-2
votes
0answers
9 views

Mathematic Theory of Computation [on hold]

Let Sigma be an alphabet. Use the principle of induction on Sigma^*, to prove that |w v| = |w| + |v| for all w, v in Sigma^.
1
vote
0answers
10 views

Asymptotics of the multipartition function

Recall that the multipartition function $p_k(n)$ counts the number of $k$-tuples of partitions $\lambda^1,\ldots,\lambda^k$ of numbers $a_1,\ldots,a_k$ with $a_1+\cdots+a_k=n$. It has a generating ...
3
votes
0answers
33 views

Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"? Or is there something else that states ...
0
votes
0answers
17 views

A 2D random walk on a lattice of equilateral triangles [on hold]

Calculate the characteristic ratio C∞ for a random chain on a two-dimensional lattice made of equilateral triangles with side a. At each step, a walk has five choices (it cannot double back).
1
vote
0answers
24 views

How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...
-1
votes
0answers
38 views

Lawvere algebraic theory as presentation-invariant [on hold]

We can read in a lot of papers, included Lawvere's PhD thesis, that algebraic theories are "an invariant notion of which the usual formalism with operations and equations may be regarded as a ...
0
votes
0answers
46 views

In search of a preprint by Litherland

I've seen the following citation a lot: "R. LITHERLAND: A formula for the Casson-Gordon invariants of a knot, preprint." I can't seem to find a corresponding publication. [Added in edit: apparently ...
0
votes
0answers
20 views

double Sum with a function of 3 variables inside. How to solve it? [on hold]

I am studying time scheduling problem solving with linear programming. I am reading paragraph 2.5 from this book (pages 32-35) and I am trying to solve it with Java. I have some questions about ...
4
votes
1answer
79 views

Establishing Duality in Tannakian Categories

I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid. However, I find the definition of rigid categories somewhat difficult. I don't ...
3
votes
0answers
32 views

Replacing functors by topologically or simplicially enriched functors

I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, ...
-2
votes
0answers
37 views

C^*-algebras and bidual [on hold]

Let $A$ be a C$^*$-algebra, $x \in A$ a non-zero positive element of $A$, $r(x)$ the range projection of $x$ in $A^{**}$ and $e = 1 - r(x)$. How do I show that exist a projection $f$ in $A^{**}$ such ...
0
votes
0answers
44 views

Motivation of $a_p$ for non-CM elliptic curves [on hold]

For an elliptic curve $E$ without CM let $\overline{E}$ be the good reduction of $E$ modulo $p$ prime. The value $a_p = p+ 1 - \mathbb{F}_p$ is referenced by DDT on p.19 and Ribet on p.5 . However ...
0
votes
0answers
37 views

Simple example game with 3 players [on hold]

I am currently writing an algorithm to compute different things as nash equilibria, dominated strategies etc for normal-form games. Since I am now trying to extend it to an infinite amount of players, ...
2
votes
1answer
35 views

Limits at infinity of fellow-travelling sequences in Teichmuller space,

I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling. Suppose that we have closed surface of genus $g\geq 2$, and ...
1
vote
0answers
61 views

local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal). Is there a sufficient condition for $A$ to be noetherian? For example, we know that the completion ...
1
vote
1answer
55 views

Non-closability of an operator

Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by $$T(u_1,u_2) := (u_1' + au_2',0)$$ on $\textrm{Dom} \,T := ...
2
votes
2answers
76 views

Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$. Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...
5
votes
2answers
111 views

Is every connected reductive group over a local field already defined over a global field?

Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...
2
votes
0answers
38 views

Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...
-1
votes
0answers
39 views

Largest subsequence with average $\geq \delta$ [on hold]

I found a solution in Largest subarray with average $\geq$ k. However, I fail to understand the algorithm correctly. Let us consider the following sequence: 12, -10, 10, -10, 12, -9 Let us pick ...
1
vote
0answers
39 views

Density of smooth functions in Sobolev space, respecting nonnegative traces

I am looking for a reference for the following result (if it is true): Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be sufficiently regular. Let ...
0
votes
0answers
43 views

Reducedness of scheme theoretic fibers of toric morphisms

Let's consider $X=X(\Delta_X)$ and $Y=Y(\Delta_Y)$ two complete $\mathbb{Q}$-factorial toric varieties over an algebraically closed field of characteristic zero, and let $f:X\to Y$ be a flat toric ...
0
votes
0answers
33 views

Extending and contracting an ideal by a faithfully flat homomorphism [migrated]

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $. This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry ...
2
votes
1answer
40 views

Largeness, generic, random points

As presented in Oxtoby's book ( http://link.springer.com/book/10.1007%2F978-1-4615-9964-7 ), there are two notions of largeness for subspace $Y$ of a given space $X$: Topology: $X$ is a topological ...
3
votes
0answers
25 views

Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...
0
votes
0answers
99 views

Fermat's Theorem on p = a^2 + b^2 [migrated]

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4. I heard that such a criterion could be possible for a given integer $n$ like $p = a^2 + n b^2$ ...
1
vote
0answers
24 views

Constructing pointwise-or independent binary matrices

Let $M$ be an $m \times n$ matrix such that every $M_{ij} \in \{0,1\}$. We say that $M$ is `$\lor$-irreducible' if (i) no row is a pointwise-or of other rows, and (ii) no column is a pointwise-or of ...
0
votes
0answers
58 views

Proof for this expression for Dottie number [on hold]

Neither Wolfram's mathworld nor Wikipedia mentions any series expanion for the cosine's fixed point. Therefore I am asking for a proof for this Kapteyn series representation of Dottie number: ...
2
votes
0answers
26 views

Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved? $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { ...
4
votes
1answer
88 views

Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case. Question: Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
0
votes
1answer
44 views

Fixed point property for the projectivization of manifold of fixed rank matrices

Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$. Does $PM$ satisfy fixed point property?
-1
votes
0answers
42 views

$\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...
2
votes
1answer
122 views

Manifold with corners [on hold]

Iam looking at the following situation of a manifold $Z$ with corners. More specifically a product of a smooth manifold X with a standard $k$-simplex $\Delta^k$. I wish to study certain formulas for ...
0
votes
0answers
47 views

Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
-1
votes
0answers
8 views

Partition of function into pieces for interpolation needs [migrated]

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...
1
vote
1answer
83 views

Weak topology on a pre-Hilbert Space

Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state. Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product ...
-2
votes
0answers
20 views

Is Markov Chain Sampled at stopping times a Markov chain? [on hold]

Given a Markov hain $\{X_n\}$ and $T_n$ be an increasing sequence of stopping times, is $\{X_{T_n}\}$ Markov chain ?
0
votes
0answers
31 views

Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by \begin{align} Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t}, \end{align} that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...
8
votes
1answer
170 views

A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves

Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to ...
0
votes
0answers
30 views

Hadamard product and matrix inverse [on hold]

Is there any relation between normal matrix inverse and the Hadamard product? Suppose we have a matrix $ M $ and its eigen/singular value decomposition. Can we say anything about the inverse of $ N = ...
1
vote
0answers
92 views

Experimentation with partial Euler products

Richard Mathar $[1]\& [2]$ shows that \begin{align} &\zeta_{2}(s)\equiv\prod_{\Omega(n)=2}^{}\left(\dfrac{1}{1 - n^{s}}\right)^{-1}=\exp \left(\sum _{k=1}^n \frac{P(k s)^2+P(2 k s)}{2 ...
2
votes
1answer
80 views

Relation between intersection multiplicities

Consider $(f_1,\dots,f_n), (g_1,\dots,g_n)\in \mathbb{C}[z_1,\dots,z_n]\ $ such that: i) $\{f_1=\dots=f_n=0\}= \{g_1=\dots=g_n=0\}=\{0\}\in \mathbb{C}^n\ $ and ii) $f_1g_1+\dots+f_ng_n\equiv0$. ...
2
votes
1answer
70 views

Central limit theorem with degenerate covariance matrix

Are there known generalisations of the central limit theorem for several random variables when the covariance matrix is degenerate? The usual proof of CLT based on characteristic functions (see e.g. ...
-3
votes
0answers
42 views

Circle packing II I need the solution/answer to this [on hold]

Let R(a, b, c) be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths a, b and c. Let S(n) be the average value of R(a, b, c) over all integer triplets (a, ...
2
votes
1answer
76 views

a question about Brooks' Theorem for $\Delta =4$

From Brooks' Theorem, we know that if a graph $G$ satisfies that $\Delta (G)=4$ and there is no $5$-clique in $G$, then $\chi (G)\leq 4$. And it is easy to find a counterexample to the following: ...
2
votes
1answer
70 views

Optimization over symmetric polynomials

Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that (i) $0 ...
1
vote
0answers
80 views

Fixed area, largest mass — is there a name?

Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as ...
12
votes
2answers
446 views

What other books are like these?

A certain class of books is defined as follows: (1) the book was kept for years in a cafe or mathematics library; (2) the primary contents are research problems and comments, handwritten by resident ...
-2
votes
0answers
76 views

Reference Request for Finite Axiom Of Choice [on hold]

I am looking for a book that would contain the kind of proofs that were given in the answers to this question: Finite axiom of choice: how do you prove it from just ZF? because I want to quote them in ...
0
votes
0answers
70 views

Combinatorial support set in CRT

Is there a function $g(s)$ such that if there is a set of numbers $\{r_i\}_{i=1}^m$ such that $r_i\bmod p_j\in\{0,1\}$ at every prime in $\{p_j\}_{j=1}^n$ such that $2^t\bmod p_j\neq1$ at every ...

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