-2
votes
0answers
73 views

Analytic formula to evaluate the exact value of solid angle subtended by an ellipse at any arbitrary point lying on the vertical axis [closed]

`Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center. Standard equation ...
0
votes
0answers
72 views

Relating overlapping simplicial complexes

Let $X$ be a simplicial complex and let $A,B\subset X$ be subcomplexes such that $C=A\cap B$ is a non-empty simplicial complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$, ...
1
vote
1answer
103 views

Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E (i.e. line bundles). Let $$ n_1:= ...
0
votes
0answers
64 views

Integral geometry and curvature of surfaces

I'll stick to 2-surfaces in $\mathbb{R}^3$ for simplicity. Higher dimensions generalizations welcome. In classical integral geometry, we may obtain up to a scaling factor, for example, the surface ...
0
votes
0answers
29 views

Find all integers x, y, and z such that 1/x + 1/y = 1/z [migrated]

Characterize all positive integers x, y, and z such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. For example, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$
0
votes
0answers
23 views

contiguity and infinite complexes

Two simplicial maps $f: K^{(n)}\to L$, $g: K\to L$ are called contiguous if for every simplex $\tau\in K^{(n)}$ and it carrier $\sigma$ we have that $f(\tau)\cup g(\sigma)$ is a simplex in $L$. It ...
0
votes
1answer
127 views

A question about open subsets of Hilbert space whose complements are compact sets

Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other ...
18
votes
6answers
961 views

Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ...
5
votes
1answer
133 views

Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
6
votes
0answers
157 views

Polynomial growth without Gromov's theorem

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies ...
5
votes
1answer
109 views

Foliations of Lorentzian manifolds by Spacelike Hypersurfaces

Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...
1
vote
1answer
127 views

Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...
1
vote
1answer
168 views

How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?

Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and $\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of points of $M$, where $d\mu|_p$ fails to be ...
0
votes
1answer
136 views

Function field of the Jacobian of genus 2 curve over $\mathbb{F}_q$

I have been trying to build the function field of the jacobian of a genus 2 smooth curve over a finite field, but I am having problems making it explicit, I need to work with another curve with points ...
5
votes
3answers
123 views

Product of binary Boolean operators

I asked this question a day ago on math.stackoverflow but figured it could have an interest here. I'm interested in the set $\mathcal{P}_N$ of boolean functions of boolean variables $p_1, p_2, ...
0
votes
0answers
63 views

Pro-constructible subset of scheme intersects very dense subsets?

Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$? If $Y$ is just constructible, this is true. ...
0
votes
0answers
56 views

Characterization of the stable manifold [closed]

Assume we study a (finite dimensional) differential system $$ x'(t)=f(x(t)), \quad x(t) \in \mathbb R^n, $$ for a smooth function $f$ and such that $0$ is an equilibrium point. Thus, we have existence ...
-1
votes
0answers
33 views

Combinatorial optimization problem [closed]

Suppose I have a population, divided into 6 known classes. I get a feasible solution when I select 2 elements from each class (so, 12 in total). For every feasible solution, I can compute a "cost". ...
-2
votes
0answers
52 views

Good upper bound for an alternanting series [closed]

Someone know a good upper bound for the partial sums of $S=\sum(-1)^{n+1}\sqrt{n}$? I mean how fast is the growth of this sum?
-4
votes
0answers
57 views

Can a non-compact manifold be embedded? [closed]

Can a non-compact smooth manifold be embedded into another smooth manifold? Moreover, Can we get a diffeomorphism between tow non-compact manifolds ? The first part of the question is about smooth ...
0
votes
0answers
67 views

Rolling map as a diffeomorphism?

Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map) $$(\Phi\gamma)(t) = ...
0
votes
0answers
77 views

When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...
0
votes
0answers
157 views

Multiplication Map, Is it invariant?

Let $\pi:X\rightarrow Z$ a double cover of an elliptic curve with genus $g\geq 3$. Choose a general rank 2 and degree -1 vector bundle $F$ on $Z$, let $E=\pi^*F$ and fix $x\in X$. The involution $i$ ...
3
votes
1answer
78 views

Upper bounds on elements of a matrix

During my research I have come across matrices this type $$C=B\left(B^T B\right)^{-1}B^T\ ,$$ where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
-5
votes
0answers
28 views

what is the computational complexity for finding SVD and pseudo inverse? [closed]

For a given MxN matrix A and A is full rank matrix (rank=N and M>N),what is the computational complexity for finding SVD and pseudo inverse ?Which one will be having low complexity?
0
votes
0answers
18 views

Deriving inequalities from a polynomially-bounded derivative

In this paper (p. 2, definition/remark) the following notion of ‘polynomial growth’ is defined for a non-negative real function $g(x)$ and a real constant $b\in(0;1)$: There exist positive ...
10
votes
1answer
220 views

Piecewise linear (PL) structures on $\mathbf R^4$

One can read in Wikipedia that the 4-dimensional affine space $\mathbf R^4$ has uncountably many piecewise linear structures (in contrast with other dimensions, where it has exactly one). A reference ...
3
votes
1answer
798 views

Who coined “mob” and “clan” and why these words?

A mob is a word used for a topological semigroup which is a Hausdorff space. A clan is a compact connected mob with a two-sided identity element. Who used these words with these meanings first and ...
-1
votes
0answers
43 views

Hyperbolic paraboloid, Analytic Geometry [closed]

Please don't ban this question, I just need some advice on how to find the equation of the tangent plane on a point of the hyperbolic paraboloid which is perpendicular to a certain plane say ...
1
vote
1answer
166 views

Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
1
vote
1answer
54 views

An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$

In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...
-1
votes
0answers
75 views

Represention of the element of a group: A Confusion [closed]

I am reading a paper "FAST ALGORITHMS FOR CALCULATION OF GIBBS DERIVATIVES ON FINITE GROUPS" by R. S. Stankovic (Approx. Theory & its AppL 7:2, June 1991). In Section 2, following is written about ...
1
vote
0answers
96 views

Averages of $L(s,\chi)$

Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol. What is the abscissa of convergence of the double Dirichlet series ? $$ \sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 ...
-4
votes
0answers
20 views

Expression in theta notation [closed]

Am I right that the theta notation for the following expression is: $n^2+(n^3/2)$ = theta ($gn^3$) as $n^2/2$ is the low order term
0
votes
0answers
109 views

On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...
0
votes
0answers
61 views

A simple question about a resolution of a conifers singularity

Let $X$ be a conifold defined by the equation $xy-zw=0$ in $\mathbb{C}^4$ and $\tilde{X}$ its crepant resolution, which is isomorphic to $\mathcal{O}_{\mathbb{P^1}}(-1)^{\oplus 2}$. Then there is a ...
7
votes
1answer
132 views

When is the tensor product of two graphs planar?

Given two graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$, the tensor product of $G$ and $H$ is the graph $G \times H = (V,E)$, where $V=V_1 \times V_2$ is the Cartesian product of the $V_i$ and $ (u,v) \ E \ ...
1
vote
1answer
92 views

configuration spaces of real projective space

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring $$ H^*(F(\mathbb{R}P^n,k);R)$$ is obtained for any ...
0
votes
0answers
42 views

To prove ideals are coprime [closed]

Let $K=\mathbb Q(\sqrt{-m}) $ be a quadratic field. Let $ O_K$ be ring of algebraic integers. Let $\alpha=a+b\sqrt{-m}\in O_{K} $ with gcd$(a,b)=1 .$ Then how to show that $\langle \alpha \rangle $ ...
0
votes
1answer
62 views

Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold

Let $M$ be an $m$-dimensional simply connected Riemannian manifold that is not geodesically complete. Suppose $M$ has constant sectional curvature. Because the curvature is constant, locally $M$ ...
-1
votes
0answers
25 views

Linear combinations of columns of matrices [closed]

Suppose E is a 4 × 3 matrix with columns ⃗c1, ⃗c2, ⃗c3 and rows ⃗r1, ⃗r2, ⃗r3, ⃗r4. Let ⃗v be a 3 x 1 matrix that = [2, -1, 1] How could we express E⃗v as a linear combination of ⃗c1, ⃗c2, ⃗c3? Now ...
0
votes
0answers
16 views

About uniform convergence of empirical distribution [migrated]

By Glivenko–Cantelli theorem, we know that the empirical cdf converges to the true cdf uniformly, i.e., $\underset{x\in\mathbb{R}}{\sup}|F_n(x)-F(x)|\overset{a.s.}{\longrightarrow}0$, where ...
3
votes
0answers
94 views

When does a map in the stable homotopy group gets killed when smashed with cone of itself?

Consider an element $e \in \pi_n^s(S^0)$, in the stable homotopy groups of sphere. Let $C$ denote the spectrum which is cone of $e$, i.e. $C$ fits in the cofiber sequence $$ S^n \to S^0 \to C.$$ ...
1
vote
1answer
124 views

Noetherianess of a finite module over a noethrian ring without Axiom of Choice

All rings are assumed to be commutative with 1. We say a module over a ring is strictly noetherian if every non-empty set of submodules has a maximal member. We say a ring is strictly noetherian if it ...
1
vote
1answer
91 views

Q(\sqrt{-l_0}) satisfies Heegner hypothesis for an Elliptic curve of conductor C implies C is a square

Trying to understand the proof of Corollary 2.3 in the following paper, http://arxiv.org/pdf/1312.3884.pdf I would like to be able to justify that the root number of the quadratic twist ...
1
vote
1answer
28 views

About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
2
votes
1answer
210 views

What happens to the cohomology ring after a “flip-flop”?

I've been trying to understand what happens to the cohomology ring (say with coefficients in $\mathbb{R}$) of a smooth complex projective manifold after blowing up along a smooth complex submanifold. ...
-2
votes
0answers
72 views

Prove the contrapositive: (p => q) => (~q => ~p) using only FITCH rules of inference [closed]

I stumbled upon this problem when taking Stanford's Intro to Logic Course. Wikipedia shows some proofs to the contraposition ( http://en.wikipedia.org/wiki/Contraposition ) but I couldn't fit any of ...
8
votes
3answers
210 views

How do small central extensions drop the dimension of a faithful representation?

Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied... I am interested in the phenomenon ...
0
votes
0answers
70 views

Integer points on Elliptic Curves [closed]

It's easy to prove that equation y^2=x(x-a)(x-b) with a and b integer has integer solution, other than (0,0), (a,0) and (b,0), if a and b jointly admits the representation a=(r-s)r (1) b=(r-t)t ...

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