# All Questions

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### Positivity of a certain matrix expression

Let $A$,$X_1$, and $X_2$ be $n\times n$ non-negative definite real matrices such that $X_1 > 0$ and $X_2 >0$. Also let us assume that $A+X_1 \leq I$ and $A+X_2$ are diagonal matrices. My ...
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### How many expressions can be formed with $k$ commutative and associative functions?

This is a generalization of a question I posted to Math.SE here. Suppose we have $k$ binary functions $f_1, f_2, \ldots, f_k$, all of which are commutative and associative, and $n$ indeterminates ...
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### A Small Discussion I want to have with Professional Mathematicians [on hold]

My professor is a really learned man, having written and published more than 18 research papers within 20 years. Hence I greatly admire him and value his opinion. He is, of course, a PhD in ...
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### An elementary question about the Goldbach conjecture [on hold]

My daughter asked to me a question whether the following can be considered as an elementary approach for tackling the Goldbach conjecture. I submit to the research professionals on MO this naive ...
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### minimal entropy approximation of a discrete random variable

Let $X$ be a $\mathbb{N}$-valued random variable. Define $$H^\epsilon_n(X) = \inf_f H(f(X))$$ where $f$ runs over all functions $\mathbb{N} \to \mathbb{N}$ such that $\Pr(f(X)\neq X)<\epsilon$ ...
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### How to show validity in classical logic?

Firstly, I would like to know what does it mean to be a valid expression in classical logic. Secondly, How do we show validity of a formula (in sequent calculus) such as: (∀x A → ∃B) → ∃x(A → B) As ...
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The following open problem was shown to me by Maxim Kontsevich. I state it in a different but equivalent form. Let $a(n)$ be the sequence at http://oeis.org/A131868, that is, $$a(n) ... 0answers 52 views ### book about string theory a la Von Neumann Can we summarize string theory (in its actual state) in some principles and fundamental equations like electromagnetism, general relativity, quantum mechanics and classical mechanics ? I am looking ... 0answers 13 views ### Signs associated to self-dual simple objects in a fusion category Every self-dual simple object X in a fusion category can canonically be assigned a number a, from its "snake" associator element: The square of a equals Muger's "squared dimension" of X, an ... 0answers 58 views ### Algebraic Curves: Exercise 2.17 (William Fulton) [on hold] Let V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2, e \overline{X}, \overline {Y} the residues of X,Y in A(V) its coordinate ring; let z= \dfrac{\overline{Y}}{\overline{X}}\in K(V). Find the pole sets ... 0answers 22 views ### Bound from distinct integer summation We want to find r positive integers \{a_i\}_{i=1}^r such that of atmost (2s+1)^r values obtained from$$\sum_{i=1}s_ia_i$$where s_i\in\{-s,-s+1,\dots,0,\dots,s-1,s\}, we insist on some ... 0answers 5 views ### Bits of precision matrix reconstruction We have a real rank r matrix M\in\{0,1\}^{n\times n}. Suppose we have diagonalized using LMR=D. I want to recover a real matrix \widetilde{M} such that maximum absolute entry of ... 1answer 98 views ### Direct image of structural sheaf I am sorry if my question is not of high level!! Let \pi:X\rightarrow Y be a double cover where X and Y are projective smooth curves. Is it true that R^1\pi_*\mathcal O_X=0 ? Why ? Thanks ... 1answer 151 views ### Formal group law over \mathbb{F}_p Let p be a prime. For each n > 0 there is a unique 1-dimensional commutative formal group law F over \mathbf{Z}, F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]], whose logarithm function ... 0answers 19 views ### Enumerating matrices function of ranks Is there an expression/approximate expression for number of real matrices M\in\{0,1\}^{n\times n} of rank r\leq n? 2answers 110 views ### how to calculate the sum of remainders of N? I'm trying to sum the remainders when dividing N by numbers from 1 up to N$$\sum_{i = 1}^{N} N \bmod i$$It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ... 0answers 62 views ### Weyl group representation Let G be a reductive p-adic group. Let W be a weyl group. if x,y in W I want to know in which case we have x y x^-1 = y ? in case if y(θ)=θ where θ is a subset of simple roots, and x is the longest ... 0answers 9 views ### Statistics of strongly connected components in random directed graphs I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for. ... 0answers 54 views ### The most general splitting of a field extension This question has been posted here on math.stackexchange, but I felt it was maybe better to post it here. Take L/K an extension of the field K. I have questions on how we can "split" the ... 0answers 28 views ### format of grading Witt Lie Algebra Let W(n,m) be generalized Jacobson-Witt algebra over a field of characteristic p>3, according to the grading of W(n,m) , we know that it inherit the grading from A(n,m) as follows: ... 0answers 39 views ### Question regarding a theorem of Erdos and Renyi on B_2(g) sequence Let S \subset \mathbb{N}. We say S is of type B_2(g) if the number of representation of the form n = s_1 + s_2 \ (s_1 \leq s_2) is bounded by g for every n \in \mathbb{N}. Let S(n) be ... 0answers 65 views ### endomorphisms algebra of a real representation Let G be a finite group. Given a real irredcible representation of G, we know that its endomorphisms algebra is a division algbra and hence is the real, complex or quaternion algebra. Is there a ... 0answers 81 views ### What is the calculus based proof for 0.(9)=1? [on hold] 0.(9)=0.9 repeating infinitely. I have heard of simple proofs but I was curious of the calculus based proof. Thanks in advance 0answers 61 views ### Polinominal equations [on hold] Explain why it is possible that polynomial has no real solutions. Use reasoning to expand your explanation to find the general characteristics of polynomials that have no real solutions 0answers 35 views ### solution to a parabolic PDE I'm reading a paper where the following parabolic PDE is considered: u_t(x,t)=u_{xx}(x,t)+b(x)u_x(x,t)+\lambda(x)u(x,t), with boundary conditions u_x(0,t)=qu(0,t) \text{ and } u(1,t)=\int_0^1 ... 0answers 33 views ### Find subset of collection of sets whose intersection has minimum average value Let a_1,\ldots,a_n>0, and let S_1,\ldots,S_d\subset\{1,\ldots,n\} (all non-empty). For any I\subseteq\{1,\ldots,d\}, define S(I)=\bigcap_{i\in I} S_i. Given some 1\leq s < d, consider ... 0answers 35 views ### A cohomology associated with a codimension one foliation(2) What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension *? Moreover what is the description of this cohomology for ... 1answer 140 views ### Sum of two squares - Number of steps in Fermat descent If a prime p can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an x such that x^2 \equiv −1 \mod p. Is there a possibility to say ... 1answer 56 views ### How to compute the Expectation of the random variable using Taylor Series expansion I don't know how to solve the following expression:  = nm^2 E \bigg[\frac{ \exp(\theta) {(\log(R))}^2}{N(R,x)}\bigg] \hskip 5 pt Eq(4)  which I have explained below. R follows Poisson ... 1answer 54 views ### Are there 2-connected regular graphs whose maximum matching leaves 3 vertices uncovered? I'd like to use Corollary 5 of a paper by Hell & Kirkpatrick on graph packings to obtain an NP-hardness result. They want a 2-vertex-connected graph F such that every matching in F leaves at ... 1answer 76 views ### A multinomial-type sum over compositions of an integer I find myself needing to compute (or asymptotically estimate) the following sum over the 2^{S-1} compositions of S. I am hoping an expert in combinatorics (I am a computer scientist) will ... 0answers 110 views ### Ext of Skyscraper sheaf [on hold] Let X be projective curve over the complex number field; and let \mathbb C_p be the skyscraper sheaf whose fiber aver p\in X is \mathbb C and 0 otherwise. How could we prove that ... 0answers 34 views ### Supremum of positve kernel Let A(x,y)\geq 0 \forall x,y be a positive kernel of a bounded, positive operator A. How does one prove that$$\sup_{x,y}A(x,y)=\sup_x A(x,x)?$$0answers 19 views ### Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil Given a Hermitian indefinite pencil (A-\lambda B) where both A=A^H and B=B^H \in \mathbb{C}^{n\times n} are possibly indefinite, it is straightforward to show that the eigenvalues are either ... 1answer 22 views ### Reorder rows and columns to find as close to block diagonal structure in a non-symmetric sparse matrix I have a sparse matrix with no apparent structure and am wanting to reorder the rows and columns in such a way that the matrix becomes as close to block diagonal as possible. I am using R and have ... 0answers 122 views ### Twisting sheaf of Serre I'm sorry if my question is rather trivial, but I can't figure it out.. Given A a ring and P=Proj(A[X_0,\cdots,X_n]), I know that \oplus_n H^0(P,\mathcal{O}(n))=A[X_0,\cdots,X_n]. This equality ... 3answers 238 views ### Sets of points containing permutations - a Ramsey-type question The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with ... 0answers 15 views ### Finite elements W^{1,\infty} error estimates Are there finite element method setups that provide error estimates in the W^{1,\infty} norm (i.e., bounds on \|u'_h - u'\|_\infty)? Which families of elements can be used for implementing them? 0answers 35 views ### connectedness of coincidence set Consider the following obstacle problem in the whole domain \mathbb{R}^n min{\Delta u, u-\phi}=0 with prescribed boundary value \lim_{|x|\rightarrow\infty}u(x)=0 and \phi (can be assumed ... 0answers 92 views ### Under which conditions the inclusion of a sub-simplicial set of the nerve of a category is a Joyal equivalence? Let i: X \to N\mathcal C be a monomorphism in the category of simplicial sets, with C a category and NC its nerve. I am looking for sufficient conditions (and not too difficult to check) under ... 1answer 54 views ### Discrete Taylor's Formula in n dimensions [on hold] I am searching for discrete form of Taylor's formula in n dimensions. Please share the appropriate resources. 0answers 139 views ### Polynomial existence over finite field Denote \mathcal{F_n} as collection of multiaffine polynomials f\in\Bbb F_2[x_1,\dots,x_n]. Denote total degree of f\in\mathcal{F_n} as deg(f) (note deg(f)\leq n). Denote ... 1answer 91 views ### Combinatorics problem involving counting the number of certain substrings I'm not sure if this question is suited for MO, but it does seem quite challenging to me, and is required for a research problem in chemistry I'm working on. I did try getting help from elsewhere ... 0answers 21 views ### A curious example envolving moment's convergence Let \{X_n\} be a random variable sequence and X\sim N(0,\sigma). In general, the convergence E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k) doesn't implie that E(X_n^{k+1}) ... 0answers 34 views ### An embedding of modules by tensor product over a Noetherian domain I have a problem on Ring theory. I would like to prove or disprove the following statement: Let R be a Noetherian domain. Then by the Goldie theorem R have Q as a full ring of quotients and Q ... 0answers 139 views ### Conjugation of the quotient of SL(n,\mathbb{C}) by a finite subgroup EDITED Let G={SL}_{n,{\mathbb{C}}}, the special linear group over {\mathbb{C}}. Let H\subset G be a finite subgroup. Set X=G/H be the corresponding homogeneous space, it is a complex variety. ... 0answers 103 views ### Can anyone solve a stochastic differential equation - related to neuroscience research? I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ... 0answers 29 views ### Helmholtz boundary value problem in 2D I want to solve the Helmholtz equation in 2D with constant nonhomogeneities:$$\nabla^2w-\lambda w=C$$and with Dirichlet boundary conditions such that$$w(0,0)=0 ...
Let $X$ be a complex space. We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$. We say that $X$ is Kobayashi hyperbolic if the Kobayashi ...
### Prove that a Graph is connected using eigen values $\lambda$ [on hold]
Prove that for a graph is connected if and only if $\lambda_{max}$ > $\lambda_{1}$ Prove that for a $d$-regular graph $\lambda_{\max} = \lambda_1 = \cdots = \lambda_{k-1}$ if and only if the graph ...