1
vote
0answers
20 views

Simultaneous root of polynomials — must it exist by continuity?

Suppose we have $n$ polynomials in $n$ variables $p_1, \dots, p_n$ and $n$ scalars $y_1, \dots, y_n$ which are in the range $[0,1]$. These polynomials have all positive coefficients. We want to find a ...
1
vote
0answers
5 views

Why do rotationally ordered configurations have well defined distributon function?

Let $u=(u_{j})_{j \in \mathbb{Z}}$ where $u_{j}\in \mathbb{R}$ for all $j \in \mathbb{Z}$ be a rotationally ordered configuration i.e. $S_{n,m}u>u$ or $S_{n,m}u<u$ or $S_{n,m}u=u$ where ...
-1
votes
0answers
49 views

Is Mumford's statement about the representability of some functor wrong?

I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface. It is a statement about the representability of some functor. It is stated on page 108 and says the ...
2
votes
1answer
45 views

A variation on Bulgarian solitare

It appears that a variation on Bulgarian solitare has a fixed point regardless of the starting $n$. For example, let $n=69$, and consider this partition: $$ (8,8,7,7,5,5,5,5,5,4,3,3,2,2) $$ In ...
0
votes
0answers
28 views

Definition of J-function of cotangent bundle of flag variety

Braverman-Maulik-Okounkov showed that the Givental's $J$-function of the cotangent bundle of the complete flag variety becomes an eigenfunction of trigonometric Calogero-Moser Hamiltonian. Recently, I ...
1
vote
0answers
17 views

Controling mixed derivatives

This is a cross-post from Math.SE since the question got nothing (but upvotes) even after offering a decent bounty. If it is too trivial or in other ways not suited for this site, please let me know ...
1
vote
0answers
35 views

Reference on calculation of 2nd cohomology group

Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...
2
votes
0answers
18 views

Inverse Fourier of $\omega^{-1+{\rm i}\alpha} u(\omega-1)$

Let $\alpha$ be an arbitrary real number and define \begin{align} \widehat{f}(\omega)=\left\{\begin{array}{ll} \omega^{-1+{\rm i}\alpha}, & \omega>1,\\ 0, & \textrm{otherwise}. \end{array} ...
0
votes
0answers
76 views

Higher Homotopy Groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
5
votes
0answers
63 views

Nonunital $E_\infty$-rings

An elementary fact of algebra is that the category of nonunital commutative rings is equivalent to that of $\mathbb{Z}$-augmented unital commutative rings, the equivalence being given by forming ...
1
vote
2answers
44 views

How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...
3
votes
1answer
164 views

An identity for elementary symmetric functions

Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...
4
votes
3answers
144 views

Characterising the adjoint representation of SU(N)

One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an ...
1
vote
1answer
77 views

An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
-6
votes
0answers
24 views

No of rotations by a series of connected cog wheels [on hold]

A machine has 4 cog wheels in connection. The largest wheel has 242 teeth and the others have 66,48 and 28 respectively. How many rotations must the largest wheel make before each of the wheel is back ...
-6
votes
0answers
30 views

Digits <---> Numbers [on hold]

What is the sum of all four digit numbers that are formed using each of the digits 1,2,3 and 4 exactly once? (ans provided is 66,660) Pls explain how...
1
vote
0answers
29 views

The convergence of Matrix factorization

I'm trying to prove the convergence of Matrix factorization. The problem is described below. $|X-WH|^2 + |H|_2^2 +|W|_2^2$. My optimization steps are using Alternating least squares which update H ...
1
vote
1answer
55 views

On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$. Let $\chi_v$ be a unramified unitary character of $F_v$. Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...
0
votes
1answer
62 views

Hilbert scheme of a closed subscheme

Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over ...
8
votes
1answer
445 views

Is there a higher Grothendieck ring?

Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor ...
-3
votes
0answers
25 views

Approximation of non-Lipschitz (but continuous) functions by Lipschitz functions [on hold]

Is there any algorithm to approximate non-Lipschitz (but continuous) functions by Lipschitz functions ?
0
votes
0answers
10 views

Zeros of non-lipschitz functions (when noisy estimates are available only)

Given noisy (martingale difference) of a Lipschitz continuous function $f$ it is known how to compute zeros of it. It is the stochastic approximation approach (by Borkar, Kushner and Yin etc.). Is ...
3
votes
1answer
59 views

Fixed point relation $\ Fix\ $ for pairs of manifolds

First the classical definition: a topological space $X$ has the fixed point property (fpp) $\ \Leftarrow:\Rightarrow\ $ for every continuous $\ f : X\rightarrow X\ $ there exists $\ p\in X\ $ such ...
3
votes
0answers
66 views

Inverse cohomological isomorphisms

Let $\ M'\ M''\ $ be simply-connected Hausdorff compact manifolds (possibly with boundary for another variant of the question). Let $\ f:M'\rightarrow M''\ $ be a continuous function which induces an ...
-2
votes
0answers
35 views

Infinite subset of a closed set [on hold]

This statement is true or false: If $F$ is an infinite subset of a closed set $E$, then $F$ has a limit point in $E$? The original one is: if $E$ is an infinite subset of a compact set $K$, then $E$ ...
1
vote
0answers
20 views

Non-finitely based varieties and pseudovarieties

The variety of semigroups defined by $B=\{xyzt=xzyt, yx^ny=xyx^{n-2}yx:n=2,3,\ldots\}$ is non-finitely based (Perkins). Is the pseudovariety defined by $B$ also non-finitely based? More generally, if ...
1
vote
1answer
77 views

Does projective duality preserve arithmetic-Cohen-Macaulay-ness?

Let $V$ be a vector space over $\mathbb{C}$. Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring ...
1
vote
0answers
48 views

Finitely generated groups non-embeddable into $L_1(0,1)$

I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups: (1) Heisenberg group ...
-1
votes
0answers
19 views

BNF grammar for given problem [on hold]

I have a problem (a) Give a grammar using BNF rules to construct a program in the language "witless". A witless program must follow the rules: The program must start and end with the word 'endstart' ...
3
votes
0answers
58 views

Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...
1
vote
0answers
29 views

Ostaszewski space's construction Lemma

I'm studying the Ostaszewski's article "On Countably Compact, Perfectly Normal Spaces". I'll add some context. Lemma 1.2 says the following: Let $X$ be a locally compact, zero-dimensional and ...
4
votes
2answers
160 views

Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint"). I know that all the eigenvalues of matrices in this ...
3
votes
0answers
43 views

Limits and colimits in a 2-category vs. in an infinity category: the non- (2,1)-case

Suppose that $\mathscr{C}$ is a $2$-category (or more generally a bicategory) which is not a $\left(2,1\right)$-category. Is there any relation between limits and colimits in $\mathscr{C}$ (in the ...
3
votes
0answers
79 views

Examples of unproven but likely true existential sentence (in the sense of incompleteness)

Some examples of universal statements that are unproven but likely true include the Riemann hypothesis (all non-trivial zeros of the zeta function have real part 1/2) and the Goldbach conjecture (all ...
2
votes
1answer
42 views

Criteria for abstract polytopes to be convex polytope

Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
0
votes
1answer
42 views

Gaussian expectation of a exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation, $$ E\left[ \exp(\mathbf{xx}^\top)\right]$$ where $\exp(\cdot)$ is element-wise exponential function (not ...
0
votes
0answers
66 views

Collatz property implying infinite “fall below” trajectories, is it known?

(this was discovered analyzing Collatz empirically.) a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value. consider ...
-1
votes
0answers
32 views

On a sum statistically independent of its term [on hold]

Suppose $U$ and $V$ are two non-degenerate random variables, say real-valued for simplicity. Suppose further that their sum, $U+V$, and one term, $U$, are statistically independent. This happens when ...
12
votes
7answers
520 views

Where to find (personal) motivation [on hold]

I think it would be appropriate to make this question CW... It is likely that this question will not survive here on MO for long, but I do hope that the community gives it a chance. I also hope to ...
1
vote
1answer
28 views

Bi-invariant one forms on compact Lie groups

I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact ...
0
votes
0answers
7 views

Complex parameters in the Ritz procedure

I am using the Ritz procedure to write a trial function as the superposition of other admissible functions, with the coefficients being unknown variational parameters to be determined. The variational ...
1
vote
0answers
71 views

Does the canonical morphism commute with the inverse image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
0
votes
0answers
53 views

A question about subgroups [on hold]

Is there a group $G$ and a non-abelian subgroup $H$ of $G$ such that $[G:H]=2$, $|Z(H)|>1$ and $C_G(H)=2|Z(H)|$?
1
vote
1answer
27 views

Concavity of the solution of a parametric implicit function

Suppose $F(x,y;k)=f(x,y)+kg(x,y)=0$ uniquely defines the solution $y(x;k)$ for $x\in \mathbb{D}$, a compact domain, and $0\leq k \leq 1$ is a parameter. We know that for $k=0,1$, $y(x;0)$ and $y(x,1)$ ...
0
votes
0answers
80 views

Examples of etale group schemes [on hold]

What are important examples of etale group schemes over some field $F$, apart from finite group schemes?
0
votes
1answer
52 views

Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
1
vote
0answers
75 views

What is known about the Brauer group of an arithmetic surface?

Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat. What is known ...
2
votes
3answers
104 views

Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
2
votes
1answer
89 views

Degree of a smooth curve in an abelian variety

Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$. Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in ...
5
votes
0answers
82 views

Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying $b$ is bilinear, ...

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