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0
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0answers
301 views

Divisor function inequality

I have been reading a paper on the Goldbach conjecture found at http://people.exeter.ac.uk/pt224/Goldbach.pdf. At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then $$\sum_{w\leq ...
11
votes
5answers
8k views

Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional. Is this true in general for ...
1
vote
1answer
340 views

Is there a complex analog of this sharpened Cauchy Inequality?

Let $x$ and $y$ be two points on the unit sphere $S^{n-1}$ in Euclidean space ${\mathbb{R}}^n$. Suppose that the angle $\theta$ between the points $x$ and $y$ is acute, so that the dot product $x\cdot ...
7
votes
2answers
501 views

Is this ergodic inequality true?

Is anything similar to the following inequality true, $\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$ where $A_n f = ...
11
votes
5answers
2k views

Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
0
votes
1answer
274 views

Complex version of Farkas' lemma

It is well known result of linear optimization theory that, if a finite set $S$ of linear inequalities in real variables $x_1,x_2, \ldots ,x_n$ implies a linear inequality $i$ in $x_1,x_2, \ldots ...
1
vote
1answer
389 views

Coefficient bounds of an inequality

Hello, Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences ...
9
votes
2answers
1k views

Question on eigenvalue square root subadditivity

ORIGINAL QUESTION Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a $2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller eigenvalue of a $2\times2$ matrix. Is it true ...
1
vote
0answers
99 views

Linear complementarity problem - Classification

For Linear complementarity problems (LCP) like $\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$ $\mathbf{z} \ge \mathbf{0}$ $\mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q}) = 0$ there exists a vast amount ...
7
votes
1answer
383 views

Characterizing intersection of zero sets of elementary symmetric polynomials on R^n

Stated simply, the question is: Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb ...
7
votes
1answer
464 views

Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$: $\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...
5
votes
0answers
441 views

Any similar inequality in literature?

I got the following inequality: $B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...
2
votes
2answers
367 views

About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$. I am considering the generalized question for ...
3
votes
1answer
729 views

Cauchy-like inequality for Kronecker (tensor) product

General question first: upper/lower bound a sum of Kronecker products by its components. More specifically, how is $$ \Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$ bounded by the operator ...
4
votes
0answers
200 views

Applying the amplification trick + probabilistic method on connected graphs

First of all, I apologise if my question is not very specific. I am not really looking for a concrete answer but rather some hints and pointers what could be used/applied in this situation. A concrete ...
2
votes
3answers
664 views

a “reverse Hadamard inequality”

Is there an inequality of the form $|\det(A)| \geq F(v_1, \ldots, v_n)$ for a real $n\times n$-matrix $A$ with columns $v_i$, $F \geq 0$?
12
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1answer
912 views

An $L^0$ Khintchine inequality

Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...
4
votes
1answer
537 views

“Less than” formula for complete theory of the rationals

Does anyone know the name/author of the model theory paper where it's proven that you can define the $ < $ relation on $ (\mathbb{Q}, +, \cdot, 1, 0) $?
4
votes
3answers
589 views

Is the componentwise square-root of a positive-definite matrix also pos.-def.?

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and $B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$. Is $B$ positive-definite whenever $A$ is? In other words: ...
1
vote
1answer
1k views

How to estimate derivatives of multivariate polynomial near a manifold

I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value ...
2
votes
4answers
663 views

Proving triangle inequality for an affine invariant distance on $Sym_n^+$

In the paper Riemannian framework for tensor computing, by Pennec et al., on page 46 the authors state a "distance" function on the manifold of positive definite matrices $Sym_n^+$ given by ...
6
votes
5answers
516 views

Binary operations compatible with the usual order on the reals

An officemate passes along the following natural-seeming question: Say that a binary operation $\oplus$ is compatible with the usual order $\leq$ on $\mathbb{R}$ if for any $w, x, y, z$ in ...
16
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6answers
3k views

Applications of Hardy's inequality

Every so often I would encounter Hardy's inequality: Theorem 1 (Hardy's inequality). If $p>1$, $a_n \geq 0$, and $A_n=a_1+a_2+\cdots+a_n$, then $$\sum_{n=1}^\infty ...
3
votes
1answer
874 views

Inequality on probability distributions

I would like to know if the following inequality is satisfied by all probability distributions (or at least some class of probability distributions) for all integer $n \geq 2$. $\int_0^{\infty} ...
1
vote
1answer
319 views

Statistical inequality

Let $X$ be a finite discrete variable and $X\ge0$. Is it true that $$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$ where ...
3
votes
2answers
702 views

Tight bound for sum of entries of the inverse of a nonnegative matrix

While playing around with certain non-negative matrices, I got stuck at the following question. Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...
5
votes
2answers
397 views

Sturm chain analogue for exponential polynomials?

I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form $f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real). My first question is: is there an algorithm for ...
0
votes
1answer
607 views

Probability space analogue of Cauchy-Schwarz inequality

Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that: $$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j ...
7
votes
2answers
762 views

Generalization of the positive semidefinite Grothendieck inequality

In a recent paper, S. Khot and A. Naor show a natural generalization of the positive semidefinite Grothendieck's inequality. Grothendieck showed that there exists a constant $K > 0$ such that for ...
2
votes
5answers
3k views

What is the angle between two complex vectors?

Let $x, y\in R^n$ and $x, y$ are nonzero, it is well known $\frac{x^Ty}{\parallel x\parallel_2\parallel y\parallel_2}(\parallel x\parallel_2+\parallel y\parallel_2)\le \parallel x+y\parallel_2$. How ...
4
votes
2answers
579 views

Logarithm of AM/GM ratio: $\sqrt{\log((x+y)/(2\sqrt{xy}))}$

Recently, while playing around with infinite-divisibility, i arrived at the following metric: $$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$ defined for positive reals $x$ and $y$. ...
0
votes
3answers
424 views

Why does this inequality hold?

Hi people. Can you help me realize why this is true? I can tell you that $P_i$ and $P_j$ are probabilities, i.e. $0 \leq P_i, P_j \leq 1$. $\displaystyle \sum_{i=1}^\infty \sum_{j=1}^\infty ijP_iP_j ...
-1
votes
1answer
404 views

Little conjecture about sums of reciprocals

Given a finite list $x_i$ of $N$ positive reals, it seems that $\sum_{i=1}^N x_i = \sum_{i=1}^N x_i {}^{-1} \Rightarrow \sum_{i=1}^N x_i \geq N$. Can anyone give me a proof?
11
votes
5answers
2k views

subadditive implies concave

Let $f:R_+\to R_+$ be smooth on $(0,\infty)$, increasing, $f(0)=0$ and $\lim_{x\to\infty}=\infty$. Assume also that $f$ is subadditive: $f(x+y)\le f(x)+f(y)$ for all $x,y\ge 0$. Must $f$ be concave? ...
1
vote
1answer
163 views

Maximal inequality over two indices

In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like: P[$\max_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$] in the background of ...
1
vote
0answers
931 views

Inequality concerning absolute value of a polynomial

Let $$f(z) = (1-1/t) z^w + z/t - 1$$ with integers $t\geq2$ and $w\geq2$.Let $r=1+1/(tw^3)$. How do I show $$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$ for any ...
3
votes
1answer
874 views

Proving inequalities over algebraic structures

I've been looking at proof techniques in formal systems like Coq and Agda recently, and encountered the newring tactic described here for proving equalities over ...
3
votes
1answer
396 views

Analogues of the Golden-Thompson inequality

Are there any analogues of the Golden-Thompson Inequality for moment generating functions? The Golden-Thompson Inequality asserts the following: If $A$ and $B$ are two $n \times n$ Hermitian matrices, ...
5
votes
1answer
263 views

Denominators in the solution to Hilbert's XVII

Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...
5
votes
2answers
964 views

Inequality involving probability measures [closed]

I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck. An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
3
votes
2answers
402 views

Positivity of a finite sum

Let $i$, $k$ be integers such that $2 \leq i \leq k$. I would like to show that the sum $$ \sum_{j=1}^{i-1} \frac{(-1)^{j-1}(i-j)^k}{(i-j)! (j-1)!} $$ is positive. I have carried out extensive ...
11
votes
2answers
744 views

Positivity of sequences via generating series

There are different ways of showing that a given sequence $a_0,a_1,a_2,\dots$ of integers, say, is nonnegative. For example, one can show that $a_n$ count something, or express $a_n$ as a (multiple) ...
1
vote
1answer
635 views

Inequality constraints, probability distributions, and integer partitions

I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus, $a + b + c = 10$ ...
4
votes
1answer
322 views

A plausible positivity

After getting stuck with the previous positivity (it probably sounds too complex), I would like to give a version of the problem which is of most interest to me. Consider a sequence of real numbers ...
6
votes
1answer
302 views

Positivity of “harmonic” summation

The settings for the problem are as follows. Given a real number $\alpha\in[0,1]$, consider a sequence of real (positive, negative and zero) numbers $a_1,a_2,\dots,a_n,\dots$ satisfying (1) $a_1=1$, ...
4
votes
1answer
472 views

Salem Inequality

I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" ...
61
votes
15answers
5k views

Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
4
votes
2answers
583 views

Another plausible inequality.

I come across the following problem in my study. Consider in the real field. Let $ 0\le x\le1 $, $a_1^2+a_2^2=b_1^2+b_2^2=1$.Is it true $ ...
0
votes
2answers
545 views

Linear algebra inequality

I'm wondering (hoping) if an inequality is true. Please can anyone help me? Let $V$ be a complex vector space $dim_{\mathbb{C}}(V)=n$ with a hermitian scalar product $h$. Let $v,a, b \in V$. Is it ...
3
votes
2answers
898 views

Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n. Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?