4
votes
0answers
91 views

No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible? Topological irreducible: it is not homemorphic to ...
5
votes
1answer
157 views

Consistency of the nonrigidity of $P(\omega_1)/NS$

Is it consistent with ZFC that there exists an automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$ which is not the identity?
0
votes
0answers
95 views

Defining Global Choice in terms of strong limit cardinals over $ZF$

In his answer to user33038's mathoverflow question "What axioms are stronger than the Axiom of choice?", Prof. Hamkins writes: "What's more, the axiom of choice is equivalent over $ZF$ to the ...
1
vote
0answers
90 views

When is a conformal class equal to a conformal orbit?

Let $(M,g)$ be a Riemannian manifold of dimension $n$. Let $\text{conf}(M,g)$ denote the conformal group, i.e. the subgroup of diffeomorphisms of $M$ that acts by conformal transformations relative to ...
6
votes
1answer
211 views

Differential geometry without the Hausdorff condition or the second axiom of countability

I would like to know how the standard differential geometry of manifolds would change if we didn't assume the Hausdorff condition and/or the second axiom of countability. There are some simple things ...
-2
votes
0answers
35 views

Global minimization. How? [closed]

I know it's impossible to have an algorithm that finds the global minimum (without a brute force approach), for a general problem. I also understand that the efficacy of the flavour of minimization ...
3
votes
0answers
81 views

Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular: Loops are okay. An infinite set of vertexes is okay. Furthermore, I will tend to identify each digraph with its underlying ...
0
votes
0answers
28 views

Positive-definite and positive semi-definite matrixes sum [closed]

I'm doing an exercise of numerical analysis that ask me to demonstrate a particular sum of matrixes. From Wikipedia, I know that: M and N are two matrixes: if M is positive definite and r > 0 is ...
0
votes
1answer
111 views

perfect modules over polynomial algebra

This may be obvious. My question is short: $R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...
14
votes
2answers
458 views

Matrix equation $XAXBXC=I$

Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution ...
1
vote
0answers
35 views

Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional $$ F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1 $$ where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
1
vote
1answer
67 views

Sum of two surjective operators

It is well-known that the sum of two surjective operators isn't (in general) a surjective operator (for example consider $A+(-A)$). When it happens that the sum of two surjective operators is still ...
9
votes
3answers
338 views

Minimum size of the union of sets

I came accross this combinatorial problem in my computer science research. You are given a collection of k sets $S_1,...,S_k$ such that for any $i \neq j$, $ \vert S_i \setminus S_j \vert \geq p$ ...
4
votes
1answer
129 views

Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness

It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...
1
vote
1answer
108 views

Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions

I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it. Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of ...
2
votes
0answers
79 views

Possible argument against Height bound hypothesis

From this paper. $f(x,y)$ is polynomial with integer coefficients. $s(f)$ is its size, the sum of the logarithms of the absolute values of the nonzero coefficients, defined on p. 6. From p. 7. ...
1
vote
0answers
67 views

Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$. Let ...
2
votes
1answer
179 views

Trace of a Product of Finitely Many Matrices with Cosine Entry

Can someone help me prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & -m \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} 2 & ...
3
votes
0answers
82 views

Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen: TVS's $V$ of ...
6
votes
1answer
93 views

Closed leaves of a foliation

Let $M$ be a differentiable manifold of dimension $n + k$, let $\Delta$ be an $n$-dimensional integrable distribution (à la Frobenius), let $N$ be an $n$-dimensional connected integral manifold of ...
3
votes
0answers
212 views

Existence of a block design

Let $\ell$ be an integer parameter. I want to ask the existence of the following design: There is a universal constant $\beta < 1$ such that for all sufficiently large $\ell$, the following holds: ...
2
votes
0answers
104 views

Euler's totient function relative function

For the $\sigma$ function, the ratio $\sigma(m)/m$ is known as the abundancy index. Is there any special name for $\phi(m)/m$ with $\phi$ the Euler's totient function ?
-4
votes
0answers
40 views

Commutator of a matrix as matrix multiplication [closed]

I want to find whether two square matrix A and B are commmutative as a multiplication either A is a contant multiplication of the identity matrix or the matrix B can be expressed as p(A) where p(x) is ...
2
votes
1answer
153 views

Schubert calculus and Pieri's formula

In the lecture notes Grassmannians: the first example of a moduli space. MIT Open Course Ware. page 7: Are there any formal publications (books/papers) where I can find the formula?
3
votes
0answers
129 views

C$^*$-algebras isomorphic after tensoring

If $\mathfrak S$ denotes the set of all non-zero C$^*$-algebras (up to $*$-isomorphism) of some bounded cardinality, for instance separable, then $(\mathfrak S, \otimes_\textrm{min})$ and $(\mathfrak ...
1
vote
2answers
111 views

Is there a version of the Titchmarsh Convolution theorem to find singular support?

Okay, some terminology, correct me if I'm wrong. Singular support - the set on which a distribution fails to be smooth. In this case a piecewise function. Is there a name for $f*f*f$? The ...
9
votes
2answers
616 views

How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...
-2
votes
0answers
32 views

Taking a 3d to 2d point [closed]

I'm trying to create a billboard type effect (orienting an object in 1/2 axis' excluding the other) using this code. ...
2
votes
0answers
35 views

Fractional Sobolev spaces and extension by zero

The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero to whole $\mathbb{R}^n$ ...
7
votes
0answers
111 views
+50

Counting the size of the largest sets of independent strings

This question derives from a PPCG coding challenge I posed previously but despite asking on math.se and offering a bounty, no progress has been made. For a given positive integer $n$, consider all ...
3
votes
2answers
132 views

Vanishing of sheaf cohomology with compact support

Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions. 1) Under what sufficient conditions on $F$ for any compact subset ...
3
votes
2answers
77 views

Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality ...
12
votes
1answer
250 views

Applications of Lubotzky's linearity theorem?

Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
0
votes
0answers
45 views

Hermitian Matrices over Quaternions with Rank at most k [migrated]

The set of Hermitian matrices of the form: $X+iY+jW+kZ$ with $X,Y,Z,W \in \mathbb{C}^{M x M}$. $X$ symmetric, and $Y,Z,W$ skew-symmetric, with $rank(X+iY+jW+kZ)\leq{k}$, has what dimension as a ...
1
vote
0answers
227 views

Which mathematics journal can I submit an article which is short? [closed]

My situation is as follows: I have found a new proof for a theorem related to root system and Weyl arrangements. The theorem was proved in my teacher's paper published at J. Eur. Math. Soc.. Recently ...
5
votes
1answer
152 views

group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...
4
votes
2answers
241 views

Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by $$ DX = \left\{ p \in [0,1]^X \ ...
-4
votes
1answer
146 views

What's the name of this theorem? [closed]

I would like to know the name of a theorem that states that if a continuous variable (I.E. y) takes a positive (negative) value for x(i) and a negative (positive) value for x(j), it is sure that y has ...
0
votes
0answers
35 views

Quotient of cumulative binomial distribution functions

Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient $$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$ where $F$ denotes ...
6
votes
1answer
158 views

How does one identify flow lines on a vector bundle with those on the base in Morse theory?

In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function ...
0
votes
0answers
83 views

complex dynamic system and affine algebraic variety

Let $M^n$ be a $n$-dimensional noncompact complex manifold. In "The density property for complex manifolds and geometric structures II", Dror Varolin showed that some open set of $M$ is ...
-2
votes
0answers
33 views

Having the highest value in a interval appear less often [closed]

I have an array of size 5. And initially in each index, they are initialized with the value 1. so it looks like this : 1 1 1 1 1 Every iteration, I get a decimal value between 0.0 and 1.0 At the ...
2
votes
1answer
192 views

Any representation is a sub representation of direct sum of regular representation

I need a reference for the following statement: Let G be a linear algebraic group over algebraically closed field k. Let V be a finite dimensional G-module. The V is sub representation of k[G]^n for ...
3
votes
1answer
92 views

Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants. Let $Q(R) := Q(R^2,R) = B(0, R) \times ...
0
votes
0answers
64 views

Is this factoring algorithm sufficiently efficient for some integers of special kind?

Basically the question is if $m$ is factored over the integers, can it be relatively efficiently factored over $\mathbb{Z}[\sqrt{n}]$ where $n$ is not factored, but might be of special form? Suppose ...
1
vote
0answers
52 views

Finding all feasible solutions

Let $u$ be a $n_{max} \times m$ matrix. Let $z$ be a $n_{max} \times s_{max} \times n_{max}$ cube. Let $w$ be a $n_{max} \times 1$ vector. All the three matrices can have values from the set $\{ 0, ...
1
vote
1answer
52 views

$G$-CW complex structure of universal a $\mathcal{F}$-space

Let $G$ be a finite group and $H$ be an abelian subgroup of $G$. Let $\mathcal{F}$ be a family of all subgroups of $H$ , i.e. $\mathcal{F}= \{K : K \leq H\}$ Define universal $\mathcal{F}$-space ...
3
votes
0answers
96 views

do there exist finite simple characteristic quotients of the free group of rank 2?

Let $F_2$ be the free group of rank 2. Let $Aut^+(F_2)$ be the subgroup of $Aut(F_2)$ consisting of automorphisms of determinant 1 under abelianization. Do there exist maximal normal finite index ...
3
votes
1answer
165 views

May open sentences be eliminated?

Saul Kripke famously invoked a free logic to avoid validating the Barcan Formula and its converse. In that context he adduced a generality interpretation of free variables. The converse of the Barcan ...
4
votes
0answers
85 views

Integer sum of distinct reciprocals with no integer subset sum

Question $\def\nn{\mathbb{N}}$ For any $n \in \nn^+$, is there a finite set $S \subset \nn^+$ such that $\sum_{k \in S} \frac{1}{k} = n$ but $\sum_{k \in T} \frac{1}{k} \notin \nn^+$ for any $T ...

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