# All Questions

**1**

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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

**0**answers

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**

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**0**answers

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

**1**answer

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

**2**answers

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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**

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**0**answers

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**

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**0**answers

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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**

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**0**answers

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

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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**

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**0**answers

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

**1**answer

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

**3**answers

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**

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**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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**

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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

**1**answer

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**

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**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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**

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**0**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**2**answers

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 ...