All Questions

90 views

Reference request: Ebin

I'm after the paper The manifold of Riemanian metrics by D. Ebin. A link to the reference is: http://www.ams.org/mathscinet-getitem?mr=0267604 The paper seems to be very hard to track down. Can ...
17 views

convex optimization [migrated]

Attached below is a convex problem. I just start learning this and is kind of confused of this question. I notice that the dom of W is convex, tr(WQ) is convex, the composition of convex functions ...
56 views

Properties of a specific antichain of a lattice formed by the cartesian product of finite ordered sets

Introduction Let $X$ be a poset of all $n$-tuples, $x = (x_1, x_2, ..., x_n)$, where $0 \leq x_i \leq m_i - 1$ for $i = 1, ..., n$ together with the relation $x \prec y$ defined so that for ...
189 views

Connectedness of moduli of vector bundles

Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? ...
347 views

Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$. Probable primes are the union of the primes and base two pseudoprimes. This definition is much ...
68 views

Undecidability of the existential theory

Do you know if I can find the proof that the existential theory of $\mathbb{Z}$ with the structure of addition , divisibility and the relation $(\exists s \in \mathbb{Z})m=np^s$ is undecidable, ...
74 views

Counting number of primorial factors

Denote $$P(n)=\prod_{p\in\mathsf{Primes}\leq n}p$$ signifying $n^{\mbox{th}}$ primorial. We know that $P(n)$ has approximately $n/\log2$ bits ...
399 views

33 views

66 views

cohomology ring of base-point-preserving maps on the 3-sphere

I find that $\text{Map}_*(S^3;S^3)=\Omega^3S^3$. I want to find the cohomology ring of $H^*(\Omega^3S^3;\mathbb{Z}_2)$. In the paper On configuration spaces, their homology, and Lie groups, I find ...
53 views

convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...
Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints? $x_1+x_2+...+x_n = 1$ $a_1 \le x_1 \le b_1$ $a_2 \le x_2 \le b_2$ $...$ $a_n \le ... 0answers 63 views Eigenvalues of a random matrix [on hold] For test cases i generated a random real uniform distributed matrix with entries from the intervall$[0,1]$. Here is the MATLAB Code i used ... 16answers 22k views Mathematical software wish list Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course$\LaTeX$extensively. When I chat with colleagues about such software tools, very often someone has ... 0answers 155 views Probability that an integer contains no$1\bmod 4$prime factor$n$represents integer variable. What is the probability that and integer contains at most$r(n)$prime factors of form$1\bmod 4$where$r(n)$is a function of$\omega(n)$(number of distinct prime ... 0answers 87 views Fundamental Group of SL_2 [on hold] I am thinking whether there is a simple criterion or visible method to know the fundamental group of SL_2(R), or SL_2(F) with an arbitrary field F. Because SL_2(R) is already a 3-dimensional ... 2answers 148 views Relationship between$H_*(X, A)$and$H_*(Y \cup_f X, Y)$?$\pi_*(X, A)$and$\pi_*(Y \cup_f X, Y)$? Let$A$be a subcomplex of a CW complex$X$, let$Y$be a CW complex, and let$f: A \to Y$be a cellular map. What is the relationship between$H_*(X, A)$and$H_*(Y \cup_f X, Y)$? Is there a similar ... 1answer 102 views Anosov representations and boundaries of (harmonic) maps Let$\Sigma_g$be a closed hyperbolic surface and$\rho\colon\pi_1\Sigma_g\to G$an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a$\rho$-equivariant ... 0answers 38 views eigenvalues of cycle and its complement [on hold] I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose$\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then,$\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$Is it ... 1answer 138 views Laplace-Beltrami and averaging For a Riemannian manifold$M$with metric$g$and Laplace-Beltrami operator$-\Delta_{g}$, what conditions on$M$guarantee that$-\Delta_{g} u(x)$measures the difference between$u(x)$and the ... 0answers 181 views Conjectured new primality test for Mersenne numbers How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let$M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$... 1answer 34 views floating point representation via the perspective of TTE/computable analysis Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ... 2answers 134 views Counting number of$2\times 2$unimodular matrices of particular type From set of numbers from$\Bbb S=\{0,1,\dots,m\}$, how many distinct$3\times 3$unimodular matrices parametrized by$(a,b,c,d,e,f)\in\Bbb S^6$of following type can one form? \begin{bmatrix} a^2 ... 3answers 138 views Probability of random geodesics on the half-sphere intersecting 4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ... 0answers 33 views Obtaining z-transform of a multivariate nonlinear difference equation [on hold] I need to obtain the z-transform of difference equations that are as follows: My problem however is multivariate and looks like this: x[k+1]=ay[k]+ ((x[k])^2)(y[k]) ... 1answer 90 views Commuting ODE's implies existence of nonzero vanishing two variable polynomial? Write$\partial := d/dt$, fix$m, n > 0$, and let$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ... 1answer 106 views What are some useful invariants for distinguishing between random graph models? Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as: The Erdos-Renyi model The Stochastic Block model The Watts-Strogatz model The Barabasi-Alber model ... 1answer 115 views Parametrized Atiyah-Singer index theorem Let$M$be any smooth manifold (could be unorientable - I think). Let$E,F \to M$be two complex vector bundles. Let$S$be any compact space, and let$D_s:E\to F,s\in S$be a continuous family ... 5answers 1k views Why should we care about “higher infinities” outside of set theory? Let's say you are a prospective mathematician with some addled ideas about cardinality. If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :) If you ... 1answer 645 views A problem in elementary geometry Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle: On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise, ... 1answer 97 views Need an explanation of a deduction When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ... 1answer 814 views Remark on Fermat's Last Theorem by Darmon, Diamond and Taylor In their paper, Darmon, Diamond and Taylor remarked the following : (the previous paragraph of Section 2.2 (p. 55), https://www.math.wisc.edu/~boston/ddt.pdf) If$\rho : G \rightarrow ...
Can anyone suggest a reference for (left) Kan extensions of pseudofunctors? In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors \$\mathscr A \xrightarrow{G} \mathscr ...