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1
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0answers
83 views

Bound a sum of a serie defined by a recursive integer function

I'm using a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, that is defined as \begin{equation} f(n)=\lceil \log(f(n-1)) \rceil +f(n-1) \end{equation} where $f(1)=F\in \mathbb{N}$, and ...
0
votes
0answers
70 views

Looking for tighter bounds

I have to solve an equation which is $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where $$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$ The $z_i$ are all positive ...
15
votes
1answer
848 views

Quantitative lower bounds related to Zhang's theorem on bounded gaps

Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ ...
1
vote
0answers
108 views

Can this be bounded [closed]

Let us denote by $a_{k}$ the supremum of the minimum of the angle formed by the line segments between any $k$ points in the unit triangle. Is $\sqrt{k}a_{k}$ bounded as $k\rightarrow\infty$?
3
votes
2answers
446 views

Lower bound for Euler's totient for almost all integers

Let $\varphi(n)$ be the Euler's totient function. It is well know that $\liminf_{n \to \infty} \frac{\varphi(n)}{n / \log \log n} = e^{-\gamma}$, so that for $\varepsilon > 0$ it results ...
3
votes
2answers
123 views

Techniques for showing optimality of given packing

There are some natural packing problems that have been asked in mathematics. Some of them are: 1)How many balls can be placed with in a cube? 2)How many equidistant points can be place on the ...
2
votes
2answers
311 views

Lower bounds on derivative around zero set of a positive smooth function

As part of a different problem, I came across the following simplified question, for which I cannot exhibit a proof nor a counterexample. Note that the assumptions of smoothness and strict positivity ...
0
votes
1answer
152 views

Good lower bound on matching in bipartite graph

Suppose a bipartite graph $G=(V_1 \cup V_2, E)$ is given, and one is interested in matching vertices $V_1$ to vertices $V_2$. Assume Hall's condition does not hold, so a perfect matching does not ...
4
votes
1answer
278 views

Simple lower bounds for Bell numbers (number of set partitions)?

The $n$-th Bell number $B_n$ represents the number of distinct partitions of a set with $n$ distinguished elements. It can be expressed as the infinite sum $B_n = (1/e)\sum_{k=1}^{\infty} (k^n/k!)$, ...
2
votes
0answers
79 views

Lower asymptotic bounds for the derivative of Laguerre polynomials

Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there exists a strictly ...
6
votes
0answers
277 views

Bounding the probability that a random variable is maximal

Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for ...
1
vote
0answers
100 views

Bounding the Schur's complement of similiar matrices

Assume the following: • $L\leq K$ . • $\Gamma\in M_{K,L}$ is a $L$ rank ${ 0,1} $ matrix, without identical rows or the zeros row. • $N\in M_{K,K}$ is a diagonal matrix, whose diagonal is a ...
6
votes
2answers
328 views

bound for zeros of a polynomial with bounded integer coefficients

Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)? ...
2
votes
2answers
324 views

lower bound for $\Re\zeta(1+it)$

Hi is there any lower bound for $\Re\zeta(1+it)$. I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$. If it is true, is there any reference to prove it. thanks
1
vote
0answers
87 views

Bounding a q-expansion on a bounded open subset of the complex upper-half plane

Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $q:\tau\mapsto \exp(\pi i \tau)$ be the nome on $\mathbf{H}$. Suppose that there are integers $a_j$ ...
1
vote
0answers
201 views

Geometric/Analytic techniques for constructive and asymptotic bounds in the Lee metric

Slight extension of cross posting from http://cstheory.stackexchange.com/questions/7408/lee-metric-gilbert-varshamov-and-hamming-bounds-for-larger-relative-distance-rang (closed there) The following ...
3
votes
2answers
899 views

tight bounds on probability of sum of laplace random variables.

Are there tight upper and lower bounds on the density of the sum of $n$ i.i.d laplace random variables that depend on $n$ and the individual laplacian densities?
0
votes
0answers
346 views

Lower bound over a concave function

Hi everyone, I will be to grateful if help me find a tight lower bound $g(x)$ over the following concave function: $$f(x) = \sqrt{1+4x} -1 + \log(\sqrt{1+4x}-1) - \log(2x) \geq g(x),$$ where $x \geq ...
3
votes
2answers
572 views

Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature

A friend in physics asked this question, and I didn't know the answer. Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...
4
votes
1answer
541 views

Best lower bound for off-diagonal Ramsey numbers

What are the current best lower bounds for off-diagonal Ramsey numbers $R(k,l)$ with $l$ of order unity and asking for asymptotic behavior for large $k$, such as $R(k,4)$, $R(k,5)$, and so on? ...
1
vote
1answer
280 views

Lower bound on the convergence rate of a specific Markov chain

I have a Markov chain $\mathbf{A} = (A_0, A_1, \ldots)$ with state space $\{0, \ldots, n\}$ which converges towards a stationary distribution $\pi$. There are a lot of well-known results on ...
14
votes
6answers
3k views

Lower bounds for chromatic number of a graph

I am trying to find a good lower bound for chromatic number of one family of graphs. I'm curious what are the known lower bounds for chromatic number. There are two obvious: $\chi(G) \geq \omega(G)$ ...
2
votes
2answers
1k views

Proofs of Lower Bounds for Ramsey Numbers?

As a sort of dual question to this question, I am wondering what proofs people know of lower bounds on Ramsey numbers $R(k, k)$. I know of two proofs: there is Erdos's beautiful probabilistic ...
5
votes
1answer
237 views

Feasibility of linear programs

It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?
8
votes
0answers
538 views

Lower bounds for linear forms of logarithms (a la Baker)?

Let $\lambda_1$, $\lambda_2$, and $a$ be three fixed complex algebraic numbers. For a given integer $n$, write $\Theta(n) = \arg(a \lambda_1^n + \lambda_2^n)$. Assuming $\Theta(n)$ is not zero, I am ...
13
votes
4answers
1k views

Super-linear time complexity lower bounds for any natural problem in NP?

Do we know any problem in NP which has a super-linear time complexity lower bound? Ideally, we would like to show that 3SAT has super-polynomial lower bounds, but I guess we're far away from that. I'd ...