Fuzzy BCK—代数

在集合论中关于差的运算及在命题演算中关于→的运算均具有以下性质: 1) (x*y)*(x*z)≤(z*y); 2) x*(x*y)≤y; 3) x≤x, 4) 0≤x; 5) 若x≤y,y≤x则x=y; 6) x≤y x*y=0. 在集合论中,“*”表示两个集合之差的运算,“≤”表示两个集合之间的包含“(?)”关系,“0”表示空集,“=”表示两个集合相等.在命题演算中,“*”表示两个命题之间的     

R0-代数中一种混合运算的性质及L*系统的完备性

研究了王国俊教授建立的模糊命题演算的形式演绎系统L*及与之在语义上相关的R0-代数,讨论了R0-代数中混合运算():a()b= (a→()b)的性质,并以此为工具利用Petr Hajek证明Lukasiewicz模糊命题演算系统关于语义ΩL完备性的方法证明了L*系统关于语义ΩW的完备性.     

BL代数的等价刻画及更多性质

研究了与H(a)jek的模糊命题演算系统BL相对应的BL代数,提出了仅涉及运算*和→的NBL代数概念并探讨了其有关性质,证明了BL代数与N-BL代数是等价的,进而得到了BL代数更多的性质.     

基础模糊命题演算系统BL~*的改进系统

基础模糊命题演算系统BL*是一个和基础命题演算系统BL相对独立的命题演算系统。命题演算系统L*是系统BL*的扩张,但不是系统BL的扩张。通过对系统BL*及其它模糊命题演算系统的研究,本文对BL*系统进行了修正,进一步改进了BL*系统中的公理体系。     

基础R0-代数与基础L*系统

研究了王国俊教授建立的模糊命题演算的形式演绎系统L^*和与之在语义上相匹配的R0-代数，以及：Petr Hajek建立的模糊命题演算系统BL和BL-代数，提出了基础R0-代数和基础L^*系统的观点，讨论了基础L^*代数与BL代数，基础L^*系统与BL系统之间．的相互关系及相对独立性，讨论了基础L^*系统关于基础风一代数的完备性问题，证明了MV-代数是特殊的基础R0-代数，指出了Lukasiewicz模糊命题演算系统是基础L^*系统的扩张，最后作为基础R0-代数与基础L^*系统的一个应用，证明了L^*系统关于语义Ωw的完备性，并在将模糊命题演算系统中的推演证明转化为相应逻辑代数中的代数运算方面作了一些尝试．     

制约逻辑C_m系统与相干逻辑R系统的等价性

C_m系统是制约逻辑的命题演算系统,是由浦其堇、邹君璧和林邦瑾三人在业师沈有鼎先生的具体指导下给出的,于一九七九年见世。R系统是相干逻辑(relevance logic)的命题演算系统,是美国逻辑学家安特逊和贝尔纳普于1959年建立的。本文的目的是证明这两个系统等价。 相干逻辑是关于相干蕴涵的逻辑。所谓相干蕴涵就是顾及命题在内容上的联系的一种联结词,它反映了可用命题变项之共同出现表示的内容上的联系;也就是说,A相干蕴涵B的必要条件是:     

模糊逻辑(ξ)*和NM的公理系统的简化

从王国俊教授提出的模糊命题演算形式系统(ξ)*、(ξ)0*的性质以及它们与F.Esteva和L.Godo提出的MTL、IMTL和NM的关系出发,借助代数方法证明了(ξ)*和NM中的公理(L10*)和(NM)可以由一条只含一个命题变元且形式更为简单的公理模式(L*w)代替.这一结果简化了(ξ )*和NM的公理系统.     

MV代数定义的蕴涵简化形式

通过对MV代数和Lukasiewicz命题演算系统的研究,我们对MV代数的定义进行了简化,并讨论了MV代数和其它代数之间的关系。主要结果是:(1)从蕴涵角度出发,给出了MV代数的两种简化定义;(2)提出了弱格蕴涵代数的概念,并证明了它与BR0代数等价;(3)证明了弱格蕴涵代数是正则Fuzzy蕴涵代数。     

中介逻辑的命题演算系统MP的语义解释及可靠性、完备性

给出中介逻辑的命题演算系统MP~*(见〔1〕)的语义解释如下: 1.MP~*的语义解释是指所有原子命题到集合T={0,l,～}的一个映射,记为σ,也称σ为一个指派。 2.合式公式(以下简称公式)A在σ下的值记为σ(A),被归纳定义为: (i)若A是原子命题,则σ(A)已被定义;     

原始递归算术的新系統(Ⅰ)

引言递归算术可以说是不借助于逻辑(命题演算,谓词演算及集合论),而直接建立关于自然数的公理系统的理论,由于它不借助于集合论,故它不受集合论誖论的牵累,其不矛盾性应该无容怀疑。由于它不借助于命题演算及谓词演算,它是第一个无须借助于逻辑的数学系统,它的成就证明了逻辑主义者那么勉强地把数学化归于逻辑(包括集合论),实在     

Closed‐form approximations for spread options in Lévy markets

We provide new closed‐form approximations for the pricing of spread options in three specific instances of exponential Lévy markets, ie, when log‐returns are modeled as Brownian motions (Black‐Scholes model), variance gamma processes (VG model), or normal inverse Gaussian processes (NIG model). For the specific case of exchange options (spread options with zero strike), we generalize the well‐known Margrabe formula (1978) that is valid in a Black‐Scholes model to the VG model under a homogeneity assumption.     

Translations from natural deduction to sequent calculus

Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut‐free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut‐free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination.     

Investigating statistical approximations with computer graphics

This paper describes the use of interactive computer graphics software to explore the notion of the approximation of one statistical distribution by another. In particular, the authors illustrate how such programs can be used to investigate the normal approximation to the binomial distribution, the Poisson approximation to the binomial distribution, the convergence of t‐distributions to the normal distribution, and the limiting behaviour of the chi‐squared distribution. The discussions include possible methods using such programs, either in a classroom setting on a demonstration basis or on an individual student basis for independent study in a computer laboratory setting.     

Numerical range and functional calculus in Hilbert space

We prove an inequality related to polynomial functions of a square matrix, involving the numerical range of the matrix. We also show extensions valid for bounded and also unbounded operators in Hilbert spaces, which allow the development of a functional calculus.     

On Commutative and Nonassociative Syntactic Calculi and Categorial Grammars

Two axiomatizations of the nonassociative and commutative Lambek syntactic calculus are given and their equivalence is proved. The first axiomatization employs Permutation as the only structural rule, the second one, with no Permutation rule, employs only unidirectional types. It is also shown that in the case of the Ajdukiewicz calculus an analogous equivalence is valid only in the case of a restricted set of formulas. Unidirectional axiomatizations are employed in order to establish the generative power of categorial grammars based on the nonassociative and commutative Lambek calculus with product. Those grammars produce CF-languages of finite degree generated by CF-grammars closed with respect to permutations.     

Normal form of derivations in the nonassociative and commutative lambek calculus with product

We show that derivations in the nonassociative and commutative Lambek calculus with product can be transformed to a normal form as it is the case with derivations in noncommutative calculi. As an application we obtain that the class of languages generated by categorial grammars based on the nonassociative and commutative Lambek calculus with product is included in the class of CF-languages. MSC: 68Q50, 03D15, 03B65.     

Mathematical design of a highway exit curve

A highway exit curve is designed under the assumption that the tangential and normal components of the acceleration of the vehicle remain constant throughout the path. Using fundamental principles of physics and calculus, the differential equation determining the curve function is derived. The equation and initial conditions are cast into a dimensionless form first for universality of the results. It is found that the curves are effected by only one dimensionless parameter which is the ratio of the tangential acceleration to the normal acceleration. For no tangential acceleration, the equation can be solved analytically yielding a circular arc solution as expected. For nonzero tangential acceleration, the function is complicated and no closed-form solutions exist for the differential equation. The equation is solved numerically for various acceleration ratios. Discussions for applications to highway exits are given.     

Boundary conditions at a thin membrane for normal diffusion,classical subdiffusion,and slow subdiffusion processes

We consider three different diffusion processes in a system with a thin membrane: normal diffusion, classical subdiffusion, and slow subdiffusion. We conduct the considerations following the rule: If a diffusion equation is derived from a certain theoretical model, boundary conditions at a thin membrane should also be derived from this model with additional assumptions taking into account selective properties of the membrane. To derive diffusion equations and boundary conditions at a thin membrane, we use a particle random walk model in one-dimensional membrane system in which space and time variables are discrete. Then we move from discrete to continuous variables. We show that the boundary conditions depend on both selective properties of the membrane and a type of diffusion in the system.     

A classification of Taylor towers of functors of spaces and spectra

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad.     

A characterization of those categories whose internal logic is Hilbert's ε-calculus

We characterize categories whose internal logic is Hilbert's ε-calculus as those categories which have a proper factorization system satisfying the axiom of choice and in which every non-initial object is injective. We provide an example of such a category where the law of excluded middle is not valid.