Some examples and counterexamples of convergence of σ-algebras and filtrations

We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of σ-algebras to filtrations generated by different discretizations of a single process. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 295–306, July–September, 2000. Translated by V. Mackevičius     

Geometry and Topology of Spaces of Quasi-constant Curvature

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Atoms of Lattices of Supernilpotent and Special Radicals

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Characterizations of Ellipsoid and Sphere

The purpose of this paper is to characterize the ellipsoids in the unimodularaffine space of dimension 3 by the affine principal curvatures and the spheres inthe space of constant curvature of dimension 3 by the principal curvatures. Let A~3 be the unimodular affine space of dimension 3,x:M→A~3 be a closed,locally strongly convex C~5 surface.Denote the equiaffine principal curvatures of Mby λ_1,λ_2 and let     

The Supremum and Infimum of the Set of Fuzzy Numbers and Its Application

In 1986,Goetschel and Voxman represented a fuzzy number by two usualfunctions,which promote further development of the theory and application of fuzzynumbers.For example,we can embbed the fuzzy number space E~1 into a concreteBanach space by using this representation theorem(see[2—4]).     

Structure and Application of Algebraic Spline Curve and Surface

Algebraic-geometry foundations for non-linear spline functions are estab-lished.The structure of algebraic spline curves and surfaces represented by implicitforms are investigated.The necessary and sufficient conditions for smooth connection ofalgebraic curves(or surfaces)are presented.     

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.     

Various disconnectivities of spaces and projectabilities of ?-groups

Arch denotes the category of archimedean ?-groups and ?-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable ?-group and demonstrate that the class of αcc-projectable ?-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W—the category of Arch-objects with designated weak unit and ?-homomorphisms that preserve the weak unit—and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete ?-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).     

The Entropy and Amount of Information of Fuzzy Events

In this paper,the entropy of random experiments with results of fuzzy events and the mutualinformation between two experiments of such kind are defined,therefore,the Shannon entropy andinformation are extended to the fuzzy situations.The principal properties of entropy and informationof fuzzy events are discussed.Among other things,we find that the entropy of a random experimentwith fuzzy outcomes is the sum of random entropy and fuzzy entropy.The connection betweenShannon entropy and De Luca Entropy is established.     

Generllzation of Direct Product and Generation of Sets of Orthogonal Functions

Two new kinds of direct product of matrices are defined. Their properties are investigated. Direct products of matrix and set of continuous functions are also defined. Many complete sets of orthogonal functions, such as those sets given by Walsh [2], Paley [3], Chrestenson[4], and Watari [5], may be generated by these newkinds of direct product. Direct products are also applicable to the generation of sets of piecewise orthogonal functions.     

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on ε of the length of ε-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered in Elezovi?, ?ubrini? and ?upanovi? (2007) [5] in the differentiable case, and related to the box dimension of the orbit.Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity of fixed points and the dependence on ε of the length of ε-neighborhoods of orbits in non-differentiable cases.Applications include in particular Poincaré maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the ε-neighborhood of one orbit of the Poincaré map (for example numerically), and by comparing it to the appropriate scale.     

On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation     

Ramanujan and extensions and contractions of continued fractions

If a continued fraction K _n=1^∞ a _n /b _n is known to converge but its limit is not easy to determine, it may be easier to use an extension of K _n=1^∞ a _n /b _n to find the limit. By an extension of K _n=1^∞ a _n /b _n we mean a continued fraction K _n=1^∞ c _n /d _n whose odd or even part is K _n=1^∞ a _n /b _n . One can then possibly find the limit in one of three ways:

Inversion of Analytic Characteristic Functions and?Infinite Convolutions of Exponential and Laplace Densities

We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, which in turn is an infinite linear combination of exponential or Laplace densities. These results are applied to several examples.     

Geometry of ℑ-Spaces and Properties of Reversing Operators

Mathematical Notes -     

MacMahon partition analysis and the Poincaré series of the algebras of invariants of ternary and quaternary forms

By using MacMahon partition analysis technique, the Poincaré series for the algebras of invariants of the ternary and quaternary forms of small degrees are calculated.     

Uniform and L-convergence of Logarithmic Means of Walsh-Fourier Series

The （NSrlund） logarithmic means of the Fourier series of the integrable function f is： 1/lnn-1∑k=1Sk（f）/n-k, where ln：=n-1∑k=11/k. In this paper we discuss some convergence and divergence properties of this logarithmic means of the Walsh-Fourier series of functions in the uniform, and in the L^1 Lebesgue norm. Among others, as an application of our divergence results we give a negative answer to a question of Móricz concerning the convergence of logarithmic means in norm.