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This monograph is devoted to the proof of two related results. The first one asserts that if The second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform

In this paper we introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as “tractable cases” of a general theory. As an outcome of this, we provide extensions of known results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be “almost” studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions. Supplementary Information The online version contains the Armenian language version of the article available at  https://doi.org/10.1007/s10958-023-06674-w .

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.

We establish the fundamental factorization theorem for operators between Banach spaces un der the condition of the variable exponent p(·)-completely boundedness. Assuming X, Y and ˜X, ˜Y be separable Banach spaces, we show that the variable exponent p(·)-completely bounded mapping Λ : S → LB (˜X, ˜Y) , S ⊂ LB(X, Y) can be presented in the form Λ(a) = θ2ˆπ(a)θ1, where θ1 : ˜X → V/N, θ2 : ˘V/˘N → ˜Y,and ˆπ(a) : V/N → ˘V/ ˘N suchthat∥θ1∥∥θ2∥ ≤ ∥Λ∥p(·)−cb ,thereverse is also true.

In this study, we demonstrate the existence of solutions to an anisotropic elliptic problem featuring a singularity, where the non-homogeneous term is characterized by a non-negative Radon measure μ. The model problem is { − ∑ i = 1 N ∂ i ( | ∂ i u | p i − 2 ∂ i u ) = f ( e u − 1 ) γ + μ in   Ω , u = 0 on   ∂ Ω , u > 0 in   Ω , in Ω, u = 0 on ∂Ω, u > 0 in Ω, where Ω is a bounded domain in ℝ N , γ > 0, f ∈ L¹(Ω) and 2 < p₁ ≤ p₂ ≤ . . . ≤ pN . The primary goal of this work is to establish the existence of solutions based on the values of γ.

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux, more specifically, for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x = x ( t ). We prove that this problem admits global Radon measure solutions for all kinds of initial data. The over-compressing condition on the discontinuity x = x ( t ) is not enough to ensure the uniqueness of the solution. However, there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x = x ( t ) + 0, in addition to the full adhesion condition on its left-side. As an application, we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas. In particular, we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas. This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

We investigate the wave structure and new phenomena of the Riemann problems of isentropic compressible Euler equations with discontinuous flux in momentum caused by different equations of states, including pressureless flow and Chaplygin gas. Specifically, we focus on solutions within the class of Radon measures. To resolve the discontinuous flux, we introduce a delta shock that admits mass concentration between the pressureless flow on the left and Chaplygin gas on the right. By exploring both the classical and singular Riemann problems, we find that a global delta shock solution exists, satisfying the over-compressing condition. This finding is a generalization of classical theories on Riemann problems. In particular, we demonstrate that a vacuum left state and right Chaplygin gas can always be connected by a global delta shock satisfying the over-compressing condition. For singular Riemann problems, influenced by initial velocity, we observe that for some initial data, the composite wave comprises contact discontinuities, vacuum, and a local delta shock satisfying the over-compressing condition. Through a detailed analysis of the intricate interactions between contact discontinuities and delta shocks, we show that this local solution can be extended globally.

We develop elements of a general dilation theory for operator-valued measures. Hilbert space operator-valued measures are closely related to bounded linear maps on abelian von Neumann algebras, and some of our results include new dilation results for bounded linear maps that are not necessarily completely bounded, and from domain algebras that are not necessarily abelian. In the non-cb case the dilation space often needs to be a Banach space. We give applications to both the discrete and the continuous frame theory. There are natural associations between the theory of frames (including continuous frames and framings), the theory of operator-valued measures on sigma-algebras of sets, and the theory of continuous linear maps between

For subharmonic functions in there is an associated Radon measure that is used to represent locally as an integral up to an additive harmonic function. We prove that the total measure is finite if and only if the subharmonic function has harmonic majorant outside a compact set. However, this relationship does not hold in higher dimensions.