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We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in Kim and Pass (SIAM J Math Anal 46:1538–1550, 2014; SIAM J Math Anal 46:1538–1550, 2014). In addition, it is satisfied by the special cost functions in our earlier work (Pass and Vargas-Jiménez in SIAM J Math Anal 53:4386–4400, 2021; Monge solutions and uniqueness in multi-marginal optimal transport via graph theory. arXiv:2104.09488, 2021), when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here, including a class of examples arising in robust risk management problems; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals. [ABSTRACT FROM AUTHOR]

Let A be a general expansive matrix and X a ball quasi-Banach function space on Rn, which supports both a Fefferman-Stein vector-valued maximal inequality and the boundedness of the powered Hardy-Littlewood maximal operator on its associate space. The authors first introduce the Hardy space HXA (Rn), associated with both A and X, via the nontangential grand maximal function, and then establish its various equivalent characterizations, respectively, in terms of radial and nontangential maximal functions, (finite) atoms, and molecules. As an application, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from HXA (Rn) to X or to HXA (Rn) itself via first establishing some boundedness criteria of linear operators on HXA (Rn). All these results have a wide range of generality and, particularly, even when they are applied to the Morrey space and the Orlicz-slice space, the obtained results are also new. The novelties of this article exist in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of quasi-norm ∥ ⋅ ∥ X, the authors embed X into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of the anisotropic weighted Lebesgue space. [ABSTRACT FROM AUTHOR]

We consider Non-autonomous Conformal Iterated Function Systems (NCIFS) in the overlapping case. We focus on the parameterized families of NCIFS that satisfy the transversality condition, and give a dimension formula for the generalized invariant measure for a.e. parameter. We also give a sufficient condition that the generalized invariant measure is absolutely continuous for a.e. parameter. [ABSTRACT FROM AUTHOR]

Given a homeomorphism f:X→Yf:X\to Y between QQ-dimensional spaces X,YX,Y, we show that ff satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that ff belongs to the Sobolev class Nloc1,p(X;Y){N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y), where 1≤p≤Q1\le p\le Q, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors QQ-regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f∈Nloc1,Q(X;Y)f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion hf{h}_{f} belonging to L∞(X){L}^{\infty }\left(X).

This article, titled "Sub-Riemannian Coarea Formula for Classes of Noncontact Mappings of Carnot Groups," discusses the metric properties of level sets of noncontact mappings on Carnot groups. The author explores the possibility of extracting "sub-Riemannian" properties for such mappings and presents the sub-Riemannian analogs of differentiability for classes of mappings. The article also introduces definitions and theorems related to the sub-Riemannian metric properties of noncontact mappings. The results provide insights into the structure of the kernel of the differential and the Hausdorff dimension of level sets. The article concludes with the coarea formula for open sets not containing characteristic points. This research was supported by the Mathematical Center in Akademgorodok. [Extracted from the article]

In this paper, we consider a random one-dimensional Schrödinger operator H ω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize H ω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that H ω is actually self-adjoint with probability one. When the white noise in H ω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of H ω . We shall further prove that the negative part of the spectrum of H ω is discrete, with no accumulation points other than 0. [ABSTRACT FROM AUTHOR]

In the present paper, we have developed the theory of weak null-additivity and absolute continuity in the realm of nonadditive measures on difference posets, and afterward proved two variants of the Lebesgue decomposition theorem in the context. A number of examples are constructed to illustrate the theory. [ABSTRACT FROM AUTHOR]

This article offers an examination of the Japanese postwar writer Takeda Taijun's concept of destruction (metsubō) and his view of the sekai, or "world," as a theoretical tool against Japanese imperialism, and potentially also for a transnational history. It first examines Takeda's wartime book-length essay entitled Sima Qian: The World of Shiji (Shiba Sen: Shiki no sekai, 1943) to highlight Takeda's conceptualization of space (kūkan) that anchors his figuration of the sekai. According to Takeda's reading of Shiji, the structure of the sekai suggests a world without a hegemonic center that could survive its fate of destruction, which is considered by many Takeda scholars to be a prophesy of the bankruptcy of Japanese imperialism. It then points out the focus on the totality of the sekai in Takeda's Sima Qian, in which the writer looks at history only from a retrospective standpoint and as a bystander. After the first section follows a reading of Takeda's short story 'Judgment' ('Shinpan,' 1947) and his last work Shanghai Firefly (Shanhai no hotaru, 1976), in which this article not only observes the trajectory of the evolution of Takeda's concept of the sekai as an absolute continuity that feeds on the absolute destruction of its parts, but also recognizes how Takeda's concern of the individual in the sekai has been developed over the years. Finally, it discusses how a transnational history might be possible based on a scrutiny and critique of Takeda's concept of the sekai. [ABSTRACT FROM AUTHOR]

This paper deals with a sharp version of mid-point type inequality in connection with fractional integrals of real valued absolutely continuous functions as a generalization and refinement of non-sharp classical mid-point inequality which is presented by the Riemann integrals of differentiable real valued functions whose derivative absolute values are convex. Some special functions, numerical means and a mid-point type formula are considered to discuss about some applications of main results. [ABSTRACT FROM AUTHOR]

In this paper the extension of the weighted Montgomery identity is established by using the integral formula of Pecaric, Matic and Ujevc. Further, by using this extended weighted Montgomery identity for functions whose derivatives of order n-1 are absolutely continunous functions, new inequalities of the weighted Hermite-Hadamard type are obtained. Also, applications of these results are given for various types of weight function. [ABSTRACT FROM AUTHOR]

Paolo de Lucia passed away on March 18, 2022 in Nyon, Switzerland. He was undoubtedly one of the most respected experts on Real Analysis and Measure Theory, but most of us miss him first of all as a scientific role model, colleague and fatherly friend. Last year, some of his coauthors, students and friends attended the 'Abstract Measure Theory and Applications: a symposium remembering Paolo de Lucia's contributions', held in Naples on March 30–31, 2023, and contributed orally with lectures and speeches in the section 'Memories'. For this reason, this year we wish to commemorate him by transcribing our two contributions and keeping them unchanged, albeit without the familiar context of the symposium, because unchanged is the sorrow for his loss and the weight of his absence for those who knew him. So, despite an unbridgeable distance, we can imagine him blushing slightly, smiling and saying ironically "At last, for me even a jurist agrees to publish in a mathematics journal". [ABSTRACT FROM AUTHOR]

In various stereological problems a n-dimensional convex body is intersected with an (n−1)-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the (n−1)-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.

A family of bivariate copulas given by: for v+2u

We study the main features of homeomorphisms between two Finsler manifolds satisfying the Rohde condition, i.e. the difference between the moduli of geodesic rings in the image and its preimage is uniformly bounded from below and above. Some equivalent conditions for such a characterization are established. Also we investigate the absolute continuity and the relations to the class of mappings with finite are distortion. As applications, the boundary correspondence results for homeomorphisms satisfying the Rohde condition between regular domains are also presented. [ABSTRACT FROM AUTHOR]

Ando studied Lebesgue decomposition for (bounded) positive operators, where a notion known as the maximal absolutely continuous part plays a crucial role. Based on technique involving unbounded operators we study certain expressions for the maximal absolutely continuous part, Radon–Nikodym type results, behavior of the maximal absolutely continuous part under the tensor product operation and characterization of mutual absolute continuity and so on. [ABSTRACT FROM AUTHOR]

In various stereological problems an n-dimensional convex body is intersected with an (n−1)-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the (n−1)-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to prove a result of convergence in variation for a family of multidimensional sampling-Kantorovich operators in the case of averaged-type kernels. The setting in which we work is that one of BV-spaces in the sense of Tonelli.

We develop a new method to prove absolute continuity of the integrated density of states of quasiperiodic operators, leading to the absolute continuity result for frequency-independent analytic perturbations of the non-critical almost Mathieu operator under arithmetic conditions on frequency. [ABSTRACT FROM AUTHOR]

We prove the existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games with absolutely continuous information and a Bayesian potential that is upper semi-continuous in actions for any realization of the players' types. In particular, all Bayesian potential games with finitely many actions and absolutely continuous information possess a pure-strategy Bayesian Nash equilibrium. [ABSTRACT FROM AUTHOR]

Change of measures has been an effective method in stochastic control and analysis; in continuous-time control this follows Girsanov's theorem applied to both fully observed and partially observed models, in decentralized stochastic control (or stochastic dynamic team theory) this is known as Witsenhausen's static reduction, and in discrete-time classical stochastic control Borkar has considered this method for partially observed Markov Decision processes (POMDPs) generalizing Fleming and Pardoux's approach in continuous-time. This method allows for equivalent optimal stochastic control or filtering in a new probability space where the measurements form an independent exogenous process in both discrete-time and continuous-time and the Radon–Nikodym derivative (between the true measure and the reference measure formed via the independent measurement process) is pushed to the cost or dynamics. However, for this to be applicable, an absolute continuity condition is necessary. This raises the following question: can we perturb any discrete-time sequential stochastic control problem by adding some arbitrarily small additive (e.g. Gaussian or otherwise) noise to the measurements to make the system measurements absolutely continuous, so that a change-of-measure (or static reduction) can be applicable with arbitrarily small error in the optimal cost? That is, are all sequential stochastic (single-agent or decentralized multi-agent) problems ϵ -away from being static reducible as far as optimal cost is concerned, for any ϵ > 0 ? We show that this is possible when the cost function is bounded and continuous in controllers' actions and the action spaces are convex. We also note that the solution and the cost obtained for the perturbed system is realizable (under a randomized policy) for the original model. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient. [ABSTRACT FROM AUTHOR]

The concept of a quasi-absolutely continuous function on a closed interval of functions is introduced and a condition for the compactness of a set in the space of quasi-absolutely continuous functions on a closed interval is obtained.

Consider a sequence of Gibbsian line ensembles, whose lowest labeled curves (i.e., the edge) have tight one-point marginals. Then, given certain technical assumptions on the nature of the Gibbs property and underlying random walk measure, we prove that the entire spatial process of the edge is tight. We then apply this black-box theory to the log-gamma polymer Gibbsian line ensemble, which we construct. The edge of this line ensemble is the transversal free energy process for the polymer, and our theorem implies tightness with the ubiquitous KPZ class 2/3 exponent, as well as Brownian absolute continuity of all the subsequential limits. A key technical innovation which fuels our general result is the construction of a continuous grand monotone coupling of Gibbsian line ensembles with respect to their boundary data (entrance and exit values, and bounding curves). Continuous means that the Gibbs measure varies continuously with respect to varying the boundary data, grand means that all uncountably many boundary data measures are coupled to the same probability space, and monotone means that raising the values of the boundary data likewise raises the associated measure. This result applies to a general class of Gibbsian line ensembles where the underlying random walk measure is discrete time, continuous valued and log-convex, and the interaction Hamiltonian is nearest neighbor and convex. [ABSTRACT FROM AUTHOR]

According to the Jordan Decomposition Theorem, for any function f : [ 0 , 1 ] → R of bounded variation there exists a pair of nondecreasing functions f 1 , f 2 : [ 0 , 1 ] → R with f = f 1 − f 2. Such a pair is called a Jordan decomposition of f. Ko (1991) and Zheng and Rettinger (2005) have analyzed computability-theoretic and complexity-theoretic aspects of this theorem. We complement their observations by three new results. One of our results says that there exists a polynomial time computable, absolutely continuous function f : [ 0 , 1 ] → R with polynomial modulus of absolute continuity such that, on the one hand, the so-called standard Jordan decomposition of f is computable but, on the other hand, neither the standard Jordan decomposition of f nor any other Jordan decomposition of f can be computed easily, say, in polynomial time, or in exponential time. • A function with a non-computable standard Jordan decomposition can have a polytime computable Jordan decomposition. • A computable but not polytime computable standard Jordan decomposition allows a polytime computable Jordan decomposition. • There is a PAC function without polytime computable Jordan decomposition but with computable standard Jordan decomposition. [ABSTRACT FROM AUTHOR]

Let μ be a planar self-similar measure with similarity dimension exceeding 1, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the orthogonal projections πe#(μ) are absolutely continuous for all e ε S¹ \ E, where the exceptional set E has zero Hausdorff dimension. The result is obtained from a more general framework which applies to certain parametrised families of self-similar measures on the real line. Our results extend the previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with μ have a common contraction ratio. [ABSTRACT FROM AUTHOR]

We consider the point-to-point log-gamma polymer of length 2N in a half-space with i.i.d. Gamma-1(2θ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\text {Gamma}}^{-1}(2\theta )$$\end{document} distributed bulk weights and i.i.d. Gamma-1(α+θ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\text {Gamma}}^{-1}(\alpha +\theta )$$\end{document} distributed boundary weights for θ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta >0$$\end{document} and α>-θ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >-\theta $$\end{document}. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when α=N-1/3μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha =N^{-1/3}\mu $$\end{document} for μ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu \in \mathbb {R}$$\end{document} fixed (critical regime) and when α>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >0$$\end{document} is fixed (supercritical regime). In particular, in these two regimes, we show that after appropriate centering, the free energy process with spatial coordinate scaled by N2/3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{2/3}$$\end{document} and fluctuations scaled by N1/3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{1/3}$$\end{document} is tight. These regimes correspond to a polymer measure which is not pinned at the boundary. This is the first instance of establishing the 2/3 transversal exponent for a positive temperature half-space model, and the first instance of the 1/3 fluctuation exponent besides precisely at the boundary where recent work of Imamura et al. (Solvable models in the KPZ class: approach through periodic and free boundary Schur measures. arXiv:2204.08420. 2022) applies and also gives the exact one-point fluctuation distribution (our methods do not access exact fluctuation distributions). Our proof relies on two inputs—the relationship between the half-space log-gamma polymer and half-space Whittaker process (facilitated by the geometric RSK correspondence as initiated in Corwin et al. (Duke Math J 163(3):513–563, 2014), O’Connell et al. (Invent Math 197(2):361–416, 2014), and an identity in Barraquand and Wang (Int Math Res Not 2023:11877, 2022) which relates the point-to-line half-space partition function to the full-space partition function for the log-gamma polymer. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated in Corwin and Hammond (Invent Math 195(2):441–508, 2014), a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. [ABSTRACT FROM AUTHOR]

As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T1, ..., Tr. In any case, we assume that T1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics.

We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the random weights. We also obtain sharp bounds for the lower Hausdorff and upper packing dimensions of the limiting measure. When the original measure is a Gibbs measure associated with a potential of certain modulus of continuity (weaker than Hölder), all our results are sharp. This improves results previously obtained by Kahane and Peyrière, Ben Nasr, and Fan. We exploit our results to derive dimension estimates and absolute continuity for some random fractal measures. [ABSTRACT FROM AUTHOR]

Given disjoint countable dense subsets and of the half-line , there exists a flow preserving a sigma-finite measure and such that all automorphisms with have simple singular spectrum and all automorphisms with have Lebesgue spectrum of countable multiplicity. [ABSTRACT FROM AUTHOR]

In this article, we investigate critical C³-homeomorphisms T of a circle with single cubic critical point xcr, and "golden mean" rotation number ρ = √5-1/2. Let μ be the unique probability invariant measure of T. Fix θ Ⲉ (0,1) and for each n ≥ 1 define cn := cn(θ) such that μ ([xcr, cn]) = θ.μ([xcr, Tqnxcr]), where qn is the first return time of the linear rotation Tρ. We prove the absolute continuity of limit distributions for different values of parameter θ of rescaled hitting times. [ABSTRACT FROM AUTHOR]

We study the distortion features of homeomorphisms of Sobolev class Wloc-1,1 admitting integrability for p-outer dilatation. We show that such mappings belong to Wloc1,n-1 are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower Q-homeomorphisms with appropriate measurable functions Q. This allows us to derive various distortion results like Lipschitz, Holder, logarithmic Holder continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R n + 1 be an open set whose boundary may be composed of pieces of different dimensions. Assume that Ω satisfies the quantitative openness and connectedness, and there exist doubling measures m on Ω and μ on ∂Ω with appropriate size conditions. Let L u = − div (A ∇ u) be a real (not necessarily symmetric) degenerate elliptic operator in Ω. Write ω L for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) ω L ∈ A ∞ (μ) , (ii) the Dirichlet problem for L is solvable in L p (μ) for some p ∈ (1 , ∞) , (iii) every bounded null solution of L satisfies Carleson measure estimates with respect to μ , (iv) the conical square function is controlled by the non-tangential maximal function in L q (μ) for all q ∈ (0 , ∞) for any null solution of L , and (v) the Dirichlet problem for L is solvable in BMO (μ). On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of ω L with respect to μ in terms of local L 2 (μ) estimates of the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness μ -almost everywhere of the truncated conical square function for any bounded null solution of L. [ABSTRACT FROM AUTHOR]

We show that the spectrum of a Schr\"odinger operator on $\mathbb{R}^n$,$n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiplehas a product structure with one Euclidean direction, and with a periodicelectric potential in $L^{n/2}_{\text{loc}}(\mathbb{R}^n)$, is purelyabsolutely continuous. Previously known results in the case of a general metricare obtained in [12], see also [8], under the assumption that the metric, aswell as the potential, are reflection symmetric.

We continue the study of the fractional variation following the distributional approach developed in the previous works Bruè et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space B V α , p (R n) of L p functions, with p ∈ [ 1 , + ∞ ] , possessing finite fractional variation of order α ∈ (0 , 1) . Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a B V α , p function. [ABSTRACT FROM AUTHOR]

We study the interplay of mappings of finite area distortion (FAD) with finitely bi-Lipschitz mappings, ring and lower Q-homeomorphisms, and absolutely continuous homeomorphisms of the class AC Λ n , p on Riemannian manifolds. Some additional relations to the hyper Q-homeomorphisms, η -quasisymmetric and ω -quasimöbius mappings are also established. As applications of the above results, we provide several extension conditions to the weakly flat and strongly accessible boundaries under FAD-homeomorphisms. [ABSTRACT FROM AUTHOR]

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators L 1 and L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of L 1 is equivalent to the same property for L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which L 2 is either the transpose of L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex. [ABSTRACT FROM AUTHOR]

We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact. [ABSTRACT FROM AUTHOR]

This paper, a keynote lecture delivered at the Fellowship for Biblical Studies Conference, 30 September, 2021, reports on the author's pursuit over many decades of how Paul saw the connection between Christ's death and human salvation. It formulates twelve "theses" in this regard, stressing the significance of apocalyptic in Paul's theology and of the last judgment as background to his proclamation of the gospel. It also asserts the centrality of Paul's view of Christ in "Adamic" terms and the necessity of seeing the absolute continuity between Father and Son in assessing the soteriological function of Christ's death. [ABSTRACT FROM AUTHOR]

This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty. [ABSTRACT FROM AUTHOR]

Large‐scale Gaussian process (GP) regression is infeasible for large training data due to cubic scaling of flops and quadratic storage involved in working with covariance matrices. Remedies in recent literature focus on divide‐and‐conquer, for example, partitioning into subproblems and inducing functional (and thus computational) independence. Such approximations can be speedy, accurate, and sometimes even more flexible than ordinary GPs. However, a big downside is loss of continuity at partition boundaries. Modern methods like local approximate GPs (LAGPs) imply effectively infinite partitioning and are thus both good and bad in this regard. Model averaging, an alternative to divide‐and‐conquer, can maintain absolute continuity but often over‐smooths, diminishing accuracy. Here we propose putting LAGP‐like methods into a local experts‐like framework, blending partition‐based speed with model‐averaging continuity, as a flagship example of what we call precision aggregated local models (PALM). Using K LAGPs, each selecting n from N total data pairs, our scheme is at most cubic in n, quadratic in K, and linear in N. Extensive empirical illustration shows how PALM is at least as accurate as LAGP, can be much faster, and furnishes continuous predictions. Finally, we propose sequential updating scheme that greedily refines a PALM predictor up to a computational budget. [ABSTRACT FROM AUTHOR]

Some differentiability results from the paper of D.Ş. Marinescu & M. Monea [7] on delta-convex mappings, obtained for real functions, are extended for mappings with values in a normed linear space. In this way, we are nearing the completion of studies established in papers [2], [5] and [7].

We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that if the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt [12] , we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function. [ABSTRACT FROM AUTHOR]

We prove an irreducible decomposition for Markov processes associated with quasi-regular symmetric Dirichlet forms or local semi-Dirichlet forms under the absolute continuity condition of transition probability with respect to the underlying measure. We do not assume the conservativeness nor the existence of invariant measures for the processes. As applications, we establish a concrete expression for Chacon–Ornstein type ratio ergodic theorem for such Markov processes and show a compactness of semi-groups under the Green-tightness of measures in the framework of symmetric resolvent strong Feller processes without irreducibility. [ABSTRACT FROM AUTHOR]