Questions tagged [tag-removed]

This tag is to be used only when re-tagging highly(!) off-topic questions where none of the actual tags would make sense; all actual tags the questioner has used are removed and something is needed to have some tag, which is enforced by the software, so this tag is used. However note that this tag is also used in the process of moderators deleting tags. Thus a question having this tag does most of the time not mean that somebody found it off-topic.

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-3 votes
1 answer
158 views

How to find the content of a sphone [closed]

I need to know how to find the contents of a sphone; however I have not been able to find an equation for it online. I noted that the equation for a cone is 1/3(h)(A base). So I thought that perhaps ...
0 votes
1 answer
253 views

Background on the functional equation $F(x+1)+F(x)=f(x)‎$ [closed]

In the theory of indefinite sums, anti-differences and finite calculus, ‎the following ‎difference ‎functional ‎equation ‎and ‎its ‎solutions ‎are ‎very ‎important: ‎$$‎\bigtriangleup ‎F(x):=F(x+1)-...
2 votes
1 answer
340 views

Submodularity property of trace of inverse matrix

$\newcommand{\tr}{\operatorname{tr}}$Does submodularity property hold for the trace of a positive-definite hermitian matrix? I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ ...
  • 121
8 votes
2 answers
570 views

Is there a closed-form solution for $\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$?

I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$, $$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$ I ...
  • 493
2 votes
1 answer
129 views

Exact solution of a particular system of non-linear equations (re-formulated to matrix equation)

I have this system of $n$ non-linear equations in $n$ unknowns, arising out of my research problem. Given that $x_0=1$, I have to solve for $(x_1,x_2,\ldots,x_n).$ $$\sum_{i=0}^n x_i^2+2\sum_{j=1}^n\...
  • 121
1 vote
3 answers
622 views

When does $f^{-1}=\frac{1}{f}$ with $f$ a function mapping $\mathbb{R}^{*}$ to $\mathbb{R}$?

In mathematics, an inverse function is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ ...
15 votes
3 answers
3k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
3 votes
1 answer
130 views

Trying to solve this non-linear differential equation

I have a second order differential equation given by: $x''(t) = \displaystyle\frac{\exp(-\frac{x(t)^2}{4t})}{A \sqrt{t}}$ I would like to be able to obtain an analytic solution to this equation, ...
  • 151
1 vote
1 answer
454 views

Exact solution to nonlinear differential equation sought

I am looking for an exact solution to equation: $w''=aw^{-1/3}+b[f(y)]w^{-5/3}$, where $w=w(y), f(y)$ - arbitrary function (in this case $y^n$ with arbitrary $n$); $a,b$ - constants. Of course I can ...
  • 11
7 votes
2 answers
421 views

Fundamental lemma: why is the transfer factor a power of q

Let $k$ be a finite field of sufficiently large characteristic, $F = k((t))$ and $\mathfrak{o} = k[[t]]$. Let $G$ be a reductive algebraic group defined over $\mathfrak{o}$. Roughly stated, for sake ...
  • 1,793
5 votes
2 answers
1k views

Equation between the two branches of the lambert w function

My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$? For example the two square roots $r_1(y)$ and $r_2(y)$ of ...
6 votes
1 answer
2k views

Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $...
3 votes
0 answers
1k views

What is the inverse kernel of this integral transform?

I am looking for the associated inverse kernel to the integral transform $T$ defined by $(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$ whose kernel is $K(u,t) = \frac{...
2 votes
1 answer
236 views

Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$. We can generalize the notion of indecomposable from groups to inclusion of groups as ...
6 votes
0 answers
440 views

Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
  • 21.8k
2 votes
1 answer
436 views

How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let: $\begin{eqnarray} p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\ G(x,y) &=& c_k\...
2 votes
1 answer
371 views

radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke: Teorem. Let $f : U ⊂ \mathbb{R}^n ...
6 votes
0 answers
248 views

Other applications of the 'increment' approach

I would like to hear about other instances of the so-called 'increments' approach, first used by Roth to prove that a subset of $\mathbb{N}$ of positive upper density contained infinitely many ...
10 votes
0 answers
336 views

Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...
2 votes
1 answer
295 views

Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$ I have tried various substitutions, simplifications ...
  • 173
7 votes
3 answers
1k views

When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say). So far I had the following idea:...
  • 79
16 votes
1 answer
656 views

Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
  • 163
2 votes
2 answers
548 views

Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site. Let $S$ be some orientable surface obtained by removing ...
  • 9,220
4 votes
1 answer
480 views

Topological characterisation for a (closed irreducible) hyperbolic 3-manifold

Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. ...
0 votes
1 answer
265 views

Fibration in the 3 torus.

The Hopf fibration $S^1\rightarrow S^3\rightarrow S^2$ gives a decomposition of $S^3$ into 2-tori and to circles, so that the tori are foliated by circles of slope 1. If you take the region between ...
6 votes
1 answer
374 views

Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$. Q: Does this imply that $U$ is homeomorphic to $U'$? In the case where the $\pi_1$'s are trivial then ...
4 votes
1 answer
1k views

Exact solutions to nonlinear Klein-Gordon equation

The nonlinear pde $$ \partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0 $$ has the exact solution $$ \phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i) $$ ...
  • 1,622
6 votes
1 answer
1k views

Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...
2 votes
1 answer
242 views

Finding a good ordering of $\mathbb{Q}$

Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result. From a research question I am working on I have simplified the example/...
1 vote
1 answer
150 views

Approximating rational generating functions

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0}^\infty x_iX^i$ is ...
3 votes
0 answers
141 views

P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
  • 31
0 votes
1 answer
240 views

Are period domains ever contractible

Which simply-connected period domains are contractible? Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety? Are these contractible?
  • 19
1 vote
1 answer
321 views

Name convention for the composition of the preimage of a function and the function itself

Hi, given a function $f:X \rightarrow Y$, not necessarily invertible, is there a conventional name for the function $$g_f := f^{-1} \circ f:X \rightarrow \mathcal{P}(X),$$ where $\mathcal{P}(X)$ ...
  • 207
5 votes
1 answer
846 views

Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...
2 votes
1 answer
626 views

Continuous functions on path-connected subsets

Let $X$ be a topological space, and $PX$ the space of all paths on $X$. Then let $G\subset X$ be a path-connected subset and $p\in G$ a point. Let $\sigma:G\rightarrow PX$ be a continuous function ...
5 votes
1 answer
1k views

Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
  • 51
-2 votes
1 answer
2k views

sections of tensor product bundle ( tensor product of two vector bundles ) [closed]

Suppose we have a smooth manifold M and E--->M is a vector bundle. A connection on E is a linear map from the set of all smooth section on E into the set of smooth sections of the tensor product of E ...
  • 155
3 votes
1 answer
226 views

Spectral synthesis for central functions on locally compact groups

There is a large literature on harmonic analysis on locally compact group, that I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...
  • 25k
2 votes
1 answer
264 views

symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation $v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...
's user avatar
1 vote
2 answers
575 views

Limit with theorem of dominated convergence

Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$) I have to calculate this limit $$\lim_{|x-y|\to 0}\int_{\...
  • 25
-1 votes
1 answer
180 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} f(x)=...
  • 25
3 votes
0 answers
359 views

Extension divergence-free, curl-converging vector field

Hi. Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
  • 2,120
4 votes
1 answer
717 views

Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$$...
3 votes
2 answers
381 views

Homotopy Equivalences and Induced Correspondences between Fibre Bundles

Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...
  • 4,810
2 votes
1 answer
562 views

A uniformity with a countable base is a pseudometric uniformity.

I need a proof for this proposition: If a uniformity $\mathfrak U$ on $X$ has a countable fundamental system of entourages, then it can be defined by a pseudometric on $X$. which is the ...
2 votes
1 answer
77 views

Computing a point of refraction

Oddball question: say I want to travel from $(a, b)$ where $b > 0$ to $(c, d)$ where $d < 0$ using the shortest path, where I can travel at velocity $v_1$ in the upper half-plane and at velocity ...
  • 285
1 vote
1 answer
266 views

unramified base change in characteristic p > 0?

Hi, Suppose that $E/F$ is a unramified extension of local fields of characteristic zero. Let $G = GL_n$. Then it is well-known (due to Clozel?) that base change of tempered representations from $G(F)$...
  • 2,732
1 vote
1 answer
424 views

Is this min not less than a min

Let $\mathbf{D}$ be the unit disk, is $$\inf_{\begin{array}{c} v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\ v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right) \end{array}}\max_{0\le i,j,k\le4}\...
  • 27
-1 votes
1 answer
212 views

Group or manifold ? [closed]

I have a question in seeing this $$U(n)=\frac{U(n)}{U(n-1)} * \frac{U(n-1)}{U(n-2)}*\cdots *\frac{U(2)}{U(1)}*U(1)$$ So, group U(n) is written as product of quotient spaces. Is quotient space, for ...
  • 9
3 votes
1 answer
1k views

How do minimal polynomials relate?

How does the idea of a "minimal polynomial" for a matrix (i.e. for a matrix $A$, the polynomial, $\mu (x)$, of least degree, such that $\mu (A) =0$) relate the the "minimal polynomial" for some ...
  • 103

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