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In this paper, by using umbral calculus and umbral algebra methods, we derive several interesting identities and relations related to the modified and unification of the Bernoulli, Euler and Genocchi polynomials and numbers and the generalized (β-) Stirling numbers of the second kind. We also give some applications and remarks related to these numbers and polynomials. [ABSTRACT FROM AUTHOR]

Fatou's lemma is a classic fact in real analysis stating that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit. This paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space. The necessary and sufficient condition, under which this inequality holds for a sequence of finite measures converging in total variation, is provided. This statement is called the uniform Fatou lemma, and it holds under the minor assumption that all the integrals in the inequality are well-defined. The uniform Fatou lemma improves the classic Fatou lemma in the following directions: the uniform Fatou lemma states a more precise inequality, it provides the necessary and sufficient condition, and it deals with variable measures. Various corollaries of the uniform Fatou lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou lemma may indeed provide a more accurate inequality than the classic Fatou lemma; (b) the uniform Fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence. [ABSTRACT FROM AUTHOR]

In this article we discuss a variational aspect of a system of two equations arising in photometric stereo. Assuming that this system has two distinct solutions, we construct a minimizing sequence for any function lying on a segment joining these two solutions. In general, the system in question may have one, two, four or more than four solutions. [ABSTRACT FROM AUTHOR]

In this paper we have introduced the concept of geometric convergence of a sequence and determined the necessary and sufficient condition under which convergence follows from geometric convergence of a sequence in multiplicative sense. Corollaries allow this condition to be replaced by multiplicative analogues of Schmidt type slow oscillation condition or Landau type two-sided condition. [ABSTRACT FROM AUTHOR]

We develop integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite-dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano–Jordan-like integration domains. [ABSTRACT FROM AUTHOR]

Tangent fibrations generate a "multi-oored tower", while raising from one of its oors to the next one, one practically reiterates the previously performed actions. In this way, the "tower" admits a ladder-shaped structure. Raising to the first oors sufaces for iteratively performing the subsequent steps. The paper mainly studies the tangent functor. We describe the structure of multiple vector bundle which naturally appears on the oors, tangent maps, sector-forms, the lift of vector elds to upper oors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant di erentiation. Finally, we will point out within the oors special subbundles - the osculating bundles, which play an essential role in classical theories. [ABSTRACT FROM AUTHOR]

We obtain a vector-valued subordination principle for gα,gβ-regularized resolvent families which unified and improves various previous results in the literature. As a consequence, we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus, and stable Lévy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results. [ABSTRACT FROM AUTHOR]

This paper presents the dynamic response of Bernoulli-Euler homogeneous isotropic fractionally-damped simplysupported beam. The beam is appendaged to single-degree-of-freedom (SDOF) fractionally-damped N-oscillators and subjected to a moving load with a uniform velocity. The damping characteristics of the beam and oscillators are formally described in terms of fractional derivative of an arbitrary order. In the analysis, the initial conditions are assumed to be homogeneous, and the Laplace transform is combined with the used decomposition method to find the solution of the handled problem. Subsequently, curves are plotted to show the vibration of the utilized beam under different sets of parameters including different orders of fractional derivatives for both of the beam and oscillators. The numerical results obtained in this paper show that the dynamic response decreases as the number of absorbers attached to the beam increases, as both the damping-ratios of absorbers and beam increase the dynamic response decreases, and there are some critical values of fractional derivatives which are different from unity at which the beam has less dynamic response than that obtained for the full-order derivatives model. In addition, the obtained results show very good agreement with special case studies that were previously published in literature. [ABSTRACT FROM AUTHOR]

Cauchy summation formula plays a central role in application of character calculus to many problems, from AGT-implied Nekrasov decomposition of conformal blocks to topological-vertex decompositions of link invariants. We briefly review the equivalence between Cauchy formula and expressibility of skew characters through the Littlewood-Richardson coefficients. As not-quite-a-trivial illustration we consider how this equivalence works in the case of plane partitions - at the simplest truly interesting level of just four boxes. [ABSTRACT FROM AUTHOR]