All Questions

My question is just a ``I don't understand what goes on in X of paper Y" so I don't know if I can post it; on the other hand it is research. I posted it in stackexchange but it received no ...

My name is J. Calvin Smith. I graduated in 1979 with a Bachelor of Arts in Mathematics from Georgia College in Milledgeville, Georgia. My Federal career (1979-2012) in the US Department of Defense led ...

Given an n-dimensional hypercube, join each vertex to obtain a complete graph. Then the number of subgraphs of four coplanar vertices is given by $6^n/8 -4^{n-1} +2^{n-3}$ Example: a 2D square has ...

The (left) regular representation of a finite group $G$ is theThe (left) regular representation of a finite group $G$ is the action on itself by left multiplication, $g\cdot h = gh$. The adjoint ...

I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\...

I was wondering if someone could tell me if the function x|x| where f(x)-x^2 if x<=0 and f(x)=x^2 for x>0 is differentiable at all values of x, particularly at x=0. if this is true, how would ...

Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$. By Koszul resolution I mean $$\ldots \to A \...

Let $G$ be a graph s.t. any two cycles $C_1, C_2 \subseteq G$ either have a common vertex or $G$ has an edge joining a vertex in $C_1$ to a vertex of $C_2$. Equivalently: for every cycle $C$ the graph ...

Let $R$ be an Artin algebra and let $0 \to A \to B \to C \to 0$ be an Auslander-Reiten sequence of finitely generated left $R$-modules. Is it always true that the projective cover of $B$ equals to the ...

OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326 For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...

What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far. In [Lei2001], Leinster ...

I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any. I'm looking at the description of a short-term position in ...

It's known that ZF proves that for every set $x$ there exists a set $H_x$ of all sets that are hereditarily strictly subnumerous to $x$. [see here]. Now is the following principle equivalent with ...

This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...

I was wondering if the following general problem has a standard solution. Let $X, Y, Z$ be CW-complexes and $f: X \to Y, g: X \to Z$ be continuous functions. Is there a criterion to say when it does ...

According to many books (and also to Stack project, see https://stacks.math.columbia.edu/tag/01RI ), a morphism $f\colon X\to Y$ between schemes is said to be dominant if the image is dense in $Y$. ...

I found this intriguing remark at the end of Woodin's Supercompact cardinals, sets of reals, and weakly homogeneous trees (1988): The assertion that every set of reals, in $L(\mathbb{R})$, is the ...

This is a cross-post. Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...

Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...

By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$ ...

Let $H=(V,E)$ be a hypergraph that is a finite Fano plane, that is, $V$ is a finite set and $E$ has the following properties: for $e_1\neq e_2\in E$ we have $|e_1|=|e_2|$, as well as $|e_1\cap e_2|=1$...

Note: I have asked this question on another forum as we well, but have not received a response. https://math.stackexchange.com/questions/3878100/confusion-about-exponential-versus-inverse-exponential-...

I am looking for a proof of Generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem on one configuration as follows: Let four points $A, B, C, D$ in the plain, the perpendicular of $...

It is probably an easy question, but somehow I am stuck. Question Is the following statement true? If yes, how to prove it? Suppose that $f\in C^1(\mathbb{R}^n)$ is convex and $$ \langle\nabla f(x)-\...

Let $M$ be a von Neumann algebra. Suppose $T: S(M_{*})\cap M_{*}^+ \rightarrow S(M_{*})\cap M_{*}^+$ is a surjective isometry between normal state spaces of $M$ , then there exists a Jordan $*$-...

Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is ...

We know: AB = AC = BC Location of "D" in not specified, but we know it's on BC bow. Now prove that: AD = BD + CD ...

What the title says. I need 2 answers using the methods listed above. Got an exam on monday, need to know how to do this

Let $A,B(a,b,c,d)\in\mathsf{GL}(4,\mathbb{Z})$ be given by $$A=\begin{pmatrix} I_2 & \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \\0& -I_2\end{pmatrix},\quad B(a,b,c,d)=\begin{pmatrix} -2-b ...

For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that ...

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...

Recall that a Hopf manifold is a quotient $\mathbb C^n\setminus 0$ by a free action of $\mathbb Z$ where the generator is acting by a holomorphic contraction. Question 1. Is it true that any ...

When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even. Question: For every $p$, does ...

Let $\Omega$ be a domain in $\mathbb{C}^n$. Let $\mathbb{D}$ denote the open unit disc in $\mathbb{C}$. Let $C_b(\Omega)$ and $C_b(\mathbb{D})$ denote the space of all bounded continuous complex ...

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}^k$ and $0\leq a_i,b_i\leq 1$ be in the probability $n$-simplex. Define the finite measures $\mu=\sum_{i=1}^n a_i \delta_{x_i}$ and $\nu=\sum_{i=1}^n b_i ...

Probably it's a simple question but I don't know how to prove that $\mathbb{Q}$($\sqrt{2}$) isn't isomorphic to $\mathbb{Q}$($\sqrt{7}$) like fields. I know that they are like $\mathbb{Q}$-Vector ...

If $V \hookrightarrow W$ and $W \hookrightarrow V$ are injective linear maps, then is there an isomorphism $V \cong W$? If we assume the axiom of choice, the answer is yes: use the fact that every ...

I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form $$ \frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...

In Galois extensions of structured ring spectra, Rognes introduces the notion of a faithful $G$-Galois extension of ring spectra. Let me recall what this means: We have a commutative ring spectrum $R$ ...

We are given a set $S$ of $n$ real numbers in $[0,1]$, with $0,1\in S$, and a value $\alpha\in(0,1)$. For each ordered triplet $(i,j,k)$ of values contained in $S$ (with $i\le j \le k$), we define its ...

could some one help me with an algorithm to evaluate a go board after a player has played his turn.the board is represented as a vector with a 1 in a dimension showing player 1's prices, and a 0.5 ...

It is well known that univariate heavy tailed distribution can be defined by looking at its tail behavior, which is $x$ has heavy tail with index $\alpha$ if and only if $$f(x)\sim |x|^{-\alpha}$$ ...

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy. My first problem is, essentially, about ...

Let $(X_n)_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}_+^d,\,\sum_{1\leq i\leq n} x_i=1\big\rbrace$. Assuming ...

Can you help me explain the following 2 sequences that can be periodic and aperiodic sequence look like? What does the cap arrow mean in sequence first item 1 and 3? Consider the following sequence ...

I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...

Let $L_i:X\rightarrow [0,\infty)$ be continuous (objective) functions defined on a metric space $X$ and suppose that $$ L_1(x)\leq L_2(x)\leq L_3(x)\qquad (\forall x \in X). $$ Here, I imagine that $...

This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general? For instance, McMullen's polytope algebra is ...

I am looking for examples of normal complex spaces $X$ which locally around a singular point are homeomorphic to a smooth complex manifold. The only example I know is a curve with a cusp, but this is ...

I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$ Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...