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1 vote
2 answers
114 views

For a given value of $n$ and $m$, find $\text{fib}(n)$ $\text{mod } m$ where $n$ is very huge. (Pisano Period) [closed]

Input Integers $'n'$ (up to $10^{14}$) and $'m'$(up to $10^3$) Output $\text{Fib}(n)$ $\text{modulo}$ $m$ My questions For example : Why $\text{fib}(n=2015)$ $\text{mod}$ $3$ is equivalent to $\...
9 votes
1 answer
381 views

Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?

Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact ...
1 vote
0 answers
60 views

Splitting formulas for spectral flows

I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...
3 votes
0 answers
235 views

Motive to motive via Langlands

I have been tying myself in knots trying to straighten out this speculative circle of connections, no doubt because I am just a novice in these matters, so I thought I'd pop the question here and ...
2 votes
2 answers
419 views

A recurrence formula for the Legendre function $P_\mu^\nu(x)$

Im looking for a recurrence formula of type: $$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$ where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...
25 votes
4 answers
5k views

Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...
6 votes
1 answer
361 views

Maximum size of $k$-Sidon set over $\mathbb{F}_2^n.$

Fix $k \in \mathbb{N}$, $k \ge 2.$ Does there exist a subset $A \subset \mathbb{F}_2^n$ such that $|A| \ge c 2^{n/k}$ with some absolutely positive constant $c,$ and satisfying $$ a_1 + a_2 + \...
3 votes
1 answer
252 views

Hochschild homology of acyclic complex

Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic. Question: Does this imply that the Hochschild homology $HH_*(A)$ also vanishes ...
1 vote
0 answers
95 views

Determining the behavior of a contraction mapping with undefined points

Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
1 vote
1 answer
182 views

Approximating expectation of the trace of inverse of a Gaussian random matrix combination

In order to characterize the performances of MIMO systems that depend directly on the distribution of the eigenvalues of random Hermitian matrix so I would like to feature the quality of some ...
4 votes
0 answers
124 views

What is a fast way to multiply a Schubert polynomial by an elementary symmetric polynomial (specifically $x_1\cdots x_k$)?

What is a computationally fast way to get the coefficients of Schubert polynomials in the expansion of the product of a Schubert polynomial and an elementary symmetric polynomial? I know "fast" is ...
0 votes
1 answer
79 views

difference between: Measurable multifunction integrably bounded and Measurable multifunction integrable

I read the article "Komlós Theorem for Unbounded Random Sets" by G. KRUPA (MSN), but I did not understand the difference between: Measurable multifunction integrably bounded, Measurable multifunction ...
2 votes
2 answers
149 views

Can we say that : $ (A-B)\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $

Let $X$ be a separable Banach space and $A,B$ are closed convex subsets of $X$ such that $B\subset A$ and $$ A\cap\overline{B}(0,r) \text{ and } B\cap\overline{B}(0,r) \text{ are weakly compact, } \...
1 vote
0 answers
295 views

Eigenvalues of symmetric tridiagonal matrices with identical off diagonal elements

Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric) $$ \begin{...
11 votes
3 answers
2k views

How to cite a MathSciNet review

I want to cite a MathSciNet review. Is there a standard format for that? (I couldn't find anything)
7 votes
0 answers
122 views

The bidual of the space of divergence-free vector fields

Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
-1 votes
1 answer
67 views

Can we extract an injective envelope from a monomorphism?

Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we ...
2 votes
0 answers
98 views

Polytopes with large dihedral angles

The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$. The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
12 votes
1 answer
486 views

Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants

It is well known that any smooth curve $C\in |\mathcal{O}_{\mathbf{P}^1\times\mathbf{P}^1}(2,2)| $ has geometric genus equal to 1, so its isomorphism class is determined by its $j$-invariant. ...
4 votes
1 answer
326 views

Quotient of Coxeter Group II

My last question on the quotients of the group $$H := \langle a, b, c \ | \ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19} \rangle$$ couldn't be completely answered, because the finiteness of the ...
15 votes
1 answer
1k views

Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, ...
2 votes
1 answer
847 views

Bounds on the second derivative of a natural cubic spline in terms of the data

Suppose we have real numbers $x_1 < \cdots < x_n$ and $v_1, \ldots, v_n$. Let $f$ be the natural cubic spline such that $f(x_i) = v_i$. Is there a simple explicit bound on $\|f''\|_\infty$ in ...
3 votes
0 answers
158 views

Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
6 votes
1 answer
738 views

Quotient of Coxeter group

Since the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ seems to have resisted attacks of some powerful programs, I will turn to a group that seems to be a bit easier to analyse....
2 votes
0 answers
121 views

Rings whose finitely-generated modules are co-hopfian

Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent: For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$; For ...
1 vote
1 answer
307 views

How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]

There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results. Is there any method, which ...
6 votes
2 answers
747 views

Gaussian measure on function spaces

I'm reading this classic work and I'd like to get deeper inside some of its techniques. In particular, the authors state: "We construct a Gaussian measure $d\mu_{0}(\phi)$ on a measure space of ...
1 vote
1 answer
2k views

Lipschitz continuity of multivariable function in expected value

Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e., $$ \| h(x,y,w) - h(x',y',w) \|_2 \le L_h (...
5 votes
0 answers
187 views

Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
10 votes
2 answers
763 views

Cohomology vanishing for tensor powers of tangent bundle on the flag variety

Let $X$ denote the flag variety of a semi-simple group $G$ (in characteristic 0) and let $T_X$ denote its tangent bundle. I would like to ask the following question(s): 1) Is it true that for any $n\...
0 votes
0 answers
57 views

maximum modulus of polynomials

Suppose $p(z)$ is a polynomial of degree $n$ having no zeros in $|z|<1.$ Then for any $R\geq1$ we have a result $ \max_{|z|=R}|p(z)|\leq \frac{1+R^n}{2}max_{|z|=1}|p(z)|,\;\;R\geq 1.$ I could not ...
1 vote
0 answers
198 views

The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$. We regard $X$ as an algebraic variety over $\Bbb C$. Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...
78 votes
11 answers
20k views

How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...
5 votes
0 answers
298 views

Hopf fibration extended to bundle over $\mathbb{C}^2$

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
7 votes
1 answer
610 views

Possible application of divergence Theorem?

suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e. $$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$ then, for every ...
2 votes
1 answer
13k views

Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?

In this question on MSE, Enzo Creti asks for a prime number formed by concatenating the Mersenne numbers $2^n-1$ and $2^{n-1}-1$, for example, 40952047. For all residues modulo 7, he found primes ...
8 votes
4 answers
866 views

Self-defining structures

The relations $R$ in abstract graphs (with genuinely propertyless vertices) cannot be defined because there is nothing the relations can base on: they have to be presupposed. But consider derived ...
0 votes
1 answer
77 views

Reduce ergodicity to the ergodicity of the coordinate process

Let $(E,\mathcal E,\lambda)$ be a probability space and $\lambda$ be a measurable map on $(E,\mathcal E)$ with $\lambda\circ\tau^{-1}=\lambda$. I would like to show that $\tau$ is $\lambda$-ergodic ...
2 votes
0 answers
61 views

Separating a Riemann-Hilbert problem

Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is $$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$ where $g_j(x)$ are Holder continuous functions and $...
3 votes
1 answer
260 views

$\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties

Reading M. Hindry and J. H. Silverman (Diophantine Geometry-An Introduction), I find the claim that $\mathcal{M}_g$ and $\mathcal{A}_g$ have natural structures as quasi-projective varieties. Mumford ...
1 vote
1 answer
195 views

The functional equation $f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$

Incidentally, I came across the following functional equation $$f(xy) - 2 f(\frac{x+y}{2}) + f(x+y- x\cdot y) = 0$$ that is to hold for all $x,y\in \mathbb R$. Is there a neat way to find all ...
1 vote
1 answer
269 views

Subschemes of projective varieties

I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective variety any zero-locus $X$ of homogeneous polynomials in ...
3 votes
1 answer
373 views

Projection formula for flat morphisms

Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\...
2 votes
1 answer
266 views

complemented subspace of the direct sum of two Banach spaces

When I was reading a paper, I saw something like: If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
5 votes
1 answer
306 views

Ricci flow proof of isoperimetric inequality

It is well-known in geometric analysis that one can use curve-shortening flow to prove the isoperimetric inequality (where the general result requires curve-shortening flow for non-convex curves). I ...
1 vote
0 answers
107 views

Relationship between Quasicoherent sheaves and $\mathbb A^1$-fpqc modules over an fpqc stack

In what follows, assume several universes for simplicity. Let $X$ be a stack in groupoids on the fpqc site of small affine schemes $\mathbf{Aff}_{\text{fpqc}}$. We can define $\mathbf{QCoh}(X)$ ...
1 vote
1 answer
142 views

Geometric meaning of colocalization of modules?

Let $A$ be a commutative ring and $S\subset A$ a subset. A localization of $A$ at $S$ is defined as a ring morphsim $A\to A[S^{-1}]$ which is initial with respect to inverting $S$. Similarly, a ...
28 votes
2 answers
2k views

Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here. The question is this: Today ...
3 votes
1 answer
87 views

Space of holomorphic functions multiplied by smooth functions taking real values

Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
56 votes
6 answers
5k views

What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...

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