On the positive radial solutions of some semilinear elliptic equations on ℝ n

We study the existence and behavior of positive radial solutions of the equationu + f(u) = 0 in ⁿ . This equation arises in various problems in applied mathematics, e.g. in the study of phase transitions, nuclear cores and more recently in population genetics and solitary waves. The important model casef(u) = – u + u ^p, p>1, describes for instance the pressure distribution in a van der Waals fluid. In this case, we obtain fairly complete knowledge of all positive radial solutions.Supported in part by an NSF grant and a research grant from the Graduate School of the University of Minnesota.     

On the existence of pair positive–negative solutions for resonance problems

This paper deals with a class of semilinear elliptic Dirichlet boundary value problems at resonance. We introduce a sufficient Landesman–Lazer condition for the existence of pair positive–negative solutions. Furthermore, developing the fibering method in the framework of the Leray–Schauder degree theory we can prove the existence of branches for positive and negative solutions.     

On the Generalized Ideal Asymptotically Statistical Equivalent of Order α for Functions

Ukrainian Mathematical Journal - We introduce new definitions related to the notions of asymptotically Iλ -statistical equivalent of order α to multiple L and strongly Iλ...     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

On the law of the iterated logarithm for the discrepancy of 〈n k x〉

By a well known result of Philipp (1975), the discrepancy D _N (ω) of the sequence (n _k ω) _k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n _{k + 1}/n _k ≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n _k ), for subexpenentially growing (n _k ), but in general the behavior of (n _k ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (n _k ), in this paper we prove a law of the iterated logarithm for the discrepancy D _N (ω) for subexponentially growing (n _k ) without number theoretic assumptions. C. Aistleitner, Research supported by FWF grant S9603-N13. I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961. Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria     

Correction to: On positive integers n with σl(2n + 1) < σl(2n)

Periodica Mathematica Hungarica -     

On the positive nonoscillatory solutions of the difference equation x_(n+1)= α +(x_(n-K)/x_(n-m))~p

The aim of this paper is to show that the following difference equation:xn+1=α+(xn-k/xn-m)p,n=0,1,2…,where α＞-1,p＞0,k,m ∈ N are fixed, 0≤m＜k,x-k,x-k+1,…,x-m,…,x-1,x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α+1.It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.     

On the Existence of Positive Solution for n OrderSingular Boundary Value Problems

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｅｘｉｓｔｅｎｃｅｆｏｒｔｈｅｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｈａｖｅｂｅｅｎｗｉｄｅｌｙｓｔｕｄｉｅｄｒｅｃｅｎｔｌｙ．Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｗｉｌｌｄｉｓｃｕｓｓｔｈｅＢＶＰｏｆｔｈｅｆｏｌｌｏｗｉｎｇｃｏｎｄｉｔｉｏｎｓ：ｕ（ｎ）＋ａ（ｔ）ｆ（ｕ）＝０，０＜ｔ＜１，ｕ（ｋ）（０）＝０，０ｋｎ－２（１．１）ｕ（１）＝０，　ｗｈｅｒｅ（ｈ１）ａ：（０，１）→（０，∞）ｉｓａｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎ．（ｈ２）ｆ：［０，∞）→［…     

On the primality of 2h·3n+1

We consider the primality test of Williams and Zarnke for rational integers of the form 2h·3ⁿ+1. We give an algebraic proof of the test, and we resolve a sign ambiguity. We also show that the conditions of the original test can be relaxed, especially if h is divisible by a power of 2.     

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

On the future infimum of positive self-similar Markov processes†

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont, L. and Pardo, J.C. (Prépublication (L'université de Paris 6), 2005). These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan, D., Lewis, T.M. and Li, W.V. (On the future infima of some transient processes, Probability Theory and Related Fields, 99, 337–360, 1994).     

On the directional Lelong–Demailly numbers of positive currents

In this paper, we study the existence of the directional Lelong–Demailly numbers of positive plurisubharmonic or plurisuperharmonic currents. We prove the independence of these numbers of the system of coordinates. Moreover, these numbers will be given by locally integrable functions.     

On the optimality of periodic barrier strategies for a spectrally positive Lévy process

We study the optimal dividend problem in the dual model where dividend payments can only be made at the jump times of an independent Poisson process. In this context, Avanzi et al. (2014) solved the case with i.i.d. hyperexponential jumps; they showed the optimality of a (periodic) barrier strategy where dividends are paid at dividend-decision times if and only if the surplus is above some level. In this paper, we generalize the results for a general spectrally positive Lévy process with additional terminal payoff/penalty at ruin, and also solve the case with classical bail-outs so that the surplus is restricted to be nonnegative. The optimal strategies as well as the value functions are concisely written in terms of the scale function. Numerical results are also given.     

On the smooth transfer conjecture of Jacquet–Rallis for n=3

We obtain a relative Shalika germ expansion of orbital integrals appeared in the relative trace formulae Jacquet?CRallis when n=3. This is the first example where there are infinitely many nilpotent orbits. As an application we can prove the smooth transfer conjecture of Jacquet?CRallis for n=3.     

On some properties of the positive m-Hessian operators in C 2(Ω)

We consider the p-Hessian operator T _p [u] := tr _p u _xx on the set ${\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}$ and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in ${C^2(\bar{\Omega})}$ for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals I _p [u; f] in ${C^2(\bar{\Omega})}$ and we derive some applications.