On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Neighbor Connectivity of Two Kinds of Cayley Graphs

In this paper, we determine the neighbor connectivity κNB of two kinds of Cayley graphs: alternating group networks AN _n and star graphs S _n ; and give the exact values of edge neighbor connectivity λ_NB of ANn and Cayley graphs generated by transposition trees Γ _n . Those are κ_NB(AN _n ) = n^?1, λ_NB(AN _n ) = n^?2 and κ_NB(S _n ) = λ_NB(Γ _n ) = n^?1.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6: Suppose that λ₁, λ₂,..., λ₅ be nonzero real numbers not all of the same sign, satisfying that λ₁/λ₂ is irrational, and suppose that η is a real number. In this paper, for any ε > 0; we consider the inequality |λ₁p₁ + λ₂p ₂ ² + λ₃p ₃ ³ + λ₄p ₄ ⁴ + λ₅p ₅ ^k + η | < (max p_j)^-σ(k)+ε has infinitely many solutions in prime variables p₁, p₂,...,p₅, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.     

Normal functions and a class of associated boundary functions

Letμ′ be the family of non-empty closed subsets of the Riemann sphere and Λ the family of continuous curves λ with values in the unit disk and lim _t→1 |λ(t)|=1. A meromorphic functionf in |z|<1 induces a mapping\(\hat f\) from Λ intoμ′ by setting\(\hat f\left( \lambda \right)\) equal to the cluster set off on λ. The authors show that if\(\hat f\) is continuous then existence of an asymptotic value ate ^iθ implies the existence of an angular limit. Further if the spherical derivative off iso(1/(1?|z|)) then\(\hat f\) is constant on every open disk in the space Λ.     

Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

It is shown that every echelon space λ_∞(A), with A an arbitrary Köthe matrix, is a Grothendieck space with the Dunford-Pettis property. Since λ_∞(A) is Montel if and only if it coincides with λ₀(A), this identifies an extensive class of non-normable, non-Montel Fréchet spaces having these two properties. Even though the canonical unit vectors in λ_∞(A) fail to form an unconditional basis whenever λ_∞(A) ≠ λ₀(A), it is shown, nevertheless, that in this case λ_∞(A) still admits unconditional Schauder decompositions (provided it satisfies the density condition). This is in complete contrast to the Banach space setting, where Schauder decompositions never exist. Consequences for spectral measures are also given.     

Empirical examination of Edgeworth series

The material presented is based on a numerical investigation that was made for five types of probability approximations which involve the first seven terms of the Edgeworth series expansion for the distribution of a continuous random variableT. For each approximation, the probability expressions considered in the investigation were Pr(T≦t), Pr(?t≦T≦t) and Pr(?t+1≦T≦t), whereT has zero mean, unit variance, and specified central momentsμ ₃,μ ₄,μ ₅. Computations were made for thoset values in the set ?4.00(0.25) 4.00 that are pertinent for the probability expression being considered and for all combinations of the following values forμ ₃,μ ₄,μ ₅∶μ ₃=?2.0, ?1.0, ?0.5,0.0, 0.5, 1.0, 2.0;μ ₄=1, 2, 3, 5, 10;μ ₅=0.0, 3μ ₃?6.0, 3μ ₃, 3μ ₃+6.0. The principal results of this paper consist of a specification (for each approximation, probability expression, andμ ₃,μ ₄,μ ₅ combination) of limits ont such that within these limits the computed values of the probability expression are meaningful; that is, satisfy required monotonicity properties as a function oft and are neither negative nor greater than unity. Also the values of Pr(T≦0) and of Pr(?1.75≦T≦1.75) are listed for the cases considered. These results indicate that the types of approximations investigated are of doubtful usefulness for the situations examined; that is, for cases where the third and higher order moments of the random variable considered differ substantially from those for the normal variable having the same mean and variance.     

A new class for large sets of almost Hamilton cycle decompositions

A k-cycle system of order v with index λ, denoted by CS(v, k, λ), is a collection A of k-cycles (blocks) of K _v such that each edge in K _v appears in exactly λ blocks of A. A large set of CS(v, k, λ)s is a partition of the set of all k-cycles of K _v into CS(v, k, λ)_s, and is denoted by LCS(v, k, λ). A (v ?1)-cycle in K _v is called almost Hamilton. The completion of the existence problem for LCS(v, v?1, λ) depends only on one case: all v ≥ 4 for λ = 2. In this paper, it is shown that there exists an LCS(v, v ? 1, 2) for all v ≡ 2 (mod 4), v ≥ 6.     

Functional estimation of activity criticality indices and sensitivity analysis of expected project completion time

For a PERT network, a new method is developed for estimating the criticality index of activity i (ACI _i ) as a function of the expected duration of activity i (μ _i ) and for the sensitivity analysis of the expected project completion time (μ _T ) with respect to μ _i . The proposed method evaluates the frequency of activity i being on the critical path, and thereby its ACI _i using Monte Carlo simulation or a Taguchi orthogonal array experiment at several values of μ _i , fits a logistic regression model for estimating ACI _i as a function of μ _i , and then, using the estimated ACI _i function, evaluates the amount of change in μ _T when μ _i is changed by a given amount. Unlike the previous works, the proposed method models ACI _i as a nonlinear (ie, logistic) function of μ _i , which can be used to estimate the amount of change in μ _T for a variety of changes in μ _i . Computational results indicate that the performance of the proposed method is comparable to that of direct Monte Carlo simulation.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Graphs with small diameter determined by their <Emphasis Type="Italic">D</Emphasis>-spectra

Let G be a connected graph with vertex set V(G) = {v₁, v₂,..., v _n }. The distance matrix D(G) = (d _ij )_n×n is the matrix indexed by the vertices of G, where d _ij denotes the distance between the vertices v _i and v _j . Suppose that λ₁(D) ≥ λ₂(D) ≥... ≥ λ _n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.     

Phase transition for SIR model with random transition rates on complete graphs

We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C _n with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d. copies of a positive random variable ρ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists λ _c > 0 such that when λ < λ _c ; the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → +∞ while when λ > λ _c ; there exist c(λ) > 0 and b(λ) > 0 such that for each n ≥ 1 with probability p ≥ b(λ); the proportion r_∞ ≥ c(λ): Furthermore, we prove that λ _c is the inverse of the production of the mean of ρ and the mean of the inverse of ξ.     

The exceptional set for Diophantine inequality with unlike powers of prime variables

Suppose that λ₁, λ₂, λ₃, λ₄ are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ₁/λ₂ is algebraic and irrational. Denote by E(V,N, δ) the number of v ∈ V with v ≤ N such that the inequality

$$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$

has no solution in primes p₁, p₂, p₃, p₄. We show that

$$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$

for any ? > 0.

     

The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces

Let(M, θ) be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ_1 of the Kohn-Laplacian □_b on(M, θ). In the present paper, we give a sharp upper bound for λ_1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ_1 when the pseudohermitian structure θ is volume-normalized.     

Direct and inverse asymptotic scattering problems for Dirac-Krein systems

The asymptotic scattering matrix s _ε(λ) for a Dirac-Krein system with signature matrix J = diag{ I _p ,-I _p }, integrable potential, and the boundary condition u ₁(0, λ) = u ₂(0, λ)ε(λ) with a coefficient ε(λ) that belongs to the Schur class of holomorphic contractive p × p matrix-valued functions in the open upper half-plane is defined. The inverse asymptotic scattering problem for a given s _ε is analyzed by Krein’s method. Earlier studies by Krein and others focused on the case in which ε = I _p (or a constant unitary matrix).     

X-ray spectroscopic study of chemical bonding in MnSe2 and CoSe2

X-ray absorption study of two intermetallic compounds MnSe₂ and CoSe₂ has been carried out using a Cauchois type bent crystal spectrograph. The metal K absorption edges in both the compounds are found to shift toward the high energy side with respect to the discontinuities in the pure metals. On the other hand, the Se K absorption edge in both these compounds is found to shift toward the lower energies. Emission study of the compound MnSe₂ has shown that the Mn Kβ ₅ band in this compound is shifted toward the high energy side with respect to that in the pure metal. From the magnetic data and the results obtained in this work it is possible to obtain the chemical bonding pictures in these compounds. For MnSe₂ the bondings sp ³ d ² for the metal atom andsp ³ for the metalloid atom have been suggested. For CoSe₂ the bondings appear to bed ² sp ³ for the metal atom andsp ³ for the metalloid atom. These bondings are compatible with the pyrite type structure of these compounds. It is possible to explain the electrical behaviour of the compounds on the basis of these bonding pictures.     

Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}

Consider the n×n matrix with (i, j)’th entry gcd (i, j). Its largest eigenvalue λ_n and sum of entries s_n satisfy λ_n > s_n/n. Because s_n cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that λ_n > 6π^?2nlogn for all n. If n is large enough, this follows from F.Balatoni (1969).     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

On nonstationary radiation fields in an infinite one-dimensional homogeneous medium

This paper considers nonstationary monochromatic radiative transfer in an infinite onedimensional homogeneous medium. The medium is considered to be illuminated by a momentary isotropic point energy source. The optical properties of the medium are characterized by the absorption coefficient α, the single-scattering albedo λ, the mean time t ₁ of photon stay in the absorbed state, and the mean time t ₂ of its stay on the path between two consecutive scatterings. The exact solution of the nonstationary radiative transfer equation has been obtained for the case t ₁ = t ₂. Asymptotic expressions have been derived for the source function, for the average intensity, and for radiation flux when points of the medium are located at large optical distances from the power source |τ| ? 1 and for small absorption of light in the medium (1 ? λ ? 1), assuming that t ₁ ? t ₂, t ₁ ? t ₂, or t ₁ = t ₂. These expressions are more precise than the ones previously known.     

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.