2
votes
2answers
23 views

Complexity of Turing Machine behavior

If one looks at the code for a Turing Machine (TM) with $q$ states and, let's say, $2$ symbols, they all look pretty much the same: A list of $5$-tuples: $$ < state, symbol{-}read, ...
0
votes
1answer
18 views

Relationship between Multiplicative Ergodic Theorems

One version of Oseledets' Multiplicative Ergodic Theorem states that if $\sigma$ is an ergodic measure-preserving transformation of a space $(\Omega,\mathbb P)$ and if $A\colon\Omega\to GL(d,\mathbb ...
0
votes
0answers
33 views

Calculation of cardinality of Jacobians

The problem of calculation the number of rational points on curves over finite fields is $\#P$-complete - "Counting curves and their projections". Is it true for calculation of number of rational ...
1
vote
0answers
24 views

The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
0
votes
0answers
24 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
-3
votes
0answers
56 views

Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...
-4
votes
0answers
63 views

Isomorphism between $\mathbb R^3$ and the the Heisenberg group [on hold]

What is the isomorphism between $\mathbb R^3$ and $\mathbb C \times \mathbb R$ as a group (the Heisenberg group), provided with the law $$(z,t).(w,s) = (z+w, t+s+\Im m(z\bar{w})\, ); \quad z,w\in ...
0
votes
0answers
40 views

sum over all integer partitions, of the product of the factorials of the terms

I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms. Thanks,
1
vote
0answers
41 views

Why is the kernel of an algebraic Hecke character open in the ideles?

I've been reading about algebraic Hecke characters, and how one obtains one dimensional $p$-adic representations from them. I have a question about why the kernel of a map defined on the ideles is ...
6
votes
0answers
52 views

Does every ring of integers sit inside a monogenic ring of integers?

Given a number field $K/\mathbf{Q}$ whose ring of integers $\mathcal{O}_K$ is, in general, not of the form $\mathbf{Z}[\alpha]$ (not monogenic), does there exist an extension $L/K$ which has ...
3
votes
0answers
40 views

Bott-type vanishing results for the weighted Grassmannian wGr(2,5)

If $G=Gr(k,n)$ denotes the Grassmannian of k-dimensional subspaces in $V= \mathbb C^n$, representation theory gives us a Bott-type result for the cohomology groups $H^q(G, \Omega^p(k))$ of the twisted ...
3
votes
0answers
48 views

Number of critical points of a smooth function

Are there any examples of closed manifolds with the property that the minimal number of critical points (possibly degenerate) of a smooth function on this manifold is strictly bigger than the stable ...
3
votes
0answers
24 views

Model structure on DG-algebras with flat cofibrants

Let $k$ be a commutative ring, and consider the category of associative non-positive DG-algebras over $k$ (thus, $A^i = 0$ for $i>0$, and the differential has degree $+1$). Is there a closed model ...
-4
votes
0answers
22 views

Linear programming formulation with conditional constraints? [on hold]

The problem is below: Max f(x1,x2,k) = R*(x1+x2) - C*Q + k s.t., if Q > x1+x2 then k = e*(Q-(x1+x2)) else k = -s*((x1+x2)-Q) in which R, C, Q, e, s are ...
0
votes
0answers
79 views

Existence of a solution of a system of polynomial equations [on hold]

Consider a non-linear operator $\mathbb{F}:\mathbb{R}^n \longrightarrow \mathbb{R}^n$, $\mathbb{F}(x_1, x_2, \cdots, x_n)=(f_1(x_1, x_2, \cdots, x_n), f_2(x_1, x_2, \cdots, x_n), \cdots, f_n(x_1, x_2, ...
1
vote
2answers
262 views

Perverse sheaves and tensor product

If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so ...
7
votes
0answers
121 views

Swan K-theory of Z/4

Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...
3
votes
0answers
39 views

Maslov class of a diagonal

Let $(M,\omega)$ be a symplectic manifold. Which condition on $M$ guarantees that the diagonal of $(M \times M, (\omega,-\omega))$ has a vanishing Maslov class? $H^1(M,\mathbb{Z})=0$ is enough, but I ...
3
votes
1answer
65 views

Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite. $P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$, and when $A$ ...
-4
votes
0answers
41 views

Relationship in arcsin series [on hold]

Does any relationship exist in the series $(2/\pi) \arcsin \sqrt{1/2}$, $(2/\pi)\arcsin\sqrt{(1/2)^2}$, $(2/\pi)\arcsin\sqrt{(1/2)^3}$, $(2/\pi)\arcsin\sqrt{(1/2)^4}$ etc?
3
votes
0answers
76 views

An $\mathsf{SL(n,F)}$ decomposition problem

Given $n\times n$ square matrix $A\in\Bbb F^{n\times n}_{}$ where $\Bbb F$ is a field is there an easy way to test there is NO decomposition $A=B+C+D$ where $B,C,D\in\mathsf{SL}(n,{\Bbb F})$ are ...
1
vote
1answer
113 views

Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in Math Stack Exchange. I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
0
votes
0answers
23 views

positive operator surjectivity

In nonlinear analysis and monotone operator theory it is well known that if the operator $A$ is a maximal monotone and strongly monotone on a real Hilbert space $H$, then $A$ is surjective. This can ...
-7
votes
0answers
70 views

Golden Ratio & Fibonacci - Two-Beamed problem by Charles de Gaulle (13 unit squares) [on hold]

I don't even know where to begin... Here is the question: http://i.imgur.com/hxtDXst.jpg You are required to find the lengths of PB and BQ. I have already discussed a little over at; ...
4
votes
2answers
217 views

Picard groups of quartic K3 surfaces

Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors ...
3
votes
1answer
68 views

Distribution of decomposition types of primes in non-Galois extensions of number fields

Let $L/K$ be an extension of number fields. If $p$ is a prime of $K$ that is unramified in $L/K$ and $(f_1, \dots, f_r)$ is a partition of $n = [L:K]$, say that $p$ has "decomposition type" $(f_1, ...
1
vote
0answers
23 views

Motivation for some operators in the dyadic model of Navier Stokes equation

What's the motivation for cascade operator $C_{u}, C_{d}$ (and then the dyadic version of operator $B=(u\cdot \nabla)u$), which is $(C_{u}(u,v))_{Q} = 2^{\frac{5}{2}j}u_{\tilde{Q}}v_{\tilde{Q}}$, ...
1
vote
0answers
7 views

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...
0
votes
0answers
84 views

Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true. Since posting the question, ...
1
vote
0answers
60 views

Does the link of a hypersurface singularity determine its analytic type?

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...
-5
votes
0answers
37 views

Formula for getting a value that doubles the amount of the previous value? [on hold]

I am new to Math overflow. I have a question that I cannot seem to answer whatever formula I try. I don't know how to explain it so I'll just graph it: Let 'x' be an increasing number. x = y ...
1
vote
0answers
40 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
0
votes
1answer
42 views

Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates an analyitc ...
29
votes
3answers
2k views

Is it possible to have a research career while checking the proof of every theorem that you cite?

A colleague raised the above question with me; more precisely he said: Suppose that a mathematician were resolved not to publish any theorems unless she had checked the proof of every theorem ...
0
votes
0answers
14 views

Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous). I'm interested in the function ...
-6
votes
0answers
23 views

Discrete Math proof problem, unsure where to start [on hold]

Let { m1, m2, ....., mk } be pairwise relatively prime positive integers. Prove that there cannot be more than one solution to the system of congruence's $$ \langle x ≡ ai (mod mi) \rangle $$ in ...
1
vote
0answers
61 views

connectedness of semi algebraic sets

We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...
1
vote
0answers
50 views

Moving a result from the unconditional to the conditional

I'm generally wary when lifting a result stated unconditionally to a situation where I'm conditioning on a random variable. Consider the following classical result in weak convergence: Theorem. Let ...
0
votes
0answers
31 views

On Schrijver Lower bound

Shrijver lower bound gives number of perfect matchings on a $k$-regular bipartite graphs as $\Big(\frac{(k-1)^{k-1}}{k^{k-2}}\Big)^n$. What is the corresponding lower bound for min-degree $k$ and ...
4
votes
1answer
58 views

Algebraic $K_1$ group for a $C^*$-algebra

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: ...
1
vote
0answers
38 views

First variation on double integral [on hold]

Currently I am trying to fully understand the paper of munk1921. In the derivation of the minimum induced drag theorem it is at one point stated (p.378) that in order to minimize drag the following ...
-6
votes
0answers
45 views

Example of infinite field of characteristic prime is not algebraically closed field [on hold]

I know that if $F$ is an algebraically closed field, then $F$ is infinite. The converse is not true, so what is the example of an infinite field of characteristic prime $p>0$ not algebraically ...
-5
votes
0answers
73 views

Rational power Napier number [on hold]

Help me with the following question. Prove that $2^e$ irrational, where $e$ is the Napier number.
1
vote
1answer
98 views

Relation between Harmonic vector field and Harmonic 1-form

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function $$E(X)=\int_M\|dX\|dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$ ...
-7
votes
0answers
34 views

Proof about a measure zero set [on hold]

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. Also, show that ...
-6
votes
0answers
146 views

Great Mathematicians Without a PhD [on hold]

While listing to some music, I was wondering which great mathematicians did not have or do not have a PhD. This is a very subjective question, since "great" is not formally defined. But to describe it ...
20
votes
1answer
503 views

Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?

$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic. Is this folklore, or is it credited to someone? ...
-8
votes
0answers
129 views

Polynomial expansion for $\frac {\sin x}{x}$ and roots [on hold]

Consider the polynomial expansion for $$\frac {\sin x}{x} = p(x)= 1-\frac {x^2}{3!}+\frac {x^4}{5!}–\frac {x^6}{7!} + \cdots$$ $p(x) = \prod(x-a_i)$ for $i = 1 →∞$ where the $a_i$ are the roots of ...
4
votes
1answer
76 views

Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
1
vote
0answers
41 views

Decomposition of flat homogeneous Kahler manifolds

In a paper I am reading, it is claimed that a flat homogeneous Kahler manifold is a Kahler product $\mathbb C ^k \times T_1 \times \cdots \times T_s $ where $\mathbb C ^k $ is considered with its ...

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