ali
1993
Gain scheduling via linear fractional transformations. Zbl 0792.93043
Packard, Andy
1994
Zero-sum stochastic differential games and backward equations. Zbl 0877.93125
1995
Stability analysis for continuous systems with two additive time-varying delay components. Zbl 1120.93362
Lam, James; Gao, Huijun; Wang, Changhong
2007
Global stabilization by output feedback: Examples and counterexamples. Zbl 0816.93068
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1994
Local feedback stabilization and bifurcation control. I. Hopf bifurcation. Zbl 0587.93049
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A Lyapunov–Krasovskii methodology for ISS and iISS of time-delay systems. Zbl 1120.93361
Pepe, P.; Jiang, Z.-P.
2006
Group consensus in multi-agent systems with switching topologies and communication delays. Zbl 1197.93096
Yu, Junyan; Wang, Long
2010
Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems. Zbl 0948.93056
Lin, W.; Qian, C.
2000
Stability radius for structured perturbations and the algebraic Riccati equation. Zbl 0626.93054
Hinrichsen, Diederich; Pritchard, A. J.
1986
Stability radii of linear systems. Zbl 0631.93064
Hinrichsen, Diederich; Pritchard, A. J.
1986
Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Zbl 0634.93066
Zhou, Kemin; Khargonekar, Pramod P.
1988
Linear output feedback with dynamic high gain for nonlinear systems. Zbl 1157.93494
Praly, L.; Jiang, Z. P.
2004
Stabilization of a class of nonlinear systems by a smooth feedback control. Zbl 0569.93056
Aeyels, Dirk
1985
Stability analysis of systems with time-varying delay via relaxed integral inequalities. Zbl 1338.93290
Zhang, Chuan-Ke; He, Yong; Jiang, L.; Wu, Min; Zeng, Hong-Bing
2016
An algebraic Riccati equation approach to $$H^{\infty}$$ optimization. Zbl 0666.93025
Zhou, Kemin; Khargonekar, Pramod P.
1988
Robust energy-to-peak filter design for stochastic time-delay systems. Zbl 1129.93538
Gao, Huijun; Lam, James; Wang, Changhong
2006
Middleton, R. H.; Goodwin, G. C.
1988
Backstepping observers for a class of parabolic PDEs. Zbl 1129.93334
Smyshlyaev, Andrey; Krstic, Miroslav
2005
Hierarchy of LMI conditions for the stability analysis of time-delay systems. Zbl 1330.93211
Seuret, A.; Gouaisbaut, F.
2015
Decentralized finite-time sliding mode estimators and their applications in decentralized finite-time formation tracking. Zbl 1207.93103
Cao, Yongcan; Ren, Wei; Meng, Ziyang
2010
Auxiliary model-based least-squares identification methods for Hammerstein output-error systems. Zbl 1130.93055
Ding, Feng; Shi, Yang; Chen, Tongwen
2007
A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Zbl 0986.93063
Bhat, Sanjay P.; Bernstein, Dennis S.
2000
Stability theory for nonnegative and compartmental dynamical systems with time delay. Zbl 1157.34352
2004
Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Zbl 0684.93065
Molchanov, A. P.; Pyatnitskij, E. S.
1989
Gradient based iterative algorithm for solving coupled matrix equations. Zbl 1159.93323
Zhou, Bin; Duan, Guang-Ren; Li, Zhao-Yan
2009
Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach. Zbl 1129.93547
Chen, Wu-Hua; Guan, Zhi-Hong; Lu, Xiaomei
2005
Output feedback control of switched nonlinear systems using multiple Lyapunov functions. Zbl 1129.93497
El-Farra, Nael H.; Mhaskar, Prashant; Christofides, Panagiotis D.
2005
New results and examples in nonlinear feedback stabilization. Zbl 0684.93059
Byrnes, Christopher I.; Isidori, Alberto
1989
On the regularization of Zeno hybrid automata. Zbl 0948.93031
Johansson, K. H.; Egerstedt, M.; Lygeros, J.; Sastry, S.
1999
On the convergence of an extended state observer for nonlinear systems with uncertainty. Zbl 1225.93056
Guo, Bao-Zhu; Zhao, Zhi-Liang
2011
Sliding mode control of a discrete system. Zbl 0692.93043
Furuta, Katsuhisa
1990
On a state space approach to nonlinear $$H_ \infty$$ control. Zbl 0737.93018
Van der Schaft, A. J.
1991
Robust stability of uncertain stochastic differential delay equations. Zbl 0909.93054
Mao, Xuerong; Koroleva, Natalia; Rodkina, Alexandra
1998
Finite-time control for robot manipulators. Zbl 0994.93041
Hong, Yiguang; Xu, Yangsheng; Huang, Jie
2002
On the linear equivalents of nonlinear systems. Zbl 0482.93041
Su, Renjeng
1982
Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift. Zbl 0744.93084
Pomet, Jean-Baptiste
1992
Barrier Lyapunov functions for the output tracking control of constrained nonlinear switched systems. Zbl 1281.93092
Niu, Ben; Zhao, Jun
2013
Forward completeness, unboundedness observability, and their Lyapunov characterizations. Zbl 0986.93036
Angeli, David; Sontag, Eduardo D.
1999
Adding an integrator for the stabilization problem. Zbl 0747.93072
Coron, Jean-Michel; Praly, Laurent
1991
Robust consensus tracking of a class of second-order multi-agent dynamic systems. Zbl 1250.93009
Hu, Guoqiang
2012
Stabilization of nonlinear systems in the plane. Zbl 0666.93103
Kawski, Matthias
1989
Nonlinear observer design using Lyapunov’s auxiliary theorem. Zbl 0909.93002
Kazantzis, Nikolaos; Kravaris, Costas
1998
Self-tuning control based on multi-innovation stochastic gradient parameter estimation. Zbl 1154.93040
Zhang, Jiabo; Ding, Feng; Shi, Yang
2009
Distributed finite-time tracking control for multi-agent systems: an observer-based approach. Zbl 1257.93011
Zhao, Yu; Duan, Zhisheng; Wen, Guanghui; Zhang, Yanjiao
2013
Global stabilizability and observability imply semi-global stabilizability by output feedback. Zbl 0820.93054
Teel, Andrew; Praly, Laurent
1994
Balancing for nonlinear systems. Zbl 0785.93042
Scherpen, J. M. A.
1993
Leader-follower cooperative attitude control of multiple rigid bodies. Zbl 1161.93002
Dimarogonas, Dimos V.; Tsiotras, Panagiotis; Kyriakopoulos, Kostas J.
2009
A lifting technique for linear periodic systems with applications to sampled-data control. Zbl 0747.93057
Bamieh, Bassam; Pearson, J. Boyd; Francis, Bruce A.; Tannenbaum, Allen
1991
On the decay rate of Hankel singular values and related issues. Zbl 1003.93024
Antoulas, A. C.; Sorensen, D. C.; Zhou, Y.
2002
Adaptive rational Krylov subspaces for large-scale dynamical systems. Zbl 1236.93035
Druskin, V.; Simoncini, V.
2011
Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Zbl 0977.93034
Penzl, T.
2000
Second-order consensus for multi-agent systems with switching topology and communication delay. Zbl 1225.93020
Qin, Jiahu; Gao, Huijun; Zheng, Wei Xing
2011
Terminal sliding mode control design for uncertain dynamic systems. Zbl 0909.93005
Wu, Yuqiang; Yu, Xinghuo; Man, Zhihong
1998
Remote stabilization over fading channels. Zbl 1129.93498
Elia, Nicola
2005
Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Zbl 0815.93034
Becker, G.; Packard, A.
1994
Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance. Zbl 0948.93053
Deng, H.; Krstić, M.
2000
A converse Lyapunov theorem for discrete-time systems with disturbances. Zbl 0987.93072
Jiang, Zhong-Ping; Wang, Yuan
2002
Finite-time stability and stabilization of time-delay systems. Zbl 1140.93447
Moulay, Emmanuel; Dambrine, Michel; Yeganefar, Nima; Perruquetti, Wilfrid
2008
A positive real condition for global stabilization of nonlinear systems. Zbl 0684.93066
Kokotovic, P. V.; Sussmann, H. J.
1989
Controllability and stabilizability of switched linear-systems. Zbl 1134.93403
Xie, Guangming; Wang, Long
2003
Robust stability and stabilisation of 2d discrete state-delayed systems. Zbl 1157.93472
Paszke, Wojciech; Lam, James; Gałkowski, Krzysztof; Xu, Shengyuan; Lin, Zhiping
2004
Equilibria and steering laws for planar formations. Zbl 1157.93406
Justh, E. W.; Krishnaprasad, P. S.
2004
Input-state incidence matrix of Boolean control networks and its applications. Zbl 1217.93026
Zhao, Yin; Qi, Hongsheng; Cheng, Daizhan
2010
Passive linear continuous-time systems: characterization through structure. Zbl 1454.93097
Lewkowicz, Izchak
2021
Stability analysis for a class of stochastic delay nonlinear systems driven by $$G$$-Brownian motion. Zbl 1447.93369
Zhu, Quanxin; Huang, Tingwen
2020
Global adaptive stabilization of stochastic high-order switched nonlinear non-lower triangular systems. Zbl 1433.93115
Niu, Ben; Liu, Ming; Li, Ang
2020
Funnel control in the presence of infinite-dimensional internal dynamics. Zbl 1447.93163
Berger, Thomas; Puche, Marc; Schwenninger, Felix L.
2020
Stability analysis of impulsive stochastic delayed differential systems with unbounded delays. Zbl 1433.93118
Hu, Wei; Zhu, Quanxin
2020
A closed-loop saddle point for zero-sum linear-quadratic stochastic differential games with mean-field type. Zbl 1433.91013
Tian, Ran; Yu, Zhiyong; Zhang, Rucheng
2020
Longtime behavior of a class of stochastic tumor-immune systems. Zbl 1453.92083
Tuong, T. D.; Nguyen, N. N.; Yin, G.
2020
Probabilistic error analysis for some approximation schemes to optimal control problems. Zbl 1441.93347
Picarelli, Athena; Reisinger, Christoph
2020
Intermittent stochastic stabilization based on discrete-time observation with time delay. Zbl 1441.93324
Liu, Lei; Wu, Zhaojing
2020
Single-boundary control of the two-phase Stefan system. Zbl 1433.93050
Koga, Shumon; Krstic, Miroslav
2020
A lower bound on the dimension of minimal positive realizations for discrete time systems. Zbl 1433.93030
Benvenuti, Luca
2020
Stochastic Nicholson-type delay system with regime switching. Zbl 1433.93128
Wang, Wentao; Chen, Wei
2020
Shape turnpike for linear parabolic PDE models. Zbl 1451.93170
Lance, Gontran; Trélat, Emmanuel; Zuazua, Enrique
2020
ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains. Zbl 1454.93246
Zheng, Jun; Zhu, Guchuan
2020
Repetitive process based stochastic iterative learning control design for linear dynamics. Zbl 1441.93354
Pakshin, Pavel; Emelianova, Julia; Rogers, Eric; Gałkowski, Krzysztof
2020
Anti-windup for model-reference adaptive control schemes with rate-limits. Zbl 1441.93153
Turner, Matthew C.; Sofrony, Jorge; Prempain, Emmanuel
2020
Prescribed-time decentralized regulation of uncertain nonlinear multi-agent systems via output feedback. Zbl 1441.93002
Chen, Xiandong; Zhang, Xianfu; Liu, Qingrong
2020
Control law in analytic expression of a system coupled by reaction-diffusion equation. Zbl 1441.93126
Liu, Xinglan; Xie, Chengkang
2020
Approximating optimal finite horizon feedback by model predictive control. Zbl 1447.93087
Dontchev, A. L.; Kolmanovsky, I. V.; Krastanov, M. I.; Veliov, V. M.; Vuong, P. T.
2020
Stabilizability properties of a linearized water waves system. Zbl 1440.93204
Su, Pei; Tucsnak, Marius; Weiss, George
2020
Sampled-data containment control for double-integrator agents with dynamic leaders with nonzero inputs. Zbl 1447.93206
Ding, Yong; Ren, Wei
2020
Lipschitz regularity of the minimum time function of differential inclusions with state constraints. Zbl 1447.93203
Aubin-Frankowski, Pierre-Cyril
2020
Memoryless linear feedback control for a class of upper-triangular systems with large delays in the state and input. Zbl 1447.93117
Zhao, Congran; Lin, Wei
2020
Stabilization of constrained switched systems via multiple Lyapunov R-functions. Zbl 1447.93315
Wu, Feiyue; Lian, Jie
2020
Stability analysis of discrete-time semi-Markov jump linear systems with partly unknown semi-Markov kernel. Zbl 1447.93364
Wang, Bao; Zhu, Quanxin
2020
Same decay rate of second order evolution equations with or without delay. Zbl 1443.35012
Bayili, Gilbert; Aissa, Akram Ben; Nicaise, Serge
2020
A simple finite-time distributed observer design for linear time-invariant systems. Zbl 1447.93128
Silm, Haik; Efimov, Denis; Michiels, Wim; Ushirobira, Rosane; Richard, Jean-Pierre
2020
Cooperative output regulation of singular multi-agent systems under adaptive distributed protocol and general entirety method. Zbl 1436.93009
Liu, Xiaofan; Xie, Yongfang; Li, Fanbiao; Gui, Weihua
2020
Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control. Zbl 1436.93123
Lhachemi, Hugo; Shorten, Robert; Prieur, Christophe
2020
Stability analysis of perturbed infinite-dimensional sampled-data systems. Zbl 1436.93118
Wakaiki, Masashi; Yamamoto, Yutaka
2020
Boundary control of partial differential equations using frequency domain optimization techniques. Zbl 1433.93049
Apkarian, P.; Noll, D.
2020
Strict Lyapunov-Krasovskiĭ functionals for undirected networks of Euler-Lagrange systems with time-varying delays. Zbl 1433.93013
Nuño, Emmanuel; Sarras, Ioannis; Loría, Antonio; Maghenem, Mohamed; Cruz-Zavala, Emmanuel; Panteley, Elena
2020
A filtering problem with uncertainty in observation. Zbl 1433.93138
Ji, Shaolin; Kong, Chuiliu; Sun, Chuanfeng
2020
On construction of Lyapunov functions for scalar linear time-varying systems. Zbl 1433.93121
Zhou, Bin; Tian, Yang; Lam, James
2020
Event-triggered control for a class of switched uncertain nonlinear systems. Zbl 1433.93082
Lian, Jie; Li, Can
2020
A new condition for stability of switched linear systems under restricted minimum dwell time switching. Zbl 1433.93096
Kundu, Atreyee
2020
Controllability-Gramian submatrices for a network consensus model. Zbl 1433.93020
Roy, Sandip; Xue, Mengran
2020
Semi-global incremental input-to-state stability of discrete-time Lur’e systems. Zbl 1433.93119
Gilmore, Max E.; Guiver, Chris; Logemann, Hartmut
2020
Bipartite consensus for a network of wave equations with time-varying disturbances. Zbl 1433.93123
Chen, Yining; Zuo, Zhiqiang; Wang, Yijing
2020
Global convergence of the EM algorithm for ARX models with uncertain communication channels. Zbl 1433.93135
Chen, Jing; Huang, Biao; Zhu, Quanmin; Liu, Yanjun; Li, Lun
2020
An $$L_T^2$$-error bound for time-limited balanced truncation. Zbl 1433.93023
Redmann, Martin
2020
Identification of port-Hamiltonian systems from frequency response data. Zbl 1451.93063
Benner, Peter; Goyal, Pawan; Van Dooren, Paul
2020
Luenberger-like interval observer design for discrete-time descriptor linear system. Zbl 1425.93057
Guo, Shenghui; Jiang, Bin; Zhu, Fanglai; Wang, Zhenhua
2019
Second-order consensus of hybrid multi-agent systems. Zbl 1425.93256
Zheng, Yuanshi; Zhao, Qi; Ma, Jingying; Wang, Long
2019
Sampled-data observers for 1-D parabolic PDEs with non-local outputs. Zbl 1427.93090
Karafyllis, Iasson; Ahmed-Ali, Tarek; Giri, Fouad
2019
Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay. Zbl 1408.93136
Fridman, Emilia; Shaikhet, Leonid
2019
Stabilizing Boolean networks by optimal event-triggered feedback control. Zbl 1425.93230
Zhu, Qunxi; Lin, Wei
2019
A quadratic matrix inequality based PID controller design for LPV systems. Zbl 1425.93086
Wang, Yan; Rajamani, Rajesh; Zemouche, Ali
2019
Conditions for fixed-time stability and stabilization of continuous autonomous systems. Zbl 1425.93218
Lopez-Ramirez, Francisco; Efimov, Denis; Polyakov, Andrey; Perruquetti, Wilfrid
2019
Fault deviation estimation and integral sliding mode control design for Lipschitz nonlinear systems. Zbl 1408.93040
Gao, Yabin; Liu, Jianxing; Sun, Guanghui; Liu, Ming; Wu, Ligang
2019
On event-triggered control for integral input-to-state stable systems. Zbl 1408.93124
Yu, Hao; Hao, Fei; Chen, Xia
2019
Interval estimation for discrete-time linear systems: a two-step method. Zbl 1408.93130
Tang, Wentao; Wang, Zhenhua; Shen, Yi
2019
Time-varying low gain feedback for linear systems with unknown input delay. Zbl 1408.93105
Wei, Yusheng; Lin, Zongli
2019
An explicit mapping from linear first order hyperbolic PDEs to difference systems. Zbl 1408.93117
Auriol, Jean; Di Meglio, Florent
2019
On the limits of stabilizability for networks of strings. Zbl 1425.93229
Gugat, Martin; Gerster, Stephan
2019
Weak Feller property of non-linear filters. Zbl 1428.93109
Kara, Ali Devran; Saldi, Naci; Yüksel, Serdar
2019
PI boundary control of linear hyperbolic balance laws with stabilization of ARZ traffic flow models. Zbl 1408.93108
Zhang, Liguo; Prieur, Christophe; Qiao, Junfei
2019
An approach to multidimensional Fornasini-Marchesini state-space model realization w.r.t. columns of transfer matrices. Zbl 1408.93045
Zhao, Dongdong; Yan, Shi; Matsushita, Shinya; Xu, Li
2019
Power control for multi-sensor remote state estimation over interference channel. Zbl 1425.93196
Li, Yuzhe; Chen, Chung Shue; Wong, Wing Shing
2019
Infinite-time exact observability of Volterra systems in Hilbert spaces. Zbl 1425.93056
Chen, Jian-Hua
2019
Finite-time balanced truncation for linear systems via shifted Legendre polynomials. Zbl 1425.93067
Xiao, Zhi-Hua; Jiang, Yao-Lin; Qi, Zhen-Zhong
2019
Model-invariant viability kernel approximation. Zbl 1425.93029
Yousefi, Mahdi; van Heusden, Klaske; Mitchell, Ian M.; Dumont, Guy A.
2019
A neuroadaptive architecture for model reference control of uncertain dynamical systems with performance guarantees. Zbl 1425.93152
Arabi, Ehsan; Yucelen, Tansel; Gruenwald, Benjamin C.; Fravolini, Mario; Balakrishnan, Sivasubramanya; Nguyen, Nhan T.
2019
Output synchronization of nonlinear heterogeneous multi-agent systems with switching networks. Zbl 1425.93244
Khan, Gulam Dastagir; Chen, Zhiyong; Meng, Haofei
2019
Non-equilibrium dynamic games and cyber-physical security: a cognitive hierarchy approach. Zbl 1426.91042
Kanellopoulos, Aris; Vamvoudakis, Kyriakos G.
2019
New criteria of input-to-state stability for nonlinear switched stochastic delayed systems with asynchronous switching. Zbl 1425.93245
Zhang, Meng; Zhu, Quanxin
2019
Distributed Kalman filtering for sensor network with balanced topology. Zbl 1425.93283
Li, Chaoyong; Dong, Hangning; Li, Jianqing; Wang, Feng
2019
Finite-time internal stabilization of a linear 1-D transport equation. Zbl 1427.93198
Zhang, Christophe
2019
A game theoretic approach to multi-channel transmission scheduling for multiple linear systems under DoS attacks. Zbl 1427.91017
Zhang, Junhui; Sun, Jitao
2019
Adaptive event-triggered distributed model predictive control for multi-agent systems. Zbl 1428.93061
Zhan, Jingyuan; Hu, Yanjie; Li, Xiang
2019
PDE output feedback control of LTI systems with uncertain multi-input delays, plant parameters and ODE state. Zbl 1408.93109
Zhu, Yang; Krstic, Miroslav; Su, Hongye
2019
Synthesis of minimally restrictive optimal stability-enforcing supervisors for nondeterministic discrete event systems. Zbl 1408.93138
Han, Xiaoguang; Chen, Zengqiang; Su, Rong
2019
Stabilization of discrete time stochastic system with input delay and control dependent noise. Zbl 1408.93143
Tan, Cheng; Yang, Lin; Zhang, Fangfang; Zhang, Zhengqiang; Wong, Wing Shing
2019
Sufficient conditions for dispersiveness of invariant control affine systems on the Heisenberg group. Zbl 1408.93027
Souza, Josiney A.
2019
Attitude observer on the special orthogonal group with Earth velocity estimation. Zbl 1425.93055
Batista, Pedro; Silvestre, Carlos; Oliveira, Paulo
2019
Stabilizing predictive control with persistence of excitation for constrained linear systems. Zbl 1425.93242
Vicente, Bernardo A. Hernandez; Trodden, Paul A.
2019
The viability kernel of dynamical systems with mixed constraints: a level-set approach. Zbl 1425.93173
Gajardo, Pedro; Hermosilla, Cristopher
2019
Deterministic continuous-time virtual reference feedback tuning (VRFT) with application to PID design. Zbl 1425.93105
Formentin, Simone; Campi, Marco C.; Carè, Algo; Savaresi, Sergio M.
2019
Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks. Zbl 1425.93231
Ammari, Kaïs; Crépeau, Emmanuelle
2019
Distributed output-feedback model predictive control for multi-agent consensus. Zbl 1425.93249
Copp, David A.; Vamvoudakis, Kyriakos G.; Hespanha, João P.
2019
MPC-based defense strategy for distributed networked control systems under DoS attacks. Zbl 1425.93263
Yang, Hongjiu; Li, Ying; Dai, Li; Xia, Yuanqing
2019
Well-posedness of infinite-dimensional linear systems with nonlinear feedback. Zbl 1425.93106
Hastir, Anthony; Califano, Federico; Zwart, Hans
2019
Distributed multi-step subgradient optimization for multi-agent system. Zbl 1425.93252
Li, Chaoyong; Chen, Sai; Li, Jianqing; Wang, Feng
2019
Control of biological models with hysteresis. Zbl 1425.92164
Timoshin, Sergey A.; Aiki, Toyohiko
2019
Distributed anti-windup approach for consensus tracking of second-order multi-agent systems with input saturation. Zbl 1425.93250
Fu, Junjie; Lv, Yuezu; Huang, Tingwen
2019
Co-design of aperiodic sampled-data min-jumping rules for linear impulsive, switched impulsive and sampled-data systems. Zbl 1425.93178
Briat, Corentin
2019
Lyapunov matrices for neutral time-delay systems with exponential kernel. Zbl 1425.93213
Aliseyko, A. N.
2019
Robust predictor based control of state multiplicative noisy retarded systems. Zbl 1425.93089
Gershon, E.; Shaked, U.
2019
Computation of the Lyapunov matrix for periodic time-delay systems and its application to robust stability analysis. Zbl 1425.93224
Gomez, Marco A.; Egorov, Alexey V.; Mondié, Sabine; Zhabko, Alexey P.
2019
Stability in distribution of stochastic functional differential equations. Zbl 1425.93299
Wang, Ya; Wu, Fuke; Mao, Xuerong
2019
Moment bounds and ergodicity of switching diffusion systems involving two-time-scale Markov chains. Zbl 1425.93260
Li, Xiaoyue; Wang, Rui; Yin, George
2019
Stability analysis for stochastic impulsive switched time-delay systems with asynchronous impulses and switches. Zbl 1427.93268
Ren, Wei; Xiong, Junlin
2019
Adaptive state observers using dynamic regressor extension and mixing. Zbl 1427.93108
Pyrkin, Anton; Bobtsov, Alexey; Ortega, Romeo; Vedyakov, Alexey; Aranovskiy, Stanislav
2019
Asynchronous control for switched systems by using persistent dwell time modeling. Zbl 1427.93099
Shi, Shuang; Shi, Zhenpeng; Fei, Zhongyang
2019
Discrete-time port-Hamiltonian systems: a definition based on symplectic integration. Zbl 1427.93133
Kotyczka, Paul; Lefèvre, Laurent
2019
From internal to pointwise control for the 1D heat equation and minimal control time. Zbl 1427.93091
Letrouit, Cyril
2019
An integral sliding mode approach to distributed control of coupled networks with measurement quantization. Zbl 1427.93048
Xiong, Yongyang; Gao, Yabin; Yang, Liu; Wu, Ligang
2019
Stabilization of reaction-diffusions PDE with delayed distributed actuation. Zbl 1427.93092
Qi, Jie; Krstic, Miroslav; Wang, Shanshan
2019
Observers to the aid of “strictification” of Lyapunov functions. Zbl 1428.93100
Praly, Laurent
2019
Event-triggered $$\mathcal{L}_\infty$$ control for network-based switched linear systems with transmission delay. Zbl 1428.93055
Qi, Yiwen; Liu, Yanhui; Fu, Jun; Zeng, Pengyu
2019
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167 Lam, James 162 Shi, Peng 142 Xu, Shengyuan 132 Krstić, Miroslav 115 Cao, Jinde 100 Wang, Zidong 95 Zhao, Jun 92 Karimi, Hamid Reza 91 Zhang, Qingling 90 Fridman, Emilia 90 Park, Juhyun (Jessie) 89 Duan, Guangren 89 Ortega, Romeo S. 88 Lin, Zongli 83 Yang, Guanghong 82 Wang, Long 80 Teel, Andrew Richard 79 Zhong, Shou-Ming 78 Jiang, Zhong-Ping 77 Guo, Bao-Zhu 72 Feng, Gang 72 Su, Hongye 71 Xie, Lihua 70 Ding, Feng 70 Gao, Huijun 69 Jia, Yingmin 69 Zhou, Bin 67 Mao, Xuerong 67 Wen, Changyun 67 Zheng, Wei Xing 63 Anderson, Brian David Outram 63 Benner, Peter 63 Chen, Guanrong 63 Efimov, Denis V. 63 Xiang, Zhengrong 60 Nešić, Dragan 59 Duan, Zhisheng 59 Karafyllis, Iasson 59 Sontag, Eduardo D. 58 Petersen, Ian Richard 55 Jiang, Bin 55 Nijmeijer, Henk 54 Chen, Bing 54 Fei, Shumin 54 Ge, Shuzhi Sam 54 Liu, Yungang 54 Saberi, Ali 54 Zhu, Quanxin 53 Haddad, Wassim Michael 53 Liu, Xinzhi 53 Zou, Yun 52 Wu, Min 52 Xia, Yuanqing 52 Zhang, Weihai 51 Wu, Yuqiang 49 Fradkov, Aleksandr L’vovich 49 Lin, Chong 47 Guan, Zhihong 47 He, Yong 47 Li, Shihua 47 Tornambè, Antonio 47 Wu, Zhen 46 Mazenc, Frédéric 46 van der Schaft, Arjan J. 46 Xie, Xuejun 46 Zhao, Xudong 45 Astolfi, Alessandro 45 Ho, Daniel W. C. 45 Liu, Fei 45 Sun, Ji Tao 43 Savkin, Andrey V. 42 Hajarian, Masoud 42 Han, Qinglong 42 Mahmoud, Magdi Sadik Mostafa 42 Shi, Yang 41 Cheng, Daizhan 41 Hong, Yiguang 41 Lewis, Frank Leroy 41 Perruquetti, Wilfrid 40 Chen, Tongwen 40 Deng, Feiqi 40 Hammami, Mohamed Ali 40 Lin, Yan 40 Vaidyanathan, Sundarapandian 40 Xie, Guangming 39 Partington, Jonathan R. 39 Peres, Pedro L. D. 38 Fu, Minyue 38 Geromel, Jose Claudio 38 Li, Xiaodi 38 Wang, Junmin 38 Wang, Qingguo 37 Colaneri, Patrizio 37 Fan, Shengjun 37 Huang, Lin 37 Liu, Xiaoping 37 Wu, Aiguo 37 Xu, Gen-Qi 37 Zhang, Baoyong 36 de la Sen, Manuel ...and 20,693 more Authors
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#### Cited in 699 Journals
3,635 Automatica 2,827 Systems & Control Letters 1,799 International Journal of Control 1,531 Journal of the Franklin Institute 825 International Journal of Robust and Nonlinear Control 622 Mathematical Problems in Engineering 589 Applied Mathematics and Computation 498 European Journal of Control 457 International Journal of Systems Science 440 Nonlinear Dynamics 420 Linear Algebra and its Applications 413 Asian Journal of Control 349 SIAM Journal on Control and Optimization 327 International Journal of Systems Science. Principles and Applications of Systems and Integration 297 Nonlinear Analysis. Hybrid Systems 292 Journal of Mathematical Analysis and Applications 266 MCSS. Mathematics of Control, Signals, and Systems 255 Circuits, Systems, and Signal Processing 245 Information Sciences 234 Journal of Optimization Theory and Applications 224 International Journal of Adaptive Control and Signal Processing 220 Abstract and Applied Analysis 183 Kybernetika 181 Automation and Remote Control 171 Journal of Systems Science and Complexity 151 Fuzzy Sets and Systems 148 Complexity 146 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 144 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 138 Chaos, Solitons and Fractals 138 Stochastic Processes and their Applications 136 Advances in Difference Equations 134 Communications in Nonlinear Science and Numerical Simulation 130 Journal of Computational and Applied Mathematics 127 Computers & Mathematics with Applications 126 Applied Mathematics Letters 120 Optimal Control Applications & Methods 119 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 117 Applied Mathematical Modelling 114 Journal of Differential Equations 109 Stochastic Analysis and Applications 109 Journal of Control Theory and Applications 100 Multidimensional Systems and Signal Processing 100 Discrete Dynamics in Nature and Society 81 Applied Mathematics and Optimization 80 Journal of Control Science and Engineering 79 Statistics & Probability Letters 78 Mathematics and Computers in Simulation 77 International Journal of Applied Mathematics and Computer Science 75 Journal of Dynamical and Control Systems 74 Journal of Vibration and Control 73 Journal of Applied Mathematics 67 Mathematical and Computer Modelling 60 Neural Networks 59 Nonlinear Analysis. Real World Applications 56 Discrete and Continuous Dynamical Systems. Series B 55 Chaos 54 Mathematical Control and Related Fields 51 The Annals of Applied Probability 49 Discrete Event Dynamic Systems 47 Comptes Rendus. Mathématique. Académie des Sciences, Paris 46 Journal of Mathematical Sciences (New York) 41 Acta Applicandae Mathematicae 41 Discrete and Continuous Dynamical Systems 39 Mathematical Methods in the Applied Sciences 39 Journal of Applied Mathematics and Computing 38 Mathematical Biosciences 38 Numerical Functional Analysis and Optimization 37 Differential Equations 36 Acta Mathematicae Applicatae Sinica. English Series 34 Optimization 34 Signal Processing 34 Computational Optimization and Applications 34 Computational and Applied Mathematics 34 Science in China. Series F 33 Applicable Analysis 33 Integral Equations and Operator Theory 33 SIAM Journal on Scientific Computing 33 Stochastics and Dynamics 33 Control Theory and Technology 32 Numerical Algorithms 32 International Journal of Computer Mathematics 32 Journal of Difference Equations and Applications 31 Stochastics 30 Linear and Multilinear Algebra 30 SIAM Journal on Matrix Analysis and Applications 30 Nonlinear Analysis. Theory, Methods & Applications 29 Applied Numerical Mathematics 29 European Journal of Operational Research 28 Random Operators and Stochastic Equations 27 Journal of Mathematical Physics 27 Numerische Mathematik 27 Journal of Inequalities and Applications 26 The Annals of Probability 26 Physica D 26 Set-Valued and Variational Analysis 25 Journal of Functional Analysis 25 Journal of Nonlinear Science 25 Numerical Linear Algebra with Applications 24 International Journal of General Systems ...and 599 more Journals
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#### Cited in 62 Fields
20,403 Systems theory; control (93-XX) 2,583 Ordinary differential equations (34-XX) 2,512 Probability theory and stochastic processes (60-XX) 1,718 Calculus of variations and optimal control; optimization (49-XX) 1,482 Computer science (68-XX) 1,426 Operations research, mathematical programming (90-XX) 1,281 Partial differential equations (35-XX) 1,247 Numerical analysis (65-XX) 1,078 Linear and multilinear algebra; matrix theory (15-XX) 840 Biology and other natural sciences (92-XX) 837 Mechanics of particles and systems (70-XX) 815 Dynamical systems and ergodic theory (37-XX) 683 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 555 Information and communication theory, circuits (94-XX) 432 Operator theory (47-XX) 310 Statistics (62-XX) 274 Mechanics of deformable solids (74-XX) 247 Combinatorics (05-XX) 221 Difference and functional equations (39-XX) 197 Fluid mechanics (76-XX) 164 Real functions (26-XX) 105 Functions of a complex variable (30-XX) 95 Functional analysis (46-XX) 94 Approximations and expansions (41-XX) 89 Quantum theory (81-XX) 81 Integral equations (45-XX) 75 Global analysis, analysis on manifolds (58-XX) 62 Commutative algebra (13-XX) 52 Differential geometry (53-XX) 46 Optics, electromagnetic theory (78-XX) 43 Mathematical logic and foundations (03-XX) 43 Harmonic analysis on Euclidean spaces (42-XX) 41 Topological groups, Lie groups (22-XX) 40 Classical thermodynamics, heat transfer (80-XX) 39 Nonassociative rings and algebras (17-XX) 37 Associative rings and algebras (16-XX) 35 Statistical mechanics, structure of matter (82-XX) 27 Algebraic geometry (14-XX) 23 Number theory (11-XX) 23 Measure and integration (28-XX) 23 General topology (54-XX) 22 Manifolds and cell complexes (57-XX) 21 General and overarching topics; collections (00-XX) 18 Field theory and polynomials (12-XX) 18 Convex and discrete geometry (52-XX) 18 Geophysics (86-XX) 16 History and biography (01-XX) 16 Special functions (33-XX) 14 Integral transforms, operational calculus (44-XX) 13 Group theory and generalizations (20-XX) 13 Several complex variables and analytic spaces (32-XX) 11 Relativity and gravitational theory (83-XX) 8 Order, lattices, ordered algebraic structures (06-XX) 6 Abstract harmonic analysis (43-XX) 6 Algebraic topology (55-XX) 5 Sequences, series, summability (40-XX) 4 Potential theory (31-XX) 3 Category theory; homological algebra (18-XX) 3 $$K$$-theory (19-XX) 3 Astronomy and astrophysics (85-XX) 2 Geometry (51-XX) 1 Mathematics education (97-XX)
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2021-09-19 21:33:24
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https://ug.costsproject.org/10493-heat-exchange.html
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# Heat exchange
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## Cross and mixed flow guidance
With cross-flow heat transport, one of the fluids flows through pipes and channels that are perpendicular to the direction of flow of the surrounding second fluid. Calculating this cross-flow heat transport is considerably more difficult than calculating the direct and counter-flow, since the temperature conditions of the external fluid change, for example in a heat exchanger with several rows of tubes, as it flows past each row of tubes. For this reason, the mean logarithmic temperature difference becomes$ΔTm$as with cocurrent or countercurrent, taking into account a correction factor$εk$calculated for cross flow.
## Agitator selection (process engineering)
The choice of stirrer depends on the stirring task, the viscosity of the medium, the shear strength (whether the medium has to be spared) or, conversely, on the stirrer power available or required for the process.
A distinction is made between the following essential stirring tasks, which in practice have to be carried out individually, simultaneously or in a chronological sequence one after the other in a stirred tank:
• Homogenization: The equalization of differences in concentration of different media that can be mixed with one another
• Dispersing liquid / liquid: The stirring of insoluble media into another fluid
• Dispersing liquid / gaseous: The stirring of a gas phase into a liquid phase, e.g. during hydrogenation
• Heat transfer, i.e. the targeted introduction of large amounts of heat into a fluid
• Suspending: The fluidizing and mixing of solids in a liquid phase
• Emulsification: The stirring of a liquid phase into a second liquid
For each of these stirring tasks, certain types of stirrers are better or worse suited. Since in practice, as described above, stirring tasks rarely occur in isolation, a compromise must often be made between the "optimal" stirring element for the most important process step and the general suitability for process steps that also occur but are less relevant for the yield and quality of the product . In some cases, several agitators, which can be operated independently of one another, are installed in one agitator tank in order to operate the best agitator for several process steps.
Multi-stage stirrers are often used in practice, as shown, for example, in the illustration on the right. While the lower level realizes a high power input with high shear and a residual quantity stirring function, the upper stirring element is suitable as an axial conveyor to achieve a good homogenization of the container contents.
## Prévost's theorem
Of the Prévost's theorem is a concept of physics and is used in thermodynamics.
Pierre Prévost recognized in 1809 that the heat exchange between two different hot bodies A and B in a closed system proceeds as follows: The warmer body A radiates a certain amount $S_A$ of radiant energy onto the colder body B. At the same time, however, the body A also receives a smaller amount $S_B$ from it. $S_A$ also represents the radiation energy absorbed by B. Since A emits more energy than it receives, it cools down slowly, while conversely B warms up until both have the same temperature. In this dynamic state of equilibrium, the exchanged heat quantities $S_A$ and $S_B$ are the same.
The designation Prévost's theorem or Prévost's theory of heat exchange only has historical significance, since the described connection forms the natural basis of the radiation laws today.
## The correction for the heat exchange between a calorimeter and the environment
Freiburg i. Br., December 20, 1942. Thermochemical Research Center, University Medical Clinic.
Freiburg i. Br., December 20, 1942. Thermochemical Research Center, University Medical Clinic.
### Abstract
King and Grover have asserted that the correction for heat exchange with the environment can only be calculated in rare cases on the basis of Newton's law of cooling. B. more complicated approaches would have to be made when working with the calorimetric bomb. That is denied. Even if the test conditions vary greatly, the simple cooling law “integration with the slide rule” means that matching values are obtained. Even if the measurement accuracy is increased (platinum resistance thermometer), the simple law still applies to temperature differences of 8. Adiabatic and compensatory work, where the correction is completely or almost completely omitted, has delivered the same values as poikilothermal measurements with correction according to Newton.
## Heat exchange - chemistry and physics
Stefan Pietrusky, Learning Level Up
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When does a heat exchange occur? Which body gives off heat and which body absorbs heat? How can one make use of the heat exchange? What does the basic law of heat exchange say? When will there be no more heat exchange? You can find all the important information about heat exchange simply explained in this content.
## Richmann's rule of mixing
the Richmann's rule of mixing is a rule for determining the Mixing temperaturethat occurs when two (or more) bodies of different temperatures are brought together. It is named after its discoverer Georg Wilhelm Richmann. & # 911 & # 93
Under the condition that there is no change in the aggregate state and the system of the bodies is closed (in particular only heat exchange between the bodies is possible), the following applies:
The resolved formula according to the mixture temperature:
$T_ = frac cdot c_ <1> cdot T_ <1> + m_ <2> cdot c_ <2> cdot T_ <2>> cdot c_ <1> + m_ <2> cdot c_ <2>>$
• m1, m2 stands for the mass of bodies 1 and 2,
• c1, c2 stands for the specific heat capacity of bodies 1 and 2,
• T1 stands for the temperature of the body 1, which gives off heat, i.e. the warmer one is,
• T2 stands for the temperature of body 2, which absorbs heat, i.e. the colder one,
• Tm represents the common temperature of both bodies after mixing.
After recognizing the conservation of energy, the mixing rule could be derived from the conservation of thermal energy.
Calorimetric measurements are carried out in a calorimeter. In most cases, heat is added to or withdrawn from the calorimeter and the temperature change is observed. If a thermometer is used, it should be a Beckmann thermometer or a digital thermometer (reading accuracy up to 0.01 K). There are many different calorimeters. They can be divided into the following groups.
### Anisothermal calorimeters
The calorimeter is thermally insulated from the environment. The heat exchange takes place with a liquid (liquid calorimeter) or with a metal (metal block calorimeter). This type of device is the most common in calorimetry. With clean work, accuracies of up to 0.01% are possible. This procedure is used when the heat exchange takes a maximum of 20 minutes.
#### Liquid calorimeter
It consists of a double-walled copper container, the space between which is filled with water and is intended to ensure a temperature-constant environment in the inner calorimeter. The calorimeter vessel made of thin sheet metal is placed on a heat-insulated surface. Ordinary water is used as the calorimeter liquid, but other liquids can also be used. A stirrer, the speed of which must remain constant, ensures better heat exchange. The change in temperature is measured with a thermometer. see also: Bomb calorimeter to determine the calorific value.
With these devices, the temperature difference between the calorimeter liquid and the vessel jacket is constantly compensated by heating or cooling. Both processes must take place at the same speed. The slower the heat transfer to the calorimeter, the easier it is to achieve this (20 to 60 minutes).
### Isothermal calorimeter
With these devices, the amount of heat is taken from certain substances, which undergo a phase change in the process. The temperatures therefore remain constant during the experiment. These devices are also known as phase change calorimeters. They are used for slow reactions that take several hours.
#### Ice calorimeter
For measurements of heat quantities at 0 ° C, this calorimeter is one of the most accurate. In the picture is the room b with dist. Filled with water and is in contact with the mercury at the bottom of the vessel. The mercury is in the capillary to the point m filled. Around the bottom of the pipe a An ice coat is created by placing a cold mixture in a thin-walled sample tube in a introduces. The entire calorimeter is protected from external temperature influences by an ice pack and additional insulation. The amount of heat to be measured is taken from the room V fed and given there to the ice coat, which partially melts. As a result, a certain change in volume occurs, which is a measure of the amount of heat given off and from the displacement of the mercury thread in the capillary m can be calculated.
#### Condensation calorimeter
This calorimeter, often also called a steam calorimeter, is mainly used to determine the specific heat capacity of a substance between 100 ° C and 20 ° C. Steam is used as the condensing gas. The body K to be examined is suspended from a sensitive scale by means of a fine wire and is located inside the calorimeter. If one suddenly introduces saturated water vapor, which has been freed from dripping liquid, into this space, a certain amount of vapor will condense on the initially cold body until the body has assumed the temperature of the vapor. An amount of heat of ΔQ = r · m has passed to the body (ΔQ = amount of heat r = heat of condensation m = mass of condensed steam).
A thin-walled platinum bowl attached to the bottom of the body protects against water dripping. The buoyancy that occurs due to the steam flow must be taken into account. The method can provide very precise values.
#### Heat exchange calorimeter
In the case of reactions that extend over several hours to a few months, a quick and complete exchange of heat with the environment is ensured. The speed is measured as a function of time.
## Description Heat flow (convection)
Anton bathes in a lake on a beautiful summer day. Then he is frightened in the water. I explain to him that he was startled by a flow of heat. We first repeat the terms “thermal energy” and “heat”. I would also like to remind you of heat sources as energy converters with a few examples. Although we have almost understood it, let's now explain what the heat flow has to do with it. Now we also understand how warm and cold air move in a heated room. In the penultimate section I give you an explanation of the Gulf Stream. Finally, we will discuss how a cooling tower works in a power plant. Have fun watching the video!
### Transcript Heat flow (convection)
Hello and welcome. This video is called “Heat Flow (Convection)”. You already know heat, energy converters, thermal energy. Afterwards you can explain what heat flow, convection is. And you can describe examples of it. 1. The shock of bathing. It's a hot summer day. Anton bathes in a swimming lake. “Oh, that's nice. Refreshing and not too cold. Oops, what's that? Someone poured cold water into the lake? " “No, you came into a cold current. Stupid current. " - “Oh, you know, even if it was cold, that was a flow of heat. It is also called convection. " - "That sounds interesting, tell me more about it." - "Willingly." 2. Thermal energy and heat. What is it? I've brewed a huge tea here. He's here in the air. The tea is warmer than the air. The tea gives off heat to the air. The ability of a body to give off heat to the colder environment is called thermal energy. The symbol E is used for thistherm. Let's say we have a warm body. Around him is the environment. The environment should be cold. Then heat is transferred from the body to the environment. This shows that the body has thermal energy at its disposal. 3. Heat sources as energy converters. You can already see that we are repeating a little. The red light lamp is a heat source and an energy converter. Try Anton once. It's nice and warm. Electrical energy is converted into heat. And there is also light. The sun is also a source of heat. It converts nuclear energy into heat. And the earth is also a source of heat. It converts light energy from the sun into heat. One last example. A thermal power plant. Here chemical energy is converted into heat. 4. Heat transfer by heat flow. One also says convection. The candle heats air. The warm air rises now. This is because their density is relatively small. On the way up, the air cools down a little. Heat is transferred from warmer to cooler air. The cold air now goes down again. Because their density is relatively large. Once at the bottom, heat is transferred from the candle to air and so on. The process starts all over again. In liquids and gases, the heat can be transferred by heat flow, convection. Thermal energy is carried along with the flowing gas or the flowing liquid. 5. We heat. This is our room. And that's the heater. The heater warms the air and the air becomes lighter, the density decreases. Now the air rises. Right Anton, the warm air doesn't last long upstairs. The air is still moving a bit up in the room. Exactly. It gives off heat to the colder environment and thus becomes cold itself. Its density increases, it becomes heavier and it sinks to the ground. The cold air now moves to the heater. That's right Anton. This happens because warm air is carried away by the heater itself. Heat is exchanged upstairs in the room, between warm and cold air. There is also an exchange of heat between the warm heating and the cold air. 6. Ocean currents. There are different ocean currents in the world's oceans. The most important ocean current for us Europeans is the Gulf Stream. The Gulf Stream is an ocean current from America to Europe. It's relatively cold in Europe. The cold water sinks to the bottom. Right Anton. Because its density is quite large. This creates a pull between Europe and America. It is warm at the beginning of the Gulf Stream. The warm water reaches Europe via the suction. Besides, America is pushing it away. Because its density becomes smaller, it expands. The cold is moving to America. A gradual warming takes place, the movement also comes about through a suction, because the warm water moves from America to Europe. As you can see, for this to work there must be very deep water, like in the ocean. 7. Cooling towers. Cooling towers are of great importance in industry. For the most part, they are used in power plants. Their function is to dissipate heat. And something like this is what a cooling tower looks like there. It has the shape of a tapered hollow cylinder. He stands at a height of one meter, above the ground, on pillars. Below the tower is a basin with cooling water. Distribution pipes are attached at a suitable distance above. There are nozzles on these. Hot water is sprayed from the nozzles into the water basin. This causes the air above the water basin to heat up. The air expands and rises together with water vapor in the cooling tower. There is suction. Cold air is now sucked in at the edges of the cooling tower. This is called the chimney effect. So-called droplet separators are located on the inside of the cooling tower. Part of the water vapor condenses and flows back into the tower. This is called trickling. The rest of the water vapor leaves the tower and forms clouds. The water returns to the earth as rain. That was another film by Andre Otto. Also many thanks to Anton. Bye. I wish you all well and good luck. Bye.
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2022-05-22 04:38:32
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https://davesquared.net/2007/04/converting-values-for-datacolumn-to.html
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# Converting values for a DataColumn to an Array
I was asked this question today: is there an easy way (i.e. single method call) to convert the row values for a particular DataColumn of a DataTable into a String array. I don’t necessarily recommend doing this, but yes:
String[] rowValuesForColumn =
Array.ConvertAll<DataRow, String>(
dataTable.Select(),
delegate(DataRow row) { return (String) row[columnName]; }
);
You can obviously package all this into a generic method to convert to any type of array.
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2019-02-20 18:19:29
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https://www.zbmath.org/?q=ai%3Abahmanian.m-amin+se%3A00000919+in%3A00335600
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# zbMATH — the first resource for mathematics
Connected Baranyai’s theorem. (English) Zbl 1324.05053
Let $$\lambda K_{n}^{h}$$ be the complete $$h$$-uniform hypergraph with $$n$$ vertices and every edge of multiplicity $$\lambda$$ and $$r_1+r_2+\dots +r_k=\lambda {{n-1}\choose{h-1}}$$. Then, an $$(r_1,r_2,\dots,r_k)$$-factorization of $$\lambda K_{n}^{h}$$ is a partition of the edge set of $$\lambda K_{n}^{h}$$ into factors $$F_1,F_2,\dots,F_k$$, where $$F_i$$ is $$r_i$$-regular. When $$r_1=r_2=\dots=r_k=r$$, then the factorization is called an $$r$$-factorization.
The well known Baranyai’s theorem states that $$K_{n}^{h}$$ can be decomposed into edge-disjoint $$r$$-regular factors if and only if $$h$$ divides $$rn$$ and $$r$$ divides $${n-1}\choose{h-1}$$.
The author generalizes Baranyai’s result by proving that $$\lambda K_{n}^{h}$$ has an $$(r_1,r_2,\dots ,r_k)$$-factorization if and only if $$h$$ divides $$r_i n$$ for all $$r_i$$, $$1\leq i\leq k,$$ and $$\sum_{i=1}^{k}r_i=\lambda{{n-1}\choose{h-1}}$$. Moreover, the result is strengthening Baranyai’s theorem because it further asserts that whenever $$r_i\geq 2$$, the $$r_i$$-regular factor $$F_i$$ can be guaranteed to be connected.
##### MSC:
05C15 Coloring of graphs and hypergraphs 05C40 Connectivity 05C51 Graph designs and isomorphic decomposition 05C65 Hypergraphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05B40 Combinatorial aspects of packing and covering 05B05 Combinatorial aspects of block designs
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##### References:
[1] Bahmanian, M A; Rodger, C A, Multiply balanced edge colorings of multigraphs, J. Graph Theory, 70, 297-317, (2012) · Zbl 1244.05085 [2] Bahmanian, M A, Detachments of amalgamated 3-uniform hypergrpahs I: factorization consequences, J. Combin. Designs, 20, 527-549, (2012) · Zbl 1258.05087 [3] Bahmanian, M A, Detachments of hypergraphs I: the berge-Johnson problem, Combin. Probab. Comput., 21, 483-495, (2012) · Zbl 1247.05161 [4] Bahmanian, M A; Rodger, C A, Embedding edge-colorings into Hamiltonian decompositions, Graphs and Combinatorics, 29, 747-755, (2013) · Zbl 1268.05050 [5] Bahmanian, M A; Rodger, C A, Extending partial edge-colorings of complete 3-uniform hypergraphs to r-factorizations, J. Graph Theory, 73, 216-224, (2013) · Zbl 1264.05088 [6] Baranyai, Zs, On the factorization of the complete uniform hypergraph, Colloq. Math. Soc. Janos Bolyai, 10, 91-108, (1975) · Zbl 0306.05137 [7] Hilton, A J W, Hamiltonian decompositions of complete graphs, J. Combin. Theory B, 36, 125-134, (1984) · Zbl 0542.05044 [8] Bahmanian, M A; Rodger, C A, Multiply balanced edge colorings of multigraphs, J. Graph Theory, 70, 297-317, (2012) · Zbl 1244.05085 [9] Bahmanian, M A, Detachments of amalgamated 3-uniform hypergrpahs I: factorization consequences, J. Combin. Designs, 20, 527-549, (2012) · Zbl 1258.05087 [10] Bahmanian, M A, Detachments of hypergraphs I: the berge-Johnson problem, Combin. Probab. Comput., 21, 483-495, (2012) · Zbl 1247.05161 [11] Bahmanian, M A; Rodger, C A, Embedding edge-colorings into Hamiltonian decompositions, Graphs and Combinatorics, 29, 747-755, (2013) · Zbl 1268.05050 [12] Bahmanian, M A; Rodger, C A, Extending partial edge-colorings of complete 3-uniform hypergraphs to r-factorizations, J. Graph Theory, 73, 216-224, (2013) · Zbl 1264.05088 [13] Baranyai, Zs, On the factorization of the complete uniform hypergraph, Colloq. Math. Soc. Janos Bolyai, 10, 91-108, (1975) · Zbl 0306.05137 [14] Hilton, A J W, Hamiltonian decompositions of complete graphs, J. Combin. Theory B, 36, 125-134, (1984) · Zbl 0542.05044 [15] Hilton, A J W; Rodger, C A, Hamilton decompositions of complete regular $$s$$-partite graphs, Discrete Math., 58, 63-78, (1986) · Zbl 0593.05047 [16] Johnson, M, Amalgamations of factorizations of complete graphs, J. Combin. Theory B, 97, 597-611, (2007) · Zbl 1153.05055 [17] Katona, G O H, Rényi and the combinatorial search problems, Studia Sci. Math. Hungar., 26, 363-378, (1991) · Zbl 0788.68004 [18] E. Lucas: Récréations Mathématiques, Vol. 2, Gauthiers Villars, Paris, 1883. · Zbl 1244.05085 [19] Nash-Williams, C St J A, Amalgamations of almost regular edge-colourings of simple graphs, J. Combin. Theory B, 43, 322-342, (1987) · Zbl 0654.05031
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.
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2021-01-23 21:48:51
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https://zbmath.org/authors/?q=ai%3Azhou.jiang.1
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## Zhou, Jiang
Compute Distance To:
Author ID: zhou.jiang.1 Published as: Zhou, Jiang
Documents Indexed: 96 Publications since 1993 Co-Authors: 24 Co-Authors with 44 Joint Publications 1,171 Co-Co-Authors
all top 5
### Co-Authors
3 single-authored 41 Bu, Changjiang 15 Sun, Lizhu 9 Wang, Wenzhe 7 Wei, Yimin 7 Wu, Lan 3 Yao, Hongmei 2 Dorignac, Jérôme 2 Huang, Shaobin 2 Li, Haifeng 2 Li, Hongbo 2 Wang, Songbai 2 Wei, Yuanpeng 2 Zhao, Yichun 2 Zhou, Xiuqing 1 Akay, Hasan U. 1 Bai, Yang 1 Bu, Tianyi 1 Chen, Jinyang 1 Chien, Stanley Y.-P. 1 Coatrieux, Jean-Louis 1 Coiffet, Philippe 1 Dai, Dong 1 Ecer, Akin 1 Fan, Yamin 1 Feng, Xinlei 1 Gao, Wei 1 Gao, Weidong 1 Gu, Weiqi 1 Hall, Frank J. 1 Han, Guo-Niu 1 He, Shuibing 1 Huang, Heng 1 Huang, Hongbing 1 Huang, Lihong 1 Irani, Shahrukh A. 1 Jing, Guangming 1 Jordaan, Kerstin 1 Kikura, Hiroshige 1 Kou, Supeng 1 Lai, Hong-Jian 1 Li, Rongwu 1 Li, Zhongshan 1 Liu, Xiaogang 1 Long, Bingsong 1 Lu, Yuan 1 Luo, Limin 1 Ma, Liang 1 Qi, Guangxiao 1 Sanctuary, B. C. 1 Shen, Jihong 1 Shu, Huazhong 1 Skrynnikov, N. R. 1 Tsuzuki, Nobuo 1 Udai, T. K. 1 Van Dam, Edwin Robert 1 Wang, Hui 1 Wang, Shi-Zhu 1 Wang, Zhongyu 1 Wu, Wanyu 1 Wu, Ya-Jie 1 Xie, Wei 1 Xie, Xishun 1 Yan, Bo 1 Yan, Hong 1 Ye, Yingyu 1 Yu, Lijun 1 Yu, Shuang 1 Zagrodny, Chris 1 Zhang, Houkun 1 Zheng, Baodong
all top 5
### Serials
15 Linear Algebra and its Applications 6 Linear and Multilinear Algebra 4 International Journal of Control 4 The Electronic Journal of Combinatorics 3 Discrete Applied Mathematics 3 Discrete Mathematics 3 Physics Letters. A 3 International Journal of Systems Science. Principles and Applications of Systems and Integration 2 Indian Journal of Pure & Applied Mathematics 2 Ars Combinatoria 2 Insurance Mathematics & Economics 2 Statistics & Probability Letters 2 Physica D 2 Filomat 2 Frontiers of Mathematics in China 1 Acta Mechanica 1 Analysis Mathematica 1 Applicable Analysis 1 International Journal of Heat and Mass Transfer 1 Chaos, Solitons and Fractals 1 Czechoslovak Mathematical Journal 1 IEEE Transactions on Automatic Control 1 IEEE Transactions on Computers 1 Journal of Applied Probability 1 Journal of Computational and Applied Mathematics 1 Advances in Applied Mathematics 1 International Journal of Production Research 1 Graphs and Combinatorics 1 Journal of Theoretical Probability 1 Mathematical and Computer Modelling 1 Journal of Robotic Systems 1 CAD. Computer-Aided Design 1 Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences 1 Applied Mathematical Modelling 1 Journal of Physics A: Mathematical and General 1 Pattern Recognition 1 Journal of Algebraic Combinatorics 1 Turkish Journal of Mathematics 1 Bulletin des Sciences Mathématiques 1 Integral Transforms and Special Functions 1 ELA. The Electronic Journal of Linear Algebra 1 Positivity 1 Journal of Inequalities and Applications 1 Mechanism and Machine Theory 1 Communications in Nonlinear Science and Numerical Simulation 1 Physica Scripta 1 Journal of Northwest Normal University. Natural Science 1 Journal of the Australian Mathematical Society 1 Journal of Applied Mathematics 1 MATCH - Communications in Mathematical and in Computer Chemistry 1 International Journal of Computational Methods 1 Applicable Analysis and Discrete Mathematics 1 Advances in Differential Equations and Control Processes 1 Annals of Functional Analysis
all top 5
### Fields
39 Combinatorics (05-XX) 25 Linear and multilinear algebra; matrix theory (15-XX) 10 Systems theory; control (93-XX) 9 Harmonic analysis on Euclidean spaces (42-XX) 6 Probability theory and stochastic processes (60-XX) 5 Functional analysis (46-XX) 4 Computer science (68-XX) 4 Fluid mechanics (76-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Real functions (26-XX) 3 Ordinary differential equations (34-XX) 3 Numerical analysis (65-XX) 3 Mechanics of particles and systems (70-XX) 3 Quantum theory (81-XX) 3 Biology and other natural sciences (92-XX) 3 Information and communication theory, circuits (94-XX) 2 Partial differential equations (35-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Number theory (11-XX) 1 Group theory and generalizations (20-XX) 1 Functions of a complex variable (30-XX) 1 Special functions (33-XX) 1 Difference and functional equations (39-XX) 1 Operator theory (47-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Mechanics of deformable solids (74-XX) 1 Optics, electromagnetic theory (78-XX) 1 Operations research, mathematical programming (90-XX)
### Citations contained in zbMATH Open
71 Publications have been cited 502 times in 253 Documents Cited by Year
Brualdi-type eigenvalue inclusion sets of tensors. Zbl 1320.15019
Bu, Changjiang; Wei, Yuanpeng; Sun, Lizhu; Zhou, Jiang
2015
The inverse, rank and product of tensors. Zbl 1286.15030
Bu, Changjiang; Zhang, Xu; Zhou, Jiang; Wang, Wenzhe; Wei, Yimin
2014
Some results on resistance distances and resistance matrices. Zbl 1306.05046
Sun, Lizhu; Wang, Wenzhe; Zhou, Jiang; Bu, Changjiang
2015
Resistance distance in subdivision-vertex join and subdivision-edge join of graphs. Zbl 1295.05096
Bu, Changjiang; Yan, Bo; Zhou, Xiuqing; Zhou, Jiang
2014
Resistance distance and Kirchhoff index of $$R$$-vertex join and $$R$$-edge join of two graphs. Zbl 1315.05129
Liu, Xiaogang; Zhou, Jiang; Bu, Changjiang
2015
Some spectral properties of uniform hypergraphs. Zbl 1302.05114
Zhou, Jiang; Sun, Lizhu; Wang, Wenzhe; Bu, Changjiang
2014
Pointwise frequency responses framework for stability analysis in periodically time-varying systems. Zbl 1358.93144
Zhou, J.; Qian, H. M.
2017
Some results on Kirchhoff index and degree-Kirchhoff index. Zbl 1461.05058
Huang, Shaobin; Zhou, Jiang; Bu, Changjiang
2016
Signless Laplacian spectral characterization of the cones over some regular graphs. Zbl 1241.05069
Bu, Changjiang; Zhou, Jiang
2012
Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra. Zbl 1242.05160
Bu, Changjiang; Zhou, Jiang
2012
Group inverse for block matrices and some related sign analysis. Zbl 1246.15009
Zhou, Jiang; Bu, Changjiang; Wei, Yimin
2012
Laplacian spectral characterization of some graphs obtained by product operation. Zbl 1242.05170
Zhou, Jiang; Bu, Changjiang
2012
Principal eigenvectors and spectral radii of uniform hypergraphs. Zbl 1436.05062
Li, Haifeng; Zhou, Jiang; Bu, Changjiang
2018
Some characterizations of $$M$$-tensors via digraphs. Zbl 1333.15014
Zhou, Jiang; Sun, Lizhu; Wei, Yuanpeng; Bu, Changjiang
2016
On the resistance matrix of a graph. Zbl 1333.05191
Zhou, Jiang; Wang, Zhongyu; Bu, Changjiang
2016
A note on block representations of the group inverse of Laplacian matrices. Zbl 1250.15003
Bu, Changjiang; Sun, Lizhu; Zhou, Jiang; Wei, Yimin
2012
Spectral characterization of line graphs of starlike trees. Zbl 1272.05169
Zhou, Jiang; Bu, Changjiang
2013
Some inequalities for the Hadamard product of tensors. Zbl 1391.65090
Sun, Lizhu; Zheng, Baodong; Zhou, Jiang; Yan, Hong
2018
Some block matrices with signed Drazin inverses. Zbl 1259.15008
Zhou, Jiang; Bu, Changjiang; Wei, Yimin
2012
Occupation times of hyper-exponential jump diffusion processes with application to price step options. Zbl 1333.60177
Wu, Lan; Zhou, Jiang
2016
Occupation times of Lévy-driven Ornstein-Uhlenbeck processes with two-sided exponential jumps and applications. Zbl 1377.60056
Zhou, Jiang; Wu, Lan; Bai, Yang
2017
E-cospectral hypergraphs and some hypergraphs determined by their spectra. Zbl 1300.05195
Bu, Changjiang; Zhou, Jiang; Wei, Yimin
2014
Resistance characterizations of equiarboreal graphs. Zbl 1370.05087
Zhou, Jiang; Sun, Lizhu; Bu, Changjiang
2017
The time of deducting fees for variable annuities under the state-dependent fee structure. Zbl 1314.91149
Zhou, Jiang; Wu, Lan
2015
Stability analysis of sampled-data systems via open-/closed-loop characteristic polynomials contraposition. Zbl 1371.93172
Zhou, J.; Qian, H. M.
2017
On the nullity of connected graphs with least eigenvalue at least $$-2$$. Zbl 1313.05260
Zhou, Jiang; Sun, Lizhu; Yao, Hongmei; Bu, Changjiang
2013
Spectral characterizations of the corona of a cycle and two isolated vertices. Zbl 1298.05196
Bu, Changjiang; Zhou, Jiang; Li, Hongbo; Wang, Wenzhe
2014
Some results for the periodicity and perfect state transfer. Zbl 1229.05205
Zhou, Jiang; Bu, Changjiang; Shen, Jihong
2011
Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs. Zbl 1361.05072
Bu, Changjiang; Fan, Yamin; Zhou, Jiang
2016
$$l^{k,s}$$-singular values and spectral radius of partially symmetric rectangular tensors. Zbl 1362.15021
Yao, Hongmei; Long, Bingsong; Bu, Changjiang; Zhou, Jiang
2016
Signless Laplacian spectral characterization of graphs with isolated vertices. Zbl 1462.05228
Huang, Shaobin; Zhou, Jiang; Bu, Changjiang
2016
Image reconstruction from limited range projections using orthogonal moments. Zbl 1118.68183
Shu, H. Z.; Zhou, J.; Han, G. N.; Luo, L. M.; Coatrieux, J. L.
2007
Some results on the Drazin inverse of anti-triangular matrices. Zbl 1303.15004
Bu, Changjiang; Sun, Lizhu; Zhou, Jiang; Wei, Yimin
2013
Spectral determination of some chemical graphs. Zbl 1289.05271
Bu, Changjiang; Zhou, Jiang; Li, Hongbo
2012
A note on the multiplicities of graph eigenvalues. Zbl 1283.05163
Bu, Changjiang; Zhang, Xu; Zhou, Jiang
2014
Fast synchronization of directionally coupled chaotic systems. Zbl 1349.34244
Cheng, S.; Ji, J. C.; Zhou, J.
2013
Monotonicity of zeros of polynomials orthogonal with respect to an even weight function. Zbl 1297.33009
Jordaan, K.; Wang, H.; Zhou, J.
2014
Line star sets for Laplacian eigenvalues. Zbl 1292.05183
Zhou, Jiang; Sun, Lizhu; Wang, Wenzhe; Bu, Changjiang
2014
On matrices whose Moore-Penrose inverses are ray unique. Zbl 1387.15035
Bu, Changjiang; Gu, Weiqi; Zhou, Jiang; Wei, Yimin
2016
Communication with spatial periodic chaos synchronization. Zbl 1123.94372
Zhou, J.; Huang, Hongbing; Qi, Guangxiao; Yang, P.; Xie, Xishun
2005
Minimum (maximum) rank of sign pattern tensors and sign nonsingular tensors. Zbl 1321.15007
Bu, Changjiang; Wang, Wenzhe; Sun, Lizhu; Zhou, Jiang
2015
An argument-principle-based framework for structural and spectral characteristics in linear dynamical systems. Zbl 1380.93084
Zhou, J.; Qian, H. M.; Lu, X. B.
2017
The enumeration of spanning tree of weighted graphs. Zbl 1470.05082
Zhou, Jiang; Bu, Changjiang
2021
Buckling analysis of a plate with built-in rectangular delamination by strip distributed transfer function method. Zbl 1071.74022
Li, D.; Tang, G.; Zhou, J.; Lei, Y.
2005
Infinite-time and finite-time synchronization of coupled harmonic oscillators. Zbl 1262.34057
Cheng, S.; Ji, J. C.; Zhou, J.
2011
Inverse Perron values and connectivity of a uniform hypergraph. Zbl 1402.05134
Bu, Changjiang; Li, Haifeng; Zhou, Jiang
2018
Rank conditions for sign patterns that allow diagonalizability. Zbl 1435.05117
Feng, Xin-Lei; Gao, Wei; Hall, Frank J.; Jing, Guangming; Li, Zhongshan; Zagrodny, Chris; Zhou, Jiang
2020
Valuing equity-linked death benefits with a threshold expense strategy. Zbl 1318.91128
Zhou, Jiang; Wu, Lan
2015
Singular value inclusion sets of rectangular tensors. Zbl 1423.15026
Yao, Hongmei; Zhang, Can; Liu, Lei; Zhou, Jiang; Bu, Changjiang
2019
State transfer and star complements in graphs. Zbl 1297.05154
Zhou, Jiang; Bu, Changjiang
2014
Occupation times of refracted double exponential jump diffusion processes. Zbl 1397.60111
Zhou, Jiang; Wu, Lan
2015
Optimal mechanism design using interior-point methods. Zbl 1047.70556
Zhang, Xiong; Zhou, J.; Ye, Yingyu
2000
Multiresolution filtering with application to image segmentation. Zbl 1185.94025
Zhou, J.; Fang, X.; Ghosh, B. K.
1996
Modeling of transport phenomena in hybrid laser-MIG keyhole welding. Zbl 1144.80362
Zhou, J.; Tsai, H. L.
2008
The distribution of refracted Lévy processes with jumps having rational Laplace transforms. Zbl 1400.60070
Zhou, Jiang; Wu, Lan
2017
Spectral properties of general hypergraphs. Zbl 1401.05207
Sun, Lizhu; Zhou, Jiang; Bu, Changjiang
2019
Laplacian spectral characterization of some graph join. Zbl 1408.05083
Sun, Lizhu; Wang, Wenzhe; Zhou, Jiang; Bu, Changjiang
2015
Occupation times of general Lévy processes. Zbl 1408.60037
Wu, Lan; Zhou, Jiang; Yu, Shuang
2017
On block triangular matrices with signed Drazin inverse. Zbl 1349.15084
Bu, Changjiang; Wang, Wenzhe; Zhou, Jiang; Sun, Lizhu
2014
The direct and inverse problem for an inclusion within a heat-conducting layered medium. Zbl 1361.65074
Guo, J.; Yan, G.; Zhou, J.
2017
Edge-disjoint spanning trees and forests of graphs. Zbl 1465.05086
Zhou, Jiang; Bu, Changjiang; Lai, Hong-Jian
2021
Eigenvalues and clique partitions of graphs. Zbl 1468.05175
Zhou, Jiang; Bu, Changjiang
2021
A gradient algorithm for stereo matching without correspondence. Zbl 0863.68115
Zhou, J.; Ghosh, B. K.
1996
Classification of cyclic initial states and geometric phase for the $$\text{spin-}j$$ system. Zbl 0843.58127
Skrynnikov, N. R.; Zhou, J.; Sanctuary, B. C.
1994
FSMT: A feature solid-modelling tool for feature-based design and manufacture. Zbl 0795.68193
Duan, W.; Zhou, J.; Lai, K.
1993
Existence, properties and trajectory specification of generalised multi-agent flocking. Zbl 1416.93037
Zhou, J.; Wang, C.; Qian, H. M.
2019
Complex scaling circle criteria for TI. Zbl 1416.93165
Zhou, J.
2019
Discrete breathers in nonlinear Schrödinger hypercubic lattices with arbitrary power nonlinearity. Zbl 1153.39307
Dorignac, J.; Zhou, J.; Campbell, D. K.
2008
Production flow analysis and simplification toolkit (PFAST). Zbl 0944.90553
Irani, S. A.; Zhang, H.; Zhou, J.; Huang, H.; Udai, T. K.; Subramanian, S.
2000
Fluid simulation using an adaptive semi-discrete central-upwind scheme. Zbl 1198.76073
Cai, L.; Zhou, J.; Feng, J.-H.; Xie, W.-X.
2007
Stability analysis and stabilisation in linear continuous-time periodic systems by complex scaling. Zbl 1453.93196
Zhou, J.
2020
The enumeration of spanning tree of weighted graphs. Zbl 1470.05082
Zhou, Jiang; Bu, Changjiang
2021
Edge-disjoint spanning trees and forests of graphs. Zbl 1465.05086
Zhou, Jiang; Bu, Changjiang; Lai, Hong-Jian
2021
Eigenvalues and clique partitions of graphs. Zbl 1468.05175
Zhou, Jiang; Bu, Changjiang
2021
Rank conditions for sign patterns that allow diagonalizability. Zbl 1435.05117
Feng, Xin-Lei; Gao, Wei; Hall, Frank J.; Jing, Guangming; Li, Zhongshan; Zagrodny, Chris; Zhou, Jiang
2020
Stability analysis and stabilisation in linear continuous-time periodic systems by complex scaling. Zbl 1453.93196
Zhou, J.
2020
Singular value inclusion sets of rectangular tensors. Zbl 1423.15026
Yao, Hongmei; Zhang, Can; Liu, Lei; Zhou, Jiang; Bu, Changjiang
2019
Spectral properties of general hypergraphs. Zbl 1401.05207
Sun, Lizhu; Zhou, Jiang; Bu, Changjiang
2019
Existence, properties and trajectory specification of generalised multi-agent flocking. Zbl 1416.93037
Zhou, J.; Wang, C.; Qian, H. M.
2019
Complex scaling circle criteria for TI. Zbl 1416.93165
Zhou, J.
2019
Principal eigenvectors and spectral radii of uniform hypergraphs. Zbl 1436.05062
Li, Haifeng; Zhou, Jiang; Bu, Changjiang
2018
Some inequalities for the Hadamard product of tensors. Zbl 1391.65090
Sun, Lizhu; Zheng, Baodong; Zhou, Jiang; Yan, Hong
2018
Inverse Perron values and connectivity of a uniform hypergraph. Zbl 1402.05134
Bu, Changjiang; Li, Haifeng; Zhou, Jiang
2018
Pointwise frequency responses framework for stability analysis in periodically time-varying systems. Zbl 1358.93144
Zhou, J.; Qian, H. M.
2017
Occupation times of Lévy-driven Ornstein-Uhlenbeck processes with two-sided exponential jumps and applications. Zbl 1377.60056
Zhou, Jiang; Wu, Lan; Bai, Yang
2017
Resistance characterizations of equiarboreal graphs. Zbl 1370.05087
Zhou, Jiang; Sun, Lizhu; Bu, Changjiang
2017
Stability analysis of sampled-data systems via open-/closed-loop characteristic polynomials contraposition. Zbl 1371.93172
Zhou, J.; Qian, H. M.
2017
An argument-principle-based framework for structural and spectral characteristics in linear dynamical systems. Zbl 1380.93084
Zhou, J.; Qian, H. M.; Lu, X. B.
2017
The distribution of refracted Lévy processes with jumps having rational Laplace transforms. Zbl 1400.60070
Zhou, Jiang; Wu, Lan
2017
Occupation times of general Lévy processes. Zbl 1408.60037
Wu, Lan; Zhou, Jiang; Yu, Shuang
2017
The direct and inverse problem for an inclusion within a heat-conducting layered medium. Zbl 1361.65074
Guo, J.; Yan, G.; Zhou, J.
2017
Some results on Kirchhoff index and degree-Kirchhoff index. Zbl 1461.05058
Huang, Shaobin; Zhou, Jiang; Bu, Changjiang
2016
Some characterizations of $$M$$-tensors via digraphs. Zbl 1333.15014
Zhou, Jiang; Sun, Lizhu; Wei, Yuanpeng; Bu, Changjiang
2016
On the resistance matrix of a graph. Zbl 1333.05191
Zhou, Jiang; Wang, Zhongyu; Bu, Changjiang
2016
Occupation times of hyper-exponential jump diffusion processes with application to price step options. Zbl 1333.60177
Wu, Lan; Zhou, Jiang
2016
Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs. Zbl 1361.05072
Bu, Changjiang; Fan, Yamin; Zhou, Jiang
2016
$$l^{k,s}$$-singular values and spectral radius of partially symmetric rectangular tensors. Zbl 1362.15021
Yao, Hongmei; Long, Bingsong; Bu, Changjiang; Zhou, Jiang
2016
Signless Laplacian spectral characterization of graphs with isolated vertices. Zbl 1462.05228
Huang, Shaobin; Zhou, Jiang; Bu, Changjiang
2016
On matrices whose Moore-Penrose inverses are ray unique. Zbl 1387.15035
Bu, Changjiang; Gu, Weiqi; Zhou, Jiang; Wei, Yimin
2016
Brualdi-type eigenvalue inclusion sets of tensors. Zbl 1320.15019
Bu, Changjiang; Wei, Yuanpeng; Sun, Lizhu; Zhou, Jiang
2015
Some results on resistance distances and resistance matrices. Zbl 1306.05046
Sun, Lizhu; Wang, Wenzhe; Zhou, Jiang; Bu, Changjiang
2015
Resistance distance and Kirchhoff index of $$R$$-vertex join and $$R$$-edge join of two graphs. Zbl 1315.05129
Liu, Xiaogang; Zhou, Jiang; Bu, Changjiang
2015
The time of deducting fees for variable annuities under the state-dependent fee structure. Zbl 1314.91149
Zhou, Jiang; Wu, Lan
2015
Minimum (maximum) rank of sign pattern tensors and sign nonsingular tensors. Zbl 1321.15007
Bu, Changjiang; Wang, Wenzhe; Sun, Lizhu; Zhou, Jiang
2015
Valuing equity-linked death benefits with a threshold expense strategy. Zbl 1318.91128
Zhou, Jiang; Wu, Lan
2015
Occupation times of refracted double exponential jump diffusion processes. Zbl 1397.60111
Zhou, Jiang; Wu, Lan
2015
Laplacian spectral characterization of some graph join. Zbl 1408.05083
Sun, Lizhu; Wang, Wenzhe; Zhou, Jiang; Bu, Changjiang
2015
The inverse, rank and product of tensors. Zbl 1286.15030
Bu, Changjiang; Zhang, Xu; Zhou, Jiang; Wang, Wenzhe; Wei, Yimin
2014
Resistance distance in subdivision-vertex join and subdivision-edge join of graphs. Zbl 1295.05096
Bu, Changjiang; Yan, Bo; Zhou, Xiuqing; Zhou, Jiang
2014
Some spectral properties of uniform hypergraphs. Zbl 1302.05114
Zhou, Jiang; Sun, Lizhu; Wang, Wenzhe; Bu, Changjiang
2014
E-cospectral hypergraphs and some hypergraphs determined by their spectra. Zbl 1300.05195
Bu, Changjiang; Zhou, Jiang; Wei, Yimin
2014
Spectral characterizations of the corona of a cycle and two isolated vertices. Zbl 1298.05196
Bu, Changjiang; Zhou, Jiang; Li, Hongbo; Wang, Wenzhe
2014
A note on the multiplicities of graph eigenvalues. Zbl 1283.05163
Bu, Changjiang; Zhang, Xu; Zhou, Jiang
2014
Monotonicity of zeros of polynomials orthogonal with respect to an even weight function. Zbl 1297.33009
Jordaan, K.; Wang, H.; Zhou, J.
2014
Line star sets for Laplacian eigenvalues. Zbl 1292.05183
Zhou, Jiang; Sun, Lizhu; Wang, Wenzhe; Bu, Changjiang
2014
State transfer and star complements in graphs. Zbl 1297.05154
Zhou, Jiang; Bu, Changjiang
2014
On block triangular matrices with signed Drazin inverse. Zbl 1349.15084
Bu, Changjiang; Wang, Wenzhe; Zhou, Jiang; Sun, Lizhu
2014
Spectral characterization of line graphs of starlike trees. Zbl 1272.05169
Zhou, Jiang; Bu, Changjiang
2013
On the nullity of connected graphs with least eigenvalue at least $$-2$$. Zbl 1313.05260
Zhou, Jiang; Sun, Lizhu; Yao, Hongmei; Bu, Changjiang
2013
Some results on the Drazin inverse of anti-triangular matrices. Zbl 1303.15004
Bu, Changjiang; Sun, Lizhu; Zhou, Jiang; Wei, Yimin
2013
Fast synchronization of directionally coupled chaotic systems. Zbl 1349.34244
Cheng, S.; Ji, J. C.; Zhou, J.
2013
Signless Laplacian spectral characterization of the cones over some regular graphs. Zbl 1241.05069
Bu, Changjiang; Zhou, Jiang
2012
Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra. Zbl 1242.05160
Bu, Changjiang; Zhou, Jiang
2012
Group inverse for block matrices and some related sign analysis. Zbl 1246.15009
Zhou, Jiang; Bu, Changjiang; Wei, Yimin
2012
Laplacian spectral characterization of some graphs obtained by product operation. Zbl 1242.05170
Zhou, Jiang; Bu, Changjiang
2012
A note on block representations of the group inverse of Laplacian matrices. Zbl 1250.15003
Bu, Changjiang; Sun, Lizhu; Zhou, Jiang; Wei, Yimin
2012
Some block matrices with signed Drazin inverses. Zbl 1259.15008
Zhou, Jiang; Bu, Changjiang; Wei, Yimin
2012
Spectral determination of some chemical graphs. Zbl 1289.05271
Bu, Changjiang; Zhou, Jiang; Li, Hongbo
2012
Some results for the periodicity and perfect state transfer. Zbl 1229.05205
Zhou, Jiang; Bu, Changjiang; Shen, Jihong
2011
Infinite-time and finite-time synchronization of coupled harmonic oscillators. Zbl 1262.34057
Cheng, S.; Ji, J. C.; Zhou, J.
2011
Modeling of transport phenomena in hybrid laser-MIG keyhole welding. Zbl 1144.80362
Zhou, J.; Tsai, H. L.
2008
Discrete breathers in nonlinear Schrödinger hypercubic lattices with arbitrary power nonlinearity. Zbl 1153.39307
Dorignac, J.; Zhou, J.; Campbell, D. K.
2008
Image reconstruction from limited range projections using orthogonal moments. Zbl 1118.68183
Shu, H. Z.; Zhou, J.; Han, G. N.; Luo, L. M.; Coatrieux, J. L.
2007
Fluid simulation using an adaptive semi-discrete central-upwind scheme. Zbl 1198.76073
Cai, L.; Zhou, J.; Feng, J.-H.; Xie, W.-X.
2007
Communication with spatial periodic chaos synchronization. Zbl 1123.94372
Zhou, J.; Huang, Hongbing; Qi, Guangxiao; Yang, P.; Xie, Xishun
2005
Buckling analysis of a plate with built-in rectangular delamination by strip distributed transfer function method. Zbl 1071.74022
Li, D.; Tang, G.; Zhou, J.; Lei, Y.
2005
Optimal mechanism design using interior-point methods. Zbl 1047.70556
Zhang, Xiong; Zhou, J.; Ye, Yingyu
2000
Production flow analysis and simplification toolkit (PFAST). Zbl 0944.90553
Irani, S. A.; Zhang, H.; Zhou, J.; Huang, H.; Udai, T. K.; Subramanian, S.
2000
Multiresolution filtering with application to image segmentation. Zbl 1185.94025
Zhou, J.; Fang, X.; Ghosh, B. K.
1996
A gradient algorithm for stereo matching without correspondence. Zbl 0863.68115
Zhou, J.; Ghosh, B. K.
1996
Classification of cyclic initial states and geometric phase for the $$\text{spin-}j$$ system. Zbl 0843.58127
Skrynnikov, N. R.; Zhou, J.; Sanctuary, B. C.
1994
FSMT: A feature solid-modelling tool for feature-based design and manufacture. Zbl 0795.68193
Duan, W.; Zhou, J.; Lai, K.
1993
all top 5
### Cited by 348 Authors
41 Bu, Changjiang 28 Zhou, Jiang 16 Sun, Lizhu 16 Wei, Yimin 12 Wang, Ligong 9 He, Jun 9 Liu, Jia-bao 8 Liu, Yanmin 8 Qi, Liqun 7 Li, Chaoqian 7 Wang, Wenzhe 7 Wang, Yiju 6 Cui, Jingjing 6 Huang, Zhengge 6 Liu, Xiaogang 6 Zeydi Abdian, Ali 5 Deng, Chunli 5 Fan, Yizheng 5 Kang, Liying 5 Pan, Xiangfeng 5 Stanimirović, Predrag S. 5 Xu, Guangjun 5 Yao, Hongmei 5 Zhao, Jianxing 4 Carmona, Ángeles 4 Duan, Cunxiang 4 Li, Haifeng 4 Li, Shuchao 4 Ma, Haifeng 4 Mitjana, Margarida 4 Shan, Erfang 4 Xu, Yangyang 4 Xu, Zhong 4 Zhang, Yuan 4 Zheng, Baodong 4 Zheng, Bing 4 Zhou, Bo 3 Cao, Jinde 3 Chen, Haibin 3 Das, Kinkar Chandra 3 He, Weihua 3 Li, Xihe 3 Li, Yaotang 3 Liu, Qun 3 Lu, Pengli 3 Monsó, Enric 3 Sang, Caili 3 Shao, Jiayu 3 Sharafdini, Reza 3 Xiao, Peng 3 Yuan, Xiying 3 Zhao, Ruijuan 3 Zhu, Zhongxun 2 Atik, Fouzul 2 Bao, Yanhong 2 Bapat, Ravindra Bhalchandra 2 Behmaram, Afshin 2 Bozorgmanesh, Hassan 2 Cardoso, Kauê 2 Chang, An 2 Chen, Yannan 2 Cooper, Joshua N. 2 Devriendt, Karel 2 Fath-Tabar, Gholam Hossein 2 Gao, Ke 2 Guo, Haiyan 2 Hajarian, Masoud 2 He, Lili 2 Hou, Yaoping 2 Huang, Jing 2 Huang, Qiongxiang 2 Ji, Jun 2 Katsikis, Vasilios N. 2 Lei, Xiaoqiang 2 Li, Honghai 2 Li, Qishun 2 Li, Wen 2 Lin, Hongying 2 Liu, Fenjin 2 Liu, Lele 2 Liu, Qilong 2 Liu, Xiaoji 2 Miao, Yun 2 Mosić, Dijana 2 Palacios, Jose Luis 2 Rajesh Kannan, M. 2 Rakshith, B. R. 2 Rehman, Masood Ur 2 Shen, Jihong 2 Song, Chuanning 2 Song, Yisheng 2 Tian, Junkang 2 Wang, Jianfeng 2 Wang, Jing 2 Wang, Yanan 2 Wang, Yi 2 Wang, Yue 2 Xu, Qingxiang 2 Yan, Weigen 2 Yang, Yujun ...and 248 more Authors
all top 5
### Cited in 81 Serials
39 Linear Algebra and its Applications 22 Discrete Applied Mathematics 22 Linear and Multilinear Algebra 12 Applied Mathematics and Computation 12 Frontiers of Mathematics in China 10 Discrete Mathematics 7 Computational and Applied Mathematics 6 Indian Journal of Pure & Applied Mathematics 5 Bulletin of the Iranian Mathematical Society 5 The Electronic Journal of Combinatorics 5 Journal of Combinatorial Optimization 5 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 5 Discrete Mathematics, Algorithms and Applications 4 Filomat 4 Journal of Inequalities and Applications 4 Journal of Industrial and Management Optimization 3 Computers & Mathematics with Applications 3 Open Mathematics 2 Journal of Computational and Applied Mathematics 2 Applied Mathematics Letters 2 Journal of Scientific Computing 2 Journal of Algebraic Combinatorics 2 SIAM Journal on Scientific Computing 2 Discussiones Mathematicae. Graph Theory 2 ELA. The Electronic Journal of Linear Algebra 2 Mathematical Problems in Engineering 2 Taiwanese Journal of Mathematics 2 Discrete Dynamics in Nature and Society 2 Acta Mathematica Sinica. English Series 2 Journal of Applied Mathematics 2 AKCE International Journal of Graphs and Combinatorics 2 Iranian Journal of Mathematical Chemistry 2 Mathematics 2 Communications on Applied Mathematics and Computation 1 Journal of Mathematical Analysis and Applications 1 Physica A 1 Ukrainian Mathematical Journal 1 Acta Mathematica Vietnamica 1 Calcolo 1 Czechoslovak Mathematical Journal 1 Journal of Combinatorial Theory. Series A 1 Journal of Optimization Theory and Applications 1 Proceedings of the American Mathematical Society 1 Revista de la Unión Matemática Argentina 1 Theoretical Computer Science 1 Transactions of the American Mathematical Society 1 Advances in Applied Mathematics 1 Mathematica Numerica Sinica 1 Applied Numerical Mathematics 1 Graphs and Combinatorics 1 Machine Learning 1 Neural Computation 1 Japan Journal of Industrial and Applied Mathematics 1 Acta Mathematica Universitatis Comenianae. New Series 1 Applications of Mathematics 1 Numerical Algorithms 1 Aequationes Mathematicae 1 SIAM Review 1 ETNA. Electronic Transactions on Numerical Analysis 1 Journal of Mathematical Chemistry 1 Honam Mathematical Journal 1 Journal of Discrete Mathematical Sciences & Cryptography 1 Journal of the Australian Mathematical Society 1 Journal of Algebra and its Applications 1 Applicable Analysis and Discrete Mathematics 1 Banach Journal of Mathematical Analysis 1 Advances in Mathematical Physics 1 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 Numerical Algebra, Control and Optimization 1 Journal of Mahani Mathematical Research Center 1 Journal of Mathematical Research with Applications 1 Transactions on Combinatorics 1 Carpathian Mathematical Publications 1 Journal of Mathematics 1 East Asian Journal on Applied Mathematics 1 Special Matrices 1 Journal of Mathematical Modeling 1 Sahand Communications in Mathematical Analysis 1 Journal of Algebra, Combinatorics, Discrete Structures and Applications 1 AIMS Mathematics 1 Journal of Algebraic Systems
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### Cited in 25 Fields
153 Combinatorics (05-XX) 143 Linear and multilinear algebra; matrix theory (15-XX) 35 Numerical analysis (65-XX) 9 Operations research, mathematical programming (90-XX) 6 Biology and other natural sciences (92-XX) 4 Differential geometry (53-XX) 4 Computer science (68-XX) 3 Ordinary differential equations (34-XX) 3 Operator theory (47-XX) 3 Probability theory and stochastic processes (60-XX) 3 Quantum theory (81-XX) 3 Information and communication theory, circuits (94-XX) 2 Commutative algebra (13-XX) 2 Potential theory (31-XX) 2 Convex and discrete geometry (52-XX) 1 General and overarching topics; collections (00-XX) 1 Field theory and polynomials (12-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Functional analysis (46-XX) 1 Geometry (51-XX) 1 Optics, electromagnetic theory (78-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Systems theory; control (93-XX)
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2022-09-28 14:04:51
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|
https://docs.openvino.ai/2022.1/omz_demos_text_spotting_demo_python.html
|
# Text Spotting Python* Demo¶
This demo shows how to run Text Spotting models. Text Spotting models allow us to simultaneously detect and recognize text.
Note
Only batch size of 1 is supported.
## How It Works¶
The demo application expects a text spotting model that is split into three parts. Every model part must be in the Intermediate Representation (IR) format.
First model is Mask-RCNN like text detector with the following constraints:
• Two inputs: im_data for input image and im_info for meta-information about the image (actual height, width and scale).
• At least five outputs including:
• boxes with absolute bounding box coordinates of the input image
• scores with confidence scores for all bounding boxes
• classes with object class IDs for all bounding boxes
• raw_masks with fixed-size segmentation heat maps for all classes of all bounding boxes
• text_features with text features which are fed to Text Recognition Head further
Second model is Text Recognition Encoder that takes text_features as input and produces encoded text.
Third model is Text Recognition Decoder that takes encoded text from Text Recognition Encoder , previous symbol and hidden state. On the first step special Start Of Sequence (SOS) symbol and zero hidden state are fed to Text Recognition Decoder. The decoder produces symbols distribution, current hidden state each step until End Of Sequence (EOS) symbol is generated.
Examples of valid inputs to specify with a command-line argument -i are a path to a video file or a numeric ID of a web camera.
The demo workflow is the following:
1. The demo application reads frames from the provided input, resizes them to fit into the input image blob of the network (im_data).
2. The im_info input blob passes resulting resolution and scale of a pre-processed image to the network to perform inference of Mask-RCNN-like text detector.
3. The Text Recognition Encoder takes input from the text detector and produces output.
4. The Text Recognition Decoder takes output from the Text Recognition Encoder output as input and produces output.
5. The demo visualizes the resulting text spotting results. Certain command-line options affect the visualization:
• If you specify --show_boxes and --show_scores arguments, bounding boxes and confidence scores are also shown.
• By default, tracking is used to show text instance with the same color throughout the whole video. It assumes more or less static scene with instances in two frames being a part of the same track if intersection over union of the masks is greater than the 0.5 threshold. To disable tracking, specify the --no_track argument.
Note
By default, Open Model Zoo demos expect input with BGR channels order. If you trained your model to work with RGB order, you need to manually rearrange the default channels order in the demo application or reconvert your model using the Model Optimizer tool with the --reverse_input_channels argument specified. For more information about the argument, refer to When to Reverse Input Channels section of Embedding Preprocessing Computation.
## Preparing to Run¶
For demo input image or video files, refer to the section Media Files Available for Demos in the Open Model Zoo Demos Overview. The list of models supported by the demo is in <omz_dir>/demos/text_spotting_demo/python/models.lst file. This file can be used as a parameter for Model Downloader and Converter to download and, if necessary, convert models to OpenVINO IR format (*.xml + *.bin).
omz_downloader --list models.lst
An example of using the Model Converter:
omz_converter --list models.lst
### Supported Models¶
• text-spotting-0005-detector
• text-spotting-0005-recognizer-decoder
• text-spotting-0005-recognizer-encoder
Note
Refer to the tables Intel’s Pre-Trained Models Device Support and Public Pre-Trained Models Device Support for the details on models inference support at different devices.
## Running¶
Run the application with the -h option to see the following usage message:
usage: text_spotting_demo.py [-h] -m_m "<path>" -m_te "<path>" -m_td "<path>"
-i INPUT [--loop] [-o OUTPUT]
[-limit OUTPUT_LIMIT] [-d "<device>"]
[--delay "<num>"] [-pt "<num>"] [-a ALPHABET]
[--trd_input_prev_symbol TRD_INPUT_PREV_SYMBOL]
[--trd_input_prev_hidden TRD_INPUT_PREV_HIDDEN]
[--trd_input_encoder_outputs TRD_INPUT_ENCODER_OUTPUTS]
[--trd_output_symbols_distr TRD_OUTPUT_SYMBOLS_DISTR]
[--trd_output_cur_hidden TRD_OUTPUT_CUR_HIDDEN]
[-trt "<num>"] [--keep_aspect_ratio] [--no_track]
[--show_scores] [--show_boxes] [-r] [--no_show]
[-u UTILIZATION_MONITORS]
Options:
-h, --help Show this help message and exit.
Required. Path to an .xml file with a trained Mask-
RCNN model with additional text features output.
-m_te "<path>", --text_enc_model "<path>"
Required. Path to an .xml file with a trained text
recognition model (encoder part).
-m_td "<path>", --text_dec_model "<path>"
Required. Path to an .xml file with a trained text
recognition model (decoder part).
-i INPUT, --input INPUT
Required. An input to process. The input must be a single image,
a folder of images, video file or camera id.
--loop Optional. Enable reading the input in a loop.
-o OUTPUT, --output OUTPUT
Optional. Name of the output file(s) to save.
-limit OUTPUT_LIMIT, --output_limit OUTPUT_LIMIT
Optional. Number of frames to store in output.
If 0 is set, all frames are stored.
-d "<device>", --device "<device>"
Optional. Specify the target device to infer on, i.e. CPU, GPU.
The demo will look for a suitable plugin for device specified
(by default, it is CPU). Please refer to OpenVINO documentation
for the list of devices supported by the model.
--delay "<num>" Optional. Interval in milliseconds of waiting for a
key to be pressed.
-pt "<num>", --prob_threshold "<num>"
Optional. Probability threshold for detections
filtering.
-a ALPHABET, --alphabet ALPHABET
Optional. Alphabet that is used for decoding.
--trd_input_prev_symbol TRD_INPUT_PREV_SYMBOL
Optional. Name of previous symbol input node to text
--trd_input_prev_hidden TRD_INPUT_PREV_HIDDEN
Optional. Name of previous hidden input node to text
--trd_input_encoder_outputs TRD_INPUT_ENCODER_OUTPUTS
Optional. Name of encoder outputs input node to text
--trd_output_symbols_distr TRD_OUTPUT_SYMBOLS_DISTR
Optional. Name of symbols distribution output node
from text recognition head decoder part.
--trd_output_cur_hidden TRD_OUTPUT_CUR_HIDDEN
Optional. Name of current hidden output node from text
-trt "<num>", --tr_threshold "<num>"
Optional. Text recognition confidence threshold.
--keep_aspect_ratio Optional. Force image resize to keep aspect ratio.
--no_track Optional. Disable tracking.
--show_scores Optional. Show detection scores.
--show_boxes Optional. Show bounding boxes.
-r, --raw_output_message
Optional. Output inference results raw values.
--no_show Optional. Don't show output
-u UTILIZATION_MONITORS, --utilization_monitors UTILIZATION_MONITORS
Optional. List of monitors to show initially.
Running the application with an empty list of options yields the short version of the usage message and an error message.
To run the demo, please provide paths to the model in the IR format and to an input with images:
python3 text_spotting_demo.py \
-m_m <path_to_model>/text-spotting-0005-detector.xml \
-m_te <path_to_model>/text-spotting-0005-recognizer-encoder.xml \
-m_td <path_to_model>/text-spotting-0005-recognizer-decoder.xml \
-i 0
> NOTE : If you provide a single image as an input, the demo processes and renders it quickly, then exits. To continuously visualize inference results on the screen, apply the loop option, which enforces processing a single image in a loop.
You can save processed results to a Motion JPEG AVI file or separate JPEG or PNG files using the -o option:
• To save processed results in an AVI file, specify the name of the output file with avi extension, for example: -o output.avi.
• To save processed results as images, specify the template name of the output image file with jpg or png extension, for example: -o output_%03d.jpg. The actual file names are constructed from the template at runtime by replacing regular expression %03d with the frame number, resulting in the following: output_000.jpg, output_001.jpg, and so on. To avoid disk space overrun in case of continuous input stream, like camera, you can limit the amount of data stored in the output file(s) with the limit option. The default value is 1000. To change it, you can apply the -limit N option, where N is the number of frames to store.
> NOTE : Windows* systems may not have the Motion JPEG codec installed by default. If this is the case, you can download OpenCV FFMPEG back end using the PowerShell script provided with the OpenVINO install package and located at <INSTALL_DIR>/opencv/ffmpeg-download.ps1. The script should be run with administrative privileges if OpenVINO is installed in a system protected folder (this is a typical case). Alternatively, you can save results as images.
## Demo Output¶
The application uses OpenCV to display resulting text instances. The demo reports
• FPS : average rate of video frame processing (frames per second).
• Latency : average time required to process one frame (from reading the frame to displaying the results). You can use both of these metrics to measure application-level performance.
|
2023-03-29 22:48:42
|
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|
https://fellowhuman.com/blog/2015/12/13/cmc-n-2015-at-asilomar/
|
# fellowhuman
## CMC-N 2015 at Asilomar
Here are my notes on the 2015 CMC-North conference, plus a little neat graphical stuff (and doodles of the rooms some of the talks were held in).
### Friday Keynote: Practices for discussion
By Matt Larson. I took this largely as a pep talk about the current state of math education and our challenges in communicating our process and intention to parents; much of it was directed at issues that (thankfully) I don’t often have to deal with in my school’s culture. However, it also introduced me to the NCTM Principles to Actions document, whose most useful feature for me (as far as I can tell) is a set of “Mathematics Teaching Practices”:
• Establish mathematics goals
• Implement tasks that promote reasoning and problem solving
• Use and connect representations
• Facilitate meaningful mathematical discourse
• Pose purposeful questions
• Build procedural fluency from conceptual understanding
• Support productive struggle
• Elicit and use evidence of student thinking
Some other points were good reminders:
• Tracking is an obstacle to equity.
• A school’s best teachers should be assigned to its lowest-level classes.
• We (U.S. teachers) spend too little time planning lessons — cf. Chinese practice of putting many hours in to prep one 45-minute lesson — and should collaboratively plan one lesson per unit to counter this habit.
• Co-planning of lessons correlates strongly with better learning outcomes.
• Ask questions that make students struggle; then, avoid the temptation to jump in and rescue them when they express frustration.
Two book pointers: Words that Work and 5 Practices for Orchestrating Productive Mathematics Discussions. We should get a department copy of that second book, I think!
### Jessica Balli: Perseverance in exploration
This talk introduced Bongard problems as a tool for teaching mathematical thinking. Named after a Russian AI researcher (who published a book of them), each of these is two sets of six images; the problem is to determine a rule that includes all of one set and excludes all of the other.
Example uses of them:
• warm-up at beginning of class, with each student writing a rule and then sharing
• show first two images of each set, and share rules; then show second two, and revise rules; then show third pair, and revise rules. (She presented a template she’d devised for stating & revising rules.)
Importantly, it’s all about the discourse around the puzzles: getting all the rules out into the open, examining different correct (and incorrect) ways of stating a rule, critiquing rules, suggesting improvements to rules, and so on.
### Anna Blinstein: Writing in math
I have students do a lot of journaling already, so I was curious how someone else does it. Main points:
• Decide whether you’re having students write for learning & assessment, or for metacognition, SEL, and rapport-building. This informs your choice of prompts.
• It’s helpful to ask (in journal prompts) about non-math happenings in your students’ lives too. There’s a lot you can learn that will usefully inform your teaching.
### Julie Yu: The math of mirrors
Good presenter!
The warm-up was an exercise in which each of us taped together two small rectangular acrylic mirrors so they were hinged on one side. Julie had prepared a sheet that included angles looking like this:
The prompt was, “For each picture, predict how many arrows you will see if you place the mirrors on the two sides of the angle.”
Then we could confirm using the mirrors, producing results like this:
There was a second sheet that worked similarly, but with more interesting patterns:
We were invited to sketch what we expected to see — more challenging than it looked at first! — and, again, check with the mirrors:
The rest of this presentation went towards exploring the question, “How large does a mirror need to be for you to see your whole body in it, without wasted space above, below, or on either side?” This is a fun problem, already familiar to me from Harold Jacobs’ excellent geometry book. We’ll assume this is a wall-mounted mirror, and therefore that the mirror is perpendicular to the ground (and parallel to you, the viewer).
Warning: spoilers ahead. If you want to solve the problem, come back to this post later.
So, Julie had a volunteer stand in front of a full-length wall mirror, at a length equal to the volunteer’s height, h — which became our unit of measure. On the mirror, we taped a rectangular border around the portion of the mirror necessary for our volunteer to see herself. So, the situation looked like this:
Our volunteer is HF, looking at the mirror TB, seeing her reflection XY. Her eye level is at point I.
At this point in the session, a lot of folks were surprised at just how small the necessary rectangle of mirror really was.
Next question: If she stands twice as far from the mirror, will we need our mirror to be smaller, larger, or the same size? Here’s a diagram:
And the answer is...the same size! So, to all appearances, the mirror length is independent of the viewer’s distance from the mirror. I’ll leave the proof as an exercise; the most obvious method (to me, at least) involves similar triangles.
The session concluded with a look at cylindrical and conical mirrors — simple mylar sheets wrapped around soda cans, or taped into cone shape — and a fun exercise in which we placed beans around a cylinder to make their reflection appear as a straight line.
Actually, what was most memorable to me about this talk was the use of clotheslines as number lines, and in particular, the idea of running two clotheslines in parallel as a dynamic representation of an equation. He’s summarized this well on his blog, so I won’t retread it here.
Also — and since I’m basing this on my notes, some fifteen days later, I’m probably deviating from what he said — he offered one form of activity for recurring use: presenting wrong answers and asking the students to discuss what thinking led to them and where the misconceptions lie. For example, we discussed some exponential mix-ups he offered:
• $$2^5 = 10$$
• $$(-2)^3 = 8$$
• $$-6^2 = 36$$
• $$37^0 = 0$$
• $$100^{1/2} = 50$$
• $$7^{-2} = -49$$
• $$(y^3)^5 = y^8$$
• $$\frac{x^5}{x^9} = x^4$$
### Lew Douglas & Henri Picciotto: Transformations
This session discussed how to organize a coherent geometry curriculum around transformations, with side tracks into using 1-D translation, reflection, and dilation to study integer arithmetic (on the number line) and using rotation to study complex numbers. The main thesis was (roughly) that a transformational approach can eliminate some of the pedantic details common to traditional geometry, and lead to a more logical, consistent, and unified curriculum.
The geometry discussion included axioms for construction...
1. Two lines intersect in at most one point.
2. A circle and a line intersect in at most two points.
3. Two circles intersect in at most two points.
...and one transformation axiom, reflection preserves distance and angle measure. The speakers also walked through some transformation-based proofs of statements such as:
• For all points P and Q, there exists a reflection mapping P to Q.
• A point P is equidistant from points A and B iff it lies on the perpendicular bisector of AB.
• The SSS congruence theorem.
I was mildly bugged by their defining rotation and translation in terms of reflection — I’ve done that in the past, but I’m uneasy about it because it replaces a meaning-based definition with a mathematically opportunistic one that’s less meaningful. If we define rotation and translation more meaningfully, it’s easier to extract facts from the definitions. For example, this is how I’d define rotation:
• Rotating a point $$P$$ around a point $$C$$ by angle $$x$$ produces a point $$P’$$, with $$PC = P’C$$ and $$\angle PCP’ = x$$.
• A rotation is a transformation that rotates every point by the same angle about the same center.
From this definition we can prove that a rotation is equivalent to two reflections through intersecting lines — which to me is a beautifully interesting mathematical fact that’s incidental to the definition, not central to the nature of reflection. (Also, it serves as another illustration of bootstrapping a complex definition by defining an operation on a point and then “lifting” that operation to the plane.)
I do think comparing the above definition of rotation with the one that uses reflection is a great discussion to have with students, however; we often fail to devote much time to comparing pros and cons of definitions, but students need to learn that mathematical definitions aren’t chosen by fiat, and such comparisons are one way to get that point across.
Also worth noting: focusing on transformation opens different avenues for proving basic facts (such as “Vertical angles are equal”), and at least for the examples they showed, the proofs look possibly simpler to me. I see a connection to the imperative/declarative dichotomy in programming. After thinking about that distinction (and the controversy around it in CS0/CS1 courses) for years, I’ve concluded that
• simple tasks are easier to explain crudely in imperative terms
• complex tasks are vastly easier to understand declaratively
• declarative thinking, along with the language of definitions, is a major stumbling block for many math and CS students — who must push through it in order to become clear thinkers
Here, transformations may let us cast the geometric and algebraic ideas in imperative terms that the students are more intuitively comfortable with, and it’s quite possible to drill down to axioms and equations to get rigorous proofs that challenge students to think declaratively.
### Sunday Keynotes
#### Grace Kelemanik on the SMPs
This speaker discussed using the eight Standards for Mathematical Practice — particularly, how eight is too many to think about on a daily basis, and how to use them to manageably guide instruction. Her strategy is to focus on SMPs #2 (re quantitative thinking), #7 (re structural thinking), and #8 (re looking for repeated reasoning):
• Quantitative reasoning focuses on identifying quantities and relationships in the problem.
• Structural reasoning focuses on identifying and grouping meaningful chunks of structure within the problem; changing elements of the problem into a more convenient form; and connecting problem elements to familiar structures. Slogan: Chunk, change, connect.
• Looking for repeated reasoning is not about number patterns; it is about finding repetition in the processes you use in the problem — typically, processes like counting, calculating, or constructing. Often identified by phrases like “Every time, I...” or “It always...” or “It keeps doing X when...”. Slogan: Privilege the process.
She suggests teaching these three ways of thinking, and using them to guide (and interpret the results of) the “What do we notice?” and “What would you like to know?” questions that we so often use in solving a problem. The ways of thinking are rooted in what we observe — about both the problem and our way of responding to it — and so our discussions in class should bring out those observations and connect them to the thinking we’d like to impart to our students.
#### Steve Leinwand
Engaging, high-energy talk. The slides (pptx) are available on his web site. Main take-aways:
• In contrast to the previous speaker, he contends SMPs #1-4 are the important parts — especially #3, about constructing and critiquing arguments.
• We absolutely must observe and coach each other, and assess how many of the NCTM’s Mathematical Teaching Practices we are meeting in class.
• “It is unreasonable and unprofessional to expect professionals to change by more than 10% a year.” But...
• “It is unreasonable and unprofessional for us to be changing by much less than 10% a year.” (10% being about 6-10 min per class, or 18 days a year, or 1 unit a year.)
• Frequent cumulative review is essential for retention.
• Exit slips are essential for assessing progress and guiding next day’s instruciton.
• A “2-4-2” model of HW: Two (or maybe three) problems on a new skill, four cumulative-review problems (one each from past day, week, and month, plus one more), and two that require showing work or explanation — mathematical reasoning and justification for that reasoning.
• Questioning: Work on increasing the DoK (depth of knowledge) elicited. Handy reminder posters: “Why?” “How do you know?” “Convince us!” “Please explain.” “Draw a picture.” These are reminders for the teacher, not the students!
• Problems: Don’t state a question. Give a situation, and ask the students for questions.
• Crucial actions: struggle, explore, share, justify, compare, debrief, and consolidate.
• The teacher is most useful for organizing and consolidating knowledge at the end — not for presenting it at the beginning.
### The logo
One last item: I liked this year’s conference logo — based on the Spiral of Theodorus — and felt compelled to figure out how to code it up myself between talks:
I couldn’t get the color gradient quite right: where mine passes through light green and cyan, the actual conference logo passes through dark green and violet shades. I haven’t thought much about computing color gradients, but it could make for a great project — especially considering that color schemes are modeled geometrically with each color corresponding to a point in a cube, cylinder, spiral, or other geometric object. I’ll have to keep that in mind for the computational geometry course I hope to eventually get to teach.
The other neat thing about writing code to generate the picture is that now I can see what more extreme cases look like — say, with about a thousand triangles:
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2021-04-18 11:58:33
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/?level=2366&weight=2&char_order=1&atkin_lehner_string=%2B-%2B
|
## Results (displaying all 8 matches)
Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 7 13
2366.2.a.a $$1$$ $$18.893$$ $$\Q$$ None $$-1$$ $$-2$$ $$-3$$ $$1$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}-2q^{3}+q^{4}-3q^{5}+2q^{6}+q^{7}+\cdots$$
2366.2.a.d $$1$$ $$18.893$$ $$\Q$$ None $$-1$$ $$0$$ $$-2$$ $$1$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots$$
2366.2.a.f $$1$$ $$18.893$$ $$\Q$$ None $$-1$$ $$1$$ $$3$$ $$1$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}+q^{7}+\cdots$$
2366.2.a.g $$1$$ $$18.893$$ $$\Q$$ None $$-1$$ $$3$$ $$-3$$ $$1$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+3q^{3}+q^{4}-3q^{5}-3q^{6}+q^{7}+\cdots$$
2366.2.a.h $$1$$ $$18.893$$ $$\Q$$ None $$-1$$ $$3$$ $$4$$ $$1$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+3q^{3}+q^{4}+4q^{5}-3q^{6}+q^{7}+\cdots$$
2366.2.a.r $$2$$ $$18.893$$ $$\Q(\sqrt{3})$$ None $$-2$$ $$0$$ $$-4$$ $$2$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+\beta q^{3}+q^{4}+(-2-\beta )q^{5}-\beta q^{6}+\cdots$$
2366.2.a.u $$3$$ $$18.893$$ $$\Q(\zeta_{14})^+$$ None $$-3$$ $$-5$$ $$-2$$ $$3$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+(-2-\beta _{2})q^{3}+q^{4}-2\beta _{1}q^{5}+\cdots$$
2366.2.a.w $$3$$ $$18.893$$ $$\Q(\zeta_{14})^+$$ None $$-3$$ $$0$$ $$8$$ $$3$$ $$+$$ $$-$$ $$+$$ $$q-q^{2}+(-\beta _{1}-\beta _{2})q^{3}+q^{4}+(3+\beta _{2})q^{5}+\cdots$$
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2020-08-12 04:37:23
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https://earthscience.stackexchange.com/questions/4385/the-climatological-variance-of-albedo-soil-moisture-in-weather-models/4409
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# The climatological variance of albedo, soil moisture in weather models
Looking at WRF currently but I am sure this applies to most weather models there is a file (specifically LANDUSE.TBL) that specifies USGS derived albedo, soil moisture and other parameters on a biannual basis - one for summer and one for winter.
Why would not these parameters vary on a more seasonal basis and/or on a annual basis? Why are these these variables assumed to be constant? Would not changing vegetation or land cover modifications affect these values on a regional basis as well as their impact on regional climate?
Also would it make sense to model these values on a regional basis after deriving them from remote sensing as indicated in the below mentioned article http://web.maths.unsw.edu.au/~jasone/publications/evansetal2012a.pdf
• Sometimes, uncertainties are so large that the most general assumption is to assume something to remain constant. It might be what you get if you average many models that disagree widely about amplitude and phase of the seasonal cycle. I can't say if that is what's going on here, though. – gerrit Feb 9 '15 at 16:53
• @gerrit - how about snow cover over large areas i.e. regional scales ? Would that not vary seasonally ? – gansub Feb 10 '15 at 1:39
• It certainly should. I really don't know enough to answer your question, I hope others do. – gerrit Feb 10 '15 at 4:20
• Most of those values (except surface roughness) will get overwritten with the VEGPARM.TBL values if you use Noah or the RUC LSM. Noah uses monthly values as far as I know. RUC might write values on its own... not really sure if it uses the VEGPARM.TBL or not. There are many groups that use the monthly MODIS-derived values. – farrenthorpe Feb 11 '15 at 22:43
• @gansub model development is done such that you can run the model independently before you start adding more complex options to ingest observations. A "summer" and "winter" lookup table is provided for users, but I don't think anyone in the WRF community would argue those defaults are appropriate for the best retrospective weather simulations. Forecasting can use old satellite data (MODIS land-surface data is several days if not weeks old by the time it is processed) but that has error just as a seasonal average would. – farrenthorpe Feb 12 '15 at 17:31
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2019-12-10 19:17:57
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https://plus.cobiss.net/cobiss/si/sl/bib/2048213779
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VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
• On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets [Elektronski vir]
Eichfelder, Gabriele ; Povh, Janez, 1973-
In the paper we prove that any nonconvex quadratic problem over some set KRn with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of ... K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246267, 2003). Our result also generalizes the well-known completely positive representation result from Burer (Math Program 120:479495, 2009), which is actually a special instance of our result with K=Rn+ .
Vir: Optimization letters [Elektronski vir]. - ISSN 1862-4480 (Vol. 7, issue 6, 2013, str. 1373-1386)
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2022-08-12 19:11:07
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https://mathoverflow.net/questions/404340/does-the-existence-of-a-landau-siegel-zero-imply-the-existence-of-a-complex-zero
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# Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?
The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the critical strip off the critical line?
This question came to my mind considering the sequence of trivial zeros in decreasing order for zeta as a $$2$$-periodic signal whose Fourier transform would be a $$1/2$$-periodic signal made of Dirac peaks (the former future physicist in me is speaking, sorry), which if we compactify partially the critical strip by identifying the vertical lines of real parts $$0$$ and $$1$$ becomes a single Dirac peak which is supported on $$1/2$$, hence the real part of the non trivial zeros under RH. So my idea is that adding a Landau-Siegel zero would create a non periodic signal made by the decreasing sequence of real zeros, whose Fourier transform would not be periodic either, suggesting the existence of a complex non trivial zero in the critical strip off the critical line.
So would the existence of a Landau-Siegel zero create such a havoc that the analogue of RH for the considered Dirichlet L-function would fail completely?
• No such implication is known. Our current state of knowledge permits that GRH could hold with the exception of a single Siegel zero for a single primitive Dirichlet L-function. Sep 19 at 20:14
• As Wojowu notes this is not known. @Wojowu: that said, the definition of Landau-Siegel zeros only makes sense in the context of an infinite collection, so "a single Siegel zero for a single primitive Dirichlet L-function" doesn't make too much sense. Sep 19 at 20:46
• @MarkLewko That's true, I suppose I meant a single real zero of such an L-function. But even an infinite family like usually considered in this context isn't known to have any such implications as far as I'm aware. Sep 19 at 20:58
• Just curious, why did this question got 3 downvotes? Sep 20 at 1:59
• A pedantic note: "Single (or only) real zero" here means both the exceptional zero and its reflection under $s\rightarrow 1-s$. Sep 20 at 13:37
• I think their phrase of "$L$-functions under consideration" includes the $L$-function of a specific elliptic curve (and its twist by the exceptional character $\chi$). So it is not a Dirichlet $L$-function necessarily, that has the nonreal zero off the critical line. Sep 20 at 13:06
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2021-10-16 23:38:13
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https://stats.stackexchange.com/questions/387152/wilcoxon-signed-rank-test-or-one-sample-t-test/387164
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# Wilcoxon signed-rank test or One Sample t-test?
I conducted a training session and for each of my users, I measured pre and post score (before and after training session). Each of my users (n=28) is measured twice, so my data is paired:
Data:
user pre-score post-score
1 10 30
....
Hypothesis: $$H_0: \mu_{Post-Pre} \leq 0\\ H_1: \mu_{Post-Pre} > 0$$
In order to check the hypothesis, whether the post-score is significantly higher than the pre-score (whether the training had a positive effect on users' score), I would conduct a paired t-test (t.test(Post, Pre, mu=0, conf.level=0.95, alt="greater") in R).
Problem: The post data is not normally distributed. That's why I would go with a non-parametric test, i.e. Wilcoxon signed-rank, instead of the paired t-test.
However, if a calculate the difference between the pairs (Post-Pre), I end up with one sample which is normally distributed. So I would have to test the following hypothesis in a one sample t-test (t.test(Post-Pre, mu=0, conf.level=0.95, alt="greater") in R): $$H_0: \mu_{Diff.} \leq 0\\ H_1: \mu_{Diff.} > 0\\$$
Question: Which method is appropriate? Should I use the non-parametric paired Wilcoxon text because my post data is not normally distributed? Or can I work with the difference of the pairs (one sample) which is normally distributed and use a one sample t-test?
• The one sample t-test assumes that the differerences are normally distributed, not the scores. – Jeremy Miles Jan 14 '19 at 17:54
• So, both samples (pre and post) each must follow a normal distribution? If one does not (like my post data), I'd use a non-parametric test? – Ioannis K. Jan 14 '19 at 18:07
• Only the differences would need to be drawn from a normal population (or something sufficiently close to a normal population). Normality of the differences is used in deriving the t-distribution (you need it for the distribution of the numerator and the distribution of the denominator and for the two being independent). Outside that, you may be able to justify an asymptotic normal distribution for the statistic (via CLT and Slutsky's theorem). – Glen_b -Reinstate Monica Jan 15 '19 at 0:04
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2020-02-17 13:18:57
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https://www.tutorialspoint.com/difference-between-resistance-and-reluctance
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# Difference between Resistance and Reluctance
In electrical engineering, there are two important terms resistance and reluctance related to electromagnetic circuits such as motor, generator, transformer etc. Where, the resistance is the parameter of circuit which is regarded to electric current, whereas the reluctance is the parameter regarded to the magnetic flux in the circuit.
In this article, we will highlight all the significant differences between resistance and reluctance by considering different parameters such as basic definition, unit, denotation, affected quantity, etc. Let's start with some basics of Resistance and Reluctance so that it becomes easy to understand the differences between them.
## What is Resistance?
Resistance or electrical resistance is defined as the measure of opposition offered by a substance in the path of electric current. The resistance is denoted by the symbol ‘R’ and is measured in Ohms (Ω).
When an electric current flows through a material, the material opposes the flow of the current and results in the rise in temperature. Therefore, the resistance is nothing but electric friction. In the nature, every matter has a certain amount of resistance. The value of the resistance depends on the type of material. Since, in electrical and electronics engineering, we come across three types of materials namely conductor, semiconductor and insulator.
The conductors (or metals) offer least resistance in the flow of current, while the insulators possess highest resistance in the path of current flow. The semiconductors are those materials whose resistance lies between the resistance of conductors and insulators.
The experimental formula of the resistance of a wire is given by the following expression,
$$R\:=\:\rho\frac{l}{A}$$
Where, ρ is the resistivity or specific resistance, l is the length of wire and A is the area of cross-section of the wire.
Therefore, from this equation, we can state the factors affecting the resistance of a material or wire, which are as follows:
• The resistance of a wire is directly proportional to the length of wire.
• The resistance is inversely proportional to area of cross-section of the wire.
• The resistance depends on the nature of material.
In practice, the resistance of a material is used to design different types of electrical devices such as electric iron, line insulators, heater, bulb, etc.
## What is Reluctance?
The measure of opposition offered by a substance in the path of flow of magnetic flux is called the reluctance of the material. The reluctance is denoted by the symbol ‘S’ and is measured in ampere-turn per weber (AT/Wb) or 1/Henry. The reluctance is also known as magnetic reluctance, magnetic resistance or magnetic insulation.
The magnetic reluctance is the physical quantity which is related to a magnetic or electromagnetic circuit. The reluctance of the magnetic circuit depends on the length and area of the circuit. Where, it is directly proportional to the length and inversely proportional to the area of cross-section. Therefore, we can express the reluctance of a magnetic circuit as
$$S\:=\:\frac{l}{\mu\:A}\:=\:\frac{l}{\mu_o\:\mu_r\:A}$$
Where, l is the mean length of magnetic circuit, A is the area of cross-section and μ is the permeability of the material.
The magnetic reluctance finds applications in the electromagnetic devices such as transformers, generators, motors, measuring instruments, etc. It reduces the effect of magnetic saturation in a magnetic circuit.
## Difference between Resistance and Reluctance
Both Resistance and Reluctance are Analogous quantities related to electromagnetic circuits, however, there are several noticeable differences between Resistance and Reluctance that are given in the following table:
Basis of Difference Resistance Reluctance
Definition The measure of opposition in the path of flow of electric current offered by a substance is known as resistance. The measure of opposition in the path of flow of magnetic flux offered by the substance is known as reluctance.
Alternate name Resistance is also called electric friction. Reluctance is also known as magnetic resistance or magnetic friction.
Denotation Resistance is usually denoted by the symbol ‘R’. Reluctance is usually denoted by the symbol ‘S’.
Regarded circuit Resistance is the circuit parameter related to the electric circuits. Reluctance is the circuit parameter related to the magnetic circuits.
Formula The resistance of any conductor wire is calculated by using the following formula: $$R\:=\:\frac{\rho\:l}{A}$$ The reluctance of a magnetic circuit is calculated by using the following formula: $$S\:=\:\frac{l}{\mu\:A}$$
Relation with current The following expression shows the relation between resistance and current − $$R\:=\:\frac{V}{I}$$
Where, V is voltage and I is current.
The relation between reluctance and current is given by, $$S\:=\:\frac{NI}{\phi}$$
Where, N is the number of turn in electromagnetic coil and φ is the magnetic flux.
Unit of measurement Resistance is measured in Ohm (Ω). It may also be measured in volt per ampere (V/A). $$1\Omega\:=\:1VA^{-1}$$ Reluctance is measured in Ampere- Turns per Weber (AT/Wb) or 1/Henry.
Primary function The main function of the electric resistance is to limit the electric current flowing in the circuit. The main function of the magnetic reluctance is to limit the value of magnetic flux flowing in a magnetic circuit.
Reciprocal The reciprocal of resistance is called the conductance (G). Which is the measure of ease in the flow of electric current. The reciprocal of reluctance is called permeance (P). Which is the measure of ease in the flow of magnetic flux.
Effect of temperature The resistance of a material greatly changes with the change in temperature. The reluctance of the material comparatively less affected by the change in temperature.
Change with AC and DC Resistance of a material does not change with the AC or pulsating DC. Reluctance of a material changes with the AC and pulsating DC. Where, it pulsates at the same frequency as that of the supply.
Value in air medium Resistance of air is ideally infinity. Which means no electric current can flow through air. Reluctance of air is very high, but not infinity. Which means air allows the flow of magnetic flux.
Loss Resistance opposes the flow of electric current and results in the loss of power in the form heat. Reluctance opposes the flow of magnetic flux, but instead of dissipating it in the form of heat, it stores in the form of magnetic field.
Applications Resistance is used to design many electrical appliances such as electric iron, heater, bulb, etc. Reluctance is used in various electromagnetic devices like transformer, generator, motor, etc. It is used to reduce the magnetic saturation of a magnetic core.
## Conclusion
The most significant difference that you should note here is that Resistance is the measure of opposition in the flow of electric current, while Reluctance is the measure of opposition in the flow of magnetic flux.
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2023-03-31 17:18:49
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https://eventuallyalmosteverywhere.wordpress.com/tag/simple-random-walk/
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# DGFF 4 – Properties of the Green’s function
I’m at UBC this month for the PIMS probability summer school. One of the long courses is being given by Marek Biskup about the Discrete Gaussian Free Field (notes and outline here) so this seems like a good moment to revive the sequence of posts about the DGFF. Here’s DGFF1, DGFF2, DGFF3 from November.
The first draft of this post was about the maximum of the DGFF in a large box $V_N$, and also about the Green’s function $G^{V_N}(x,y)$, which specifies the covariance structure of the DGFF. This first draft also became too long, so I’m splitting it into two somewhat shorter ones. As we’ll see, some understanding and standard estimates of the Green’s function is enough to say quite a bit about the maximum. In this first post, we’ll explore some ‘low-hanging fruit’ concerning the Green’s function, as defined through a simple random walk, which are useful, but rarely explained in the DGFF literature.
Symmetry of Green’s function
We start with one of these low-hanging fruit. If $G^{V_N}$ is to be a covariance matrix, it has to be symmetric. In the first post, showing that the definition of the DGFF as a random field with given Hamiltonian is equivalent to $\mathcal{N}(0,G^{V_N})$ certainly can be viewed as a proof of symmetry. However, it would be satisfying if there was a direct argument in the language of the definition of the Green’s function.
To make this self-contained, recall the random walk definition of $G^{V_N}(x,y)$. Let $(S_m)_{m\ge 0}$ be simple random walk on $V_N$, and $\mathbb{P}_x,\,\mathbb{E}_x$ denote starting the random walk at $x\in V_N$. As usual, let $\tau_y,\,\tau_A$ denote the hitting time of a vertex y or a set A respectively. Then
$G^{V_N}(x,y):= \mathbb{E}_x \left[ \sum_{m=0}^{\tau_{\partial V_N}}1_{(S_m=y) }\right].$
That is, $G^{V_N}(x,y)$ is the expected number of visits to y by a random walk from x, before it exits $V_N$.
Let’s drop the superscript for now, as everything should hold for a more general subset of the lattice. I don’t think it’s immediately obvious at the level of Markov chains why G(x,y)=G(y,x). In particular, it’s not the case that
$\mathbb{P}_x(\tau_y < \tau_{D^c}) = \mathbb{P}_y(\tau_x <\tau_{D^c}),$
and it feels that we can’t map between paths $x \to \partial D$ and $y\to \partial D$ in a way that preserves the number of visits to y and x, respectively. However, we can argue that for any m
$\mathbb{P}_x(S_m=y, \tau_{D^c}>m) = \mathbb{P}_y(S_m=x, \tau_{D^c}>m),$
by looking at the suitable paths of $(S_m)$. That is, if we have a path $x=S_0,S_1,\ldots,S_m=y$ that stays within D, then the probability of seeing this path starting from x and its reverse direction starting from y are equal. Why? Because
$\mathbb{P}_x(S_0=x,S_1=v_1,\ldots,S_{m-1}=v_{m-1},S_m=y) = \prod_{\ell=0}^{m-1} \frac{1}{\mathrm{deg}(v_\ell)},$
and
$\mathbb{P}_y(S_0=y,S_1=v_{m-1},\ldots,S_{m-1}=v_1, S_m=x) = \prod_{\ell=0}^{m-1} \frac{1}{\mathrm{deg}(v_{m-\ell})} = \prod_{\ell=1}^m \frac{1}{\mathrm{deg}(v_\ell)}.$
Since $D\subset \mathbb{Z}^d$ and x,y are in the interior of D, we must have $\mathrm{deg}(x)=\mathrm{deg}(y)$, and so these two expressions are equal. Summing over all such two-way paths, and then all m gives the result.
Fixing one argument
We now focus on $G^D(\cdot,y)$, where the second argument is fixed. This is the solution to the Poisson equation
$\Delta G^D(\cdot,y) = -\delta_y(\cdot),\quad G^D(x,y)=0,\; \forall x\in \partial D.$
To see this, can use a standard hitting probability argument (as here) with the Markov property. This is harmonic in $D\backslash \{y\}$, and since we know
$G^D(y,y)= \frac{1}{\mathbb{P}_y(\text{RW hits }\partial D\text{ before returning to }y)},$
this uniquely specifies $G^D(\cdot,y)$. Anyway, since harmonic functions achieve their maxima at the boundary, we have $G(y,y)\ge G(x,y)$ for all $x\in D$. We can also see this from the SRW definition as
$G(x,y)=G(y,x) = \mathbb{P}_y (\tau_x < \tau_{\partial D} ) G(x,x) \le G(x,x).$
Changing the domain
Now we want to consider nested domains $D\subset E$, and compare $G^D(\cdot,\cdot)$ and $G^E(\cdot,\cdot)$ on DxD. The idea is that for SRW started from $x\in D$, we have $\tau_{\partial D}\le \tau_{\partial E}$, since one boundary is contained within the other. From this, we get
$G^D(x,y)\le G^E(x,y),\quad \forall x,y\in D,$
and we will use the particular case y=x.
For example, if $x\in V_N$, the box with width N, then the box with width 2N centred on x contains the whole of $V_N$. So, if we set $\bar {V}_{2N}:= [-N,N]^d$, then with reference to the diagram, we have
$G^{V_N}(x,x)\le G^{\bar{V}_{2N}}(0,0),\quad x\in V_N.$
As we’ll see when we study the maximum of the DGFF on $V_N$, uniform control over the pointwise variance will be a useful tool.
Maximising the Green’s function
The idea of bounding $G^{V_N}(x,x)$ by $G^{\bar V_{2N}}(0,0)$ for any $x\in V_N$ is clever and useful. But a more direct approach would be to find the value of x that maximises $G^{V_N}(x,x)$. We would conjecture that when $V_N$ has a central vertex, then this is the maximiser.
We can prove this directly from the definition of the Green’s function in terms of random walk occupation times. Let’s assume we are working with $\bar{V}_N$ for even N, so that 0 is the central vertex. Again, since
$G^D(x,x)=\frac{1}{\mathbb{P}_x(\text{RW hits }\partial D\text{ before returning to }x)},$ (*)
it would suffice to show that this probability is minimised when x=0. This feels right, since 0 is furthest from the boundary. Other points are closer to the boundary in some directions but further in others, so we can’t condition on the maximum distance from its start point achieved by an excursion of SRW (we’re vertex-transitive, so these look the same from all starting points), as even allowing for the four possible rotations, for an excursion of diameter slightly larger than N, starting at the centre is maximally bad.
However, intuitively it does feel as if being closer to the boundary makes you more likely to escape earlier. In fact, with a bit more care, we can couple the SRW started from 0 and the SRW started from $r=(r^x,r^y)\ne 0$ such that the latter always exits first. For convenience we’ll assume also that $r^x,r^y$ are both even.
I couldn’t find any reference to this, so I don’t know whether it’s well-known or not. The following argument involves projecting into each axis, and doing separate couplings for transitions in the x-direction and transitions in the y-direction. We assume WLOG that x is in the upper-right quadrant as shown. Then, let $0=S_0,S_1,S_2,\ldots$ be SRW started from 0, and we will construct $r=R_0,R_1,R_2,\ldots$ on the same probability space as $(S_m)_{m\ge 0}$ as follows. For every m, we set the increment $R_{m+1}-R_m$ to be $\pm(S_{m+1}-S_m)$. It remains to specify the sign, which will be determined by the direction of the S-increment, and a pair of stopping times. The marginal is therefore again an SRW, started from r. Temporarily, we use the unusual notation $S_m= (S^x_m,S^y_m)$ for the coordinates of $S_m$.
So, if $S_{m+1}-S_m=(1,0), (-1,0)$, ie S moves left or right, then we set
$R_{m+1}-R_m = \begin{cases} -(S_{m+1}-S_m) &\quad \text{if }mT^x.\end{cases}$ (*)
where $T^x:= \min\{m\,:\, R^x_m=S^x_m\}$. That is, $R^x$ moves in the opposing direction to $S^x$ until the first time when they are equal (hence the parity requirement), and then they move together. WLOG assume that $r^x>0$. Then suppose $S^x_m=\pm N$ and such m is minimal. Then by construction, if $m\ge T^x$, then $R^x_m=\pm N$ also. If $m, then we must have $S^x_m=-N$, and so since $R^x$‘s trajectory is a mirror image of $S^x$‘s, in fact $R^x_m = N+r^x>N$, so $R^x$ hit +N first. In both cases, we see that $R^x$ hits $\pm N$ at the same time or before $S^x$.
In other words, when $S^x_m$ has non-negative x coordinate, the lazy random walk $R^x$ follows the same trajectory as $S^x$, and when it has negative x coordinate, the $R^x$ mirrors $S^x$. At some time, it may happen that $S^x_m= R^x_m=0$ (recall the parity condition on r). Call this time $T^x$. We then adjust the description of the coupling so that (*) is the mechanism for $m, and then for $m\ge T^x$, we take $S^x_m=R^x_m$.
Similarly, if $S_{m+1}-S_m =(0,1), (0,-1)$, ie S moves up or down, then we set
$R_{m+1}-R_m = \begin{cases} -(S_{m+1}-S_m)&\quad \text{ if }m
with corresponding definition of the stopping time $T^y$.
This completes the coupling, and by considering $T^x\wedge T^y$, we have shown what that the exit time for the walk started from zero dominates the exit time for walk started from r. Recall that so far we are in the case where the box has even width and $r=(r^x,r^y)$ has even coordinates.
This exit time comparison isn’t exactly what we need to compare $G^N(0,0)$ and $G^N(x,x)$. It’s worth remarking at this stage that if all we cared about was the Green’s function on the integer line [-N,N], we would have an easier argument, as by the harmonic property of $G(\cdot,y)$
$G^{[-N,N]}(0,r)=\frac{N-r}{N}G^{[-N,N]}(0,0),$
$G^{[-N,N]}(r,0) = \frac{N}{N+r}G^{[-N,N]}(r,r),$
and so $G(0,0)>G(r,r)$ follows by symmetry. To lift from 1D to 2D directly, we need a bit more than this. It’s possible that S returns in both x- and y- coordinates more often than R, but never at the same time. Fortunately, the coupling we defined slightly earlier does give us a bit more control.
Let $\tau^x(S), \tau^x(R)$ be the first times that $S^x, R^x$ hit $\pm N$. Under this coupling, for any $m\ge 0$
$\mathbb{P}(S^x_m=0, m
since these events are literally equal. Since we showed that $\tau^x(R)\le \tau^x(S)$ almost surely, we can further deduce
$\mathbb{P}(S^x_m=0,m
$=\mathbb{P}(R^x_m=r^x, m
To address the corresponding events for which $m\ge T^x$, we apply the strong Markov property at $T^x$, to obtain SRW $Z_m$ started from r/2, and let $\tau_{-N},\tau_{+N}$ be the hitting times of $-N,+N$ respectively and $\tau_{\pm N}=\tau_{-N}\wedge \tau_{+N}$. It will now suffice to prove that
$\mathbb{P}(Z_m=0, m< \tau_{\pm N}) \ge \mathbb{P}(Z_m=r,m<\tau_{\pm N}),$ (**)
as then we can apply the law of total probability and sum over values of $T^x$ and $m\ge 0$.
To prove this result, we consider the following bijection between trajectories of length m from r/2 to {0,r}. We decompose the trajectories into excursions away from r/2, and then a final meander from r/2 to {0,r} that stays on the same side of r/2. We construct the new trajectory by preserving all the initial excursions, but reversing all the steps of the final meander. So if the original trajectory ended up at 0, the image ends up at r. Trivially, the initial excursions in the image only hit $\pm N$ if the excursions in the original trajectory did this too. But it’s also easy to see, by a similar argument to the coupling at the start of this section, that if the original trajectory ends at r and does not hit $\pm N$, then so does the image. However, the converse is not true. So we conclude (**), and thus
$\mathbb{P}(S_m^x=0) \ge \mathbb{P}(R_m^x=0)$
for all m by combining everything we have seen so far. And so we can now lift to a statement about $S_m$ itself, that is considering both coordinates separately.
The remaining cases for r require a little more care over the definition of $T^x$, though the same projection argument works, for fundamentally the same reason. (Note that in the above argument, if $S^x_m=-N$ and $m, then in fact $R^x_m\ge N+2$, and so it’s not hard to convince yourself that a sensible adjustment to the stopping time will allow a corresponding result with $R^x_m\ge N+1$ in the odd $r^x$ case.) The case for N odd is harder, since in one dimension there are two median sites, and it’s clear by symmetry that we can’t couple them such that RW from one always exits at least as early as RW from the other. However, the distributions of exit times started from these two sites are the same (by symmetry), and so although we can’t find a coupling, we can use similar stopping times to obtain a result in probability.
In the next post, we’ll see how to apply this uniform bound on $G^{V_N}(x,x)$ to control the maximum of the DGFF on $V_N$. In particular, we address how the positive correlations of DGFF influence the behaviour of the maximum by comparison with independent Gaussians at each site.
# Random walks conditioned to stay positive
In this post, I’m going to discuss some of the literature concerning the question of conditioning a simple random walk to lie above a line with fixed gradient. A special case of this situation is conditioning to stay non-negative. Some notation first. Let $(S_n)_{n\ge 0}$ be a random walk with IID increments, with distribution X. Take $\mu$ to be the expectation of these increments, and we’ll assume that the variance $\sigma^2$ is finite, though at times we may need to enforce slightly stronger regularity conditions.
(Although simple symmetric random walk is a good example for asymptotic heuristics, in general we also assume that if the increments are discrete they don’t have parity-based support, or any other arithmetic property that prevents local limit theorems holding.)
We will investigate the probability that $S_n\ge 0$ for n=0,1,…,N, particularly for large N. For ease of notation we write $T=\inf\{n\ge 0\,:\, S_n<0\}$ for the hitting time of the negative half-plane. Thus we are interested in $S_n$ conditioned on T>N, or T=N, mindful that these might not be the same. We will also discuss briefly to what extent we can condition on $T=\infty$.
In the first paragraph, I said that this is a special case of conditioning SRW to lie above a line with fixed gradient. Fortunately, all the content of the general case is contained in the special case. We can repose the question of $S_n$ conditioned to stay above $n\alpha$ until step N by the question of $S_n-n\alpha$ (which, naturally, has drift $\mu-\alpha$) conditioned to stay non-negative until step N, by a direct coupling.
Applications
Simple random walk is a perfectly interesting object to study in its own right, and this is a perfectly natural question to ask about it. But lots of probabilistic models can be studied via naturally embedded SRWs, and it’s worth pointing out a couple of applications to other probabilistic settings (one of which is the reason I was investigating this literature).
In many circumstances, we can desribe random trees and random graphs by an embedded random walk, such as an exploration process, as described in several posts during my PhD, such as here and here. The exploration process of a Galton-Watson branching tree is a particularly good example, since the exploration process really is simple random walk, unlike in, for example, the Erdos-Renyi random graph G(N,p), where the increments are only approximately IID. In this setting, the increments are given by the offspring distribution minus one, and the hitting time of -1 is the total population size of the branching process. So if the expectation of the offspring distribution is at most 1, then the event that the size of the tree is large is an atypical event, corresponding to delayed extinction. Whereas if the expectation is greater than one, then it is an event with limiting positive probability. Indeed, with positive probability the exploration process never hits -1, corresponding to survival of the branching tree. There are plenty of interesting questions about the structure of a branching process tree conditional on having atypically large size, including the spine decomposition of Kesten [KS], but the methods described in this post can be used to quantify the probability, or at least the scale of the probability of this atypical event.
In my current research, I’m studying a random walk embedded in a construction of the infinite-volume DGFF pinned at zero, as introduced by Biskup and Louidor [BL]. The random walk controls the gross behaviour of the field on annuli with dyadically-growing radii. Anyway, in this setting the random walk has Gaussian increments. (In fact, there is a complication because the increments aren’t exactly IID, but that’s definitely not a problem at this level of exposition.) The overall field is decomposed as a sum of the random walk, plus independent DGFFs with Dirichlet boundary conditions on each of the annuli, plus asymptotically negligible corrections from a ‘binding field’. Conditioning that this pinned field be non-negative up to the Kth annulus corresponds to conditioning the random walk to stay above the magnitude of the minimum of each successive annular DGFF. (These minima are random, but tightly concentrated around their expectations.)
Conditioning on $\{T > N\}$
When we condition on $\{T>N\}$, obviously the resulting distribution (of the process) is a mixture of the distributions we obtain by conditioning on each of $\{T=N+1\}, \{T=N+2\},\ldots$. Shortly, we’ll condition on $\{T=N\}$ itself, but first it’s worth establishing how to relate the two options. That is, conditional on $\{T>N\}$, what is the distribution of T?
Firstly, when $\mu>0$, this event always has positive probability, since $\mathbb{P}(T=\infty)>0$. So as $N\rightarrow\infty$, the distribution of the process conditional on $\{T>N\}$ converges to the distribution of the process conditional on survival. So we’ll ignore this for now.
In the case $\mu\le 0$, everything is encapsulated in the tail of the probabilities $\mathbb{P}(T=N)$, and these tails are qualitatively different in the cases $\mu=0$ and $\mu<0$.
When $\mu=0$, then $\mathbb{P}(T=N)$ decays polynomially in N. In the special case where $S_n$ is simple symmetric random walk (and N has the correct parity), we can check this just by an application of Stirling’s formula to count paths with this property. By contrast, when $\mu<0$, even demanding $S_N=-1$ is a large deviations event in the sense of Cramer’s theorem, and so the probability decays exponentially with N. Mogulskii’s theorem gives a large deviation principle for random walks to lie above a line defined on the scale N. The crucial fact here is that the probabilistic cost of staying positive until N has the same exponent as the probabilistic cost of being positive at N. Heuristically, we think of spreading the non-expected behaviour of the increments uniformly through the process, at only polynomial cost once we’ve specified the multiset of values taken by the increments. So, when $\mu<0$, we have
$\mathbb{P}(T\ge(1+\epsilon)N) \ll \mathbb{P}(T= N).$
Therefore, conditioning on $\{T\ge N\}$ in fact concentrates T on N+o(N). Whereas by contrast, when $\mu=0$, conditioning on $\{T\ge N\}$ gives a nontrivial limit in distribution for T/N, supported on $[1,\infty)$.
A related problem is the value taken by $S_N$, conditional on {T>N}. It’s a related problem because the event {T>N} depends only on the process up to time N, and so given the value of $S_N$, even with the conditioning, after time N, the process is just an unconditioned RW. This is a classic application of the Markov property, beloved in several guises by undergraduate probability exam designers.
Anyway, Iglehart [Ig2] shows an invariance principle for $S_N | T>N$ when $\mu<0$, without scaling. That is $S_N=\Theta(1)$, though the limiting distribution depends on the increment distribution in a sense that is best described through Laplace transforms. If we start a RW with negative drift from height O(1), then it hits zero in time O(1), so in fact this shows that conditonal on $\{T\ge N\}$, we have T= N +O(1) with high probability. When $\mu=0$, we have fluctuations on a scale $\sqrt{N}$, as shown earlier by Iglehart [Ig1]. Again, thinking about the central limit theorem, this fits the asymptotic description of T conditioned on T>N.
Conditioning on $T=N$
In the case $\mu=0$, conditioning on T=N gives
$\left[\frac{1}{\sqrt{N}}S(\lfloor Nt\rfloor ) ,t\in[0,1] \right] \Rightarrow W^+(t),$ (*)
where $W^+$ is a standard Brownian excursion on [0,1]. This is shown roughly simultaneously in [Ka] and [DIM]. This is similar to Donsker’s theorem for the unconditioned random walk, which converges after rescaling to Brownian motion in this sense, or Brownian bridge if you condition on $S_N=0$. Skorohod’s proof for Brownian bridge [Sk] approximates the event $\{S_N=0\}$ by $\{S_N\in[-\epsilon \sqrt{N},+\epsilon \sqrt{N}]\}$, since the probability of this event is bounded away from zero. Similarly, but with more technicalities, a proof of convergence conditional on T=N can approximate by $\{S_m\ge 0, m\in[\delta N,(1-\delta)N], S_N\in [-\epsilon \sqrt{N},+\epsilon\sqrt{N}]\}$. The technicalities here emerge since T, the first return time to zero, is not continuous as a function of continuous functions. (Imagine a sequence of processes $f^N$ for which $f^N(x)\ge 0$ on [0,1] and $f^N(\frac12)=\frac{1}{N}$.)
Once you condition on $T=N$, the mean $\mu$ doesn’t really matter for this scaling limit. That is, so long as variance is finite, for any $\mu\in\mathbb{R}$, the same result (*) holds, although a different proof is in general necessary. See [BD] and references for details. However, this is particularly clear in the case where the increments are Gaussian. In this setting, we don’t actually need to take a scaling limit. The distribution of Gaussian *random walk bridge* doesn’t depend on the mean of the increments. This is related to the fact that a linear transformation of a Gaussian is Gaussian, and can be seen by examining the joint density function directly.
Conditioning on $T=\infty$
When $\mu>0$, the event $\{T=\infty\}$ occurs with positive probability, so it is well-defined to condition on it. When $\mu\le 0$, this is not the case, and so we have to be more careful.
First, an observation. Just for clarity, let’s take $\mu<0$, and condition on $\{T>N\}$, and look at the distribution of $S_{\epsilon N}$, where $\epsilon>0$ is small. This is approximately given by
$\frac{S_{\epsilon N}}{\sqrt{N}}\stackrel{d}{\approx}W^+(\epsilon).$
Now take $\epsilon\rightarrow\infty$ and consider the RHS. If instead of the Brownian excursion $W^+$, we instead had Brownian motion, we could specify the distribution exactly. But in fact, we can construct Brownian excursion as the solution to an SDE:
$\mathrm{d}W^+(t) = \left[\frac{1}{W^+(t)} - \frac{W^+(t)}{1-t}\right] \mathrm{d}t + \mathrm{d}B(t),\quad t\in(0,1)$ (**)
for B a standard Brownian motion. I might return in the next post to why this is valid. For now, note that the first drift term pushes the excursion away from zero, while the second term brings it back to zero as $t\rightarrow 1$.
From this, the second drift term is essentially negligible if we care about scaling $W^+(\epsilon)$ as $\epsilon\rightarrow 0$, and we can say that $W^+(\epsilon)=\Theta(\sqrt{\epsilon})$.
So, returning to the random walk, we have
$\frac{S_{\epsilon N}}{\sqrt{\epsilon N}}\stackrel{d}{\approx} \frac{W^+(\epsilon)}{\sqrt{\epsilon}} = \Theta(1).$
At a heuristic level, it’s tempting to try ‘taking $N\rightarrow\infty$ while fixing $\epsilon N$‘, to conclude that there is a well-defined scaling limit for the RW conditioned to stay positive forever. But we came up with this estimate by taking $N\rightarrow\infty$ and then $\epsilon\rightarrow 0$ in that order. So while the heuristic might be convincing, this is not the outline of a valid argument in any way. However, the SDE representation of $W^+$ in the $\epsilon\rightarrow 0$ regime is useful. If we drop the second drift term in (**), we define the three-dimensional Bessel process, which (again, possibly the subject of a new post) is the correct scaling limit we should be aiming for.
Finally, it’s worth observing that the limit $\{T=\infty\}=\lim_{N\rightarrow\infty} \{T>N\}$ is a monotone limit, and so further tools are available. In particular, if we know that the trajectories of the random walk satisfy the FKG property, then we can define this limit directly. It feels intuitively clear that random walks should satisfy the FKG inequality (in the sense that if a RW is large somewhere, it’s more likely to be large somewhere else). You can do a covariance calculation easily, but a standard way to show the FKG inequality applies is by verifying the FKG lattice condition, and unless I’m missing something, this is clear (though a bit annoying to check) when the increments are Gaussian, but not in general. Even so, defining this monotone limit does not tell you that it is non-degenerate (ie almost-surely finite), for which some separate estimates would be required.
A final remark: in a recent post, I talked about the Skorohod embedding, as a way to construct any centered random walk where the increments have finite variance as a stopped Brownian motion. One approach to conditioning a random walk to lie above some discrete function is to condition the corresponding Brownian motion to lie above some continuous extension of that function. This is a slightly stronger conditioning, and so any approach of this kind must quantify how much stronger. In Section 4 of [BL], the authors do this for the random walk associated with the DGFF conditioned to lie above a polylogarithmic curve.
References
[BD] – Bertoin, Doney – 1994 – On conditioning a random walk to stay nonnegative
[BL] – Biskup, Louidor – 2016 – Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field
[DIM] – Durrett, Iglehart, Miller – 1977 – Weak convergence to Brownian meander and Brownian excursion
[Ig1] – Iglehart – 1974 – Functional central limit theorems for random walks conditioned to stay positive
[Ig2] – Iglehart – 1974 – Random walks with negative drift conditioned to stay positive
[Ka] – Kaigh – 1976 – An invariance principle for random walk conditioned by a late return to zero
[KS] – Kesten, Stigum – 1966 – A limit theorem for multidimensional Galton-Watson processes
[Sk] – Skorohod – 1955 – Limit theorems for stochastic processes with independent increments
# Skorohod embedding
Background
Suppose we are given a standard Brownian motion $(B_t)$, and a stopping time T. Then, so long as T satisfies one of the regularity conditions under which the Optional Stopping Theorem applies, we know that $\mathbb{E}[B_T]=0$. (See here for a less formal introduction to OST.) Furthermore, since $B_t^2-t$ is a martingale, $\mathbb{E}[B_T^2]=\mathbb{E}[T]$, so if the latter is finite, so is the former.
Now, using the strong Markov property of Brownian motion, we can come up with a sequence of stopping times $0=T_0, T_1, T_2,\ldots$ such that the increments $T_k-T_{k-1}$ are IID with the same distribution as T. Then $0,B_{T_1},B_{T_2},\ldots$ is a centered random walk. By taking T to be the hitting time of $\{-1,+1\}$, it is easy to see that we can embed simple random walk in a Brownian motion using this approach.
Embedding simple random walk in Brownian motion.
The Skorohod embedding question asks: can all centered random walks be constructed in this fashion, by stopping Brownian motion at a sequence of stopping time? With the strong Markov property, it immediately reduces the question of whether all centered finite-variance distributions X can be expressed as $B_T$ for some integrable stopping time T.
The answer to this question is yes, and much of what follows is drawn from, or at least prompted by Obloj’s survey paper which details the problem and rich history of the many approaches to its solution over the past seventy years.
Applications and related things
The relationship between random walks and Brownian motion is a rich one. Donsker’s invariance principle asserts that Brownian motion appears as the scaling limit of a random walk. Indeed, one can construct Brownian motion itself as the limit of a sequence of consistent random walks with normal increments on an increasingly dense set of times. Furthermore, random walks are martingales, and we know that continuous, local martingales can be expressed as a (stochastically) time-changed Brownian motion, from the Dubins-Schwarz theorem.
The Skorohod embedding theorem can be used to prove results about random walks with general distribution by proving the corresponding result for Brownian motion, and checking that the construction of the sequence of stopping times has the right properties to allow the result to be carried back to the original setting. It obviously also gives a coupling between a individual random walk and a Brownian motion which may be useful in some contexts, as well as a coupling between any pair of random walks. This is useful in proving results for random walks which are much easier for special cases of the distribution. For example, when the increments are Gaussian, or when there are combinatorial approaches to a problem about simple random walk. At the moment no aspect of this blog schedule is guaranteed, but I plan to talk about the law of the iterated logarithm shortly, whose proof is approachable in both of these settings, as well as for Brownian motion, and Skorohod embedding provides the route to the general proof.
At the end, we will briefly compare some other ways to couple a random walk and a Brownian motion.
One thing we could do is sample a copy of X independently from the Brownian motion, then declare $T= \tau_{X}:= \inf\{t\ge 0: B_t=X\}$, the hitting time of (random value) X. But recall that unfortunately $\tau_x$ has infinite expectation for all non-zero x, so this doesn’t fit the conditions required to use OST.
Skorohod’s original method is described in Section 3.1 of Obloj’s notes linked above. The method is roughly to pair up positive values taken by X appropriately with negative values taken by X in a clever way. If we have a positive value b and a negative value a, then $\tau_{a,b}$, the first hitting time of $\mathbb{R}\backslash (a,b)$ is integrable. Then we choose one of these positive-negative pairs according to the projection of the distribution of X onto the pairings, and let T be the hitting time of this pair of values. The probability of hitting b conditional on hitting {a,b} is easy to compute (it’s $\frac{-a}{b-a}$) so we need to have chosen our pairs so that the ‘probability’ of hitting b (ie the density) comes out right. In particular, this method has to start from continuous distributions X, and treat atoms in the distribution of X separately.
The case where the distribution X is symmetric (that is $X\stackrel{d}=-X$) is particularly clear, as then the pairs should be $(-x,x)$.
However, it feels like there is enough randomness in Brownian motion already, and subsequent authors showed that indeed it wasn’t necessary to introduce extra randomness to provide a solution.
One might ask whether it’s possible to generate the distribution on the set of pairs (as above) out of the Brownian motion itself, but independently from all the hitting times. It feels like it might be possible to make the distribution on the pairs measurable with respect to
$\mathcal{F}_{0+} = \bigcap\limits_{t>0} \mathcal{F}_t,$
the sigma-algebra of events determined by limiting behaviour as $t\rightarrow 0$ (which is independent of hitting times). But of course, unfortunately $\mathcal{F}_{0+}$ has a zero-one law, so it’s not possible to embed non-trivial distributions there.
Dubins solution
The exemplar for solutions without extra randomness is due to Dubins, shortly after Skorohod’s original argument. The idea is to express the distribution X as the almost sure limit of a martingale. We first use the hitting time of a pair of points to ‘decide’ whether we will end up positive or negative, and then given this information look at the hitting time (after this first time) of two subsequent points to ‘decide’ which of four regions of the real interval we end up in.
I’m going to use different notation to Obloj, corresponding more closely with how I ended up thinking about this method. We let
$a_+:= \mathbb{E}[X \,|\, X>0], \quad a_- := \mathbb{E}[X\,|\, X<0],$ (*)
and take $T_1 = \tau_{\{a_-,a_+\}}$. We need to check that
$\mathbb{P}\left( B_{T_1}=a_+\right) = \mathbb{P}\left(X>0\right),$
for this to have a chance of working. But we know that
$\mathbb{P}\left( B_{T_1}=a_+\right) = \frac{a_+}{a_+-a_-},$
and we can also attack the other side using (*) and the fact that $\mathbb{E}[X]=0$, using the law of total expectation:
$0=\mathbb{E}[X]=\mathbb{E}[X\,|\, X>0] \mathbb{P}(X>0) + \mathbb{E}[X\,|\,X<0]\mathbb{P}(X<0) = a_+ \mathbb{P}(X>0) + a_- \left(1-\mathbb{P}(X>0) \right),$
$\Rightarrow\quad \mathbb{P}(X>0)=\frac{a_+}{a_+-a_-}.$
Now we define
$a_{++}=\mathbb{E}[X \,|\, X>a_+],\quad a_{+-}=\mathbb{E}[X\,|\, 0
and similarly $a_{-+},a_{--}$. So then, conditional on $B_{T_1}=a_+$, we take
$T_2:= \inf_{t\ge T_1}\left\{ B_t\not\in (a_{+-},a_{++}) \right\},$
and similarly conditional on $B_{T_1}=a_-$. By an identical argument to the one we have just deployed, we have $\mathbb{E}\left[B_{T_2} \,|\,\mathcal{F}_{T_1} \right] = B_{T_1}$ almost surely. So, although the $a_{+-+}$ notation now starts to get very unwieldy, it’s clear we can keep going in this way to get a sequence of stopping times $0=T_0,T_1,T_2,\ldots$ where $B_{T_n}$ determines which of the $2^n$ regions of the real line any limit $\lim_{m\rightarrow\infty} B_{T_m}$ should lie in.
A bit of work is required to check that the almost sure limit $T_n\rightarrow T$ is almost surely finite, but once we have this, it is clear that $B_{T_n}\rightarrow B_T$ almost surely, and $B_T$ has the distribution required.
We want to know how close we can make this coupling between a centered random walk with variance 1, and a standard Brownian motion. Here, ‘close’ means uniformly close in probability. For large times, the typical difference between one of the stopping times $0,T_1,T_2,\ldots$ in the Skorohod embedding and its expectation (recall $\mathbb{E}[T_k]=k$) is $\sqrt{n}$. So, constructing the random walk $S_0,S_1,S_2,\ldots$ from the Brownian motion via Skorohod embedding leads to
$\left |S_k - B_k \right| = \omega(n^{1/4}),$
for most values of $k\le n$. Strassen (1966) shows that the true scale of the maximum
$\max_{k\le n} \left| S_k - B_k \right|$
is slightly larger than this, with some extra powers of $\log n$ and $\log\log n$ as one would expect.
The Komlos-Major-Tusnady coupling is a way to do a lot better than this, in the setting where the distribution of the increments has a finite MGF near 0. Then, there exists a coupling of the random walk and the Brownian motion such that
$\max_{k\le n}\left|S_k- B_k\right| = O(\log n).$
That is, there exists C such that
$\left[\max_{k\le n} \left |S_k-B_k\right| - C\log n\right] \vee 0$
is a tight family of distributions, indeed with uniform exponential tail. To avoid digressing infinitely far from my original plan to discuss the proof of the law of iterated logarithm for general distributions, I’ll stop here. I found it hard to find much coverage of the KMT result apart from the challenging original paper, and many versions expressed in the language of empirical processes, which are similar to random walks in many ways relevant to convergence and this coupling, but not for Skorohod embedding. So, here is a link to some slides from a talk by Chatterjee which I found helpful in getting a sense of the history, and some of the modern approaches to this type of normal approximation problem.
# DGFF 1 – The discrete Gaussian free field from scratch
I’ve moved to Haifa in northern Israel to start a post-doc in the probability group at the Technion, and now that my thesis is finished I want to start blogging again. The past couple of weeks have been occupied with finding an apartment and learning about the Discrete Gaussian Free Field. All questions about the apartment are solved, but fortunately lots remain open about the DGFF, so I thought I’d write some background about this object and methods which have been used to study it.
Background – Random walk bridge
When we think of a random walk, we usually think of the index as time, normally going forwards. So for a random walk bridge, we might assume $Z_0=0$, and then condition on $Z_N=0$, thinking of this as a demand that the process has returned to zero at the future time. In some applications, this is the ideal intuition, but in others, it is more useful to think of the random walk bridge
$(0=Z_0,Z_1,\ldots,Z_{N-1},Z_N=0),$
as a random height function indexed by [0,N], where the probability of a given path decomposes naturally into a product depending on the N increments, up to a normalising constant.
Naturally, we are interested in the asymptotic behaviour of such a random walk bridge when $N\rightarrow\infty$. So long as the step distribution has finite variance, a conditioned version of Donsker’s theorem shows that the rescaled random walk bridge converges in distribution to Brownian bridge. Note that Brownian bridge
$(B^{\mathrm{br}}_t, t\in[0,1])$
can be constructed either by conditioning a standard Brownian motion B to return to zero at time one (modulo some technicalities – this event has zero probability), or by applying an appropriate (random) linear shift
$B^{\mathrm{br}}(t):= B(t) - tB(1).$ (*)
It is not too hard to calculate the distribution of $B^{\mathrm{br}}(t)$ for each $t\in[0,1]$, and with a bit more work, one can calculate the joint distribution of $(B^{\mathrm{br}}(s),B^{\mathrm{br}}(t))$. In particular, the joint distribution is multivariate Gaussian, and so everything depends on the covariance ‘matrix’ (which here is indexed by [0,1]).
So if we return to a random walk bridge what should the step distribution be? Simple symmetric RW is a natural choice, as then lots of the quantities we might want to consider boil down to combinatorial calculations. Cleverness and Stirling’s formula can often get us useful asymptotics. But there are lots of inconveniences, not least the requirement to be careful about parity (N has to be even for a start unless you make the walk lazy, in which case the combinatorics becomes harder), and even if these can be overcome in a given calculation, it would be better not to have this.
The claim is that the random walk with Gaussian increments is by far the easiest to analyse asymptotically. As a further heuristic, think about the statement of the central limit theorem in the case where the underlying distribution is normal: it’s true but obvious. [Indeed, it’s my favourite piece of advice to anyone taking second year probability exams to check that your proposed statement of CLT does actually work for $N(\mu,\sigma^2)$…] More concretely, if a RW has Gaussian increments, then the path $(Z_1,\ldots,Z_N)$ is a multivariate normal, or a Gaussian process with finite index set. In particular, covariances define the distribution. It remains a Gaussian process after conditioning on $Z_N=0$, and the linear tilting argument at (*) remains true here, and can indeed be applied to turn any boundary conditions into any other boundary conditions.
The discrete Gaussian free field
We know how to generalise the domain of a random walk to higher dimensions. But what generalising the index to higher dimension? So now there is definitely no arrow of time, and the notion of a random height function above $\mathbb{Z}^2$ (or a subset of it) is helpful, for which a scaling limit might be a random surface rather than Brownian motion.
Because we can’t well-order $\mathbb{Z}^d$, it’s harder to define any such random object on the entire lattice immediately, so we start with compact connected subsets, with zero boundary conditions, as in the one-dimensional case of random walk bridge. Formally, let D be a finite subset of $\mathbb{Z}^d$, and the boundary $\partial D$ those elements of $D^c$ which are adjacent to an element of D, and let $\bar D:= D\cup \partial D$.
Then, the discrete Gaussian free field on D is a random real vector $h^D=(h^D_x: x\in \bar D)$, with probability density proportional to
$\mathbf{1}\{h^D_x=0, x\in\partial D\}\exp\left ( - \frac{1}{4d} \sum_{x\sim y}(h^D_x - h^D_y)^2 \right),$ (1)
where we write $x\sim y$ if that x,y are adjacent in $\bar D$. We won’t at any stage worry much about the partition function which normalises this pdf. Note also that $\frac{1}{4d}$ is just a convenient choice of constant, which corresponds to one of the canonical choices for the discrete Laplacian. Adjusting this constant is the same as uniformly rescaling the values taken by the field.
The immediate interpretation of (1) is that the values taken by the field at vertices which are close to each other are positively correlated. Furthermore, the form of the density is Gaussian. Concretely, if the values of $h^D$ are fixed everywhere except one vertex $x\in D$, then the conditional distribution of $h^D_x$ is Gaussian. Later, or in subsequent posts, we will heavily develop this idea. Alternatively, we could if we really wanted describe the model in terms of independent Gaussians describing the ‘increment’ along each edge in D (which we should direct), subject to a very large number of conditions, namely that the sum of increments along any directed cycle is zero. This latter description might be more useful if you wanted to define a DGFF on a more sparse graph, but won’t be useful in what follows.
Note that we can rearrange the Laplacian in (1) in terms of the transition kernel p( ) of the simple random walk of D to obtain
$\exp\left( -\frac12 (h^D)^T (\mathbf{P}-\mathbf{1})h^D \right),$
where $P_{x,y}=p(y-x)$ is the transition matrix of SRW on D. In particular, this means that the free field is Gaussian, and we can extract the covariances via
$\mathrm{Cov}(h^D_x,h^D_y) = \left[ (\mathbf{1}-\mathbf{P})^{-1}\right]_{x,y}$
$= \left[\sum_{n\ge 0} \mathbf{P}^n\right]_{x,y} = \sum_{n\ge 0} \mathbb{P}_x\left[X_n=y,\tau_{\partial D}>n\right],$
where, under $\mathbb{P}_x$, $(X_0,X_1,\ldots)$ is simple random walk started from x.
This final quantity records the expected number of visits to y before leaving the domain D, for a random walk started at x, and is called the Green’s function.
In summary, the DGFF on D is the centred Gaussian random vector indexed by $\bar D$ with covariance given by the Green’s function $G_D(x,y)$.
How many of these equivalences carries over to more general D-indexed random fields is discussed in the survey paper by Velenik. But it’s worth emphasising that having the covariance given by the Green’s function as in the definition we’ve just given is a very nice property, as there are lots of pre-existing tools for calculating these. By contrast, it’s hard to think of a natural model for an integer-valued surface of this kind, as an analogue to SRW.
[Though definitely not impossible. The nicest example I’ve heard of is for height functions of large uniform domino tilings within their ‘arctic circle’, which have GFF asymptotics. See this paper by Kenyon.]
A continuous limit?
We motivated the discussion of random walk bridge by the limit object, namely Brownian bridge. Part of the reason why the DGFF is more interesting than Gaussian random walk bridge, is that the limit object, the (continuum) Gaussian free field is hard to define classically in two dimensions.
We might suppose that the DGFF in $V_N$, the square box of width N has some scaling limit as $N\rightarrow\infty$. However, for fixed $x,y\in [0,1]^2$, (and taking integer parts component-wise), well-known asymptotics for SRW in a large square lattice (more on this soon hopefully) assert that
$\mathrm{Cov}(h^{V_N}_{\lfloor Nx \rfloor},h^{V_N}_{\lfloor Ny\rfloor}) \sim \log |x-y|,$ (2)
and so any scaling limit will rescale only the square domain, not the height (since there is no N on the RHS of (2)). However, then the variance of the proposed limit is infinite everywhere.
So the GFF does not exist as a random height function on $[0,1]^2$, with the consequence that a) more care is needed over its abstract definition; b) the DGFF in 2D on a large square is an interesting object, since it does exist in this sense.
What makes it ‘free’?
This seemed like a natural question to ask, but I’ve received various answers. Some sources seem to suggest that having zero boundary condition is free. Other sources refer to the Hamiltonian (that is the term inside the exponential function at (1) ) as free since it depends only on the increments between values. If the Hamiltonian also depends on the heights themselves, for example via the addition of a $\sum_{x} \Psi(h^D_x)$ term, then for suitable choice of function $\Psi$, this is interpreted as a model where the particles have mass. The physical interpretation of these more general Gibbs measures is discussed widely, and I’m not very comfortable with it all at the moment, but aim to come back to it later, when hopefully I will be more comfortable.
# Fair games and the martingale strategy III
Gambler’s Ruin
Continuing directly from the previous post, the nicest example of the optional stopping theorem we developed there is to example a simple random walk constrained between two values, say 0 and N. This represents an idealised gambling situation, where the gambler stops playing either when they reach some pre-agreed profit, or when they go bankrupt. We assume that we start at level k, for k = 1,2,…,N-1.
Naturally, we want to know the probabilities of winning (ie getting to N) and losing (ie going bankrupt). We could set this up by conditioning on the first step. Let $p_k$ be the probability of winning starting from level k. Then we must have
$p_k= \frac12 p_{k+1}+\frac12 p_{k-1},\quad k=1,\ldots,N-1,$ (*)
with the obvious boundary conditions $p_0=0, p_N=1$. In an ideal world, we just know how to solve second order difference equations like (*). Well, actually it isn’t too hard, because we can see from (*) directly that
$p_{k+1}-p_k = p_k-p_{k-1},$
and so $p_k$ is a linear function of k, and so $p_k = k/N$ follows pretty much immediately.
But, we can also use OST profitably. Let T be the time at which we first hit 0 or N. It’s intuitively clear that this should have finite expectation, since the problems you might encounter with just the hitting time of a single level shouldn’t apply. Or you can consider the expected number of steps before you see N ups or downs in a row, which certainly provides an upper bound on T. This random number of steps is sort of geometric (at least, can be upper bounded by a geometric RV) and so has finite expectation. So can apply OST to X at T, and we have
$\mathbb{E}[X_T] = N\cdot \mathbb{P}(X_T=N) + 0 \cdot \mathbb{P}(X_T=0) = \mathbb{E}[X_0]=k,$
from which we also derive $p_k=k/N$.
The reason we talk about gambler’s ruin is by considering the limit $N\rightarrow\infty$ with k fixed. After a moment’s thought, it’s clear we can’t really talk about stopping the process when we hit infinity, since that won’t happen at any finite time. But we can ask what’s the probability that we eventually hit zero. Then, if we imagine a barrier at level N, the probability that we hit 0 at some point is bounded below by the probability that we hit 0 before we hit level N (given that we know we hit either zero or level N with probability one), and this is $\frac{N-k}{N}$, and by choosing N large enough, we can make this as close to 1 as we want. So the only consistent option is that the probability of hitting 0 at some point is one. Hence gambler’s ruin. With probability one, ruin will occur. There’s probably a moral lesson hiding there not especially subtly.
So the deal here seems to be that if you just care about your average, it doesn’t matter how to choose to play a sequence of fair games. But what if you care about something other than your average? In any real setting, we maybe care about slightly more than this. Suppose I offer you a bet on a coin toss: you get £3 if it comes up heads, and I get £1 if it comes up tails. Sounds like a good bet, since on average you gain a pound. But what about if you get £10,003 if it comes up heads and I get £10,001 if it comes up tails? I’m guessing you’re probably not quite so keen now.
But if you were an international bank, you might have fewer reservations about the second option. My intention is not to discuss whether our valuation of money is linear here, but merely to offer motivation for the financial option I’m about to propose. The point is that we are generally risk-averse (well, most of us, most of the time) and so we are scared of possible large losses, even when there is the possibility of large profits to balance it out.
Let’s assume we have our simple random walk, and for definiteness let’s say it starts at £1. Suppose (eg as a very niche birthday present) we have the following opportunity: at any point between now and time t=5, we have the right to buy one unit of the stock for £2.
We want to work out how much this opportunity, which from now on I’m going to call an option, is worth on average. Note that now it does seem that when we choose to cash in the option will have an effect on our return, and so we will have to include this in the analysis.
Note that, once we’ve bought a unit of the stock, we have an asset which is following a simple random walk (ie sequential fair games) and so from this point on its expected value remains unchanged. So in terms of expectation, we might as well sell the stock at the same moment we buy it. So if we cash in the option when the stock is currently worth £X, we will on average have a return of £(X-2). This means that we’ll only ever consider exercising our option if the current value of the stock is greater than £2. This narrows down our strategy slightly.
This sort of option minimises the risk of a large loss, since the worst thing that happens is that you never choose to exercise your option. So if you actually paid for the right to have this option, that cost is the largest amount you can lose. In the trading world, this type of opportunity is called an American option.
The trick here is to work backwards in time, thinking about strategies. If at time t=4, the stock is worth £1, then the best that can happen is that it’s worth £2 at time t=5, and this still gains you no wealth overall. Similarly if it’s worth £0 at time t=3. So we’ve identified a region where, if the stock value enters this region, we might as well rip up our contract, because we definitely aren’t going to gain anything. Remember now that we’ve also said you won’t ever cash in if the stock’s value is at most £2, because you don’t gain anything on average.
Now suppose that the stock has value £3 at time t=4. There’s no danger of it ever getting back below £2 during the lifetime of the option, so from now on your potential return is following the trajectory of a simple random walk, ie a fair game. So on average, it makes no difference whether you cash in now, or wait until t=5, or some combination of the two. The same argument holds if the stock has value £4 at time t=3 or time t=4, and so we can identify a region where you might as well cash in.
What about the final region? If the stock value is greater than £2, but not yet in the definitely-cash-in area, what should you do? Well, if you think about it, the value of the stock is a fair game. But your return should be better than that, because the stock price doesn’t take account of the fact that you wouldn’t buy in (and make a loss overall) if the value drops below £2. So at this stage, your future options are better than playing a fair game, and so it doesn’t make sense (in terms of maximising your *average*) to cash in.
Now we can actually work backwards in time to establish how much any starting value is worth under this optimal strategy. We can fill in the values in the ‘doomed’ area (ie all zeros) and on the ‘cash in now’ area (ie current value minus 2), and construct backwards using the fact that we have a random walk.
The final answer ends up being 7/16 if the stock had value £1 at time 0. Note that the main point here is that working out the qualitative form of the strategy was the non-trivial part. Once we’d done that, everything was fairly straightforward. I claim that this was a reasonably fun adjustment to the original problem, but have minimal idea whether pricing options is in general an interesting thing to do.
Anyway, I hope that provided an interesting overview to some of the topics of interest within the question of how to choose strategies for games based on random processes.
# Fair games and the martingale strategy I
I went back to my school a couple of weeks ago and gave a talk. I felt I’d given various incarnations of a talk on card-shuffling too many times, so it was time for a new topic. The following post (and time allowing, one or two more) is pretty much what I said.
The Martingale Strategy
Suppose we bet repeatedly on the outcome of tossing a fair coin. Since it’s November, my heart is set on buying an ice cream that costs £1, so my aim is to win this amount from our game. My strategy is this:
First, I bet £1. If I win, then that’s great, because I now have made exactly enough profit to buy the ice cream. If I lose, then I play again, and this time I bet £2. Again, if I win, then my total profit is £2-£1 = £1, so I stop playing and buy the ice cream. If I lose, then I play a third time, again doubling my stake. So if I win for the first time on the seventh go, my overall profit will be
£64 – (£1+£2+£4+£8+£16+£32) = £1,
and it’s clear that this can be continued and I will eventually win a round, and at this point my total profit will be £1. So I will always eventually be able to buy my ice cream.
But, there’s nothing special about the value £1, so I could replace the words ‘ice cream’ with ‘private tropical island’, so why am I still here in the UK on a wet Monday when I could be on my beach lounger?
There are some fairly obvious reasons why the strategy I’ve described is not actually a fail-safe way to make a profit. For a start, although with probability one a head will come up eventually, there is a small positive chance that the first 200 rolls will all be tails. At this point, I would have accrued a debt of roughly $2^{200}$ pounds, and this is slightly more than the number of atoms in the universe. All this for an ice cream?
So there are major problems carrying out this strategy in a finite world. And of course, it’s no good if we stop after a very large but finite number of turns, because then there’s always this very small chance that we’ve made a very large loss, which is bad, partly because we can’t have the ice cream, but also because it exactly cancels out the chance of making our £1 profit, and so our overall average profit is exactly zero.
Though I’ve set this up in an intentionally glib fashion, as so often is the case, we might have stumbled across an interesting mathematical idea. That is, if we play a fair game a finite number of times, we have a fair game overall, meaning our overall average profit is zero. But if we are allowed to play a potentially infinite number of times, then it’s not clear how to define our overall ‘average’ profit, since we feel it ought to be zero, as an extension of the finite case, but also might be positive, because it ends up being £1 with probability one.
It’s tempting at this stage to start writing statements like
$1 \times 1 + (-\infty) \times 0=0 ,$
to justify why this might have come about, where we consider the infinitely unlikely event that is infinitely costly. But this is only convincing at the most superficial level, and so it makes more sense to think a bit more carefully about under exactly what circumstances we can extend our observation about the overall fairness of a finite sequence of individual fair games.
A second example
The previous example was based upon a series of coin tosses, and we can use exactly the same source of randomness to produce a simple random walk. This is a process that goes up or down by 1 in each time step, where each option happens with probability ½, independently of the history.
We could avoid the requirement to deal with very large bets by always staking £1, and then cashing in the first time we have a profit of £1. Then, if we start the random walk at zero, it models our profit, and we stop the first time it gets to 1. It’s not obvious whether we hit 1 with probability one. Let’s show this.
In order to hit some positive value k, the random walk must pass through 1, 2, and so on, up to (k-1) and then finally k. So $\mathbb{P}(\text{hit k}) = [\mathbb{P}(\text{hit 1})]^k$. And similarly for negative values. Also, the probability that we return to zero is the same as the probability that we ever hit 1, since after one time-step they are literally the same problem (after symmetry). So, if the probability of hitting 1 is p<1, then the number of visits to zero is geometric (supported on 1,2,3,…) with parameter p, and so
$\mathbb{E}[\text{visits to k}] = \mathbb{E}[\text{visits to zero}] \times \mathbb{P}(\text{hit k})=(1+1/p) \times p^{|k|} = (p+1)p^{|k|-1}.$
Thus, when we sum over all values of k, we are summing a pair of geometric series with exponent <1, and so we get a finite answer. But if the expected number of visits to anywhere (ie the sum across all places) is finite, this is clearly ridiculous, since we are running the process for an infinite time, and at each time-step we must be somewhere! So we must in fact have p=1, and thus another potential counter-example to the claim that a sequence of fair games can sometimes be unfair.
We might have exactly the same set of practical objections, such as this method requiring arbitrarily large liquidity (even though it doesn’t grow exponentially fast so doesn’t seem so bad).
What will actually turn out to be useful is that although the bets are now small, the average time until we hit 1 is actually infinite. Remember that, even though most things we see in real life don’t have this property, it is completely possible for a random variable to take finite values yet have infinite expectation.
Notes on the Martingale Strategy
There’s no reason why the originally proposed strategy had to be based upon fair coin tosses. This strategy might work in a more general setting, where the chance of winning on a given turn is not ½, or is not even constant. So long as at each stage you bet exactly enough that, if you win, you recoup all your losses so far, and one extra pound, this has the same overall effect.
Of course, we need to check that we do eventually win a round, which is not guaranteed if the probability of winning (conditional on not having yet won) decays sufficiently fast. If we let $p_k$ be the probability of winning on turn k, given that we haven’t previously won, then we require that the probability of never winning $\prod_{k\ge 1}(1-p_k)=0$. By taking logs and taking care of the approximations, it can be seen that the divergence or otherwise of $\sum p_k$ determines which way this falls.
In the next post, we’ll talk about how the two problems encountered here, namely allowing large increments, and considering a stopping time with infinite expectation are exactly the two cases where something can go wrong. We’ll also talk about a slightly different setting, where the choice of when to stop playing becomes a bit more dynamic and complicated.
# Ornstein-Uhlenbeck Process
A large part of my summer has been spent proving some technical results pertaining to the convergence of some functionals of a critical Frozen Percolation process. This has been worthwhile, but hasn’t involved a large amount of reading around anything in particular, which has probably contributed to the lack of posts in recent months. Perhaps a mixture of that and general laziness?
Anyway, it turns out that the limit of the discrete processes under consideration is the Ornstein-Uhlenbeck process. The sense in which this limit holds (or at least, for now, is conjectured to hold) is something for another article. However, I thought it would be worth writing a bit about this particular process and why it is interesting.
The O-U process is described by the SDE
$dX_t=-\beta (X_t-\mu)dt+\sigma dW_t,$
where W is a standard Brownian motion. We think of $\mu$ as the ‘mean’. The extent to which this behaves as a mean will be discussed shortly. The process is then mean-reverting, in the sense that the drift is directed against deviations of the process away from this mean. The parameter $\beta$ measures the extent of this mean reversion, while as usual $\sigma$ controls the magnitude of the Brownian noise.
The motivation for considering mean-reverting processes is considerable. One measure of this is how many equations with articles on Wikipedia turn out to be precisely this Ornstein-Uhlenbeck process with different context or notation. In most cases, the motivation arises because Brownian motion is for some reason unsuitable to take as a canonical random process. We will see why the O-U process is somehow the next most canonical choice for a random process.
In physics, it is sometimes unsatisfactory to model the trajectory of a particle with Brownian motion (even though this motivated the name…) as the velocities are undefined (see this post from ages ago), or infinite, depending on your definition of velocity. Using the Ornstein-Uhlenbeck process to model the velocity of a particle is often a satisfactory alternative. It is not unreasonable that there should be a mean velocity, presumably zero. The mean reversion models a frictional force from the underlying medium, while the Brownian noise describes random collisions with similar particles.
In financial applications, the Ornstein-Uhlenbeck model has been applied, apparently under the title of the Vasicek model since the 70s to describe quantities such as interest rates where there is some underlying reason to ban indefinite growth, and require mean reversion. Another setting might be a commodity which, because of external driving factors, has over the relevant time-scale well-defined mean value, around which mean-reverting fluctuations on the observed time-scale can be described. As with other financial models, it is undesirable for a process to take negative values. This can be fixed by taking a positive mean, then setting the volatility to be state dependent, decaying to zero as the state tends to zero, so for small values, the positive drift dominates. I don’t fully understand why patching this aspect is significantly more important than patching any other non-realistic properties of the model, but the resulting SDE is, at least in one particular case where the volatility is $\sqrt{X_t}$, called the Cox-Ingersoll-Ross model.
Anyway, a mathematical reason to pay particular attention to this Ornstein-Uhlenbeck process is the following. It is the unique family of continuous Markov processes to have a stationary Gaussian distribution. It is the mean-reverting property that is key. There is no chance of Brownian motion having any stationary distribution, let alone a Gaussian one. If this isn’t clear, you can convince yourself by thinking of the stationary distribution of SRW on $\mathbb{Z}$. Since the process is space-homogeneous, the only stationary measure is the uniform measure.
I want to focus on one particular property of the O-U process, through which some other aspects will be illuminated. If we take $\sigma=\beta$ and let $\beta\rightarrow\infty$, then the stationary processes converge to white noise.
First though, we should note this is perhaps the easiest SDE to solve explicitly. We consider $X_t e^{\theta t}$, and applying Ito’s lemma rapidly gives
$X_t=\mu + (X_0-\mu)e^{-\beta t}+\sigma\int_0^t e^{-\beta(t-s)}dW_s.$
W is Gaussian so the distribution of $X_t$ conditional on $X_0=x_0$ is also Gaussian, and since W is centred we can read off the expectation. Applying the Ito isometry then gives the variance. In conclusion:
$X_t\stackrel{d}{=}\mathcal{N}(\mu+(x_0-\mu)e^{-\beta t}, \frac{\sigma^2}{\beta}(1-e^{-2\beta t})).$
In particular, note that the variation has no dependence on $x_0$. So as t grows to infinity, this converges to $\mathcal{N}(\mu, \frac{\sigma^2}{\beta})$. This is, unsurprisingly, the stationary distribution of the process.
To address the white noise convergence, we need to consider $\text{Cov}(X_0,X_t)$ in stationarity. Let’s assume WLOG that $\mu=0$ so most of the expectations will vanish. We obtain
$\text{Cov}(X_0,X_t)=\mathbb{E}[X_0X_t]=\mathbb{E}_{x_0}\left[\mathbb{E}[X_t| X_0=x_0]\right]=\mathbb{E}[X_0^2 e^{-\beta t}]= \frac{sigma^2}{2\beta}2^{-\beta t}.$
If we want, the Chapman-Kolmogorov equations work particularly nicely here, and we are able to derive a PDE for the evolution of the density function, though obviously this is very related to the result above. This PDE is known as the Fokker-Planck equation.
So, in particular, when $\sigma=\beta\rightarrow \infty$, this covariance tends to 0. I’m not purporting that this constitutes a proof that the Ornstein-Uhlenbeck processes converge as processes to white noise. It’s not obvious how to define process convergence, not least because there’s flexibility about how to view white noise as a process. One doesn’t really want to define the value of white noise at a particular time, but you can consider the covariance of integrals of white noise over disjoint intervals as a limit, in similar way to convergence of finite dimensional distributions.
The fact that taking $\beta=0$ gives Brownian motion, and this case gives white noise, intermediate versions of the Ornstein-Uhlenbeck process are sometimes referred to as coloured noise.
Finally, the Ornstein-Uhlenbeck process emerges as the scaling limit of mean-reverting discrete Markov chains, analogous to Brownian motion as the scaling limit of simple random walk. One particularly nice example is the Ehrenfest Urn model. We have two urns, and 2N balls. In each time step one of the 2N balls is chosen uniformly at random, and it is moved to the other urn. So a ball is more likely to be removed from an urn with more than N balls. We can view this as a model for molecules in, say a room, with a slightly porous division between them, eg a small hole. More complicated interface models in higher dimensions lead to fascinating PDEs, such as the famous KPZ equation, which are the subject of much ongoing interest in this area.
This result can be an application of the theory of convergence of Markov chains to SDEs pioneered by Stroock and Varadhan, about which more may follow very soon. In any case, it turns out that the fluctuations in the Ehrenfest Urn model are on the scale of $\sqrt{n}$, unsurprisingly, and are given by a centred Ornstein-Uhlenbeck process.
Investigating this has reminded me how much I’ve forgotten, or perhaps how little I ever knew, about the technicalities of stochastic processes are their convergence results, so next up will probably be a summary of all the useful definitions and properties for this sort of analysis.
# Hitting Probabilities for Markov Chains
This continues my previous post on popular questions in second year exams. In the interest of keeping it under 2,500 words I’m starting a new article.
In a previous post I’ve spoken about the two types of Markov chain convergence, in particular, considering when they apply. Normally the ergodic theorem can be used to treat the case where the chain is periodic, so the transition probabilities do not converge to a stationary distribution, but do have limit points – one at zero corresponding to the off-period transitions, and one non-zero. With equal care, the case where the chain is not irreducible can also be treated.
A favourite question for examiners concerns hitting probabilities and expected hitting times of a set A. Note these are unlikely to come up simultaneously. Unless the hitting probability is 1, the expected hitting time is infinite! In both cases, we use the law of total probability to derive a family of equations satisfied by the probabilities/times. The only difference is that for hitting times, we add +1 on the right hand side, as we advance one time-step to use the law of total probability.
The case of hitting probabilities is perhaps more interesting. We have:
$h_i^A = 1,\; i\in A, \quad h_i^A=\sum_{j\in S}p_{ij}h_j^A,\; i\not\in A.$
There are two main cases of interest: where the chain is finite but has multiple closed communicating classes, and where the chain is infinite, so even though it is irreducible, a trajectory might diverge before hitting 0.
For the case of a finite non-irreducible Markov chain, this is fairly manageable, by solving backwards from states where we know the values. Although of course you could ask about the hitting probability of an open state, the most natural question is to consider the probability of ending up in a particular closed class. Then we know that the hitting probability starting from site in the closed class A is 1, and the probability starting from any site in a different closed class is 0. To find the remaining values, we can work backwards one step at a time if the set of possible transitions is sparse enough, or just solve the simultaneous equations for $\{h_i^A: i\text{ open}\}$.
We therefore care mainly about an infinite state-space that might be transient. Typically this might be some sort of birth-and-death chain on the positive integers. In many cases, the hitting probability equations can be reduced to a quadratic recurrence relation which can be solved, normally ending up with the form
$h_i=A+B\lambda^i$,
where $\lambda$ might well be q/p or similar if the chain is symmetric. If the chain is bounded, typically you might know $h_0=1, h_N=0$ or similar, and so you can solve two simultaneous equations to find A and B. For the unbounded case you might often only have one condition, so you have to rely instead on the result that the hitting probabilities are the minimal solution to the family of equations. Note that you will always have $h^i_i=1$, but with no conditions, $h^i_j\equiv 1$ is always a family of solutions.
It is not clear a priori what it means to be a minimal solution. Certainly it is not clear why one solution might be pointwise smaller than another, but in the case given above, it makes sense. Supposing that $\lambda<1$, and A+B=1 say, then as we vary the parameters, the resulting set of ‘probabilities’ does indeed vary monotically pointwise.
Why is this true? Why should the minimum solution give the true hitting probability values? To see this, take the equations, and every time an $h_i^A$ appears on the right-hand side, substitute in using the equations. So we obtain, for $i\not\in A$,
$h_i^A=\sum_{j\in A}p_{ij}+\sum_{j\not\in A} p_{ij}h_j^A,$
and after a further iteration
$h_i^A=\sum_{j_1\in A}p_{ij_1}+\sum_{j_1\not\in A, j_2\in A}p_{ij_1}p_{j_1j_2}+\sum_{j_1,j_2\not\in A}p_{ij_1}p_{j_1j_2}h_{j_2}^A.$
So we see on the RHS the probability of getting from i to A in one step, and in two steps, and if keep iterating, we will get a large sum corresponding to the probability of getting from i to A in 1 or 2 or … or N steps, plus an extra term. Note that the extra term does not have to correspond to the probability of not hitting A by time N. After all, we do not yet know that $(h_{i}^A)$ as defined by the equations gives the hitting probabilities. However, we know that the probability of hitting A within N steps converges to the probability of hitting A at all, since the sequence is increasing and bounded, so if we take a limit of both sides, we get $h_i^A$ on the left, and something at least as large as the hitting probability starting from i on the right, because of the extra positive term. The result therefore follows.
It is worth looking out for related problems that look like a hitting probability calculation. There was a nice example on one of the past papers. Consider a simple symmetric random walk on the integers modulo n, arranged clockwise in a circle. Given that you start at state 0, what is the probability that your first return to state 0 involves a clockwise journey round the circle?
Because the system is finite and irreducible, it is not particularly interesting to consider the actual hitting probabilities. Also, note that if it is convenient to do so, we can immediately reduce the problem when n is even. In two steps, the chain moves from j to j+2 and j-2 with probability ¼ each, and stays at j with probability ½. So the two step chain is exactly equivalent to the lazy version of the same dynamics on n/2.
Anyway, even though the structure is different, our approach should be the same as for the hitting probability question, which is to look one step into the future. For example, to stand a chance of working, our first two moves must both be clockwise. Thereafter, we are allowed to move anticlockwise. There is nothing special about starting at 0 in defining the original probability. We could equally well ask for the probability that starting from j, the first time we hit 0 we have moved clockwise round the circle.
The only thing that is now not obvious is how to define moving clockwise round the circle, since it is not the case that all the moves have to be clockwise to have experienced a generally clockwise journey round the circle, but we definitely don’t want to get into anything complicated like winding numbers! In fact, the easiest way to make the definition is that given the hitting time of 0 is T, we demand that the chain was at state n at time T-1.
For convenience (ie to make the equations consistent) we take $h_0=0, h_n=1$ in an obvious abuse of notation, and then
$h_j=\frac12h_{j-1}+\frac12 h_{j+1},$
from which we get
$h_j=a+bj \Rightarrow h_j=\frac{j}{n}.$
Of course, once we have this in mind, we realise that we could have cut the circle at 0 (also known as n) and unfolded it to reduce the problem precisely to symmetric gambler’s ruin. In particular, the answer to the original problem is 1/2n, which is perhaps just a little surprising – maybe by thinking about the BM approximation to simple random walk, and that BM started from zero almost certainly crosses zero infinitely many times near we might have expected this probability to decay faster. But once it is unfolded into gambler’s ruin, we have the optimal stopping martingale motivation to reassure us that this indeed looks correct.
# Sticky Brownian Motion
This follows on pretty much directly from the previous post about reflected Brownian motion. Recall that this is a process defined on the non-negative reals which looks like Brownian motion away from 0. We consider whether RBM is the only such process, and how any alternative might be constructed as a limit of discrete-time Markov processes.
One of the alternatives is called Sticky Brownian motion. This process spends more time at 0 than reflected Brownian motion. In fact it spends some positive proportion of time at 0. My main aim here is to explain why some intuitive ideas I had about how this might arise are wrong.
The first thought was to ensure that each visit to 0 last some positive measure of time. This could be achieved by staying at 0 for an Exp(1) duration, every time the process visited it. It doesn’t seem unreasonable that this might appear as the limit of a standard SRW adjusted so that on each visit to 0 the walker waits for a time given by independent geometric distributions. These distributions are memoryless, so that is fine, but by Blumenthal’s 0-1 Law, starting from 0 a Brownian motion hits zero infinitely many times before any small time t. So in fact the process described above would be identically zero as before it gets anywhere it would have to spend some amount of time at 0 given by an infinite sum of Exp(1) RVs.
We will return later to the question of why the proposed discrete-time model will still converge to reflected BM rather than anything more exotic. First though, we should discount the possibility of any intermediate level of stickiness, where the set of times spent at 0 still has measure zero, but the local time at 0 grows faster than for standard reflected BM. We can define the local time at 0 through a limit
$L_t=\lim_{\epsilon\downarrow 0}\frac{1}{2\epsilon}\text{Leb}(\{0\le s \le t: |B_t|<\epsilon\})$
of the measure of time spent very near 0, rescaled appropriately. So if the measure of the times when the process is at 0 is zero, then the local time is determined by behaviour near zero rather than by behaviour at zero. More precisely, on the interval $[-\epsilon,\epsilon]$, the process behaves like Brownian motion, except on a set of measure zero, so the local time process should look the same as that of BM itself. Note I don’t claim this as a formal proof, but I hope it is a helpful heuristic for why you can’t alter the local time process without altering the whole process.
At this stage, it seems sensible to define Sticky Brownian motion. For motivation, note that we are looking for a process which spends a positive measure of time at 0. So let’s track this as a process, say $C_t$. Then the set of times when C is increasing is sparse, as it coincides with the process being 0, but we know we cannot wait around at 0 for some interval of time without losing the Markov property. So C shares properties with the local time of a reflected BM. The only difference is that the measure of times when C is increasing is positive here, but zero for the local time.
So it makes sense to construct the extra time spent at zero from the local time of a standard reflected BM. The heuristic is that we slow down the process whenever it is at 0, so that local time becomes real time. We can also control the factor by which this slowing-down happens, so define
$\sigma(s)=\rho L(s)+s,$
where L is the local time process of an underlying reflected BM, and $\rho>0$ is a constant. So $\sigma$ is a map giving a random time-change. Unsurprisingly, we now define Sticky BM as the reflected BM with respect to this time-change. To do this formally, it is easiest to define a family of stopping times $\{\tau_t\}$, such that $\sigma(\tau_t)=t, \tau_{\sigma(s)}=s$, then if X is the reflected BM, define $Y_t=X_{\tau_t}$ for the sticky BM.
It is worth thinking about what the generator of this process should be. In particular, why should it be different to reflected BM? The key observation is that the drift of the underlying reflected BM is essentially infinite at 0. By slowing down the process at 0, this drift becomes finite. So the martingale associated with sticky BM is precisely a time-changed version of the martingale associated with the underlying reflected BM, but this time-change is precisely what is required to give a generator. We get:
$(\mathcal{L}f)(x)=\begin{cases}\frac12f''(x)&\quad x>0\\ \rho^{-1}f'(0) &\quad x=0.\end{cases}$
Now that we have the generator, it starts to become apparent how sticky BM might appear as a limit of discrete-time walks. The process must look like mean-zero, unit-variance RW everywhere except near 0, where the limiting drift should be $\rho^{-1}$. Note that when considering the limiting drift near zero, we are taking a joint limit in x and h. The order of this matters. As explained at the end of the previous article, we only need to worry about the limiting drift along sequences of $x,h$ such that $a_h(x)\rightarrow 0$. If no such sequences exist, or the limiting drift along any of these is infinite, then we actually have a reflected boundary condition.
This highlights one confusing matter about convergence of reflected processes. The boundary of the discrete-time process should converge to the boundary of the reflected process, but we also have to consider where reflective behaviour happens. Can we get sticky BM with reflection only at the boundary in the discrete-time processes? The answer turns out to be no. At the start of this article, I proposed a model of SRW with geometric waiting times whenever the origin was visiting. What is the limit of this?
The trick is to consider how long the discrete process spends near 0, after rescaling. It will spend a multiple 1/p more time at 0 itself, where p is the parameter of the geometric distribution, but no more time than expected at any point $x\in(0,\epsilon)$. But time spent in $(0,\epsilon)$ dominates time spent at 0 before this adjustment, so must also dominate it after the adjustment, so in the limit, the proportion of time spent at 0 is unchanged, and so in particular it cannot be positive.
Because of all of this, in practice it seems that most random walks we might be interested in converge (if they converge to a process at all) to a reflected SDE/diffusion etc, rather than one with sticky boundary conditions. I feel I’ve been talking a lot about Markov processes converging, so perhaps next, or at least soon, I’ll write some more technical things about exactly what conditions and methods are required to prove this.
REFERENCES
S. Varadhan – Chapter 16 from a Lecture Course at NYU can be found here.
# Reflected Brownian Motion
A standard Brownian motion is space-homogeneous, meaning that the behaviour of $B_{T+t}-B_T$ does not depend on the value of $B_T$. By Donsker’s Theorem, such a Brownian motion is also the limit in a process space of any homogeneous random walk with zero-drift and constant variance, after suitable rescaling.
In many applications, however, we are interested in real-valued continuous-time Markov processes that are defined not on the whole of the real line, but on the half-line $\mathbb{R}_{\ge 0}$. So as BM is the fundamental real-valued continuous-time Markov process, we should ask how we might adjust it so that it stays non-negative. In particular, we want to clarify uniqueness, or at least be sure we have found all the sensible ways to make this adjustment, and also to consider how Donsker’s Theorem might work in this setting.
We should consider what properties we want this non-negative BM to have. Obviously, it should be non-negative, but it is also reasonable to demand that it looks exactly like BM everywhere except near 0. But since BM has a scale-invariance property, it is essentially meaningful to say ‘near 0’, so we instead demand that it looks exactly like BM everywhere except at 0. Apart from this, the only properties we want are that it is Markov and has continuous sample paths.
A starting point is so-called reflected Brownian motion, defined by $X_t:=|B_t|$. This is very natural and very convenient for analysis, but there are some problems. Firstly, this has the property that it looks like Brownian motion everywhere except 0 only because BM is space-homogeneous but also symmetric, in the sense that $B_t\stackrel{d}{=}-B_t$. This will be untrue for essentially any other process, so as a general method for how to keep stochastic processes positive, this will be useless. My second objection is a bit more subtle. If we consider this as an SDE, we get
$dX_t=\text{sign}(B_t)dB_t.$
This is a perfectly reasonable SDE but it is undesirable, because we have a function of B as coefficient on the RHS. Ideally, increments of X would be a function of X, and the increments of B, rather than the values of B. That is, we would expect $X_{t+\delta t}-X_t$ to depend on $X_t$ and on $(B_{t+s}-B_t, 0\le s\le \delta t)$, but not on $B_t$ itself, as that means we have to keep track of extra information while constructing X.
So we need an alternative method. One idea might be to add some non-negative process to the BM so that the sum stays non-negative. If this process is deterministic and finite, there there is some positive probability that the sum will eventually be negative, so this won’t do. We are looking therefore so a process which depends on the BM. Obviously we could take $\max(-B_t,0)$, but this sum would then spend macroscopic intervals of time at 0, and these intervals would have the Raleigh distribution (for Brownian excursions) rather than the exponential distribution, hence the process given by the sum would not be memoryless and Markov.
The natural alternative is to look for an increasing process $A_t$, and then it makes sense to talk about the minimal increasing process that has the desired property. A moment’s thought suggests that $A_t=-min_{s\le t}B_t$ satisfies this. So we have the decomposition
$B_t=-A_t+S_t,$
where $S_t$ is the height of B above its running minimum. So S is an ideal alternative definition of reflecting BM. In particular, when B is away from its minimum, $dB_t=dS_t$, so this has the property that it evolves exactly as the driving Brownian motion.
What we have done is to decompose a general continuous process into the sum of a decreasing continuous process and a non-negative process. This is known as the Skorohod problem, and was the subject of much interest, even in the deterministic case. Note that process A has the property that it is locally constant almost everywhere, and is continuous, yet non-constant. Unsurprisingly, since A only changes when the underlying BM is 0, A is continuous with respect to the local time process at 0. In fact, A is the local time process of the underlying Brownian motion, by comparison with the construction by direct reflection.
One alternative approach is to look instead at the generator of the process. Recall that the generator of a process is an operator on some space of functions, with $\mathcal{L}f$ giving the infinitissimal drift of $f(X_t)$. In the case of Brownian motion, the generator $(\mathcal{L}f)(x)=\frac12 f''(x)$ for bounded smooth functions f. This is equivalent to saying that
$f(X_t)-f(X_0)-\int_0^t \frac12 f''(X_s)ds$ (*)
is a martingale. This must hold also for reflected Brownian motion, whenever x is greater than 0. Alternatively, if the function f is zero in a small neighbourhood of 0, it should have the same generator with respect to reflected BM. Indeed, for a general smooth bounded function f, we can still consider the expression (*) with respect to reflected BM. We know this expression behaves as a martingale except when X is zero. If f'(0)>0, and T is some hitting time of 0, then $f(X_{T+\delta T})-f(X_T)\ge 0$, hence the expression (*) is a submartingale. So if we restrict attention to functions with f'(0)=0, the generator remains the same. Indeed, by patching together all such intervals, it can be argued that even if f'(0) is not zero,
$f(X_t)-f(X_0)-\int_0^t \frac12 f''(X_s)ds - f'(0)A_t$
is a martingale, where A is the local time process at zero.
I was aware when I started reading about this that there was another family of processes called ‘Sticky Brownian Motion’ that shared properties with Reflected BM, in that it behaves like standard BM away from zero, but is also constrained to the non-negative reals. I think this will get too long if I also talk about that here, so that can be postponed, and for now we consider reflected BM as a limit of reflected (or other) random walks, bearing in mind that there is at least one other candidate to be the limit.
Unsurprisingly, if we have a family of random walks constrained to the non-negative reals, that are zero-drift unit-variance away from 0, then if they converge as processes, the limit is Brownian away from zero, and non-negative. Note that “away from 0” means after rescaling. So the key aspect is behaviour near zero.
What is the drift of reflected BM at 0? We might suspect it is infinite because of the form of the generator, but we can calculate it directly. Given $X_0=0$, we have:
$\frac{\mathbb{E}X_t}{t}=\frac{\mathbb{E}|B_t|}{t}=\frac{\sqrt{t}\mathbb{E}|B_1|}{t},$
so letting $t\rightarrow 0$, we see indeed that the drift is infinite at 0.
For convergence of discrete processes, we really need the generators to converge. Typically we index the discrete-time processes by the time unit h, which tends to 0, and $b_h(x),a_h(x)$ are the rescaled drift and square-drift from x. We assume that we don’t see macroscopic jumps in the limit. For the case of simple random walk reflected at 0, it doesn’t matter exactly how we construct the joint limit in h and x, as the drift is uniform on x>0, but in general this does matter. I don’t want to discuss sticky BM right now, so it’s probably easiest to be vague and say that the discrete Markov processes converge to reflected BM so long they don’t spend more time than expected near 0 in the limit, as the title ‘sticky’ might suggest.
The two ways in which this can happen is if the volatility term $a_h(x)$ is too small, in which case the process looks almost deterministic near 0, or if the drift doesn’t increase fast enough. And indeed, this leads to two conditions. The first is straightforward, if $a_h(x)$ is bounded below, in the sense that $\liminf_{h,x\rightarrow 0} a_h(x)\ge C>0$, then we have convergence to reflected BM. Alternatively, the only danger can arise down those subsequences where $a_h(x)\rightarrow 0$, so if we have that $b_h(x)\rightarrow +\infty$ whenever $h,x,a_h(x)\rightarrow 0$, then this convergence also holds.
Next time I’ll discuss what sticky BM means, what it doesn’t mean, why it isn’t easy to double the local time, and how to obtain sticky BM as a limit of discrete random walks in a similar way to the above.
REFERENCES
S. Varadhan – Chapter 16 from a Lecture Course at NYU can be found here.
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2017-08-23 11:42:52
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https://scholarship.rice.edu/handle/1911/64862/browse?rpp=20&sort_by=1&type=title&etal=-1&starts_with=B&order=ASC
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Now showing items 47-66 of 1345
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We report measurements of the nuclear modification factor RCP for charged hadrons as well as identified π+(−), K+(−), and p(¯p) for Au+Au collision energies of √sNN=7.7, 11.5, 14.5, 19.6, 27, 39, and 62.4 GeV. We observe ...
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(2018)
New measurements of directed flow for charged hadrons, characterized by the Fourier coefficient v1, are presented for transverse momenta pT, and centrality intervals in Au+Au collisions recorded by the STAR experiment for ...
• #### Beam-Energy Dependence of Charge Separation along the Magnetic Field in Au+Au Collisions at RHIC
(2014)
Local parity-odd domains are theorized to form inside a quark-gluon plasma which has been produced in high-energy heavy-ion collisions. The local parity-odd domains manifest themselves as charge separation along the magnetic ...
• #### Beam-Energy Dependence of Directed Flow of $\mathrm{\ensuremath{\Lambda}}$, $\overline{\mathrm{\ensuremath{\Lambda}}}$, ${K}^{\ifmmode\pm\else\textpm\fi{}}$, ${K}_{s}^{0}$, and $\ensuremath{\phi}$ in $\mathrm{Au}+\mathrm{Au}$ Collisions
(2018)
Rapidity-odd directed-flow measurements at midrapidity are presented for Λ, ¯Λ, K±, K0s, and ϕ at √sNN=7.7, 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV in Au+Au collisions recorded by the Solenoidal Tracker detector at the ...
• #### Beam-Energy Dependence of the Directed Flow of Protons, Antiprotons, and Pions in Au+Au Collisions
(2014)
Rapidity-odd directed flow (v1) measurements for charged pions, protons, and antiprotons near midrapidity (y=0) are reported in √sNN=7.7, 11.5, 19.6, 27, 39, 62.4, and 200 GeV Au+Au collisions as recorded by the STAR ...
• #### Beam-energy-dependent two-pion interferometry and the freeze-out eccentricity of pions measured in heavy ion collisions at the STAR detector
(2015)
We present results of analyses of two-pion interferometry in Au+Au collisions at √sNN=7.7, 11.5, 19.6, 27, 39, 62.4, and 200 GeV measured in the STAR detector as part of the BNL Relativistic Heavy Ion Collider Beam Energy ...
• #### Benchmark measurements and simulations of dose perturbations due to metallic spheres in proton beams
(2013)
Monte Carlo simulations are increasingly used for dose calculations in proton therapy due to its inherent accuracy. However, dosimetric deviations have been found using Monte Carlo code when high density materials are ...
(2006)
• #### Biomolecular dynamics: order–disorder transitions and energy landscapes
(2012)
While the energy landscape theory of protein folding is now a widely accepted view for understanding how relatively-weak molecular interactions lead to rapid and cooperative protein folding, such a framework must be extended ...
• #### Bose-Einstein Condensation of 84Sr
(2009)
We report Bose-Einstein condensation of Sr84 in an optical dipole trap. Efficient laser cooling on the narrow intercombination line and an ideal s-wave scattering length allow the creation of large condensates (N0∼3×105) ...
• #### Bose-Einstein condensation of 88Sr through sympathetic cooling with 87Sr
(2010-05-05)
We report Bose-Einstein condensation of Sr88, which has a small, negative s-wave scattering length (a88=−2a0). We overcome the poor evaporative cooling characteristics of this isotope by sympathetic cooling with Sr87 atoms. ...
• #### Bose-Einstein condensation of lithium
(1997)
Bose-Einstein condensation of 7Li has been studied in a magnetically trapped gas. Because of the effectively attractive interactions between 7Li atoms, many-body quantum theory predicts that the occupation number of the ...
• #### Bose-Einstein Condensation of Lithium: Observation of Limited Condensate Number
(1997)
Bose-Einstein condensation of 7Li has been studied in a magnetically trapped gas. Because of the effectively attractive interactions between 7Li atoms, many-body quantum theory predicts that the occupation number of the ...
• #### Bose-Einstein correlations in $pp, p\mathrm{Pb}$, and PbPb collisions at $\sqrt{{s}_{NN}}=0.9--7$ TeV
(2018)
Quantum-statistical (Bose-Einstein) two-particle correlations are measured in pp collisions at √s=0.9, 2.76, and 7 TeV, as well as in pPb and peripheral PbPb collisions at nucleon-nucleon center-of-mass energies of 5.02 ...
• #### Bose-Fermi mapping and a multibranch spin-chain model for strongly interacting quantum gases in one dimension: Dynamics and collective excitations
(2016)
We show that the wave function in one spatial sector x1<x2<⋯<xN (with xi being the coordinate of the ith particle) of a one-dimensional spinor gas with contact s-wave interaction, either bosonic or fermionic, can be mapped ...
• #### Bosonic molecules in a lattice: Unusual fluid phase from multichannel interactions
(2018)
We show that multichannel interactions significantly alter the phase diagram of ultracold bosonic molecules in an optical lattice. Most prominently, an unusual fluid region intervenes between the conventional superfluid ...
• #### Bridging quantum and classical plasmonics with a quantum-corrected model
(2012)
Electromagnetic coupling between plasmonic resonances in metallic nanoparticles allows for engineering of the optical response and generation of strong localized near-fields. Classical electrodynamics fails to describe ...
• #### Bright matter wave solitons in Bose–Einstein condensates
(2003)
We review recent experimental and theoretical work on the creation of bright matter wave solitons in Bose–Einstein condensates. In two recent experiments, solitons are formed from Bose–Einstein condensates of 7Li by ...
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2019-09-16 08:49:37
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https://www.prepanywhere.com/prep/textbooks/9-mathematics-nelson/chapters/chapter-8-measurement/materials/8-10-chapter-review
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Chapter Review
Chapter
Chapter 8
Section
Chapter Review
Solutions 28 Videos
Aryn is creating a rectangular outdoor space for her pet rabbit. Fencing material costs $15.25/m. She has$145. What dimensions give the greatest area, to the nearest tenth of a metre?
2.05mins
Q1
What is the minimum perimeter possible for a rectangle with an area of 500 cm^2?
0.48mins
Q2
Sarah has 20 m of garden edging. What are the dimensions of the rectangular garden with the greatest area can she enclose with the edging?
0.36mins
Q3
Denzel wants to rope off a 800 m^2 rectangular swimming area using the beach as one of the sides. What should the dimensions of the rectangle be in order to use the minimum amount of rope?
1.28mins
Q4
Calculate the area of the figure.
1.08mins
Q5
A field has the dimensions shown.
a) Calculate the length of one lap of the track.
b) If Alice ran 625 m, how many laps did she run?
c) Calculate the area of the field.
2.17mins
Q7
Calculate the area and perimeter of each regular polygon.
1.04mins
Q8a
A baseball diamond is a square. The distance between the bases is 27.4 m. Calculate the direct distance from first base to third base.
1.09mins
Q9
Find the length of x accurate to the nearest tenth.
0.57mins
Q10
Determine the length of the fence around the playground.
1.24mins
Q11
Calculate the surface area of the regular pyramid.
2.03mins
Q13
We want to paint the house shown below, including the door. For the roof, we want to re-shingle the entire roof. One 4L can of paint covers 35 m^2. One bundle of shingles covers 2.25 m^2
Height from the ground to peak = 5.0 m
a) How many bundles of shingles do they need for the roof? (Hint: Find the slant height of the roof first.)
b) How many cans of paint do they need?
c) One can of paint is $29.95 and one bundle of shingles is$35.99. Find the total cost of the job.
5.20mins
Q14
Determine the surface area of a square-based pyramidal candle with a base side length of 8 cm and a slant height of 10 cm.
1.54mins
Q15
Determine the height of a square-based pyramid with a base side length of 8.0 cm and a surface area of 440.0 cm^2.
1.07mins
Q16
Calculate the volume and surface area of each figure.
1.32mins
Q17a
Calculate the volume and surface area of each figure.
1.41mins
Q17b
Gum is packaged in a square-based pyramid- shaped box with a distance of 6 cm from the centre of the base to the sides and a height of 12 cm.
a) How much material was used to create the box?
b) What is the volume of the box?
2.07mins
Q18
A solid figure is said to be truncated when a portion of the bottom is cut and removed. The cut line must be parallel to the base. Many paper cups, such as the one shown here, are truncated cones. Calculate the volume of this paper cup.
0.49mins
Q19
Calculate the volume and surface area of this sphere.
0.37mins
Q20
A spherical bar of soap just fits inside its package, which is a cube with a side length of 8 cm.
a) What is the volume of the bar of soap.
b) Calculate the amount of empty space in the box.
1.23mins
Q21
A toy company makes rubber balls with a diameter of 20 cm. How much rubber would be saved per ball if the balls had a diameter' of 15 cm?
1.39mins
Q22
A square-based pyramid has a base side length of 13 cm and a height of 16 cm. What are the dimensions for a cylinder having the same volume as the pyramid if the height of the cylinder is sam as the pyramid.
1.56mins
Q23
Determine, to one decimal place, the dimensions of the rectangular square-based prism that would have the greatest volume for the surface area. Show your solution.
210 cm^2
1.34mins
Q24a
Determine, to one decimal place, the dimensions of the rectangular square-based prism that would have the greatest volume for the surface area. Show your solution.
490 cm^2
What is the greatest volume for an open-topped rectangular prism with a surface area of 101.25 cm^2?
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2021-01-27 07:27:36
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http://mathhelpforum.com/algebra/205290-steps-between-2-forms-divisons-print.html
|
# Steps between 2 forms of divisons
• Oct 14th 2012, 05:20 AM
Creephun
Steps between 2 forms of divisons
Hello!
(((l+1)^2)/((2l+1)*(2l+3))) + ((l*(l+1))/(2*(2l+1))) - Wolfram|Alpha
I can't solve, how to get the Alternate form from the Input one on the link. Please somebody help me, step-by step to understand, how do the calculator did it.
Thanks :)
• Oct 14th 2012, 06:09 AM
topsquark
Re: Steps between 2 forms of divisons
Quote:
Originally Posted by Creephun
Hello!
(((l+1)^2)/((2l+1)*(2l+3))) + ((l*(l+1))/(2*(2l+1))) - Wolfram|Alpha
I can't solve, how to get the Alternate form from the Input one on the link. Please somebody help me, step-by step to understand, how do the calculator did it.
Thanks :)
Here's a "road map" for you.
I'm going to relabel your "l" since it looks so much like a "1". I'll call it x.
It looks like
$\frac{(x + 1)^2}{(2x + 1)(2x + 3)} + \frac{x(x + 1)}{2(2x + 1)}$
I'm going to do a lot of factoring here. You can do it by the brute force method, but the factoring approach is a bit easier to see. So let's factor what we can here.
$\left [ \frac{x + 1}{2x + 1} \right ] \cdot \left [ \frac{x + 1}{2x + 3} + \frac{x}{2} \right ]$
Fractions and instructions to simplify. Let's get some common denominators and do another factorization.
$\left [ \frac{x + 1}{2(2x + 1)(2x + 3)} \right ] \cdot \left [ 2x + 2 + 2x^2 + 3x \right ]$
Combining terms and factoring the numerator (on the right side brackets) gives
$\left [ \frac{x + 1}{2(2x + 1)(2x + 3)} \right ] \cdot \left [ (x + 2)(2x + 1) \right ]$
and canceling the common terms we finally get
$\frac{(x + 1)(x + 2)}{2(2x + 3)}$
-Dan
• Oct 14th 2012, 06:48 AM
Creephun
Re: Steps between 2 forms of divisons
Thank you very much!
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2017-04-26 08:45:09
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https://homework.cpm.org/category/CCI_CT/textbook/int2/chapter/12/lesson/12.2.1/problem/12-63
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### Home > INT2 > Chapter 12 > Lesson 12.2.1 > Problem12-63
12-63.
A $30°$-$60°$-$90°$ triangle has a hypotenuse of length $2$ units. Sketch and label the triangle, including the length of each side. Then use your diagram to determine exact values for the following trig expressions.
1. $\text{tan }30°$
2. $\text{sin }30°$
3. $\text{cos }30°$
4. How is $\text{sin }60°$ related to $\text{cos }30°$? Is the relationship true for any pair of acute angles in a right triangle? Explain your reasoning.
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2020-06-06 15:24:09
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https://koasas.kaist.ac.kr/handle/10203/30077
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#### Nitric oxide reduction with carbon monoxide over copper chromite catalyst = Copper chromite 촉매상에서 일산화탄소에 의한 일산화질소의 환원반응
Cited 0 time in Cited 0 time in
• Hit : 309
Main pollutants in automobile exhaust gases can be controlled by the following reaction in a catalytic reactor. $$NO\,+\,CO\;-\;\frac{1}{2}N_2\,+\,CO_2$$ The reaction was studied in view of the effects of the operating conditions and thermal treating of the catalysts on reaction rates by using copper chromite catalysts in a integral reactor. Two types of copper chromite, normal and spinel type, were used in this experiment. The normal type catalyst contained CuO and $CuCr_2O_4$, and spinel type catalyst was almost pure $CuCr_2O_4$. After thermal treating at 800$^\circ$C for 24 hours, the activity of the spinel type slightly decreased, but that of normal type decreased markedly to 50\% of the thermally untreated one. At low CO concentration $(38)$ CO caused reaction inhibition. By analyzing the experiental data, the following rate equation based on the adsorption control model was obtained. $$r_m = \frac{4.04 \exp(-2600/RT) C_{NO}}{1+1.29 \times 10^5 \exp(-11270/RT)C_{CO}}$$ The system performances were taken at temperature range of 150 - 270$^\circ$C, 1000 to 6000 ppm of No and 1 to 5\% CO, and contact times in the order of tenths of a second.
Lee, Won-Kook이원국
Description
한국과학기술원 : 화학공학과,
Publisher
한국과학기술원
Issue Date
1979
Identifier
62433/325007 / 000771139
Language
eng
Description
학위논문 (석사) - 한국과학기술원 : 화학공학과, 1979.2, [ [iv], 80 p. ]
URI
http://hdl.handle.net/10203/30077
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2020-09-19 10:55:08
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http://www.computer.org/csdl/trans/tc/1997/01/t0118-abs.html
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Subscribe
Issue No.01 - January (1997 vol.46)
pp: 118-124
ABSTRACT
<p><b>Abstract</b>—A combined (2<it>log</it>N − 1)-stage interconnection network (denoted by <tmath>$\Delta \oplus \Delta '$</tmath>) is constructed by concatenating two Omega-equivalent networks (Δ and Δ′) with the rightmost stage of Δ and the leftmost stage of Δ′ overlapped. Benes network and the (2<it>log</it>N − 1)-stage shuffle-exchange network are two examples of such networks. Although these two networks have received intensive studies, the research on the topology of entire class of <tmath>$\Delta \oplus \Delta '$</tmath> networks is very limited so far. In this paper, we study the topological structure of <tmath>$\Delta \oplus \Delta '$</tmath> networks and propose an algorithm for determining topological equivalence between two given <tmath>$\Delta \oplus \Delta '$</tmath> networks. We also present a simplified Ω-equivalence checking algorithm as a supporting result.</p>
INDEX TERMS
Combined (2logN − 1)-stage network, equivalence, multistage interconnection network, permutation, topology, Ω-equivalent class
CITATION
Qing Hu, Xiaojun Shen, Jingyu Yang, "Topologies of Combined (2logN - 1)-Stage Interconnection Networks", IEEE Transactions on Computers, vol.46, no. 1, pp. 118-124, January 1997, doi:10.1109/12.559812
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2015-08-04 03:46:17
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https://kofzor.github.io/Learning_Value_Functions/
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# Learning Value Functions
## Value Functions
In this blog post we gonna talk about one of my favorite parts of reinforcement learning: Value Functions. The first parts will look difficult and kinda mathemetical, but they will give you a good basis to understand how the different learning algorithms are derived.
A value function maps each state to a value that corresponds with the output of the objective function. We use $V_{M}^{\pi}$ to denote the value function when following $\pi$ on $M$, and let $V_{M}^{\pi}(s)$ denote the state-value of a state $s$. In the literature $M$ is often omitted as it is clear from context. A state-value for the expected return of a policy on $M$ is defined by
where $\mathbb{E}_{\pi, M} \{ X \vert Y \}$ is the conditional expectation of $X$ given $Y$ when following $\pi$ on $M$. Thus, a state-value indicates how good it is to be in its respective state in terms of the expected return. If we want to perform well, we prefer states with high state-values.
The value function of the optimal policy, $\pi^*$ is called the optimal value function and is denoted by $V_{M}^{\pi^*}$ and in the literature usually just $V^*$. Evidently, it holds that for each state, $V^*(s)$ is equal or higher than for other policies:
## Q-functions
An action-value function or more commonly known as Q-function is a simple extension of the above that also accounts for actions. It is used to map combinations of states and actions to values. A single combination is often referred to as a state-action pair, and its value as a (policy) action-value.
We use $Q_{M}^{\pi}$ to denote the Q-function when following $\pi$ on $M$, and let $Q_{M}^{\pi}(s,a)$ denote the action-value of a state-action pair $(s,a)$. In the literature, it is common to leave out both $\pi$ and $M$. The action-value is then:
which corresponds to the idea that when you are in state $s$ and take $a$ and follow $\pi$ afterwards then the expectation is as above.
The Q-function of $\pi^*$ is called the optimal Q-function. It is usually noted as $Q^*$. Notice that state-values and action-values are related to each other as follows:
$V^{\pi}(s) = \sum_{a\in\mathbb{A}}\pi(s,a)Q^{\pi}(s, a)$ and $V^*(s) = max_{a\in\mathbb{A}}Q^*(s, a)$
### Deriving policies from Q-functions
It is very straightforward to derive policies from Q-functions as they describe how good each action is. In the literature, the most common derivation is the greedy policy of a Q-function: in each state, choose the greedy action, which is the action with the highest action-value. Notice that we can derive $\pi^*$ by using a greedy policy with respect to $Q^*$*. Also notice that $Q^*$ is sufficient but not necessary for $\pi^*$: any of the action-value actions can be changed as long as the same best actions keep the highest action-values.
Q-functions are frequently used to guide exploration and exploitation. The most common approach is to use epsilon-greedy: at each timestep, choose a greedy action with probability $1-\epsilon$ or choose a random action with probability $\epsilon$ [Sutton and Barto, 1998]. Interestingly, in the literature never considers the case that a random action might still choose the same greedy action and hence the chance of choosing the greedy action is usually higher than $1-\epsilon$. When using epsilon-greedy, one commonly starts with a high $\epsilon$ and decrease it over a time. Evidently, when $\epsilon = 0$, it is equal to the greedy policy.
Another approach I like to mention is Boltzmann exploration, where one introduces a temperature parameter $\beta$ to map action-values to action probabilities as follows:
The parameter is used to control how the difference in action-values corresponds to a difference in action-probabilities. As $\beta$ goes to zero, Boltzmann chooses greedily, and as $\beta$ goes to infinity, all actions have an equal chance. In different research fields this formula is also known as softmax.
## Bellman Equations
For MDPs, action-values (and state-values) share an interesting recursive relation between them, clarified by the so-called Bellman equation [Bellman, 1957]. The Bellman equation for an action-value is derived as follows:
The Bellman equation version for $\pi^*$ is called the Bellman optimality equation, and is formulated as
Clearly, an action-value can be alternatively defined as the expectation over immediate rewards and the action-values of successor state-action pairs. Reinforcement learning algorithms commonly exploit these recursive relations for learning state-values and action-values.
## Temporal-Difference Learning
Temporal-difference (TD) learning algorithms bootstrap value estimates by using samples that are based on other value estimates as inspired by the Bellman equations [Sutton and Barto, 1998]. We will only consider estimates for action-values by using Q-functions.
Let $Q_t$ denote a Q-function estimate at timestep $t$, where $Q_0$ is arbitrarily initialized. In general, a TD algorithm updates the action-value estimate of the state-action pair that was visited at timestep $t$, denoted by $Q_t(S_t, A_t)$, as follows
where $Q_{t+1}(S_t, A_t)$ is the updated action-value estimate, and $X_t$ is a value sample observed at $t$ and is based on different mechanisms in the literature. The learning rate $\alpha_t \in [0, 1]$ weighs off new samples with previous samples.
The above TD-learning algorithm is commonly implemented in an online manner: whenever the agent takes an action and observes reward $r_{t+1}$ and transition to next state $s_{t+1}$, a value sample $X_t$ is constructed and the relevant estimate (of the previous state-action pair) is updated. Once updated, the sample is discarded.
Below a short list of different value samples used by different RL algorithms:
Algorithm Value Sample
Q-learning $X_t = R_{t+1} + \gamma \max_{a' \in \mathbb{A}} Q_t(S_{t+1}, a')$
SARSA $X_t = R_{t+1} + \gamma Q_t(S_{t+1}, a')$
Expected SARSA $X_t = R_{t+1} + \gamma \sum_{a' \in \mathbb{A}} \pi_t(S_{t+1},a') Q_t(S_{t+1}, a')$
Double Q-learning $X_t = R_{t+1} + \gamma \max_{a' \in \mathbb{A}} Q_t^b(S_{t+1}, a')$
where $Q_t^b$ is the second Q-function, see Double Q-learning. Notice that Q-learning and Double Q-learning are off-policy: they learn about a different (implicit) policy than the behaviour policy used for interaction. SARSA and Expected SARSA are on-policy: they learn about the same policy as the agent follows.
### Bias and Variance
Despite that sample rewards and transitions are unbiased, the value samples used by TD algorithms are usually biased as they are drawn from a bootstrap distribution. Namely, estimates are used instead of the true action-values:
$Q_t(S_{t+1}, \cdot)$ instead of $Q^*(S_{t+1}, \cdot)$ or $Q^\pi(S_{t+1}, \cdot)$
However, bootstrapping has several advantages. First, we do not need to wait until we have a sample return, consisting of a(n infinite) number of sample rewards, before we can update estimates. This can tremendously improve the speed with which we learn. Second, we need to know the true action-values (or the optimal policy) to obtain unbiaased samples, which are simply not known. Third, the bootstrap distribution’s variance is smaller than the variance of the distribution over whole returns.
A second bias occurs when we change the implicit estimation policy towards an action that is perceived as better than it actually is. I.e., for Q-learning, we estimate the optimal action-value based on the maximum action-value estimate [van Hasselt, 2011] :
$max_{a \in \mathbb{A}} Q_t(S_{t+1}, a)$ instead of $Q_t(S_{t+1}, a*)$
where $a*$ denotes the optimal action in the successor state. This overestimation bias becomes apparent when we severely overestimate action-values of suboptimal actions, due to an extremely high value sample or improper initialization. As a result, other estimates can also become overestimated as value samples will be inflated due to bootstrapping, and may remain inaccurate as well as mislead action-selection for an extended period of time. Double Q-learning addresses this overestimation bias, but may suffer from an underestimation bias instead [van Hasselt, 2011]
### Learning Rates
The learning rate $\alpha_t$ is used to weigh off new value samples with previous value samples. There are many schemes available for the learning rate. For example, if the MDP is deterministic we can set the learning rate at one at all times. Usually, the MDP is stochastic and a lower value than one should be used. For instance, one can use a hyperharmonic learning rate scheme:
where $n_t(s,a)$ is the number of times $(s,a)$ has been visited by timestep $t$, and $w \in (0.5, 1]$ is a tunable parameter of the scheme. [Even-Dar and Mansour, 2003]showed that $% $ (hyperharmonic) works better than $w = 1$ (harmonic). This is because the value samples are based on other estimates that may change over time, and hence the distribution of the value samples are non-stationary as well as not independent and identically distributed (not i.i.d.).
### Convergence
Q-learning and the other above-mentioned choices are proven to converge to the optimal Q-function in the limit with probability one [van Hasselt, 2011]. Provided that a few conditions are satisfied:
• The behaviour policy guarantees that every state-action pair is infinitely often tried in the limit.
• The learning-rate satisfies the Robbin-Monro’s conditions for stochastic approximation: $\sum_{t=0}^{\infty} \alpha_t = \infty$ and $% $
• for Sarsa and Expected Sarsa, the estimation policy (and hence behaviour policy) is greedy in the limit.
Put simply, the easiest way to guarantee convergence: use a simple learning rate as mentioned above, initialize however you want, and use epsilon-greedy where $\epsilon$ is above $0$ (already satisfied by doing $\epsilon = 1/t$).
## Python Implementations
### Q-learning
For a discrete problem, the following python implementation of Q-learning works well enough:
import gym
env = gym.make("Taxi-v1")
# Q-function
# initial_Q = 0.
from collections import defaultdict
Q = defaultdict(lambda : 0.) # Q-function
n = defaultdict(lambda : 1.) # number of visits
# Extra
actionspace = range(env.action_space.n)
greedy_action = lambda s : max(actionspace, key=lambda a : Q[(s,a)])
max_q = lambda sp : max([Q[(sp,a)] for a in actionspace])
import random
epsilon = 0.1
gamma = 0.9
# Simulation
episodescores = []
for _ in range(500):
nextstate = env.reset()
currentscore = 0.
for _ in range(1000):
state = nextstate
# Epsilon-Greedy
if epsilon > random.random() :
action = env.action_space.sample()
else :
action = greedy_action(state)
nextstate, reward, done, info = env.step(action)
currentscore += reward
# Q-learning
if done :
Q[(state,action)] = Q[(state,action)] + 1./n[(state,action)] * ( reward - Q[(state,action)] )
break
else :
Q[(state,action)] = Q[(state,action)] + 1./n[(state,action)] * ( reward + gamma * max_q(nextstate) - Q[(state,action)] )
episodescores.append(currentscore)
import matplotlib.pyplot as plt
plt.plot(episodescores)
plt.xlabel('Episode')
plt.ylabel('Cumu. Reward of Episode')
plt.show()
This code resulted in the following performance:
### SARSA
Also, a simple implementation of SARSA:
import gym
env = gym.make("Taxi-v1")
# Q-function
# initial_Q = 0.
from collections import defaultdict
Q = defaultdict(lambda : 0.) # Q-function
n = defaultdict(lambda : 1.) # number of visits
# Extra
actionspace = range(env.action_space.n)
greedy_action = lambda s : max(actionspace, key=lambda a : Q[(s,a)])
import random
epsilon = 0.1
gamma = 0.9
# Simulation
episodescores = []
for _ in range(500):
state = env.reset()
currentscore = 0.
for t in range(1000):
# Epsilon-Greedy
if epsilon > random.random() :
action = env.action_space.sample()
else :
action = greedy_action(state)
# SARSA
if t > 0 : # Because previous state and action do not yet exist at t=0
Q[(prevstate,prevaction)] = Q[(prevstate,prevaction)] + 1./n[(prevstate,prevaction)] * ( reward + gamma * Q[(state, action)] - Q[(prevstate,prevaction)] )
nextstate, reward, done, info = env.step(action)
currentscore += reward
if done :
Q[(prevstate,prevaction)] = Q[(prevstate,prevaction)] + 1./n[(state,action)] * ( reward - Q[(prevstate,prevaction)] )
break
prevstate, state, prevaction = state, nextstate, action
episodescores.append(currentscore)
import matplotlib.pyplot as plt
import numpy as np
plt.plot(episodescores)
plt.xlabel('Episode')
plt.ylabel('Cumu. Reward of Episode')
plt.show()
And the resulting image. Notice that the performance does not seem to converge to the optimal, this is because SARSA is on-policy and the behaviour-policy remains epsilon-greedy with $\epsilon = 0.1$ and thus will do a random (bad) action roughly 10% of the time.
### Expected SARSA
import gym
env = gym.make("Taxi-v1")
# Q-function
# initial_Q = 0.
from collections import defaultdict
Q = defaultdict(lambda : 0.) # Q-function
n = defaultdict(lambda : 1.) # number of visits
# Extra
actionspace = range(env.action_space.n)
greedy_action = lambda s : max(actionspace, key=lambda a : Q[(s,a)])
import random
epsilon = 0.1
gamma = 0.9
# Simulation
episodescores = []
for _ in range(1000):
state = env.reset()
currentscore = 0.
for t in range(1000):
# Epsilon-Greedy
if epsilon > random.random() :
action = env.action_space.sample()
else :
action = greedy_action(state)
# Expected SARSA
if t > 0 : # Because previous state and action do not yet exist at t=0
Vpi = sum([ Q[(state, a)] * ((action==a)*(1.-epsilon) + epsilon*(1./len(actionspace))) for a in actionspace ])
Q[(prevstate,prevaction)] = Q[(prevstate,prevaction)] + 1./n[(prevstate,prevaction)] * ( reward + gamma * Vpi - Q[(prevstate,prevaction)] )
nextstate, reward, done, info = env.step(action)
currentscore += reward
if done :
Q[(prevstate,prevaction)] = Q[(prevstate,prevaction)] + 1./n[(state,action)] * ( reward - Q[(prevstate,prevaction)] )
break
prevstate, state, prevaction = state, nextstate, action
episodescores.append(currentscore)
import matplotlib.pyplot as plt
import numpy as np
plt.plot(episodescores)
plt.xlabel('Episode')
plt.ylabel('Cumu. Reward of Episode')
plt.show()
And its performance:
### Double Q-learning
Code:
import gym
env = gym.make("Taxi-v1")
# Q-function
# initial_Q = 0.
from collections import defaultdict
Q = defaultdict(lambda : 0.) # Q-function
n = defaultdict(lambda : 1.) # number of visits
# Extra
actionspace = range(env.action_space.n)
greedy_action_i = lambda i, s : max(actionspace, key=lambda a : Q[(i,s,a)])
max_q = lambda i, sp : max([Q[(i,sp,a)] for a in actionspace])
greedy_action = lambda s : max(actionspace, key=lambda a : (Q[(0,s,a)]+Q[(1,s,a)])/2.)
import random
epsilon = 0.1
gamma = 0.9
# Simulation
episodescores = []
for _ in range(1000):
nextstate = env.reset()
currentscore = 0.
for _ in range(1000):
state = nextstate
# Epsilon-Greedy
if epsilon > random.random() :
action = env.action_space.sample()
else :
action = greedy_action(state)
nextstate, reward, done, info = env.step(action)
currentscore += reward
# Double Q-learning
i = random.choice([0,1])
o = (i + 1) % 2
if done :
Q[(i,state,action)] = Q[(i,state,action)] + 1./n[(i,state,action)] * ( reward - Q[(i,state,action)] )
break
else :
greedy_i = greedy_action_i(i,nextstate)
Q[(i,state,action)] = Q[(i,state,action)] + 1./n[(i,state,action)] * ( reward + gamma * Q[(o,nextstate,greedy_i)] - Q[(i,state,action)] )
episodescores.append(currentscore)
import matplotlib.pyplot as plt
plt.plot(episodescores)
plt.xlabel('Episode')
plt.ylabel('Cumu. Reward of Episode')
plt.show()
And the performance:
### Afterword
We have discussed value-functions and a few simple temporal-difference learning algorithms, and demonstrated their implementation and some performance. Although SARSA and Expected SARSA look like they perform poorly, they certainly are not always worse than Q-learning. It just so happens that the Taxi environment works quite well with the chosen epsilon-greedy action-selection. In other environments, SARSA and Expected SARSA may perform much better than Q-learning, namely environments were a single mistake inflicts a devastating reward feedback. For example when walking alongside a cliff, the on-policy approach will then take into account the chance that one sometimes randomly wanders off the cliff if it walks too close along the cliff and hence induces a safer policy [Sutton and Barto, 1998].
Next topic is on Advanced Value Functions.
## References
[Bellman, 1957] R. Bellman. Dynamic Programming. Princeton University Press, 1957.
[Even-Dar and Mansour, 2003] E. Even-Dar and Y. Mansour. Learning rates for Q-learning. The Journal of Machine Learning Research, 5:1–25, 2003.
[Sutton and Barto, 1998] R.S. Sutton and A.G. Barto. Reinforcement Learning: An Introduction (Adaptive Computation and Machine Learning). The MIT Press, 1998.
[H. van Hasselt. 2011] Insights in Reinforcement Learning. PhD thesis, Utrecht University, 2011.
Written on August 21, 2016
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2022-01-22 20:55:16
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http://math.stackexchange.com/questions/594/how-do-you-prove-that-a-prime-is-the-sum-of-two-squares-iff-it-is-congruent-to-1?answertab=oldest
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How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it?
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Yes we figured that – BlueRaja - Danny Pflughoeft Jul 23 '10 at 17:27
@Akhil: It is acceptable to answer your own question. – John Gietzen Jul 23 '10 at 17:34
As long as you are open to accepting someone else's answer if it is better than your own :) – Larry Wang Jul 23 '10 at 18:06
For another interesting method of proof, see demonstrations.wolfram.com/… and the Larson article it refers to. (L. C. Larson, "A Theorem about Primes Proved on a Chessboard," Mathematics Magazine, 50(2), 1977 pp. 69–74.) – Doug Chatham Aug 9 '10 at 19:02
Let $p$ be a prime congruent to 1 mod 4. Then to write $p = x^2 + y^2$ for $x,y$ integers is the same as writing $p = (x+iy)(x-iy) = N(x+iy)$ for $N$ the norm.
It is well-known that the ring of Gaussian integers $\mathbb{Z}[i]$ is a principal ideal domain, even a euclidean domain. Now I claim that $p$ is not prime in $\mathbb{Z}[i]$. To determine how a prime $p$ of $\mathbb{Z}$ splits in $\mathbb{Z}[i]$ is equivalent to determining how the polynomial $X^2+1$ splits modulo $p$.
First off, $-1$ is a quadratic residue modulo $p$ because $p \equiv 1 \mod 4$. Consequently, there is $t \in \mathbb{Z}$ with $t^2 \equiv -1 \mod p$, so $X^2+1$ splits modulo $p$, and $p$ does not remain prime in $\mathbb{Z}[i]$. (Another way of seeing this is to note that if $p$ remained prime, then we'd have $p \mid (t+i)(t-i)$, which means that $p \mid t+i$ or $t \mid t-i$.)
Anyway, as a result there is a non-unit $x+iy$ of $\mathbb{Z}[i]$ that properly divides $p$. This means that the norms properly divide as well. In particular, $N(x+iy) = x^2+y^2$ properly divides $p^2$, so is $p$ or $1$. It cannot be the latter since otherwise $x+iy$ would be a unit. So $x^2+y^2 = p$.
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"To determine how a prime $p$ of $\mathbb{Z}$ splits in $\mathbb{Z}[i]$ is equivalent to determining how the polynomial $X^2+1$ splits modulo $p$" - What theorem is that? – Casebash Jul 24 '10 at 0:08
I don't think it has a name, but basically the point is that determing how $p$ splits in $\mathbb{Z}[i] = \mathbb{Z}[X]/(X^2+1)$ is the same thing as considering the quotient by the ideal generated by $p$, i.e. $\mathbb{Z}_p[X]/(X^2+1)$. If $X^2+1$ splits modulo $p$, then this ring has two prime ideals. So essentially it reduces to properties of quotient rings. – Akhil Mathew Jul 24 '10 at 0:29
[Originally left as a reply when I didn't have enough reputation]: Sorry that I don't have enough rep yet, I just want to point out the fact Casebash's asking about in Akhil's comment goes by the name Kummer-Dedekind theorem. – Soarer Jan 13 '11 at 6:44
@Casebash: The essential point for the argument for the equivalence is that the ring $Q=\mathbb Z[X]/(p,X^2+1)$ can be viewed as a quotient ring of a Euclidean (and therefore principal) domain in two ways; saying that $Q$ is not a field (or equivalently not a domain) has repercussions, namely the generator of an ideal being reducible, in both those Euclidean domains, which repercussions are then of course equivalent. I don't think this requires the Kummer-Dedekind theorem. – Marc van Leeuwen Aug 22 '12 at 9:56
Here is another proof without complex numbers.
We start with proving that there exists $z \in \mathbb{N}$ such that $z^2 + 1 \equiv 0 \pmod p$. We do this in the same way as Akhil Mathew.
Let we have $a^2 + b^2 = pm$. Take $x$ and $y$ such that
$x \equiv a \pmod m$ and
$y \equiv b \pmod m$ and
$x, y \in [-m/2, m/2)$.
Consider $u = ax + by$ and $v = ay - bx$. Then $u^2 + v^2 = (a^2 + b^2)(x^2 + y^2)$. Moreover, $u$ and $v$ are multiples of $m$. Hence $(u/m)^2 + (v/m)^2 = p (x^2 + y^2)/m$. $(x^2 + y^2)/m$ is an integer because of the definition of $x$ and $y$ and that $a^2 + b^2 = pm$.
Also $(x^2 + y^2)/m$ is less than $m/2$.
Now we change $a$ by $u$ and $b$ by $v$ and continue this process until we get $m=1$.
Notice that this is quite efficient way to find representation of $p$ as a sum of two squares - it takes $O(\log p)$ steps to find it provided we have found $z$ such that $z^2 + 1$ is multiple of $p$.
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I don't understand your proof, could you please give me a pointer? I don't see the connection with the mentioned $z$, and I fail to see what the new $m$ is at the end of the algorithm's first step. – Weltschmerz Oct 5 '10 at 20:23
Taken From I.N. Herstein There are 2 results which i am going to use.
1. Let $p$ be a prime integer and suppose that for some integer $c$ relatively prime to $p$ we can find integers $x$ and $y$ such that $x^{2}+y^{2}=cp$. Then $p$ can be written as the sum of 2 squares.
2. If $p$ is a prime of the form $4n+1$, then we can solve the congruence $x^{2} \equiv \ -1 \ (mod) \ p$.
Now the main result. If $p$ is a prime of the form $4n+1$ then $p$ is the sum of 2 squares.
Proof. By 2 there exists and $x$ such that $x^{2} \equiv -1 \text{mod} \ p$. So $x$ can be chose such that $0 \leq x \leq (p-1)$. We can restrict the size of $p$ even further, namely to satisfy $|x| \leq \frac{p}{2}$. For if $x > p/2$ then, $y=p-x$ satisfies $y^{2} \equiv -1 \text{mod} \ p$ but $|y| \leq p/2$. Thus we may assume that we have an integer $x$ such that $|x| \leq p/2$ and $x^{2}+1$ is a multiple of $p$ say $cp$. Now $cp=x^{2}+1 \leq p^{2}/4 +1 < p^{2}$, hence $c < p$ and hence $(c,p)=1$. Invoking (1) we have $p=a^{2}+b^{2}$.
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You should cite the source from where you copied this answer. – Bill Dubuque Aug 14 '10 at 19:37
@Chandru1: Thanks for adding the reference! Please do add references like this whenever you take something from somewhere. (It is indeed the proof from I. N. Herstein's Topics in Algebra, Theorem 3.G.) – ShreevatsaR Aug 15 '10 at 19:00
Perhaps my favorite argument (other than any arguably "correct" arguments, such as the one Akhil has given, or arguments starting from the fact that $x^2 + y^2$ is the unique binary quadratic form of discriminant $-4$ up to equivalence) uses continued fractions. Suppose that $u^2 \equiv -1 \pmod{p}$, and consider the continued fraction expansion of the rational number $u/p$. Let $r/s$ be the last convergent to $u/p$ with the property that $s < \sqrt{p}$. Then, setting $x=s$ and $y = rp-us$, one has $x^2 + y^2 = p$.
Here's the argument: let $r'/s'$ be the convergent following $r/s$. Then the basic theory of continued fractions gives the estimate $|r/s - u/p| < 1/ss'$, and the right-hand side is less than $1/s\sqrt{p}$ by hypothesis. Clearing denominators gives $y < \sqrt{p}$, so that $0 < x^2 + y^2 < 2p$. On the other hand $x^2 + y^2$ is checked to be divisible by $p$ (by choice of $u$), so must be equal to $p$.
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Dear Dave, This is very nice. In fact, it must be very close to (essentially the same as?) both the proof the $\mathbb Z[i]$ is a Euclidean domain, and the reduction theory argument showing uniqueness of $x^2 + y^2$. In other words, it probably qualifies as a "correct" argument in its own right. (But perhaps lacking the surrounding theoretical infrastructure that those arguments admit, which I guess is what you are getting at with your parenthetical remark.) – Matt E Aug 9 '10 at 16:21
Dear Matt: yes, I agree. The content is there, but (it seems to me) it's hard to see that it's there unless one already knows another argument in some more theoretical context! – D. Savitt Aug 9 '10 at 16:36
Here's a convenient reference for this: Editor's Corner: The Euclidean Algorithm Strikes Again, Stan Wagon, Amer. Math. Monthly, Vol. 97, No. 2 (Feb., 1990), pp. 125-129. jstor.org/stable/2323912 – Bill Dubuque Aug 14 '10 at 19:40
There is an amazing proof of this due to Don Zagier : one-sentence proof.
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2015-04-19 00:30:46
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https://math.stackexchange.com/questions/1106429/klein-gordon-field-commutator-integral/1620301
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# Klein-Gordon field commutator integral?
Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$[\phi(x),\phi(y)]=c \int_{\mathbb{R}^3} \frac{1}{(2\pi)^3 2 \sqrt{\lvert \vec{k} \rvert^2 + \omega_0^2}}\left ( e^{-i\sqrt{\lvert \vec{k} \rvert^2 + \omega_0^2} \ (x_0-y_0)} - e^{i\sqrt{\lvert \vec{k} \rvert^2 + \omega_0^2} \ (x_0-y_0)} \right )e^{i \vec{k}\cdot (\vec{x}-\vec{y})} \ dk.$$ I want to verify that this integral is equal to $$[\phi(x),\phi(y)]=c \ \text{sgn}(x_0-y_0) \left ( i \omega_0 \theta (\tau^2) \frac{J_1(\omega_0 \tau)}{4\pi \tau} - \frac i {2\pi} \delta (\tau^2) \right )$$ where $\tau\equiv\sqrt{(x_0-y_0)^2 - \lvert \vec{x}-\vec{y} \rvert^2}$, $\theta$ is the Heavyside function, and $J_1$ is the Bessel function. This identity looks intractable to me.
Writing the integral in polar coordinates and doing the integral over the angles simplifies the expression somewhat but I still cannot derive the result using either theorems regarding the Fourier transform or substitution techniques.
• I guess you want to be integrating $k$ over $\mathbb{R}^2$? – Fabian Jan 16 '15 at 8:47
• Also there is some further mistake: as it stands the term in the brackets simply vanishes. – Fabian Jan 16 '15 at 8:51
• As you have posted it additionally on physics.stackexchange, I vote to close this question as it better fits to the other forum. – Fabian Jan 16 '15 at 9:39
• Should be migrated but it is already there. – Fabian Jan 16 '15 at 9:40
• This question was ruled off-topic at Physics.SE because it is Homework-like. The only difficulty is in computing the integral. There is no conceptual physics question to be asked here. I have changed some of the notation so it is less physics-centric and hopefully it is understandable and can be reopened. – Kevin Driscoll Jan 20 '15 at 22:59
($c\equiv 1$) The commutator can be written as the sum of the positive- and negative-frequency part of the Pauli-Jordan function: $$i[\phi(x),\phi(y)]=D^+(x-y)+D^-(x-y),$$ where \begin{align} D^\pm(x) &= \pm\frac{1}{i(2\pi)^3}\int\frac{d^3k}{2\sqrt{\mathbf k^2+m^2}} e^{\pm i\sqrt{\mathbf k^2+m^2}\,x^0-i\mathbf k\cdot\mathbf x}. \end{align} Performing the $3$-momentum integration and letting $r\equiv|\mathbf x|$, one has $$D^+(x) = -\frac{1}{8\pi^3r}\int_{-\infty}^{+\infty}\frac{e^{i\sqrt{\mathbf \rho^2+m^2}x^0+i\rho r}}{\sqrt{\rho^2+m^2}}\rho d\rho=\frac{1}{4\pi r}\frac{\partial}{\partial r}\,f(r),$$ where $$f(r) = \frac{i}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{i\sqrt{\mathbf \rho^2+m^2}x^0+i\rho r}}{\sqrt{\rho^2+m^2}}d\rho =\frac{i}{2\pi}\int_{-\infty}^{+\infty}e^{im(x^0\cosh\varphi+r\sinh\varphi)}d\varphi,$$ using $\rho = m \sinh\varphi$. Now: let $\lambda\equiv(x^0)^2-r^2$,
1) if $x^0,\lambda>0$, let $x^0=\sqrt{\lambda}\cosh\varphi_0$ and $r=\sqrt{\lambda}\sinh\varphi_0$, so that \begin{align} -f(r)=\frac{1}{i2\pi}\int_{-\infty}^{+\infty}e^{im\sqrt{\lambda}\cosh(\varphi+\varphi_0)}d\varphi=\frac{1}{2}H_0^1(m\sqrt{\lambda})=\frac{1}{2}J_0(m\sqrt{\lambda})+\frac{i}{2}Y_0(m\sqrt{\lambda}) \end{align} (see here for reference on integral representations of Bessel functions);
2) if $x^0>0,\lambda<0$, let $x^0=\sqrt{-\lambda}\sinh\varphi_0$ and $r=\sqrt{-\lambda}\cosh\varphi_0$, so that \begin{align} f(r)=\frac{i}{2\pi}\int_{-\infty}^{+\infty}e^{im\sqrt{-\lambda}\sinh(\varphi+\varphi_0)}d\varphi=\frac{i}{\pi}K_0(m\sqrt{-\lambda}); \end{align}
3) if $x^0<0,\lambda>0$, let $x^0=-\sqrt{\lambda}\cosh\varphi_0$ and $r=\sqrt{\lambda}\sinh\varphi_0$, so that \begin{align} f(r)=\frac{i}{2\pi}\int_{-\infty}^{+\infty}e^{-im\sqrt{\lambda}\cosh(\varphi-\varphi_0)}d\varphi=\frac{1}{2}H_0^2(m\sqrt{\lambda})=\frac{1}{2}J_0(m\sqrt{\lambda})-\frac{i}{2}Y_0(m\sqrt{\lambda}); \end{align}
4) if $x^0,\lambda<0$, let $x^0=-\sqrt{-\lambda}\sinh\varphi_0$ and $r=\sqrt{-\lambda}\cosh\varphi_0$, so that \begin{align} f(r)=\frac{i}{2\pi}\int_{-\infty}^{+\infty}e^{-im\sqrt{\lambda}\sinh(\varphi-\varphi_0)}d\varphi=\frac{i}{\pi}K_0(m\sqrt{-\lambda}). \end{align} From all this: let $\text{sign}(x^0)\equiv \varepsilon(x^0)$ $$f(r) = \begin{cases} \frac{1}{2i}Y_0(m\sqrt{\lambda})-\frac{\varepsilon(x^0)}{2}J_0(m\sqrt{\lambda})\text{ for }\lambda>0\\ \frac{i}{\pi}K_0(m\sqrt{-\lambda})\text{ for }\lambda<0. \end{cases}$$ Now, to take the derivative with respect to $r$, use $$\frac{1}{4\pi r}\frac{\partial}{\partial r}=-\frac{1}{2\pi}\frac{\partial}{\partial \lambda}$$ together with the fact that the function $f$ is discontinuous at $\lambda=0$ due to of $J_0(0^+)=1$ showing up for for positive $\lambda$, then: $$D^+(x)=\frac{1}{4\pi}\varepsilon(x^0)\delta(\lambda)-\frac{im}{8\pi\sqrt{\lambda}}\theta(\lambda)\left[Y_1(m\sqrt{\lambda})-i\varepsilon(x^0) J_1(m\sqrt\lambda) \right]-\frac{im}{4\pi^2\sqrt{-\lambda}}\theta(-\lambda)K_1(m\sqrt{-\lambda}).$$ Working in a similar fashion, $$D^-(x)=\frac{1}{4\pi}\varepsilon(x^0)\delta(\lambda)-\frac{im}{8\pi\sqrt{\lambda}}\theta(\lambda)\left[-Y_1(m\sqrt{\lambda})-i\varepsilon(x^0) J_1(m\sqrt\lambda) \right]+\frac{im}{4\pi^2\sqrt{-\lambda}}\theta(-\lambda)K_1(m\sqrt{-\lambda}).$$ Finally $$\boxed{ i[\phi(x), \phi(y)]=\frac{1}{2\pi}\varepsilon(x^0)\delta(\lambda)-\frac{m}{8\pi\sqrt{\lambda}}\theta(\lambda)\varepsilon(x^0) J_1(m\sqrt\lambda).}$$
|
2020-01-21 14:46:20
|
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|
https://www.nag.com/numeric/py/nagdoc_latest/naginterfaces.library.specfun.bessel_i1_scaled.html
|
# naginterfaces.library.specfun.bessel_i1_scaled¶
naginterfaces.library.specfun.bessel_i1_scaled(x)[source]
bessel_i1_scaled returns a value of the scaled modified Bessel function .
For full information please refer to the NAG Library document for s18cf
https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s18cff.html
Parameters
xfloat
The argument of the function.
Returns
si1float
The value of the function at .
Notes
bessel_i1_scaled evaluates an approximation to , where is a modified Bessel function of the first kind. The scaling factor removes most of the variation in .
The function uses the same Chebyshev expansions as bessel_i1_real(), which returns the unscaled value of .
References
NIST Digital Library of Mathematical Functions
|
2022-08-13 15:58:18
|
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|
https://www.alphacodingskills.com/python/pages/python-program-to-check-leap-year.php
|
# Python Program to Check Leap Year
A leap year is a calendar year in which an additional day is added to February month. In a leap year, the number of days in February month and the year are 29 and 366 respectively. A year that is not a leap year is called a common year. A year is said to be a leap year if
• it is divisible by 4.
• it is divisible by 4 but not divisible by 100.
• it is divisible by 4, 100 and 400.
### Example: Using conditional statements
In the below example, conditional statements are used to identify a leap year.
year = 2019
if year % 400 == 0:
print(year, "is a leap year.")
elif year % 100 == 0:
print(year, "is not a leap year.")
elif year % 4 == 0:
print(year, "is a leap year.")
else:
print(year, "is not a leap year.")
Output
2019 is not a leap year.
### Example: Using function
In the below example, a function called leapyear() is created which takes year as argument and prints whether the passed year is a leap year or not.
def leapyear(year):
if year % 400 == 0:
print(year, "is a leap year.")
elif year % 100 == 0:
print(year, "is not a leap year.")
elif year % 4 == 0:
print(year, "is a leap year.")
else:
print(year, "is not a leap year.")
leapyear(2019)
Output
2019 is not a leap year.
|
2020-03-29 18:33:31
|
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|
https://ask.sagemath.org/questions/9386/revisions/
|
# Revision history [back]
### Order of elements in group multiplication?
This is probably a very simple mistake on my part, but can anyone please explain this error in Sage?
If I do this...
sage: G = SymmetricGroup(3)
sage: H = AlternatingGroup(3)
then this works as expected:
sage: [(g*h) for h in H for g in G]
[(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3), (1,2,3), (1,2), (1,3), (1,3,2), (), (2,3), (1,3,2), (1,3), (2,3), (), (1,2,3), (1,2)]
but if I reverse the order of h and g I get an error
sage: [(h*g) for h in H for g in G]
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
/Users/toby/<ipython console> in <module>()
/Applications/Sage-4.8-OSX-64bit-10.6.app/Contents/Resources/sage/local/lib/python2.6/site-packages/sage/structure/element.so in sage.structure.element.MonoidElement.__mul__ (sage/structure/element.c:10197)()
/Applications/Sage-4.8-OSX-64bit-10.6.app/Contents/Resources/sage/local/lib/python2.6/site-packages/sage/structure/element.so in sage.structure.element.MonoidElement.__mul__ (sage/structure/element.c:10056)()
/Applications/Sage-4.8-OSX-64bit-10.6.app/Contents/Resources/sage/local/lib/python2.6/site-packages/sage/structure/coerce.so in sage.structure.coerce.CoercionModel_cache_maps.bin_op (sage/structure/coerce.c:7467)()
TypeError: unsupported operand parent(s) for '*': 'Alternating group of order 3!/2 as a permutation group' and 'Symmetric group of order 3! as a permutation group'
Can anyone explain what I am doing wrong? Thanks Toby
|
2020-11-23 23:12:02
|
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|
https://www.wptricks.com/question/add_rewrite_rule-in-hivepress-wordpress-plugin-theme/
|
## Add_rewrite_rule in HivePress (WordPress plugin/theme)
Question
I’m trying to change URL structure, to make it more SEO-friendly.
I use HivePress theme/plugin and there is a filter block? when you use it, the result gets URL for example like this
https://example.com/?_sort=&s=&post_type=hp_listing&_category=3&price%5B%5D=799&price%5B%5D=2599&old_price%5B%5D=999&old_price%5B%5D=3299
But I need it to change. I’ve tried to use add_rewrite_rule, but there is no any result.
For example,
add_rewrite_rule( '^_sort/([^/]+)/?', 'index.php?pagename=_sort$matches[1]', 'top' ); or function wpd_filter_rewrite() { add_rewrite_rule( '^_sort/([^/]+)/?', 'index.php?pagename=sort=$matches[1]',
'top'
);
}
|
2023-02-07 17:48:44
|
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|
https://socratic.org/questions/how-do-you-find-the-vertices-and-foci-of-36x-2-10y-2-360
|
# How do you find the vertices and foci of 36x^2-10y^2=360?
Nov 17, 2016
The vertices are $= \left(\sqrt{10} , 0\right)$ and $= \left(- \sqrt{10} , 0\right)$
The foci are F$= \left(\sqrt{46} , 0\right)$ and F'$= \left(- \sqrt{46} , 0\right)$
The equations of the asymptotes are
$y = \frac{6}{\sqrt{10}} x$ and $y = - \frac{6}{\sqrt{10}} x$
#### Explanation:
Let's rewrite the equation
$36 {x}^{2} - 10 {y}^{2} = 360$
Divide by 360
${x}^{2} / 10 - {y}^{2} / 36 = 1$
This is the equation of a right left hyperbola
${x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = 1$
The center is $= \left(0 , 0\right)$
The vertices are $\left(\pm a , 0\right)$
$= \left(\pm \sqrt{10} , 0\right)$
To calculate the foci, we need $c = \pm \sqrt{{a}^{2} + {b}^{2}}$
$c = \pm \sqrt{10 + 36} = \pm \sqrt{46}$
The foci are F$= \left(\sqrt{46} , 0\right)$ and F'$= \left(- \sqrt{46} , 0\right)$
The slope of the asymptotes are $= \pm \frac{b}{a}$
$= \pm \frac{6}{\sqrt{10}}$
The equations of the asymptotes are
$y = \frac{6}{\sqrt{10}} x$ and $y = - \frac{6}{\sqrt{10}} x$
graph{(x^2/10-y^2/36-1)(y-6/sqrt10x)(y+6/sqrt10x)=0 [-28.87, 28.87, -14.44, 14.44]}
|
2021-10-20 01:51:00
|
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|
https://tex.stackexchange.com/questions/142763/strikethrough-cyrillic-text-in-xetex-preserving-hyphenation
|
# Strikethrough Cyrillic text in XeTeX preserving hyphenation
I was looking for the way to create a strikeout text in XeTeX. After some googling I found two packages that supposedly could do that: ulem and soul. However, it turned out that both of them have some issues with my text.
ulem doesn't offer automatic hyphenation, which probably wouldn't be a problem if I only had couple of words to strike through, but that's not the case-I'm dealing with quite a large chunk of text and to manually hyphenate it would be difficult.
sout without additional configuration doesn't work properly with Cyrillic text in XeTeX: it only strikes out words that border punctuation marks and swallows the rest of text. It seems to work fine with LaTeX but I'd like to use Unicode fonts which would be much easier with XeTeX.
To illustrate the above, here's a small example:
\documentclass[draft=true]{scrartcl}
\usepackage[cm-default]{fontspec}
\usepackage{xltxtra}
\setmainfont{Palatino Linotype}
\usepackage{polyglossia}
\setdefaultlanguage{russian}
\usepackage{microtype}
\usepackage{ulem}
\usepackage{soulutf8}
\begin{document}
ulem:
\sout{Сшит колпак не по-колпаковски, вылит колокол не по-колоколовски.
Надо колпак переколпаковать-перевыколпаковать, надо колокол
переколоколовать-перевыколоколовать.}\\
soulutf8:
\st{Сшит колпак не по-колпаковски, вылит колокол не по-колоколовски.
Надо колпак переколпаковать-перевыколпаковать, надо колокол
переколоколовать-перевыколоколовать.}
\end{document}
It is rendered as follows:
As you see, there are line line overfulls with ulem variant and the soul variant is a complete mess. So I was wondering if there is any other way to achieve my goal.
Eventually I found this discussion: Discussion in XeTeX mailing list. It explained the soul behavior:
soul.sty defines the fix font \SOUL@tt (\font\SOUL@tt=ectt1000) and use it in various places when analyzing the input. And obviously one gets problems as soon as words ends with glyphs not existing in the font as this leads to boxes of width 0 and so soul thinks that there is no longer something to process.
And later in that thread a solution was suggested. Adding the following code makes soul work correctly with Cyrillic script:
\makeatletter
\font\SOUL@tt="Palatino Linotype"
\setbox\z@\hbox{\SOUL@tt-}
\SOUL@ttwidth\wd\z@
\makeatother
I don't completely understand the purpose of lines 3 and 4 here. I thought line 2 would be enough but with only it I'm getting a "Reconstruction failed" error message. soul documentation didn't really make it clear.
Well, as long as I achieved my goal I'm satisfied, but if someone could explain how these settings work I'd be grateful.
|
2019-04-19 06:44:02
|
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|
https://tensorlayer.readthedocs.io/en/1.6.5/modules/cost.html
|
# API - Cost¶
To make TensorLayer simple, we minimize the number of cost functions as much as we can. So we encourage you to use TensorFlow’s function. For example, you can implement L1, L2 and sum regularization by tf.nn.l2_loss, tf.contrib.layers.l1_regularizer, tf.contrib.layers.l2_regularizer and tf.contrib.layers.sum_regularizer, see TensorFlow API.
TensorLayer provides a simple way to create you own cost function. Take a MLP below for example.
network = InputLayer(x, name='input')
network = DropoutLayer(network, keep=0.8, name='drop1')
network = DenseLayer(network, n_units=800, act=tf.nn.relu, name='relu1')
network = DropoutLayer(network, keep=0.5, name='drop2')
network = DenseLayer(network, n_units=800, act=tf.nn.relu, name='relu2')
network = DropoutLayer(network, keep=0.5, name='drop3')
network = DenseLayer(network, n_units=10, act=tf.identity, name='output')
The network parameters will be [W1, b1, W2, b2, W_out, b_out], then you can apply L2 regularization on the weights matrix of first two layer as follow.
cost = tl.cost.cross_entropy(y, y_)
cost = cost + tf.contrib.layers.l2_regularizer(0.001)(network.all_params[0]) + tf.contrib.layers.l2_regularizer(0.001)(network.all_params[2])
Besides, TensorLayer provides a easy way to get all variables by a given name, so you can also apply L2 regularization on some weights as follow.
l2 = 0
for w in tl.layers.get_variables_with_name('W_conv2d', train_only=True, printable=False):
l2 += tf.contrib.layers.l2_regularizer(1e-4)(w)
cost = tl.cost.cross_entropy(y, y_) + l2
### Regularization of Weights¶
After initializing the variables, the informations of network parameters can be observed by using network.print_params().
tl.layers.initialize_global_variables(sess)
network.print_params()
param 0: (784, 800) (mean: -0.000000, median: 0.000004 std: 0.035524)
param 1: (800,) (mean: 0.000000, median: 0.000000 std: 0.000000)
param 2: (800, 800) (mean: 0.000029, median: 0.000031 std: 0.035378)
param 3: (800,) (mean: 0.000000, median: 0.000000 std: 0.000000)
param 4: (800, 10) (mean: 0.000673, median: 0.000763 std: 0.049373)
param 5: (10,) (mean: 0.000000, median: 0.000000 std: 0.000000)
num of params: 1276810
The output of network is network.outputs, then the cross entropy can be defined as follow. Besides, to regularize the weights, the network.all_params contains all parameters of the network. In this case, network.all_params = [W1, b1, W2, b2, Wout, bout] according to param 0, 1 … 5 shown by network.print_params(). Then max-norm regularization on W1 and W2 can be performed as follow.
y = network.outputs
# Alternatively, you can use tl.cost.cross_entropy(y, y_) instead.
cross_entropy = tf.reduce_mean(tf.nn.sparse_softmax_cross_entropy_with_logits(y, y_))
cost = cross_entropy
cost = cost + tl.cost.maxnorm_regularizer(1.0)(network.all_params[0]) +
tl.cost.maxnorm_regularizer(1.0)(network.all_params[2])
In addition, all TensorFlow’s regularizers like tf.contrib.layers.l2_regularizer can be used with TensorLayer.
### Regularization of Activation outputs¶
Instance method network.print_layers() prints all outputs of different layers in order. To achieve regularization on activation output, you can use network.all_layers which contains all outputs of different layers. If you want to apply L1 penalty on the activations of first hidden layer, just simply add tf.contrib.layers.l2_regularizer(lambda_l1)(network.all_layers[1]) to the cost function.
network.print_layers()
layer 0: Tensor("dropout/mul_1:0", shape=(?, 784), dtype=float32)
layer 1: Tensor("Relu:0", shape=(?, 800), dtype=float32)
layer 2: Tensor("dropout_1/mul_1:0", shape=(?, 800), dtype=float32)
layer 3: Tensor("Relu_1:0", shape=(?, 800), dtype=float32)
layer 4: Tensor("dropout_2/mul_1:0", shape=(?, 800), dtype=float32)
layer 5: Tensor("add_2:0", shape=(?, 10), dtype=float32)
cross_entropy(output, target[, name]) It is a softmax cross-entropy operation, returns the TensorFlow expression of cross-entropy of two distributions, implement softmax internally. sigmoid_cross_entropy(output, target[, name]) It is a sigmoid cross-entropy operation, see tf.nn.sigmoid_cross_entropy_with_logits. binary_cross_entropy(output, target[, …]) Computes binary cross entropy given output. mean_squared_error(output, target[, is_mean]) Return the TensorFlow expression of mean-square-error of two distributions. normalized_mean_square_error(output, target) Return the TensorFlow expression of normalized mean-square-error of two distributions. dice_coe(output, target[, loss_type, axis, …]) Soft dice (Sørensen or Jaccard) coefficient for comparing the similarity of two batch of data, usually be used for binary image segmentation i.e. dice_hard_coe(output, target[, threshold, …]) Non-differentiable Sørensen–Dice coefficient for comparing the similarity of two batch of data, usually be used for binary image segmentation i.e. iou_coe(output, target[, threshold, axis, …]) Non-differentiable Intersection over Union (IoU) for comparing the similarity of two batch of data, usually be used for evaluating binary image segmentation. cross_entropy_seq(logits, target_seqs[, …]) Returns the expression of cross-entropy of two sequences, implement softmax internally. cross_entropy_seq_with_mask(logits, …[, …]) Returns the expression of cross-entropy of two sequences, implement softmax internally. cosine_similarity(v1, v2) Cosine similarity [-1, 1], wiki. li_regularizer(scale[, scope]) li regularization removes the neurons of previous layer, i represents inputs. lo_regularizer(scale[, scope]) lo regularization removes the neurons of current layer, o represents outputs maxnorm_regularizer([scale, scope]) Max-norm regularization returns a function that can be used to apply max-norm regularization to weights. maxnorm_o_regularizer(scale, scope) Max-norm output regularization removes the neurons of current layer. maxnorm_i_regularizer(scale[, scope]) Max-norm input regularization removes the neurons of previous layer.
## Softmax cross entropy¶
tensorlayer.cost.cross_entropy(output, target, name=None)[source]
It is a softmax cross-entropy operation, returns the TensorFlow expression of cross-entropy of two distributions, implement softmax internally. See tf.nn.sparse_softmax_cross_entropy_with_logits.
Parameters: output : Tensorflow variable A distribution with shape: [batch_size, n_feature]. target : Tensorflow variable A batch of index with shape: [batch_size, ]. name : string Name of this loss.
References
• The code is borrowed from: here.
Examples
>>> ce = tl.cost.cross_entropy(y_logits, y_target_logits, 'my_loss')
## Sigmoid cross entropy¶
tensorlayer.cost.sigmoid_cross_entropy(output, target, name=None)[source]
It is a sigmoid cross-entropy operation, see tf.nn.sigmoid_cross_entropy_with_logits.
## Binary cross entropy¶
tensorlayer.cost.binary_cross_entropy(output, target, epsilon=1e-08, name='bce_loss')[source]
Computes binary cross entropy given output.
For brevity, let x = output, z = target. The binary cross entropy loss is
loss(x, z) = - sum_i (x[i] * log(z[i]) + (1 - x[i]) * log(1 - z[i]))
Parameters: output : tensor of type float32 or float64. target : tensor of the same type and shape as output. epsilon : float A small value to avoid output is zero. name : string An optional name to attach to this layer.
References
## Mean squared error¶
tensorlayer.cost.mean_squared_error(output, target, is_mean=False)[source]
Return the TensorFlow expression of mean-square-error of two distributions.
Parameters: output : 2D or 4D tensor. target : 2D or 4D tensor. is_mean : boolean, if True, use tf.reduce_mean to compute the loss of one data, otherwise, use tf.reduce_sum (default).
References
## Normalized mean square error¶
tensorlayer.cost.normalized_mean_square_error(output, target)[source]
Return the TensorFlow expression of normalized mean-square-error of two distributions.
Parameters: output : 2D or 4D tensor. target : 2D or 4D tensor.
## Dice coefficient¶
tensorlayer.cost.dice_coe(output, target, loss_type='jaccard', axis=[1, 2, 3], smooth=1e-05)[source]
Soft dice (Sørensen or Jaccard) coefficient for comparing the similarity of two batch of data, usually be used for binary image segmentation i.e. labels are binary. The coefficient between 0 to 1, 1 means totally match.
Parameters: output : tensor A distribution with shape: [batch_size, ….], (any dimensions). target : tensor A distribution with shape: [batch_size, ….], (any dimensions). loss_type : string jaccard or sorensen, default is jaccard. axis : list of integer All dimensions are reduced, default [1,2,3]. smooth : float This small value will be added to the numerator and denominator. If both output and target are empty, it makes sure dice is 1. If either output or target are empty (all pixels are background), dice = smooth/(small_value + smooth), then if smooth is very small, dice close to 0 (even the image values lower than the threshold), so in this case, higher smooth can have a higher dice.
References
Examples
>>> outputs = tl.act.pixel_wise_softmax(network.outputs)
>>> dice_loss = 1 - tl.cost.dice_coe(outputs, y_)
## Hard Dice coefficient¶
tensorlayer.cost.dice_hard_coe(output, target, threshold=0.5, axis=[1, 2, 3], smooth=1e-05)[source]
Non-differentiable Sørensen–Dice coefficient for comparing the similarity of two batch of data, usually be used for binary image segmentation i.e. labels are binary. The coefficient between 0 to 1, 1 if totally match.
Parameters: output : tensor A distribution with shape: [batch_size, ….], (any dimensions). target : tensor A distribution with shape: [batch_size, ….], (any dimensions). threshold : float The threshold value to be true. axis : list of integer All dimensions are reduced, default [1,2,3]. smooth : float This small value will be added to the numerator and denominator, see dice_coe.
References
## IOU coefficient¶
tensorlayer.cost.iou_coe(output, target, threshold=0.5, axis=[1, 2, 3], smooth=1e-05)[source]
Non-differentiable Intersection over Union (IoU) for comparing the similarity of two batch of data, usually be used for evaluating binary image segmentation. The coefficient between 0 to 1, 1 means totally match.
Parameters: output : tensor A distribution with shape: [batch_size, ….], (any dimensions). target : tensor A distribution with shape: [batch_size, ….], (any dimensions). threshold : float The threshold value to be true. axis : list of integer All dimensions are reduced, default [1,2,3]. smooth : float This small value will be added to the numerator and denominator, see dice_coe.
Notes
• IoU cannot be used as training loss, people usually use dice coefficient for training, IoU and hard-dice for evaluating.
## Cross entropy for sequence¶
tensorlayer.cost.cross_entropy_seq(logits, target_seqs, batch_size=None)[source]
Returns the expression of cross-entropy of two sequences, implement softmax internally. Normally be used for Fixed Length RNN outputs.
Parameters: logits : Tensorflow variable 2D tensor, network.outputs, [batch_size*n_steps (n_examples), number of output units] target_seqs : Tensorflow variable target : 2D tensor [batch_size, n_steps], if the number of step is dynamic, please use cross_entropy_seq_with_mask instead. batch_size : None or int. If not None, the return cost will be divided by batch_size.
Examples
>>> see PTB tutorial for more details
>>> input_data = tf.placeholder(tf.int32, [batch_size, num_steps])
>>> targets = tf.placeholder(tf.int32, [batch_size, num_steps])
>>> cost = tl.cost.cross_entropy_seq(network.outputs, targets)
## Cross entropy with mask for sequence¶
tensorlayer.cost.cross_entropy_seq_with_mask(logits, target_seqs, input_mask, return_details=False, name=None)[source]
Returns the expression of cross-entropy of two sequences, implement softmax internally. Normally be used for Dynamic RNN outputs.
Parameters: logits : network identity outputs 2D tensor, network.outputs, [batch_size, number of output units]. target_seqs : int of tensor, like word ID. [batch_size, ?] input_mask : the mask to compute loss The same size with target_seqs, normally 0 and 1. return_details : boolean If False (default), only returns the loss. If True, returns the loss, losses, weights and targets (reshape to one vetcor).
Examples
• see Image Captioning Example.
## Cosine similarity¶
tensorlayer.cost.cosine_similarity(v1, v2)[source]
Cosine similarity [-1, 1], wiki.
Parameters: v1, v2 : tensor of [batch_size, n_feature], with the same number of features. a tensor of [batch_size, ]
## Regularization functions¶
For tf.nn.l2_loss, tf.contrib.layers.l1_regularizer, tf.contrib.layers.l2_regularizer and tf.contrib.layers.sum_regularizer, see TensorFlow API.
### Maxnorm¶
tensorlayer.cost.maxnorm_regularizer(scale=1.0, scope=None)[source]
Max-norm regularization returns a function that can be used to apply max-norm regularization to weights. About max-norm: wiki.
The implementation follows TensorFlow contrib.
Parameters: scale : float A scalar multiplier Tensor. 0.0 disables the regularizer. scope: An optional scope name. A function with signature mn(weights, name=None) that apply Lo regularization. ValueError : If scale is outside of the range [0.0, 1.0] or if scale is not a float.
### Special¶
tensorlayer.cost.li_regularizer(scale, scope=None)[source]
li regularization removes the neurons of previous layer, i represents inputs.
Returns a function that can be used to apply group li regularization to weights.
The implementation follows TensorFlow contrib.
Parameters: scale : float A scalar multiplier Tensor. 0.0 disables the regularizer. scope: An optional scope name for TF12+. A function with signature li(weights, name=None) that apply Li regularization. ValueError : if scale is outside of the range [0.0, 1.0] or if scale is not a float.
tensorlayer.cost.lo_regularizer(scale, scope=None)[source]
lo regularization removes the neurons of current layer, o represents outputs
Returns a function that can be used to apply group lo regularization to weights.
The implementation follows TensorFlow contrib.
Parameters: scale : float A scalar multiplier Tensor. 0.0 disables the regularizer. scope: An optional scope name for TF12+. A function with signature lo(weights, name=None) that apply Lo regularization. ValueError : If scale is outside of the range [0.0, 1.0] or if scale is not a float.
tensorlayer.cost.maxnorm_o_regularizer(scale, scope)[source]
Max-norm output regularization removes the neurons of current layer.
Returns a function that can be used to apply max-norm regularization to each column of weight matrix.
The implementation follows TensorFlow contrib.
Parameters: scale : float A scalar multiplier Tensor. 0.0 disables the regularizer. scope: An optional scope name. A function with signature mn_o(weights, name=None) that apply Lo regularization. ValueError : If scale is outside of the range [0.0, 1.0] or if scale is not a float.
tensorlayer.cost.maxnorm_i_regularizer(scale, scope=None)[source]
Max-norm input regularization removes the neurons of previous layer.
Returns a function that can be used to apply max-norm regularization to each row of weight matrix.
The implementation follows TensorFlow contrib.
Parameters: scale : float A scalar multiplier Tensor. 0.0 disables the regularizer. scope: An optional scope name. A function with signature mn_i(weights, name=None)` that apply Lo regularization. ValueError : If scale is outside of the range [0.0, 1.0] or if scale is not a float.
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2020-03-28 15:58:57
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https://in.mathworks.com/help/mpc/ug/servomechanism-controller.html
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## Design MPC Controller for Position Servomechanism
This example shows how to design a model predictive controller for a position servomechanism using MPC Designer.
### System Model
A position servomechanism consists of a DC motor, gearbox, elastic shaft, and load.
The differential equations representing this system are
$\begin{array}{l}{\stackrel{˙}{\omega }}_{L}=-\frac{{k}_{T}}{{J}_{L}}\left({\theta }_{L}-\frac{{\theta }_{M}}{\rho }\right)-\frac{{\beta }_{L}}{{J}_{L}}{\omega }_{L}\\ {\stackrel{˙}{\omega }}_{M}=\frac{{k}_{M}}{{J}_{M}}\left(\frac{V-{k}_{M}{\omega }_{M}}{R}\right)-\frac{{\beta }_{M}{\omega }_{M}}{{J}_{M}}+\frac{{k}_{T}}{\rho {J}_{M}}\left({\theta }_{L}-\frac{{\theta }_{M}}{\rho }\right)\end{array}$
where,
• V is the applied voltage.
• T is the torque acting on the load.
• ${\omega }_{L}={\stackrel{˙}{\theta }}_{L}$ is the load angular velocity.
• ${\omega }_{M}={\stackrel{˙}{\theta }}_{M}$ is the motor shaft angular velocity.
The remaining terms are constant parameters.
Constant Parameters for Servomechanism Model
Symbol
Value (SI Units)
Definition
kT
1280.2
Torsional rigidity
kM
10
Motor constant
JM
0.5
Motor inertia
JL
50JM
ρ
20
Gear ratio
βM
0.1
Motor viscous friction coefficient
βL
25
R
20
Armature resistance
If you define the state variables as
${x}_{p}={\left[\begin{array}{cccc}{\theta }_{L}& {\omega }_{L}& {\theta }_{M}& {\omega }_{M}\end{array}\right]}^{T},$
then you can model the servomechanism as an LTI state-space system.
$\begin{array}{c}{\stackrel{˙}{x}}_{p}=\left[\begin{array}{cccc}0& 1& 0& 0\\ -\frac{{k}_{T}}{{J}_{L}}& -\frac{{\beta }_{L}}{{J}_{L}}& \frac{{k}_{T}}{\rho {J}_{L}}& 0\\ 0& 0& 0& 1\\ \frac{{k}_{T}}{\rho {J}_{M}}& 0& -\frac{{k}_{T}}{{\rho }^{2}{J}_{M}}& -\frac{{\beta }_{M}+\frac{{k}_{M}^{2}}{R}}{{J}_{M}}\end{array}\right]{x}_{p}+\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{{k}_{M}}{R{J}_{M}}\end{array}\right]V\\ {\theta }_{L}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]{x}_{p}\\ T=\left[\begin{array}{cccc}{k}_{T}& 0& -\frac{{k}_{T}}{\rho }& 0\end{array}\right]{x}_{p}\end{array}$
The controller must set the angular position of the load, θL, at a desired value by adjusting the applied voltage, V.
However, since the elastic shaft has a finite shear strength, the torque, T, must stay within the range |T| ≤ 78.5 Nm. Also, the voltage source physically limits the applied voltage to the range |V| ≤ 220 V.
### Construct Plant Model
Specify the model constants (units are in MKS).
Kt = 1280.2; % Torsional rigidity
Km = 10; % Motor constant
Jm = 0.5; % Motor inertia
Jl = 50*Jm; % Load inertia
N = 20; % Gear ratio
Bm = 0.1; % Rotor viscous friction
Bl = 25; % Load viscous friction
R = 20; % Armature resistance
Define the state-space matrices derived from the model equations.
A = [ 0 1 0 0;
-Kt/Jl -Bl/Jl Kt/(N*Jl) 0;
0 0 0 1;
Kt/(Jm*N) 0 -Kt/(Jm*N^2) -(Bm+Km^2/R)/Jm];
B = [0; 0; 0; Km/(R*Jm)];
C = [ 1 0 0 0;
Kt 0 -Kt/N 0];
D = [0; 0];
Create a state-space model.
plant = ss(A,B,C,D);
mpcDesigner
### Import Plant and Define Signal Configuration
In MPC Designer, on the MPC Designer tab, select MPC Structure.
In the Define MPC Structure By Importing dialog box, select the plant plant model, and assign the plant I/O channels to the following signal types:
• Manipulated variable — Voltage, V
• Measured output — Load angular position, θL
• Unmeasured output — Torque, T
Click .
MPC Designer imports the specified plant and creates an MPC controller and a simulation scenario:
• mpc1 — Default MPC controller created using plant as its internal model.
• scenario1 — Default simulation scenario. The results of this simulation are displayed in the Input Response and Output Response plots.
Plants, controllers and simulation scenarios are accessible via the data browser, on the on the left hand side of MPC Designer.
### Define Input and Output Channel Attributes
On the MPC Designer tab, in the Structure section, click I/O Attributes.
In the Input and Output Channel Specifications dialog box, for each input and output channel:
• Specify a meaningful Name and Unit.
• Keep the Nominal Value at its default value of 0.
• Specify a Scale Factor for normalizing the signal. Select a value that approximates the predicted operating range of the signal:
Channel NameMinimum ValueMaximum ValueScale Factor
Voltage–220 V220 V440
Torque–78.5 Nm78.5 Nm157
Click to update the channel attributes and close the dialog box.
### Modify Scenario To Simulate Angular Position Step Response
In the Scenario section, Edit Scenario drop-down list, select scenario1 to modify the default simulation scenario.
In the Simulation Scenario dialog box, keep a Simulation duration of 10 seconds.
In the Reference Signals table, keep the default configuration for the first channel. These settings create a Step change of 1 radian in the angular position setpoint at a Time of 1 second.
For the second output, in the Signal drop-down list, select Constant to keep the torque setpoint at its nominal value.
Click .
The app runs the simulation with the new scenario settings and updates the Input Response and Output Response plots.
### Specify Controller Sample Time and Horizons
On the Tuning tab, in the Horizon section, specify a Sample time of 0.1 seconds.
For the specified sample time, Ts, and a desired response time of Tr = 2 seconds, select a prediction horizon, p, such that:
${T}_{r}\approx p{T}_{s}.$
Therefore, specify a Prediction horizon of 20.
Specify a Control horizon of 5.
As you update the sample time and horizon values, the Input Response and Output Response plots update automatically. Both the input voltage and torque values exceed the constraints defined in the system model specifications.
### Specify Constraints
In the Design section, select Constraints.
In the Constraints dialog box, in the Input Constraints section, specify the Min and Max voltage values for the manipulated variable (MV).
In the Output Constraints section, specify Min and Max torque values for the unmeasured output (UO).
There are no additional constraints, that is the other constraints remain at their default maximum and minimum values, —Inf and Inf respectively
Click .
The response plots update to reflect the new constraints. In the Input Response plot, there are undesirable large changes in the input voltage.
### Specify Tuning Weights
In the Design section, select Weights.
In the Weights dialog box, in the Input Weights table, increase the manipulated variable Rate Weight.
The tuning Weight for the manipulated variable (MV) is 0. This weight indicates that the controller can allow the input voltage to vary within its constrained range. The increased Rate Weight limits the size of manipulated variable changes.
Since the control objective is for the angular position of the load to track its setpoint, the tuning Weight on the measured output is 1. There is no setpoint for the applied torque, so the controller can allow the second output to vary within its constraints. Therefore, the Weight on the unmeasured output (UO) is 0, which enables the controller to ignore the torque setpoint.
Click .
The response plots update to reflect the increased rate weight. The Input Response is smoother with smaller voltage changes.
### Examine Output Response
In the Output Response plot, right-click the Theta plot area, and select Characteristics > Peak Response.
The peak output response occurs at time of 3 seconds with a maximum overshoot of 3%. Since the reference signal step change is at 1 second, the controller has a peak time of 2 seconds.
### Improve Controller Response Time
Click and drag the Closed-Loop Performance slider to the right to produce a more Aggressive response. The further you drag the slider to the right, the faster the controller responds. Select a slider position such that the peak response occurs at 2.6 seconds.
The final controller peak time is 1.6 seconds. Reducing the response time further results in overly-aggressive input voltage changes.
### Generate and Run MATLAB Script
In the Analysis section, click the Export Controller arrow .
Under Export Controller, click Generate Script.
In the Generate MATLAB® Script dialog box, check the box next to scenario1.
Click .
The app exports a copy of the plant model, plant_C, to the MATLAB workspace, along with simulation input and reference signals.
Additionally, the app generates the following code in the MATLAB Editor.
%% create MPC controller object with sample time
mpc1 = mpc(plant_C, 0.1);
%% specify prediction horizon
mpc1.PredictionHorizon = 20;
%% specify control horizon
mpc1.ControlHorizon = 5;
%% specify nominal values for inputs and outputs
mpc1.Model.Nominal.U = 0;
mpc1.Model.Nominal.Y = [0;0];
%% specify scale factors for inputs and outputs
mpc1.MV(1).ScaleFactor = 440;
mpc1.OV(1).ScaleFactor = 6.28;
mpc1.OV(2).ScaleFactor = 157;
%% specify constraints for MV and MV Rate
mpc1.MV(1).Min = -220;
mpc1.MV(1).Max = 220;
%% specify constraints for OV
mpc1.OV(2).Min = -78.5;
mpc1.OV(2).Max = 78.5;
%% specify overall adjustment factor applied to weights
beta = 1.2712;
%% specify weights
mpc1.Weights.MV = 0*beta;
mpc1.Weights.MVRate = 0.4/beta;
mpc1.Weights.OV = [1 0]*beta;
mpc1.Weights.ECR = 100000;
%% specify simulation options
options = mpcsimopt();
options.Constraints = 'on';
options.OpenLoop = 'off';
%% run simulation
sim(mpc1, 101, mpc1_RefSignal, mpc1_MDSignal, options);
In the MATLAB Window, in the Editor tab, select Save.
Complete the Save dialog box and then click .
In the Editor tab, click Run.
The script creates the controller, mpc1, and runs the simulation scenario. The input and output responses match the simulation results from the app.
### Validate Controller Performance In Simulink
If you have a Simulink® model of your system, you can simulate your controller and validate its performance.
Open the model.
open_system('mpc_motor')
This model uses an MPC Controller block to control a servomechanism plant. The Servomechanism Model block is already configured to use the plant model from the MATLAB workspace.
The Angle reference source block creates a sinusoidal reference signal with a frequency of 0.4 rad/sec and an amplitude of π.
Double-click the MPC Controller block.
In the MPC Controller Block Parameters dialog box, specify an MPC Controller from the MATLAB workspace. Use the mpc1 controller created using the generated script.
Click .
At the MATLAB command line, specify a torque magnitude constraint variable.
tau = 78.5;
The model uses this value to plot the constraint limits on the torque output scope.
In the Simulink model window, click Run to simulate the model.
In the Angle scope, the output response, yellow, tracks the angular position setpoint, blue, closely.
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2023-02-01 12:33:16
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http://www.chegg.com/homework-help/questions-and-answers/gas-confined-tank-pressure-90-atm-temperature-290-c-two-thirds-gas-withdrawn-temperature-r-q3176606
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Gas is confined in a tank at a pressure of 9.0 atm and a temperature of 29.0�C. If two-thirds of the gas is withdrawn and the temperature is raised to 86.0�C, what is the pressure of the gas remaining in the tank?
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2016-05-31 14:27:37
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|
http://www.numdam.org/item/AIHPC_1995__12_5_599_0/
|
Solutions of Ginzburg-Landau equations and critical points of the renormalized energy
Annales de l'I.H.P. Analyse non linéaire, Volume 12 (1995) no. 5, pp. 599-622.
@article{AIHPC_1995__12_5_599_0,
author = {Hua Lin, Fang},
title = {Solutions of {Ginzburg-Landau} equations and critical points of the renormalized energy},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {599--622},
publisher = {Gauthier-Villars},
volume = {12},
number = {5},
year = {1995},
zbl = {0845.35052},
mrnumber = {1353261},
language = {en},
url = {http://www.numdam.org/item/AIHPC_1995__12_5_599_0/}
}
TY - JOUR
AU - Hua Lin, Fang
TI - Solutions of Ginzburg-Landau equations and critical points of the renormalized energy
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1995
DA - 1995///
SP - 599
EP - 622
VL - 12
IS - 5
PB - Gauthier-Villars
UR - http://www.numdam.org/item/AIHPC_1995__12_5_599_0/
UR - https://zbmath.org/?q=an%3A0845.35052
UR - https://www.ams.org/mathscinet-getitem?mr=1353261
LA - en
ID - AIHPC_1995__12_5_599_0
ER -
%0 Journal Article
%A Hua Lin, Fang
%T Solutions of Ginzburg-Landau equations and critical points of the renormalized energy
%J Annales de l'I.H.P. Analyse non linéaire
%D 1995
%P 599-622
%V 12
%N 5
%I Gauthier-Villars
%G en
%F AIHPC_1995__12_5_599_0
Hua Lin, Fang. Solutions of Ginzburg-Landau equations and critical points of the renormalized energy. Annales de l'I.H.P. Analyse non linéaire, Volume 12 (1995) no. 5, pp. 599-622. http://www.numdam.org/item/AIHPC_1995__12_5_599_0/
[BBH] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau vertices, Birkhaüser, Boston, 1994. | MR | Zbl
[BBH2] F. Bethuel, H. Brezis and F. Helein, Asymptotics for the minimization of a Ginzburg-Landau functional, Cal. variations and P.D.E., Vol. 1#2, 1993, pp. 123-148. | MR | Zbl
[BMR] H. Brezis, F. Merle and T. Riviere, Quantization effects, for -Δu = u(1-(u)2) in R2, preprint. | Zbl
[CL] Y.M. Chen and F.H. Lin, Evaluation of harmonic maps with the Dirichlet boundary condition, Comm. in Analysis and Geometry, Vol. 1#3, 1993, pp. 327-346. | MR | Zbl
[CS] Y.M. Chen and M. Struwe, Existence and partial regularity for heat flow for harmonic maps, Math. Z, Vol. 201, 1989, pp. 83-103. | EuDML | MR | Zbl
[HL] R. Hardt and F.H. Lin, Singularities for p-energy minimizing unit vector fields on planar domains, to appear in Cal. variation and P. D. E. | Zbl
[E] E. Weinan, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, preprint. | MR | Zbl
[LSU] O.A. Ladyzenskaya, N.A. Solonnikov and N.N. Uralseva, Linear and quasilinear equations of parabolic type, Translations of AMS, Mon. #23 (196X). | Zbl
[N] J. Neu, Vortex dynamics of complex scalar fields, Physics D., Vol. 43, 1990, pp. 384-406. | Zbl
[PR] L. Pismen and J. Rubinstein, Dynamics of defects, in nematics, mathematical and physical aspects, J. M. Coron et al. eds, Kluwer Pubs., 1991. | MR
[R] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of the Math. Sci., by AMS #65. | MR | Zbl
[RS] J. Rubinstein and P. Sternberg, On the Slow Motion of Vortices in the Ginzburg-Landau heat flow, preprint. | MR
[S] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals of Math., Vol. 118, 1983, pp. 527- 571. | MR | Zbl
[St] M. Struwe, On the asymptotic behavior of minimizers of the Ginsburg-Landau model in 2 dimensions, J. Diff. Int. Eqs., Vol. 7, 1994. | MR | Zbl
[St2] M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., Vol. 60, 1985, pp. 558-581. | MR | Zbl
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2023-02-01 19:28:54
|
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|
https://web2.0calc.com/questions/bob-and-alice
|
+0
# bob and alice
+2
1547
1
+197
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?
Dec 30, 2017
#1
+197
+4
Solution:
Since there are the same amounts of all the balls in Alice's bag and Bob's bag, and there is an equal chance of each ball being selected, the color of the ball that Alice puts in Bob's bag doesn't matter. Without loss of generality, let the ball Alice puts in Bob's bag be red.
For both bags to have the same contents, Bob must select one of the 2 red balls out of the 6 balls in his bag.
So the desired probability is $$\frac{2}{6}=\frac{1}{3}$$
Dec 30, 2017
#1
+197
+4
So the desired probability is $$\frac{2}{6}=\frac{1}{3}$$
|
2023-04-01 04:32:18
|
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https://proxies-free.com/real-analysis-if-f_nto-f-uniformly-then-min_abf_n-to-min_abf/
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# real analysis – If \$f_nto f\$ uniformly, then \$min_{[a,b]}f_n to min_{[a,b]}f\$
For $$f_n,f$$ continuous on $$(a,b)$$, and $$f_n to f$$ uniformly, define $$M_n=max_{(a,b)}f_n$$ and $$M=max_{(a,b)}f$$, then it was asked to show that $$M_n to M$$. And it was also asked is it also true for sequence of minima. I did the first part as follows:
$$f_n(x)-f(x)leq max_{(a,b)}lvert f_n(x)-f(x) rvert$$,
$$Rightarrow f_n(x)leq max_{(a,b)}lvert f_n(x)-f(x) rvert + f(x)$$,
Then first taking maximum on right side and then on left side, we will get,
$$M_n-Mleq max_{(a,b)}lvert f_n(x)-f(x) rvert$$.
Similarly, we can also show that,
$$M-M_nleq max_{(a,b)}lvert f_n(x)-f(x) rvert$$, which gives,
$$lvert M_n-M rvert leq max_{(a,b)}lvert f_n(x)-f(x) rvert$$.
By uniform continuity, it is evident that right hand side of above equation goes to zero, which implies $$M_n to M$$.
But I am having difficulty in showing the second part of question. I guess it is true and I thought proof will be similar but the same idea doesn’t seem to work. Any hint. Thanks.
|
2021-03-07 09:20:26
|
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|
http://tex.stackexchange.com/questions/55210/not-element-of-in-latin-modern?answertab=oldest
|
# 'Not element of' in Latin Modern
In the following example, the ∉ symbol does not look good.
\documentclass{article}
\usepackage{unicode-math}
\setmainfont{Latin Modern Roman}
\setmathfont{Latin Modern Math}
\begin{document}
This does not show: $\notin$ \\
This does not look good: $\not\in$
\end{document}
\not\in works when not loading unicode-math, but unicode-math somehow prevents it from working (?). The Unicode character 0x2209 does not seem to be included in the Latin Modern Math font. I know that I can load the Unicode character from another font, but I just want the output to look like \not\in in Computer Modern.
edit: To clarify, I just want the output to look like \not\in without using unicode-math. I want to use unicode-math for reasons not visible in this example.
-
After last Khaled's comment, here's something that seems to work
\Umathchardef\xnot="3 \symoperators "0338
\AtBeginDocument{
\renewcommand\not[1]{#1\xnot}
\renewcommand{\notin}{\not\in}
}
Then
$a\not\in b \notin c$
will work (although the placement doesn't seem to be optimal).
\documentclass{article}
\usepackage{unicode-math}
\setmainfont{Latin Modern Roman}
\setmathfont{Latin Modern Math}
\Umathchardef\xnot="3 \symoperators "0338
\AtBeginDocument{\renewcommand\not[1]{#1\xnot}
\renewcommand{\notin}{\in\xnot}}
\begin{document}
$a \not\in S_{\not\in}$
$a\in\xnot b \notin c$
$a\in b \in c$
\end{document}
A possible improvement is to say
\Umathchardef\xnot="3 \symoperators "0338
\AtBeginDocument{
\renewcommand\not[1]{#1\mathrel{\mkern1mu}\xnot}
\renewcommand{\notin}{\not\in}
}
that pushes the slash slightly to the right, so that the upper end is lined up with the terminators of the \in symbol.
Note: this definitely doesn't work with XITS Math or Asana Math, which, however, have the proper symbol.
## UPDATE
As of January 2013, the problem seems to be solved; here's the minimal example:
\documentclass{article}
\usepackage{unicode-math}
\setmainfont{Latin Modern Roman}
\setmathfont{Latin Modern Math}
\begin{document}
$a \not\in S_{\not\in}$
$a \notin S_{\notin}$
\end{document}
Both LuaLaTeX and XeLaTeX give the correct result.
-
I once considered something like that, but it does not look that good with XITS and Cambria, not sure if unicode-math should do that by default or not. – Khaled Hosny May 10 '12 at 10:30
@KhaledHosny I guess not too; just a hack for the missing symbol in Latin Modern Math. I'm trying to fix the spacing. – egreg May 10 '12 at 11:29
This works perfectly and fixes my problem! Do you think you could briefly explain the \Umathchardef command? Google is not a big help. I suppose you use it to define a command (\xnot) to correspond to a Unicode character (0x338), but I don't know what the "3 and \symoperators are for, or what the different options are here. – Semafoor May 10 '12 at 17:01
@Semafoor It's XeTeX and LuaTeX lingo for defining a relation symbol ("3), from math font \symoperators and position "338. – egreg May 10 '12 at 18:46
\documentclass{article}
\usepackage{unicode-math}
\setmainfont{Latin Modern Roman}
\setmathfont{Latin Modern Math}
\newcommand\cnot[1]{%
\mathrel{\ooalign{\hfil$#1$\hfil\cr\hfil$/$\hfil\cr}}}
\begin{document}
$a \cnot\in S$
\end{document}
Note: I don't know why Oberdiek's centernot package also fail. So I had to implement one myself.
Edit:
To make the macro to change size automatically as egreg suggested,
\def\cnot#1{\mathrel{\mathpalette\ccnot{#1}}}
\def\ccnot#1#2{\ooalign{\hfil$#1#2$\hfil\cr\hfil$#1/$\hfil\cr}} % helper macro
-
It seems that unicode-math's \not is really badly implemented, all my math fonts cannot typeset \not\in properly. – Leo Liu May 10 '12 at 5:59
@egrep: \not is zero width in most Unicode fonts too, I think the problem is because it is defined as a math accent not a relation symbol in unicode-math. – Khaled Hosny May 10 '12 at 8:31
@LeoLiu Would you implement it with \mathpalette so that the symbol changes size in subscripts and superscripts? – egreg May 10 '12 at 9:07
@egreg: not so mysterious, it is a combining mark in Unicode, so that is the logical choice, it should then be handled by the engine (I had a patch for luatex, but had to hold it back because accents break math spacing rules, I'm yet to find a sane solution for this). But it makes little difference, defining it as mathrel does not help a bit, because of the way it is designed in all fonts, try \Umathchardef\not="3 \symoperators "0338 yourself. – Khaled Hosny May 10 '12 at 9:25
@egreg, it prints here, but look closely, it will be shifted to the left away of the symbol. It doesn't combine because the engines don't handle it correctly, the old TeX approach doesn't work here because it was a CM specific Knuthian hack. – Khaled Hosny May 10 '12 at 10:04
Try \centernot\in (from the centernot package): It centers the / above whatever symbol follows.
-
|
2014-07-28 10:35:55
|
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|
https://www.swrfootball.com/news/dimazio-ranieri-is-getting-closer-and-closer-to-coaching-cagliari
|
# DiMazio: Ranieri is getting closer and closer to coaching Cagliari
Di Mazio reported on December 23 that Claudio Ranieri is getting closer and closer to becoming coach of Cagliari.
Cagliari recently sacked coach Giuseppe Rivalani and the club is in talks with Claudio Ranieri.
DiMazio said that the signs revealed by the negotiations between the two sides are positive and are getting closer to cooperation, but they still need some time to reach a final agreement, and the situation is now moving forward.
(goblin killer)
|
2023-03-24 18:22:30
|
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|
https://events.mpe.mpg.de/event/9/contributions/421/
|
# Mapping the X-ray Sky with SRG: First Results from eROSITA and ART-XC
15-20 March 2020
Garching
Europe/Berlin timezone
## Heavily-Obscured X-ray AGNs shed Light on the Connection between the Quenching of M* Galaxies and the Cosmic Web
Not scheduled
20m
Garching
Poster
### Speaker
Dr Thibaud Moutard
### Description
Well documented up to redshift z ~ 4, the bimodality between “blue/star-forming" (SF) and “red/quiescent" (Q) galaxies is the statistical expression of the so-called quenching of the star formation. On the other hand, the diversity observed among quiescent galaxies (e.g., in terms of mass or morphology) suggests that the mechanisms involved in the quenching are multiple. For instance, the processes that are at play in the quenching of low-mass galaxies may be quite different from those involved in the quenching of massive galaxies, after billion years on the star formation main sequence (Faber et al. 2007, Schawinski et al. 2014, Moutard et al. 2016b).
In particular, the fact that star formation is observed to stop earlier in more massive galaxies, on average, underlies a downsizing of the quenching that argues for the existence of mass-related quenching processes, and the stellar mass function of SF galaxies shows that such “mass quenching” operates in galaxies with stellar mass reaching the characteristic mass $M^* \sim 10^{10.6}M_\odot$ (Ilbert et al. 2010, Peng et al. 2010, Moutard et al. 2016b). While $M^*$ galaxy quenching has been shown to be quite slow (lasting 1–3.5 Gyrs; e.g., Moutard et al. 2016b, Pandya et al. 2017), several mechanisms able to halt the cold gas supply have been put forth, e.g., due to the heating of the gas via viral shocks within dark-matter halos reaching a critical mass of $M_h \sim 10^{12}M_\odot$ (e.g., Kereš et al. 2005) or via radio-loud active galactic nucleus (AGN) feedback (e.g., Best et al. 2005).
I will present the results we obtained regarding the connection between the quenching of the star formation in evolved and massive galaxies and the presence of radio-loud and X-ray AGNs, by taking advantage of the rest-frame NUV-r vs. r-K colour (or NUVrK) diagram to unambiguously identify $M^*$ quenching galaxies that are in transition in the so-called green valley between SF and Q galaxies.
I will first show that radio-loud AGNs are mostly hosted by already quenched and massive ($M_* > 10^{11}M_\odot$) galaxies, which tends to confirm that their feedback is not the primary cause for $M^*$ galaxy quenching, as suggested previously (e.g., Hickox et al. 2009). More interestingly, I will then discuss the fact that the X-ray AGNs suffering from heavy obscuration of their soft X-ray emission are mostly hosted by $M^*$ galaxies that in the process of quenching, which argues for a quenching scenario that involves mergers of (gas-poor) $M^*$ galaxies after the onset of the quenching process, i.e., a scenario where $M^*$ galaxy mergers are not the cause but rather an aftermath of the quenching process. I will finally discuss how this is consistent with a picture where $M^*$ galaxy quenching happens along cosmic filaments.
### Presentation Materials
There are no materials yet.
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|
2022-01-28 23:30:07
|
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|
http://gameknight999.wikia.com/wiki/User_blog:Swordmaster767/Help_Desk
|
## FANDOM
263 Pages
Do you need help with any wikitext, or editing? CSS or JavaScript? Templates or infoboxes?
Just post a question here, and I'll help to the best of my abilities. Here are some examples of what I can do:
$5+10=15$
This will display a random number from 1 to 5: 5
Hey, I'm a box!
Crafter
|
2017-07-24 18:33:14
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https://bio-protocol.org/bio101/r8758338
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# Also in the Article
Data Analysis
This protocol is extracted from research article:
Content Validity Evidence for the Verbal Behavior Milestones Assessment and Placement Program
J Autism Dev Disord, Jan 14, 2021;
Procedure
The following analyses were conducted using IBM SPSS version 25 (IBM Corp 2017) and Microsoft Excel (Microsoft Corporation 2018) after all data were collected. First, the frequency distribution for each item was generated to show how SMEs rated each item’s (a) domain relevance, (b) developmental age appropriateness, (c) method of measurement/prompt appropriateness, and (d) domain representation. Popham (1992) recommended that 70% of SMEs endorsing an item’s relevance to support content validity as sufficient, which is most closely approximated by nine out of 13 SMEs (69.2%). All percentages were compared against the threshold of 69.2%, which is informed by Popham (1992). The authors classified the strength of evidence based on percentages; that is, percentages 69% or greater were classified as strong, percentages between 50 and 60% as moderate, and percentages less than 50% as limited. Second, the content validity ratio (CVR, Lawshe 1975) was computed for each item as follows:
where ne is the number of SMEs rating the item as “Essential,” and N is the total number of SMEs who provided a rating. The CVR can range from − 1 to + 1, with higher scores indicating greater content validity evidence. A CVR of 0 indicates that 50% of the SMEs rated the item as “Essential.” Wilson et al. (2012) recalculated the critical values for Lawshe’s (1975) CVR based on differing levels of Type I error control and number of SMEs. The critical value for the CVR with 13 SMEs using a one-tailed test with Type I error rate of 5% is 0.456, which was used as a comparison for all CVRs in this study. The critical value of 0.456 corresponds to 10 or more SMEs out of 13 rating an item as “Essential” in order to be considered statistically significant (i.e., percentage of SME support exceeds 50%). The content validity index (CVI) was calculated as the average CVR across all items and can be interpreted as content validity evidence of the domain as a whole (Lynn 1986; Shultz et al. 2014). Additionally, the CVI was calculated for each level across domains and the test as a whole. Following Lawshe (1975), CVRs and CVIs apply only to domain relevance.
It should be noted that the critical value of CVR requires more SMEs to rate an item as “Essential” than Popham’s recommended criterion (i.e., 10 versus 9). Based on the size of the sample, each SME has considerable weight in the distribution of ratings. Ten out of 13 SMEs (76.9%) has a CVR of 0.54, and nine out of 13 SMEs (69.2%) has a CVR of 0.38. Thus, an SME endorsement rating of 69.2% is considered meaningful even though the hypothesis test of the CVR is more conservative.
The relationships between the method of measurement appropriateness ratings across different methods of measurement were estimated using Cramér’s V, which is a $χ2$-based measure of association between categorical or nominal variables (Cramér 1946; Liebetrau 1983; Rea and Parker 1992).
As noted previously, SMEs could provide recommendations for item revisions, additions, or deletions on the items within each domain. The responses provided were analyzed qualitatively using thematic analysis of the text. That is, the responses were reviewed and categorized based on thematic elements that emerged with respect to recommended changes to the instrument.
Note: The content above has been extracted from a research article, so it may not display correctly.
Q&A
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2021-11-28 14:40:00
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https://math.stackexchange.com/questions/3881395/is-this-an-example-of-a-bounded-operator-with-no-closed-range/3881497
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# Is this an example of a Bounded Operator with NO closed range?
My professor asked us for an example of a Bounded Operator with whose range is not closed, after some attempts I thought this, but I am not sure about it:
Consider $$n\in \mathbb{N}$$ and $$\mathbb{R}^n$$ with the $$\max$$ norm, i.e if $$\mathbb{R}^n\ni x = (x_1,\dots,x_n)$$ then $$|| x|| = \displaystyle \max_{i=1,\dots,n}{x_i}.$$ And the operator $$T=\mathbb{R}^n\to \mathbb{R}^n$$ such that, if we denote $$||x||=a$$ then $$T:x\mapsto(a,\dots,a) \in \mathbb{R}^{n}$$. This is all the non-negative multiples of the vector $$(1,1,…,1)$$.
Clearly $$||T(x)||=||x||$$, so $$T$$ is bounded, but I am not sure if the range is closed, for this I thought the following: We know that a set $$A$$ is closed iff $$A^{c}$$ is open. In this case $$A= T(\mathbb{R}^n)$$ so I can consider an element in the complement of the image of $$T$$, this is $$y=(y_1,\dots,y_n)$$ such that there are at least two $$y_i$$ that are different. Let's denote $$b=||y||$$ and consider an open ball $$B(y,\epsilon)$$, note that for every possible $$\epsilon$$ the element $$T(y)$$ is contained in $$B(y,\epsilon)$$, so there is no open ball in $$A^c$$ for $$y$$, so $$A^c$$ is not an open set, therefore $$A$$ is not a closed set.
Is this correct? I feel like I am missing something but I am not sure. If I am wrong, can you give me some other example of a bounded operator with no closed range? I saw a couple but most of the examples I found use the $$L^p$$ spaces that we haven't seen on class. Thanks.
• Is this a linear map? You're going to have to use infinite dimensional spaces, I'm afraid. This function does have closed range, by the way. Describe the image concretely: It's all the non-negative multiples of the vector $(1,1,\dots,1)$, which is homeomorphic to the set of non-negative real numbers. – Ted Shifrin Oct 26 at 5:16
• @TedShifrin I edited it with that aclaration on the image, thanks. But I don't get why you are afraid of infinite dimensional, I am only using $\mathbb{R}^n$. And jmm, now I am not sure if it is linear... – J.Rodriguez Oct 26 at 5:25
• Sorry for my English. Figure out why your map isn't linear, but I'm telling you that you need infinite dimensions. Linear subspaces of finite dimension are always closed. – Ted Shifrin Oct 26 at 5:30
• The term 'bounded operator' is generally used for continuous linear maps in FA. I am sure the question is about linear operators. – Kavi Rama Murthy Oct 26 at 5:47
The proof is faulty because $$T$$ is not linear ($$T(-y)=T(y)$$) and even so, it is not true that $$T(y)$$ is contained in $$B(y,\epsilon)$$. (Take $$n=2$$, $$y=(1,0)$$, then $$T(y)=(1,1)$$ is not in a small ball around $$y$$.) In fact, $$B(y,\epsilon)\cap T(\mathbb{R}^n)=\emptyset$$ for $$\epsilon$$ small enough.
The mapping $$T : \ell^2\to\ell^2$$, defined by $$T (a_n) := (a_0, a_1/2, a_2/3,...)$$, is linear and bounded, with $$\|T\|=\pi/\sqrt6$$. Its image is not closed in $$\ell^2$$.
Proof: Consider $$y_n:=(1, \frac{1}{2},..., \frac{1}{n}, 0, 0,...)=T(\underbrace{1,1,\ldots,1}_n,0,\ldots)\in T(\ell^2)$$. It converges to the sequence $$y=(1,\frac{1}{2},...)\in\ell^2$$ but $$y\notin T(\ell^2)$$ otherwise $$y=T(x)$$ implies $$x=(1,1,\ldots)\notin\ell^2$$.
• Ohh, thanks, this example I can understand, but still have a question, now I know that my proof is wrong because T is not linear, but why $B(y,\epsilon) \cap T(\mathbb{R}^n) = \varnothing$ ? In your example if I take a ball arround $y$ of radious epsilon, this contains all elements $x$ of $\mathbb{R}^n$ such that $||x||\in (1-\epsilon,1+\epsilon)$, and $||T(y)||=1$ (this is because we are not using the usual norm). Am I wrong about this? – J.Rodriguez Oct 26 at 11:56
• $B(y,\epsilon)$ in the $\infty$-norm (max-norm) has the shape of a square of side $2\epsilon$. One can find such a small square around $(1,0)$ that does not touch the line $A$ (through $(0,0)$ and $(1,1)$). I don't understand your second assertion. $B(y,\epsilon)$ does not contain $-y$ for example even though $\|-y\|=1$. – Chrystomath Oct 26 at 12:07
• Ohh sorry, I was misinterpreting something, my bad. Thanks for the good examples. – J.Rodriguez Oct 26 at 12:12
Every linear subspace of a finite dimensional space is closed so there is no hope of such an example in finite dimensions. Here is a valid example: Define $$Tf(x)=\int_0^{x} f(t)dt$$ on $$C[0,1]$$. Then $$\|T\|\leq 1$$ but the range is not closed. The range consist precisely of continuously differentiable functions vanishing at $$0$$. Take any continuous function $$f$$ vanishing at $$0$$ which is not differentiable and use Weierstrass Theorem to construct a sequence polynomials in the range of $$T$$ converging uniformly to $$f$$.
Let X be a non reflexive Banach space and $$Y$$ be a reflexive Banach space. Suppose that $$A \colon X \to Y$$ is an injective bounded linear operator. I claim that the range of $$A$$, denoted by $$R(A)$$, can't be closed in $$Y$$.
Arguing by contradiction, suppose that $$R(A)$$ is closed and thus reflexive. Then, the operator $$A \colon X \to R(A)$$ is a continuous bijection and thus (by virtue of the Open Mapping Theorem) is an isomorphism. It's not hard to check that $$X$$ has to be reflexive, since it's isomorphic to a reflexive Banach space, which leads to a contradiction.
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2020-11-24 15:06:22
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https://docs.lucedaphotonics.com/ipkisseda/ledit/logging.html
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# Logging configuration¶
Logging can be important to figure out what is happening in case something goes wrong (i.e., you can’t instantiate IPKISS components, the waveguide routing fails). By default, IPKISS logs to L-Edit GUI, which already gives you some information on what might go wrong.
## File logging¶
When something goes wrong, detailed information is written to a file log. You can open the logs from the L-Edit GUI menu: Luceda > Logs > Open Log Folder.
Luceda Logs Menu: open log folder
Both integration logs as well as standalone tool logs are kept in the same directory:
• The PCell generator and the L-Edit integration macros log to a daily rotating file log. In this way you can clean up the logs easily when they fill up your disk space.
• Some IPKISS.eda python tools log to a separate file ipkisseda.log.
### Loglevel¶
The loglevel of the file logs is debug by default. If necessary, you can change this by setting the LUCEDA_FILE_LOG_LEVEL environment variable. For instance, to set the loglevel to trace from a Windows cmd console:
set LUCEDA_FILE_LOG_LEVEL=trace
### Logfolder location¶
The location of the log folder depends on your system:
• On Windows, the default location is %APPDATA%\luceda\logs (typically C:\users\yourlogin\AppData\Roaming\luceda\logs). If that does not exist, the a logs folder in the current working folder is used.
• On Linux, the default location is \$HOME\.luceda\logs. If that does not exist, a logs folder in the current working directory is used.
You can specify an alternative directory (folder) to store the logfiles by setting the LUCEDA_LOG_DIR environment variable. For instance, from a Windows cmd console:
set LUCEDA_LOG_DIR=C:\path\to\my\logdir
If you now start L-Edit from this terminal, all log information will be written to the C:\path\to\my\logdir\ directory.
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2021-06-12 18:01:41
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https://domino.mpi-inf.mpg.de/internet/reports.nsf/c125634c000710d0c12560400034f45a/1a1d4e6f67c128b6c12560400038ce6a?OpenDocument
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max planck institut
informatik
# MPI-I-91-107
## An O(n log n log log n) algorithm for the on-line closes pair problem
### Schwarz, Christian and Smid, Michiel
MPI-I-91-107. July 1991, 21 pages. | Status: available - back from printing | Next --> Entry | Previous <-- Entry
Abstract in LaTeX format:
Let $V$ be a set of $n$ points in $k$-dimensional space.
It is shown how the closest pair in $V$ can be maintained
under insertions in
$O(\log n \log\log n)$
amortized time, using $O(n)$ space. Distances are measured in the
$L_{t}$-metric, where $1 \leq t \leq \infty$.
This gives an $O(n \log n \log\log n)$ time on-line algorithm
for computing the closest pair. The algorithm is based
on Bentley's logarithmic method for decomposable searching problems.
It uses a non-trivial extension of fractional cascading to
$k$-dimensional space. It is also shown how to extend
the method to maintain the closest pair during semi-online updates.
Then, the update time becomes $O((\log n)^{2})$, even in the worst case.
Acknowledgement:
References to related material:
12035 KBytes
Please note: If you don't have a viewer for PostScript on your platform, try to install GhostScript and GhostView
URL to this document: http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1991-107
BibTeX
@TECHREPORT{SchwarzSmid91,
AUTHOR = {Schwarz, Christian and Smid, Michiel},
TITLE = {An O(n log n log log n) algorithm for the on-line closes pair problem},
TYPE = {Research Report},
INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
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2019-10-22 14:02:50
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https://artofproblemsolving.com/wiki/index.php?title=2005_AIME_I_Problems/Problem_3&diff=next&oldid=11869
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# Difference between revisions of "2005 AIME I Problems/Problem 3"
## Problem
How many positive integers have exactly three proper divisors, each of which is less than 50?
## Solution
Suppose $n$ is such an integer. Then $n=p\cdot q$ or $n=p^3$ for distinct prime numbers $p$ and $q$.
In the first case, the three proper divisors of $n$ are 1, $p$ and $q$. Thus, we need to pick two prime numbers less than 50. There are fifteen of these (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47) so there are ${15 \choose 2} =105$ numbers of the first type.
In the second case, the three proper divisors of $n$ are 1, $p$ and $p^2$. Thus we need to pick a prime number whose square is less than 50. There are four of these (2, 3, 5 and 7) and so four numbers of the second type.
Thus there are $105+4=109$ integers that meet the given conditions.
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2022-05-26 05:31:20
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http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_NonParam_ANOVA&diff=6829&oldid=6828
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# AP Statistics Curriculum 2007 NonParam ANOVA
(Difference between revisions)
Revision as of 20:23, 2 March 2008 (view source)IvoDinov (Talk | contribs) (→Notes)← Older edit Revision as of 20:38, 2 March 2008 (view source)IvoDinov (Talk | contribs) (→Calculations)Newer edit → Line 31: Line 31: ==Calculations== ==Calculations== - # Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group membership. Assign any tied values the average of the ranks they would have received had they not been tied. + Let ''N'' be the total number of observations, then $N = \sum_{i=1}^k {n_i}$. - # The test statistic is given by: + - : $K = (N-1)\frac{\sum_{i=1}^g n_i(\bar{r}_{i\cdot} - \bar{r})^2}{\sum_{i=1}^g\sum_{j=1}^{n_i}(r_{ij} - \bar{r})^2}$, where: + - #*$n_g$ is the number of observations in group $g$ + - #*$r_{ij}$ is the rank (among all observations) of observation ''j'' from group ''i'' + - #*$N$ is the total number of observations across all groups + - #*$\bar{r}_{i\cdot} = \frac{\sum_{j=1}^{n_i}{r_{ij}}}{n_i}$, + - #*$\bar{r} =(N+1)/2$ is the average of all the $r_{ij}$. + - #*Notice that the denominator of the expression for $K$ is exactly $(N-1)N(N+1)/12$. Thus $K = \frac{12}{N(N+1)}\sum_{i=1}^g n_i(\bar{r}_{i\cdot} - \bar{r})^2$. + - # A correction for ties can be made by dividing $K$ by $1 - \frac{\sum_{i=1}^G (t_{i}^3 - t_{i})}{N^3-N}$, where G is the number of groupings of different tied ranks, and ti is the number of tied values within group i that are tied at a particular value. This correction usually makes little difference in the value of K unless there are a large number of ties. + - # Finally, the p-value is approximated by $\Pr(\chi^2_{g-1} \ge K)$. If some ni's are small (i.e., less than 5) the probability distribution of K can be quite different from this [http://en.wikipedia.org/wiki/Chi-square Chi-square distribution]. + - The null hypothesis of equal population medians would then be rejected if $K \ge \chi^2_{\alpha: g-1}$. + Let $R(X_{ij})$ denote the rank assigned to $X_{ij}$ and let $R_i$ be the sum of ranks assigned to the $i^{th}$ sample. + + : $R_i = \sum_{j=1}^{n_i} {R(X_{ij})}, i = 1, 2, ... , k$. + + The SOCR program computes $R_i$ for each sample. The test statistic is defined for the following formulation of hypotheses: + + : $H_o$: All of the k population distribution functions are identical. + : $H_1$: At least one of the populations tends to yield larger observations than at least one of the other populations. + + Suppose {$X_{i,1}, X_{i,2}, \cdots, X_{i,n_i}$} represents the values of the $i^{th}$ sample, where $1\leq i\leq k$. + + : Test statistics: + :: T = $(1/{{S}^{2}}) (\sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-}{N {(N + 1)}^{2} }) / 4$, + where + ::${{S}^{2}} = \left( \left({1/ {N - 1}}\right) \right) \sum{{R(X_{ij})}^{2}} {-} {N {\left(N + 1)\right)}^{2} } ) / 4$. + + * Note: If there are no ties, then the test statistic is reduced to: + ::$T = \left(12 / N(N+1) \right) \sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-} 3 \left(N+1\right)$. + \end{center} + + However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation. + + Multiple comparisons have to be done here. For each pair of groups, the following is computed and printed at the '''Result''' Panel. + + $|R_{i} /n_{i} -R_{j} /n_{j} | > t_{1-\alpha /2} (S^{2^{} } (N-1-T)/(N-k))^{1/2_{} } /(1/n_{i} +1/n_{j} )^{1/2_{}}$. + + The SOCR computation employs the exact method instead of the approximate one (Conover 1980), since computation is easy and fast to implement and the exact method is somewhat more accurate. ===The Kruskal-Wallis Test using SOCR Analyses=== ===The Kruskal-Wallis Test using SOCR Analyses===
## General Advance-Placement (AP) Statistics Curriculum - Means of Several Independent Samples
In this section we extend the multi-sample inference which we discussed in the ANOVA section, to the situation where the ANOVA assumptions are invalid. Hence we use a non-parametric analysis to study differences in centrality between two or more populations.
### Motivational Example
Suppose four groups of students were randomly assigned to be taught with four different techniques, and their achievement test scores were recorded. Are the distributions of test scores the same, or do they differ in location? The data is presented in the table below.
Teaching Method Method 1 Method 2 Method 3 Method 4 Index 65 75 59 94 87 69 78 89 73 83 67 80 79 81 62 88
The small sample sizes, and the lack of information about the distribution of each of the four samples, imply that ANOVA may not be appropriate for analyzing these data.
## The Kruskal-Wallis Test
Kruskal-Wallis one-way analysis of variance by ranks is a non-parametric method for testing equality of two or more population medians. Intuitively, it is identical to a one-way analysis of variance with the raw data (observed measurements) replaced by their ranks.
Since it is a non-parametric method, the Kruskal-Wallis test does not assume a normal population, unlike the analogous one-way ANOVA. However, the test does assume identically-shaped distributions for all groups, except for any difference in their centers (e.g., medians).
## Calculations
Let N be the total number of observations, then $N = \sum_{i=1}^k {n_i}$.
Let R(Xij) denote the rank assigned to Xij and let Ri be the sum of ranks assigned to the ith sample.
$R_i = \sum_{j=1}^{n_i} {R(X_{ij})}, i = 1, 2, ... , k$.
The SOCR program computes Ri for each sample. The test statistic is defined for the following formulation of hypotheses:
Ho: All of the k population distribution functions are identical.
H1: At least one of the populations tends to yield larger observations than at least one of the other populations.
Suppose {$X_{i,1}, X_{i,2}, \cdots, X_{i,n_i}$} represents the values of the ith sample, where $1\leq i\leq k$.
Test statistics:
T = $(1/{{S}^{2}}) (\sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-}{N {(N + 1)}^{2} }) / 4$,
where
${{S}^{2}} = \left( \left({1/ {N - 1}}\right) \right) \sum{{R(X_{ij})}^{2}} {-} {N {\left(N + 1)\right)}^{2} } ) / 4$.
• Note: If there are no ties, then the test statistic is reduced to:
$T = \left(12 / N(N+1) \right) \sum_{i=1}^{k} {{R_i}^{2}} / {n_i} {-} 3 \left(N+1\right)$.
\end{center}
However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation.
Multiple comparisons have to be done here. For each pair of groups, the following is computed and printed at the Result Panel.
$|R_{i} /n_{i} -R_{j} /n_{j} | > t_{1-\alpha /2} (S^{2^{} } (N-1-T)/(N-k))^{1/2_{} } /(1/n_{i} +1/n_{j} )^{1/2_{}}$.
The SOCR computation employs the exact method instead of the approximate one (Conover 1980), since computation is easy and fast to implement and the exact method is somewhat more accurate.
### The Kruskal-Wallis Test using SOCR Analyses
It is much quicker to use SOCR Analyses to compute the statistical significance of this test. This SOCR KruskalWallis Test activity may also be helpful in understanding how to use this test in SOCR.
For the teaching-methods example above, we can easily compute the statistical significance of the differences between the group medians (centers):
Clearly, there are significant differences between the group medians, even after the multiple testing correction, all groups appear different from each other.
Group Method1 vs. Group Method2: 1.0 < 5.2056
Group Method1 vs. Group Method3: 4.0 < 5.2056
Group Method1 vs. Group Method4: 6.0 > 5.2056
Group Method2 vs. Group Method3: 5.0 < 5.2056
Group Method2 vs. Group Method4: 5.0 < 5.2056
Group Method3 vs. Group Method4: 10.0 > 5.2056
TBD
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2018-03-19 00:47:36
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https://www.shaalaa.com/question-bank-solutions/discuss-treatment-water-using-bleaching-powder-drinking-water-or-municipal-water_57867
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# Discuss the treatment of water using bleaching powder - Applied Chemistry 1
Short Note
Discuss the treatment of water using bleaching powder
#### Solution
In small water-works, about 1 kg of bleaching powder per 1,000 kilolitres of water is mixed and water is allowed to stand undisturbed for several hours the chemical action
produces hypochlorous acid
Ca0Cl_2 + H_2 0 → Ca(OH)_2 + Cl_2
Cl_2 + H_2 0 → HCl + HOCl
germs + HOCl → germs are killed
The disinfecting action of bleaching powder is due to the chlorine made available by it.
Drawbacks:
1. Bleaching powder introduces calcium in water, thereby making it more hard.
2. Bleaching powder deteriorates, due to its continuous decomposition during storage. So whenever it is added, it has to be analysed for its effective chlorine content.
3. Only a calculated quantity of bleaching powder should be used, since an excess of it gives a bad taste and smell to treated-water.
Is there an error in this question or solution?
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2021-03-08 20:01:02
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https://www.numpy.org.cn/en/reference/c-api/dtype.html
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# # Data Type API
The standard array can have 24 different data types (and has some support for adding your own types). These data types all have an enumerated type, an enumerated type-character, and a corresponding array scalar Python type object (placed in a hierarchy). There are also standard C typedefs to make it easier to manipulate elements of the given data type. For the numeric types, there are also bit-width equivalent C typedefs and named typenumbers that make it easier to select the precision desired.
Warning
The names for the types in c code follows c naming conventions more closely. The Python names for these types follow Python conventions. Thus, NPY_FLOAT picks up a 32-bit float in C, but numpy.float_ in Python corresponds to a 64-bit double. The bit-width names can be used in both Python and C for clarity.
## # Enumerated Types
• NPY_TYPES
There is a list of enumerated types defined providing the basic 24 data types plus some useful generic names. Whenever the code requires a type number, one of these enumerated types is requested. The types are all called NPY_{NAME}:
• NPY_BOOL
The enumeration value for the boolean type, stored as one byte. It may only be set to the values 0 and 1.
• NPY_BYTE
• NPY_INT8
The enumeration value for an 8-bit/1-byte signed integer.
• NPY_SHORT
• NPY_INT16 The enumeration value for a 16-bit/2-byte signed integer.
• NPY_INT
• NPY_INT32
The enumeration value for a 32-bit/4-byte signed integer.
• NPY_LONG
Equivalent to either NPY_INT or NPY_LONGLONG, depending on the platform.
• NPY_LONGLONG
• NPY_INT64
The enumeration value for a 64-bit/8-byte signed integer.
• NPY_UBYTE
• NPY_UINT8
The enumeration value for an 8-bit/1-byte unsigned integer.
• NPY_USHORT
• NPY_UINT16
The enumeration value for a 16-bit/2-byte unsigned integer.
• NPY_UINT
• NPY_UINT32
The enumeration value for a 32-bit/4-byte unsigned integer.
• NPY_ULONG
Equivalent to either NPY_UINT or NPY_ULONGLONG, depending on the platform.
• NPY_ULONGLONG
• NPY_UINT64
The enumeration value for a 64-bit/8-byte unsigned integer.
• NPY_HALF
• NPY_FLOAT16
The enumeration value for a 16-bit/2-byte IEEE 754-2008 compatible floating point type.
• NPY_FLOAT
• NPY_FLOAT32
The enumeration value for a 32-bit/4-byte IEEE 754 compatible floating point type.
• NPY_DOUBLE
• NPY_FLOAT64
The enumeration value for a 64-bit/8-byte IEEE 754 compatible floating point type.
• NPY_LONGDOUBLE
The enumeration value for a platform-specific floating point type which is at least as large as NPY_DOUBLE, but larger on many platforms.
• NPY_CFLOAT
• NPY_COMPLEX64
The enumeration value for a 64-bit/8-byte complex type made up of two NPY_FLOAT values.
• NPY_CDOUBLE
• NPY_COMPLEX128
The enumeration value for a 128-bit/16-byte complex type made up of two NPY_DOUBLE values.
• NPY_CLONGDOUBLE
The enumeration value for a platform-specific complex floating point type which is made up of two NPY_LONGDOUBLE values.
• NPY_DATETIME
The enumeration value for a data type which holds dates or datetimes with a precision based on selectable date or time units.
• NPY_TIMEDELTA
The enumeration value for a data type which holds lengths of times in integers of selectable date or time units.
• NPY_STRING
The enumeration value for ASCII strings of a selectable size. The strings have a fixed maximum size within a given array.
• NPY_UNICODE
The enumeration value for UCS4 strings of a selectable size. The strings have a fixed maximum size within a given array.
• NPY_OBJECT
The enumeration value for references to arbitrary Python objects.
• NPY_VOID
Primarily used to hold struct dtypes, but can contain arbitrary binary data.
Some useful aliases of the above types are
• NPY_INTP
The enumeration value for a signed integer type which is the same size as a (void *) pointer. This is the type used by all arrays of indices.
• NPY_UINTP
The enumeration value for an unsigned integer type which is the same size as a (void *) pointer.
• NPY_MASK
The enumeration value of the type used for masks, such as with the NPY_ITER_ARRAYMASK iterator flag. This is equivalent to NPY_UINT8.
• NPY_DEFAULT_TYPE
The default type to use when no dtype is explicitly specified, for example when calling np.zero(shape). This is equivalent to NPY_DOUBLE.
Other useful related constants are
• NPY_NTYPES
The total number of built-in NumPy types. The enumeration covers the range from 0 to NPY_NTYPES-1.
• NPY_NOTYPE
A signal value guaranteed not to be a valid type enumeration number.
• NPY_USERDEF
The start of type numbers used for Custom Data types.
The various character codes indicating certain types are also part of an enumerated list. References to type characters (should they be needed at all) should always use these enumerations. The form of them is NPY_{NAME}LTR where {NAME} can be
BOOL, BYTE, UBYTE, SHORT, USHORT, INT, UINT, LONG, ULONG, LONGLONG, ULONGLONG, HALF, FLOAT, DOUBLE, LONGDOUBLE, CFLOAT, CDOUBLE, CLONGDOUBLE, DATETIME, TIMEDELTA, OBJECT, STRING, VOID
INTP, UINTP
GENBOOL, SIGNED, UNSIGNED, FLOATING, COMPLEX
The latter group of {NAME}s corresponds to letters used in the array interface typestring specification.
## # Defines
### # Max and min values for integers
• NPY_MAX_INT{bits}
• NPY_MAX_UINT{bits}
• NPY_MIN_INT{bits}
These are defined for {bits} = 8, 16, 32, 64, 128, and 256 and provide the maximum (minimum) value of the corresponding (unsigned) integer type. Note: the actual integer type may not be available on all platforms (i.e. 128-bit and 256-bit integers are rare).
• NPY_MIN_{type} This is defined for {type} = BYTE, SHORT, INT, LONG, LONGLONG, INTP
• NPY_MAX_{type}
This is defined for all defined for {type} = BYTE, UBYTE, SHORT, USHORT, INT, UINT, LONG, ULONG, LONGLONG, ULONGLONG, INTP, UINTP
### # Number of bits in data types
All NPY_SIZEOF_{CTYPE} constants have corresponding NPY_BITSOF_{CTYPE} constants defined. The NPY_BITSOF_{CTYPE} constants provide the number of bits in the data type. Specifically, the available {CTYPE}s are
### # Bit-width references to enumerated typenums
All of the numeric data types (integer, floating point, and complex) have constants that are defined to be a specific enumerated type number. Exactly which enumerated type a bit-width type refers to is platform dependent. In particular, the constants available are PyArray_{NAME}{BITS} where {NAME} is INT, UINT, FLOAT, COMPLEX and {BITS} can be 8, 16, 32, 64, 80, 96, 128, 160, 192, 256, and 512. Obviously not all bit-widths are available on all platforms for all the kinds of numeric types. Commonly 8-, 16-, 32-, 64-bit integers; 32-, 64-bit floats; and 64-, 128-bit complex types are available.
### # Integer that can hold a pointer
The constants NPY_INTP and NPY_UINTP refer to an enumerated integer type that is large enough to hold a pointer on the platform. Index arrays should always be converted to NPY_INTP , because the dimension of the array is of type npy_intp.
## # C-type names
There are standard variable types for each of the numeric data types and the bool data type. Some of these are already available in the C-specification. You can create variables in extension code with these types.
### # (Un)Signed Integer
Unsigned versions of the integers can be defined by pre-pending a ‘u’ to the front of the integer name.
• npy_(u)byte
(unsigned) char
• npy_short
short
• npy_ushort
unsigned short
• npy_uint
unsigned int
• npy_int
int
• npy_int16
16-bit integer
• npy_uint16
16-bit unsigned integer
• npy_int32
32-bit integer
• npy_uint32
32-bit unsigned integer
• npy_int64
64-bit integer
• npy_uint64
64-bit unsigned integer
• npy_(u)long
(unsigned) long int
• npy_(u)longlong
(unsigned long long int)
• npy_intp
Py_intptr_t (an integer that is the size of a pointer on the platform).
• npy_uintp
unsigned Py_intptr_t (an integer that is the size of a pointer on the platform).
### # (Complex) Floating point
• npy_half
16-bit float
• npy_(c)float
32-bit float
• npy_(c)double
64-bit double
• npy_(c)longdouble
long double
complex types are structures with .real and .imag members (in that order).
### # Bit-width names
There are also typedefs for signed integers, unsigned integers, floating point, and complex floating point types of specific bit- widths. The available type names are
npy_int{bits}, npy_uint{bits}, npy_float{bits}, and npy_complex{bits}
where {bits} is the number of bits in the type and can be 8, 16, 32, 64, 128, and 256 for integer types; 16, 32 , 64, 80, 96, 128, and 256 for floating-point types; and 32, 64, 128, 160, 192, and 512 for complex-valued types. Which bit-widths are available is platform dependent. The bolded bit-widths are usually available on all platforms.
## # Printf Formatting
For help in printing, the following strings are defined as the correct format specifier in printf and related commands.
NPY_LONGLONG_FMT, NPY_ULONGLONG_FMT, NPY_INTP_FMT, NPY_UINTP_FMT, NPY_LONGDOUBLE_FMT
Last Updated: 8/28/2019, 6:17:52 PM
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2020-01-26 17:22:47
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http://ieeexplore.ieee.org/xpl/tocresult.jsp?reload=true&isnumber=4447269&punumber=7260
|
By Topic
IEEE Microwave and Wireless Components Letters
Filter Results
Displaying Results 1 - 25 of 29
Publication Year: 2008, Page(s): C1
| PDF (41 KB)
• IEEE Microwave and Wireless Components Letters publication information
Publication Year: 2008, Page(s): C2
| PDF (41 KB)
Publication Year: 2008, Page(s):73 - 75
Cited by: Papers (50)
| | PDF (238 KB) | HTML
A broadband eight-way spatial combiner using coaxial probes and radial waveguides has been proposed and designed. The simple electromagnetic modeling for the radial waveguide power divider/combiner has been developed using equivalent-circuit method. The measured 10-dB return loss and 1-dB insertion loss bandwidth of this waveguide spatial combiner are all demonstrated to be about 8 GHz. View full abstract»
• An Iterative Unconditionally Stable LOD–FDTD Method
Publication Year: 2008, Page(s):76 - 78
Cited by: Papers (21)
| | PDF (134 KB) | HTML
We present an iterative, unconditionally stable locally-one-dimensional (LOD) finite-difference time-domain (FDTD) method based on the use of an iterative fixed-point correction to reduce the splitting error. Numerical examples are used to illustrate the gain in accuracy of the proposed method versus the conventional LOD-FDTD method and the improved computational efficiency versus the iterative al... View full abstract»
• Wide Band Metallic Waveguide With In-Line Dielectric Rods
Publication Year: 2008, Page(s):79 - 81
Cited by: Papers (5)
| | PDF (158 KB) | HTML
Conventional rectangular metallic waveguides are seldom used at frequencies higher than twice the cutoff frequency because of higher mode propagation. Single-mode propagation is available for a metallic waveguide with arrayed dielectric rods at the center of the waveguide in the frequency under twice the cutoff frequency region using the TE20 mode, and in the frequency over twice the cu... View full abstract»
• A Patterned Dielectric Support Process for High Performance Passive Fabrication
Publication Year: 2008, Page(s):82 - 84
Cited by: Papers (2)
| | PDF (647 KB) | HTML
This letter presents a micromachining process to effectively reduce substrate loss via a structure of patterned oxide/nitride fins on a silicon substrate with a resistivity of 1 Omega-cm. Experimental results demonstrate that the insertion loss of a coplanar waveguide (CPW) deposited on the structure can be lowered to the value of 4.33 dB/cm at 40 GHz. Meanwhile, an analytical model is developed t... View full abstract»
• A Dual-Band Wilkinson Power Divider
Publication Year: 2008, Page(s):85 - 87
Cited by: Papers (94)
| | PDF (116 KB) | HTML
A new scheme is proposed for the dual-band operation of the Wilkinson power divider/combiner. The dual band operation is achieved by attaching two central transmission line stubs to the conventional Wilkinson divider. It has simple structure and is suitable for distributed circuit implementation. View full abstract»
• Miniature Dual-Band Filter Using Quarter Wavelength Stepped Impedance Resonators
Publication Year: 2008, Page(s):88 - 90
Cited by: Papers (44)
| | PDF (609 KB) | HTML
A miniature dual-band filter using quarter wavelength (lambdag/4) stepped impedance resonators (SIRs) is proposed. Short and open SIRs are coupled together to realize lower and upper passbands, respectively. Miniaturization is achieved due to the use of lambdag/4 resonators and a combline coupling structure. Two transmission zeros in a mid-stopband and one in each lower and u... View full abstract»
• Design of a Wide Stopband Microstrip Bandpass Filter With Asymmetric Resonators
Publication Year: 2008, Page(s):91 - 93
Cited by: Papers (7)
| | PDF (207 KB) | HTML
In this letter, a novel microstrip bandpass filter (BPF) with asymmetric resonators is presented. With the asymmetric structure and capacitively loaded coupling, a wide bandwidth with sufficient rejection level can be achieved easily. A full-wave electromagnetic simulator IE3D is used, and the prototype of the BPF is fabricated and measured. Comparisons of simulated results and experimental data a... View full abstract»
• A 3.3 mW K-Band 0.18-$mu$ m 1P6M CMOS Active Bandpass Filter Using Complementary Current-Reuse Pair
Publication Year: 2008, Page(s):94 - 96
Cited by: Papers (9)
| | PDF (450 KB) | HTML
This letter presents a low-power active bandpass filter (BPF) at K-band fabricated by the standard 0.18 mum 1P6M CMOS technology. The proposed filter is evolved from the conventional half-wavelength resonator filter, using the complementary-conducting-strip transmission line (CCS TL) as the half-wavelength resonator. Furthermore, the complementary MOS cross-couple pair is proposed as a form of cur... View full abstract»
• Miniaturized Dual-Mode Ring Resonator Bandpass Filter With Microstrip-to-CPW Broadside-Coupled Structure
Publication Year: 2008, Page(s):97 - 99
Cited by: Papers (7)
| | PDF (370 KB) | HTML
A microstrip-to-coplanar waveguide broadside-coupled section is used to replace a 34-section of the conventional dual-mode 1-ring resonator filter for circuit area miniaturization. It is believed that it is a new idea to incorporate the broadside-coupled section into a ring resonator filter for size reduction. For design purpose, some important characteristics of the broadside-coupled lines are in... View full abstract»
• Lifetime Measurements on a High-Reliability RF-MEMS Contact Switch
Publication Year: 2008, Page(s):100 - 102
Cited by: Papers (63) | Patents (1)
| | PDF (921 KB) | HTML
Radio frequency microelectromechanical systems (RF MEMS) cantilever contact switches have been tested for lifetime. The mean cycles-to-failure measured on an ensemble of switches was 430 billion switch cycles. The longest lifetime exhibited without degradation of the switch was 914 billion switch cycles. The devices were switched at 20 kHz with an incident RF frequency of 10 GHz and an incident RF... View full abstract»
• An Electronic UWB Continuously Tunable Time-Delay System With Nanosecond Delays
Publication Year: 2008, Page(s):103 - 105
Cited by: Papers (30)
| | PDF (379 KB) | HTML
We propose and demonstrate an electronic system achieving continuously tunable time-delays with nanosecond-scale delay excursions for ultra-wideband signals. Our demonstration system yields an adjustable delay of up to 1.6 ns for input signals spanning 3 to 7 GHz. The key component is a dispersive length of microstrip line created by etching a chirped electromagnetic bandgap structure in the condu... View full abstract»
• A Full-360$^{circ}$ Reflection-Type Phase Shifter With Constant Insertion Loss
Publication Year: 2008, Page(s):106 - 108
Cited by: Papers (14)
| | PDF (456 KB) | HTML
A new reflection-type phase shifter with a full 360deg relative phase shift range and constant insertion loss is presented. This feature is obtained by incorporating a new cascaded connection of varactors into the impedance-transforming quadrature coupler. The required reactance variation of a varactor can be reduced by controlling the impedance ratio of the quadrature coupler. The implemented pha... View full abstract»
• A Novel Miniature 1–22 GHz 90$^{circ}$ MMIC Phase Shifter with Microstrip Radial Stubs
Publication Year: 2008, Page(s):109 - 111
Cited by: Papers (6)
| | PDF (406 KB) | HTML
A novel miniature ultra wide bandwidth 90 monolithic microwave integrated circuit phase shifter with microstrip radial stubs operated from 1 to 22 GHz is presented. The phase shifter exhibits a high performance. Within the whole bandwidth from 1 to 22 GHz, the phase error of the phase shifter is less than 3deg, the return losses of the different phase shift states are more than 14 dB, the insertio... View full abstract»
• Compact CPW-MS-CPW Two-Stage pHEMT Amplifier Compatible With Flip Chip Technique in V-Band Frequencies
Publication Year: 2008, Page(s):112 - 114
Cited by: Papers (1)
| | PDF (582 KB) | HTML
The V-band coplanar waveguide (CPW)-microstrip line (MS)-CPW two-stage amplifier with the flip-chip bonding technique is demonstrated using 0.15 mum AlGaAs/InGaAs pseudomorphic high electron mobility transistor technology. The CPW is used at input and output ports for flip-chip assemblies and the MS transmission line is employed in the interstage to reduce chip size. This two-stage amplifier emplo... View full abstract»
• An Approximation of Volterra Series Using Delay Envelopes, Applied to Digital Predistortion of RF Power Amplifiers With Memory Effects
Publication Year: 2008, Page(s):115 - 117
Cited by: Papers (24)
| | PDF (246 KB) | HTML
In this letter, a new model for digital predistortion (DPD) of radio frequency power amplifiers for wide-band applications is proposed. The model is based on a spline approximation of Volterra series by considering second-order cross-terms. The advantage of the spline cross-term model is a reduction in the number of model parameters. We further reduce the model order by operating on delayed envelo... View full abstract»
• A Highly Linear Wideband CMOS Low-Noise Amplifier Based on Current Amplification for Digital TV Tuner Applications
Publication Year: 2008, Page(s):118 - 120
Cited by: Papers (16) | Patents (4)
| | PDF (219 KB) | HTML
A differential wideband low-noise amplifier (LNA) based on the current amplification scheme is presented for digital TV tuners. In order to highly improve the linearity and exploit the noise cancellation, a common-gate stage with positive current feedback is integrated in parallel with a common-source stage using the current mirror amplifier. The proposed 0.18-mum CMOS LNA exhibits a power gain of... View full abstract»
• 80-GHz Tuned Amplifier in Bulk CMOS
Publication Year: 2008, Page(s):121 - 123
Cited by: Papers (15)
| | PDF (410 KB) | HTML
An 80-GHz six-stage common source tuned amplifier has been demonstrated using low leakage (higher VT) NMOS transistors of a 65-nm digital CMOS process with six metal levels. It achieves power gain of 12 dB at 80 GHz with a 3-dB bandwidth of 6 GHz, noise figures (NF's) lower than 10.5 dB at frequencies between 75 and 81 GHz with the lowest NF of 9 dB. IP1 dB is -21 ... View full abstract»
• A 86 to 108 GHz Amplifier in 90 nm CMOS
Publication Year: 2008, Page(s):124 - 126
Cited by: Papers (36)
| | PDF (836 KB) | HTML
This letter presents a CMOS amplifier with 22 GHz 3-dB bandwidth ranging from 86 to 108 GHz. The amplifier is implemented in 90 nm mixed signal/radio frequency (RF) CMOS process using three-stage cascode RF NMOS configuration. It achieves a peak gain of 17.4 dB at 91 GHz from the measured results. To our knowledge, this is the highest frequency CMOS amplifier reported to date. View full abstract»
• 5.7 GHz Gilbert I/Q Downconverter Integrated With a Passive LO Quadrature Generator and an RF Marchand Balun
Publication Year: 2008, Page(s):127 - 129
Cited by: Papers (7)
| | PDF (659 KB) | HTML
A 5.7 GHz downconversion mixer is demonstrated in this letter using 0.35 mum SiGe BiCMOS technology. A quarter-wavelength coupled line and two center-tapped transformers are utilized to generate differential quadrature LO signals. A miniaturized Marchand balun is placed before the common-base-configured RF input stage of each Gilbert mixer to generate balanced RF signals. All the reactive passive ... View full abstract»
• A CMOS K-Band Quadrature Generator
Publication Year: 2008, Page(s):130 - 132
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This letter presents a K-band quadrature signal generator in a standard 0.13 mu m CMOS process. The quadrature generator operates from 18 to 21 GHz. A maximum output power of -3.7 dBm (per I or Q channel) is achieved, and the down converted signal suppression is >25 dB at the operating bandwidth. A measured sideband rejection ratio >30 dB is achieved from 19 to 21 GHz, with a peak of >40 ... View full abstract»
• A 12-GHz Fully Integrated Cascode CMOS $LC$ VCO With $Q$-Enhancement Circuit
Publication Year: 2008, Page(s):133 - 135
Cited by: Papers (28)
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A fully integrated complementary metal oxide semiconductor (CMOS) cascode LC voltage controlled oscillator (VCO) with Q-enhancement technique has been designed for high frequency and low phase noise. The symmetrical cascode architecture is implemented with negative conductance circuit for improving phase noise performance in 0.18 mum CMOS technology. The measured phase noise is -110.8 dBc/Hz at th... View full abstract»
• Low Phase-Noise and Low-Power CMOS VCO Constructed in Current-Reused Configuration
Publication Year: 2008, Page(s):136 - 138
Cited by: Papers (19) | Patents (1)
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A Ku-band CMOS voltage-controlled oscillator (VCO) constructed in a modified current-reused configuration is presented in this letter. Two dc level shifters combined into two metal-insulator-metal capacitors are adopted to solve the transconductance and load mismatch problems of the conventional current-reused VCO for obtaining more symmetrical oscillation signals and lowering the phase noise of o... View full abstract»
• Integrated VCO With Up/Down Converter for Si-Based 60 GHz WPAN Applications
Publication Year: 2008, Page(s):139 - 141
Cited by: Papers (1)
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This letter presents the design and implementation of the largest reported bandwidth of a 60 GHz up/down converter with an integrated voltage controlled oscillator (VCO) in a low-cost 0.18 mum silicon-germanium process. The up/down conversion is achieved using the 2X sub-harmonic passive mixing with anti-parallel diode pairs. A 30 GHz cross-coupled VCO is designed, optimized and integrated with th... View full abstract»
Aims & Scope
The IEEE Microwave and Wireless Components Letters (MWCL) publishes three page papers that focus on microwave theory, techniques and applications as they relate to components, devices, circuits, biological effects, and systems involving the generation, modulation, demodulation, control, transmission, and detection of microwave signals.
Full Aims & Scope
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2017-05-26 06:45:31
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https://www.physicsforums.com/threads/maginutude-and-phase-angle-for-nyquist-plots.362820/
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# Maginutude and Phase Angle for Nyquist Plots
1. Dec 12, 2009
### EugP
Magnitude and Phase Angle for Nyquist Plots
1. The problem statement, all variables and given/known data
The assignment is to draw a Nyquist Plot of a certain transfer function. The problem is that I can't figure out how they got the angle for the plot.
$G(s) = \frac{k(s+2)}{(s+1)(s-3)}$
2. Relevant equations
$|G(j\omega)| \angle G(j\omega)$
3. The attempt at a solution
$G(j\omega) = \frac{k(j\omega+2)}{(j\omega+1)(j\omega-3)}$
From that I know that the magnitude is found like this:
$|G(j\omega)| = \frac{k\sqrt{(\omega^2+4)}}{\sqrt{(\omega^2+1)}\sqrt{(\omega^2+9)}}$
Now, the solution says that the phase angle is:
$\angle G(j\omega) = \angle \tan ^{-1} (\frac{\omega}{2})-\tan ^{-1} (\frac{\omega}{1})-(180 - \tan ^{-1} (\frac{\omega}{3}))$
What I don't understand is why there is a $180^o$ shift in the last term.
Any help would be greatly appreciated.
Last edited: Dec 12, 2009
2. Dec 16, 2009
### AqA
If you consider (jw-3) alone, its y coordinate is 'w' and its x coordinate is '-3'.
Thus (jw-3) as a directed vector from the origin makes an angle of 180-tan-1(w/3) with the positive x axis.
The angle with the positive x axis is to be considered, not just tan-1(y/x).
Last edited: Dec 16, 2009
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2017-10-18 11:38:09
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https://www.gamedev.net/forums/topic/645882-why-doesnt-my-templated-function-call-the-correct-overload/
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# Why doesn't my templated function call the correct overload?
This topic is 1602 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I have this function:
template<typename Type>
typename std::enable_if<!(std::is_arithmetic<Type>::value || std::is_enum<Type>::value), size_t>::type
Serialize(Serializer&, Type&, size_t)
{
static_assert(false, "Serialize<TYPE>() hasn't been specialized for this type.");
return size_t();
}
template<typename ElementType>
size_t Serialize(Serializer &serializer, std::vector<ElementType> &container, size_t pos)
{
//...stuff...
}
I give it a std::vector<std::pair<std::string, int>> (for testing), and it chooses the first function.
Isn't C++ supposed to choose the more-specific function? I mean, this isn't a specialization, it's an overload. Aren't overloads supposed to be chosen over specializations or templates?
As a humorous aside, I had to walk through the debugger to find which templated function was getting called (out of about twenty), because in the main template declaration, I accidentally had static_assert(true, "") instead of static_assert(false, "").
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Is it really defined that the compiler can choose more specific, non-exact matching, overloads? I mean we know that the compiler will choose the exact match of an overload if there is one available but i don't know about non-exact matching, more specific overloads :p
Assuming that it does, it would probably complicate some things. If we have a simple template function for example like:
template< typename T >
T Sum( T &v1, T &v2 )
{
return v1 + v2;
}
double Sum( double &v1, double &v2 )
{
return v1 + v2;
}
If we call Sum passing in two float values, sure it can use the double version by implicitly converting float to double but may cause a bit of a performance hit. That's just a very simple example but if we have a much much more complicated templated function like yours and the compiler would choose something non-exact, but might still work, it may choose the wrong overload with a behavior that we may not want.
I'm just putting out my thoughts. I'm not exactly sure if this makes sense it's actually been a while since i messed around with complex template stuff.
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I don't understand the exact details, but you can find some of the information here.
I believe the compiler is allowed to make one implicit cast, and I think the double overload of your function would be called, because overloads are chosen over templated functions.
*goes to test*
Uh, no, the templated version was called. I guess the overload wasn't specific enough.
Still, my overload is more specific than the base template, even if not exact. No conversion is needed either. At the very least, it should be ambiguous, shouldn't it? But it's not, it's choosing the wrong one.
Anyone have an idea on how I could make my templated vector overload work?
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One problem is by using std::vector<T> as a parameter, prevents the overload in participating in template argument deduction. You need to remove std::. The compiler cannot do template argument deduction with namespace qualified types.
Another problem is that the type T& can be a better than vector<Type> & for matching const vectors, and rvalues. You say the type is "std::vector<std::pair<std::string, int>>" but it might not be in you actual code. Matching on T& is very greedy, so I would not recommend using it for overload checking like that.
For more help, you could post the actual code that instantiates the template. Edited by King Mir
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//Convience function to read or write a file. Meant to be used with "Serialize< MyFileType >" overloads.
template<typename FileFormat>
bool SerializeToFile(FileFormat &fileStruct, const std::string &filepath)
{
Serializer serializer;
size_t size = Serialize(serializer, fileStruct);
//...saves the serializer to file and returns true or false...
}
//Overloaded Serialize function for when 'pos' isn't specified.
//This is used for any template with the format: "Serialize(Serializer&, <TYPE>, size_t pos)"
template<typename Type>
size_t Serialize(Serializer &serializer, Type &type)
{
return Serialize<Type>(serializer, type, serializer.Position);
}
//The basic declaration, for non-builtin types.
template<typename Type>
typename std::enable_if<!(std::is_arithmetic<Type>::value || std::is_enum<Type>::value), size_t>::type
Serialize(Serializer&, Type&, size_t)
{
static_assert(false, "Serialize<TYPE>() hasn't been specialized for this type.");
return size_t();
}
//The serializer for vectors that I want to get called, but that isn't getting called.
template<typename ElementType>
size_t Serialize(Serializer &serializer, std::vector<ElementType> &container, size_t pos)
{
//Serialize the size of the container.
uint16_t numElements = static_cast<uint16_t>(container.size());
pos = Serialize<uint16_t>(serializer, numElements, pos);
//Serialize each element in the container.
container.resize(numElements);
for(size_t i = 0; i < numElements; i++)
{
pos = Serialize(serializer, container[i], pos);
}
return pos;
}
//...other Serialize<> functions, including an overload for std::pair<>, and including an overload for std::string, and an overload that catches basic integer types.
How I'm calling it (copy+pasted - it's just a test usage).
std::vector<std::pair<std::string,int> > ouputVector = {{"Three",3},{"Five",5},{"Seven",7},{"ThreeFiveSeven",357}};
SerializeToFile(ouputVector, "P:/Images/Blah/SerializeTest.dat"); //Actual filepath. =P
Edited by Servant of the Lord
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I believe this is what you are looking for: on template functions, specialization, and overloading rules Edited by ApochPiQ
The WYSIWYG editor is SHIT
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Problem solved!
I converted most my specialized functions to just overloads (which make more sense anyway, as the article ApochPiQ posted points out). I also commented out the 'catch all' template with the static_assert, at least for now.
However, my problematic function was already an overload anyway, so why wasn't it working?
The code I posted right before ApochPiQ's post has a very subtle template bug in it that I kept on missing.
In the second function posted looks like this:
template<typename Type>
size_t Serialize(Serializer &serializer, Type &type)
{
return Serialize<Type>(serializer, type, serializer.Position);
}
The function I was hoping for it to call, looks like this:
template<typename ElementType>
size_t Serialize(Serializer &serializer, std::vector<ElementType> &container, size_t pos)
{
//...stuff...
}
Very subtle, at least to me.
The first function's body calls a templated function, and demands the function takes a template argument of 'Type'.
return Serialize<Type>(serializer, type, serializer.Position);
^^^^^^
I was passing std::vector<blah> into the function, which means, that function is being called as:
Serialize< std::vector<blah> > (...)
Which tries to call the function I want:
template<typename ElementType>
size_t Serialize(Serializer &serializer, std::vector<ElementType> &container, size_t pos)
But tries it as:
template<>
size_t Serialize< std::vector<blah> >(Serializer &serializer, std::vector< std::vector<blah> > &container, size_t pos)
^ 1st vector ^ 2nd vector
And that isn't a good fit, so SFINAE kicks in and it tries the other functions instead.
My function itself was fine, but the function that called it told it that 'std::vector<blah>' needed to be the first template argument, instead of letting the function try and figure it out on its own.
Thank you for all the help, gentlemen! This bug was really throwing me for a loop.
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[quote name="Servant of the Lord" post="5080903" timestamp="1374898418"
And that isn't a good fit, so SFINAE kicks in and it tries the other functions instead.[/quote]
I can see that leaving it to type deduction will correctly match the one you want but why isn't the double nested vector a good fit nonetheless? To me it doesn't look like a substitution failure, unless I'm just not seeing it?
Or is it simply that the other function is a more specific overload because it doesn't involve substituting into template args?
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std::vector<int> can't be converted to std::vector<std::vector<int>>, at least in my test here.
But it does work, if I wrap the argument in brackets like: { singleVec } (because then I am explicitly initializing the double-vector, using the single vector as the first element of the double vector).
My code above in the earlier post doesn't have it wrapped in brackets, because I wasn't intending to do std::vector< std::vector< TYPE > >
That was an accidental result of my overly micro-managing of the function call.
[b][Edit - two hours layer, while refilling the ice-trays:][/b] The reason why it won't auto-convert, is because std::vector doesn't have a constructor that takes a single element as a parameter.
You can't do:
int myInt = 357;
std::vector<int> myVec = myInt;
You have to do:
int myInt = 357;
std::vector<int> myVec(N, myInt); //Where 'N' is the number of elements you want to copy-construct using 'myInt'.
Or:
int myInt = 357;
std::vector<int> myVec = {myInt}; //Using initializer-lists to construct myVec.
The std::vector class doesn't have a single-element parameter'd constructor.
Howsoever, even if it did, that'd only work if my Serialize() function took the argument by value or by const-reference (or by r-value reference).
But my Serialize() function was taking the argument by non-const reference (because it reads or writes, depending on the nature of the 'Serializer' class passed in), so no conversion can take place, otherwise you'd be referencing a temporary, which isn't allowed with non-const references. (Example)
Edited by Servant of the Lord
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2017-12-16 03:36:15
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## Rydberg & Bohr
H-Atom ($E_{n}=-\frac{hR}{n^{2}}$)
Kristen Kim 2K
Posts: 70
Joined: Fri Sep 28, 2018 12:16 am
### Rydberg & Bohr
Can someone explain the relationship between Rydberg's equation and Bohr frequency condition?
AustinGrove3B
Posts: 53
Joined: Fri Sep 28, 2018 12:23 am
### Re: Rydberg & Bohr
Ep = E2 - E1
where Ep is the energy of the photon, E2 is the energy of the excited electron, and E1 is the energy of the electron before excitation. In concert with our knowledge of E=hv, we can mathematically iterate this phenomenon known as Bohr's Frequency Condition that solves for the frequency of incident light:
hv = (E1 - E2)
v = (E1 - E2)/h
Note that it was experimentally determined that the energy En at a given energy level is equal to -hR/n^2, where h is Planck's constant and R is the Rydberg constant (3.29*10^15 s^-1). The convention is that an electron with no relation to the atom would be at an infinitely high energy level n and thus its energy would approximate 0. Therefore, electrons in orbit are said to have negative energy since their energy is lower than that of a freed electron. Thus we can extend upon Bohr's Frequency condition as follows:
v= (-hR/(n1)^2 - -hR/(n2)^2)/h [replace E with -hR/n^2]
v= -hR(1/(n1)^2 - 1/(n2)^2)/h [factor out -hR]
v=-R(1/(n1)^2 - 1/(n2)^2) [h cancels, we are left with the Rydberg eqn]
Andonios Karas 4H
Posts: 30
Joined: Fri Sep 28, 2018 12:27 am
### Re: Rydberg & Bohr
The Bohr Frequency Condition, $\nu = \frac{\Delta E}{h}$, states that the frequency is directly related to the energy of photons emitted as electrons move down energy levels. The higher the frequency, the greater the energy in each photon.
Rydberg's Equation, En = $\frac{-hR}{n^{2}}$, is used to calculate the energy of an electron in different energy levels of a Hydrogen atom.
We can use these equations in conjunction in order to understand the energy of photons emitted as an electron moves from higher energy level to a lower energy level in an atom.
Using the Rydberg Equation, we can calculate the energy of the electron in the initial and final energy level, and find the difference to give the energy of the photon or $\Delta$ E.
Using the Bohr Frequency Condition and the $\Delta$ E from the previous calculation, we can solve for the frequency of light emitted.
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2020-01-18 01:34:50
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https://www.physicsforums.com/threads/ideals-elements-r-n.633570/
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# Homework Help: Ideals, elements r^n
1. Sep 4, 2012
### EV33
1. The problem statement, all variables and given/known data
In the process of trying to prove something else I found it would be helpful if rn$\in$I, where I is an ideal, n$\in$N, and r$\in$R and R is a ring, then r is in I.
2. Relevant equations
I is an ideal if a$\in$I and b$\in$I then a+b$\in$I, a$\in$I and r$\in$R then ar$\in$I, and I is not the empty set.
3. The attempt at a solution
Base Case: Assume r1$\in$I. Then r$\in$I.
Inductive Case: Assume that if rn$\in$I then r$\in$I for all n<n+1.
Assume rn+1$\in$I. Since rn+1= rn*r then either rn or r is in I. We only need to show the first case works since the second is trivial. If rn$\in$I then r$\in$I by the inductive hypothesis. (QED)
Is this correct?
2. Sep 4, 2012
### Dick
That would certainly be true. r^n=r(r^(n-1)). If r is in I, then r^n is in I.
3. Sep 4, 2012
### EV33
By what you wrote makes me think you were thinking of the converse of my statement. Were you just showing that it, in fact, is an iff statement?
4. Sep 4, 2012
### Dick
Yes, I was thinking of the converse, sorry. Take the example of the ring of integers Z. 9Z is an ideal. Pick r=3 and n=2. What do you say now?
Last edited: Sep 5, 2012
5. Sep 5, 2012
### EV33
Ok. So I get 32=9=0 mod 9. So 32 is an element of 9z. By definition 3 is an element of 9z. So I get that it holds for this case. That's what I was supposed to get right?
6. Sep 5, 2012
### Dick
You were supposed to get that it is a counterexample. 9Z is the ideal of integers that are divisible by 9. 3 is NOT in 9Z. 3^2 is in 9Z. So?
I think what you are missing is that the definition of ideal does NOT say that if ab is in I, then a or b is in I. And it doesn't follow from the definition either.
7. Sep 5, 2012
### Hurkyl
Staff Emeritus
This is the definition of a "radical ideal".
A prime ideal is one where rs in I implies r in I or s in I. Your proof:
only works for prime ideals. It is true that every prime ideal is radical.
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2018-09-22 02:59:48
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https://www.physicsforums.com/threads/electric-field-on-two-non-parallel-plates.634445/
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# Electric Field on two non-parallel plates
1. Sep 8, 2012
### Seydlitz
Why the electric field is in fact curved in that configuration? I'm just a little bit confused because normally if the plates were arranged parallel the electric field were perfectly straight, the electric field of infinite plane is also straight. Why then suddenly when the plates are angled the electric field became curved?
On what principle could I infer this from without calculating anything?
Thank You
Edit: By non-parallel in my case, the plates are arranged forming a V.
Last edited: Sep 8, 2012
2. Sep 9, 2012
### DarioC
The lines that represent equal field strength have to be perpendicular to the plates at the point of intersection? Is that true? If so, why is it so? Perhaps that is the key to answering your question.
DC
3. Sep 9, 2012
### Seydlitz
I'm sorry but I don't think I understand your statement. Could you rephrase it in someway, perhaps in reference to familiar examples like electric dipole or point charges?
Edit: What I know already is the electric field line must always be perpendicular to the surface of electric potential, and hence conductor but I can't get my mind to relate this concept to the non-parallel plates. Moreover, it seems the electric field cannot be solved analytically also.
---
Ahhh!
I think I know what you mean. Because of that reasoning the electric field must emanate perpendicularly to the plate. Hence, it would be impossible for it to have a straight field like if they were to be arranged in parallel. In the end the electric field will be curved. Does my crude prove by contradiction reasoning correct? :D
Last edited: Sep 9, 2012
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2018-09-21 07:31:58
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https://adotb.xyz/category/uncategorized/
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# Behind the curtain
A brief glimpse “behind the curtain”: the setup I use during zoom video classes!
The lockdown/circuit breaker sure has brought its shares of challenges in teaching and learning. Lessons are certainly slightly less effective without the physical interactive component: I can’t just look over students’ shoulder and check on their progress, and it’s definitely been harder to gauge how my explanations are faring without the feedback from non-verbal cues. Not to mention the occasional lag and technical difficulties.
# Still climbing onto the shoulders of giants
It was year 2005/2006: I was taking further maths in JC under Mr Wee when he showed some video clips about linear algebra (what we were learning about at that time) to us during lecture. In addition to the math content I also remembered how he told us about this initiative by MIT, the OpenCourseWare, where some course content (including video lectures) from the university were being uploaded and made available for free to the public. Thinking back, this was pretty remarkable (youtube was still in its infancy). I recall at that time, finding the experience pretty interesting (MIT is nerd heaven after all, so there’s the brand name recognition. Plus the video lecture was challenging but engaging: something I’d sure hope to experience when I get to university), but that’s about it. After all, who has time during JC do much more on top of academic and social life!
# Q9b discussion
### The integration part
We have $\displaystyle \int \frac{1}{(1-x^2)(1+x^2)} \; \mathrm{d}x = t + C$. Since the denominator is so complicated, partial fractions is the way to go.
$$\frac{1}{(1-x)(1+x)(1+x^2)} = \frac{A}{1-x} + \frac{B}{1+x} + \frac{Cx+D}{1+x^2}.$$
Give it a go!
# Youtube videos
I’ve been hard at work producing Youtube videos recently: check out my channel here:
My youtube channel
# New year, new notes
With every topic summarized into one page, we hope that these set of revision notes can help you in your learning. Hopefully it will be a really handy reference when you practice and work on questions.
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2022-05-22 15:00:24
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http://clay6.com/qa/4487/a-binary-operation-on-the-set-0-1-2-3-4-5-is-defined-as-f-x-left-a-b-quad-i
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Browse Questions
A binary operation * on the set (0,1,2,3,4,5} is defined as : $f(x) = \left\{ \begin{array}{l l}a+b, & \quad if { a+b < 6} \\ a+b-6, & \quad if { a+b \geq 6} \end{array} \right.$ Show that zero is the identity for this operation and each element 'a' of the set is invertible with 6-a, being the inverse of 'a'.
Toolbox:
• An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
• The element $a \in X$ is invertible if there exist $b \in X$ such that $a*b=e=b*a$
Step 1:
Given the set $X=\{0,1,2,3,4,5\}$ where the binary operation $\ast$ is defined by $a * b= \left\{ \begin{array}{1 1} a+b & \quad if\;a+b < 6\\ a+b-6 & \quad if a+b \geq 6 \end{array} \right.$
An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
To check if zero is the identity, we see that $a*0=a+0=a \qquad for\;a \in x$ and also $0*a=0+a=a \qquad for \;a \in x$
Given $a \in X, \qquad a+0 < 6\;$ and also $\;0+a < 6$
$\Rightarrow 0$ is the identify element for the given given operation
Step 2:
The element $a \in X$ is invertible if there exist $b \in X$ such that $a*b=e=b*a$
In this case, $e=0 \rightarrow a*b=0=b*a$.
$\Rightarrow a*b = \left\{ \begin{array}{1 1} a+b=0=b+a & \quad if\;a+b < 6\\ a+b-6=0=b+a-6 & \quad a+b \geq 6 \end{array} \right.$
ie $a=-b \;or\; b=6-a$
but since $a,b \in X=\{0,1,2,3,4,5\}$, $\;a \neq -b$
Hence $b=6-a\;$ is the inverse of $a$, i.e., $a^{-1}=6-a, \;\forall a \in \{1,2,3,4,5\}$
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2017-02-28 08:31:18
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https://socratic.org/questions/what-is-the-distance-between-3-1-1-and-2-0-1
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# What is the distance between (3, –1, 1) and (–2, 0, 1) ?
Feb 19, 2016
$\sqrt{26}$
#### Explanation:
Pythagorean Theorem (3D version)
$\sqrt{{\left(3 - \left(- 2\right)\right)}^{2} + {\left(- 1 - 0\right)}^{2} + {\left(1 - 1\right)}^{2}} = \sqrt{26}$
Feb 19, 2016
$\sqrt{26} \approx 5.099$
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2} + {\left(\Delta Z\right)}^{2}}$
color(white)("XXXx")=sqrt((3-(-2))^2+(-1-0)^2+(1-1)^2))
$\textcolor{w h i t e}{\text{XXXx}} = \sqrt{25 + 1 + 0} = \sqrt{26} \approx 5.099$
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2019-07-18 10:53:13
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https://www.physicsforums.com/threads/series-converge.172585/
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Series converge?
1. Jun 2, 2007
chesshaha
1. The problem statement, all variables and given/known data
series converge?
DL the img, sry, can't type it.
http://www.geocities.com/chessobeyer/math.bmp
what's the sum if converge?
2. Relevant equations
none.
3. The attempt at a solution
alternating series, but it's (-2)
Attached Files:
• math.bmp
File size:
84.2 KB
Views:
76
Last edited: Jun 2, 2007
2. Jun 2, 2007
quasar987
Are you saying it converges, or diverge?
3. Jun 2, 2007
chesshaha
I don't know where to start, I didn't know what test to use?
Can u give me a hint of which test to use?
4. Jun 2, 2007
quasar987
Based on what you said, I assume that you already understood that the series can be rewritten as
$$\sum_{n=1}^{\infty}2(-1)^{n}$$
??
I don't see which test can be used on this. But you can fall back on the very definition of convergence: A series converge if the sequence of the partial sums converge. But if you find two subsequences that converge to different values, then the sequence itself diverges. Can you find those subsequences?
5. Jun 2, 2007
chesshaha
thx, that helps alot, so the alternating series?
Can u factor out the 2? r u sure that can be done?i don't understand
6. Jun 2, 2007
chesshaha
I got the series diverge, because the 2 cancels and it left with $$\sum_{n=1}^{\infty}(-1)^{n}$$
is it right?
thx for ur help
7. Jun 2, 2007
quasar987
Ah, there is a test you can use! And it's the easiest. If $\lim_{n\rightarrow \infty} a_n\neq 0$, then the series $\sum a_n$ diverges.
8. Jun 2, 2007
chesshaha
Thanks for your help. But I still don't understand how can did u rewrite the series... to this $$\sum_{n=1}^{\infty}2(-1)^{n}$$
Sorry, I am a bit of slow, Please Explain more, Thank You!
9. Jun 2, 2007
quasar987
Ok, here is how.
$$\frac{(-2)^{n+1}}{2^n}=\frac{(-2)(-2)^n}{2^n}=(-2)\left(\frac{(-2)}{2}\right)^n=(-2)(-1)^n=2(-1)^{n+1}$$
Thus,
$$\sum_{n=0}^{\infty}\frac{(-2)^{n+1}}{2^n}=\sum_{n=0}^{\infty}2(-1)^{n+1}=\sum_{n=1}^{\infty}2(-1)^{n}$$
10. Jun 3, 2007
ice109
catching the bump up in the index is tricky. notice that the series now starts at n=1.
11. Jun 3, 2007
chesshaha
Thank you very much, this helps alot.
So the series converge, the sum is either 0 or -2, depends if it's even or odd, right?
12. Jun 3, 2007
Dick
A series cannot converge to two limits. That sort of behavior is called 'divergent'.
13. Jun 3, 2007
chesshaha
o yea, thanks.
14. Jun 4, 2007
Gib Z
Ahh I think a more appropriate word would have been is oscillating =]
15. Jun 4, 2007
Dick
Right. And the sum of i from i=0 to infinity could more accurately be said to 'increase without bound' rather than diverge.
16. Jun 4, 2007
Gib Z
Taking out the factor of 2, partial sums gives : -1, 0, -1,0,-1.... So it is bounded...
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2017-06-29 11:03:35
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https://datascience.stackexchange.com/questions/29527/which-type-auto-encoder-gives-best-results-for-text
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# Which type auto encoder gives best results for text
I did I couple of examples for auto encoders for images and they worked fine. Now I want to do an auto encoder for text that takes as input a sentence and returns the same sentence. But when I try to use the same auto encoders as the ones I used for the images I get bad results.
I guess the reason for this is that my text is sparse and I have a big vocabulary size of 500K words.
1. Do you have a link of a working example of an auto encoder for text in Keras?
2. I saw that in most papers they use cross-entropy as a loss function. How does cross-entropy calculate the loss exactly ? Does it make sense to use cross-entropy even if I do a character by character auto encoder?
• Hi so I tried the code you suggested. I can generate text but I don't understand how the autoencoder works. What I want to do is to give some text as input and have the same text as output. But in this case when I do: pred = vae.predict(test, batch_size=500) pred contains only 1s. So it doesn't make sense. Am I doing something wrong ? – sspp Mar 29 '18 at 19:07
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2020-03-28 21:55:30
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https://www.physicsforums.com/threads/differential-equation-spring-mass-system-of-driven-motion-with-damping.519875/
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# Differential equation: Spring/Mass system of driven motion with damping
1. ### TeenieBopper
29
1. The problem statement, all variables and given/known data
A 32 pound weight stretches a spring 2 feet. The mass is then released from an initial position of 1 foot below the equilibrium position. The surrounding medium offers a damping force of 8 times the instantaneous velocity. Find the equation of motion if the mass is driven by an external force of 2cos(5t).
2. Relevant equations
F=kx
m=W/g
m$\frac{d^{2}x}{dt^{2}}$+$\beta$$\frac{dx}{dt}$+kx=f(t)
3. The attempt at a solution
I found that k=16$\frac{lb}{ft}$ and m=1 slug. This gets me the following equation:
$\frac{d^{2}x}{dt^{2}}$+$\beta$$\frac{dx}{dt}$+16x=2cos(5t)
I'm at a loss for how to determine $\beta$, which is the damping force of 8 times the instantaneous velocity. I don't know how to determine instantaneous velocity. I know that once I have $\beta$, I can just use a LaPlace transform to find x(t). But $\beta$ is my stumbling block right now.
As I was writing this, it occured to me that $\frac{dx}{dt}$=instantaneous velocity and that would make $\beta$=8. That in turn makes the problem very easy to solve. Am I correct in this thinking?
We kind of rushed through this application in class the other day. Thanks in advance for any help you're able to provide.
2. ### rock.freak667
6,219
Yeah I would agree that β=8 here. Just make sure your initial conditions are correct with the proper signs.
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook
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2015-02-27 21:09:31
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http://mathhelpforum.com/advanced-algebra/110356-help-proving-norm-matrixes.html
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# Math Help - Help with proving norm matrixes
1. ## Help with proving norm matrixes
i need a help proving norm matrices for ||X|| infinity =< ||X||2
and for ||X||2=< (Sqrt(n))||X||infinity
where X is R^n
im new with norms and not really sure how to use them. any help/comments would be appreciated
thank you
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2015-09-01 20:20:27
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http://tex.stackexchange.com/questions/94134/placing-two-nodes-by-relative-positioning-in-tikz/94136
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# Placing two nodes by relative positioning in TikZ
I want to draw something like this:
So I did the following:
\begin{tikzpicture}
\node[draw] (node1) {Long long long box};
\node[draw, below left=of node1] {short} edge[->] (node1);
\node[draw, below right=of node1] {box} edge[->] (node1);
\end{tikzpicture}
Then the result is:
I want to move the two short boxes just under the long box and within the length of the long box. I also like the two arrows straight vertical. How can I do this? I want to use relative positioning.
-
Use appropiate anchors and reference points for positioning the nodes. Use intersection coordinate system to draw the lines.
This is the MWE:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document}
\begin{tikzpicture}
\node[draw] (node1) {Long long long box};
\node[draw, below=of node1.west, anchor=west] (this node) {short}
edge[->] (node1.south-|this node);
\node[draw, below=of node1.east, anchor=east] (this node) {box}
edge[->] (node1.south-|this node);
\end{tikzpicture}
\end{document}
Result:
-
Any idea, how to place the short and box nodes above the long box? – user4514 Jun 29 at 15:40
@user4514 Do you mean, changing below by above in the code? – JLDiaz Jun 29 at 20:32
Yes, stupid question. Thanks! – user4514 Jun 30 at 20:52
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2015-07-31 15:43:15
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http://activelydirect.blogspot.com/2011/03/
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## Wednesday, March 30, 2011
### Creating Dynamic User Objects in Active Directory
Account management in Active Directory comes at a cost of time and effort for an Administrator. This is especially burdensome for objects that are only needed for a limited timeframe such as testing objects. Active Directory provides a low cost method for dealing with temporary objects through the dynamic object class which provides automatic garbage collection based on a time-to-live value in seconds. This benefit does come with several limitations:
• The maximum TTL of a dynamic object is 31,556,926 seconds (1 year).
• A deleted dynamic object due to its TTL expiring does not leave a tombstone behind.
• All DCs holding replicas of dynamic objects must run on Windows Server 2003 or greater.
• Dynamic entries with TTL values are supported in all partitions except the Configuration partition and Schema partition.
• Active Directory Domain Services do not publish the optional dynamicSubtrees attribute, as described in the RFC 2589, in the root DSE object.
• Dynamic entries are handled similar to non-dynamic entries when processing search, compare, add, delete, modify, and modifyDN operations.
• There is no way to change a static entry into a dynamic entry and vice-versa.
• A non-dynamic entry cannot be added subordinate to a dynamic entry.
In the code sample below, I take the advantage of the dynamic object class to create 50 test accounts that will expire and delete themselves from the forest on May 1, 2011 at 2:30 pm. The accounts created for a hypothetical finance application testing are placed in a specific OU, provided a unique sAMAccountName (tested to ensure this) and a pseudo-random password created for them. All this information is placed in a comma separated values text file so you can pass it over to your Quality Assurance team.
And remember, this script creates objects in Active Directory! Use at your own risk! I might have the best of intentions but my skill may betray you. Test, test and further test before implementing this code in a production environment.
Function Get-LocalDomainController($objectDomain) { return ([System.DirectoryServices.ActiveDirectory.ActiveDirectorySite]::GetComputerSite()).Servers | Where-Object {$_.Domain.Name -eq $objectDomain } | ForEach-Object {$_.Name } | Select-Object -first 1
}
Function Get-ObjectADDomain($distinguishedName) { return ((($distinguishedName -replace "(.*?)DC=(.*)",'$2') -replace "DC=","") -replace ",",".") } Function Get-ActiveDirectoryObject($distinguishedName) {
return [ADSI]("LDAP://" + (Get-LocalDomainController (Get-ObjectADDomain $distinguishedName)) + "/" + ($distinguishedName -replace "/","\/"))
}
Function Get-DomainDistinguishedName($domain) { return ("dc=" +$domain.Replace(".",",dc="))
}
Function Test-sAMAccountName($sAMAccountName,$domain) {
$objectConnection = New-Object -comObject "ADODB.Connection"$objectConnection.Open("Provider=ADsDSOObject;")
$objectCommand = New-Object -comObject "ADODB.Command"$objectCommand.ActiveConnection = $objectConnection$ldapBase = ("LDAP://" + (Get-LocalDomainController $domain) + "/" + (Get-DomainDistinguishedName$domain))
$ldapAttr = "sAMAccountName"$ldapScope = "subtree"
$ldapFilter = "(&(objectClass=user)(sAMAccountName=$sAMAccountName))"
$ldapQuery= "<$ldapBase>;$ldapFilter;$ldapAttr;$ldapScope"$objectCommand.CommandText = $ldapQuery$objectRecordSet = $objectCommand.Execute() if($objectRecordSet.RecordCount -gt 0) {
$found =$true
} else {
$found =$false
}
return $found } #--------------------------------------------------------------------------------------------------# Set-Variable -name destinationOu -option Constant -value "ou=Finance Testing,dc=americas,dc=ad,dc=mycompany,dc=local" Set-Variable -name totalNumberOfAccounts -option Constant -value 50 Set-Variable -name givenName -option Constant -value "Test" Set-Variable -name sn -option Constant -value "Account" Set-Variable -name passwordSize -option Constant -value 12 #--------------------------------------------------------------------------------------------------#$randomNumber = New-Object System.Random
$accounts = @()$endDate = "05/01/2011 14:30:00"
$entryTtlSeconds = [int]((Get-Date$endDate) - (Get-Date)).TotalSeconds
#######################################################################
# Or you can do a time span instead for the account time to live i.e.,
# $entryTtlSeconds = [int](New-TimeSpan -days 90).TotalSeconds #$entryTtlSeconds = [int](New-TimeSpan -minutes 60).TotalSeconds
#######################################################################
if($entryTtlSeconds -gt 31556926) { Write-Host "Time-to-live greater than 1 year. Exiting!" -foregroundColor Red exit }$destinationOuObject = Get-ActiveDirectoryObject $destinationOu if($destinationOuObject.distinguishedName -ne $destinationOu) { Write-Host "Unable to connect to$destinationOu. Exiting!" -foregroundColor Red
exit
}
for($i = 1;$i -le $totalNumberOfAccounts;$i++) {
$accountName = ($givenName + $sn + "$i")
Write-Host "Creating $accountName..." if(Test-sAMAccountName$accountName (Get-ObjectADDomain $destinationOu)) { Write-Host "$accountName already exists, skipping..." -foregroundColor Yellow
continue
}
$userObject =$destinationOuObject.Create("user","CN=$accountName")$userObject.PutEx(2,"objectClass",@("dynamicObject","user"))
$userObject.Put("entryTTL",$entryTtlSeconds)
$userObject.Put("sAMAccountName",$accountName) # Mandatory attribute
$userObject.Put("givenName",$givenName)
$userObject.Put("sn",($sn + "$i"))$userObject.Put("displayName",($givenName + " " +$sn + "$i"))$userObject.Put("description","Account Used for Application Testing")
$userObject.Put("wWWHomePage","http://sharepointsite/projects/newfinancesystem")$userObject.Put("userPrincipalName",($accountName + "@" + (Get-ObjectADDomain$destinationOu)))
$userObject.SetInfo()$password = ""
for($x = 1;$x -le $passwordSize;$x++) {
$password += [char]($randomNumber.Next(33,126))
}
$userObject.SetPassword($password)
$userObject.Put("userAccountControl", 512)$userObject.SetInfo()
$account = New-Object -typeName PSObject Add-Member -inputObject$account -type NoteProperty -name "domain" -value (Get-ObjectADDomain $destinationOu).Split(".")[0] Add-Member -inputObject$account -type NoteProperty -name "sAMAccountName" -value ($userObject.sAMAccountName).ToString() Add-Member -inputObject$account -type NoteProperty -name "givenName" -value ($userObject.givenName).ToString() Add-Member -inputObject$account -type NoteProperty -name "sn" -value ($userObject.sn).ToString() Add-Member -inputObject$account -type NoteProperty -name "userPrincipalName" -value ($userObject.userPrincipalName).ToString() Add-Member -inputObject$account -type NoteProperty -name "password" -value $password$accounts += $account }$accounts | Export-Csv -path "$givenName$sn List.csv" -noTypeInformation
## Tuesday, March 29, 2011
### Retrieve the Fully Qualified Domain Name of a Local Active Directory Bound Computer
Here is the method I use to obtain the fully qualified domain name of a local Active Directory bound computer in my scripts.
$computerFqdn = (([System.Net.NetworkInformation.IPGlobalProperties]::GetIPGlobalProperties()).HostName + "." + ([System.Net.NetworkInformation.IPGlobalProperties]::GetIPGlobalProperties()).DomainName) ## Tuesday, March 22, 2011 ### Scheduling Tasks in PowerShell for Windows Server 2008 -- Part 3: Daily Tasks In my previous blog posts, "Scheduling Tasks in PowerShell for Windows Server 2008 -- Part 1: Monthly Task" and "Scheduling Tasks in PowerShell for Windows Server 2008 -- Part 2: Weekly Tasks", I explored the various settings required for scheduled tasks for those given date ranges. This post deals with the daily reoccurring tasks that administrators need to schedule, sometimes more than once a day. In the code sample below, I schedule a fictitious log cleanup PowerShell script to run three times a day; midnight, 8 am and 4 pm. For security reasons, after running this script, you should close the PowerShell command window as the password submitted for the Task Security Principal will remain in the command buffer allowing anyone with access to your keyboard to "up arrow" and reveal it. # Parameters to modify$taskName = "Start-TriDailyLogCleanup"
$taskWorkingDirectory = "C:\PowerShell"$taskPath = "%SystemRoot%\system32\WindowsPowerShell\v1.0\powershell.exe"
$taskArguments = "-command "$taskWorkingDirectory\$taskName.ps1""$taskAuthor = "ad\myaccount"
$taskDescription = "The Tri-Daily Log Cleanup on the servers."$taskSecurityPrincipal = "ad\maintenance"
$taskShedulerTaskFolder = "\MyTasks"$startTimes = @((Get-Date "03/23/2011 00:00:00" -Format s),(Get-Date "03/23/2011 08:00:00" -Format s),(Get-Date "03/23/2011 16:00:00" -Format s))
# Would like to use -asSecureString but RegisterTaskDefinition does not accept it
# Look over your shoulder before typing
$password = Read-Host -prompt "$taskSecurityPrincipal Password"
# The meaty parts
$taskService = New-Object -ComObject Schedule.Service$taskService.Connect()
$rootFolder =$taskService.GetFolder($taskShedulerTaskFolder)$taskDefinition = $taskService.NewTask(0)$registrationInformation = $taskDefinition.RegistrationInfo$registrationInformation.Description = $taskDescription$registrationInformation.Author = $taskAuthor$taskPrincipal = $taskDefinition.Principal$taskPrincipal.LogonType = 1
$taskPrincipal.UserID =$taskSecurityPrincipal
$taskPrincipal.RunLevel = 0$taskSettings = $taskDefinition.Settings$taskSettings.StartWhenAvailable = $true$taskSettings.RunOnlyIfNetworkAvailable = $true$taskSettings.Priority = 7
$taskSettings.ExecutionTimeLimit = "PT2H"$taskTriggers = $taskDefinition.Triggers foreach($startTime in $startTimes) {$executionTrigger = $taskTriggers.Create(2)$executionTrigger.StartBoundary = $startTime }$taskAction = $taskDefinition.Actions.Create(0)$taskAction.Path = $taskPath$taskAction.Arguments = $taskArguments$taskAction.WorkingDirectory = $taskWorkingDirectory # 6 == Task Create or Update # 1 == Password must be supplied at registration$rootFolder.RegisterTaskDefinition($taskName,$taskDefinition, 6, $taskSecurityPrincipal,$password, 1)
# Since we captured this in plain text I am going to nuke the value
# Not 100% security. Close the PowerShell command window to increase security.
## Monday, March 21, 2011
### MD5 Hash For PowerShell
For a project, I had a requirement to move selected files between two servers and provide proof that the copied matched exactly the original file. A common method to provide this proof is through a MD5 hash for each of the files to create a fingerprint. A MD5 hash provides the ability to compare two files without having to go to the trouble and slowness of comparing the files byte by byte using the algorithm created by Ron Rivest in 1991 to create a 128-bit hash. You can save the hash and use it for future reference to see if the file has changed.
In the code sample below, I use the System.Security.Cryptography.MD5CryptoServiceProvider class to create the hash in bytes then convert it to a string using the System.Bitconverter class and removing all dashes to create a 32 character representation of that hash. If you require even more bits to be present in your hash, the System.Security.Cryptography namespace provides up to 512 bits. To achieve this level of security, use System.Security.Cryptography.SHA512Managed class instead of System.Security.Cryptography.MD5CryptoServiceProvider in the function.
To test out this code, change out the file names below. Copy a file to a new location, run the function and see the results. Then open the copied file, make a minor edit and test again. It's interesting to see the changes in the returned hash.
Function Get-MD5Hash($fileName) { if([System.IO.File]::Exists($fileName)) {
$fileStream = New-Object System.IO.FileStream($fileName,[System.IO.FileMode]::Open,[System.IO.FileAccess]::Read,[System.IO.FileShare]::ReadWrite)
$MD5Hash = New-Object System.Security.Cryptography.MD5CryptoServiceProvider [byte[]]$fileByteChecksum = $MD5Hash.ComputeHash($fileStream)
$fileChecksum = ([System.Bitconverter]::ToString($fileByteChecksum)).Replace("-","")
$fileStream.Close() } else {$fileChecksum = "ERROR: $fileName Not Found" } return$fileChecksum
}
$fileName = "\\east-coast-fs.ad.mycompany.local\documents\Important Excel Spreadsheet.xlsx"$fileChecksumOne = Get-MD5Hash $fileName if($fileChecksumOne -match "^ERROR:") {
Write-Host $fileChecksumOne -foregroundColor Red exit }$fileName = "\\west-coast-fs.ad.mycompany.local\documents\Important Excel Spreadsheet.xlsx"
$fileChecksumTwo = Get-MD5Hash$fileName
if($fileChecksumTwo -match "^ERROR:") { Write-Host$fileChecksumTwo -foregroundColor Red
exit
}
if($fileChecksumOne -eq$fileChecksumTwo) {
Write-Host "Files match!"
Write-Host $fileChecksumOne -foregroundColor Green Write-Host$fileChecksumTwo -foregroundColor Green
} else {
Write-Host "Files do not match!"
Write-Host $fileChecksumOne -foregroundColor Red Write-Host$fileChecksumTwo -foregroundColor Green
}
## Thursday, March 17, 2011
### Add a New Mail Domain to Multiple Users
In my work, it is not uncommon to have new domains purchased for business units and employees immediately needing e-mail addresses with the new domain. So after the MX record is created and Exchange is updated to accept mail for the new domain, its time to target the users that require the new mail domain to be included in the proxyAddresses attribute. The proxyAddresses attribute is multivalued so you must use treat it as an array and use PutEx to append to the attribute with the new address. In the code sample below, I take a file of e-mail addresses that users already use, split off the "local-part" from the domain using the at sign as the delimiter and append the $newMailDomain constant to the local-part to create the new e-mail address for the user. When inserting the new e-mail address into the proxyAddresses attribute with PutEx, you will need to send it as an array even if it only has 1 element. To be on the safe side, which you should also do when modifying production data, a backup of all the current elements of the user's proxyAddresses attribute are dumped into a subdirectory called "backup" in a file name that matches their mail attribute (with ".txt" appended) and the script is set to not write to the directory by default. You must alter the$writeEnabled constant in order to actually update the user's object. The script also checks to see if the e-mail address you are trying to assign to the employee's account is already present and skips if so. I am using Write-Output to document the scripts behavior so you can easily pipe it out to a text file for review.
This script does not make the new address the employee's primary SMTP address. In order to do this, you would need to remove the current primary denoted by the "SMTP:" at the start of the proxyAddress and return it back with "smtp:" then add the new e-mail address with "SMTP:" at the start. I will provide a code sample in the future that will detail that swap out and how to do it safely.
And remember, this script modifies data! Use at your own risk! I might have the best of intentions but my skill may betray you. Test, test and further test before implementing this code in a production environment.
Function Get-LocalDomainController($objectDomain) { return ([System.DirectoryServices.ActiveDirectory.ActiveDirectorySite]::GetComputerSite()).Servers | Where-Object {$_.Domain.Name -eq $objectDomain } | ForEach-Object {$_.Name } | Select-Object -first 1
}
Function Get-ObjectADDomain($distinguishedName) { return ((($distinguishedName -replace "(.*?)DC=(.*)",'$2') -replace "DC=","") -replace ",",".") } Function Get-ActiveDirectoryObject($distinguishedName) {
return [ADSI]("LDAP://" + (Get-LocalDomainController (Get-ObjectADDomain $distinguishedName)) + "/" + ($distinguishedName -replace "/","\/"))
}
#--------------------------------------------------------------------------------------------------#
Set-Variable -name forestRootDn -option Constant -value ([ADSI]("LDAP://" + (([System.DirectoryServices.ActiveDirectory.Forest]::GetCurrentForest()).name) + "/rootDSE")).defaultNamingContext
Set-Variable -name adsPropertyAppend -option Constant -value 3
Set-Variable -name newMailDomain -option Constant -value "@newdomain.local"
Set-Variable -name inputFile -option Constant -value "userlist.txt"
Set-Variable -name writeEnabled -option Constant -value $false #--------------------------------------------------------------------------------------------------# if(Test-Path -path$inputFile) {
$userList = Get-Content -path$inputFile
} else {
Write-Host "Could not locate $inputFile. Exiting..." -foregroundColor Red exit } if(!(Test-Path -path "backup")) { New-Item -path "backup" -type directory | Out-Null }$objectConnection = New-Object -comObject "ADODB.Connection"
$objectCommand = New-Object -comObject "ADODB.Command"$objectConnection.Open("Provider=ADsDSOObject;")
$objectCommand.ActiveConnection =$objectConnection
foreach($user in$userList) {
$ldapBase = "GC://$forestRootDn"
$ldapAttr = "distinguishedName"$ldapScope = "subtree"
$ldapFilter = "(&(objectClass=user)(proxyAddresses=smtp:$user))"
$ldapQuery = "<$ldapBase>;$ldapFilter;$ldapAttr;$ldapScope"$objectCommand.CommandText = $ldapQuery$objectRecordSet = $objectCommand.Execute() if(!$objectRecordSet.EOF) {
while(!$objectRecordSet.EOF) {$userObject = Get-ActiveDirectoryObject $objectRecordSet.Fields.Item('distinguishedName').Value$newEmailAddress = ("smtp:" + ($user.Split("@")[0]).ToLower() +$newMailDomain)
Write-Output ($userObject.displayName).ToString() Write-Output "New Address:$newEmailAddress"
$notFound =$true
Set-Content -path ("backup\" + ($userObject.mail).ToString() + ".txt") -value$userObject.proxyAddresses
foreach($proxyAddress in$userObject.proxyAddresses) {
if($proxyAddress -eq$newEmailAddress) {
$notFound =$false
}
}
if($notFound -eq$true) {
Write-Output "Adding $newEmailAddress" if($writeEnabled -eq $true) {$userObject.PutEx($adsPropertyAppend, "proxyAddresses", @($newEmailAddress))
$userObject.SetInfo() } } else { Write-Output "Already has$newEmailAddress"
}
$objectRecordSet.MoveNext() } } else { Write-Output "Could not locate$user in the forest."
}
Write-Output ("-" * 50)
}
## Monday, March 14, 2011
### Unix Tail-like Functionality in PowerShell Revisited
My first attempt to replicate tail for PowerShell, which I wrote about in "Unix Tail-like Functionality in PowerShell", was horribly inefficient once you got past a couple dozen lines. This makes since given the method I was using -- a byte by byte reverse read of a text file, converting each byte at a time to ASCII. I knew the solution was to "wolf down" large byte chunks and process them as a whole. Using System.Text.ASCIIEncoding.GetString, I am doing that just after reading into memory multiple bytes using System.IO.FileStream.Read. With this change in methodology, I am getting to within 3% of the speed of tail in UNIX in my tests. The largest test I've performed was returning 1,000,000 lines from a 850MB log file. A Mac OS X 10.6.6 workstation performed the task in 16 seconds using tail and a Windows Server 2003 server returned in 17 seconds using this method. Good enough for me. Most of my needs are in the thousands of lines which I am able to return in hundreds of milliseconds which is perfect my monitoring scripts in Nagios. Compared to my previous attempt, this is a Lockheed SR-71 vs. a Wright Brothers Flyer. A small 5,000 tail using the old code took 5 1/2 minutes to return while this code returned the same lines in 200 milliseconds. Huge difference!
In the code sample below, I am using 10 kilobytes for that chunking. I found that number suited most of my needs. However, you can greatly increase that number for large number of lines to be returned (I used 4MB for my million line test). You can also do a little automatic tuning by altering the number of bytes using the number of lines you are seeking. One thing to be aware when passing files to this code, if you pass a file to System.IO.File/FileStream without a full path, it will not assume the file is located in the path of the executed script so Test-Path is not a valid test. Using System.IO.Directory.GetCurrentDirectory, you can find this by running the following in PowerShell:
[System.IO.Directory]::GetCurrentDirectory()
More than likely, it will point to the home directory of the profile the shell is executed under.
Also be aware that this tail-like function does not handle unicode log files. The method I am using to decode the bytes is ASCII dependent. I am not using System.Text.UnicodeEncoding yet in the code. Currently ASCII meets all my needs for reading log files but I am still interested in adding compatibility to this function. I am also assuming that all log files denote the end of a line using carriage return & line feed (CHR 13 + CHR 10) which is how majority of text files are written in Windows. UNIX & old style Macintosh text files will not work properly with this code. You will need to modify line 23 to change the delimiter for the split for those text file formats.
UPDATE: I have now finished an update that provides the "tail -f" functionality for continuously reading the updates to a text file. Read about it in my blog post, Replicating UNIX "tail -f" in PowerShell.
UPDATE: I have updated the code to handle unicode text files and non-Windows new lines. You can review the code here.
Function Read-EndOfFileByByteChunk($fileName,$totalNumberOfLines,$byteChunk) { if($totalNumberOfLines -lt 1) { $totalNumberOfLines = 1 } if($byteChunk -le 0) { $byteChunk = 10240 }$linesOfText = New-Object System.Collections.ArrayList
if([System.IO.File]::Exists($fileName)) {$fileStream = New-Object System.IO.FileStream($fileName,[System.IO.FileMode]::Open,[System.IO.FileAccess]::Read,[System.IO.FileShare]::ReadWrite)$asciiEncoding = New-Object System.Text.ASCIIEncoding
$fileSize =$fileStream.Length
$byteOffset =$byteChunk
[byte[]] $bytesRead = New-Object byte[]$byteChunk
$totalBytesProcessed = 0$lastReadAttempt = $false do { if($byteOffset -ge $fileSize) {$byteChunk = $fileSize -$totalBytesProcessed
[byte[]] $bytesRead = New-Object byte[]$byteChunk
$byteOffset =$fileSize
$lastReadAttempt =$true
}
$fileStream.Seek((-$byteOffset), [System.IO.SeekOrigin]::End) | Out-Null
$fileStream.Read($bytesRead, 0, $byteChunk) | Out-Null$chunkOfText = New-Object System.Collections.ArrayList
$chunkOfText.AddRange(([System.Text.RegularExpressions.Regex]::Split($asciiEncoding.GetString($bytesRead),"\r\n")))$firstLineLength = ($chunkOfText[0].Length)$byteOffset = ($byteOffset +$byteChunk) - ($firstLineLength) if($lastReadAttempt -eq $false -and$chunkOfText.count -lt $totalNumberOfLines) {$chunkOfText.RemoveAt(0)
}
$totalBytesProcessed += ($byteChunk - $firstLineLength)$linesOfText.InsertRange(0, $chunkOfText) } while($totalNumberOfLines -ge $linesOfText.count -and$lastReadAttempt -eq $false -and$totalBytesProcessed -lt $fileSize)$fileStream.Close()
if($linesOfText.count -gt 1) {$linesOfText.RemoveAt($linesOfText.count-1) }$deltaLines = ($linesOfText.count -$totalNumberOfLines)
if($deltaLines -gt 0) {$linesOfText.RemoveRange(0, $deltaLines) } } else {$linesOfText.Add("[ERROR] $fileName not found") | Out-Null } return$linesOfText
}
#--------------------------------------------------------------------------------------------------#
$fileName = "C:\Logs\really-huge.log" # Your really big log file$numberOfLines = 100 # Number of lines from the end of the really big log file to return
$byteChunk = 10240 # Size of bytes read per seek during the search for lines to return #################################################################################################### ## This is a possible self-tuning method you can use but will blow up memory on an enormous ## number of lines to return ##$byteChunk = $numberOfLines * 256 ####################################################################################################$lastLines = @()
$lastLines = Read-EndOfFileByByteChunk$fileName $numberOfLines$byteChunk
foreach($lineOfText in$lastLines) {
Write-Output $lineOfText } ## Thursday, March 10, 2011 ### Write Excel Spreadsheets Fast in PowerShell If you try to populate an Excel spreadsheet cell by cell using PowerShell expect it to take a long time. As I showed in a previous post, "Speed Up Reading Excel Files in PowerShell", when dealing with Excel work with comma separated value text files. The population of the cells appears to be the huge bottleneck in Excel permformace but if you let Excel load a comma separated values file, you can quickly format and manipulate the prepopulated cells. One example from my own work involves a spreadsheet with ~3,500 rows. If I populate the rows directly through Excel it can take upwards of five minutes to complete the process. If I dump the data to a comma separated values text file then import it into Excel, it takes about 2 seconds -- a huge time savings. When you are running ad hoc reports, it pays off. If you want an excuse to head down to Starbucks from your office, it doesn't. To demonstrate this technique, I will take a previous post, "BlackBerry User Report", and change the report from a comma separated values text file to Excel spreadsheet. First to make the final result look nice, the NoteProperty names are cleaned up so they make attractive headers in the spreadsheet. Swap out the following lines to achieve this. Add-Member -inputObject$blackberryUser -type NoteProperty -name "Domain" -value $domain Add-Member -inputObject$blackberryUser -type NoteProperty -name "User ID" -value $sAMAccountName Add-Member -inputObject$blackberryUser -type NoteProperty -name "First Name" -value $firstName Add-Member -inputObject$blackberryUser -type NoteProperty -name "Last Name" -value $firstName Add-Member -inputObject$blackberryUser -type NoteProperty -name "Display Name" -value $displayName Add-Member -inputObject$blackberryUser -type NoteProperty -name "E-Mail Address" -value $proxyAddress Add-Member -inputObject$blackberryUser -type NoteProperty -name "PIN" -value $pin Add-Member -inputObject$blackberryUser -type NoteProperty -name "Cell Phone Number" -value $phoneNumber Add-Member -inputObject$blackberryUser -type NoteProperty -name "Desk Phone Number" -value $telephoneNumber Add-Member -inputObject$blackberryUser -type NoteProperty -name "Street Address" -value $streetAddress Add-Member -inputObject$blackberryUser -type NoteProperty -name "City" -value $city Add-Member -inputObject$blackberryUser -type NoteProperty -name "State" -value $state Add-Member -inputObject$blackberryUser -type NoteProperty -name "Zip Code" -value $zipCode Add-Member -inputObject$blackberryUser -type NoteProperty -name "Country" -value $country Add-Member -inputObject$blackberryUser -type NoteProperty -name "BlackBerry Model" -value $modelName Add-Member -inputObject$blackberryUser -type NoteProperty -name "Carrier" -value $homeNetwork Add-Member -inputObject$blackberryUser -type NoteProperty -name "IMEI" -value $imei Add-Member -inputObject$blackberryUser -type NoteProperty -name "Password Enabled" -value $passwordEnabled Add-Member -inputObject$blackberryUser -type NoteProperty -name "Exchange Server" -value $exchangeServer Add-Member -inputObject$blackberryUser -type NoteProperty -name "BlackBerry Server" -value $blackberryServer Now that you have nice and pretty column headers in your NoteProperty, its time to write out that Excel spreadsheet. This code assumes that you have at least Excel 2007 installed on your scripting workstation/server. First the code writes out to the temp directory using a GUID (just to be fancy) for the comma separated values file's name. We create an Excel object and load that temporary file. At that point, we can save out a fairly boring Excel spreadsheet. That won't impress anyone. If you want to look "pro" when you distribute reports, having a good layout is key. You want to save out a nicely formatted spreadsheet. So you autofit all the columns to longest cell in each, populate the worksheet's name, update some of the metadata in the spreadsheet and then apply a table style. Table styles are a quick way to add color to your spreadsheet with minimal effort by creating a list object. I have found that if you want to apply your own custom style (a future blog post) there is no speed penalty as long as the data is prepopulated. And finally we perform a Save As, remind Excel that we are saved, quit Excel and do some garbage cleanup. If you just quit Excel programatically, you will find that Excel will still be in memory. $blackberryUsers = $blackberryUsers | Sort-Object "BlackBerry Server", "Last Name", "First Name"$excelFile = ("\\fileserver.ad.mydomain.local\it_reports\blackberry\" + (Get-Date -format yyyyMMdd) + "-BlackBerry User Report.xlsx")
$temporaryCsvFile = ($env:temp + "\" + ([System.Guid]::NewGuid()).ToString() + ".csv")
$blackberryUsers | Export-Csv -path$temporaryCsvFile -noTypeInformation
if(Test-Path -path $excelFile) { Remove-Item -path$excelFile }
$excelObject = New-Object -comObject Excel.Application$excelObject.Visible = $false$workbookObject = $excelObject.Workbooks.Open($temporaryCsvFile)
$workbookObject.Title = ("BlackBerry User List for " + (Get-Date -Format D))$workbookObject.Author = "Robert M. Toups, Jr."
$worksheetObject =$workbookObject.Worksheets.Item(1)
$worksheetObject.UsedRange.Columns.Autofit() | Out-Null$worksheetObject.Name = "BlackBerry Users"
$listObject =$worksheetObject.ListObjects.Add([Microsoft.Office.Interop.Excel.XlListObjectSourceType]::xlSrcRange, $worksheetObject.UsedRange,$null,[Microsoft.Office.Interop.Excel.XlYesNoGuess]::xlYes,$null)$listObject.Name = "User Table"
$listObject.TableStyle = "TableStyleMedium4" # Style Cheat Sheet in French/English: http://msdn.microsoft.com/fr-fr/library/documentformat.openxml.spreadsheet.tablestyle.aspx$workbookObject.SaveAs($excelFile,51) # http://msdn.microsoft.com/en-us/library/bb241279.aspx$workbookObject.Saved = $true$workbookObject.Close()
[System.Runtime.Interopservices.Marshal]::ReleaseComObject($workbookObject) | Out-Null$excelObject.Quit()
[System.Runtime.Interopservices.Marshal]::ReleaseComObject($excelObject) | Out-Null [System.GC]::Collect() [System.GC]::WaitForPendingFinalizers() if(Test-Path -path$temporaryCsvFile) { Remove-Item -path $temporaryCsvFile } ## Tuesday, March 8, 2011 ### Speed Up Reading Excel Files in PowerShell One of the most frustrating aspects of PowerShell for me is the extremely sluggish performance the ComObject is for Excel. I deal with spreadsheets on a daily basis and have processes to read them and compare information. I have spent hours upon hours trying to figure out how to optimize the process of reading cells. One day, I said to myself, "Why can't I read Excel documents at near the speed I can read a text file?" (in more colorful language than this). The proverbial lightbulb went off above my head. That one question lead to the code below. If I can't read Excel files at the speed I want and I can with text files, why not convert the Excel spreadsheet to a text based format? A comma separated value text file! Using PowerShell to load an Excel spreadsheet is quick. The slowdown is reading the cells/rows. So the solution is to load the Excel spreadsheet and immediately save it in the temp directory as a .csv file. PowerShell has a wonderful commandlet called Import-Csv that will allow you to read in a comma separated value text file as PSObject and associate the first row as a NoteProperty for each column in the spreadsheet. As long as your Excel spreadsheet is straight forward, this works perfectly and blazing fast. I have some processes that are now 20 times faster using this method. Function Remove-File($fileName) {
if(Test-Path -path $fileName) { Remove-Item -path$fileName }
}
#--------------------------------------------------------------------------------------------------#
$excelFile = "\\server.ad.mydomain.local\excelreports\myreport.xlsx" if(Test-Path -path$excelFile) {
$csvFile = ($env:temp + "\" + ((Get-Item -path $excelFile).name).Replace(((Get-Item -path$excelFile).extension),".csv"))
Remove-File $csvFile$excelObject = New-Object -ComObject Excel.Application
$excelObject.Visible =$false
$workbookObject =$excelObject.Workbooks.Open($excelFile)$workbookObject.SaveAs($csvFile,6) # http://msdn.microsoft.com/en-us/library/bb241279.aspx$workbookObject.Saved = $true$workbookObject.Close()
[System.Runtime.Interopservices.Marshal]::ReleaseComObject($workbookObject) | Out-Null$excelObject.Quit()
[System.Runtime.Interopservices.Marshal]::ReleaseComObject($excelObject) | Out-Null [System.GC]::Collect() [System.GC]::WaitForPendingFinalizers()$spreadsheetDataObject = Import-Csv -path $csvFile # Use the$spreadsheetDataObject for your analysis
Remove-File $csvFile } ## Monday, March 7, 2011 ### Unix Tail-like Functionality in PowerShell A common tool I use in shell scripts on Unix/Linux/Mac OS X servers is tail. While there are command-line tail conversions for Windows, I need something I can integrate into a script for reading the end of large log files, search for information and act on that result. I don't want to distribute third party software along with the script to accomplish the task. Get-Content and Select-Object are not suitable for large files. After researching the capabilities of File IO in .Net, I found that System.IO.FileStream class had just what I needed. Using this class, I read the target text file byte by byte from the end of the file until I reach a selected number of lines of text delimited by a carriage return. The amount of time it takes to obtain the data is related to the number of characters per line. It works very well in 500 lines or less in my typical log files (I tested up to 1 gigabyte) and much faster than using: Get-Content "C:\Logs\really-huge.log" | Select-Object -last 100 The code meets 95% of my needs but I am sure I can optimize it so it comes close to matching the speed of tail from the Unix distributions I commonly use. It's my first stab at tackling the problem. One interesting part of the code is that I use System.Collections.ArrayList instead of a standard PowerShell array. The reason is since I am reading the file in reverse, I need to return the data back in the proper order. The ArrayList object allows me to insert into the first element so I don't have to re-write the array in the right order after collecting the data. Also I noticed that using System.Convert to covert the bytes to a character instead of using PowerShell's native [char] was faster. Returning large number of lines, it was significant -- about .5 seconds per 100 lines. I will keep working on this to close that 5% and update this post with a link to an updated blog post in the future with the improvements. UPDATE: I have rewritten this function in a new blog post and it is lightning fast. This code is deprecated and should only be used for amusement purposes. Function Read-EndOfFile($fileName,$totalNumberOfLines) {$fileStream = New-Object System.IO.FileStream($fileName,[System.IO.FileMode]::Open,[System.IO.FileAccess]::Read,[System.IO.FileShare]::ReadWrite)$linesOfText = New-Object System.Collections.ArrayList
$byteOffset = 1$lineOfText = ""
do {
$fileStream.Seek((-$byteOffset), [System.IO.SeekOrigin]::End) | Out-Null
$byte =$fileStream.ReadByte()
if($byte -eq 13) { } elseif($byte -eq 10) {
$linesOfText.Insert(0,$lineOfText)
$lineOfText = "" } else {$lineOfText = [System.Convert]::ToChar($byte) +$lineOfText
}
$byteOffset++ } while ($linesOfText.count -le $totalNumberOfLines)$fileStream.Close()
return $linesOfText } #--------------------------------------------------------------------------------------------------#$fileName = "C:\Logs\really-huge.log" # Your really big log file
$numberOfLines = 100 # Number of lines from the end of the really big log file to return if([System.IO.File]::Exists($fileName) -and $numberOfLines -gt 0) {$lastLines = Read-EndOfFile $fileName$numberOfLines
foreach($lineOfText in$lastLines) {
Write-Output $lineOfText } } ### SharePoint 2010 and Social Collaboration Below is a webinar that I attended hosted by Gig Werks focused on the Social Collaboration features in SharePoint 2010. I found it to be a good introduction to the built-in capabilities of the product. ## Thursday, March 3, 2011 ### Quick Organizational Chart from Active Directory Building on my blog post, Export Manager's Nested Direct Reports, I quickly threw together a revision of the code that will create visual (ASCII at least) organizational chart based on the directReports attribute of an employee in Active Directory. The nesting method I am using could easy be converted to simulate the "tree" DOS command using Get-ChildObject on file systems or about any programming challenge that deals with nested objects. The one main fault of the code, is that it is not pretty when it gets to the last nested object of a parent. It leaves the "|" in front of the last nested subordinate object. A little more work and I could clean this presentation issue, but not today. Function Get-DirectReports($distinguishedName,$level) {$managerObject = Get-ActiveDirectoryObject $distinguishedName if($managerObject.directReports.count -gt 0) {
foreach($directReport in$managerObject.directReports) {
$directReportObject = Get-ActiveDirectoryObject$directReport
Write-Output (("| " * $level) + "|") Write-Output (("| " *$level) + "+-" + ($directReportObject.givenName).ToString() + " " + ($directReportObject.sn).ToString())
Get-DirectReports $directReport ($level + 1)
}
}
}
Function Get-LocalDomainController($objectDomain) { return ([System.DirectoryServices.ActiveDirectory.ActiveDirectorySite]::GetComputerSite()).Servers | Where-Object {$_.Domain.Name -eq $objectDomain } | ForEach-Object {$_.Name } | Select-Object -first 1
}
Function Get-ObjectADDomain($distinguishedName) { return ((($distinguishedName -replace "(.*?)DC=(.*)",'$2') -replace "DC=","") -replace ",",".") } Function Get-ActiveDirectoryObject($distinguishedName) {
return [ADSI]("LDAP://" + (Get-LocalDomainController (Get-ObjectADDomain $distinguishedName)) + "/" + ($distinguishedName -replace "/","\/"))
}
#--------------------------------------------------------------------------------------------------#
Set-Variable -name forestRootDn -option Constant -value ([ADSI]("LDAP://" + (([System.DirectoryServices.ActiveDirectory.Forest]::GetCurrentForest()).name) + "/rootDSE")).defaultNamingContext
#--------------------------------------------------------------------------------------------------#
$objectConnection = New-Object -comObject "ADODB.Connection"$objectCommand = New-Object -comObject "ADODB.Command"
$objectConnection.Open("Provider=ADsDSOObject;")$objectCommand.ActiveConnection = $objectConnection$manager = "the.ceo@mycompany.local"
$ldapBase = "GC://$forestRootDn"
$ldapAttr = "distinguishedName"$ldapScope = "subtree"
$ldapFilter = "(&(objectClass=user)(proxyAddresses=smtp:$manager))"
$ldapQuery = "<$ldapBase>;$ldapFilter;$ldapAttr;$ldapScope"$objectCommand.CommandText = $ldapQuery$objectRecordSet = $objectCommand.Execute() while(!$objectRecordSet.EOF) {
$firstLevelObject = Get-ActiveDirectoryObject$objectRecordSet.Fields.Item('distinguishedName').Value
$topLevelManager = (($firstLevelObject.givenName).ToString() + " " + ($firstLevelObject.sn).ToString()) Write-Output$topLevelManager
Get-DirectReports $firstLevelObject.distinguishedName 0$objectRecordSet.MoveNext()
}
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2017-09-23 01:53:19
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