# All Questions

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In a recent survey paper, Approximation of convex sets by polytopes, Bronstein claims that under Hausdorff distance $\rho_H$, for every convex body $U$, $$\rho_H(U,\mathcal{P}_n)\leq c(U) ... 0answers 13 views ### Can we give efficiently the solution of a bilinear system of equations over a finite field? Consider a finite field F and suppose we have a system of equations$$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$where \alpha=(\alpha_1,...,\alpha_s) and ... 1answer 120 views ### Decomposing (\mathbb C^n)^{\otimes m} as a representation of S_n\times S_m V=\mathbb C^n is a \mathbb CS_n-module, where S_n is the symmetric group of degree n, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ... 0answers 38 views ### u_n bounded in L^\infty(0,T;H) \cap L^2(0,T;V) implies u_n \to u strongly in L^2(0,T;H)? Let V \subset H be a dense and compact embedding. Let$$\lVert u_n\rVert_{L^\infty(0,T;H)} + \lVert u_n \rVert_{L^2(0,T;V)} < C$$where C is independent of n. It follows that eg. u_n ... 1answer 58 views ### The Jordan Plane and Enveloping Algebras Let k denote a field of characteristic 0 (assume algebraically closed for convenience). Define J=k\langle x,y|[x,y]=y^{2}\rangle. This noncommutative algebra (which can be viewed as a derivation ... 0answers 62 views ### Density of numbers whose prime factors all come from a fixed congruence class Let q be a positive integer greater than one, and let a be an integer such that \gcd(a,q) = 1. Define$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$Do we know the ... 0answers 49 views ### The Birthday Paradox [on hold] I was looking at the birthday paradox, and the many solutions. One of them that came up was the Poisson Distribution. The website I was looking at detailed the process to solve ... 0answers 28 views ### Property of summations [on hold] Suppose to have the following identity:$$ \sum_{i=i_0}^{i_1}\sum_{j=j_0}^{j_1} f(i,j) = \sum_{i=i_0}^{i_1}\sum_{j=j_0}^{j_1} f(i,j)g(i,j), $$for 'good' indexes i,j and some functions f,g. What ... 1answer 50 views ### Bertini theorem for big divisors and klt pairs Let X be a smooth projective variety and let D be a big \mathbb Q-divisor on X. Assume that for m large |mD| has no fixed components. Is there a \mathbb Q-divisor D'\equiv D so that ... 0answers 61 views ### History of preservation theorems in forcing theory For my honours thesis, I am studying a general preservation theorem using a framework provided by Shelah. I am mainly concerned about revised countable support iteration of \dot{S}-semiproper ... 0answers 18 views ### Multinomial proxy variables: Bound on probability of their sums Suppose (X_1,X_2,..X_i,..,X_b) as multinomial vector of random variables with N=\sum_{i=1}^b X_i and probabilities p_i to parametrize the X_i. Let us take the following imagination to ... 1answer 41 views ### Coarsely trivial Borel cross section for G\to G/N Let G be a locally compact group, and let N be a closed, normal subgroup, and let \pi\colon G\to G/N be the quotient homomorphism. It is known that there exists a Borel cross section, i.e., a ... 1answer 45 views ### Are Zariski connected and closed semisimple subgroups of semisimple and simply connected algebraic groups again simply connected? [on hold] Let G be a semisimple and simply connected linear algebraic group over \mathbb{C}. Let H be a connected, Zariski closed and semisimple linear algebraic \mathbb{Q}-subgroup of G. Is H a ... 0answers 26 views ### Infinite dimensional Cauchy-Lipschitz theorem From the answers to Mathoverflow Question 187722, it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation$$ \dot X=F(X), \tag{$\ast$} $$could fail to ... 0answers 26 views ### Probe permutationally matrix extreme properties Suppose given M\in\{0,1\}^{n\times n} of rank r. Assume that changing even a single 1 to 0 in M raises rank. Does it follow that M is permutationally equivalent to a block diagonal ... 1answer 195 views ### Why are 1 and -1 eigenvalues of this matrix? This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix \mathbf{A}. First, let's define two matrices: ... 0answers 35 views ### Potentiality classes and Borel reductions In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation E on a standard Borel space X, we say E ... 0answers 130 views ### Neighborly family of coins Here is a puzzle: Find 5 identical coins. Can you arrange them so that every coin is touching every other coin? The solution is here. The hint is: use the third dimension. My questions are ... 0answers 93 views ### Moving lemma for algebraic curves Let X be a smooth irreducible projective curve contained in \mathbb{P}^3 and Y be another reduced but not necessarily irreducible curve in \mathbb{P}^3. Denote by P the Hilbert polynomial of ... 0answers 13 views ### Renormalization for Transport Equations with SBD velocity field In the paper Traces and fine properties of a BD class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields B\in ... 1answer 134 views ### Blowing-up the Grassmannian at a point Does anyone know what the blow-up of the Grassmannian at a point looks like? Consider G=Gr(r,n) and V\in G. I want to understand more explicitly what Bl_V(G) should mean. Of course for affine ... 0answers 56 views ### Filmed lectures by Hassler Whitney Are there any filmed lectures by outstanding American mathematician Hassler Whitney, besides the two Einstein Chair lectures below? Old lectures, from the 1940s onwards, would be particularly ... 1answer 136 views ### On the coherence theorem for bicategories The coherence theorem for bicategories, as usually stated, reads Any bicategory B is biequivalent to a (strict) 2-category. It is possible to give an explicit construction of the ... 0answers 50 views ### Uncountably categorical theories which are interpretable in a strongly minimal Definition: Let \lambda be a cardinal. An \mathcal{L}-theory T is called \lambda-categorical whenever every two models of T of cardinality \lambda are isomorphic. Definition: An ... 0answers 69 views ### alternative way to recognize that a real symmetric quadratic form is positive A real symmetric quadratic form g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j is positive (definite) if g(x)>0 for every \mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0. It is well known that a ... 0answers 33 views ### Embedding rational simple algebras in the real quaternions [duplicate] Is there any way to embed a rational division algebra of dimension higher than 4 over its center in the real quaternions ? I think not, but I cannot prove it. 1answer 36 views ### On a reciprocal of Ostrowski theorem on Newton polytopes and factorization \newcommand\KK{\mathbb{K}}Let \KK be any field and f\in\KK[x_1,\dotsc,x_n] be a polynomial. Its support S_f is the set \{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n} has a nonzero ... 1answer 58 views ### A bijection between Lusztig series induced by inflation Context: Let \pi: \widehat{G} \rightarrow G be a surjective morphism between connected reductive groups defined over \mathbb{F}_q whose kernel is a central torus. Then \pi : \widehat{G}^F ... 0answers 15 views ### Unitary derivative and countable set Let \mathbf{r}:I\to\mathbb{R}^2, where I\subseteq\mathbb{R} is an open interval, be a continuous function that is not constant on any subinterval J\subseteq I such that at each point t\in I ... 1answer 26 views ### Degree of the negative part of a divisor Let K be an algebraically closed field (or \overline{\mathbb{C}(z)} for more requirement). And let P \in K[x,y] be an irreducible polynomial of degree m with respect to x and degree n with ... 1answer 34 views ### Graph of bounded continous functions with distance 1 Let V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\} and consider the metric that is defined for f,g\in V by$$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$We set E = \{\{f,g\}: f,g \in ... 2answers 55 views ### Completion of a single totally ordered down-set This is a follow-up question to Complete sets of incompatible totally ordered down-set in a partially ordered set. Let (P,\leq) be a partially ordered set such that for every p\in P the set ... 0answers 96 views ### Galois correspondence subgroups/subsystems In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups: By applying this result to finite groups, we get a Galois correspondence ... 0answers 27 views ### How to define the determinant of a morphism between graded Lie algebras? I have the following question. Suppose g_1 and g_2 are two finite dimensional, nilpotent, stratified Lie algebras and A:g_1\to g_2 is a morphism of the graded Lie algebra. I wonder whether there ... 0answers 39 views ### Combinatorics: Identical objects and distinct groups [on hold] I'm confused between the following 2 formulae: 1) Number of ways to put n identical objects into r distinct boxes, such that the ordering is NOT important is: (n+ r - 1) C r 2) Number of ways to ... 0answers 23 views ### Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix P\in \mathbb{R}^{n\times n} is an irreducible column stochastic matrix. P is also diagonally dominant. w \in \mathbb{R}^{n}  is a strictly positive vector satisfying w^T \mathbf{1} = 1 where ... 0answers 53 views ### A combinatorial question on ranks Denote$$\mathscr{C}[r]=\{Q\in\Bbb Z_{\geq 0,\leq 1}^{n\times n}:\mathsf{rk}(Q)= r\}.\mathscr{D}[A,t]=\{B\in\mathscr{C}[\mathsf{rk}(A)]:\mathsf{dim}(col(A)\cap col(B))\geq t\}.$$Given ... 0answers 26 views ### the choosing of an independent set in a specific k-partite graph Let k\geq2 be an integer, a graph G=(V,E) is called k-partite if V admits a partition into k classes such that every edge of G has its ends in different classes: vertices in the same class ... 1answer 137 views ### contractible configuration spaces Let F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}. It is known that F(\mathbb{R}^\infty,k) is contractible for each k. My question: is F(S^\infty,k) ... 0answers 27 views ### Skein theory: How axiomatizing a 2-box space? Let (A,+,\times, *) with an adjoint operation compatible with +, \times and *, such that (A,+,\times) and (A,+,*) are finite dimensional {\rm C}^{*}-algebras. What are the axioms on ... 0answers 78 views ### How do you categorify the cycle index series? Let F be a combinatorial species. The exponential generating series of F is defined to be$$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!}  It was observed by Baez and Dolan in their paper ...
If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for \$A \subseteq ...