# All Questions

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### Computer Program for Calculations in Tensor Algebra Quotients

Let $V$ be a finite dimensional vector space, and $S$ a finite dimensional subspace of its tensor algebra ${\cal T}(V)$ for which $X := {\cal T}(V)/<S>$ is finite dimensional, where $S$ is the ...
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### Number of Dyck paths with k returns and b peaks

The number of Dyck paths from the origin to $(2n,0)$ which touch the $x$-axis $k+2$ times ($k$ internal touches) is given by $$\frac{k}{2n-k}{2n-k \choose n}.$$ The number of Dyck paths from the ...
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### a question about the branches of $\log z$ [on hold]

Let $U$ be a connected open set in the complex plane $C$ and there exists a coninuous mapping $f:U\longrightarrow C$ such that for every $z\in U$,$f(z)^2=z$. I want to ask if there must exist a ...
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### Which homology classes from loop space?

Fix a closed connected manifold $Q$ and let $LQ$ denote its free loop space. We can get second homology classes on $Q$ by "doing things" to loops in $Q$. For instance, if we have a loop of loops, it ...
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### Chance of pulling (game) cards [on hold]

What is the chance of pulling 6 cards in the correct order? The game I play gives you 3 cards at random to play out of my 6 card deck to start with. From the 3 open cards given you pick one to ...
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### Automorphism group of a modular curve and its action on the set of cusps

Let $X$ be a modular curve, that is the compact Riemann surface obtained by adding cusps to a quotient of Poincaré half-plane $\mathbb H$ by a congruence subgroup $\Gamma$ of $SL_2(\mathbb Z)$. The ...
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### how do binary logic extend to algebra and find an algebra for 3 valued logic [on hold]

how do binary logic extend to set theory and then extend to algebra? and find an algebra for 3 valued logic? i guess no longer use add or multiplication it will use from logic 1 to logic 16 and ...
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### Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)). I'm to teach the ...
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### what is the first non-constant term in the Kronecker Limit formula?

The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...
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### Max flow with minimal requirements algo problem

While applying the algorithm to solve the max flow of the network with minimal requirements on edges, I have encountered a problem. The algorithm states: For graph G create an edge from target to ...
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### is a network a sum of its subnetworks?

I was wondering if networks/graphs are the sum of their parts. Let's say you have a 15-node network. The spectral density of that network has X kurtosis and Y skewness. You also have a 20-node ...
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### Hausdorff dimension of wandering set

I am searching some papers about the dimension of wandering set. It seems that there are more papers considering the non-wandering sets. I will appreciate if any references are recommended.
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### Action of the pure braid group on the commutator subgroup of a free group

Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...
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### Optimization - Linear or Non Linear? [on hold]

I'm a total newbie so pardon me, here is my problem. I want to find out the maximum value of a loan I can give to a customer so that the annual simple interest rate does not exceed say 25% and the ...
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### Analytical solution of diffusion PDE with Robin boundary condition

I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk with subject to Robin boundary conditions. The formulation is as follows: ...
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### Is it provable in $\mathsf{ZF}$ that there is a group structure on any set $X$? [duplicate]

Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?
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### Lie bialgebras cohomology

I am wondering if there exist a universal construction of (co)-chain complex associated to a given algebra to study deformations as Hochschild homology for associative algebras, or Chevalley-Eilenberg ...
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### How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?

Suppose we are given three algebraic numbers $\alpha,\beta,\gamma$ by presenting their minimal polynomial (degree less than $m$), the goal is to compute all positive integers $n$ such that $\alpha^n$ ...
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### Inclusion-preserving bijection between subsets of cardinality k and n-k

Let $n$ be a positive integer. A subset of $[n] := \{1,2,...,n\}$ having $k$ elements will be called a $k$-subset. For $n,k \in \mathbb{N}$ with $k \leq \lfloor n/2 \rfloor$, it is clear that one can ...
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### Bounded operator on a normed space with empty spectrum

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space. Is there any example of bounded operator acting on a ...
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### How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?

To make the question more precise: We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$. Let $\mathcal{C}$ be ...
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### About Noncommutative Geometry [on hold]

I have some questions following: 1- what is the noncommutative geometry? 2- what are the prerequisites to study noncommutative geometry? 3- what are the branches of the noncommutative geometry? ...
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### Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)

Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...
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### abstract affine representations of semisimple Lie groups

in 1933 van der Waerden proved that any abstract unitary representation of a compact semisimple Lie group is necessarily continuous. Is any kind of similar result known for abstract affine ...
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### Mathematical simulation of viscous material behaviour

I have a non linear first order differential equation of the type: $[y(t)]^n + a \frac{dy(t)}{dt} = b(t)$ where $y(t)$ is a real function, the exponent n is a real number greater than $2$, but not ...
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### Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?

Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...
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### Question on inverse normal distribution [on hold]

We we're asked the following question among many, however I'm not quite sure how to start: InvNormal(x) + InvNormal(1-x) = ? Is this homework? Absolutely, ...
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### Central limit theorems for unequal probability sampling (weak but ill-defined dependence)

Suppose we are choosing samples of size $s$ from a finite population $\{a_1, a_2, \dots , a_n\}$ where our sampling is with unequal probabilities. Construct $$S_n = \sum_{k=1}^{n} a_k$$ Under what ...
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### counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer. Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...
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### What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
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### References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
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### What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$?

EDIT: Thanks for your answers and comments. There is indeed a classical easy formula, given by Pietro Majer (with a simple nice proof) in his answer below. Given $x\in\mathbb{R}^n$, $x_i$ denotes ...
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### Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question: Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
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### Newvectors in tensor product representations

Let $\pi_p$ and $\pi_p^\prime$ two smooth admissible irreducible complex representations of ${\rm GL}_2(F)$ where $F$ is a non archimedean local field of residual characteristic $p$ of central ...
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### Do we know that 'most' finite groups are Galois groups of number fields?

The inverse Galois problem is a classical problem in mathematics and asks whether every finite group can be realized as the Galois group of a finite field extension of the rational numbers. The ...
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### Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$

Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this ...
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### Elementary embeddings with the same critical point

Question: Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j_n \circ k_n$ for ...
### another question about connected open sets in $R^2$
Before posting this question,I just asked a similar question:a question about connected open sets in $R^2$. I got several nice answers.Now I want to ask: Let $U$ be a nonempty connected open set in ...