中介逻辑的命题演算系统(Ⅱ)

This paper is a continuation of [1]、[2]、[4]. In this paper, we shall con-tinue to constitute the formal theorems and the important substituting theorem(see theorem 16 in this paper) of the propositional calculus system of mediumlogic MP. The order numbers of following formal theorems follow those in reference[4]     

中介逻辑的命题演算系统 (Ⅲ)

The paper is a continution of [1]、[2]、[4]、[5]. In this paper we shall introduceconjunction symbol"∧", disjunction symbol "∨" and equality value symbol " asdefining symbols of MP, which are read as "and"、"or"、"if and only if" respectively.With these symbols we shall go on to prove the formal theorems of MP. The list ofthe following formal theorems are in succssion of refernce.     

The Theory of Finite Models without Equal Sign

In this paper, it is the first time ever to suggest that we study the model theory of all finite structures and to put the equal sign in the same situtation as the other relations. Using formulas of infinite lengths we obtain new theorems for the preservation of model extensions, submodels, model homomorphisms and inverse homomorphisms. These kinds of theorems were discussed in Chang and Keisler＇s Model Theory, systematically for general models, but Gurevich obtained some different theorems in this direction for finite models. In our paper the old theorems manage to survive in the finite model theory. There are some differences between into homomorphisms and onto homomorphisms in preservation theorems too. We also study reduced models and minimum models. The characterization sentence of a model is given, which derives a general result for any theory T to be equivalent to a set of existential-universal sentences. Some results about completeness and model completeness are also given.     

ON THE STRUCTURE OF HOMOGENEOUS COMPLETELY REDUCIBLE MODULE

In this paper we illuminate the equivalent relations between the structures of homoge-neous completely reducible modules and vector spaces over division rings.Besides,we alsogive a method to simplify the proof of some important theorems concerning the homogeneouscompletely reducible modules in[1].     

SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE AND GAUSS MAPS

The aim of this paper is to study the relations between submanifolds with parallel mean curvature and their Gauss maps, and in virtue of them we obtain three main theorems. They show some geometrical restrictions to a submanifold of Euclidean space E~n with parallel mean curvature.     

ON THE RELIABILITY OF MEDIUM LOGIC(ML)

Medium logic (ML) is set up for the common theoritical foundation of the classical mathematics and fuzzy mathematics. It has been formalized as a new theory of logic. See note (1) and note (2). As mathematical logic the ML's construct reches the study of the informal deductive inference by means of studying the formal inference in it, and it demands the formal inference of ML reliable reflected the deductive inference. For this reason, this paper deals with its reliability. The result shows that the formal inference of M L consists with deductive inference, and that ML reliably reflects the deductive inference.     

CHARACTER, DENSITY AND CELLULARITY OF L-FUZZY TOPOLOGICAL SPACES

In this paper we generalize the notions of weight,character, density and cellularity of topological spaces to L-fuzzy topogical spaces. We prove theorems corresponding to some of usual theorems. In particular we generalize Hewitt-Marczewski-Pondiczery theorem on the density of the product of topological spaces to the case of L-fuzzy topological spaces (Theorem 4. 4). We also discuss the relations between topological spaces and its induced fuzzy topological spaces for these cardinal functions.     

The Proofs of Several Theorems about the Order Relation

In this note,we use the fundamental rulesof the mathematical logic to pro-ve several theorems concerned with the order relation in decision theory,in orderto demonstrate that this kind of method seems simpler,clearer,and more strict,and efficient to decrease mistakes.In the maentime,this process laeds to findinga contradition and a flaw in the previous statements of the theorems.     

有限群在特征为零的任意域上的表示的一些注记

Let G be a finite group and K a field of characteristic zero.It is well-known that if K is a splitting field for G,then G is abelian if and only if any irreducible representation of G has degree 1.In this paper,we generalize this result to the case that K is an arbitrary field of characteristic zero（that is,K need not be a splitting field for G）,and we also obtain the orthogonality relations of irreducible K-characters of G in this case.Our results generalize some well-known theorems.     

REGION QUALITATIVE ANALYSIS OF PREDATOR-PREY SYSTEMS ON TIME SCALES

We first investigate some basic properties of dynamic equations on time scales,and propose contained curves to describe the jump direction of the discrete points.Then we perform qualitative analysis regarding the planar predator-prey systems on time scales,thereby obtain two theorems of this system.At last,we emulate application examples to discuss the parameters of the system.     

LIMIT CYCLES BIFURCATION FOR A CLASS OF DEGENERATE SINGULARITY

In this paper, bifurcation of limit cycles for the degenerate equilibrium to a three- dimensional system is investigated. Firstly, we use formal series to calculate the focal values at the high-order critical point on center manifold. Then an example is studied, and the existence of 3 limit cycles on the center manifold is proved. In terms of high- order singularities in high-dimensional systems, our results are new.     

Existence and Uniqueness of Positive Solutions for a System of Multi-order Fractional Differential Equations

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equa-tions. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.     

New systems of generalized quasi-variational inclusions in FC-spaces and applications

In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.     

Bloch constant of holomorphic mappings on the unit polydisk of C~n

In this paper,we give a definition of Bloch mappings defined in the unit polydisk D~n, which generalizes the concept of Bloch functions defined in the unit disk D.It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables.We shall establish the corresponding distortion theorems for subfamiliesβ(K)andβ_(loc)(K) of Bloch mappings defined in the polydisk D~n,which extend the distortion theorems of Liu and Minda to higher dimensions.As an application,we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloeh mappings defined in D~n.In particular,our results reduce to the classical results of Ahlfors and Landau when n=1.     

Chaotic invariant sets of a delayed discrete neural network of two non-identical neurons

In this paper, we show that a delayed discrete Hopfield neural network of two nonidentical neurons with no self-connections can demonstrate chaotic behavior in a region away from the origin. To this end, we first transform the model, by a novel way, into an equivalent system which enjoys some nice properties. Then, we identify a chaotic invariant set for this system and show that the system within this set is topologically conjugate to the full shift map on two symbols. This confirms chaos in the sense of Devaney. Our main result is complementary to the results in Kaslik and Balint (2008) and Huang and Zou (2005), where it was shown that chaos may occur in neighborhoods of the origin for the same system. We also present some numeric simulations to demonstrate our theoretical results.     

ON THE CENTER OF GENERALIZED LIENARD SYSTEM

This paper discusses the local and global center of the generalized Lienard system, gives two theorems which extend and improve the relative results of [1]-[3]. Moreover the lemmas in this paper are also useful to the study for the oscillation, stability and the existence of limit cycles of the generalized Lienard system.     

On a problem of Skiba from the Kourovka Notebook

In this paper, we give a positive answer to a recent open problem of Skiba in Kourovka Notebook without using the odd order theorem and other deep theorems. Some of the techniques are improved.     

Stability of Cubic Functional Equation in Three Variables

In this paper, we prove a generalization of Hyers＇ theorem on the sta- bility of approximately additive mapping and a generalization of Badora＇s theorem on approximate ring homomorphism. We also obtain more general stability theorem, which gives stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems in this paper are given following essentially the Hyers-Rassias approach to the stability of the functional equations connected with Ulam＇s problem.     

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.     

BOUNDARY FEEDBACK CONTROL OF ELASTIC BEAM EQUATION WITH STRUCTURAL DAMPING AND STABILITY

In this paper, we consider the partial differential equation of an elastic beam with structuraldamping by boundary feedback control. First, we prove this closed system is well--posed; then weestablish tbe exponential stability for this elastic system by using a theorem whichbelongs to F. L.Huang; finally, we discuss the distribution and multiplicity of the spectrum of this system. Theseresults are very important and useful in practical applications.