Two Kinds of Numbers and Their Applications

C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

Numbers of the Equitype and Quasi-Equitype Decompositions of a Permutation

In the paper“The equitype and quasi-equitype decompositions of arbitrarypermutation by permutations of order 2 or 3”,we proved that any even permu-tation can be expressed as a product of two equitype involutions and any odd per-mutation can be expressed as a product of two quasi-equitype involutions. Meanwh-ile,we gave the condition on t such that an even permutation has(2~t)equitypedecompositions and the condition on t such that an odd permutation has(2;2~t)quasi-equitype decompositions.In this paper,we give the numbers of the equitypeand quasi-equitype decompositions of a permutation.The main results are     

Continuity of Some Classes of Functions Related to Degenerate Parabolic Equations and Its Applications

We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.     

Product of Uniform Distribution and Stirling Numbers of the First Kind

Let Vk=u1u2……uk, ui＇s be i.i.d - U（0, 1）, the p.d.f of 1 - Vk＋l be the GF of the unsigned Stirling numbers of the first kind s（n, k）. This paper discusses the applications of uniform distribution to combinatorial analysis and Riemann zeta function; several identities of Stifling series are established, and the Euler＇s result for ∑ Hn/n^k-l, k ≥ 3 is given a new probabilistic proof.     

The Supremum and Infimum of the Set of Fuzzy Numbers and Its Application

In 1986,Goetschel and Voxman represented a fuzzy number by two usualfunctions,which promote further development of the theory and application of fuzzynumbers.For example,we can embbed the fuzzy number space E~1 into a concreteBanach space by using this representation theorem(see[2—4]).     

Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

Let G be a simple graph with 2n vertices and a perfect matching.The forcing number f(G,M) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G.Among all perfect matchings M of G,the minimum and maximum values of f(G,M) are called the minimum and maximum forcing numbers of G,denoted by f(G) and F(G),respectively.Then f(G)≤F(G) ≤n-1.Che and Chen(2011) proposed an open problem:how to characterize the graphs G with f(G)=n-1.Lat...     

The Numbers of Permutation Groups Orbits and of the Wreath Product Orbits

ＴｈｅＮｕｍｂｅｒｓｏｆＰｅｒｍｕｔａｔｉｏｎＧｒｏｕｐｓＯｒｂｉｔｓａｎｄｏｆｔｈｅＷｒｅａｔｈＰｒｏｄｕｃｔＯｒｂｉｔｓ￥ＬｉＸｉａｎｚｈｏｎｇ；ＬｉｕＧｕａｎｇｌｉａｎｇ（ＰｕｙａｎｇＥｄｕｃａｔｉｏｎａｌＣｏｌｌｅｇｅ，Ｈｅｎａｎ，４５７００...     

Some Inequalities Related to Transience and Recurrence of Markov Processes and Their Applications

For an irreducible symmetric Markov process on a (not necessarily compact) state space associated with a symmetric Dirichlet form, we give Poincaré-type inequalities. As an application of the inequalities, we consider a time-inhomogeneous diffusion process obtained by a time-dependent drift transformation from a diffusion process and give general conditions for the transience or recurrence of some sets. As a particular case, the diffusion process appearing in the theory of simulated annealing is considered.     

Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.     

On the Relations Between the Numbers of Deficient Function and the Growth of an Algebroidal Function

Let f(z) be an algebroidal function defined by the following irreducible eq-uation A_ny~n A_(n-1)y~(n-1) … A_1y A_0=0 (1)where A_j(j=0,1,…,n) are entire functions in the finite complex plane.     

Covering Numbers of the Sporadic Simple Groups

Let G be a finite nonabelian simple group. The covering number of G is the sm-allest integer n such that C~n=G for every nontrivial conjugacy class C of G andis denoted by cn(G). The extended covering number of G, denoted by ecn(G), is     

On the Distribution of Weakly Composite Numbers

ＯｎｔｈｅＤｉｓｔｒｉｂｕｔｉｏｎｏｆＷｅａｋｌｙＣｏｍｐｏｓｉｔｅＮｕｍｂｅｒｓ王天泽，李国强ＯｎｔｈｅＤｉｓｔｒｉｂｕｔｉｏｎｏｆＷｅａｋｌｙＣｏｍｐｏｓｉｔｅＮｕｍｂｅｒｓ￥ＷａｎｇＴｉａｎｚｅ；ＬｉＧｕｏｑｉａｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭ...     

Extensions of Fréchet ?-Subdifferential Calculus and Applications

In this paper, we establish some calculus rules for the limiting Fréchet ?-subdifferentials of marginal functions and composite functions. Necessary conditions for approximate solutions of a constrained optimization problem are derived.     

Zero-free Regions for Dirichlet L-functions and the Exceptional Set of Goldbach Numbers

Ｚｅｒｏ－ｆｒｅｅＲｅｇｉｏｎｓｆｏｒＤｉｒｉｃｈｌｅｔＬ－ｆｕｎｃｔｉｏｎｓａｎｄｔｈｅＥｘｃｅｐｔｉｏｎａｌＳｅｔｏｆＧｏｌｄｂａｃｈＮｕｍｂｅｒｓＬｉＨｏｎｇｚｅ（李红泽）ＭＲ（１９９１）ＳｕｂｊｅｃｔＣｌａｓｉｆｉｃａｔｉｏｎ：１１Ｎ１３，１...     

On the Strong Laws of Large Numbers and the Rates of Convergence for B-Valued Random Variable Sequences

ＬｅｔＢｂｅａｒｅａｌｓｅｐａｒａｂｌｅＢａｎａｃｈｓｐａｃｅｗｉｔｈｎｏｒｍ‖·‖ａｎｄ｛Ｘｎ，ｎ≥１｝ａｓｅｑｕｅｎｃｅｏｆｒａｎｄｏｍｖａｒｉａｂｌｅｓｏｎｓｏｍｅｃｏｍｐｌｅｔｅｐｒｏｂａｂｉｌｉｔｙｓｐａｃｅ（Ω，Ｆ，Ｐ）ｔａｋｉｎｇｖａｌ...     

Regularity of the Laws of Shot Noise Series and of Related Processes

We investigate the regularity of shot noise series and of Poisson integrals. We give conditions for the absolute continuity of their law with respect to the Lebesgue measure and for their continuity in total-variation norm. In particular, the case of truncated series in addressed. Our method relies on a disintegration of the probability space based on a mere conditioning by the first jumps of the underlying Poisson process.     

Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences

Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n｜Sk｜,n≥1.Suppose limn→∞ESn^2/n=：σ^2〉0 and ∑n^∞=1 ρ^2/d（2^n）〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E｜X｜^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2（r-p）/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1（-1）^k/（2k＋1）^2（r-p）/（2-p）E｜Z｜^2（r-p）/2-p,where Z has a normal distribution with mean 0 and variance σ^2.     

On the Truth of a Conjecture Concerning the q-Derangement Numbers

ＯｎｔｈｅＴｒｕｔｈｏｆａＣｏｎｊｅｃｔｕｒｅＣｏｎｃｅｒｎｉｎｇｔｈｅｑ－ＤｅｒａｎｇｅｍｅｎｔＮｕｍｂｅｒｓＺｈａｎｇＸｉａｎｇｄｅ（Ｄｅｐｔ．ｏｆＡｕｔｏｍａｔｉｃＣｏｎｔｒｏｌ，ＮｏｒｔｈｅａｓｔｅｒｎＵｎｉｖ，Ｓｈｅｎｙａｎｇ１１０００６）...