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In this paper, the notion of slice Radon measures is introduced and some of their fundamental properties are established. Based on it, the classes of functions associated to slice Radon measures are studied in which some basic notions of classes of functions are proposed and their corresponding properties and relations are established depending on quaternionic Cauchy integral formula and slice regular function theory. Moreover, several examples are constructed towards these classes of quaternionic functions, and then the uniqueness related to the slice Radon measure is proved and the Radon–Riesz theorem in quaternionic version is established by which the point convergence is analyzed. On the basis of this, the quaternionic Hirsch functional calculus via quaternionic non-negative operator is developed in which the product formula under intrinsic slice hyper-holomorphic conditions is obtained and the stability of different types of quaternionic operator functions under composition is discussed. Some challenges associated with quaternionic Radon measures and the Hirsch functional calculus in the noncommutative setting are resolved. [ABSTRACT FROM AUTHOR]

Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as -convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of -convolution. [ABSTRACT FROM AUTHOR]

We provide a general expression of the Haar measure—that is, the essentially unique translation-invariant measure—on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on SO (2 , Q p) is obtained by a direct application of our general formula. As for SO (3 , Q p) and SO (4 , Q p) , instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field Q p and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory. [ABSTRACT FROM AUTHOR]

Due to the inherent ambiguity of a polygonal approximation of the surface, it is difficult to compute the major curvatures of a smooth surface with sufficient differentiability when just a polygonal approximation of the surface is given. A variety of techniques-based approaches have been presented in the past for addressing this issue. We examine the challenge of estimating curvature from a smooth surface samples. The definition of the curvature tensor for polyhedral surfaces is derived using the notion of normal cycles. This definition is very direct and novel formula in the spirit of 2003. Our concept goes beyond that of 2003, offering a more solid mathematical underpinning. When applied to a polyhedral approximation of a smooth surface, it produces a reliable and efficient approach for estimating curvature. In addition, we limit the discrepancy between the calculated curvature and the smooth surface curvature for restricted Delaunay triangulations. As a demonstration, we will run feature detection procedures on a 3D model later. [ABSTRACT FROM AUTHOR]

We consider the inviscid Burgers equation with force ∂ t u + ∂ x (u 2 / 2) = ν , where the discontinuities of initial datum u 0 are interpreted as force sources. Thence, ν is the force of shocks in a sticky dynamics of (paradoxically) non accelerated particles, whose the mass distribution field is ∂ x u . The force has its own dynamics of density field η = u - w (the experienced impulsion), where w denotes the sticky particle velocity field. Along the sticky particle trajectory t ↦ X t , the processes t ↦ η (X t , t) , u (X t , t) , w (X t , t) are backward martingales. [ABSTRACT FROM AUTHOR]

We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.

Integration logic is a logical (model theoretic) framework for studying measure and probability structures by logical means. The Daniell–Stone theorem for Daniell integrals and Riesz representation theorem are two important classical results in analysis concerning existence of measures with certain properties. Many proofs of the Riesz representation theorem can be found in the literature from elementary ones which use ordinary techniques from measure theory and analysis, to more sophisticated ones that highly employ techniques from other fields of mathematics such as nonstandard analysis. There are also a few proofs for the Daniell–Stone theorem. This paper pursues two main goals. One is to give novel and uniform proofs for both these theorems using some ideas from logic. The second and maybe even more important one is to try to reveal more the power of the logical methods in mathematics in particular measure theory, and make stronger connections between analysis and logic. Our proofs are uniform in the sense that they are based on the same general idea and rely on the application of the same technical tool from logic to measure theory, namely the logical compactness theorem. We use the setting of "integration logic" mentioned above, elaborate it and use its expressive power and a version of the compactness theorem holding in it to prove our results. The paper is mostly written for general mathematicians, in particular the people active in logic or analysis as the main audience. It is self-contained and the reader does not need to have any advanced prerequisite knowledge from logic or measure theory. [ABSTRACT FROM AUTHOR]

When X is locally compact, a quasi-integral (also called a quasi-linear functional) on C c (X) is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple quasi-integrals, i.e., quasi-integrals whose corresponding compact-finite topological measures assume exactly two values. We present equivalent conditions for a quasi-integral to be simple or almost simple. We give a criterion for repeated quasi-integration (i.e., iterated integration with respect to topological measures) to yield a quasi-linear functional. We find a criterion for a double quasi-integral to be simple or almost simple. We describe how a product of topological measures acts on open and compact sets. We show that different orders of integration in repeated quasi-integrals give the same quasi-integral if and only if the corresponding topological measures are both measures or one of the corresponding topological measures is a positive scalar multiple of a point mass. [ABSTRACT FROM AUTHOR]

This work is devoted to the generalization of previous results on Gleason measures to complex Gleason measures. We develop a functional calculus for complex measures in relation to the Nemytsky operator. Furthermore we present and discuss the interpretation of our results with applications in the field of quantum mechanics. Some concrete examples and further extensions of several theorems are also presented.

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative continuous vanishing at infinity function; and it produces a signed deficient topological measure if we integrate a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure, and their corresponding non-linear functionals are Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure μ that assumes finitely many values, there is a function f such that ∫ X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0, but ∫ X $\begin{array}{} \int\limits_X \end{array}$ (–f) dμ ≠ 0. We present different criteria for ∫ X $\begin{array}{} \int\limits_X \end{array}$ f dμ = 0. We also prove some convergence results, including a Monotone convergence theorem. [ABSTRACT FROM AUTHOR]

We show that the sequential closure of a family of probability measures on the canonical space of càdlàg paths satisfying Stricker's uniform tightness condition is a weak∗ compact set of semimartingale measures in the dual pairing of bounded continuous functions and Radon measures, that is, the dual pairing from the Riesz representation theorem under topological assumptions on the path space. Similar results are obtained for quasi- and supermartingales under analogous conditions. In particular, we give a full characterisation of the strongest topology on the Skorokhod space for which these results are true. [ABSTRACT FROM AUTHOR]

Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol'pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way. [ABSTRACT FROM AUTHOR]

DT-measures generalize the topological measures of Aarnes. We continue to study an integral of a real-valued continuous function on compact Hausdorff space with respect to a DT-measure and investigate the following: multiplicativity of this integral, $$w*$$ -topology on the family of DT-measures and density of various subfamilies of this space, and also integral as a set function. [ABSTRACT FROM AUTHOR]