On a question of Erdös and Graham

We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers x_i with 1 \leqq x_i \leqq p^e, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?_i=1^k [1/(x_i)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.     

How quadratic are the natural numbers?

Abstract. For natural numbers n we inspect all factorizations n = ab of n with a ￡ ba \le b in \Bbb N\Bbb N and denote by n=a_n b_nn=a_n b_n the most quadratic one, i.e. such that b_n - a_nb_n - a_n is minimal. Then the quotient k(n) : = a_n/b_n\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n ￡ k(n) ￡ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)^-d-e ￡ k(n) ￡ (logn)^-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log₂  2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k^*(n): = ?_{1 ￡ a ￡ b, ab=n} a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k^*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k^*(n)=W(2^{\frac 12(1-e) log n/log ₂n})\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0.     

A Remark on the Growth of the Denominators of Convergents

For log\frac1+?52 ￡ l_* ￡ l^* < ￥{\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty , let E(λ_*, λ^*) be the set {x ? [0,1): liminf_{n ? ￥}\fraclogq_n(x)n=l_*, limsup_{n ? ￥}\fraclogq_n(x)n=l^*}. \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}. It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that dimE(l_*, l^*) 3 \fracl_* -log\frac1+?522l^*\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}}     

On Clarkson's inequality,type and cotype for the Edmunds-Triebel logarithmic spaces

We show that the (p, p') Clarkson's inequality holds in the Edmunds-Triebel logarithmic spaces A_q(logA)_b,q A_{\theta}({\log}A)_{b,q} and in the Zygmund spaces L_p(logL)_b(W) L_p({\log}L)_b(\Omega) , for b ? \mathbbR b \in \mathbb{R} and for suitable 1 ￡ p ￡ 2 1 \leq p \leq 2 . As a consequence of these results we also obtain some new information about the types and the cotypes of these spaces.     

On the iterates of Euler's function

Asymptotic representations are given for the three sums ?_{n ￡ x} j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?_{n ￡ x} j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?_{n ￡ x} log j(n)/j(j(n)) ;  j\textstyle\sum\limits \limits _{n\le x}\ \log \, \varphi (n)/\varphi \bigl (\varphi (n)\bigr )\ ; \ \varphi is Euler's function.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Quermassintegrals of a random polytope in a convex body

Let K be a convex body in \mathbbRⁿ \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x₁,?, x_N x_1,\ldots, x_N independently and uniformly from K, and write C(x₁,?, x_N) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR⁺ ? \mathbbR⁺ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 ￡ i ￡ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °W_i) = ò_K ?ò_K f[W_i(C(x₁, ?, x_N))]dx_N ?dx₁ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 ￡ i ￡ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x₁,?, x_N) C(x_1,\ldots, x_N) .     

The Maximal Variation of Martingales of Probabilities and Repeated Games with Incomplete Information

The variation of a martingale $p_{0}^{k}=p_{0},\ldots,p_{k}$ of probabilities on a finite (or countable) set X is denoted $V(p_{0}^{k})$ and defined by $$ V\bigl(p_0^k\bigr)=E\Biggl(\sum_{t=1}^k\|p_t-p_{t-1}\|_1\Biggr). $$ It is shown that $V(p_{0}^{k})\leq\sqrt{2kH(p_{0})}$ , where H(p) is the entropy function H(p)=?∑ _x p(x)logp(x), and log stands for the natural logarithm. Therefore, if d is the number of elements of X, then $V(p_{0}^{k})\leq\sqrt{2k\log d}$ . It is shown that the order of magnitude of the bound $\sqrt{2k\log d}$ is tight for d≤2 ^k : there is C>0 such that for all k and d≤2 ^k , there is a martingale $p_{0}^{k}=p_{0},\ldots,p_{k}$ of probabilities on a set X with d elements, and with variation $V(p_{0}^{k})\geq C\sqrt{2k\log d}$ . An application of the first result to game theory is that the difference between v _k and lim _j v _j , where v _k is the value of the k-stage repeated game with incomplete information on one side with d states, is bounded by $\|G\|\sqrt{2k^{-1}\log d}$ (where ∥G∥ is the maximal absolute value of a stage payoff). Furthermore, it is shown that the order of magnitude of this game theory bound is tight.     

Embedding Theorem on Spaces of Homogeneous Type

In [5] the embedding theorem for the Besov spaces Bdot^\A,q_pBdot^{\A,q}_p with $\minus \varepsilon < \alpha \varepsilon $\minus \varepsilon < \alpha \varepsilon and 1 ￡ p1 \le p, q ￡ ￥q \le \infty, and Triebel-Lizorkin spaces \Fdot^\A,q_p\Fdot^{\A,q}_p with $\minus \varepsilon < \alpha \varepsilon $\minus \varepsilon < \alpha \varepsilon and 1 ￡ p1 \le p, q ￡ ￥q \le \infty, on spaces of homogeneous type was obtained. In this article the embedding theorem is generalized to the Besov spaces Bdot^\A,q_pBdot^{\A,q}_p with $p_0 < p \le \infty $p_0 < p \le \infty and 0 < q \le \infty for for p_0 < 1, and the Triebel-Lizorkin spaces, and the Triebel-Lizorkin spaces \Fdot^{\A,q}_pwith with p_1 < p < \inftyand and p_1 < q < \infty for for p_1< 1. The proofs are new even for. The proofs are new even for \mathbb{R}^n$.     

Irregular Wavelet Frames and Gabor Frames

Given g { l^\fracn2 g( l_j x - kb ) }_jezjezⁿ ,where  l_j \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L ²(R ⁿ ) are given. For a class of functions g{ e^{zrib( j,x )} g( x - l_k ) }_{jezⁿ ,kez}\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame.     

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

Numerical properties of isotrivial fibrations

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations j: X ? C{\varphi : X \longrightarrow C}, where X is a smooth, projective surface and C is a curve. In particular we prove that, if g(C) ≥ 1 and X is neither ruled nor isomorphic to a quasi-bundle, then K_X² ￡ 8 c(O_X)-2{K_X^2 \leq 8 \chi(\mathcal{O}_X)-2} ; this inequality is sharp and if equality holds then X is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that K _X is ample, we obtain K_X² ￡ 8c(O_X)-5{K_X^2 \leq 8\chi(\mathcal{O}_X)-5} and the inequality is also sharp. This improves previous results of Serrano and Tan.     

Takács’ Asymptotic Theorem and Its Applications: A Survey

The book of Lajos Takács Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let Lf(x)=-\frac1w?_i,j ?_i(a_i,j(·)?_jf)(x)+V(x)f(x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω(x)dx, where ω(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying l^-1w(x)|x|² ￡ ?ⁿ_i,j=1a_i,j(x)x_ix_j ￡ lw(x)|x|².{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for V^aL^-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator [b, V^a L^-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMO_w{b\in BMO_\omega} and 0 < α ≤ 1.     

An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem:

Exact simulation of final, minimal and maximal values of Brownian motion and jump-diffusions with applications to option pricing

We introduce a method for generating (W_x,T^(m,s),m_x,T^(m,s),M_x,T^(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , where W_x,T^(m,s)W_{x,T}^{(\mu,\sigma)} denotes the final value of a Brownian motion starting in x with drift μ and volatility σ at some final time T, m_x,T^(m,s) = inf_{0 ￡ t ￡ T}W_x,t^(m,s)m_{x,T}^{(\mu,\sigma)} = {\rm inf}_{0\leq t \leq T}W_{x,t}^{(\mu,\sigma)} and M_x,T^(m,s) = sup_{0 ￡ t ￡ T} W_x,t^(m,s)M_{x,T}^{(\mu,\sigma)} = {\rm sup}_{0\leq t \leq T} W_{x,t}^{(\mu,\sigma)} . By using the trivariate distribution of (W_x,T^(m,s),m_x,T^(m,s),M_x,T^(m,s))(W_{x,T}^{(\mu,\sigma)},m_{x,T}^{(\mu,\sigma)},M_{x,T}^{(\mu,\sigma)}) , we obtain a fast method which is unaffected by the well-known random walk approximation errors. The method is extended to jump-diffusion models. As sample applications we include Monte Carlo pricing methods for European double barrier knock-out calls with continuous reset conditions under both models. The proposed methods feature simple importance sampling techniques for variance reduction.     

Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

We study the well-posedness of the fractional differential equations with infinite delay (P ₂): D^a u(t)=Au(t)+ò^t_-￥a(t-s)Au(s)ds + f(t), (0 ￡ t ￡ 2p){D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}, where A is a closed operator in a Banach space ${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)} and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P ₂) on Lebesgue-Bochner spaces L^p(\mathbbT, X){L^p(\mathbb{T}, X)} and periodic Besov spaces B _p,q^s(\mathbbT, X){{B} _{p,q}^s(\mathbb{T}, X)} .     

Helly-type theorems for appropriate colorings of visibility sets

Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ￡ r ￡ q - 1 0 \leq r \leq q - 1 . For 1 ￡ i ￡ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 ￡ i ￡ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 ￡ j ￡ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 ￡ j ￡ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well.     

On the Mean Value of the Index of Composition of an Integer

For each integer n l(n)=[(log n)/(log g(n))]\lambda(n)={{\rm log}\, n\over{\rm log}\, \gamma(n)} be the index of composition of n, where g(n)=?_p|np\gamma(n)=\prod_{p\vert n}p . For convenience, we write ?_{x ￡ n ￡ x+?x}l(n)\sum_{x\le n\le x+\sqrt{x}}\lambda(n) and ?_{n ￡ x}l(n)\sum_{n\le x}\lambda(n) , as well as for ?_{x ￡ n ￡ x+?x}1/l(n)\sum_{x\le n\le x+\sqrt{x}}1/\lambda(n) and ?_{n ￡ x}1/l(n)\sum_{n\le x}1/\lambda(n) . Finally we study the sum of running over shifted primes.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation