text:
Eigenvalue analysis and conve... - LIBRIS

Solaris FZU gGmbH Jugend- und Umweltwerkstaetten

We shall present elements of the linear solvability theory, and then go on to the latest development: Integration by parts formulas, that are useful  6min - This video goes over three examples, covering the proper way to find definite integrals that require the application of the integration by parts formula. evaluate integrals such as. ∫ b a arctan(x)dx. Theorem (Integration by Parts). If f and g are continuous, then. ∫ fg = fg −.

stewart calculus et 5e 0534393217;7. techniques of integration; integration by parts then by equation udv=uv"! vdu ln dx/x ln xdx= ln ln let u=ln dv=xdx xln xdx. The notes of the course by Vlad Bally, co-authoredwith Lucia Caramellino, develop integration by parts formulas in an abstractsetting, extending Malliavin's work  The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's  Integration By parts. Trig Graphs And equations. Reciprocal Trigonometric functions. Trig addition formulae · Trig Ratios C 5 2 A M 4.

Related Symbolab blog posts. My Notebook, the Symbolab way. Math notebooks have been around for hundreds of years.

INTEGRAL THEORY - Avhandlingar.se

The rule for differentiating the product of two differentiable functions leads to the integration by parts formula. Let f (x) and g (x) are differentiable functions, then  This can be rearranged to give the Integration by Parts Formula : uv dx = uv − u v dx. Strategy : when trying to integrate a product, assign the name u to  Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step. Integration By Parts formula is used to find the integrals by reducing them into standard forms. Learn how to derive this formula and also get solved examples  22 Jun 2006 We obtain the integration by parts formula for the regional fractional Laplacian which are generators of symmetric α-stable processes on a  Well let's see what happens when we apply the formula without that constant. The integration by parts formula says that the integral of x e to the x dx is u x times  Integration by parts is used to integrate the product of two functions. Simple (x2 sinhx)dx.

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We try to see our integrand as and then we have. Many calc books mention the LIATE, ILATE, or DETAIL rule of thumb here. These are supposed to be memory devices to help you choose your “u” and “dv” in an integration by parts que students the standard derivation of the Integration by Parts formula as presented in [1]:.

start with the product rule:. this is the integration by parts formula.
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The plan is therefore to rewrite 1+i in  Research on partial differential equations and numerical methods. ensuring that the operators satisfy a discrete analogue of integration-by-parts known as… Flyktingmottagning och integration - PowerPoint PPT Presentation parts. start with the product rule:. this is the integration by parts formula. l. dU = -5 sin 5θ dθ, V = 1. 4 e4θ to get.

Integrating Refugees in the Swedish Labor Market: The

Integration by parts is like the reverse of the product formula: (uv) = u v + uv combined with the fundamental theorem of calculus . But in the limit, this is the integration by parts formula. Thanks to Terry Moore for fixing the formatting! 90 views  Partialintegration eller partiell integration är ett sätt att analytiskt lösa helt eller delvis baserad på material från engelskspråkiga Wikipedia, Integration by parts. The integration by parts formula will convert this integral, which you can't do directly, into a simple product minus an integral you'll know how to  So we just used the product rule to derive this formula for integration by parts, and in a lot of calculus books they do this u and v and dvd. Så använt vi bara  derivative of the other, we integrate by parts. Integration By PartsWhen an integral is a product of two functions and neither is thederivative of the other, we integrate by parts.

The key thing in integration by parts is to choose $$u$$ and $$dv$$ correctly. The acronym ILATE is good for picking $$u.$$ ILATE stands for The Integration by Parts formula may be stated as: $$\int uv' = uv - \int u'v.$$ I wonder if anyone has a clever mnemonic for the above formula. What I often do is to derive it from the Product Rule (for differentiation), but this isn't very efficient.


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# Notation for unevenly-spaced to evenly-spaced time series conversion

I have an unevenly-spaced time series. To make it evenly-spaced, I resample the time series to a larger timespan (e.g. day) and sum up all values within this time frame.

In Python (and pandas) it is a one-liner:

ts.resample('D').sum()

But what is a precise (and ideally easily unterstandable) mathematical notation for that?

• It all depends on exactly how "resample" works! There can't be a mathematical notation for a vaguely defined operation. – whuber Jul 12 '19 at 15:51
• Thanks for your comment. resample('MS').sum() sums up all values within a calendar month. Independent from the timespan (one month), I would like to express exactly this. Probably my Python example is not helpful? – Michael Dorner Jul 12 '19 at 15:57
• It's helpful insofar as it uncovers potential complications. For instance, a monthly series is not actually evenly spaced in terms of physical time: it is only evenly spaced relative to a social construct concerning how we represent time on a human scale. This suggests that something well beyond a mathematical operator is needed to express what Python is doing--but is that really what you're after? – whuber Jul 12 '19 at 16:07
• Good point - months are a really bad example. I changed it to days. Then it should be evenly-spaced time series. I actually would prefer a mathematical notion over some Python/pseudo code. – Michael Dorner Jul 12 '19 at 16:13

Let's start with a generalization. Suppose the original data are a series of values $$\mathbf{x}=(x_1, x_2, \ldots, x_n)$$ associated with times $$\mathbf{t}=(t_1, t_2, \ldots, t_n).$$ You have in mind a new set of ordered times indexed by the integers $$\mathbf Z$$ along with a function $$f(k, \mathbf{t}, \mathbf{x})$$ that determines what new value to assign to the integer $$k$$ based on the data in $$(\mathbf{t}, \mathbf{x}).$$ The new time series is just

$$(y_k) = (f(k, \mathbf{t}, \mathbf{x})),\ k\in\mathbb{Z}.$$

In the example in the question, you have partitioned an interval of time from $$t_0$$ to $$t_\infty \gt t_0$$ into (say) $$N$$ evenly spaced intervals, each of which therefore has a length $$h= (t_\infty - t_0)/N,$$ and I am presuming (as is conventional) that you wish to index them with the integers $$0, 1, 2, \ldots, N-1.$$ The interval in which any time $$t$$ lies therefore is

$$g(t) = \lfloor \frac{t-t_0}{h} \rfloor.$$

Your resampling function sums all values within each interval and therefore can be written

$$f(k,\mathbf{t},\mathbf{x}) = \sum_{\{i\,\mid\, g(t_i)=k\}}x_i.$$

(Empty sums are, by definition, zero.)

If you like, you may combine these formulas in one grand (but less immediate) one,

$$y_k = \sum_{\{i\,\mid\, \lfloor N\frac{t_i-t_0}{t_\infty-t_0} \rfloor=k\}}x_i.$$

To demonstrate how non abstract and natural this is, here is an R implementation of the last formula:

function(times, x, t.0, t.infty, N) tapply(x, floor(N * (times - t.0)/(t.infty - t.0)), sum)

It returns an array indexed by the new times with values equal to the sums within each new interval.

format: plain


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# Universal Properties: a Prelude

So I want to take some time to talk about universal properties.  I personally think they’re awesome because if you look hard enough, you start to see them everywhere in mathematics.  Especially in abstract algebra and algebraic geometry.  They admit a fairly intuitive explanation, but the actual details of their definition require a lot of work.  A lot.

Universal properties are used to define certain “special” objects in a category.  That is, a universal property, in a sense, picks out the “best possible” object in a category that satisfies a certain property.  Any other object that satisfies this property then has a morphism to/from (depending on the type of property) this “best possible” object.  The nicest thing is then that an object satisfying a universal property is unique up to unique isomorphism.  Yes, there are two “uniques” there.

For example (still working intuitively), let’s look at the kernel of a group homomorphism.  If $\varphi: G \to H$ is a group homomorphism, then we usually define the kernel to be the normal subgroup $\text{Ker}\varphi = \{ g \in G | \varphi(g) = 0\}$ of $G$.  Easy enough.  This also relies on the fact that groups are also sets.  Another way of looking at the kernel: It is the largest group $K$ that is “killed off” by $\varphi$,  i.e. if $i_K : K \to G$ is a group homomorphism, then $\varphi \circ i_k = 0_H$, the map that sends everything in $K$ to 0 in $H$.  The same idea holds for the cokernel of $\varphi$, and tons of other special objects that you’ve undoubtedly run into before.

The technical details of the universal property are in the “largest group such that…” part.  Here, this means that for any other group $K'$ and homomorphism $i_{K'} : K' \to G$ such that $\varphi \circ i_{K'} = 0_H$, there is a unique group $K$, homomorphism $i_K : K \to G$, and homomorphism $g: K' \to K$ such that $i_{K'} = i_K \circ g$.

Yes, that is a bit wordy, it’s not just you.  The point of this post (and probably the next as well), is to unravel that mess of words into something tractable.  The first step to understanding these properties is to understand initial and terminal objects.

Initial and Terminal Objects

These aren’t too bad.  Let C be some category.  An initial object in C is an object $a$ such that for any other object $b$, there is a unique morphism $a \to b$.  A simple example to keep in mind is the empty set in the category Set.  Since the empty set is a subset of every set, it admits a unique map into any set (there’s only one way to send nothing to nothing!).

A terminal object in C is an object $a$ such that for any other object $b$, there is a unique morphism $b \to a$.  Again, a simple example is any singleton set in the category of sets.  To see this, if $X$ is any set, there is a unique map that sends everything everything in $X$ to the singleton set.  I’m lying a tiny bit here (in what the “universal ” object is), and I’ll tell you later what it was.  Is “the” singleton set unique?  Of course not.  It is, however, unique up to a unique isomorphism.

The lesson here is that initial objects capture the idea of “the most efficient” or “the smallest such that” and that terminal objects capture the idea of  “the largest such that.”  The key, however, is to define them in the right category.

## Author: brianhepler

I'm a third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best to show you a world of pure imagination.

## 2 thoughts on “Universal Properties: a Prelude”

1. Joe Hannon says:

In your penultimate paragraph, I think you mean “terminal” where you have written “initial” again.


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Anomalous and topological Hall effects in epitaxial thin films of the noncollinear antiferromagnet Mn 3 Sn

@article{Taylor2020AnomalousAT,
title={Anomalous and topological Hall effects in epitaxial thin films of the noncollinear antiferromagnet Mn 3 Sn},
author={J. M. Taylor and A. Markou and E. Lesne and P. K. Sivakumar and C. Luo and F. Radu and P. Werner and C. Felser and S. Parkin},
journal={Physical Review B},
year={2020},
volume={101},
pages={094404}
}
Noncollinear antiferromagnets with a $D{0}_{19}$ ($\mathrm{space}\phantom{\rule{0.28em}{0ex}}\mathrm{group}=194$, $P{6}_{3}/mmc$) hexagonal structure have garnered much attention for their potential applications in topological spintronics. Here, we report the deposition of continuous epitaxial thin films of such a material, ${\mathrm{Mn}}_{3}\mathrm{Sn}$, and characterize their crystal structure using a combination of x-ray diffraction and transmission electron microscopy. Growth of \${\mathrm… Expand
13 Citations

References

SHOWING 1-10 OF 56 REFERENCES


text:
# What is the error in this disproof of the $\Omega$-conjecture?

I was reading about the $\Omega$-conjecture and have thought of a refutation of it, which seems too simple to not have been noticed since the $\Omega$-conjecture has been around, so i'm skeptical and want to see whether anyone can spot a flaw in it.

I will assume there is a proper class of hyper-huge cardinals. This assumption implies a proper class of Woodin cardinals and by Usuba implies the existence of the bedrock model $W$, and since $V$ is a set-generic extension of $W$ and our assumption (as well as the $\Omega$-conjecture) is invariant throughout the set-generic multiverse, we can work in $W$. Suppose for a contradiction that the $\Omega$-conjecture holds. Since $V = W$ is $\Sigma_2$ definable in any set-forcing $V[G]$, we can uniformly evaluate $\Sigma_2$-truths of $V$ in any set-forcing of $V$ by other recursively given $\Sigma_2$ sentences. Since the $\Sigma_2$-laws of the set-generic multiverse are definable in $H(\delta_0^+)$ where $\delta_0$ is the least Woodin cardinal (by the $\Omega$-conjecture), and they can be used to compute all the $\Sigma_2$ truths of $V$ (including the theory of $H(I_0^{+})$ and beyond), this violates Tarski's undefinability of truth. Thus the $\Omega$-conjecture must fail assuming our large cardinal hypothesis. $\square$

Is this argument really valid? (I'm worried 2016 will end by refuting $V = \text{Ultimate }L$)

• Uncountable_turtle, could you clarify how you are violating Tarski? After all, there is no problem with having a $\Sigma_2$ truth predicate of $V$--this doesn't violate Tarski. Also, could you clarify the role of parameters when you refer to "$\Sigma_2$-truths". Dec 31, 2016 at 12:50
• Also, why do you worry that the year will end with a refutation of $V=\text{Ultimate }L$? That's a good thing, not a bad thing! :) Dec 31, 2016 at 12:58
• I'm voting to close this question as off-topic because it asks to verify a proof / disproof. Dec 31, 2016 at 17:22
• @StefanKohl Please don't close the question; it is a fine question, and definitely on topic! I for one would like to get to the bottom of it. Dec 31, 2016 at 17:28
• @StefanKohl I know that you already retracted your close vote, so no harm no foul, but the usual application of "we don't vet proofs here" is to crackpot posts, which doesn't apply here. This seems like a pretty serious and high-level discussion. Jan 1, 2017 at 19:19

## 1 Answer

Woodin's theorem says that assuming the $\Omega$ Conjecture and the existence of a proper class of Woodin cardinals, the set $\mathcal V_\Omega$ of $\Pi_2$ sentences that hold in every universe of the generic multiverse is lightface definable over $H_{\delta^+}$ where $\delta$ is the least Woodin cardinal. You claim there is a Turing reduction from the $\Pi_2$ theory of the bedrock to $\mathcal V_\Omega$, obtaining in this way a $\Sigma_2$ definition of $\Pi_2$ truth (assuming the Ground Axiom), a contradiction. The proposed reduction sends a formula $\phi$ to the sentence $f(\phi)$ expressing "The bedrock satisfies $\phi$." Tell me if I'm misunderstanding you.

One problem: it isn't clear that $f(\phi)$ is $\Pi_2$. Note that if $W$ is a $\Sigma_2$ or $\Pi_2$ or even $\Delta_2$ inner model and $\phi$ is a $\Pi_2$ sentence, the sentence $W\vDash \phi$ is not obviously $\Pi_2$. We can write $\phi$ as $$\forall x \ (x\notin W \vee \exists y\ (y\in W\wedge \psi(x,y))$$ for some $\Delta_0$ formula $\psi$. This seems to be no simpler than $\Pi_3$ even if $W$ is $\Delta_2$.

For example, we claim that the statement $$\Psi\equiv \text{CH}\text{ fails in the bedrock}$$ is not equivalent over ZFC + the Bedrock Axiom to a $\Pi_2$ formula, even though $\neg\text{CH}$ is $\Delta_2$. (The Bedrock Axiom just asserts that the generic multiverse has a bedrock.) Suppose towards a contradiction that this statement is equivalent to a $\Pi_2$ formula, which we may assume is of the form $\forall \alpha\ (V_\alpha\vDash \psi)$. Fix a model $$M\vDash\text{the Bedrock Axiom}+\neg\text{CH} + \neg\Psi$$ (so the bedrock of the generic multiverse of $M$ satisfies $\text{CH}$). Since $M\vDash \neg \Psi$, there is some ordinal $\alpha$ of $M$ such that $M_\alpha\vDash \neg \psi$. Now pass to a class forcing extension $M[H]$ satisfying the Ground Axiom and such that $M[H]_{\alpha} = M_{\alpha}$ and $M[H]_{\omega+\omega} = M_{\omega+\omega}$. (See Reitz's thesis, Theorem 12.) We have $M[H]\vDash \neg\text{CH}$ (since $M\vDash \neg \text{CH}$ and $M[H]_{\omega+\omega} = M_{\omega+\omega}$). On the other hand since $M[H]_\alpha = M_\alpha\vDash \neg \psi$, $M[H]\vDash \exists \alpha\ (V_\alpha\vDash \neg \psi)$. Thus $M[H]\vDash \neg \Psi,$ so the bedrock of $M[H]$ satisfies $\text{CH}$, which contradicts the fact that $M[H]$ is its own bedrock.


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Network: Global | Joburg Libraries | MOOCs | StartUpTribe | Zimbabwe | Donate

MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB

1 like 0 dislike
30 views
Does a big difference between P(Y) and P[Y | do(X=x)] imply that X has a strong causal effect on Y?
| 30 views

0 like 0 dislike
$P(Y)$ and $P[Y | \text{do(X=x)}]$ represent the probability distribution of $Y$ and $Y$ given that there is an intervention on $X$. So the larger the difference between the outcome as a result of the intervention  and no action is an indicator of the impact of the intervention.

Yes it indicates a strong causal effect on $Y$.

Source: https://www.stat.cmu.edu/~larry/=sml/Causation.pdf
by Diamond (67,402 points)

0 like 0 dislike


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## CryptoDB

### Paper: Scalable Pseudorandom Quantum States

Authors: Zvika Brakerski , Weizmann Institute of Science Omri Shmueli , Tel Aviv University DOI: http://dx.doi.org/10.1007/978-3-030-56880-1_15 (login may be required) Search ePrint Search Google CRYPTO 2020 Efficiently sampling a quantum state that is hard to distinguish from a truly random quantum state is an elementary task in quantum information theory that has both computational and physical uses. This is often referred to as pseudorandom (quantum) state generator, or PRS generator for short. In existing constructions of PRS generators, security scales with the number of qubits in the states, i.e.\ the (statistical) security parameter for an $n$-qubit PRS is roughly $n$. Perhaps counter-intuitively, $n$-qubit PRS are not known to imply $k$-qubit PRS even for \$k
##### BibTeX
@inproceedings{crypto-2020-30409,
title={Scalable Pseudorandom Quantum States},
publisher={Springer-Verlag},
doi={http://dx.doi.org/10.1007/978-3-030-56880-1_15},
author={Zvika Brakerski and Omri Shmueli},
year=2020
}


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## mardi 18 mars 2014

### Plotting VTK files with matplotlib and python-vtk

Paraview is good. But pictures produced by the software is in general raster (maybe I can't use it right but even if I build with gl2ps I can't find a way to export scene to PS or PDF). So usually I used sample utility to write a slice of interest to a RAW file, then load it with numpy and then use tricontourf to produce color map of a value. And then I can save it as a PDF (or EPS, or PS) so the figure can be scaled without quality loss.

Everything is good (though there's certain problems with matplotlib's triangulation sometimes) but in this case I had a hole in a flow field. So if I have a file with (x, y, value) tuples, the hole will be also triangulated with a value equal to zero and a final tricontourf figure will be quite far from the original flow field. Also as OpenFOAM's sample utility will anyway create a triangulation of the mesh, why do it again. So I've decided to try plotting VTK file with a triangular grid. And it was more or less easy. Though there's almost no documentation on python-vtk module. Well, I guess this module is just a wrapper around C++ so one need to write C++-python instead of python.

Surely this script can be improved to handle vectors and scalars automatically, to handle polymesh files in more general way (currently I suppose, all polygons in the mesh are triangles).
Here is a script for loading a vector field and plotting its X component:

#!/usr/bin/env python

import numpy as N
import vtk
import matplotlib.pyplot as P

if not os.path.exists(filename): return None

cells = data.GetPolys()
triangles = cells.GetData()
points = data.GetPoints()
point_data = data.GetPointData()
Udata = point_data.GetArray(0)

ntri = triangles.GetNumberOfTuples()/4
npts = points.GetNumberOfPoints()
nvls = Udata.GetNumberOfTuples()

tri = N.zeros((ntri, 3))
x = N.zeros(npts)
y = N.zeros(npts)
ux = N.zeros(nvls)
uy = N.zeros(nvls)

for i in xrange(0, ntri):
tri[i, 0] = triangles.GetTuple(4*i + 1)[0]
tri[i, 1] = triangles.GetTuple(4*i + 2)[0]
tri[i, 2] = triangles.GetTuple(4*i + 3)[0]

for i in xrange(points.GetNumberOfPoints()):
pt = points.GetPoint(i)
x[i] = pt[0]
y[i] = pt[1]

for i in xrange(0, Udata.GetNumberOfTuples()):
U = Udata.GetTuple(i)
ux[i] = U[0]
uy[i] = U[1]

return (x, y, tri, ux, uy)

P.clf()
levels = N.linspace(-0.5, 1.5, 64)
x, y, tri, ux, uy = load_velocity('U-file.vtk')
P.tricontour(x, y, tri, ux, levels, linestyles='-',
colors='black', linewidths=0.5)
P.tricontour(x, y, tri, ux, levels)
P.gca().set_aspect('equal')
P.minorticks_on()
P.gca().set_xticklabels([])
P.gca().set_yticklabels([])
P.title('$U_x$ / Gamma')
P.savefig("Ux.png", dpi=300, bbox_inches='tight')
P.savefig("Ux.pdf", bbox_inches='tight')

(Note: updated version of the script adapted for VTK 6.2.0 is posted here)

And as a result of this script we'll get this nice picture:

And here is a link to the genereated PDF.

### OpenFOAM, residuals, and vortex street

This post appeared due to the question on cfd-online.com OpenFOAM forum. It has been started as yet another post on Kármán vortex street but then (around message #60) original poster revealed an article he was trying to compare with, so I've decided to check influence of different discretisation schemes and convergence criterions on final results.

## Geometry and flow conditions

The first version of geometry and boundary conditions can be found in post #11, though they slightly differ from the geometry provided in the article (and as a result, from geometry used in simulations).

So we've got a cylinder with diameter D placed in the channel with a length 16D and a width 8D. At the inlet we have velocity 1 m/s, at the outlet zero pressure and zero gradient in velocity. Initially velocity inside the domain set to 1 m/s. With D = 1 m and fluid kinematic viscosity 100 m2/s, Reynolds number is equal to 100.

## First attempt

Well, the first variant of the simulation was just a simulation for its own sake (even Re was 200). Just to produce neat pictures. And these videos:

Flow develops, vortices go.

## Second attempt

After reading the original article I've decided to check different meshes, discretisation schemes, and convergence criteria to learn if there's any difference. In the article authors were developing new method in FEM framework, so theirs original mesh was triangular with increased density near a centerline of the channel. Now I think there's a need for another set of simulations: with and without increase of mesh density near the center-line of the channel.

### Meshes

I've fallen in love with Gmsh for generating meshes, so these two meshed are produced with it. As Gmsh now has transfinite algorithm, it is more or less like blockMesh with GUI, variables and functions in a mesh file, more convenient way of grading description. Much like in blockMesh one needs to define vertices and then can use GUI to describe lines, surfaces and volumes. Then again switch to text editor for definition of transfinite lines, progressions (in the sense of Gmsh), etc. I will omit iterations during the mesh creation process and just provide GEO files with screenshots of the mesh:

"Coarse" mesh

Coarse cylinder-2D.geo

"Fine" mesh

Fine cylinder-2D.geo

The only difference between coarse and fine mesh is twice increased density in case of "Fine" mesh. So a number of cells in the fine mesh is 4 times higher than in coarse mesh.

### fvSchemes

Typical fvSchemes file is provided below. The only line that is changed from simulation to simulation is div(phi,U) Gauss vanLeerV;, I've chosen to compare three different second order discretisation schemes with limiters: van Leer, QUICK, and Gamma.

/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  1.6                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
version     2.0;
format      ascii;
class       dictionary;
location    "system";
object      fvSchemes;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

ddtSchemes
{
default         CrankNicolson 1.0;
}

{
default         Gauss linear;
}

divSchemes
{
default         none;
div(phi,U)      Gauss vanLeerV;
}

laplacianSchemes
{
default         none;
laplacian(nuEff,U) Gauss linear corrected;
laplacian((1|A(U)),p) Gauss linear corrected;
}

interpolationSchemes
{
default         linear;
}

{
default         corrected;
}

fluxRequired
{
default         no;
p               ;
}

// ************************************************************************* //

### fvSolution

In case of fvSolution file I changed PIMPLE dictionary:
PIMPLE
{
momentumPredictor yes;
nCorrectors       2;
nOuterCorrectors  20;
nNonOrthogonalCorrectors 1;

residualControl
{
"(U|p)"
{
tolerance 1e-2;
relTol 0;
}
}
}

Keeping everything else the same, I changed tolerance in residualControl dictionary to see if it influence results. I've checked values 1e-2, 1e-4, and 1e-6. This value defines the moment when PIMPLE loop will exit from outer corrector cycle. And usually in the beginning of a simulation number of outer correctors is around 4 (for the case of residual 1e-6 - 8), during the time this number falls to 2 (or 4 for the case of 1e-6). I don't know why all tutorial cases use fixed number of outer corrector iterations, sometimes it can lead to completely incorrect results. Maybe OpenFOAM decided that residualControl is "advanced" technique.

Also there're lots of scary posts on relaxation, so I've decided also to check the influence of relaxation factors on results of a simulation. So instead of standard

relaxationFactors
{
fields
{
}
equations
{
"U.*" 1.0;
"p.*" 1.0;
}
}

I used

relaxationFactors
{
fields
{
}
equations
{
"U.*" 0.7;
"p.*" 0.3;
}
}

### Results

Color contours usually provide only schematic view of the flow, so further (surely after these pictures) I will be comparing oscillations of force coefficients. Sometimes it is possible to plot isolines of pressure and velocity on the same picture (when the phases of the oscillating flow do not differ much). But now, the color contours (time is 150 s):

Pressure
van Leer

QUICK

Gamma

Though values seem to be more or less the same, vortices are shifted in time. The same thing is for velocity components:

Ux
van Leer

QUICK

Gamma

Uy
van Leer

QUICK

Gamma

#### Discretisation schemes comparison

I've fixed final residual in fvSolution at 1e-6 and just changed discretisation scheme. On the graphs X-axis is time, Y-axis is value. All simulations were done on the 'fine' mesh.

There is phase shift between graphs, I've moved them to compare oscillation amplitude. There is difference but it's not so big.

And as there is phase shift between different scheme I will omit comparison of the pressure and velocity contours. While there is difference between Cdrag oscillation amplitudes, Clift oscillations go rather synchronously:

#### Final residuals comparison

After that I've decided to check sensitivity of a discretisation scheme on final residual. I've fixed the scheme and run simulations for different final residuals: 1e-2, 1e-4, and 1e-6.
While van Leer and Gamma showed almost no dependence (though there is slight shift when going from 1e-2 to 1e-4 for Gamma):

For QUICK switch from 1e-2 to 1e-4 leads to quite large time shift between oscillations:

But even if the time shift in oscillations seems to be not so big, positions of the vortices are strongly depend on this shift. Below is overlays of Ux isolines for van Leer and Gamma schemes at 150 s:

If for van Leer scheme contours are almost the same, for Gamma results with residual 1e-2 are sufficiently shifted.

#### Mesh density comparison

Also compared results acquired using meshes with different densities. Number of cells in the meshes differ 4 times as I've reduced densities twice in X and Y directions.
Gamma showed just decrease of Cdrag and Clift values on fine mesh, while with QUICK and van Leer schemes there's also phase shift between results on different meshes.

Cdrag

Clift

#### To relax or not

Finally I've decided to check if a relaxation is really crucial to results. I've fixed the discretisation scheme (van Leer, though now I think maybe it was better to choose Gamma), the final residual in PIMPLE dictionary and just modified relaxation part as shown in the beginning of the post. Surely due to the relaxation solution converges slowly, i.e. instead of 4-5 outer corrector iterations 9-11 iterations was necessary. Also there is a time shift between results.

### Conclusion

So it goes.

As usual case files (if you'd like to reproduce results) are on bitbucket.org.

## samedi 1 mars 2014

### Introduction

After I've done simulations of vortex street in 2D, I kept thinking that it would be nice to do more or less the same thing in 3D. And after watching the movie about vortex flow meters by Endress+Hauser, I've decided it would be really funny to put something in a tube and watch vortices going by.

First of all we need to decide on operational parameters of the system. Let's assume that we have 10 m long tube with diameter of 1 m and let's install something at 2 m from inlet plane. Further calculations are based upon the article from Wikipedia about Kármán vortex street.
Re = uD/ν

where u is flow velocity, D is characteristic length, and ν is fluid viscosity. We'll get vortex street for Re values larger than 90. If we take, for example, water with nu around 1000000 it's not hard to get large enough Reynolds number. But then a question of the distance between vortices arises. We also want our vortices inside the tube. Vortex shedding frequency is
f = 0.198*u/d*(1 - 19.7/Re)
where d is cylinder diameter (correct for 250 < Re < 10^5). After playing with tube and cylinder sizes, fluid viscosity and inflow velocities (here is a script) I've decided to go with the following:
D = 1 m
d = 0.2 m
u = 1 m/s
ν = 0.0001 m^2/s

### Meshing

For meshing I've used Gmsh and snappyHexMesh. I.e. I've created hexagonal mesh of the tube, then I've created triangular surface mesh for the cylinder and sphere, a nd finally I've used snappyHexMesh to cut small cylinder (or sphere) tube.

Mesh for tube is based upon my previous post. The addition is a parametrization of the GEO file, i.e. one need to set tube leng th and its diameter; gmsh will recalculate positions of the points in the mesh. GEO files for cutting shapes (cylinder, sphere) are even simpler cause we need only triangular surface mesh so there's no need to split a shape into volumes with 6 sides (this operation necessary for transfinite algorithm to work).

One of the problems I've encountered during "cutting" procedure is the need to use more or less cubical mesh for sHM to work. In case of sphere it wan't actually a problem, cause it lies inside "inner" part of the mesh which is rather regular. So one can have graded mesh near the walls of the tube. That was not so in case on cylinder. After several unsuccessful attempts to cut a cylinder hole in graded mesh (though sHM tried to do it but wasn't able to generate acceptable quality mesh, usually non-orthogonality on the new mesh was too high), I've removed grading.
snappyHexMeshDict is quite simple, first I define the surface:
geometry
{
sphere.stl
{
type triSurfaceMesh;
name sphere;
}
};

then I refine mesh around the surface
refinementRegions
{
sphere
{
mode distance;
levels ((0.05 3) (0.1 2) (0.2 1));
}
}

refinement is based upon distance from the mesh. Other parameters in the dictionary is just a copy from tutorial.

So after running mesh conversion (from gmsh to OpenFOAM) and then running snappyHexMesh, I've got the following meshes for the simulation:

Initial tube mesh

Mesh with internal cylinder

Mesh with internal sphere

Unlike previous time I've decided not to compare turbulence models and just selected realizable k-epsilon.

### Flow movies

After running the simulations I've got the following flow movies:

The cylinder in the tube, vorticity

The cylinder in the tube, velocity

The sphere in the tube, vorticity

The sphere in the tube, velocity

As expected after initial flow development I've got quite nice vortices going along the tube.

### Probes

Since the movies are not quite enough I've added probes along the axis of the tube to record values of velocity, pressure, and turbulent viscosity during the simulations (this is a fragment of controlDict):

functions
{
probes
{
type            probes;
functionObjectLibs ("libsampling.so");
outputControl   timeStep;
outputInterval  1;
probeLocations
(
(2.47 0 0)
(2.90 0 0)
(3.5 0 0)
(5 0 0)
(6 0 0)
);
fields
(
U
p
nut
);
}
}

Below is a graphs of the pressure evolution at the location of the first probe (click on the figure for a larger version):

For internal cylinder

For internal sphere

Though evolution has periodic structure but the frequency differs from estimations.

### How to run the cases on your own

I've put all necessary files into repository on BitBucket. So you can clone the repo or just download archive of the default branch. There is Cases folder where case files reside. Every case directory has Allprepare script which will do all the necessary utility invocations:

#!/bin/sh
cd ${0%/*} || exit 1 .$WM_PROJECT_DIR/bin/tools/RunFunctions

gmsh - tube.geo
gmsh - cylinder.geo
sed -i -e "s/Created by Gmsh/cylinder/" cylinder.stl
mv cylinder.stl constant/triSurface
gmshToFoam tube.msh
cp -r 0.org 0
changeDictionary
surfaceFeatureExtract
snappyHexMesh -overwrite

The script assumes that you have gmsh in your \$PATH and first it creates mesh for tube and cylinder, then corrects STL file created by Gmsh, then places STL files to the directory where sHM will look for it, converts tube mesh to OpenFOAM format, corrects boundary type for walls, and finally runs sHM for final mesh creation. sHM procedure is rather lengthy and needs RAM as the limit on the number of cells is 1500000.


text:
# 995 search results for "latex"

## Power-laws: choose your x and y variables carefully

November 16, 2011
By

This is a follow-up of the post Power of running world records As suggested by Andrew, plotting running world records could benefit from a change of variables. More exactly the use of different variables sheds light on a well-known sports result provided in a 2000 Nature paper by Sandra Savaglio and Vincenzo

## Lambert’s W function and the generalised logarithm

November 16, 2011
By
$Lambert’s W function and the generalised logarithm$

Yesterday I ran into an equation that was a sum of an exponential and a linear term: It doesn’t take long to figure out that there is no analytical solution, and so I set out to write some crappy numerical code. After wasting some time with a fixed point iteration that did not really work,

## Black-Litterman Model

November 15, 2011
By

The Black-Litterman Model was created by Fisher Black and Robert Litterman in 1992 to resolve shortcomings of traditional Markovitz mean-variance asset allocation model. It addresses following two items: Lack of diversification of portfolios on the mean-variance efficient frontier. Instability of portfolios on the mean-variance efficient frontier: small changes in the input assumptions often lead to

## Grrr…

November 15, 2011
By

I’ve been working through Gelman et al.’s otherwise excellent Bayesian Data Analysis and it’s going reasonably well. My statistics is a little bit rusty so it’s taken time to work through all of the exercises and really understand what’s going on. But I...

## Conference in Lyon on climate change and insurance

November 14, 2011
By

I will be in Lyon next Monday to give a talk on "Modeling heat-waves: return period for non-stationary extremes" in a workshop entitled "Changement climatique et gestion des risques". An interesting reference might be some pages from Le Monde (201...

## Create your own Beamer template

November 14, 2011
By

For the past couple of days, I had been searching for a tutorial that would show how to create a custom Beamer template. I found some great resources and some really great customized templates (I have listed the ones that I referred to below) but none ...

## 5 Interesting Free Books for R from beginner to experts

November 12, 2011
By

Always new software language in one technical activity is difficult, normally a good documentation can help, these are three book to use R software for beginner and for experts: ·         “Introduction to the R Project for Statistical Computing for Use at the ITC” by David Rossiter (PDF, 2010-11-21).

## Managing a Local R Repository

November 10, 2011
By

I will be teaching a workshop on R and LaTeX at NEAIR in just under a month. One of the issues I will encounter is a lack of Internet access. I also work with restricted data from NCES which requires the computer to be secured including no network access. As such, I need to manage software from removable

## Managing a Local R Repository

November 10, 2011
By

I will be teaching a workshop on R and LaTeX at NEAIR in just under a month. One of the issues I will encounter is a lack of Internet access. I also work with restricted data from NCES which requires the computer to be secured including no network a...


text:
zbMATH — the first resource for mathematics

Harmonic forms on compact symplectic 2-step nilmanifolds. (English) Zbl 1033.53078
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 257-270 (2003).
If $$(M,\omega)$$ is a $$2m$$-dimensional symplectic manifold, one can define the space $$\mathcal{H}_{\omega}^k(M)$$ of all harmonic $$k$$-forms on $$M$$ and the symplectic harmonic $$k$$-th cohomology group $$H_{\omega\text{-}hr}^k(M):=\mathcal{H}_{\omega}^k(M)/B^k(M)\cap\mathcal{H}_{\omega}^k(M)$$ [see J. Brylinski, J. Differ. Geom. 28, 93–114 (1988; Zbl 0634.58029)]. Assume that $$M$$ is a $$2$$-step compact nilmanifold $$G/\Gamma$$ (i.e. $$G$$ is a simply connected $$2m$$-dimensional Lie group, whose Lie algebra $$\mathfrak{g}$$ is $$2$$-step nilpotent and $$\Gamma$$ is a discrete subgroup of $$G$$ such that $$G/\Gamma$$ is compact) and $$\omega$$ a $$G$$-invariant symplectic form on $$G/\Gamma$$. Let $$\mathcal{H}_{\omega}^k(\mathfrak{g})$$ be the space of all $$G$$-invariant harmonic $$k$$-forms on $$G/\Gamma$$ and $$H_{\omega\text{-}hr}^k(\mathfrak{g}):=\mathcal{H}_{\omega}^k(\mathfrak{g})/B^k(\mathfrak{g})\cap\mathcal{H}_{\omega}^k(\mathfrak{g})$$. The authors prove that $$B^3(\mathfrak{g})\subset \mathcal{H}_{\omega}^3(\mathfrak{g})$$ and some properties concerning $$\dim H_{\omega\text{-}hr}^{2m-2}(\mathfrak{g})$$ and $$\dim H_{\omega\text{-}hr}^{2m-1}(\mathfrak{g})$$. In particular, the dimension of $$H_{\omega\text{-}hr}^{2m-1}(\mathfrak{g})$$ does not depend on $$\omega$$ [see also the second author, Osaka J. Math. 39, 363–381 (2002; Zbl 1012.53076)]. Some examples are considered. Among them a $$2$$-step compact nilmanifold $$G/\Gamma\;(m\geq 6)$$ admitting symplectic structures $$\omega$$ such that the dimension of $$H_{\omega\text{-}hr}^3(\mathfrak{g})$$ varies [see also D. Yan, Adv. Math. 120, 143–154 (1996; Zbl 0872.58002)], for a question raised by Khesin and McDuff].
For the entire collection see [Zbl 1008.00022].

MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension


text:
NCERT Solutions for Class 9 Maths chapter 10 Exercise 10.5 – Circles – Free PDF Download
Hey, are you a class 9 Student and Looking for Ways to Download NCERT Solutions for Class 9 Maths chapter 10 Exercise 10.5? If Yes then you are at the right place.

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So, without wasting more time Let’s start.

### Download The PDF of NCERT Solutions for Class 9 Maths chapter 10 Exercise 10.5 “Circles”

#### All Questions of Chapter 10 Exercise 10.5

Once you complete the chapter 10 then you can revise Ex. 10.5 by solving following questions

Q1. In Fig. A, B and C are three points on a circle with centre $\mathrm{O}$ such that $\angle \mathrm{BOC}=30^{\circ}$ and $\angle \mathrm{AOB}=60^{\circ} .$ If $\mathrm{D}$ is a point on the circle other than the arc $\angle \mathrm{ABC}$, find $\angle \mathrm{ADC}$.

Q2. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Q3. In figure, $\angle \mathrm{PQR}=100^{\circ}$, where $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ are points on a circle with centre $\mathrm{O}$. Find $\angle \mathrm{OPR}$.

Q4. In fig. $\angle \mathrm{ABC}=69^{\circ}, \angle \mathrm{ACB}=31^{\circ}$, find $\angle \mathrm{BDC}$.

Q5. In figure, $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are four points on a circle. AC and BD intersect at a point $\mathrm{E}$ such that $\angle \mathrm{BEC}=130^{\circ}$ and $\angle \mathrm{ECD}=20^{\circ} .$ Find $\angle \mathrm{BAC}$.

Q6. $\quad \mathrm{ABCD}$ is a cyclic quadrilateral whose diagonals intersect at a point $\mathrm{E}$. If $\angle \mathrm{DBC}=70^{\circ}$, $\angle \mathrm{BAC}=$ is $30^{\circ}$, find $\angle \mathrm{BCD}$. Further, if $\mathrm{AB}=\mathrm{BC}$, find $\angle \mathrm{ECD}$.

Q7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Q8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Q9. Two circles intersect at two points $\mathrm{B}$ and $\mathrm{C}$. Through $\mathrm{B}$, two line segment $\mathrm{ABD}$ and $\mathrm{PBQ}$ are drawn to intersect the circles at $\mathrm{A}, \mathrm{D}$ and $\mathrm{P}, \mathrm{Q}$ respectively. Prove that $\angle \mathrm{ACP}=\angle \mathrm{QCD}$.

Q10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Q11. $\mathrm{ABC}$ and $\mathrm{ADC}$ are two right triangles with common hypotenuse $\mathrm{AC}$. Prove that $\angle \mathrm{CAD}=\angle \mathrm{CBD}$.

Q12. Prove that a cyclic parallelogram is a rectangle.


text:
## ~ Abstracts ~

Transferring representations between finite reductive groups

Suppose $\tilde{G}$ is a connected reductive group over a finite field $k$, and $\Gamma$ is a finite group acting on $\tilde{G}$, preserving a Borel-torus pair. Then the connected part $G$ of the group of $\Gamma$-fixed points of $\tilde{G}$ is reductive, and there is a natural map from (packets of) representations of $G(k)$ to those of $\tilde{G}(k)$. I will discuss this map, its motivation in the study of $p$-adic base change, prospects for refining it, and a generalization: the pair of groups $(\tilde{G},G)$ must satisfy some axioms, but $G$ need not be a fixed-point subgroup of $\tilde{G}$, nor even a subgroup at all.

« Back...

On the local Langlands correspondence and Arthur conjecture for orthogonal groups
Hiraku Atobe, Kyoto University, Japan

I will state precisely the local Langlands correspondence for orthogonal groups established by Arthur. As an application, I will discuss on the local Gross-Prasad conjecture for orthogonal groups. Also, I will explain the Arthur multiplicity formula for even orthogonal groups. This is a joint work with Wee Teck Gan.

« Back...

A partition of the set of enhanced Langlands parameters of a reductive p-adic group
Anne Marie Aubert, Institut de Mathématiques de Jussieu, France

Let $F$ be a local non-archimedean field, and let $G$ be the group of $F$-points of a connected reductive algebraic group defined over $F$. We will first introduce a notion of cuspidality for the enhanced Langlands parameters of $G$ (that is, pairs formed by a Langlands parameter $\phi$ and an irreducible character $\eta$ of a certain component group attached to $\phi$). The cuspidal pairs $(\phi,\eta)$ are expected to correspond to the supercuspidal irreducible representations of $G$ via the local Langlands correspondence. It is proved in [3] that it is indeed the case for split classical groups. We will next describe a construction of a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig (see [1]). We will used this map to obtain a partition of the set of enhanced Langlands parameters for $G$, which reflects the partition of the smooth dual of $G$ giving by the Bernstein Center. Finally, we will explain the expecte role of that partition towards a proof of the local Langlands correspondence in some cases [2]. It is joint work with Ahmed Moussaoui and Maarten Solleveld.

References:
[1] A.-M. Aubert, A. Moussaoui, M. Solleveld, Generalizations of the Springer correspondence and cuspidal Langlands parameters, arXiv:1511.05335.
[2] A.-M. Aubert, A. Moussaoui, M. Solleveld, Graded Hecke algebras for disconnected reductive groups, Work in progress.
[3] A. Moussaoui, Centre de Bernstein enrichi pour les groupes classiques, arXiv:1511.02521.

« Back...

Unipotent representations for complex groups
Dan Barbasch, Cornell University, USA

In this talk I will describe the unipotent representations for complex groups and how they fit as building blocks of the unitary dual of groups of classical type. For exceptional groups I will give a set of parameters which conjecturally platy the same role as in the classical cases.

« Back...

Tutorial on Hecke algebras
Dan Ciubotaru, University of Oxford, UK

The aim of these lectures is to offer an overview of harmonic analysis for affine Hecke algebras and the relations with smooth representations of reductive p-adic groups. I will explain first how affine Hecke algebras appear naturally in the world of representations of p-adic groups:
a) as Iwahori-Hecke algebras, and
b) as endomorphism algebras of certain projective generators.

I will show how certain important questions in harmonic analysis for p-adic groups can be translated to affine Hecke algebras: e.g., the Plancherel formula and the determination of the unitary dual. I will explain the classification and construction of discrete series modules for affine Hecke algebras with unequal parameters, the calculation of formal degrees, the density and trace Paley-Wiener theorems in this setting, and present some elements of the theory of Dirac operators for graded affine Hecke algebras.

The material for these lectures will be based on works by Bernstein, Kazhdan-Lusztig, Barbasch-Moy, Reeder, Opdam, Solleveld, Kato, Heiermann, Waldspurger. The parts from my own work that are relevant for these lectures are joint with Barbasch, He, Kato, Opdam, Trapa.

« Back...

On the explicit description of local theta correspondence for real reductive dual pairs
Xiang Fan, Peking University, China

In this talk, a brief survey is given on the problem to determine the local theta correspondence explicitly in terms of Langlands parameters for real reductive groups in dual pairs. The results achieved in the past three decades on this problem will be reviewed, together with some recent progress.

« Back...

On the Moy-Prasad filtration and supercuspidal representations
Jessica Fintzen, Harvard University, USA

Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes and generalizes the paper of the speaker and Romano, which treats the case of an absolutely simple split reductive group.

In addition, we will present a general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p. This provides a tool to transfer results about the Moy-Prasad filtration from one prime to arbitrary primes and also yields a new description of the Moy-Prasad filtration representations.

« Back...

Applications of the Deligne-Kazhdan philosophy to the Langlands correspondence for split classical groups
Radhika Ganapathy, Tata Institute of Fundamental Research, India

The Deligne-Kazhdan theory loosely says that the complex representation theory of Galois groups and split reductive groups over a local field of characteristic p can be viewed as the limit, as the ramification index tends to infinity, of the representation theory of these groups over local fields of characteristic 0. In this talk, I will explain how this method can be used to prove the local Langlands correspondence for split classical groups over a local field of characteristic p (with some restrictions on the characteristic) using the corresponding result of Arthur in characteristic 0. This is joint work with Sandeep Varma.

« Back...

Recent applications of a couple of theorems on holonomic distributions
Dmitry Gourevitch, The Weizmann Institute of Science, Israel

First I will formulate the Bernstein-Kashiwara theorem about the finite dimensionality of the space of solutions of a holonomic system of PDEs. Then I will present the following recent applications of this result:

a) A bound for the dimension of the space of equivariant distributions on an algebraic G-manifold with finitely many orbits (where G is a real algebraic group).

b) Hausdorffness of the zero homology of the Lie algebra of G with coefficients in the space of Schwartz functions on such a manifold.

c) Equivalence of the Casselman comparison conjecture and the automatic continuity conjecture.

d) A version of the Kobayashi-Oshima-Krötz-Schlichtkrull bounds for multiplicities of irreducible representations.

If time permits I will also present our recent easy proof for meromorphic continuation of intertwining operators on relative principal series on reductive spherical spaces, using an old analytic continuation result of Bernstein.

The talk is based on joint works with A. Aizenbud, B. Krötz, A. Minchenko, G. Liu, S. Sahi and E. Sayag.

« Back...

Max Gurevich, The Weizmann Institute of Science, Israel

In the area of relative p-adic harmonic analysis there is much interest in a description of the representations of a reductive group G which can be embedded inside the space of smooth functions on a homogeneous space G/H.

A related question is whether such an embedding can be realized in a canonical form such as an H-integral over a (smooth) matrix coefficient. In a joint work with Omer Offen we treated the symmetric case, i.e., when H is the fixed point group of an involution. As part of the answer we provide a precise criterion for such integrability, which reduces in the group case to Casselman?s known square-integrability criterion.

« Back...

Non-Siegel degenerate Eisenstein series
Marcela Hanzer, University of Zagreb, Croatia

In my talk I will elaborate on the work on the degenerate Eisenstein series for symplectic groups I've done recently. First, I'll mention the results in the Siegel case--the location of poles was already known by the work of Kudla and Rallis (using theta correspondence), but we give the explicit description of the images. Then I will address the case of non-Siegel Eisenstein series, which was not dealt with in depth before, and, using combinatorial approach, describe the results.

« Back...

On the unramified spherical automorphic spectrum
Volker Heiermann, Université d'Aix-Marseille, France

For a split connected reductive group G defined over a number field F , we compute the part of the spherical automorphic spectrum which is supported by the cuspidal data containing (T, 1), where T is a maximal split torus and 1 is the trivial automorphic character. The proof uses residue distributions introduced by E. Opdam and G. Heckman in the study of graded affine Hecke algebras, and a result by M. Reeder on the weight spaces of the (anti)spherical discrete series representations of affine Hecke algebras. Both of these ingredients are of a purely local nature. For many special cases of reductive groups G similar results have been established by various authors. The main feature of the present proof is the fact that it is uniform and general.
Joint work with M. De Martino et E. Opdam.

« Back...

Small representations of finite groups
Roger Howe, Yale University, USA

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of values of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group.

In particular, the representations of small dimension tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of some of these facts. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, this aspect remains obscure. This talk will discuss a conjectural method for constructing the small representations of finite classical groups. The method is closely related to the theory of theta series, and has an analog over local field.

« Back...

A homomorphism of Harish-Chandra for polar representations
Jing-Song Huang, Hong Kong University of Science and Technology, Hong Kong

A polar representation of a complex reductive algebraic group has a Cartan subspace just like an adjoint representation. We describe a homomorphism on invariant differential operators on a polar representation extending the Harish-Chandra homomorphism for an adjoint representation. Then we discuss extension of the works by Wallach, Hotta-Kashiwara and Levasseur-Stafford in connection with the Springer correspondence.

« Back...

The automorphic discrete spectrum of Mp(2n)
Atsushi Ichino, Kyoto University, Japan

In his 1973 paper, Shimura established a lifting from half-integral weight modular forms to integral weight modular forms. After that, Waldspurger studied this in the framework of automorphic representations and classified the automorphic discrete spectrum of the metaplectic group Mp(2), which is a nonlinear double cover of SL(2), in terms of that of PGL(2). We discuss a generalization of his classification to the metaplectic group Mp(2n) of higher rank. This is joint work with Wee Teck Gan.

« Back...

Simple epipelagic local Galois representations
Naoki Imai, University of Tokyo, Japan

We discuss a geometric construction of the n-dimensional irreducible representations of the Weil group of a non-archimedean local field, which we call simple epipelagic representations. We give also an explicit description of the local Langlands correspondences for GL(n) in the simple epipelagic representation cases.

« Back...

Automorphic L-functions and the Langlands functoriality
Dihua Jiang, University of Minnesota, USA

Automorphic L-functions are invariants attached to automorphic representations. The analytic properties of automorphic L-functions are expected to have impacts to the Langlands functoriality, which is one of the fundamental structures of automorphic representations. In this talk, we will review some known results about the relation between the poles of L-funcitons and the Langlands functoriality, and discuss with some details about the relation between the central value of L-functions and certain types of the Langlands functoriality, which is new and is based on my joint work with Lei Zhang.

« Back...

Doubling global constructions for tensor product L-functions
Eyal Kaplan, The Ohio State University, USA

In the first part of the talk I will present a joint work with Cai, Friedberg and Ginzburg.
In a series of constructions, we apply the "doubling method" from the theory of automorphic forms to covering groups. We obtain partial tensor product L-functions attached to generalized Shimura lifts, which may be defined in a natural way since at almost all places the representations are unramified principal series.

In the second part, I will describe a joint work with Jan Mollers. We present a novel integral representation for a quotient of global automorphic L-functions, the symmetric square over the exterior square. The construction involves the study of local and global aspects of a new model for double covers of general linear groups, the metaplectic Shalika model. In particular, we prove uniqueness results over p-adic and Archimedean fields, and a new Casselman-Shalika type formula.

« Back...

Some computations of the GL(n) tensor product algebra
Sangjib Kim, Korea University, Korea

I will present some computational results regarding the 'GL(n) tensor product algebra' constructed and investigated by Howe, Jackson, Lee, Tan, and Willenbring, encoding the decomposition of tensor products of finite dimensional irreducible representations of the complex general linear group GL(n). This is a joint work with Donggyun Kim.

« Back...

Alpha-determinants and the Alon-Tarsi conjecture
Kazufumi Kimoto, University of the Ryukyus, Japan

Alpha-determinant of a square matrix $A$ is defined by replacing the sign of a permutation in the definition of $\det A$ to a certain power of a parameter conventionally denoted by $\alpha$. This interpolates the the ordinary determinant and the permanent. When we consider the natural action of the general linear Lie algebra $gl_n$ on the algebra of polynomials in the variables $x_{ij}$ ($1\le i,j\le n$), each of the determinant and permanent of the matrix $X=(x_{ij})$ generates an irreducible module. Sho Matsumoto and Masato Wakayama (2006) studied the cyclic $gl_n$-module generated by the alpha determinant and determine its irreducible decomposition explicitly. This result leads us to the study of the alpha-determinants with special parameters for which the structure of the cyclic module degenerates from the generic case; Using them, we can construct a polynomial function, which we call the wreath determinant, for particular type of rectangular matrices, say $n$ by $kn$, which is relative invariant under the natural left translation by $GL_n$. In the talk, we survey the representation-theoretic results on the alpha-determinants, the wreath determinants. We also explain the relation between the wreath determinants and the Alon-Tarsi conjecture on Latin squares.

« Back...

Symmetry breaking operators for real reductive groups
Toshiyuki Kobayashi, The University of Tokyo, Japan

Branching problems ask how irreducible representations of groups G decompose when restricted to subgroups G'.We present a program on branching problems, from abstract feature to concrete construction of symmetry breaking operators.

As an abstract feature, we provide a geometric criterion on the pair of reductive groups for the multiplicities of the branching laws to be always of uniformly bounded (or more weakly, to be finite) by using analysis on (real) spherical varieties.

As a concrete construction of symmetry breaking operators (SBOs), we explain an idea of the F-method in constructing differential SBOs. Finally, we discuss some classification results of all non-local and local SBOs by an example.

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Some aspects for representation theory of GLn over non-archimedean field
Erez Lapid, The Weizmann Institute of Science, Israel

This is a joint work with Alberto Mínguez.

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Recent development on global Gan-Gross-Prasad conjecture and arithmetic applications
Yifeng Liu, Northwestern University, USA

In this talk, we will summarize major recent development on the global aspect of the Gan-Gross-Prasad conjecture, which relates the central special values of L-functions and the automorphic periods. After that, we will survey its application in number theory and arithmetic geometry, some known and some speculated.

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Tits' construction of exceptional Lie algebras and Moy-Prasad filtrations
Hung Yean Loke, National University of Singapore

In this talk, I will first quickly review Tits' construction of exceptional Lie algebras using Jordan algebras and octonion algebras.

Next I will explain how this will lead to embeddings of buildings of exceptional groups.

Finally we recall Gan and Yu results in which they identify the building of split $F_4$ (resp. building of $G_2$) with certain set of lattice functions on the Jordan algebra (resp. octonion algebra). We will use these lattice functions and Tits' construction to obtain Moy-Prasad filtrations of the split Lie algebra of type $E_8$.

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K-type, moment map and local theta correspondence
Jiajun Ma, National University of Singapore

In this talk, I will present two joint works with Loke, both use moment maps as an essential tool. One studies the K-types of the (full) theta lifting in the Archimedean case, which leads to the correspondence of associated cycles; Another studies refined minimal K-types in the p-adic case, which leads to an explicit description of the correspondence between (tamely ramified) supercuspidal representations.

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First occurrence indices of tempered representations of metaplectic groups
Ivan Matic, Josip Juraj Strossmayer University Of Osijek, Croatia

We will discuss a determination of the first occurrence indices of tempered representations of metaplectic groups in terms of the first occurrence indices of their partial cuspidal support, in the metaplectic-odd-orthogonal reductive dual pair over a non-archimedean local field.

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Exceptional poles of some L factors for GL(n)
Nadir Matringe, Université de Poitiers, France

For F a nonarchimedean local field, some L factors of GL(n,F) (for example L factors of pairs) have an integral representation which naturally involves the mirabolic subgroup. Cogdell and Piatetski-Shapiro defined the notion of an exceptional pole for such factors.

For generic representations, the occurrence of such a pole at zero should characterise the existence of an appropriate local period on the representation. As a consequence, classifying distinguished representations in terms of their cuspidal support allows one to obtain the inductivity relation of such factors. We will explain this.

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On special Bessel periods and the Gross-Prasad conjecture for SO(2n+1) x SO(2)
Kazuki Morimoto, Kyoto University, Japan

I will talk about one particular case of the global Gross-Prasad conjecture. I will show that if an irreducible cuspidal automorphic representation $\pi$ of $SO(2n+1)$, which is generic at almost all places, admits special Bessel periods corresponding to a quadratic extension $E$, then $L(1 \slash 2, \pi) L(1 \slash 2, \pi \times \chi)$ does not vanish where $\chi$ is the quadratic character corresponding to the quadratic extension $E$. In particular, when $n=2$ and $\pi$ corresponds to a Siegel modular form for $Sp(4, \mathbb{Z})$, I will prove the equivalence between the non-vanishig of the special Bessel period and that of the corresponding central $L$-value. This is a joint work with Masaaki Furusawa.

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Gamma factor and converse theorem
Chufeng Nien, National Cheng Kung University, Taiwan

This talk is about gamma factor, converse theorem for cuspidal representations of GL(n) over finite fields and related problems.

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Conormal variety on exotic Grassmannian
Kyo Nishiyama, Aoyama Gakuin University, Japan

Let $V = \C^n$ be a complex vector space and $W = V \oplus V$ with the standard symplectic strucure, and consider the Grassmannian $\Gr_n(W)$ of $n$-dimensional spaces in $W$.

Let us put $X = \Gr_n(W) \times V$ with the diagonal action of $K = GL(V)$. We call $X$ an exotic Grassmannian (or we may call it "enhanced" Grassmannian instead). Note that $(G=Sp(W), K=GL(V))$ is a symmetric pair, and we are in the situation of double flag variety in the sense of Ochiai-N (2011).

Our goal is to consider a moment map for $K$-variety $X$; defining a conormal variety and an "exotic" nilpotent variety $\ExoticNullcone_X$; and then establishing a "Steinberg-Springer" theory for them (cf. Henderson-Trapa (2012)).

In the talk, I will describe the correspondence between the $K$-orbits in $X$ and exotic nilpotent orbits in $\ExoticNullcone_X$.
We can also play the same game for the "exotic" Lagrangian Grassmannian.

This is a joint work in progress with Lucas Fresse.

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Classifying $A_{\mathfrak{q}}(\lambda)$ modules by their Dirac cohomology
Pavle Pandzic, University of Zagreb, Croatia

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.

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The character and the wave front set correspondence in the stable range
Tomasz Przebinda, University of Oklahoma, USA

We relate the distribution characters and the wave front sets of unitary representation for real dual pairs of type I in the stable range.

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Exceptional Theta Correspondences for D_4
Gordan Savin, University of Utah, USA

We report on a recent progress, joint with Gan, on the exceptional theta correspondences for dual pairs involving a triality D_4.

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The local Jacquet-Langlands correspondence: Endo-classes and congruences
Vincent Sécherre, University of Versailles Saint-Quentin, France

Let F be a non-Archimedean locally compact field of residue characteristic a prime number p, and let G be an inner form of the general linear group GL(n,F). Attached to any complex discrete series representation of G, there is an invariant called is endo-class, arising when restricting the representation to certain pro-p-subgroups of G. It is conjectured that the local Jacquet-Langlands correspondence preserves endo-classes. We prove that two discrete series representations of G have the same endo-class if and only if they are related via a chain of congruence relations mod finitely many prime numbers different from p. Thanks to recent work with A. Minguez on the modular Jacquet-Langlands correspondence, this allows to reduce the aforementioned conjecture to the case where the discrete series representation of G and its transfer to GL(n,F) are both cuspidal. This is joint work with Shaun Stevens.

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Motives with Galois group of type G_2 - construction of Gross and Savin revisited
Sug Woo Shin, The University of California, Berkeley, USA

Serre asked whether there exists a motive (over Q) with Galois group G_2. Put it in another way, the question is to find (a compatible family of) ell-adic Galois representations whose image has Zariski closure G_2. This has been answered affirmatively since 2010 by Dettweiler and Reiter, Khare-Larsen-Savin, Yun, and Patrikis (including generalizations to exceptional groups other than G_2). In this talk I revisit the construction of Gross-Savin (which was conditional when proposed in 1998) which aims to realize such a motive in the cohomology of a Siegel modular variety of genus 3 via exceptional theta correspondence between G_2 and PGSp_6. Then I will explain that the construction is now unconditional due to my recent work with Arno Kret on the construction of GSpin(2n+1)-valued Galois representations in the cohomology of Siegel modular varieties.

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Kazhdan's orthogonality conjecture for real reductive groups
Binyong Sun, Chinese Academy of Sciences, China

We prove that Kazhdan's orthogonality conjecture holds for all real reductive groups G in Harish-Chandra's class, namely the Euler-Poincare pairing of two Harish-Chandra modules of G is equal to the elliptic pairing of the corresponding global characters. This is a joint work with Jing-Song Huang.

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Hecke algebra correspondences for the metaplectic group
Shuichiro Takeda, University of Missouri, USA

We define a certain Iwahori Hecke algebra on the metaplectic group Mp_{2n} over a non-archimedean local field by using a minimal type of the Weil representation, and show that this Hecke algebra is isomorphic to the Iwahori Hecek algebra on SO_{2n+1}. This isomorphism implies an equivalence of the Bernstein component containing the even Weil representation of Mp_{2n} and the Bernstein component containing the trivial representation of SO_{2n+1}. Our result generalizes an earlier result by Gan-Savin, which shows the same equivalence with the assumption that the residue field of the local field is odd. This is a joint work with A. Wood.

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Some genuine small representations of a nonlinear double cover and the model orbit

Let G be the real points of a simply laced, simply connected complex Lie group, and G~ be the nonlinear two-fold cover of G. We will discuss a set of small genuine representations of G~, denoted by Lift(C), which can be obtained from the trivial representation of G by a lifting operator. The representations in Lift(C) can be characterized by the following properties: (a) the infinitesimal character is \rho/2; (b) they have maximal tau-invariant; (c) they have a particular associated variety \O. We will use the split real group of type D_{2n} as an example to exhibit the following further consequences: (1) all representations in Lift(C) are parametrized by pairs (central character, real form of \O); (2) The K-structure of the regular functions on a real form of \O is multiplicity free and matches up with the K-types of the small representations in Lift(C) attached to that real orbit.

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Pro-p Iwahori Hecke algebra, inverse Satake transform and change of weight in characteristic p
Marie-France Vigneras, Institut de Mathématiques de Jussieu, France

Let C be an algebraically closed field of positive characteristic p and let G be a reductive p-adic group. The classification of the admissible irreducible representations of G over C has been reduced to the classification of the supercuspidal ones by Abe, Henniart, Herzig and V. The proof uses the (injective) general Satake transform and the change of weight (proved using the pro-p Iwahori Hecke algebra of G). The image of the general Satake transform was an open question.

With N. Abe and Fl. Herzig, using the pro-p Iwahori Hecke algebra, we can now describe the image of the general Satake transform and give an explicit formula for its inverse. The explicit inverse Satake transform implies easily the change of weight.

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Langlands parameters and finite-dimensional representations
David Vogan, Massachusetts Institute of Technology, USA

A standard way to describe the representation theory of reductive groups over local fields is to say that compact groups are easy, so we should try to understand representations in terms of their restrictions to compact subgroups. I will argue that compact groups are in fact {\it not} that easy, and that we should try to understand their representations as restrictions of representations of noncompact reductive groups, and therefore in terms of arithmetic and Langlands dual groups. I'll explain how this looks for maximal compact subgroups of real reductive groups (which are more or less understood) and for finite Chevalley groups (which George Lusztig understands, but almost nobody else does).

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Waldspurger formula over function fields
Fu-Tsun Wei, Institute of Mathematics Academia Sinica, Taiwan

In this talk, I will present a function field version of the Waldspurger formula for the central critical value of Rankin-Selberg convolution L-functions. This formula says that the central critical value in question shows up in the ratio of "global period integral" to the product of "local period integrals". Applications of this formula will be discussed if time allows.

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L-groups for double covers of Chevalley-Steinberg groups
Martin Weissman, Yale-NUS College

I have recently proposed an L-group for Brylinski-Deligne covers of quasisplit reductive groups over local and global fields. In this talk, I will focus on the crucial case of double covers of quasisplit simply-connected absolutely simple groups (e.g., Sp(4), Spin(6,6), SU(2,1) ), with most examples over the real numbers. In this case, no familiarity with gerbes or with Brylinsnki-Deligne extensions is required. The L-group can be used for a choice-free parameterisation of discrete series representations over R, and of unramified representations over p-adic fields.

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Towards homotopy methods in representation theory
Tian An Wong, The City University of New York, USA

Recently J. Bernstein has suggested considering sheaves on algebraic stacks BG as a way of studying representations of all pure inner forms of G at once, over a local field. Motivated by Morel and Voevodsky's homotopy theory of schemes, I will indicate certain possibilities of extending Bernstein's suggestion, namely through the introductio of simplicial sheaves. One hopes that this will lead to a model category and other higher categorical structures, as in the case of Morel and Voevodsky.

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On the lifting of Hilbert cusp forms
Shunsuke Yamana, Kyoto University, Japan

Starting from a Hilbert cusp form, I will construct holomorphic cusp forms on certain Hermitian symmetric domains of higher degree.

In the case of Siegel cusp forms, this is a generalization of the Saito-Kurokawa lifting from degree two to higher degrees and the Ikeda lifting from the rational number field to totally real number fields.

This is a joint work with Tamotsu Ikeda.

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Tutorial on automorphic descent
Lei Zhang, National University of Singapore

These courses are intended to give a brief introduction of Arthur's endoscopic classification of automorphic representations of classical groups and explicit module constructions via automorphic descent. More precisely, the endoscopic classification parametrises those automorphic representations by the certain self-dual automorphic representations of general linear groups. On the other hand, in order to enhance our understanding of those representations, a natural question is how to construct concrete modules corresponding to Arthur's parameters. One idea is to use the structure of Fourier coefficients inspired by the idea of Roger Howe on the notion of rank and wave front set. By using certain Fourier coefficients of residual representations, the automorphic descent method of Ginzburg-Rallis-Soudry constructs generic cuspidal automorphic representations. To extend their method into great generality, Jiang and I formulate a general construction of cuspidal automorphic representations in generic Arthur packets.

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Shimura operators, Okounkov polynomials and their positivity
Genkai Zhang, Chalmers University of Technology, Sweden

Shimura defined a remarkable family of invariant differential operators on Hermitian symmetric spaces. These operators are formally nonnegative and form a basis for all invariant differential operators. We prove that the Harish-Chandra homomorphisms of the Shimura operators are the Okounkov polynomials. We determine the set of real parameters of spherical representations for which all eigenvalues of Shimura operators are nonnegative for the groups $SU(2, N)$. For rank one groups $SU(n, 1)$ this set is precisely the set of complementary series, and we prove that for $SU(2, N)$ it is a bigger set. This answers partly some question of Shimura.
Joint work with Siddhartha Sahi.

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text:
# Error when trying to color and underline

Here is my piece of code

\documentclass{article}
\usepackage{color,soul}
\begin{document}
\begin{quote}
\tt\
\ul{{\color{red}text}}.
\end{quote}
\end{document}

This simple piece of code generates a word "text" in red with an underline. However, if you just add a space after the word "text" or a space followed by a word, like

\documentclass{article}
\usepackage{color,soul}
\begin{document}
\begin{quote}
\tt\
\ul{{\color{red}text text}}.
\end{quote}
\end{document}

it will generate an error like

! Package soul Error: Reconstruction failed.

See the soul package documentation for explanation.
Type  H <return>  for immediate help.
...

l.9 \ul{{\color{red}text text}}
.

Can somebody tell me why is that? How can I do both color and underline but I can control the color of the underline?

• Try {\color{red}text} text inside \ul
– user31729
Jun 16, 2017 at 0:51
• ... and don't use the deprecated \tt command.
– user31729
Jun 16, 2017 at 1:00
• Your code compiles without problem, but it's clear that your example code differs from the error.
– Werner
Jun 16, 2017 at 1:07
• Can you fix your code to replicate the error? Otherwise this question is unclear.
– Werner
Jun 16, 2017 at 2:00
• I just added the piece that generates the error message. Jun 16, 2017 at 13:44

Grouping with \textcolor works:

\documentclass{article}
\usepackage{color,soul}
\begin{document}
\begin{quote}
\ttfamily
\ul{{\textcolor{red}{text}}}.
\end{quote}
\end{document}

The color of the underline can be controlled with \setulcolor:

\documentclass{article}
\usepackage{color,soul}
\begin{document}
\begin{quote}
\ttfamily
\setulcolor{cyan}
\ul{{\color{red}text}}.
\end{quote}
\end{document}

Or rewritten to move the color command out of \ul's argument:

\documentclass{article}
\usepackage{color,soul}
\begin{document}
\begin{quote}
\ttfamily
\setulcolor{cyan}%
\textcolor{red}{\ul{text}}.
\end{quote}
\end{document}

• In this case it also works using \ul{{\color{red}text}}...
– Werner
Jun 16, 2017 at 1:09


text:
# Why n and what does it mean?

Maybe a bit too early to ask, but I give it a shot. I'm trying to study mathematics again during my free time. Now I tried to figure out what exactly is meant with the multiple. Well this formula makes it quite easy to understand for me, but it's not totally clear yet:

a = n x b

So if I understand the formula correctly, for 15 = 3 * 5 15 is a multiple of 5, and for 15 = 5 * 3 15 is a multiple of 3, so 3 and 5 can both be a multiple of 15. But what I don't understand, why n? Why not just a = b x c? I cannot really find an answer for this either, in my study book they just start with formulas like:

an x a m = a n+m

without any further explanation why to use n or m. Hopefully somebody has an explanation for me

• What you call a variable doesn't matter. The underlying math is the same. However, there is some convention, like $n$ and $m$ are usually integers. $x,y$ are usually real numbers. – Newb Sep 23 '16 at 10:15
• haha woops, typo. But ok, so it doesn't really matter, but how n and m for integers, and x and y for real numbers. Thought they are the same, both rational numbers right? – Erik van de Ven Sep 23 '16 at 10:18
• Just found a nice explanation about the difference between real numbers and integers by the way: youtube.com/watch?v=gtiJTRd_DHM Didn't knew real numbers consists of integers as well as rational numbers and integers are, well, just integers. Now it's absolutely clear. – Erik van de Ven Sep 23 '16 at 15:11

Why not? Any letter or symbols work. You could have emoticons such that

• n and m are commonly used for integers and x and y is usually more just about everything. However in all contexts the type is made clear by for example saying $n,m\in\mathbb{Z}$ and so on, however one can use any letter for anything as long as the representation and source is made clear. In my work I have used n and m for well....everything and anything, wether it is integer or not just because of comfort on the keyboard. – Zelos Malum Sep 23 '16 at 10:23
• @ErikvandeVen There definitely are some more common conventions, e.g. $n$ and $m$ are often integers. But this is far from universal, so you always have to make sure what the variables are defined as in the text. – Eff Sep 23 '16 at 10:23
The author probably wanted to stress that he was talking about integer multiples of $b$. $n$ is commonly used to denote natural numbers. Of course, $a$ and $b$ are also naturals here, but the formula could also work with $a,b$ rational or real.
It is also pretty common to work with letters in the same alphabetic range to denote a similar role/type. For instance, a polynomial will have coefficients $a,b,c,d,\cdots$. For the same reason, $i,j,k,l,m,n$ often denote integers (indexes or counts). $x,y,z,t,u,v,w$ often denote unknowns or variables.