Questions tagged [upper-bounds]
The upper-bounds tag has no summary.
50 questions
2
votes
0
answers
114
views
Is this Knapsack looking recurrence always periodic?
Let $(a_n)_{n\geq 1}$ be a sequence of integers such that for some constant $N$ and all $n>N$ we have $$ a_n = -\max_{i+j=n}(a_i + a_j). $$
Is it true that any such sequence is eventually periodic?
...
- 163
2
votes
1
answer
125
views
Interpolation between $L^\infty$ and Lipschitz that pinches Dini on $\mathbb{T}^2$
Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ be the flat $2$D torus with geodesic distance $d(\cdot,\cdot)$.
For a bounded function $\rho:\mathbb{T}^2\to\mathbb{R}$, define the modulus of continuity
$$...
- 277
3
votes
1
answer
211
views
Bounds on Banach density of zeros of functions of exponential type
It is well known that the upper asymptotic (or natural) density of the set of zeros of a function $f$ of exponential type $\tau$ is constrained. Let $R = \{|z_n|\}$ be the set of absolute values of ...
- 167
1
vote
0
answers
147
views
Entire function bounded on a dense enough sequence
A theorem of Duffin and Schaeffer says (quoting from R. P. Boas, Entire Functions, Theorem 10.5.1):
Let $f(z)$ be regular in $|\arg z| \leq \alpha \leq \pi / 2$ and let
$h(\theta) \leq a |\cos \theta|...
- 167
3
votes
1
answer
187
views
Equivalence between sum and integral of regular functions over positive real axis
Let $f$ be a function regular in $\Re z \geq 0$ whose indicator function satisfies:
$$h(\theta) = \limsup_{r \to \infty} \frac{\log |f(r e^{i \theta})|}{r} \leq c < \pi$$
in this half plane. Let $(\...
- 167
1
vote
1
answer
190
views
Bound on norm of $H^2(D)$ functions outside the unit disk
Let $H^2(D)$ be the Hardy space of analytic functions on the unit disk $D$ with finite norm:
$$
\|f\|_{H^2(D)} = \sup_{r < 1} \left( \int_0^{2 \pi} |f(r e^{i \theta})|^2 \, d\theta \right)^{1/2}.
$$...
- 167
1
vote
1
answer
199
views
Fejér-Riesz inequality for $H^p$ on the unit disk for more general curves
Let us consider the Hardy space $H_p$ on the unit disk, and a function $f \in H_p$. There's an inequality by Fejér and Riesz stating that:
\begin{equation}
\int_{-1}^{1} |f(x)|^p \, dx \leq \frac{1}{2}...
- 167
0
votes
0
answers
109
views
Uniform entropy bounds for unions of VC subgraph classes
I'm working with VC subgraph classes of functions, say $\mathcal{F}$ and $\mathcal{G}$, which are both uniformly bounded and admit envelopes $F$ and $G$, respectively. I came across a useful lemma (...
- 55
3
votes
1
answer
141
views
Moment generating function of an inverse of a random variable
Suppose $X$ is a binomial random variable. Is there literature on how to bound the moment generating function or have an explicit expression on $m(t)$.i.e. $E[\exp(\frac{t}{1+X})]$?
Many thanks.
- 39
3
votes
1
answer
376
views
An extended version of McDiarmid's inequality
I am working in statistics and during my research, I encountered the need to establish concentration inequalities. McDiarmid's inequality seems very relevant and closely related to my problem, but ...
- 185
0
votes
1
answer
230
views
Bounds on polynomials with bounded coefficients and bounded norm in an interval
Let $p$ be a polynomial of degree $N$ with complex coefficients $a_n$ satisfying $|a_n| < 1/n$ for $n \geq 1$ and whose $L^2$ (or $L^{\infty}$) norm in a given interval $[x_1, x_2]$ with $x_1 > ...
- 167
6
votes
1
answer
418
views
Bessel function $J_\nu(x)$ asymptotics for $\nu\approx x$
If one considers the Bessel function $J_\nu(x)$ with a fixed $x$ and changing $\nu$, one gets a graph which is oscillating rapidly until $\nu\approx x$ and then a quick exponential decay. However, I ...
- 3,808
0
votes
1
answer
100
views
Proving bounds on likelihood ratio involving mixture without MLR assumption
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 23
1
vote
1
answer
246
views
Riemann sum error bounds without using derivatives or the Mean Value Theorem?
I know that if a function $f$ is continuously differentiable on $[a,b]$, one can bound the difference between its integral and a Riemann sum using either the Mean Value Theorem or Taylor expansions. ...
- 37
2
votes
0
answers
93
views
I want to know if this probability can be expressed using a formula
I have an array of length m, and there are n people who will choose positions in this array. Each person can randomly select k positions. I want to find the probability that after the selection, there ...
- 31
4
votes
2
answers
200
views
The number of distinct sparse graphs
I am interested in estimating the number of non-isomorphic simple graphs on $n$ vertices with $O(n)$ edges. Specifically, I am wondering whether it is correct that the number of such graphs is at ...
- 43
1
vote
0
answers
220
views
Expected differences between order statistics
Let $X_1,\dots,X_n$ and $Y_1,\dots,Y_n$ be independent samples drawn uniformly from the unit interval $[0,1]$, and sort them so that $0\leq X_1\leq\cdots\leq X_n\leq1$ and similarly for the $Y_i$'s. ...
- 4,131
7
votes
0
answers
146
views
Explicit global bounds for the Bessel functions $J_n(z)$
For $n \in \mathbb{Z}$ and $d \in \mathbb{Z}_{\ge 0}$, let $J^{(d)}_n(z)$ denote the $d$-th derivative of the ordinary Bessel function. The asymptotic expansion
$$J_n(z) = \sqrt{\frac{2}{\pi z}} \left(...
- 2,602
0
votes
0
answers
164
views
Minimum-weight bipartite matching on the unit interval
This is related to another question I just asked 1, after running some experiments. Let $X$ and $Y$ be two independent uniform sets of points in the unit interval, with $|X| = 2|Y|$. By running ...
- 4,131
5
votes
0
answers
240
views
Minimum weight maximum matching in the unit interval
Let $X$ and $Y$ denote two sets of $m$ and $n$ points distributed uniformly at random in the unit interval. When $m$ and $n$ are both large, is there a bound for the expected cost of a minimum-weight ...
- 4,131
2
votes
1
answer
145
views
Proving bound on expectation of likelihood ratio involving mixtures
Let $p$ be a Lebesgue density function with infinite support (i.e. $p(x)>0 \forall x\in \mathbb{R}$ and $\int p(x) dx = 1$). Moreover, assume that $p$ is even (i.e. $p(x) = p(-x)$) and unimodal: $p(...
- 23
3
votes
1
answer
151
views
Which compact Lie groups have an upper bound on the dimension of irreducible continuous representations?
To fix notation, if $G$ is a compact Lie group, $Rep(G)$ denotes the set of continuous irreducible unitary representations of G, and $\widehat{G}$ denotes the quotient $Rep(G)/\sim$, which identifies ...
0
votes
2
answers
170
views
Upper bounds on quotients of binomial coefficients
Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$.
Define $f\colon\mathbb{N}\to[0,1]$
$$
f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}},
$$
where
$$
m = \Big\lfloor{\frac{n}{\lceil\gamma ...
2
votes
0
answers
131
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 103
0
votes
1
answer
137
views
Numerically bounding a Exponential-Trigonometric Integral [closed]
I am having some trouble with this (undergrad) problem. The Twitter account I found this from was deleted so I unfortunately have not found an answer.
I have tried decomposing into Riemann sum and ...
- 13
0
votes
0
answers
117
views
Bounding the coefficients of a polynomial written as the sum of powers of linear forms
Crosspost: Just a heads up, I've posted this question on math.stackexchange as well. I have made new attempts at solving it though, and figured I'd ask here since it's more of a research-level ...
- 141
28
votes
3
answers
2k
views
How large can $\mathbf{P}[X_1 + X_2 + X_3 < 2 X_4]$ get?
Let $\mu$ be a probability measure on $[0,\infty)$ and $X_1, \dots, X_4 \sim \mu$ independent. Then what can be said about the probability that $X_1 + X_2 + X_3 < 2 X_4$?
More precisely, what is ...
- 6,886
3
votes
1
answer
162
views
Upper bound of $-r’’_\nu(x)/r’_\nu(x)$ where $r_\nu(x)=I_{\nu+1}(x)/I_\nu(x)$
Please see the bottom of the post for an updated version of the question.
Original Question. I’m looking for an analytical proof of an upper bound for $-r’’_\nu(x)/r’_\nu(x)$ for $\nu \ge 0$ and $x \...
- 31
1
vote
0
answers
111
views
Counting the number of local minima of a function that is the sum of square roots of cosines
Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows
$$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$
where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-...
- 21
2
votes
1
answer
525
views
A maximal inequality
Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. symmetric random variables, with $-1\leq X_i\leq 1$, $\mathbb{E}(X_i) =0$, $\mathbb{E}(X_i^2) = 1$. We have that:
$$
P\left(\bigcap_{k = 1}^{n}\frac{|\sum_{i = ...
- 63
2
votes
0
answers
163
views
Bound from above and from below the probability that a 1-D centered random walk remains at each step inside a square root boundary
Let $W_n = \sum_{i = 1}^{n}X_i$ be a random walk on $\mathbb{R}$, where the increments $X_i$ are i.i.d., symmetric around the origin ($X\sim -X$), such that $-1\leq |X(\omega)| \leq 1$ $\forall\omega\...
- 63
2
votes
0
answers
174
views
Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $
Background
The Norwegian mathematician and astronomer Carl Størmer did important work on the equation
$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$
...
- 5,065
0
votes
1
answer
254
views
Numerical integration with integrable singularity
Suppose I have a numerical estimation of discrete samples of a smooth function $C(t)$ at $t = a, \dots, T = Na$ and I want to (numerically) compute the integral of $f(t) = \frac{C(t)}{\sqrt{t}}$. In ...
- 33
2
votes
1
answer
134
views
Difference of probabilities of two random vectors lying in the same set
Suppose I have to random vectors:
$$\mathbf{z} = (z_1, \dots, z_n)^T, \quad \mathbf{v} = (v_1, \dots, v_n)^T$$
and set $A \subset \mathbb{R}^n$.
I want to find an upper bound $B$ for the following ...
- 33
5
votes
3
answers
453
views
A closed form (or tight upper bound) for $\sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j}$
I'm seeking a closed-form expression to the sum
$$ \sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j} $$
where for positive integers $m$ and $k$, we know $m \gg k$. Loosely, $k \sim \log(m)$.
When $k=...
- 173
2
votes
0
answers
66
views
Error bounds for a Romberg-style improvement of a non-linear approximation
I have a (possibly non-linear) functional $F$, which I want to numerically approximate by a (typically non-linear) $\widehat{F}_h$. For a suitable class of functions, I have asymptotic error behaviour
...
- 3,788
1
vote
1
answer
545
views
Conjectured upper bound on the maximum value of the absolute value of the Möbius function in the poset of multiplicative partitions under refinement
PRELIMINARIES:
Consider the poset $(\mathcal{P}_n, \leq_r)$ of the (unordered) multiplicative partitions of $n$ partially ordered under refinement (for all $\lambda, \lambda’ \in \mathcal{P}_n$, we ...
- 415
5
votes
1
answer
812
views
Is there an upper bound on the number of representations as a sum of squares?
I am interested in finding upper bounds for the Sum of squares function defined as $r_k(n) = |\{(a_1, a_2, \ldots, a_k) \in \mathbb{Z}^k \ : \ n = a_1^2 + a_2^2 + \cdots + a_k^2\}|$ whenever the ...
- 313
1
vote
0
answers
139
views
What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?
Motivation
In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
- 5,065
10
votes
3
answers
648
views
Bounding the $n$-th derivatives of $\frac{1-\cos(x)}{x^2}$
Define the smooth map $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) := \frac{1-\cos(x)}{x^2} = -\sum\limits_{k=1}^\infty \frac{(-1)^k}{(2k)!} x^{2k-2}$.
I am looking for a nice bound on $|f^{(n)}(x)...
- 1,389
2
votes
1
answer
346
views
Simplified upper bounds for moment-generating function of symmetrised random variable
Let $X$ be a nonnegative random variable such that $\mathbf{E} \left[ \exp X \right] < \infty$. For $\theta \leqslant 1$, an appropriate application of Jensen's inequality, yields that
\begin{align}...
- 949
4
votes
1
answer
401
views
Is this approximation for $\pi$ enough to make this value converge? And how to find an upper bound for it
Update:
\begin{align*}
|I_n-J_n| = (\pi-S_n)\sum_{k=0}^n |\frac{a_kp_k(\ln\pi)}{\ln^{k+1}\pi}|
\end{align*}
and
\begin{align*}
|I_n| = \sum_{k=0}^n | \frac{a_k\pi p_k(\ln\pi)}{\ln^{k+1}\pi}
-\sum_{k=...
- 531
1
vote
1
answer
134
views
Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers
This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
- 583
6
votes
2
answers
373
views
Expectation of the inner product of a subset of two random orthonormal vectors
Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, ...
- 673
0
votes
1
answer
137
views
Convergence in expectation of a discontinuous function
Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
- 11
0
votes
0
answers
328
views
Is it possible to bound Mertens function $M(n)$ from an inclusion-exclusion formulation?
In this post I proposed a formulation of Mertens function $M(n)$ using the inclusion-exclusion principle, as follows:
$$M(n)=-\pi\left(n\right)+\left(\sum_{p_{i}<\sqrt{n}}\pi\left(\lfloor\frac{n}{...
- 179
2
votes
1
answer
252
views
Bound the probability that a point belongs to a set
Let $(a_k)_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that
$$
(1) \quad \Pr\Big (\lim_{n\rightarrow +\infty}d(\frac{1}{n}\sum_{k=1}^n 1_{[a_k = b]}, [v_\ell(b,\...
- 108
0
votes
1
answer
292
views
Bound the expectation of an average
Let $(a_n)_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu_l(b) \le P[a_n = b\mid a_1,\ldots,a_{n-1}] \le \nu_u(b)$ almost surely for every $n \ge 1$ and $b \in ...
- 108
2
votes
0
answers
231
views
Tools to prove lower bounds in analytic number theory
Perron's formula and related methods are used to relate statements such as the Riemann hypothesis to upper bounds of functions occurring in analytic number theory. For example, Perron's formula is ...
- 1,068
8
votes
1
answer
8k
views
Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive
Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
- 83