On the Projections of the Mutual Multifractal Rényi Dimensions

In this paper, we compare the mutual multifractal Rényi dimensions to the mutual multifractal Hausdorff and pre-packing dimensions. We also provide a relationship between the mutual multifractal Rényi dimensions of orthogonal projections of a couple of measures $(\mu,\nu)$ in $\mathbb{R}^n$. As an application, we study the mutual multifractal analysis of the projections of measures.     

On a nonlocal problem of A. A. Dezin for the Lavrent’ev-Bitsadze equation

In a special rectangular domain, for a second-order linear equation of mixed type with discontinuous coefficients and with the Lavrent’ev-Bitsadze operator in the leading part, we prove an extremum principle and existence and uniqueness theorems for the solution of a nonlocal problem stated by A.A. Dezin in his report at the Joint Soviet-American Symposium on Partial Differential Equations (Novosibirsk, 1963).     

Robust mean variance optimization problem under Rényi divergence information

In this paper, we consider the robust mean variance optimization problem where the probability distribution of assets’ returns is multivariate normal and the uncertain mean and covariance are controlled by a constraint involving Rényi divergence. We present the closed-form solutions for the robust mean variance optimization problem and find that the choice of order parameter which is related to the Rényi divergence measure will not impact optimal portfolio strategy under the cases that the mean vector and the covariance matrix are uncertain, respectively. Moreover, we obtain the closed-form solution for the robust mean variance optimization problem under the case that the mean vector and the covariance matrix are both uncertain. We illustrate the efficiency of our results with an example.     

An Extension of the Rényi Formula

In this paper, as a natural extension of the Rényi formula which counts labeled connected unicyclic graphs, we present a formula for the number of labeled(k + 1)-uniform(p, q)-unicycles as follows:U(k+1)p, q={p!/2[(k-1)!]~q·∑qt=2(q~(q-t-1)· sgn(tk- 2))/(q- t)!, p = qk,0, p≠qk,where k, p, q are positive integers.     

On a Problem of A.C. Woods

ＯｎａＰｒｏｂｌｅｍｏｆＡ．Ｃ．ＷｏｏｄｓＺｈａｎｇＷｅｎｐｅｎｇ（张文鹏）（ＮｏｒｔｈｗｅｓｔＵｎｉυｅｒｓｉｔｙ，Ｘｉ＇ａｎ，Ｓｈａａｎｘｉ，７１００６９）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＰａｎＣｈｅｎｇｂｉａｏＲｅｃｅｉｖｅｄＪａｎ．１１，１９９４...     

On a Generalization of Rédei’s Theorem

In 1970 Rédei and Megyesi proved that a set of p points in AG(2,p), p prime, is a line, or it determines at least directions. In 81 Lovász and Schrijver characterized the case of equality. Here we prove that the number of determined directions cannot be between and . The upper bound obtained is one less than the smallest known example.     

On general versions of Erdős-Rényi laws

Summary Rather general versions of the Erds-Rényi [6] new law of large numbers have recently been given by S. Csörg [5] for sequences of rv's which have stationary and independent increments and satisfy a first order large deviation theorem. It is shown that Csörg's results can be extended to cover also situations of stochastic processes where stationarity and independence of increments are not generally available, but for randomly chosen subsequences of the process. Examples demonstrate that the main result can be applied, for instance, to waiting-times in G/G/1 queuing models or cumulative processes in renewal theory, where Erds-Rényi type laws cannot be derived from Csörg's theorems.     

On a problem of Erdős

Let s≥2 be an integer. Denote by f ₁(s) the least integer so that every integer l>f ₁(s) is the sum of s distinct primes. Erd?s proved that f ₁(s)<p ₁+p ₂+?+p _s +Cslogs, where p _i is the ith prime and C is an absolute constant. In this paper, we prove that f ₁(s)=p ₁+p ₂+?+p _s +(1+o(1))slogs=p ₂+p ₃+?+p _s+1+o(slogs). This answers a question posed by P. Erd?s.     

Limiting behavior of relative Rényi entropy in a non-regular location shift family

We calculate the limiting behavior of relative Rényi entropy between adjacent two probability distribution in a non-regular location-shift family which is generated by a probability distribution whose support is an interval or a half-line. This limit can be regarded as a generalization of Fisher information, and seems closely related to information geometry and large deviation theory.     

Rényi’s residual entropy: A quantile approach

In the present paper, we introduce a quantile based Rényi’s entropy function and its residual version. We study certain properties and applications of the measure. Unlike the residual Rényi’s entropy function, the quantile version uniquely determines the distribution.     

On a problem of Nicole Brillou?t-Belluot

We solve the problem posed by Nicole Brillou?t-Belluot (During the Forty-ninth International Symposium on Functional Equations, 2011) of determining all continuous bijections f : I → I satisfying $$f(x)f^{-1}(x) = x^2 \quad{\rm for\, every}\, x \in I,$$ where I is an arbitrary subinterval of the real line.     

On exceptional sets in Erdős–Rényi limit theorem revisited

For \(x\in [0,1],\) the run-length function \(r_n(x)\) is defined as the length of the longest run of 1’s amongst the first n dyadic digits in the dyadic expansion of x. Let H denote the set of monotonically increasing functions \(\varphi :\mathbb {N}\rightarrow (0,+\infty )\) with \(\lim _{n\rightarrow \infty }\varphi (n)=+\infty \). For any \(\varphi \in H\), we prove that the set

$$\begin{aligned} E_{\max }^\varphi =\left\{ x\in [0,1]:\liminf \limits _{n\rightarrow \infty }\frac{r_n(x)}{\varphi (n)}=0, \limsup \limits _{n\rightarrow \infty }\frac{r_n(x)}{\varphi (n)}=+\infty \right\} \end{aligned}$$

either has Hausdorff dimension one and is residual in [0, 1] or is empty. The result solves a conjecture posed in Li and Wu (J Math Anal Appl 436:355–365, 2016) affirmatively.

     

On a problem of Sierpiński,Ⅱ

For any integer s≥ 2, let μsbe the least integer so that every integer l μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpi′nski asked for the determination of μs. Let pibe the i-th prime and let μs= p2 + p3 + + ps+1+ cs. Recently, the authors solved this problem. In particular,we have(1) cs=-2 if and only if s = 2;(2) the set of integers s with cs= 1100 has asymptotic density one;(3) cs∈ A for all s ≥ 3, where A is an explicit set with A ■[2, 1100] and |A| = 125. In this paper, we prove that,(1) for every a ∈ A, there exists an index s with cs= a;(2) under Dickson's conjecture, for every a ∈ A,there are infinitely many s with cs= a. We also point out that recent progress on small gaps between primes can be applied to this problem.     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.