# All Questions

**2**

votes

**2**answers

19 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

**1**answer

17 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

**0**answers

32 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

**0**answers

23 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

**0**answers

23 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

**0**answers

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

**0**answers

62 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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

45 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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

216 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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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 ...