Incomplete group divisible designs with block size four and general index

In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h) u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u 1) ≡ 0 (mod 3), λ(g h)(u 1) ≡ 0 (mod 3), and λu(u 1)(g 2 h 2 ) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved.     

P 3-factorization of complete multipartite graphs

In this note it is shown that a necessary and sufficient condition for the existence of a P3-factorizatlon of complete multipartite graph λK, is (1) m≥3, (2) mn≡0(mod 3) and (3)λ(m-1)n≡0(mod 4).     

EMBEDDINGS OF SIMPLE TWO-FOLD BALANCED INCOMPLETE BLOCK DESIGNS WITH BLOCK SIZE FOUR

The necessary and sufficient conditions for the existence of simple incomplete block design (v, w; 4,2)-IPBDs are determined. As a consequence, the necessary and sufficient conditions for the embeddings of simple two-fold balanced incomplete block designs with block size 4 are also determined.     

On necessary and sufficient conditions for the self-normalized central limit theorem In Honor of Professor Chuanrong Lu on His 85th Birthday

Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n~2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary.     

The spectrum of cyclic (3, λ)-GDD of type g~v

In this paper, it is shown that the necessary conditions for the existence of a ( gv, {g, 3 α }, 3, λ)-DF in Z gv for α∈ {0, 1, 2} are also sufficient with two exceptions of (v, g, λ, α) = (9, 1, 1, 1), (9, 1, 2, 2). Finally, the existence spectrum of a cyclic (3, λ)-GDD of type g v is determined.     

On K_(1,k)-factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n,which partition the set of edges of λKm,n.In this paper,it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n,whenever k is any positive integer,is that(1) m ≤ kn,(2) n ≤ km,(3) km-n ≡ kn-m ≡ 0(mod(k2-1)) and(4) λ(km-n)(kn-m) ≡ 0(mod k(k -1)(k2 -1)(m n)).     

The Spectrum of （gv,g, 3, λ）-dDF in Zgv

In this paper, several recursive constructions for directed difference family and perfect directed difference family are presented by means of difference matrix and incomplete difference matrix. Finally the necessary and sufficient conditions for the existence of a （gv, g, 3, λ）-directed difference family in Zgv are established. As a consequence, the necessary and sufficient conditions for the existence of a cyclic directed group divisible design with block size three and type gv are obtained.     

The spectrum of path factorization of bipartite multigraphs

LetλK_(m,n)be a bipartite multigraph with two partite sets having m and n vertices, respectively.A P_v-factorization ofλK_(m,n)is a set of edge-disjoint P_v-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).When v is an even number,Ushio,Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P_v-factorization ofλK_(m,n).When v is an odd number,we have proposed a conjecture.Very recently,we have proved that the conjecture is true when v=4k-1.In this paper we shall show that the conjecture is true when v = 4k 1,and then the conjecture is true.That is,we will prove that the necessary and sufficient conditions for the existence of a P_(4k 1)-factorization ofλK_(m,n)are(1)2km≤(2k 1)n,(2)2kn≤(2k 1)m,(3)m n≡0(mod 4k 1),(4)λ(4k 1)mn/[4k(m n)]is an integer.     

SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let ? denote a smooth, bounded domain in R^N(N≥2). Suppose that g is a nondecreasing C¹ positive function and assume that b(x) is continuous and nonnegative in?, and that it may be singular on ??. In this paper, we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem ?_pu = b(x)g(u) for x∈?, u(x)→+∞ as dist(x, ??)→0.The estimates of such solutions are also investigated. Moreover, when b has strong sin...     

P_(4k-1)-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of Pv-factorization of Km,n.When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P4k-1-factorization of Km,n is (1) (2k - 1)m ≤ 2kn, (2) (2k -1)n≤2km, (3) m n ≡ 0 (mod 4k - 1), (4) (4k -1)mn/[2(2k -1)(m n)] is an integer.     

POSITIVE SOLUTIONS OF HIGHER ORDER SUBLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we discuss higher order sublinear functional differential equation x~(n)(t)+p(t)x~v(g(t))=0 t≥t_o where andSome necessary and sufficient conditions for existence of positive solution of the equation are established.     

The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.     

Boundary value problems of discrete generalized Emden-Fowler equation

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.     

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.     

On the Nonlinear Matrix Equation X+ A *f1（X）A +B*f2（X）B=Q

In this paper, nonlinear matrix equations of the form X + A*f1 (X)A + B*f2 (X)B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.     

The spectrum of (gv, g, 3, λ)-DF in Z_(gv)

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)- difference family in Zgv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type gv without short orbits is determined.     

On （g, f）-Uniform Graphs

A graph G is called a (g, f)-uniform graph if for each edge of G, there is a (g, f)-factor containing it and another (g, f)-factor excluding it. In this paper a necessary and sufficient condition for a graph to be a (g, f)-uniform graph is given and some applications of this condition are discussed. In particular, some simple sufficient conditions for a graph to be an [a, b]-uniform graph are obtained for a≤b.     

On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas

With the aid of Mullin-Rota's substitution rule, we show that the Sheffertype differential operators together with the delta operators ? and D could be used to construct a pair of expansion formulas that imply a wide variety of summation formulas in the discrete analysis and combinatorics. A convergence theorem is established for a fruitful source formula that implies more than 20 noted classical fomulas and identities as consequences. Numerous new formulas are also presented as illustrative examples. Finally, it is shown that a kind of lifting process can be used to produce certain chains of(∞~m) degree formulas for m ≥ 3 with m ≡ 1(mod 2) and m ≡ 1(mod3), respectively.     

Quasi-d-Koszul Modules and Applications

Let R be a Noetherian semiperfect algebra. A necessary and sufficient condition for a finitely generated R-module to be quasi-d-Koszul is given, which improves one of the main results in [1]. The authors also give a necessary and sufficient condition for the Minimal Horseshoe Lemma holding in mod（R）. As an application, it is proved that the ＂Minimal Horseshoe Lemma＂ is true in the category of quasi-d-Koszul modules under certain conditions.     

Regular n-simplices in R~n with vertices in Z~n

In this paper the Ⅰ and Ⅱ regular n-simplices are introduced. We prove that the sufficient and necessary conditions for existence of an Ⅰ regular n-simplex in Rn are that if n is even then n = 4m(m + 1), and if n is odd then n = 4m + 1 with that n + 1 can be expressed as a sum of two integral squares or n = 4m - 1, and that the sufficient and necessary condition for existence of a Ⅱ regular n-simplex in Rn is n = 2m2 - 1 or n = 4m(m+1)(m 6 N). The connection between regulars-simplex in Rn and combinational design is given.