Groups with some arithmetic conditions on real class sizes

Let G be a finite group. An x∈G is a real element if x and x ^?1 are conjugate in G. For x∈G, the conjugacy class x ^G is said to be a real conjugacy class if every element of x ^G is real. We show that if 4 divides no real conjugacy class sizes of a finite group G, then G is solvable. We also study the structure of such groups in detail. This generalizes several results in the literature.     

A system of cubic diophantine equations

In this paper we solve x³ + y + 1 ? xyz = 0 completely and study a pair of simultaneous cubic diophantine equations (1) x | y³ + 1 and y | x³ + 1, where x and y are positive integers. The main result in this paper is that there exist an infinite number of sequences such that x and y satisfy (1) if and only if they are consecutive terms of one of these sequences.     

Rational and algebraic approximations of algebraic numbers and their application

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and ε =x ₀ denotes the fundamental solution ofx ²?Dy ²=?1, thenx ²+1=Dy ⁴ is solvable if and only ify ₀=A ², where A is an integer. Moreover, if ε>64, thenx ²+1=Dy ⁴ has at most one positive integral solution (x, y).     

A nonoscillation theorem for superlinear Emden-Fowler equations with near-critical coefficients

We are interested in the oscillatory behavior of solutions of the Emden-Fowler equation y^″+a(x)|y|^γ−1y=0, γ>1, where a(x) is a positive continuous function on (0,∞). In the special case when the coefficient a(x) is a power of x, i.e. a(x)=xα for some constant α, the value α^∗=−(γ+3)/2 plays a critical role: The equation has both oscillatory and nonoscillatory solutions if α>α^∗, while all solutions are nonoscillatory if α<α^∗. When a(x) is close to the critical exponent, one of the known results is that if a(x)=x−(γ+3)/2log−σ(x), where σ>0, then all solutions are nonoscillatory. In this paper, this result is further extended to include a class of coefficients in which the above condition with log(x) can be replaced by loglog(x), or logloglog(x) and so on.     

Periodic behavior of a nonlinear third order vibrating system

A theorem is proved to show that the third order differential equation x^‴+f(t,x,x^′,x^″)=0 has nontrivial solutions characterized by x^′(0)=x^′(τ)=0 when x,x^′,x^″ and f(t,x,x^′,x^″) are bounded. A second condition is introduced to prove the existence of periodic solution for this equation. It is shown that the equation has a τ-periodic solution if f(t,x,x^′,x^″) is an even function with respect to x^′. The existence and periodicity conditions would be applied to third order systems such as viscoelastic mechanical vibration isolator system. The concepts of Green’s function and the Schauder’s fixed-point theorem have been used for proving the third-order-existence theorem.     

Conjugate cone characterization of positive definite and semidefinite matrices

Positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in Rⁿ. These results are obtained by generalizing Moreau's polar decomposition to a conjugate decomposition. Some typical results are: The matrix A is positive definite if and only if for some closed convex cone K, A is positive definite on K and (A+A^T)^?1 exists and is semidefinite on the polar cone K°. The matrix A is positive semidefinite if and only if for some closed convex cone K such that either K is polyhedral or (A+A^T)(K) is closed, A is positive semidefinite on both K and the conjugate cone K^A={s∣x^T(A+ A^T)s?0?x∈K}, and (A+A^T)x=0 for all x in K such that x^TAx=0.     

Kolmogorov Vector Fields with Robustly Permanent Subsystems

The following results are proven. All subsystems of a dissipative Kolmogorov vector field ?_i = x_if_i(x) are robustly permanent if and only if the external Lyapunov exponents are positive for every ergodic probability measure μ with support in the boundary of the nonnegative orthant. If the vector field is also totally competitive, its carrying simplex is C¹. Applying these results to dissipative Lotka-Volterra systems, robust permanence of all subsystems is equivalent to every equilibrium x* satisfying f_i(x* > 0 whenever x_i^* = 0. If in addition the Lotka-Volterra system is totally competitive, then its carrying simplex is C¹.     

Generalized exponential dichotomy and global linearization

Hartman's linearization theorem says that if all eigenvalues of matrix A have no zero real part and f(x) is small Lipschitzian, then nonlinear system x^′=Ax+f(x) and its linear system x^′=Ax are topologically equivalent. In 1970s Palmer extended the theorem to nonautonomous systems. In this paper we extend Hartman's theorem to the systems with generalized exponential dichotomy.     

Stability of difference approximations to differential equations

Consider the differential equation (1) ? = f(x) in a Banach space and let x¹ be an equilibrium. The basic question treated is the following: if x¹ is asymptotically stable and if (2) x_{k + 1} = x_k + h?(x_k, h) is a one-step method, with fixed step size h, for integrating (1), then does the sequence x_k converge to x¹? It is shown that uniform asymptotic stability of (1) implies stability of (2) and that exponential asymptotic stability of (1) implies asymptotic stability of (2).     

The recognition of the class of indecomposable digraphs under low hemimorphy

Given a digraph G=(V,A), the subdigraph of G induced by a subset X of V is denoted by G[X]. With each digraph G=(V,A) is associated its dual G^?=(V,A^?) defined as follows: for any x,y∈V, (x,y)∈A^? if (y,x)∈A. Two digraphs G and H are hemimorphic if G is isomorphic to H or to H^?. Given k>0, the digraphs G=(V,A) and H=(V,B) are k-hemimorphic if for every X⊆V, with |X|≤k, G[X] and H[X] are hemimorphic. A class C of digraphs is k-recognizable if every digraph k-hemimorphic to a digraph of C belongs to C. In another vein, given a digraph G=(V,A), a subset X of V is an interval of G provided that for a,b∈X and x∈V−X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For example, 0?, {x}, where x∈V, and V are intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

The rank-width of the square grid

Rank-width is a graph width parameter introduced by Oum and Seymour. It is known that a class of graphs has bounded rank-width if, and only if, it has bounded clique-width, and that the rank-width of G is less than or equal to its branch-width.The n×nsquare grid, denoted by G_n,n, is a graph on the vertex set {1,2,…,n}×{1,2,…,n}, where a vertex (x,y) is connected by an edge to a vertex (x^′,y^′) if and only if |x−x^′|+|y−y^′|=1.We prove that the rank-width of G_n,n is equal to n−1, thus solving an open problem of Oum.     

A quadratically convergent method for minimizing a sum of euclidean norms

We consider the problem of minimizing a sum of Euclidean norms. \(F(x) = \sum\nolimits_{i = 1}^m {||r_i } (x)||\) here the residuals {r _i(x)} are affine functions fromR ⁿ toR ¹ (n≥1≥2,m>-2). This arises in a number of applications, including single-and multi-facility location problems. The functionF is, in general, not differentiable atx if at least oner _i (x) is zero. Computational methods described in the literature converge quite slowly if the solution is at such a point. We present a new method which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along whichF is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method recently described by Calamai and Conn. The new method has quadratic convergence to a solutionx under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at^* x, thenF is differentiable at^* x and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, sayr _i ^* x), is zero, then, as the iteration proceeds, the Hessian ofF becomes extremely ill-conditioned. It is proved that this illconditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold (x?r _i (x)=0}. If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence for Newton's method applies. If several residuals are zero at^* x, several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results.     

On the numerical characterization of the reachability cone for an essentially nonnegative matrix

Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x₀=x(0), t?0 is x(t)=e^tAx₀. The problem of identifying the initial points x₀ for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x₀ are exactly those vectors for which the iterates x^(k)=(I+hA)^kx₀ become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x^(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x^(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x^(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described.     

On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

We construct a functor AC(?, ?) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)?dimensional; and (iii) For a path connected space X, π ₁(X, x) is trivial if and only if π ₂(AC(X, x)) is trivial. As a corollary, AC(S ¹, x) is a 2-dimensional nonaspherical cell-like Peano continuum.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Local rotundity structure of Calderón-Lozanovskii spaces

General results saying that a point x of the unit sphere S(E) of a Köthe space E is an extreme point (a strongly extreme point) [an SU-point] of B(E) if and only if ‖x‖ is an extreme point (a strongly extreme point) [an SU-point] of B(E⁺) and ‖x‖ is a UM-point (a ULUM-point) [nothing more] of E are proved. These results are applied to get criteria for extreme points and SU-points of the unit ball in Caldern-Lozanovski spaces which refer to problem XII from [5]. Strongly extreme points in these spaces are also discussed.     

Local monotonicity structure of Calderón-Lozanovskiǐ spaces

We first prove that if x is an element on the unit sphere of arbitrary Köthe space E, x is strickly positive μ-a.e. and x is an LM-point, then x is an UM-point. Criteria for lower and upper monotone points in Calderón-Lozanovskiǐ spaces E_? are presented. Points of lower local uniform monotonicity and upper local uniform monotonicity in E_? are also considered. Some sufficient conditions and necessary conditions for these properties of a given point x in S(E_?⁺) are given.     

Planar zero divisor graphs of partially ordered sets

We associate a graph G ^?(P) to a partially ordered set (poset, briefly) with the least element?0, as an undirected graph with vertex set P ^?=P?{0} and, for two distinct vertices x and y, x is adjacent to?y in?G ^?(P) if and only if {x,y} ^? ={0}, where, for a subset?S of?P, S ^? is the set of all elements x∈P with x≦s for all s∈S. We study some basic properties of?G ^?(P). Also, we completely investigate the planarity of?G ^?(P).     

A subadditive property of the gamma function

Let Δ=min_x?0Γ(2x)/Γ(x) and . We prove that the function x?(Γ(x))^α is subadditive on (0,∞) if and only if α∗?α?0.