The mathematics of the grand unified theory

This paper surveys the mathematics of GUT and highlights the author’s new real number system which is a continuum, non-Archimedean and non-Hausdorff, but its subspace of decimals is countably infinite, discrete, Archimedean and Hausdorff. The paper also proves Goldbach’s conjecture in and provides an overview of GUT and qualitative and computational models of many of the presently ill-defined physical concepts, e.g., gravity.     

The unified approach in a semi-infinite atmosphere

The unified approach of invariant imbedding is extended to the case of a semi-infinite atmosphere. The integral equation satisfied by the reflection function from a semi-infinite atmosphere is derived by virtue of the approach. For stationary and non-stationary radiation fields, the equations obtained in this article coincide with those already obtained.     

The short range expansion

Let V_i be short range potential and λ_i(ε) analytic functions. We show that the Hamiltonians H_ε = −Δ + ε⁻²∑_{i = l}ⁿλ_i(ε)V_i((· − x_i)/ε converge in the strong resolvent sense to the point interactions as ε → 0, and if V_i have compact support then the eigenvalues and resonances of H_ε, which remains bounded as ε → 0, are analytic in ε in a complex neighborhood of zero. We compute in closed form the eigenvalues and resonances of H_ε to the first order in ε.     

Hybrid unified theory

We shall investigate, in this paper, theories related to mind, consciousness, life, evolution, existence, non-existence, cosmology, quantum fields, quantum jump, Big Bang, and higher reality. A hybrid unification between ancient and modern discoveries is discussed with a view to shedding some light on the existing theories and strengthening them.     

The further unified theory for modifications of g-closed sets

We introduce a new set called mng-closedwhich is defined on a set with two families of sets satisfying some minimal conditions. This set enables us to unify modifications of g-closed sets due to Levine [19].      

THE CONVERGENCE FOR NODAL EXPANSION METHOD

In this paper,we prove the convergence of the nodal expansion method,a new numerical method for partial differentisl equations and provide the error estimates of approximation solution.     

The Euler-Maclaurin expansion and finite-part integrals

In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional. Received June 11, 1997 / Revised version received December 15, 1997     

The normodistributive expansion of a category

Summary Starting from a categoryC, verifying certain hypotheses, we define a categoryC ^♯ called normodistributive expansion ofC. The categoryC ^♯is orthoquaternary, then canonical isomorphisms between subquotients ofC ^♯ are composable. This is a generalization of the construction of the distributive expansion ε^♯ of an exact catagory ε, which can be applied to a larger class of categories (including the category of groups).

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Riassunto Partendo da una categoriaC, che verifica certe ipotesi, si costruisce una categoriaC ^♯ detta espansione normodistributiva diC. La categoriaC ^♯ risulta essere orto-quaternaria, quindi gli isomorfismi canonici fra i suoi subquozienti sono componibili. Si è così generalizzato il passaggio da una categoria esatta ε alla sua espansione distributiva ε^♯, al caso di una famiglia più vasta di catagorie comprendente, fra le altre, quella (non esatta) dei gruppi.

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Lavoro eseguito nell’ambito dei gruppi di ricerca matematici del C.N.R.     

The asymptotic expansion of the Feynman integral

We obtain an expansion of in powers of a small parameter, wherex _t is a random process satisfying the stochastic differential equation .Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 28–37, 1986.     

The Pontryagin maximum principle and a unified theory of dynamic optimization

The Pontryagin maximum principle is the central result of optimal control theory. In the half-century since its appearance, the underlying theorem has been generalized, strengthened, extended, proved and reinterpreted in a variety of ways. We review in this article one of the principal approaches to obtaining the maximum principle in a powerful and unified context, focusing upon recent results that represent the culmination of over thirty years of progress using the methodology of nonsmooth analysis. We illustrate the novel features of this theory, as well as its versatility, by introducing a far-reaching new theorem that bears upon the currently active subject of mixed constraints in optimal control.     

The asymptotic expansion of certain canonical integrals

Integrals of the form $\text{I(λ) = ∝}{}_{\mathrm{R}}{}^2\text{?(}\text{x}\text{?}\text{)}\text{exp}\text{{iλφ(}\text{x}\text{?}\text{)} d}\text{x}\text{?}$, common in diffraction theory, are usually evaluated asymptotically for large λ using the stationary phase technique. At caustic points the classical technique is not applicable. A computation-oriented algorithm is developed for determining the asymptotic expansion of such integrals with $\text{φ(}\text{x}\text{)}$ a Thom umbilic, the canonical form appropriate at caustic points for most cases of physical interest.     

The complete asymptotic expansion for Baskakov operators

In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn （f;x） in the form of all coefficients of n^-k, k = 0, 1... being calculated explic- itly in terms of Stirling number of the first and second kind and another number G（i, p）. As a corollary, we also get the Voronovskaja-type result for the operators.     

The expansion theorem for the deviation integra

The so-called deviation integral (functional) describes the logarithmic asymptotics of the probabilities of large deviations for random walks generated by sums of random variables or vectors. Here an important role is played by the expansion theorem for the deviation integral in which, for an arbitrary function of bounded variation, the deviation integral is represented as the sum of suitable integrals of the absolutely continuous, singular, and discrete components composing this function. The expansion theorem for the deviation integral was proved by A. A. Borovkov and the author in [9] under some simplifying assumptions. In this article, we waive these assumptions and prove the expansion theorem in the general form.     

Towards a unified bifurcation theory

Bifurcation theories for the instability of slowly evolving systems have been developed in various disciplines, and a first step is here taken towards some desirable unification. A modern account of the authors' general branching theory for discrete systems is first presented, some new features being the introduction of principal imperfections and the delineation of the important semi-symmetric points of bifurcation. This theory, embedded in a perturbation approach ideal for quantitative analysis, is complementary to the far-reaching qualitative catastrophe theory of René Thom which offers a profound topological classification of instability phenomena. For this reason, we present here a detailed correlation of the two theories. Also presented in the paper is a survey of some fields of application ranging from classical fields such as hydrodynamics, through thermodynamics, crystallography and cosmology, to the newer domains of biology and psychology.