All Questions
1,487 questions with no upvoted or accepted answers
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116
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Growth estimates for polynomials with natural coefficients
Suppose $p(x)$ is a degree $m$ polynomial whose coefficients are natural numbers. Suppose further that we have $p(1)=n$, $p(2)\leq nm$ and $p(3)=n^2$, and assume that $m\leq \log n$. So $p$ only grows ...
- 21
2
votes
0
answers
60
views
Gronwall type inequality involving iterated integrals
Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have:
$$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$
where $\kappa, C >...
- 31
2
votes
0
answers
73
views
Closed set containing infinite arithmetic progressions of ANY gap
Let $A\subseteq [0,\infty)$ be a set containing infinite arithmetic progressions of ANY gap, that is, for any $d>0$, there is $t>0$ such that $t+kd\in A$ for all $k\in \mathbb N$.
Molter and ...
- 751
2
votes
0
answers
76
views
Bounding algebraic numbers away from the Gaussian integers
Let $\alpha$ be an algebraic number with degree $\leq d$ and (absolute multiplicative) height $\leq H$. Then we can say a couple of things about such $\alpha$:
(1) We know the set of all $\alpha$ ...
- 31
2
votes
0
answers
87
views
Conditions for $B$ that make $ADB + (ADB)^T$ positive (semi-)definite
I am trying to find conditions under which this dynamical system converges. At the end of the day, we have something like
$$
0 \leq x^TD_0W_0D_1W_1D_2 \dots W_ND_NCx
$$
With $D_i$ matrices that are ...
- 121
2
votes
0
answers
58
views
Absolute continuity of DOS measure for Schrödinger operators
Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure.
I would like to ...
- 21
2
votes
0
answers
114
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Is there an explicit version of Morse Lemma used in stationary phase method?
In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
- 3,625
2
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0
answers
1k
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Is there an infinite product like this for $\cos x$?
There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
2
votes
0
answers
119
views
question about sequences and series (complex analysis may be only elementary real analysis)
I would like to have your help about the proof of the following statement:
If the sequences of complex numbers $\{F_N\}_{N \in \mathbb{N}},\{G_N\}_{N \in \mathbb{N}}$ have the following properties:
(...
- 51
2
votes
0
answers
46
views
increasing inter-class distances results in decreasing linear regression error
Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set.
Define $\mathbf{...
- 183
2
votes
0
answers
135
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Can we get rid of this test function?
I have a real-valued function $f$ defined on a ball $B$ of $\mathbb{R}^{N}$, $N\geq2$. I have found a constant $M>0$ such that for all $x\in B$ and $B(x,R)$ (ball of center $x$ and radius $R>...
- 411
2
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0
answers
142
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Self-adjointness on Banach spaces
Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded ...
- 501
2
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0
answers
65
views
Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
- 623
2
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0
answers
210
views
A sum with integer parts
Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^...
- 593
2
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0
answers
147
views
Interchanging limit and infinite product in Euler product for Dedekind function s=1
For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation
$$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
- 21
2
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0
answers
79
views
One-dimensional integral equation uniquely solvable?
I recently met a question similar to this one and I would like to post it here, because I basically found nothing:
We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
- 329
2
votes
0
answers
86
views
when is the average of a function with Gaussian inputs bounded away from zero
Define a function $\phi(x):\mathbb{R}\rightarrow\mathbb{R}$. Consider the expected value function defined as follows
\begin{align*}
\mu(\beta)=E[g\phi
(\beta g)]\quad with \quad g\sim\mathcal{N}(0,1)\...
- 363
2
votes
0
answers
60
views
Finding a function in contour integration involving Riemann mapping
Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
2
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0
answers
122
views
Which functions $f: \mathbb{R} \to \mathbb{R}$ is injective over some subinterval of $(x,y)$ whenever $x<y$ and $f(x) \ne f(y)$?
Under what conditions on a function $f: \mathbb{R} \to \mathbb{R}$ can we say that given any real numbers $x,y$ with $x<y$ if $f(x) \ne f(y)$ then there is a sub-interval $S_{(x,y)}$ of $(x,y)$ ...
- 850
2
votes
0
answers
269
views
Implicit Function Theorem, parametrized - how can we get uniform domains? (from math.se)
(This question is a duplicate from here)
Consider a family of continously differentiable functions $F_r\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (where $r\in[0,1]$). For every parameter $r$, we ...
- 141
2
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0
answers
192
views
Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
...
- 623
2
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answers
518
views
When will the upper regularization of a bounded function not defined?
Suppose $E$ is a compact metric space.
A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$
For any real-valued ...
- 623
2
votes
0
answers
73
views
A question on groupoids and measurable fields of Hilbert spaces
Suppose that we have the following data:
$ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and
range maps denoted by $ s $ and $ r $ respectively.
$ (\lambda^{x})_{x \in \...
- 1,396
2
votes
0
answers
226
views
degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
2
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0
answers
195
views
A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 505
2
votes
0
answers
115
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Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 305
2
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0
answers
341
views
Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 305
2
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0
answers
571
views
Integrating a product of integrals involving Bessel functions
I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable.
I ...
- 161
2
votes
0
answers
139
views
Existence of solution of a variational inequality
Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \...
- 21
2
votes
0
answers
92
views
Estimating the size of a subset of $\mathbb{R}^N$
This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
- 2,480
2
votes
0
answers
100
views
Roots of a partially holomorphic function
Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\...
- 16.2k
2
votes
0
answers
186
views
Is this simple oscillatory integral operator uniformly bounded on $L^2$?
Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let
$$T_\lambda f(t)=\int \frac{\...
- 171
2
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0
answers
228
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Integrating an n-fold Cauchy product of a Fourier series
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
- 161
2
votes
0
answers
45
views
Maximizing the sum of a decreasing function over a separated set
Fix $d>0$. Let $f:[0,\infty)\to(0,\infty)$ be a decreasing function of $x$ for $x\geq d$. Let $S_d\subset\mathbb{R}^n$ represent a set of points containing the origin such that the (Euclidean) ...
- 21
2
votes
0
answers
63
views
Sensitivity of a function against its random arguments
Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
- 482
2
votes
0
answers
385
views
(Quasi) convexity of separately convex homogeneous functions
Consider a function $f:\mathbb{R}^n_{\geq 0}\rightarrow \mathbb{R}$ that is separately convex, i.e. such that $\frac{d^2f}{dx_i^2}\geq 0$ for all $i\in \{1,\dots n\}$. Assume also that $f$ is ...
- 389
2
votes
0
answers
275
views
Smoothness of coefficients of remainder term in Taylor expansion
Given a $C^{k}$ function $f:\mathbb{R}^d\to\mathbb{R},$ we can use Taylor's theorem to write it as
$$f(x)=\sum_{|\alpha|\le k-1} c_\alpha x^\alpha + R(x),$$
where $R$ is $C^k$ and can be expressed ...
- 203
2
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0
answers
279
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Can a bounded open set in $R^n$ be always approximated from outside with a finite union of dyadic cubes?
Suppose we have a bounded open set $S$ in $R^n$. Consider the collection of closed dyadic cubes $C_k$'s (https://en.wikipedia.org/wiki/Dyadic_cubes). I was wondering if there always exists a finite ...
- 131
2
votes
0
answers
202
views
Universal chord theorem for curves
Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
- 121
2
votes
0
answers
183
views
Fourier series and regular distribution
Assume you have a distribution $K$ on $\mathbb{T}$, the torus, such that $\sum_{n=-\infty}^{\infty} |K(e_n)|^2$ is finite, where $e_n := e^{in\cdot}$ are the Fourier basis. Does this imply that the ...
- 95
2
votes
0
answers
60
views
A question about Kolmogorov Superpositions
D.A. Sprecher showed (https://www.researchgate.net/profile/David_Sprecher2/publication/243052898_A_Representation_Theorem_for_Continuous_Functions_of_Several_Variables/links/554929f20cf2ebfd8e3ad956....
- 371
2
votes
0
answers
254
views
Prove this function is increasing
I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...
- 21
2
votes
0
answers
110
views
If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
user76025
2
votes
0
answers
86
views
I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
2
votes
0
answers
55
views
Does there exist $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?
Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions.
We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
- 85
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
2
votes
0
answers
67
views
On two functions with isodirectional gradients
Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$
\begin{equation}
(\...
- 1,461
2
votes
0
answers
67
views
How much must a curve bend to intersect another curve twice?
Suppose $c_1$ and $c_2$ are segments of smooth plane curves. To be concrete, say $c_1$ and $c_2$ are graphs of smooth functions $f_i:[a_i,b_i]\to \mathbb R$, $i=1,2$. If the curves were lines, then ...
- 131