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The aim of this paper is to compare the two more standard geometrical formulations of gauge systems: the so-calledpresymplectic formulation and the formulation bygroup actions. We summarize the main features involved in them and prove that, at least locally, every presymplectic formulation can be interpreted in terms of group actions. The converse is also proved.

In this paper our aim is to give refinements of Jensen?s type inequalities for the convex function defined on the co-ordinates of the bidimensional interval in the plane.

In this paper we study a class of sub-spaces of loop spaces which have appeared in the calculus of variations. Generalizing a result of Smale, we show that the space of loops tangent to a distribution satisfying Hδrmander's condition is weakly homotopic to the space of all loops. If the distribution is fat, we resolve the end point map from the space of horizontal paths. This resolution has two applications: (1) the proof that the cut-locus on an analytic fat Carnot-Caratheodory manifold is sub-analytic; (2) a study of the singularity of the horizontal loop space. At the end we study the geometry of left-invarian t Carnot-Caratheodory metrics on fact nilpotent groups.

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.

In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested.

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For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are singular, singularity types of pedal curves depend on singularity types of the n-th curvature function germs and the locations of pedal points. In this paper, we investigate sigularity types of pedal curves in such cases.

The main result of this paper is a quantified version of Ingham's Tauberian theorem for bounded vector-valued sequences rather than functions. It gives an estimate on the rate of decay of such a sequence in terms of the behaviour of a certain boundary function, with the quality of the estimate depending on the degree of smoothness this boundary function is assumed to possess. The result is then used to give a new proof of the quantified Katznelson-Tzafriri theorem recently obtained in [21].

THERE appeared in your pages some three years ago (vol. xv. p. 515) an article of mine “On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent.” In a succeeding number a correspondent (Mr. Monro, of Barnet) propounded a query which may be shortly stated as follows:— “How does it happen that the perpendicular on a geodesic from a point moving along another geodesic changes sign without passing through either the value zero (0) or the value infinity (∝)?” The problem here suggested is a peculiarly knotty one. In the case of the Euclidian plane the perpendicular of course changes sign by passing through the value ∝, while in the case of a spherical surface it is equally obvious that the perpendicular passes through zero, since the two geodesies intersect twice. But what are we to say of the strange hybrid surface which formed the subject-matter of my paper? Your correspondent appeared to insinuate that the problem was insoluble, and that the definition of the surface must therefore involve a logical contradiction. For a while I was greatly puzzled by this unforeseen difficulty, but after a little thought came to the conclusion that the perpendicular changes sign by passing through the value l/2√−1, where l is positive and represents the absolute length of a complete geodesic. In other words, I conceived that the sign of the perpendicular changed from + to − by a continuous variation of the real numbers a and b in the complex number a + b√- 1. I conceived a to diminish continuously till, passing through o, it became − -a, while b at the same time increased with simple harmonic motion from o to a maximum, and then decreased from a maximum to o.