A COMPREHENSIVE PROOF OF THE GREENBERGER-HORNE-ZEILINGER THEOREM FOR THE FOUR-QUBIT SYSTEM

Greenberger-Horne-Zeilinger(GHZ)theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those ob- servables assume values that refute the attempt to assign values only required to have them by the local realism of Einstein,Podolsky,and Rosen(EPR).It is known that for a three-qubit system.there is only one form of the GHZ-Mermin-like argument with equiva- lence up to a local unitary transformation,which is exactly Mermin's version of the GHZ theorem.This article for a four-qubit system,which was originally studied by GHZ,the authors show that there are nine distinct forms of the GHZ-Mermin-like argument.The proof is obtained using certain geometric invariants to characterize the sets of mutually commuting nonlocal spin observables on the four-qubit system.It is proved that there are at most nine elements(except for a different sign)in a set of mutually commuting nonlocal spin observables in the four-qubit system,and each GHZ-Mermin-like argument involves a set of at least five mutually commuting four-qubit nonlocal spin observables with a GHZ state as a common eigenstate in GHZ's theorem.Therefore,we present a complete construction of the GHZ theorem for the four-qubit system.     

关于严格蕴涵系统的布尔值模型（续）

The reference [4] proved the consistency of S1 and S2 among Lewis' five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S3, S4 and S5 in Lewis five strict implication systems is not decided. This paper makes use of the properties: (1) the equivalence of the modal systems S3 and P3, S4 and P4; (2) the modal systems P3 and P4 all contained the modal axiom T(□p→p); (3) the modal axiom T is correspondence to the reflexive property in VB. Hence, the paper proves: (a) ‖As31‖=1; (b) ‖As41‖=1; (c) ‖As51‖=1 in the model (V^B,R,‖ ‖)(where B is a complete Boolean algebra, R is reflexive property in V^B).Therefore, the paper finally proves that the Boolean-valued model V^B of the ZFC axiom system in set theory is also a Boolean-valued model V^B of the ZFC axiom system in set theory is also a Boolean-valued model of Lewis the strict implication system S3, S4 and S5.     

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton(gμHS2) system with viscous terms.Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative(gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative(gμHS2) system, we show that the semi-group of the solution operator {S(t)}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S(t)}t≥0 possesses a global attractor in the Sobolev space H~2(S) × H~2(S).     

On a Boolean-valued Model of the Strict Implication System(Continuous)

The reference [4] proved the consistency of S1 and S2 among Lewis' five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S3, S4 and S5 in Lewis' five strict implication systems is not decided. This paper makes use of the properties: (1) the equivalence of the modal systems S3 and P3, S4 and P4; (2) the modal systems P3 and P4 all contained the modal axiom T(□p → p); (3) the modal axiom T is correspondence to the reflexive property in VB. Hence, the paper proves: (a) ‖As31‖ = 1; (b) ‖AS41‖ = 1; (c) ‖AS5l‖ = 1 in the model (where B is a complete Boolean algebra, R is reflexive property in VB). Therefore, the paper finally proves that the Boolean-valued model VB of the ZFC axiom system in set theory is also a Boolean-valued model of Lewis' the strict implication system S3, S4 and S5.     

INTERACTION OF STRONG AND WEAK SINGULARITIES FOR HYPERBOLIC SYSTEM OF CONSERVATION LAWS IN MULTIDIMENSIONAL SPACE

In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.     

Deductive Reasoning

We use inductive reasoning in everyday life. Many of the conjectures that come from this kind of thinking seem highly likely, although we can never be absolutely certain that they are true. Another method of reasoning, called deductive reasoning, or deduction, can be used to prove that some conjectures are true. Deductive reasoning is the process of proving a specific conclusion from one or more general statements. A conclusion that is proved true by deductive reasoning is called a theorem.     

THE LIOUVILLE THEOREM INVOLVING QUANTUM EFFECT

In this article we have shown, if the wave packets are used to describe the dynamical states of particles in a many-particle system, we can get a set of Langevintype Eq. (4.1) instead of the classical canonical eqs. of Hamilton (1.1). At the same time a diffusion-type Liouville theorem involving quantum effect (4.9) is resulted instead of the classical Liouville Eq. (1.3). it is shown that the diffusion-type Liouville eq. should cause the phase mixing in phase space and the entropy increasing in time for an isolated system.     

THE OPTIMAL STOPPING PROBLEM OF A CONTINUOUS PARAMETER PROCESS

Suppose X = (Xr, Fr, t ∈ R+) be an optional reward process with ( Fr) satisfying usual conditions. In this paper, we correct the proof of existence about Snell envelope in [4] and the proof of an important lemma (Lemma 4. 6) in [5], and give a proof of existence about Snell envelope under certain conditions, i. e. EZx- < ∞ and Z is upper-semi-continuous on the right (USCR) or there is a stopping rule (SR)τ ≤σ such that EZx-∞ for any stopping rule σ . At the same time, we prove a four-repeated limit theorem when Z is continuous on the right. The character and the uniqueness of the optimal stopping time (OST) or optimal stopping rule (OSR) are discussed.     

ON THE EXISTENCE OF PRODUCT STOCHASTIC MEASURES

In this paper, we prove that the existence of product stochastic measures depends on the axiomsystem of set theory: If one aecepts the axiom of choice, the answer is negative, and we give acounter-example where the product stochastic measure doesn't exist; but in the Solovay model (onekind of set theory which refuses the axiom of choice),the answer is positive, and we give a proof.     

解线约束凸规划的势下降算法

A potential reduction algorithm is proposed for optimization of a convex function subject to linear constraints. At each step of the algorithm,a system of linear equations is solved toget a search direction and the Armijo‘s rule is used to determine a stepsize. It is proved that thealgorithm is globally convergent. Computational results are reported.     

格值命题逻辑系统L（X）（Ⅱ）

本文讨论以格蕴涵代数为真值域的格值命题逻辑系统Ｌ（Ｘ）的语法问题，给出了系统的公理和推演规则，并证明了系统的可靠性定理，演绎定理及协调性定理。     

命题公式集F(S)的基于R0-算子的16类分划

利用R0-蕴涵算子对命题公式集F(S)进行分类,得出了F(S)的—个16类分划,并证明了这种分类关于非运算是同余分类.最后讨论了各类关于MP运算与HS运算的封闭性.     

利用插值法证明推广的柯西中值定理

借助插值的思想 ,首先给出函数 f( x)的泰勒公式的行列式表达式 ,推广了柯西中值定理 .据此拉格朗日中值定理、泰勒公式、罗必塔法则均是该结论的推论 ,从而对经典的中值定理、泰勒公式、罗必塔法则给出了统一证明     

命题公式集F(S)的基于R0-算子的16类分划

利用R₀-蕴涵算子对命题公式集F(S)进行分类,得出了F(S)的一个16类分划,并证明了这种分类关于非运算是同余分类.最后讨论了各类关于MP运算与HS运算的封闭性.     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

Two systems for proving tautologies,based on the split method

We consider two calculi in which all propositional tautologies, and only these, are provable. Despite their simple structure (each calculus has one axiom and one rule of inference), we may obtain linear estimates for the length of deduction in them for various frequently occurring classes of formulas, whose tautology is recognized in polynomial time. We study the characteristics of formulas which influence the length of deduction in these systems. In particular, one of these characteristics is the degree of symmetry defined by automorphisms of the formulas.     

Representations of polyadic-like equality algebras

Assuming the usual finite axiom schema of polyadic equality algebras, axiom (P10) is changed to a stronger version. It is proved that infinite dimensional, polyadic equality algebras satisfying the resulting system of axioms are representable. The foregoing stronger axiom is not given with a first order schema. The latter is to be expected knowing the negative results for the Halmos schema axiomatizability of the representable, infinite dimensional, polyadic equality algebras. Furthermore, Halmos’ well-known classical theorem that “locally finite polyadic equality algebras of infinite dimension α are representable” is generalized for locally-\({\mathfrak{m}}\) polyadic equality algebras, where \({\mathfrak{m}}\) is an arbitrary infinite cardinal and \({\mathfrak{m}}\) < α. Also, a neat embedding theorem is proved for the foregoing classes of polyadic-like equality algebras (a neat embedding theorem does not exists for polyadic equality algebras).     

A note on Spector's quantifier-free rule of extensionality

In this note we show that the so-called weakly extensional arithmetic in all finite types, which is based on a quantifier-free rule of extensionality due to C. Spector and which is of significance in the context of G?del"s functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for Π⁰ ₁-axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept fixed) was known. Received: 11 March 1999 / Published online: 21 December 2000     

The equivalence of Side-Angle-Side with Side-Side-Side and the general triangle inequality in the absolute plane

We show that if we start with an absolute plane, remove Side-Angle-Side as an axiom, and replace it with Side-Side-Side together with a weakened version of the triangle inequality as new axioms, then the new resulting axiom system is also an absolute plane. In particular, we show that Side-Angle-Side still holds in the new axiom system. In addition, we give a new proof of the two-circle theorem which does not depend on Side-Angle-Side, but instead uses Side-Side-Side and the weakened version of the triangle inequality.