# All Questions

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### The norm of a separable Banach space can be determined by countable continuous linear functionals?

Recently I'm reading Stochastic Equations in Infinite Dimensions, a result is used many times. It is If $E$ is a separable Banach spaces, then there is a sequence $\{ \phi_n \}$ in its dual ...
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### Groups such that all finite-dim representations are finitely presented

Let $G$ be an infinite group. What sorts of finiteness properties can I put on $G$ to ensure the following holds for all $M$? Let $M$ be a finite-dimensional vector space over $\mathbb{Q}$ upon ...
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### Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$

Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always ...
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### Is the domination number of a combinatorial design determined by the design parameters?

Let $D$ be a $(v,k,\lambda)$-design. By the domination number of $D$ I mean the domination number $\gamma(L(D))$ of the bipartite incidence graph of $D$. Is $\gamma(L(D))$ determined only by ...
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### A Bundle associated to a submanifold of $\mathbb{R}^{n}-\{0\}$

Edit: According to the comment of Rayan Budney I revised the question as follows: Let $M$ be a submanifold of $\mathbb{R}^{n}-\{0\}$. We define a $n-1$ dimensional bundle ...
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### Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [on hold]

Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n. We know that $f\equiv 0$. It's call Hausdorff theorem. This theorem is wrong on $\mathbb{R^+}$, a ...
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### proof that the schwarz map defined as ratios of gauss hypergeometric functions is univalent

The ratio of two linearly independent solutions of the Guass hypergeometric differential equation defines a map from the upper half plane to a Schwarz triangle. Everything I read tells me that this ...
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### Solving recurrence relation in 2 variables [on hold]

I wanted to know how to solve recurrence in 2 variables say m,n (suppose matrix). T(m,n) = 3T(m/2,n/2) + T(m/2) + T(n/2) Thanks
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### Integral elements of quaternion algebras with predescribed properties

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
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### ${p^km \choose p^k} \equiv m \pmod p$ [migrated]

Let $p$ be a prime number and $m, k$ two positive integers. Then ${p^km \choose p^k} \equiv m \pmod p$. I've been trying to demonstrate this lemme all the day. Have you got any suggestion? Thank you ...
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### A combinatorial problem - counting the solutions

Consider a square. There are 16 ways to paint its sides with two colors. For convenience, we will represent one color with a blank side, and the other - with a line drawn from the squares' center to ...
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### How to define the 'sample size' for an AICc calculation [on hold]

I'm calculating AICc values based on the formula: AICc = 2k - 2ln(SSE/n) + (2k(k + 1))/(n - k - 1) Where: k = number of parameters n = sample size SSE = sum of squared error from the model fit. I'm ...
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### Invariant measures on a compact metric space

I'm dealing with a continuous flow on a compact metric space $X$, and $\mu$, $\nu$ are two invariant Borel probability measures on $X$. If I know that $\mu(A)=\nu(A)$ for all the invariant Borel ...
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### Integer point in a non-empty polytope

I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...
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### Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded Lie algebra" as explained first in ...
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### Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?

A tournament is an orientation of a complete graph. A feedback arc set is a set of arcs in a digraph whose removal leave the digraph acyclic. The feedback arc set problem consists in finding a ...
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### Reference for statement that almost every $n$-element partial order has trivial automorphism group

I'm looking for a reference for the statement that almost every partial order on $n$ elements has trivial automorphism group. I've been told that this is a folklore result. Does anyone know of a ...
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### Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse. Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...
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### why do automorphisms preserve ample divisors?

Let $X \hookrightarrow \mathbb{P}$ be a smooth hypersurface inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$. ...
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### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

R is a local Noeth. What is geometric interpretation and of 1- Gorenstein rings 2- Complete intersections 3- regular rings? how can I realize differences by geometric interpretation
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### Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
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### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tg+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
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### Who made the famous error in calculation that 'wasted' the final years of his life?

Sorry, I am merely a Middle School maths teacher at an Australian secondary school. I remember reading years ago about a famous mathematician (18th or 19th Century?) who calculated table upon table of ...
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### Relation of elements in a set [on hold]

This is an elementary question. Membership is obviously the fundamental relation in set theory. But what is the relation--if there is one--of elements within a set? That is, suppose I have the ...
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### on a family of CM Hodge structures

I am having a look at the paper "Hodge structures of type $(n, 0, \ldots, 0, n)$" by Totaro. At the very beginning he says that a Hodge structure of type $(1, 0, 1)$ has always complex multiplication. ...
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