1
vote
0answers
9 views

Random processes with smooth paths

Is there any prototypical example of a Random process with smooth paths? I imagine one can simple integrate each path of a Brownian motion and get a $C^{\frac32-\epsilon}$ path. It's easy to ...
0
votes
0answers
4 views

Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property. Here is the definition of f.p.s.p.("map" means ...
1
vote
0answers
45 views

Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for the smooth ...
1
vote
1answer
75 views

Completing class-sized Fields

Let's say that an ordered Field is a class (proper or not) which satisfies the axioms of ordered fields. We work in NBG set theory with global choice. Let's say that an ordered Field is real closed ...
3
votes
2answers
159 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
-1
votes
0answers
15 views

Linear transformation from n-dimensinal vector space to Rn [on hold]

Suppose: U is a real n-dimensional vector space, and B = {$u_1$, $u_2$,...,$u_n$} be a basis for U, let $T: U \to R^n$ be the linear transformation defined by $$ T(u) = [u]_B $$ How to prove: ...
0
votes
0answers
21 views

Functions satisfying simultaneous conditions

Are there functions $f(x,y),g(x,y)$ that satisfy at some $x_1,x_2\in\Bbb R$ and $\alpha\neq0$ $$\sum_{y=1}^\infty f(x_1,y)=\sum_{y=1}^\infty g(x_1,y)=0$$ $$\sum_{y=1}^\infty ...
0
votes
1answer
127 views

Field extension and nilpotent element

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...
1
vote
1answer
28 views

Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...
-4
votes
0answers
25 views

Volume and Surface Area Algebra Rational Functions [on hold]

A frozen yogurt cone has a volume of 10 cubic inches. The surface area of a cone exluding the base is S = pi r sqrt(r^2+h^2), where r is the radius of the base and h is the height. Find the ...
14
votes
1answer
777 views

Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path ...
1
vote
0answers
62 views

First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...
9
votes
2answers
223 views

From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed. Let $\mathfrak{g}$ be a semisimple lie algebra. ...
0
votes
0answers
41 views

Surniversal spaces

Basic background On one hand there is a complete result: $\,\ $for every non-negative integer $n$ there exists an $n$-dimensional compact metric space $M^n$ such that it contains a homeomorphic ...
2
votes
0answers
30 views

Two questions on the James $p$-space $J_{p}(1<p<\infty)$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that ...
1
vote
0answers
30 views

Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations. Let $A$ be an ...
-1
votes
0answers
26 views

Variable modification of the functional equation for the Riemann Zeta-function

Be $$f_x(s):=\frac{\zeta(x+1-s)}{\zeta(x+s)}-\frac{\Gamma(x+\frac{s}{2})}{\Gamma(x+\frac{1-s}{2})}\pi^{\frac{1}{2}-s}$$ with $x\in\mathbb{R}_0^+$, $s_k(x)$ and $s_{-k}(x)$ the zeros of $f_x(s)$ with ...
4
votes
1answer
102 views

Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
1
vote
0answers
50 views

Base change and geometrically generic reduced fiber

Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...
3
votes
0answers
71 views

What is the integral cohomology of an Enriques surface over a finite field?

Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, ...
-3
votes
0answers
128 views

Maps from $S^3$ to $S^3$ [on hold]

As a physicist, I apologize for imprecise language. I am interested in maps from $S^3$ to $S^3$ (identical to the group $SU(2)$). Since $S^3$ is threedimensional, there is some similarity to maps ...
3
votes
1answer
112 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
8
votes
2answers
219 views

Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$ Which in our case is an isomorphism since $G$ ...
1
vote
1answer
86 views

Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...
2
votes
0answers
30 views

Are rays in Carnot groups straight?

A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...
1
vote
0answers
42 views

Question about continuity in the “complete Skorohod Topology”?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process" https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf In one of the exercises, exercise 8.9 ...
0
votes
0answers
49 views

An efficient algorithm for computing all semigroups of order n [on hold]

I attached two papers which give an algorithm for computing all semigroups of order n=3, and n=5. I understood the first(table 3) and second(table 4) steps of algorithm, but I can't understand the ...
7
votes
1answer
237 views

Choosing two-colorable subgraph in a triangulation

Consider a planar graph which is a triangulation. Is it possible to find a two-colorable subgraph which has common edge with every face of our graph? It is known that such spanning tree not always ...
4
votes
0answers
86 views

A criterion for orbits of complex reductive group to be closed

I have some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following: Let $G=K_{\Bbb C}$ be a ...
3
votes
0answers
30 views

Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
-2
votes
0answers
25 views

$k$-th diagonal element of an inverse matrix $(\textbf{H}^{\dagger}\textbf{H})^{-1}$ [on hold]

Let $\textbf{W}=\textbf{H}^{\dagger}\textbf{H}$ with $\{\cdot\}^{\dagger}$ being conjugate transpose operator. In some materials I have read so far, there is a common statement that the $k$-th ...
1
vote
0answers
23 views

Perturbations on SVD decompostions

Given a symmetric matrix $A\in \mathbb{R}^{n\times n}$, with all the entries greater than zero $A_{i,j}>0$ with rank $k<n$, we can calculate its SVD decomposition: $$ A = USU' $$ from which we ...
0
votes
0answers
53 views

inverse of operator

I want to calculate the inverse of the operator $T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial ...
1
vote
1answer
72 views

Analogue of fundamental theorem of real surfaces for complex surfaces

Is there an analogue of the fundamental theorem of surfaces for complex surfaces? If I know only differentiable functions $E,F,G,e,f,g$ (coefficients of the first and second fundamental forms) where ...
21
votes
3answers
2k views

Adapting arguments and plagiarism

I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in ...
0
votes
0answers
30 views

how to measure a bidrectional relationship effect on third variable [on hold]

Sorry that my question was unclear: I decide to determine if there is a relationship between two variables (gross national income, X and enrollment, Y) in Country A, between 2000-2007 My results ...
0
votes
0answers
21 views

How to prove prime avoidance for graded cases?

Let $R$ be a nonnegatively graded ring such that $R_0$ is local with infinite residue field. Let $I,J_1,...,J_s$ be a homogeneous ideals of $R$ such that $I$ is not contained in $J_i$. Please prove ...
5
votes
2answers
85 views

References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
1
vote
1answer
149 views

Monodromy theorem of degeneration of smooth projective varieties to non-reduced central fiber

Given a family $\pi: \mathcal{X}\rightarrow\Delta$ smooth away from $0\in\Delta$, Where $\mathcal{X}$ is a smooth complex manifold, $\Delta$ is a small disk, the general fiber of $\pi$ is smooth ...
1
vote
0answers
36 views

On a sequence of integers

I recall a well-known theorem due to Minkowski. Theorem. If $\theta$ is irrational and $\alpha$ is not of the form $\alpha = m\theta+n$ for some integers $m$ and $n$, then there are infinitely many ...
1
vote
1answer
65 views

Size of connected components of a graph via its spectrum

I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is: Is there a way to understand the size of each connected ...
0
votes
0answers
22 views

Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number?

A number $n \in \mathbb{N}$ is said to be superperfect if $$\sigma(\sigma(n)) = 2n.$$ A number $m \in \mathbb{N}$ is said to be almost perfect if $$\sigma(m) = 2m - 1.$$ Here is my question: Is ...
1
vote
0answers
73 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$? $$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$ Example of this situation is $A=C(X)$ where $X$ is the ...
2
votes
0answers
62 views

Dimension of curvature invariants

EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group. Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
2
votes
1answer
47 views

Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
18
votes
0answers
310 views

Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...
5
votes
0answers
98 views

Are there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism $$\varphi:\Omega\otimes\mathbb{Q}\to R$$ where $R$ is any ...
2
votes
3answers
204 views

Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$. Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$ $$f(a;b):=\prod\limits_{k=1}^\infty ...
21
votes
2answers
1k views

Mathematicians with Aphantasia (Inability to Visualize Things in One's Mind)

Are there any mathematicians with aphantasia? If so, could they please elaborate upon what their experience with mathematics is like? I realize that this question probably falls outside of the scope ...
9
votes
2answers
291 views

Lattice n-gons with ordered side lengths 1,2,3,…,n

Consider the octagon in the Cartesian plane with vertices at (0,0), (1,0), (1,2), (4,2), (4,6), (7,2), (7,8), and (0,8). Are there other (infinitely many) polygons, such as this, lying entirely in ...

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