2
votes
1answer
179 views

Irreducible representation of Heisenberg group with characteristic 2?

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than ...
1
vote
1answer
65 views

Markov chain Monte Carlo: why is non-reversible MC MC not as popular?

I am new to methods for simulating Markov chains in order to sample from the target, unknown distribution. After a couple days of reading, I found out that even though people have realized that ...
2
votes
0answers
27 views

When is the solution to a n initial value problem matrix differential equation invertible? [migrated]

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
0
votes
0answers
11 views

Hello, everyone, I want to ask you a question about a proof in the Terence Tao's Real Analysis notes [migrated]

everyone. I am using Terence Tao's Real Analysis notes to self learn Analysis 1. There is one thing in the proof of Theorem 27 (Least Upper Bound Property) in “week 2 note” that I don’t understand ...
0
votes
0answers
54 views

Why does the Fourier transform of bounded domain, e.g., [-1 1], differs from that of complete domain, e.g., real line. [on hold]

Recently, I asked a question about proving the positive semidefiniteness of following kernel function. $$ k(x, x') = 1 - 2|x-x'|$$ When using the Bochner's theorem, showing psd of the kernel ...
0
votes
1answer
136 views

Composition of a transcendental function with a rational function [on hold]

The problem is: let $f: \mathbb{R}\to \mathbb{R}$ be an analytic transcendental function and let $\psi(x)=\frac{x}{2(1+x^2)}$. Is the function $f(\psi(x))$ transcendental?
1
vote
1answer
95 views

Representations of parabolic subgroups of the general linear group over the complex numbers

In all that follows, we are working over $\mathbb{C}$. Let $B \subseteq P \subseteq {\rm GL}(n)$ be a parabolic subgroup. Can you say anything in general about the representations of $P$? I suspect ...
-3
votes
0answers
69 views

biproduct and tensorial product [on hold]

Let $\mathcal{C}$ be a monoidal abelian category. Let A,B, C $\in$ $\mathcal{C}$.There is an isomorphisms between this objects? (A$\bigoplus$B)$\otimes$C; A$\otimes$(B $\bigoplus$C)
0
votes
0answers
33 views

Sufficient Conditions For Monotonic Decreasing of Multivariate Function

I found the following theorem on sufficient conditions for decreasing monotonicity of an absolutely continuous function $\phi : \mathbb{R} \rightarrow \mathbb{R}$, I would like to know if it is ...
1
vote
0answers
97 views

Standard name for a Monoid/Semigroup with a+b <= a, b?

I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice? For instance, for positive reals a, b define a |+| b === 1/((1/a) + (1/b)), you ...
8
votes
1answer
764 views

What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
1
vote
0answers
40 views

“embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...
4
votes
0answers
112 views

Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients. $$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$ Monomials $x^k$ are mapped to $n ...
1
vote
2answers
48 views

Largest eigenvalue adjacency matrix-link deletion

Let G be a connected undirected graph and G\e be a graph obtained by removing a random link e from the graph G. Let $\lambda_1(A(G))$ be the largest eigenvalue of the adjacency matrix of graph G. Is ...
6
votes
1answer
167 views

When is the profinite completion a pro-$p$ group?

My research area is mainly pro-$p$ groups and profinite groups. However, in the last few year I became also interested in discrete groups. Therefore, it seems to me a natural problem to look for ...
7
votes
1answer
160 views

Block Matrix determinant

Consider the $k \times k$ block matrix: $ C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \ddots & ...
0
votes
1answer
72 views

Absolute irreducibility of a symmetric square?

This is a question I received today by email, which somebody more experienced with finite group representations can probably answer directly. Take $F:=\mathbb{F}_q$ for some prime power $q$, so ...
1
vote
0answers
42 views

Minimal density hitting set for k-length arithmetic progressions

This is problem which came up in the process of designing a game. Thus, I don't know any previous work relevant to the problem. Fix a small set $D$ of a natural numbers. For example, $D=\{1,2,3\}$. ...
16
votes
4answers
566 views
+50

What arrangement of unit cubes minimizes surface area?

Question A. How does one arrange $n$ unit cubes to minimize surface area? Question B. How does one arrange $n$ unit cubes to form a rectangular prism of minimal surface area? Various curricular ...
8
votes
0answers
115 views

Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy? To be more specific, in Which forcings ...
4
votes
1answer
207 views

“Is it possible to give a restricted set-theoretical definition of addition of natural numbers in terms of successor?” [Tarski]

In his paper "Restricted set-theoretical defintions in arithmetic" Raphael Robinson cites a problem posed by Tarski: Is it possible to give a restricted set-theoretical definition of addition of ...
1
vote
1answer
73 views

Results for resolution of equations in polynomial ring

Is there any reference for resoluttion of equations in a polynomial ring, such as $x^2+y^2=z^2$ in $\mathbb{C}[t]$? Thanks!
5
votes
1answer
207 views

Which axioms of ZF are used for finite choice?

Apologies if this is a silly question, not an expert in set theory but just wondering about it. ZF implies finite choice. But let's suppose one wanted to work without it. The thinking here is being ...
18
votes
1answer
491 views

$f(x)$ is irreducible but $f(x^n)$ is reducible

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible (over $\mathbb{Z}[x]$)?
1
vote
1answer
228 views

questions on steenrod algebra

I plan to give a talk on Steenrod algebra in a student seminar. but there are some questions that I didn't find an answer to, and it seems to me that I'm missing something: if the algebra of ...
-3
votes
3answers
100 views

if V(f) is irreducible, then how to show that the polynomial f itself is irreducible? [on hold]

V(f) is the zero locus of the polynomial f in the polynomial ring k[x1, x2, ..., xn] with k an algebraically closed field. If V(f) is irreducible, then how to show that 'f' is irreducible?
-2
votes
0answers
53 views

prime number sequence and the tendency [on hold]

before I ask.. I'm not good at english. (because I'm not an English.) so you may..not be able to understand what I want to say. 1> definite new sequences prime number sequence {Pn} = 2 3 5 7 11 13 ...
0
votes
2answers
176 views

Weil restriction

I've already asked a similar question in SE, without success, so I've decided to post here a more general version of my question. Let $f: Y \to X$ be a finite etale morphism of smooth proper ...
5
votes
2answers
201 views

What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
0
votes
1answer
329 views

Counterexample to Pólya's conjecture

It is known that Polya's conjecture is false and the smallest counter-example is about $10^9$. Assuming that we are searching for a counter-example not knowing that it exists. What useful information ...
1
vote
0answers
80 views

Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com: http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map Question: In this partially Tait's colored map, using ...
2
votes
0answers
83 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
1
vote
2answers
234 views

Asymptotics on prime divisors

Somewhat inspired from the Zsigmondy's theorem is my question. Suppose Let $a_{1}> a_{2}> \cdots > a_{k}$ be nonzero integers, with $k \geq 2$. Let $a(n) : = a_{1}^{n}+\cdots+a_{k}^{n}$ for ...
3
votes
1answer
126 views

Rational homogenous spaces and symmetric spaces

What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...
3
votes
0answers
45 views

Polynomials connected with Gale's condition and cyclic polytopes

I was reading about cyclic polytopes and (I think naturally) became interested in polynomials $f(X) \in \mathbb{R}[X]$ which have a factorization of the form $f(X)=g(X)g(X+1)$ for some $g(X) \in ...
2
votes
2answers
143 views

What is an extragradient method?

I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia ...
0
votes
0answers
57 views

reference help indecomposable representations of SL(2,R)

Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible ...
1
vote
0answers
45 views

symmetric theta structures and arithmetic subgroups

A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface. ...
0
votes
1answer
76 views

A special class of random variables

I'm interested in classes C of $R^1$-valued random variables which possess the following properties: 1) the sum of two independent random variables from class C belongs to class C; 2) for any ...
2
votes
1answer
86 views

How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$ $k(x, x') = (1 - 2|x-x'|)$ is a positive semidefinite function? It turns out to be psd function when ...
0
votes
0answers
62 views

A certain Acyclic Partition of a digraph

Has the following object been defined in the literature? What is it called? And what literature studies it? Are there other characterizations of this? What properties are known? Let $G$ be a directed ...
3
votes
0answers
51 views

Assumption of equal prior message probabilities in the standard proofs of the converse of Shannon's theorem

One of the first steps in the standard proofs for the (weak) converse of the Shannon's theorem (a.k.a. noisy-channel coding theorem) for the discrete memoryless sources is the assumption that messages ...
2
votes
1answer
168 views

What does the defect of a block measure?

In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or ...
3
votes
1answer
196 views

Classes in $H^3(G; \mathbb{Z})$ that restrict to zero on abelian subgroups

Let $G$ be a finite $p$-group. Is it possible to have a nonzero class in $H^3(G; \mathbb{Z})$ that restricts to zero in $H^3(A; \mathbb{Z})$ for every abelian subgroup $A \subset G$? If so, what is a ...
2
votes
2answers
108 views

Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map $$ N = \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0 ...
6
votes
2answers
308 views

Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...
1
vote
1answer
65 views

Algorithm for Polynomial Reduction in a Quotient Ring

Any reference or suggestion for the following problem would be greatly appreciated. I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...
4
votes
1answer
91 views

When are the congruence lattices nicer?

This is a purely idle question, but one I'm increasingly interested the more thought I put into it: For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a ...
-1
votes
1answer
54 views

Solution to simple first-order partial differential equations [on hold]

Is there a general solution for first-order partial differential equations of the form $$m(x) \partial_x f(x,y) = n(y) \partial_y f(x,y)$$ for given $m(x),n(y)$ and reasonable boundary conditions ...
0
votes
0answers
162 views

PBW proof proposal

One version of the PBW theorem states: $\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. I am curious if this is a possible proof for the PBW theorem, part is taken ...

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