0
votes
0answers
8 views

Difference in the Four Color Theorem

How is proving that any planar graph with maximum degree of four has a four coloring, different from proving the four color theorem? If they are different, then how would one prove it?
-2
votes
1answer
25 views

Adherent value of sin(sqrt(n)) [on hold]

I have been struggling against this question for several hours and I really need some help. It is from my undergraduate real analysis course. Your time and help is greatly appreciated. I got struck ...
1
vote
0answers
23 views

Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
0
votes
0answers
19 views

Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove? It seems like it should be easy ...
1
vote
1answer
34 views

Is there a generalization for the discrete fourier transform whereby eigenvalues are other roots of unity?

The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions. https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors Is ...
-2
votes
0answers
14 views

Loxodrome loop on a surface of revolution

Prove that there can be a closed loxodrommic curve on a surface of revolution only when it is doubly connected.
-1
votes
0answers
21 views

A sequence of continuous functions that converge to the characteristic set of a G-Delta Set [on hold]

If $\mathcal{O} \subset I$ is an open subset of an interval then I know we can find a sequence of continuous functions $f_j \in C(I)$ such that $0 \le f_j(x) \nearrow \chi_{\mathcal{O}}(x)$ for all $x ...
0
votes
0answers
23 views

Markov Chains and Simple Machine Learning

Suppose I have a large training set consisting of many strings of symbols. $TS = \{Str_0, Str_1, ..., Str_n\}$ $Str_i = \{Sym_0 ... Sym_{len}\}$ These strings of symbols are each generated by the ...
-1
votes
0answers
16 views

Correlation between probability of an event in the domains A and B and the event in an event AUB [on hold]

Given the event $E$ and finite sets $A$ and $B$ such that $P(E)=p_1$ in the domain $A$ and $P(E)=p_2$ in the domain $B$, then what can we say about $P(E)$ in the domain $A\cup B$?
1
vote
1answer
218 views

Use of infinitude of primes in the Green-Tao theorem

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...
5
votes
1answer
95 views

Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...
0
votes
0answers
38 views

canonical decomposition of a linear map

I have a linear map $A : {\mathbb F}^k \to {\mathbb F}^{k+m}$ where ${\mathbb F}$ is some field; (you can impose restrictions on ${\mathbb F}$ if it helps). I'd like to decompose $A$ as a product of ...
1
vote
0answers
47 views

Intuitive understanding of the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = ...
-2
votes
0answers
24 views

Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in L2: ‎‎‎‎‎$‎$‎‎‎‎d‎‎{R}(‎A‎,B)=‎\sqrt{‎{‎‎‎‎\paralle‎l ‎P‎{R(A)}‎‎- ‎‎‎‎‎‎ ...
0
votes
1answer
62 views

Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$. Now, let ...
1
vote
0answers
34 views

End points of continua

Whyburn (1942) defined an end point x of a continuum X to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines end point locally. ...
-1
votes
0answers
32 views

Metrics and Measures on a Category of Cats : a cauchy complete category of categories

Is there a suitable way to restrict the functors between objects (and I suppose the objects themselves) of a category of categories such that we have a cauchy complete category. Can we find a complete ...
-6
votes
0answers
39 views

Need a help solving a rational integral [on hold]

Today I have spent all my day solving this integral, but no result yet. So I need your help. Will be very thankful.
2
votes
0answers
34 views

Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions: All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix); all principal minors are $>1$, ...
2
votes
1answer
135 views

A decreasing sequence involving the divisor function?

Define $N_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function. It seems that $u_k = \dfrac{\sigma(N_k)}{N_k \log\log N_k}$ is a decreasing function ? By ...
-3
votes
0answers
18 views

Square wave in the limit of infinite frequency [on hold]

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity? Does it depend on how the function is written down (e.g. defined as ...
3
votes
2answers
128 views

Definition of the differential of the Cone of a morphism of complexes [on hold]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$. The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...
0
votes
0answers
29 views

CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...
2
votes
1answer
88 views

Symplectic group over integers and finite fields

For $H=\left( \begin{smallmatrix} 0 & I_n \\ -I_n & 0 \end{smallmatrix} \right)$ and a commutative ring $F$, the symplectic group $Sp(2n,F)$ is the set of all matrices $M\in F^{2n\times 2n}$ ...
8
votes
0answers
67 views

Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces $$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction. Richard ...
2
votes
1answer
43 views

If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?

Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact. Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$. Is then $u \in L^\infty(0,T;X_1)$? To apply ...
0
votes
0answers
118 views

Has the attempt of proof of the Frankl conjecture by Vladimir Blinovsky been checked? [on hold]

I found his article in arxiv: http://arxiv.org/pdf/1507.01270.pdf. But i didn't find any response to the article and as I'm an undergraduate I have no knowledge to judge if this approach is promising. ...
-2
votes
0answers
46 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first? [migrated]

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
4
votes
1answer
72 views

A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable, locally connected and has finite topological dimension, yet fails to be locally compact?
2
votes
1answer
77 views

Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
11
votes
0answers
186 views

Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...
-4
votes
0answers
22 views

Integral of cos(x) wrt t when integral of sin(x) wrt t is known [on hold]

If $\int_0^T sin(\theta) dt = A$, where $\theta$ is a variable, A is constant. Then can we find out $\int_0^T cos(\theta) dt$ = ?
0
votes
0answers
49 views

smoothness of boundary under Riemann mapping

Suppose there is a smooth Jordan curve separating the complex plane. For complicity, assume the curve is given by a graph $(x, \phi(x))$, where $\phi(x)$ is smooth, bounded, and derivatives are ...
0
votes
0answers
7 views

Frobenius series of Fuchsian PDEs

I'm interested in the analyticity of Frobenius-like series solutions to a PDE in $z=(z_1,\ldots ,z_N)\in\mathbb{C}^N$ with regular singular behavior at $z_\alpha=0$ for all $\alpha=1,\ldots, N$. For ...
1
vote
0answers
30 views

Fundamental system of neighborhoods. Self similar set

I have been reading the text Analysis on Fractals of Jun Kigami. There is a theorem about the fundamental system of neighborhoods of a point in a self similar set. It is stated as follows Let ...
0
votes
0answers
13 views

Standard term for parametrisation where heights of parameters and values are correlated

Suppose an algebraic variety over Q, or subvariety of one, has a parametrization, also over Q. Clearly an infinite number of birationally equivalent parametrisations can be obtained from this. But ...
7
votes
4answers
206 views

List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...
0
votes
0answers
44 views

an example for fundamental group of graph of groups [on hold]

suppose we have a graph $X$ with the vertex set $\left\lbrace v_1,v_2,v_3 \right\rbrace $ and the edge set $\left\lbrace e_1,e_2,e_3 \right\rbrace $ like a triangle. let $(\Gamma,X)$ be a graph of ...
1
vote
0answers
18 views

Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous? [on hold]

There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve. To construct one example of such a function ...
1
vote
0answers
41 views

Solvable Lie algebra whose nilradical is not characteristic

It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p > 0 need not be characteristic (that is, invariant under all derivations of the algebra), but ...
0
votes
0answers
28 views

HNN extension group with finitely generated base

Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The ...
-2
votes
0answers
26 views

Algorithm to rate board of tic-tac-toe [on hold]

At start I want to say that im programmer and I don't want anyone to write me code, just to help me what I can use. Is there any algorith which I can use to rate a board of tic-tac-toe ? What I want ...
5
votes
1answer
217 views

Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of ...
3
votes
0answers
30 views

Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
5
votes
0answers
73 views

Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
1
vote
0answers
11 views

Construction of point process having same pair correlations as GUE

The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately) $$ g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + ...
4
votes
1answer
76 views

Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
3
votes
2answers
46 views

$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von Neumann ...
4
votes
0answers
39 views

When is a crossed-product algebra a division algebra?

Let $L/K$ be a finite Galois extension with Galois group $G$. For every 2-cocycle $\gamma$ of $G$ with values in $L^\times$ there is the crossed-product $K$-algebra $$S(L,G,\gamma) = \bigoplus_{g\in ...
-3
votes
0answers
18 views

Get angle of Trajectory of a projectile [on hold]

Formula1 Since a view hours I'm desperately trying to solve this equation after alpha. I can't use Formula2 because my launch starts at the height h. Thanks for your guys guidance and help.

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