Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.     

The computation of the adjoint generator for a singular neutral equation

We compute the adjoint generator for the infinitesimal generator A of a C₀-semigroup. The operator A is associated with a particular scalar nonatomic neutral equation.     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

In this paper, we consider the second-order nonlinear differential equation

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.     

Extension of Newman's formula for the Sylvester equation to the generic case

The Sylvester “inivilator” equation AX - XB=C is solved in finite terms when A and B are quite general. Newman's formula for the case of A and B periodic is recovered by a limiting process. Some applications are discussed.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Ranks of Solutions of the Matrix Equation AXB = C

The purpose of this article is to solve two problems related to solutions of a consistent complex matrix equation AXB = C : (I) the maximal and minimal ranks of solution to AXB = C , and (II) the maximal and minimal ranks of two real matrices X₀ and X₁ in solution X = X₀ + iX₁ to AXB = C . As applications, the maximal and minimal ranks of two real matrices C and D in generalized inverse (A + iB)^- = C + iD of a complex matrix A + iB are also examined.     

Block matrices over a valuation ring

A result of W. E. Roth, generalized by W. H. Gustafson connecting the matrix equation AX - XB - C, with block diagonal matrices, is adapted to matrices over a complete Valuation Ring. A necessary and sufficient condition for the similarity of a full matrix to a block diagonal matrix is proved.     

The countability index of the endomorphism semigroup of a vector space

The countability index C(S) of a semigroup S is the least positive integer n, if such an integer exists, with the property that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists. C(S) is defined to be infinite. Let V be a vector space over a field F and denote by End V the endomorphism semigroup of V. In the two main results, it is determined precisely when C(End V)=2 and when C(End V)=x SpecificallyC(End V)=2 if and only if V is infinite dimensional or dim V=1 and F is finite and C(End V)=x if and only if F is infinite and dim V is an integer N≥1.     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1for some C = C^-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A^-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

Enumeration of irreducible binary words

Overlap free words over two letters are called irreducible binary words. Let d(n) denote the number of irreducible binary words of length n. In this paper we show that there are positive constants C₁ and C₂ such that C₁n^1.155<d(n)<C₂n^1.587 holds for all n>0.     

Decompositions of positive self-dual boolean functions

A coterie, which is used to realize mutual exclusion in distributed systems, is a family C of subsets such that any pair of subsets in C has at least one element in common, and such that no subset in C contains any other subset in C. Associate with a family of subsets C a positive Boolean function f_c such that f_c(x) = 1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f_c is dual-minor, and is a non-dominated (ND) coterie if and only if f_c is self-dual. We study in this paper the decomposition of a positive self-dual function into smaller positive self-dual functions, as it explains how to represent and how to construct the corresponding ND coterie. A key step is how to decompose a positive dual-minor function f into a conjunction of positive self-dual functions f₁,f₂,…, f_k. In addition to the general condition for this decomposition, we clarify the condition for the decomposition into two functions f₁, and f₂, and introduce the concept of canonical decomposition. Then we present an algorithm that determines a minimal canonical decomposition, and a very simple algorithm that usually gives a decomposition close to minimal. The decomposition of a general self-dual function is also discussed.     

Cycle-saturated graphs of minimum size

A graph G is called C_k-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in C_k-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + c₁n/k and n + c₂n/k for some positive constants c₁ and C₂. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.     

How many maxima can there be?

Let C be a planar region. Choose n points p₁,,p_nI.I.D. from the uniform distribution over C. Let M^C_n be the number of these points that are maximal. If C is convex it is known that either E(M^C_n)=Θ(√n)> or E(M^C_n)=O(log n). In this paper we will show that, for general C, there is very little that can be said, a-priori, about E(M^C_n). More specifically we will show that if g is a member of a large class of functions then there is always a region C such that E(M^C_n)=Θ(g(n)). This class contains, for example, all monotically increasing functions of the form g(n)= nln^βn, where 0<<1 and β0. This class also contains nondecreasing functions like g(n)=ln^*n. The results in this paper remain valid in higher dimensions.     

On the stability of the vertical rotation of a solid suspended on a rod

The problem of the motion of a dynamically symmetric solid suspended from a fixed point by a weightless rod and two ball and socket joints one of which is fixed at the fixed point O', and the other is on the body axis of symmetry at the point O is considered. The question of the stability of the uniform body rotation when points O' and O, and the body centre of inertia C lie on the same vertical, and at the same time point O may be either above or below point O', and point C either above or below point O, is discussed. An analysis of the necessary and sufficient conditions for stability is given. The set of all the system's parameters is reduced to three independent dimensionless parameters L, Ω and β, and in the plane (L, Ω), for fixed values of β, the regions for which the unperturbed rotation is steady, or steady to a first approximation, or non-steady are indicated. The regions for which the body rotation is steady to a first approximation when the point O is situated higher than the point O', and the point C lies higher or lower than the point O are determined.

The sufficient conditions for the vertical rotation of a dynamically symmetric body suspended on a filament were obtained in /1/ and investigated for the cases where in non-perturbed motion the point C is below point O, when points C and O coincide, and when the length of the filament is zero (Lagrange gyroscope). In /2/ an analysis is given of the sufficient conditions for stability obtained in /1/, and also the necessary conditions for the cases where in a non-perturbed motion point C is located above point O.     

Note on a paper of thompson: the congruence numerical range

Given n×n Complex matrices A, Cdefine the C-congruence numerical range of A to be the set [ILM0001]. R.C. Thompson has characterized R_C(A) when [ILM0002] are fixed complex numbers. In this note. we obtain some analogous results about R_t(A) when C is skew symmmetric and a simple proof of the result of Thompson is given.Moreover, we characterize a certain set of partial off diagonals under congruence unitary transformation.     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.     

Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD²u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C^∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.     

Linear circulant matricesover a finite field

Rings of polynomials R_N = Z_p[x]/x^N − 1 which are isomorphic to Z_P^N are studied, where p is prime and N is an integer. If I is an ideal in R_N, the code K whose vectors constitute the isomorphic image of I is a linear cyclic code. If I is a principle ideal and K contains only the trivial cycle 0 and one nontrivial cycle of maximal least period N, then the code words of K/ 0 obtained by removing the zero vector can be arranged in an order which constitutes a linear circulant matrix, C. The distribution of the elements of C is such that it forms the cyclic core of a generalized Hadamard matrix over the additive group of Z_Pp. A necessary condition that C = K/ 0 be linear circulant is that for each row vector v of C, the periodic infinite sequence a(v) produced by cycling the elements of v be period invariant under an arbitrary permutation of the elements of the first period. The necessary and sufficient condition that C be linear circulant is that the dual ideal generated by the parity check polynomial h(χ) of K be maximal (a nontrivial, prime ideal of R_N), with N = p^k − 1 and k = deg (h(χ)).