Unbalanced bipartite factorizations of complete bipartite graphs

We construct a new infinite family of factorizations of complete bipartite graphs by factors all of whose components are copies of a (fixed) complete bipartite graph K_p,q. There are simple necessary conditions for such factorizations to exist. The family constructed here demonstrates sufficiency in many new cases. In particular, the conditions are always sufficient when q=p+1.     

Coordinates of R[x,y]: Constructions and Classifications

Let R be a PID. We construct and classify all coordinates of R[x, y] of the form p ₂ y + Q ₂(p ₁ x + Q ₁(y)) with p ₁, p ₂ ∈ qt(R) and Q ₁, Q ₂ ∈ qt(R)[y]. From this construction (with R = K[z]) we obtain nontame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms.     

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

A complete classification of bifurcation diagrams of a p -Laplacian Dirichlet problem

We study the bifurcation diagrams of (classical) positive solutions u with |u |_∞ ∈ (0, ∞) of the p -Laplacian Dirichlet problem (φ_p (u ′(x)))′ + λf_q (u (x))) = 0, –1 ≤ x ≤ 1, u (–1) = 0 = u (1), where p > 1, φ_p (y) = |y |^{p –2} y, (φ_p (u ′))′ is the one-dimensional p -Laplacian, λ > 0 is a bifurcation parameter, and the nonlinearity f_q (u) = |1 – u |^q is defined on [0, ∞) with constant q > 0. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Criteria for selection of response variables and the asymptotic properties in a multivariate calibration

Summary Let a set ofp responsesy=(y ₁,...y _p )′ has a multivariate linear regression on a set ofq explanatory variablesx=(x ₁,...x _q )′. Our aim is to select the most informative subset of responses for making inferences about an unknownx from an observedy. Under normality ony, two selection methods, based on the asymptotic mean squared error and on the Akaike's information criterion, are proposed by Fujikoshi and Nishii (1986,Hiroshima Math. J.,16, 269–277). In this paper, under a mild condition we will derive the cross-validation criterion and obtain the asymptotic properties of the three procedures.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

Approximating Data in \mathfrak{R}^{n} by a Quadratic Underestimator with Specified Hessian Minimum and Maximum Eigenvalues

The problem of approximating m data points (x _i , y _i ) in , with a quadratic function q(x, p) with s parameters, m ≥ s, is considered. The parameter vector is to be determined so as to satisfy three conditions: (1) q(x, p) must underestimate all m data points, i.e. q(x _i , p) ≤ y _i , i=1,...,m. (2) The error of the approximation is to be minimized in the L1 norm. (3) The eigenvalues of H are to satisfy specified lower and upper bounds, where H is the Hessian of q(x, p) with respect to x. This is called the Quadratic Underestimator with Bounds on Eigenvalues (QUBE) problem. An algorithm for its solution (QUBE algorithm) is given and justified, and computational results presented. The QUBE algorithm has application to finding the global minimum of a basin (or funnel) shaped function with a large number of local minima. Such problems arise in computational biology where it is desired to find the global minimum of an energy surface, in order to predict native protein-ligand docking geometry (drug design) or protein structure. Computational results for a simulated docking energy surface, with n=15, are presented. It is shown that specifying a small condition number for H improves the ability of the underestimator to correctly predict the global minimum point.     

Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t ?=?div(|?u ^m | ^p?2?u ^m )?+?a∫_Ω u ^q (y,?t)dy with a, q, m?>?0, p?>?1, m(p???1)?R ^N (N?>?2). More precisely speaking, it is shown that if q?>?m(p???1), any non-negative solution with small initial data vanishes in finite time, and if 0?q?m(p???1), there exists a solution which is positive in Ω for all t?>?0. For the critical case q?=?m(p???1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ?=?∫?_Ωφ ^p?1(x)dx and φ is the unique positive solution of the elliptic problem ?div(|?φ| ^p?2?φ)?=?1, x?∈?Ω; φ(x)?=?0, x?∈??Ω.     

Characterization of a Hilbert vector lattice by the metric projection onto its positive cone

If H is a real Hilbert space, K is a closed, generating cone therein and P_K is the metric projection onto K, then the following two conditions 1 and 2 are equivalent:

1. (i) P_K is isotone: y−xKP_K(y)−P_K(x)K and (ii) P_K is subadditive: P_K(x)+P_K(y)−P_K(x+y)K, x, yH, and

2. H ordered by K: (i) is a vector lattice; (ii) ||x||=|| |x| ||, xH, and (iii) xK, y−xK imply ||x||||y||.

     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).     

Approximation of the vth Root of N

We present a class of functions g_K(w), K ≥ 2, for which the recursive sequences w_{n + 1} = g_K(w_n) converge to N^1/v with relative error . Newton's method results when K = 2. The coefficients of the g_K(w) form a triangle, which is Pascal's for v = 2. In this case, if w₁ = x₁/y₁, where x₁, y₁ is the first positive solution of Pell's equation x² ? Ny² = 1, then w_{n + 1} = x_{n + 1}/y_{n + 1} is the Kⁿpth or 2Kⁿpth convergent of the continued fraction for , its period p being even or odd.     

One-dimensional Tilings Using Tiles with Two Gap Lengths

Let n,p,k,q,l be positive integers with n=k+l+1. Let x₁,x₂, . . . ,x_n be a sequence of positive integers with x₁<x₂<···<x_n. A set {x₁,x₂, . . . ,x_n} is called a set of type (p,k;q,l) if the set of differences {x₂–x₁,x₃–x₂, . . . ,x_n–x_n–1} equals {p, . . . ,p,q, . . . ,q} as a multiset, where p and q appear k and l times, respectively. Among other results, it is shown that for any p,k,q, there exists a finite interval I in the set of integers such that I is partitioned into sets of type (p,k;q,1).     

Quasiminimizers in one dimension: integrability of the derivative,inverse function and obstacle problems

It is shown that a K-quasiminimizer u for the one-dimensional p-Dirichlet integral is a K′-quasiminimizer for the q-Dirichlet integral, 1  ≤  q  <  p ₁(p, K), where p ₁(p, K) > p; the exact value for p ₁(p, K) is obtained. The inverse function of a non-constant u is also K′′-quasiminimizer for the s-Dirichlet integral and the range of the exponent s is specified. Connections between quasiminimizers, superminimizers and solutions to obstacle problems are studied.     

Divided differences of multivariate implicit functions

Under general conditions, the equation g(x ¹,…,x ^q ,y)=0 implicitly defines y locally as a function of x ¹,…,x ^q . In this article, we express divided differences of y in terms of divided differences of g, generalizing a recent formula for the case where y is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as plane trees. Through this connection we prove as a corollary a formula for derivatives of y in terms of derivatives of g.     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

Contributo allo studio del problema di hughes sui gruppi

Sunto. Hughes ha avanzato la seguente congettura: se G è un gruppo, p un numero primo, e H_p il sottogruppo generato dagli elementi di G che non hanno periodo p, si presenta uno dei tre casi seguenti: H_p=1; H_p=G; [G: H_p]=p. Vari autori hanno provato l'esattezza di tale congettura per larghe classi di gruppi. In questa Nota si dimostra che tale congettura è esatta per i p-gruppi finiti in cui ogni sottogruppo generato da tre elementi, due dei quali di periodo p, abbia classe ≤p; essa è quindi esatta in particolare per i p-gruppi di classe ≤p.

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Summary Hughes advanced the following conjecture: “let G be a group, p a prime and H_p the subgroup generated by elements of G having order p; then H_p=1 or H_p=G or [G: H_p]=p„. Several authors have proved this conjecture for various classes of groups. The Author proves Hughes conjecture for finite p-groups such that every subgroup {x ₁ , x _{2, y}} generated by two elements x ₁ , x ₂ having order p, and by a thirth element y, has class ≤p; in particular, the conjecture is proved for finite p-groups having class ≤p.

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A Enrico Bompiani in occasione del suo Giubileo scientifico.     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

On the Eigenvalue Accumulation of Sturm-Liouville Problems Depending Nonlinearly on the Spectral Parameter

A nonlinear spectral problem for a Sturm-Liouville equation-(p(x, λ)y'(x, λ))' + q(x, λ) y(x, λ) = 0 on a finite interval [a, b] with λ-dependent boundary conditions is considered. The spectral parameter λ is varying in an interval ∧ and p(x, λ), q(x, A) are real, continuous functions on [a, b] × ∧ Some criteria to the eigenvalue accumulation at the endpoints of A will be established. The results are applied to concrete problems arising in magnetohydrodynamics.