All Questions
13,927 questions
0
votes
1
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219
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Finding closed form expression for the roots of $f(x) = \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]}$
Let us define function $f:[0~ 2\pi] \rightarrow R$ as follows:
\begin{align}
f(x)\triangleq \sum_{i=1}^K \frac{\alpha_i \gamma_i \sin(x-\theta_i)}{1+\gamma_i[1+\cos(x-\theta_i) ]},
\end{align}
...
- 105
0
votes
1
answer
269
views
Limit of eigenvalues of a matrix perturbation sequence
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
- 135
0
votes
1
answer
203
views
For $B=\int \lambda d E_\lambda $ and $X$ commutes with every $E_\lambda $, why $BX$ is positive and self-adjoint?
Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E_\lambda $, then why ...
- 617
0
votes
1
answer
142
views
A property of compact topological space via certain $C^*$ embedding in operator algebras
Assume that $A$ is a unital $C^*$ algebra. Is there a $C^*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^*$ subalgebra of $B(H)$?
If not, is the answer affirmative when $A$ is ...
- 366
0
votes
1
answer
221
views
A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded
We know every weakly open subset of an infinite-dimensional Banach vector space X is unbounded.
Now, Read's space $R$ (an infinite-dimensional Banach space) has the property:
there is $ρ >0$ such ...
- 93
0
votes
1
answer
136
views
When are Weighted $\mathcal{L}^p$-Spaces Topologically Isomorphic?
Let $X$ be a topological space and $\mu$ be the Borel measure on $X$. Suppose $W_1$ and $W_2$ are continuous, non-negative functions from $X$ into the real numbers such that, for all integers $p > ...
- 263
0
votes
1
answer
237
views
continuity of b-metric
A b-metric is defined similar to a metric in which the triangle inequality is replaced by the inequality
$$d(x,z)\leq s\Big[d(x,y)+d(x,z)\Big]\quad\forall\ x,y,z$$
where $s\geq1$.
There is an example ...
- 149
0
votes
2
answers
653
views
Poincaré inequality on annular regions
It is well known that given a function $f \in L^p(B_R)$ such that $|\{x \in B_R: f(x) = 0\}|>0$, the following Poincaré inequality holds:
$$ \int_{B_R} \left(\frac{|f|}{R}\right)^p \ dx \leq c \...
- 455
0
votes
1
answer
186
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
0
votes
1
answer
68
views
Connected $T_2$-spaces with nowhere dense covering number $3$
This is a special case of a question that has not been answered so far.
If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the ...
- 48.9k
0
votes
1
answer
127
views
Product topology and order convergence topology
Let $(P,\leq)$ be a poset. We define the order convergence topology, denoted by $\tau_o(P)$. By a set filter $\mathcal{F}$ on $P$ we mean a collection of subsets of $P$ such that:
$\emptyset \notin \...
- 48.9k
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
0
votes
1
answer
82
views
Maximal discrete subsets of connected $T_2$-spaces
If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.
Is there an infinite connected $T_2$-space $(X,\tau)$ ...
- 48.9k
0
votes
1
answer
103
views
Suppose that $a \mu = \mu a$ for all $a$ in $C^*$-algebra $A$. Then $\mu \in Z(A^{**})$
Let $A$ is a $C^*$-algebra and $\mu \in A^{**}$. Suppose that $a \mu = \mu a$ for all $a \in A$. Then $\mu \in Z(A^{**})$.
- 19
0
votes
1
answer
111
views
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
If $A$ is a $C^*$-algebra, then $H^1 (A, D(A)) = \{ 0 \}$ (first cohomology group )?
We don't know that is an open problem or it has counterexample...
- 19
0
votes
1
answer
152
views
Transitivity of the Cuntz sub-equivalence
Let $A$ be a $C^*$-algebra and $a,b \in A$ positive elements. We define a relation (Cuntz sub-equivalence) by saying
$$a\lesssim b: \Leftrightarrow \exists\, (r_n)_{n\in\mathbb{N}}\subset{A}\text{ ...
- 1
0
votes
1
answer
317
views
Some questions related to the unitary operators
A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...
- 5,529
0
votes
1
answer
55
views
On 1-iso maps and subsets of the unit circle
Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...
- 97
0
votes
1
answer
843
views
$C^{\infty}_{loc}$-convergence - right definition
Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f_{n}$ converges to $f$ in $C^{\infty}_{loc}(\Omega)$. ...
- 35
0
votes
1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 175
0
votes
1
answer
482
views
Complement of a finite union of convex sets
Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...
- 2,933
0
votes
1
answer
126
views
Nonstable $K$-theory question
Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$.
Question A: Is $GL_1(C(Y))\stackrel{\pi}...
0
votes
1
answer
153
views
Path-connected Hausdorff interval topologies
Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$?
(This is a follow-up ...
- 48.9k
0
votes
1
answer
214
views
Sobolev chain rule on non-compact manifolds
Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions $...
- 1
0
votes
1
answer
149
views
Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?
Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank ...
0
votes
1
answer
622
views
Is there any way to compare between diagonals of a resolvent and a Cauchy transform?
Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
- 617
0
votes
1
answer
152
views
When can two Cauchy transforms intersect?
Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...
- 1,893
0
votes
1
answer
91
views
Intersection of complements of connected components
Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^*\in X\setminus S$.
Let $E$ be the connected component of $X\setminus S$ that contains $x^*$. Let ${\cal C}$ be the ...
- 48.9k
0
votes
1
answer
264
views
Banach space dual to $L^\infty(I,H^1(M))$
What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus.
Any references that show ...
- 1,594
0
votes
1
answer
333
views
Sobolev's lemma on manifolds
Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying $...
- 3
0
votes
1
answer
375
views
About an integral equation
I would like to obtain $g$ by solving the following integral equation
$$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$
where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+
...
- 123
0
votes
1
answer
461
views
Orthogonal projection
Let $G$ be an operator with compact resolvent on a Hilbert space $H$ such that
$\ker G \neq \{0\}$.
Further let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} := G+P$.
My question is: ...
0
votes
1
answer
143
views
Complementation in tensor products
This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. Reasonable means that $\|x\...
- 13
0
votes
1
answer
123
views
Scalar Measure Associated to a Positive Operator Valued Measure
Suppose that $X$ is a compact metric space, $\mathcal{H}$ is a Hilbert space, and that $A: \mathcal{B}(X) \rightarrow \mathcal{B}(\mathcal{H})$ is a Positive Operator Valued Measure (POVM), where $\...
- 59
0
votes
1
answer
195
views
Existence of bounded $n-$th derivative of the solution of differential equation
This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)...
- 133
0
votes
1
answer
222
views
Behavior of the integral of products of probability densities
Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := \...
- 65
0
votes
1
answer
107
views
Topology : Study on Separation Properties [closed]
I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example
- 115
0
votes
1
answer
791
views
Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?
Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$.
It follows that for almost all $t$, $u_n(t)$ is bounded in $L^\...
- 266
0
votes
1
answer
130
views
Positive inuequality of Laplacian with a variable coefficient [closed]
Let $0 < a_0 \leq a(x)$ be a smooth function on $\mathbb{T}=[0,2\pi]$, $a(0)=a(2\pi)$, whether it holds that
$$
\int_\mathbb{T} a(x)|\partial_x \varphi|^2 \geq \int_\mathbb{T}\frac{\partial^2_x a }...
- 425
0
votes
1
answer
243
views
Dense subsets on set space
Let $X$ be a metric space, and $\mathscr{B}$ the $\sigma$-algebra generated by open sets of $X$. Can we find a countable dense subsets of the metric space $(\mathscr{B},d)$ with the metric $d(A,B)=m(A\...
- 97
0
votes
2
answers
2k
views
Spectral decomposition of compact operators
Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...
- 101
0
votes
1
answer
182
views
Rational points in the Alexandroff line
Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line.
Is $X$ a metrizable space?
- 366
0
votes
1
answer
475
views
uniqueness for Poisson equation in R^d with mildly regular data
I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
- 5,380
0
votes
1
answer
111
views
Is a weakly separable group always Lindelöf?
By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
- 135
0
votes
1
answer
169
views
Exponential Convexity Results [closed]
$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all ...
- 153
0
votes
1
answer
294
views
Exponential Convexity
$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all ...
- 153
0
votes
1
answer
92
views
Finite dimension implies regularity
Let $\mathscr{D}'(\mathbb R)$ be the set of distributions on $\mathbb R$ and $X$ be a linear subspace of $\mathscr{D}'(\mathbb R)$, which is closed under translations, i.e., if $\varphi\in X$ and $h\...
- 2,933
0
votes
1
answer
529
views
Bounded approximate identity and kernel of algebra homomorphism
Let $\cal A$ be a Banach algebra with a bounded approximate identity, when a closed two sided ideal on it has a bounded approximate identity?
In particular, let $\cal B$ be a Banach algebra with a ...
- 565
0
votes
1
answer
265
views
find a weak solution in an intersection of Sobolev spaces
In
using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces
the weak solution for
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
was discussed,...
- 111
0
votes
1
answer
157
views
An intermediate functional space
is there an intermediate functional space $Y$ such that the Sobolev space $W^{1,N}$ compactly embeds in $Y$ and $Y$ continuously embeds in $L^N$, where $N$ is the ambient space dimension?
Thank you
- 1