# All Questions

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### averages over integer points of the sphere

A paper of William Duke sketches a proof that integer points on the sphere are equidistributed. $$V_N = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = N \}$$ Up to reflections across the $x$, $y$ ...
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### Hypergraphs that can be represented by simply closed curves respecting edge intersection

Good evening. Hypergraphs can be drawn by representing each vertex as a point in the plane and each hyperedge as a closed curve that contains the points corresponding to the vertices that belong to ...
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### Reference that contains examples of absolutely indecomposable representations of quivers over a finite field

Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...
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### $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts: Lemma Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two progressivley measurable processes, ...
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### Dihedral extension of 2-adic number field

Sorry if the question is too long and maybe elementary. I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension ...
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### Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$? If ...
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### Converse to Weil Restriction of Scalars

Let $k$ be a field of characteristic zero (I'm only interested in number fields), and let $\mathbb{G}_{/k}$ be a linear algebraic group defined over $k$ which is almost $k$-simple (all normal ...
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### Degenerate linear recurrence sequences

Let $(u_n)_{n \geq 0}$ be a linear recurrence given by $$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$ where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall ...
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### Diffusion in a bounded domain

Let us consider an $\mathbb{R}^d$ diffusion $$dX_t = dW_t +\mu(X_t)dt.$$ Let further $D\subset \mathbb{R}^d$ be a bounded connected open domain. By $Y^D$ we denote the diffusion $X$ restricted to ...
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### Integral Domains [migrated]

I have to proof that ($\ N_A,1_A,T)$ are a Peano system where $\ T:N_A\rightarrow N_a$ $\ x \mapsto x+1_A , x \in N_A$ and $\ N_A= \{n 1_A|n\epsilon N \}$ where N are the natural numbers and A is an ...
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### $H^1$ convergence of eigenfunctions of Schrödinger operators

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$H^D=-\Delta + V$$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
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### Asymptotic sequence and asymptotic expansion [on hold]

If $(f_k(x))$ is an asymptotic sequence as x to infinity and $\phi=a_0f_0+a_1f_1+a_2f_2+...$ (equality) where $a_i$ are constant. Is $a_0f_0+a_1f_1+a_2f_2+...$ an asymptotic expansion of $\phi$?
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### Good covering of a sphere

Consider a sphere $S_r(0)$ with center at zero and radius $r$ in the Hamming space $\{0,1\}^n$. We will be interested in covering this sphere with balls of radius $\rho < r$. We know that there ...
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### The right expansion of a square root matrix

I have some issues in finding an asymptotic expansion for a square root matrix and I have already posted a question (Asymptotic expansion square root matrix). Somebody redirected me to a post where ...
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### Spectral Sequences of Parametrized Spectra

I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up: Suppose that I have a parametrized spectra ...
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### Intersection and union of torsion classes

One of the main result in Cassidy, C., M. Hébert, and G. M. Kelly. "Reflective subcategories, localizations and factorization systems." Journal of the Australian Mathematical Society (Series A) ...
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### “For sufficiently large” vs. “For all sufficiently large” [on hold]

A purely grammatical question: Do people generally prefer: "For sufficiently large x,..." or "For all sufficiently large x,..." or not care? Or might you use either according to context? The meaning ...
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### Category of Gödel Codings? [Reference Request]

Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...
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### Global existence for infinite dimensional ODE

Let us consider the ODE $\hskip3pt \dot x=F(t,x)\hskip3pt$ in an infinite-dimensional Banach space $E$, where the flux $F$ is defined and continous from the whole $\mathbb R\times E$ into $E$. ...
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### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in). I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...
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### dimension of a scheme and degree of an L-function [on hold]

I try to understand correctly the notion of scheme, as Serre in the third volume of his Oeuvres defines zeta and L-functions in this context. What seems interesting to me is that he states a theorem ...
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### Deligne-Lusztig and Character sheaves

Consider: $G$ - a nice group ($GL_n$) over a finite field $F$. $X$ - the flag variety. Consider a nice $G$-equivariant $l$-adic sheaf $\mathcal{M}$ on $X \times X$, equipped with Weil structure. Fix ...
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### Stalks of higher direct image under open embedding

Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...
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### Errata for Getzler-Kapranov “Cyclic operads and Cyclic homology”

Do you know if anyone made an errata for the Getzler-Kapranov paper "Cyclic operads and Cyclic homology"? I was trying to read it, and I have found (at least I think I did) quite a few typos not all ...
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### inverse of a matrix expression

$X_i = (I - P(I - t_i^Tt_i))^{-1}$ where P is a $N\times N$ matrix and $t_i$ is vector of $N$ elements. Is there a way to simplify this expression in order to calculate the inverse matrix only once ...
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### Encoding and Transforming Data for a Logistic Regression

When running a logistic regression, the result of the regression is a value that could fall in $(-\infty, \infty)$. You run it through the logistic function and get a value in $(0, 1)$. So far, so ...
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### Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$. Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...
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### Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional. It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is: ...
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### Motivic integration in positive characteristic: how much is known?

It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
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### On quasi-equivalence of norms [on hold]

Two norms $p(v),q(v)$ are equivalent if there exist two real constants $c,C$, with $c > 0$ such that for every vector $v$ in $V$, one has that: $c q(v) ≤ p(v) ≤ C q(v)$. If $V$ is discrete module ...
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### does log liklihood can take a value greater than 1??or its between 0 and 1? [migrated]

please anybody who know the answer reply to my question..i got the log likelihood probability value as -34.82 so I am not getting whether the answer which i have got is right or not..
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### vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a ...
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### Linear intersection number and coloring (not chromatic) number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
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### Independence Number of Graphs

Suppose $\alpha(G)\leq\alpha(H)$ where $G$ and $H$ are graphs, and $\alpha(.)$ is the independence number of graph. Is the following statement true? $\alpha(G\boxtimes G) \leq \alpha(H\boxtimes H)$ ...
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### Product of “prime” topological spaces

We call a topological space $(X,\tau)$ product-decomposable if there is an index set $I$ and subsets $X_i\subseteq X$ for $i\in I$ such that $|X_i| > 1$ and $X \cong \prod_{i\in I} X_i$ where each ...
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### Algebra II Probability [on hold]

I can't figure out multiple selection probability for pickles. Three cards are selected from a standard 52 deck. What is the probability of selecting a king, queen, or red card?
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### A book about periodic functions [on hold]

Can anyone give me suggestions for new books about property of periodic functions? Thanks a lot.
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### mobius transformation [on hold]

For $|z|<1$ define $f(z)$ by $$f(z)=\exp({-i\log[i(1+z/1-z)]}^{1/2})$$. (a) Show that $f$ maps the unit disk $D$ conformally onto an annulus $G$. (b) Find all Mobius transformations $S(z)$ that ...
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### Upper bound for chi-square divergence in terms of KL divergence

In my research I need an upper bound for chi-square divergence in terms KL divergence which works for general alphabets. To make this precise, note that for two probability measures $P$ and $Q$ ...
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### On Descartes / spoof odd perfect numbers

Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and ...
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### When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$. When $S$ is an analytic space, there is an exact sequence 0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...
Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...