Recent zbMATH articles in MSC 28https://www.zbmath.org/atom/cc/282021-04-16T16:22:00+00:00WerkzeugEqui-Riemann and equi-Riemann type integrable functions with values in a Banach space.https://www.zbmath.org/1456.260112021-04-16T16:22:00+00:00"Mondal, Pratikshan"https://www.zbmath.org/authors/?q=ai:mondal.pratikshan"Dey, Lakshmi Kanta"https://www.zbmath.org/authors/?q=ai:dey.lakshmi-kanta"Jaker Ali, Sk."https://www.zbmath.org/authors/?q=ai:jaker-ali.skSummary: In this paper we study equi-Riemann and equi-Riemann-type integrability of a collection of functions defined on a closed interval of \(\mathbb{R}\) with values in a Banach space. We obtain some properties of such collections and interrelations among them. Moreover we establish equi-integrability of different types of collections of functions. Finally, we obtain relations among equi-Riemann integrability with other properties of a collection of functions.On volume and surface densities of dynamical germ-grain models with ellipsoidal growth: a rigorous approach with applications to materials science.https://www.zbmath.org/1456.600452021-04-16T16:22:00+00:00"Villa, Elena"https://www.zbmath.org/authors/?q=ai:villa.elena"Rios, Paulo R."https://www.zbmath.org/authors/?q=ai:rios.paulo-rSummary: Many engineering materials of interest are polycrystals: an aggregate of many crystals with size usually below \(100\mu\)m. Those small crystals are called the grains of the polycrystal, and are often equiaxed. However, because of processing, the grain shape may become anisotropic; for instance, during recrystallization or phase transformations, the new grains may grow in the form of ellipsoids. Heavily anisotropic grains may result from a process, such as rolling, and they may have most of their interfacial area parallel to the rolling plane. Therefore, to a first approximation, these heavily deformed grains may be approximated by random parallel planes; as a consequence, the nucleation process may be assumed to take place on random parallel planes. The case of nucleation on random parallel planes and subsequent ellipsoidal growth is also possible. In this paper we model such situations employing time dependent germ grain processes with ellipsoidal growth. We provide explicit formulas for the mean volume and surface densities and related quantities. The known results for the spherical growth follow here as a particular case. Although this work has been done bearing applications to Materials Science in mind, the results obtained here may be applied to nucleation and growth reactions in general. Moreover, a generalization of the so called mean value property, crucial in finding explicit analytical formulas in the paper, is also provided as a further result in the Appendix A.Scattering matrices in the \(\mathfrak{sl}( 3)\) twisted Yangian.https://www.zbmath.org/1456.820272021-04-16T16:22:00+00:00"Avan, Jean"https://www.zbmath.org/authors/?q=ai:avan.jean"Doikou, Anastasia"https://www.zbmath.org/authors/?q=ai:doikou.anastasia"Karaiskos, Nikos"https://www.zbmath.org/authors/?q=ai:karaiskos.nikosLebesgue integral.https://www.zbmath.org/1456.260022021-04-16T16:22:00+00:00"Florescu, Liviu C."https://www.zbmath.org/authors/?q=ai:florescu.liviu-cPublisher's description: This book presents a compact and self-contained introduction to the theory of measure and integration. The introduction into this theory is as necessary (because of its multiple applications) as difficult for the uninitiated. Most measure theory treaties involve a large amount of prerequisites and present crucial theoretical challenges. By taking on another approach, this textbook provides less experienced readers with material that allows an easy access to the definition and main properties of the Lebesgue integral.
The book will be welcomed by upper undergraduate/early graduate students who wish to better understand certain concepts and results of probability theory, statistics, economic equilibrium theory, game theory, etc., where the Lebesgue integral makes its presence felt throughout. The book can also be useful to students in the faculties of mathematics, physics, computer science, engineering, life sciences, as an introduction to a more in-depth study of measure theory.Box dimension of fractal attractors and their numerical computation.https://www.zbmath.org/1456.280062021-04-16T16:22:00+00:00"Freiberg, Uta"https://www.zbmath.org/authors/?q=ai:freiberg.uta-renata"Kohl, Stefan"https://www.zbmath.org/authors/?q=ai:kohl.stefanThe authors provide a general algorithmic approach to calculate the box dimension of Lorenz attractors. The algorithm behaves very well on self-similar fractals and fractal graphs of deterministic as well as random functions; however, in the case of the Lorenz attractor the algorithm seems to fail. The reason for this phenomenon is discussed, the notion of local box dimension is introduced, and the algorithm reveals a hidden multifractal structure of the Lorenz attractor.
Reviewer: George Stoica (Saint John)Generalized transforms and generalized convolution products associated with Gaussian paths on function space.https://www.zbmath.org/1456.600832021-04-16T16:22:00+00:00"Chang, Seung Jun"https://www.zbmath.org/authors/?q=ai:chang.seung-jun"Choi, Jae Gil"https://www.zbmath.org/authors/?q=ai:choi.jae-gilSummary: In this paper we define a more general convolution product (associated with Gaussian processes) of functionals on the function space \(C_{a,b}[0,T]\). The function space \(C_{a,b}[0,T]\) is induced by a generalized Brownian motion process. Thus the Gaussian processes used in this paper are non-centered processes. We then develop the fundamental relationships between the generalized Fourier-Feynman transform associated with the Gaussian process and the convolution product.Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite-dimensional convex domains.https://www.zbmath.org/1456.280082021-04-16T16:22:00+00:00"Angiuli, L."https://www.zbmath.org/authors/?q=ai:angiuli.luciana"Ferrari, S."https://www.zbmath.org/authors/?q=ai:ferrari.stefania|ferrari.simone|ferrari.stefano|ferrari.sara|ferrari.silvia-l-de-paula|ferrari.silvia"Pallara, D."https://www.zbmath.org/authors/?q=ai:pallara.diegoSummary: Let \(X\) be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure \(\gamma\), and let \(\lambda _1\) be the maximum eigenvalue of the covariance operator associated with \(\gamma\). The associated Cameron-Martin space is denoted by \(H\). For a sufficiently regular convex function \(U:X\rightarrow{{\mathbb{R}}}\) and a convex set \(\Omega \subseteq X\), we set \(\nu :=\mathrm{e}^{-U}\gamma\) and we consider the semigroup \((T_\Omega (t))_{t\ge 0}\) generated by the self-adjoint operator defined via the quadratic form \[(\varphi ,\psi )\mapsto \int _\Omega{\left\langle D_H\varphi ,D_H\psi \right\rangle }_H \mathrm{d}\nu , \] where \(\varphi ,\psi\) belong to \(D^{1,2}(\Omega ,\nu )\), the Sobolev space defined as the domain of the closure in \(L^2(\Omega ,\nu )\) of \(D_H\), the gradient operator along the directions of \(H\). A suitable approximation procedure allows us to prove some pointwise gradient estimates for \((T_{\Omega }(t))_{t\ge 0}\). In particular, we show that \[ |D_H T_{\Omega }(t)f|_H^p\le \mathrm{e}^{- p \lambda _1^{-1} t}(T_{\Omega }(t)|D_H f|^p_H), \quad \, t>0,\ \nu \text{-a.e. in }{\Omega }, \] for any \(p\in [1,+\infty )\) and \(f\in D^{1,p}({\Omega },\nu )\). We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in \({\Omega }\) for the measure \(\nu\) and some improving summability properties for \((T_\Omega (t))_{t\ge 0}\). In addition, we prove that if \(f\) belongs to \(L^p(\Omega ,\nu )\) for some \(p\in (1,\infty )\), then \[ |D_H T_\Omega (t)f|^p_H \le K_p t^{-\frac{p}{2}} T_\Omega (t)|f|^p,\quad \, t>0,\ \nu \text{-a.e. in }\Omega , \] where \(K_p\) is a positive constant depending only on \(p\). Finally, we investigate on the asymptotic behaviour of the semigroup \((T_{\Omega }(t))_{t\ge 0}\) as \(t\) goes to infinity.Limiting probability measures.https://www.zbmath.org/1456.280092021-04-16T16:22:00+00:00"Alam, Irfan"https://www.zbmath.org/authors/?q=ai:alam.irfanSummary: The coordinates along any fixed direction(s), of points on the sphere \(S^{n-1}(\sqrt{n})\), roughly follow a standard Gaussian distribution as \(n\) approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann-Lebesgue lemma as a by-product of this theory. We finally show that for any function \(f:\mathbb{R}^k \to \mathbb{R}\) with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of \(f\) and its Gaussian mean are obtained in order to complete the above proof.Cauchy transforms of self-similar measures: starlikeness and univalence.https://www.zbmath.org/1456.280052021-04-16T16:22:00+00:00"Dong, Xin-Han"https://www.zbmath.org/authors/?q=ai:dong.xinhan"Lau, Ka-Sing"https://www.zbmath.org/authors/?q=ai:lau.kasing"Wu, Hai-Hua"https://www.zbmath.org/authors/?q=ai:wu.haihuaSummary: For the contractive iterated function system \( S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})}\) with \( 0<\rho <1, k=0,\cdots , m-1\), we let \( K\subset \mathbb{C}\) be the attractor, and let \( \mu \) be a self-similar measure defined by \( \mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1}\). We consider the Cauchy transform \( F\) of \( \mu \). It is known that the image of \( F\) at a small neighborhood of the boundary of \( K\) has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of \( F\) away from \( K\); it has nice geometric and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.Generally covariant \(N\)-particle dynamics.https://www.zbmath.org/1456.530542021-04-16T16:22:00+00:00"Miller, Tomasz"https://www.zbmath.org/authors/?q=ai:miller.tomasz"Eckstein, Michał"https://www.zbmath.org/authors/?q=ai:eckstein.michal"Horodecki, Paweł"https://www.zbmath.org/authors/?q=ai:horodecki.pawel"Horodecki, Ryszard"https://www.zbmath.org/authors/?q=ai:horodecki.ryszardSummary: A simultaneous description of the dynamics of multiple particles requires a configuration space approach with an external time parameter. This is in stark contrast with the relativistic paradigm, where time is but a coordinate chosen by an observer. Here we show, however, that the two attitudes towards modelling \(N\)-particle dynamics can be conciliated within a generally covariant framework. To this end we construct an `\(N\)-particle configuration spacetime' \(\mathcal{M}_{(N)} \), starting from a globally hyperbolic spacetime \(\mathcal{M}\) with a chosen smooth splitting into time and space components. The dynamics of multi-particle systems is modelled at the level of Borel probability measures over \(\mathcal{M}_{(N)}\) with the help of the global time parameter. We prove that with any time-evolution of measures, which respects the \(N\)-particle causal structure of \(\mathcal{M}_{(N)}\), one can associate a single measure on the Polish space of `\(N\)-particle wordlines'. The latter is a splitting-independent object, from which one can extract the evolution of measures for any other global observer on \(\mathcal{M}\). An additional asset of the adopted measure-theoretic framework is the possibility to model the dynamics of indistinguishable entities, such as quantum particles. As an application we show that the multi-photon and multi-fermion Schrödinger equations, although explicitly dependent on the choice of an external time-parameter, are in fact fully compatible with the causal structure of the Minkowski spacetime.Entropy production of nonequilibrium steady states with irreversible transitions.https://www.zbmath.org/1456.370122021-04-16T16:22:00+00:00"Zeraati, Somayeh"https://www.zbmath.org/authors/?q=ai:zeraati.somayeh"Jafarpour, Farhad H."https://www.zbmath.org/authors/?q=ai:jafarpour.farhad-h"Hinrichsen, Haye"https://www.zbmath.org/authors/?q=ai:hinrichsen.hayeMultifractal detrended moving average analysis of global temperature records.https://www.zbmath.org/1456.860092021-04-16T16:22:00+00:00"Mali, Provash"https://www.zbmath.org/authors/?q=ai:mali.provashOn the quantitative isoperimetric inequality in the plane.https://www.zbmath.org/1456.490342021-04-16T16:22:00+00:00"Bianchini, Chiara"https://www.zbmath.org/authors/?q=ai:bianchini.chiara"Croce, Gisella"https://www.zbmath.org/authors/?q=ai:croce.gisella"Henrot, Antoine"https://www.zbmath.org/authors/?q=ai:henrot.antoineSummary: In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set \(\Omega\), different from a ball, which minimizes the ratio \(\delta(\Omega)/\lambda^{2}(\Omega)\), where \(\delta\) is the isoperimetric deficit and \(\lambda\) the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.The dimensional Brunn-Minkowski inequality in Gauss space.https://www.zbmath.org/1456.520112021-04-16T16:22:00+00:00"Eskenazis, Alexandros"https://www.zbmath.org/authors/?q=ai:eskenazis.alexandros"Moschidis, Georgios"https://www.zbmath.org/authors/?q=ai:moschidis.georgiosThe authors prove the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure, thus settling a problem raised in [\textit{R. J. Gardner} and \textit{A. Zvavitch}, Trans. Am. Math. Soc. 362, No. 10, 5333--5353 (2010; Zbl 1205.52002)]. They also settle the case when equality holds.
Reviewer: George Stoica (Saint John)The strong uniqueness property of invariant measures in infinite dimensional topological vector spaces.https://www.zbmath.org/1456.280012021-04-16T16:22:00+00:00"Khachidze, Marika"https://www.zbmath.org/authors/?q=ai:khachidze.marika"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-pIn [\textit{A. B. Kharazishvili}, Bull. Acad. Sci. GSSR 114, No. 1, 41--48 (1984)] a normalized \(\sigma\)-finite metrically transitive Borel measure \(\chi\) in \(\mathbb{R}^\omega\) is constructed which is invariant with respect to
\[G=c_{00}:=\{x \in \mathbb{R}^\omega : \ \hbox{supp}~ x < \omega\}\]
Let \(s_0\) be the central symmetry of \(\mathbb{R}^\omega\) with respect to the origin and let \(S_\omega\) be the group generated by \(s_0\) and \(G\). In the present paper a \(\sigma\)-finite Borel measure on \(\mathbb{R}^\omega\) is costructed which is invariant with respect to the group \(S_\omega\).
Reviewer: Daniele Puglisi (Catania)A heterogeneous QUALIFLEX method with criteria interaction for multi-criteria group decision making.https://www.zbmath.org/1456.900862021-04-16T16:22:00+00:00"Liang, Yingying"https://www.zbmath.org/authors/?q=ai:liang.yingying"Qin, Jindong"https://www.zbmath.org/authors/?q=ai:qin.jindong"Martínez, Luis"https://www.zbmath.org/authors/?q=ai:martinez.luis-manuel|martinez.luis-a"Liu, Jun"https://www.zbmath.org/authors/?q=ai:liu.jun.3|liu.jun.1|liu.jun|liu.jun.5|liu.jun.4|liu.jun.2Summary: Green supplier selection complexity involves such problems which usually need to offer the management of dependencies among criteria in the selection process. Meanwhile, heterogeneous contexts are adopted based on different criteria, which drives to propose a novel multi-criteria group decision making (MCGDM) method. To solve this problem, this paper proposes a new MCGDM approach for heterogeneous information and dependent criteria based on the QUALItative FLEXible (QUALIFLEX) method and Choquet integral. For managing dependency among criteria, a new graphical representation of criteria interaction is presented and the identification of fuzzy measure is then obtained considering group consensus reaching. The multi-criteria heterogeneous QUALIFLEX method with regard to dependency of criteria is finally applied to a green supplier selection problem and a comparative analysis is performed to illustrate its feasibility and effectiveness.Approximating the Ising model on fractal lattices of dimension less than two.https://www.zbmath.org/1456.821042021-04-16T16:22:00+00:00"Codello, Alessandro"https://www.zbmath.org/authors/?q=ai:codello.alessandro"Drach, Vincent"https://www.zbmath.org/authors/?q=ai:drach.vincent"Hietanen, Ari"https://www.zbmath.org/authors/?q=ai:hietanen.ariCoincidences of the concave integral and the pan-integral.https://www.zbmath.org/1456.280102021-04-16T16:22:00+00:00"Ouyang, Yao"https://www.zbmath.org/authors/?q=ai:ouyang.yao"Li, Jun"https://www.zbmath.org/authors/?q=ai:li.jun.7"Mesiar, Radko"https://www.zbmath.org/authors/?q=ai:mesiar.radkoSummary: In this note, we discuss when the concave integral coincides with the pan- integral with respect to the standard arithmetic operations \(+\) and \(\cdot\). The subadditivity of the underlying monotone measure is a sufficient condition for this equality. We show also another sufficient condition, which, in the case of finite spaces, is necessary, too.Corrigendum to: ``Local dimensions of measures of finite type III -- measures that are not equicontractive''.https://www.zbmath.org/1456.280072021-04-16T16:22:00+00:00"Hare, Kathryn E."https://www.zbmath.org/authors/?q=ai:hare.kathryn-e"Hare, Kevin G."https://www.zbmath.org/authors/?q=ai:hare.kevin-g"Simms, Grant"https://www.zbmath.org/authors/?q=ai:simms.grantFrom the text: In the authors's paper [ibid. 458, No. 2, 1653--1677 (2018; Zbl 1376.28008)], Definition 3.5 was incorrectly stated. The correct definition was used, or bypassed for all proofs and calculations, and all results of the paper remain valid. We correct it here only to alleviate possible confusion.Vitali-Hahn-Saks property in coverings of sets algebras.https://www.zbmath.org/1456.280032021-04-16T16:22:00+00:00"López-Alfonso, S."https://www.zbmath.org/authors/?q=ai:lopez-alfonso.salvadorLet \(ba(\mathcal{A})\) denote the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra \(\mathcal{A}\) of subsets of \(\Omega\). A subset \(\mathcal{B}\) of \(\mathcal{A}\) is a Nikodým set for \(ba(\mathcal{A})\) if each \(\mathcal{B}\)-pointwise bounded subset \(M\) of \(ba(\mathcal{A})\) is uniformly bounded on \(\mathcal{A}\); while \(\mathcal{B}\) is a strong Nikodým set for \(ba(\mathcal{A})\) if each increasing covering \((\mathcal{B}_m )_{m=1}^\infty\) of \(\mathcal{B}\) contains a \(\mathcal{B}_n\) which is a Nikodým set for \(ba(\mathcal{A})\). The subset \(\mathcal{B}\) has (VHS) property if \(\mathcal{B}\) is a Nikodým set for \(ba(\mathcal{A})\) and for every \(\mu\in ba(\mathcal{A})\) and sequence \((\mu_m)_{m=1}^\infty \subseteq ba(\mathcal{A})\) such that \(\lim_{m\rightarrow \infty} \mu_m(B) = \mu(B)\), for each \(B \in \mathcal{B}\), then \((\mu_m)_{m=1}^\infty\) weakly converges to \(\mu\). It is proved that if \((\mathcal{B}_m)_{m=1}^\infty\) is an increasing covering of \(\mathcal{A}\) that has (VHS) property and there exist a \(\mathcal{B}_p\) which is a Nikodým set for \(ba(\mathcal{A})\), then there exists \(q \geq p\) such that \(\mathcal{B}_q\) has (VHS) property. In particular, if \((\mathcal{B}_m )_{m=1}^\infty\) is an increasing covering of a \(\sigma\)-algebra then some \(\mathcal{B}_q\) has (VHS) property. In 2013 Valdivia proved that every \(\sigma\)-algebra has strong Nikodým property. Another proof of this result is presented, here. Some related questions with respect to the problem if Nikodým property in an algebra implies strong Nikodým property are also considered.
Reviewer: Daniele Puglisi (Catania)A measure and orientation preserving homeomorphism with approximate Jacobian equal -1 almost everywhere.https://www.zbmath.org/1456.370022021-04-16T16:22:00+00:00"Goldstein, Paweł"https://www.zbmath.org/authors/?q=ai:goldstein.pawel"Hajłasz, Piotr"https://www.zbmath.org/authors/?q=ai:hajlasz.piotrSummary: We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit \(n\)-dimensional cube onto itself, whose Jacobian is equal to \(-1\) almost everywhere. Moreover, we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.Regularity of area minimizing currents mod \(p\).https://www.zbmath.org/1456.490322021-04-16T16:22:00+00:00"De Lellis, Camillo"https://www.zbmath.org/authors/?q=ai:de-lellis.camillo"Hirsch, Jonas"https://www.zbmath.org/authors/?q=ai:hirsch.jonas"Marchese, Andrea"https://www.zbmath.org/authors/?q=ai:marchese.andrea"Stuvard, Salvatore"https://www.zbmath.org/authors/?q=ai:stuvard.salvatoreRegularity of area minimizing currents mod(p) have had an important role in several classical problems of geometric measure theory and mathematical physics. Several authors studied the regularity of area [\textit{C. De Lellis} and \textit{E.N. Spadaro}, Ann. of Math. (2), 183, No. 2, 499--575 (2016; Zbl 1345.49052); \textit{C. De Lellis} and \textit{E.N. Spadaro}, Geom. Funct. Anal., 24, No. 6, 1831--1884 (2014; Zbl 1307.49043); \textit{C. De Lellis} and \textit{E. N. Spadaro}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 4, 1239--1269 (2015; Zbl 1343.49073); \textit{T. De Pauw} and \textit{R. Hardt}, Am. J. Math., 134, No. 1, 1--69 (2012; Zbl 1252.49070); \textit{T. De Pauw} and \textit{R. Hardt}, J. Math. Anal. Appl., 418, No. 2, 1047--1061 (2014; Zbl 1347.49073); \textit{L. Simon}, J. Diff. Geom. 38, No. 3, 585--652 (1993; Zbl 0819.53029); \textit{L. Spolaor}, Adv. Math., 350, 747--815 (2019; Zbl 1440.49048)].
The principal objective in this paper is to establish a first general partial regularity theorem for area minimizing currents mod(p), for every p, in any dimension and codimension. More precisely, the authors prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current mod(p) cannot be larger than \(m-1\).
Reviewer: Lakehal Belarbi (Mostaganem)Bose-Fermi duality and entanglement entropies.https://www.zbmath.org/1456.810692021-04-16T16:22:00+00:00"Headrick, Matthew"https://www.zbmath.org/authors/?q=ai:headrick.matthew"Lawrence, Albion"https://www.zbmath.org/authors/?q=ai:lawrence.albion"Roberts, Matthew"https://www.zbmath.org/authors/?q=ai:roberts.matthew-mHausdorff dimension of limsup sets of rectangles in the Heisenberg group.https://www.zbmath.org/1456.600382021-04-16T16:22:00+00:00"Ekström, Fredrik"https://www.zbmath.org/authors/?q=ai:ekstrom.fredrik"Järvenpää, Esa"https://www.zbmath.org/authors/?q=ai:jarvenpaa.esa"Järvenpää, Maarit"https://www.zbmath.org/authors/?q=ai:jarvenpaa.maaritThe main findings of the paper refer to computing the almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group, in terms of directed singular value functions.
Reviewer: George Stoica (Saint John)Existence of similar point configurations in thin subsets of \(\mathbb{R}^d\).https://www.zbmath.org/1456.520202021-04-16T16:22:00+00:00"Greenleaf, Allan"https://www.zbmath.org/authors/?q=ai:greenleaf.allan"Iosevich, Alex"https://www.zbmath.org/authors/?q=ai:iosevich.alex"Mkrtchyan, Sevak"https://www.zbmath.org/authors/?q=ai:mkrtchyan.sevakSummary: We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff dimension in Euclidean space. Let \(d \ge 2\) and \(E \subset \mathbb{R}^d\) be a compact set. For \(k \ge 1\), define
\[
\Delta_k(E) = \left\{\left(|x^1 - x^2|, \ldots, |x^i - x^j|, \ldots, |x^k - x^{k+1}|\right) : \left\{x^i\right\}_{i = 1}^{k + 1} \subset E \right\} \subset \mathbb{R}^{k(k+1)/2},
\]
the \((k+1)\)-point configuration set of \(E\). For \(k \le d\), this is (up to permutations) the set of congruences of \((k+1)\)-point configurations in \(E\); for \(k > d\), it is the edge-length set of \((k + 1)\)-graphs whose vertices are in \(E\). Previous works by a number of authors have found values \(s_{k, d} < d\) so that if the Hausdorff dimension of \(E\) satisfies \(\dim_{\mathcal H}(E) > s_{k, d} \), then \(\Delta_k(E)\) has positive Lebesgue measure. In this paper we study more refined properties of \(\Delta_k(E)\), namely the existence of similar or multi-similar configurations. For \(r \in \mathbb{R}, r > 0\), let
\[
\Delta_k^r(E) := \{\mathbf{t}\in \Delta_k (E) : r \mathbf{t} \in \Delta_k (E)\} \subset \Delta_k (E).
\]
We show that if \(\dim_{\mathcal H}(E) > s_{k, d}\), for a natural measure \(\nu_k\) on \(\Delta_k(E)\), one has \(\nu_k \left(\delta^r_k(E)\right)\) all \(r \in \mathbb{R}_+\). Thus, in \(E\) there exist many pairs of \((k + 1)\)-point configurations which are similar by the scaling factor \(r\). We extend this to show the existence of multi-similar configurations of any multiplicity. These results can be viewed as variants and extensions, for compact thin sets, of theorems of Furstenberg, Katznelson and Weiss [7], Bourgain [2] and Ziegler [11] for sets of positive density in \(\mathbb{R}^d\).On the computational complexity of the Dirichlet problem for Poisson's equation.https://www.zbmath.org/1456.030692021-04-16T16:22:00+00:00"Kawamura, Akitoshi"https://www.zbmath.org/authors/?q=ai:kawamura.akitoshi"Steinberg, Florian"https://www.zbmath.org/authors/?q=ai:steinberg.florian"Ziegler, Martin"https://www.zbmath.org/authors/?q=ai:ziegler.martinSummary: The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense `complete' for the complexity class \(\#\mathcal{P}\) and thus as hard or easy as parametric Riemann integration [\textit{H. Friedman}, Adv. Math. 53, 80--98 (1984; Zbl 0563.03023); \textit{K.-I Ko}, Complexity theory of real functions. Boston, MA etc.: Birkhäuser (1991; Zbl 0791.03019)].Multivalued strong laws of large numbers for triangular arrays with gap topology.https://www.zbmath.org/1456.600682021-04-16T16:22:00+00:00"Giap, Duong Xuan"https://www.zbmath.org/authors/?q=ai:giap.duong-xuan"van Huan, Nguyen"https://www.zbmath.org/authors/?q=ai:van-huan.nguyen"Ngoc, Bui Nguyen Tram"https://www.zbmath.org/authors/?q=ai:ngoc.bui-nguyen-tram"van Quang, Nguyen"https://www.zbmath.org/authors/?q=ai:quang.nguyen-vanSummary: We state some strong laws of large numbers for triangular arrays of random sets in separable Banach spaces with the gap topology and with or without compactly uniformly integrable condition.Pulling self-interacting linear polymers on a family of fractal lattices embedded in three-dimensional space.https://www.zbmath.org/1456.829502021-04-16T16:22:00+00:00"Elezović-Hadžić, S."https://www.zbmath.org/authors/?q=ai:elezovic-hadzic.suncica"Živić, I."https://www.zbmath.org/authors/?q=ai:zivic.ivanDynamic scaling behaviors of the discrete growth models on fractal substrates.https://www.zbmath.org/1456.829822021-04-16T16:22:00+00:00"Xun, Zhipeng"https://www.zbmath.org/authors/?q=ai:xun.zhipeng"Zhang, Yongwei"https://www.zbmath.org/authors/?q=ai:zhang.yongwei"Li, Yan"https://www.zbmath.org/authors/?q=ai:li.yan.6|li.yan.1|li.yan.8|li.yan.4|li.yan.5|li.yan|li.yan.3|li.yan.2|li.yan.10|li.yan.7|li.yan.9"Xia, Hui"https://www.zbmath.org/authors/?q=ai:xia.hui"Hao, Dapeng"https://www.zbmath.org/authors/?q=ai:hao.da-peng"Tang, Gang"https://www.zbmath.org/authors/?q=ai:tang.gangA. Kharazishvili's some results of on the structure of pathological functions.https://www.zbmath.org/1456.260032021-04-16T16:22:00+00:00"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-p"Pantsulaia, Gogi"https://www.zbmath.org/authors/?q=ai:pantsulaia.gogi-rauliThe authors present a brief survey of A. Kharazishvili's works devoted to real-valued functions with strange, pathological and paradoxical structural properties, e.g. absolutely non-measurable functions, Sierpiński-Zygmund functions, sup-measurable and weakly sup-measurable functions of two real variables, and non-measurable functions of two real variables for which both iterated integrals exist.
Reviewer: George Stoica (Saint John)Characterization of sets of singular rotations for a class of differentiation bases.https://www.zbmath.org/1456.280022021-04-16T16:22:00+00:00"Oniani, G."https://www.zbmath.org/authors/?q=ai:oniani.giorgi-gigla"Chubinidze, K."https://www.zbmath.org/authors/?q=ai:chubinidze.kakha-aSummary: We study the dependence of differential properties of an indefinite integral on rotations of the coordinate system. Namely, the following problem is studied: For a summable function \(f\) of what kind can be the set of rotations \(\gamma\) for which \(\int f\) is not differentiable with respect to the \(\gamma \)-rotation of a given basis \(B\)? The result obtained in the paper implies a solution of the problem for any homothecy invariant differentiation basis \(B\) of two-dimensional intervals which has symmetric structure.Rough basin boundaries in high dimension: can we classify them experimentally?https://www.zbmath.org/1456.370962021-04-16T16:22:00+00:00"Bódai, Tamás"https://www.zbmath.org/authors/?q=ai:bodai.tamas"Lucarini, Valerio"https://www.zbmath.org/authors/?q=ai:lucarini.valerioSummary: We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent \(\lambda_x\) on the nonattracting set is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a partial co-dimension, and, so, it can be matched with a Lyapunov exponent. We show in 2D noninvertible- and 3D invertible-minimal models, that, formally, it cannot be matched with \(\lambda_x\). Rather, the partial dimension \(D_0^{(x)}\) that \(\lambda_x\) is associated with in the case of rough boundaries is trivially unity. Further results hint that the latter holds also in higher dimensions. This is a peculiar feature of rough fractals. Yet, \(D_0^{(x)}\) cannot be measured via the uncertainty exponent along a line that traverses the boundary. Consequently, one cannot determine whether the boundary is a rough or a filamentary fractal by measuring fractal dimensions. Instead, one needs to measure both the maximal and cross-boundary Lyapunov exponents numerically or experimentally.
{\copyright 2020 American Institute of Physics}Number theory problems related to the spectrum of Cantor-type measures with consecutive digits.https://www.zbmath.org/1456.280042021-04-16T16:22:00+00:00"Ai, Wen-Hui"https://www.zbmath.org/authors/?q=ai:ai.wenhuiThe present paper deals with the scaling of spectra of a class of self-similar measures with consecutive digits and extends results obtained in some researches of other authors.
A brief survey of this paper is devoted to introduce a new area of research for a harmonic analysis on fractals, to construct the
first singular, nonatomic, spectral measure by Jorgensen and Pedersen, and to classes of fractal measures (defined by self-similarity) which have spectra, i.e., admit \(L^2\)-orthogonal Fourier expansions. Also, some related investigations are considered.
The author gives the following description of the present research:
``For integers \(p,b \ge 2\), let \(D=\{0,1,\dots , b-1\}\) be a set of consecutive digits. It is known that the Cantor measure \(\mu_{pb, D}\) generated by the iterated function system \(\{(pb)^{-1}(x+d)\}_{x\in\mathbb R, d\in D}\) is a spectral measure with spectrum
\[
\Lambda (pb, S)=\left\{\sum^{\text{finite}} _{j=0}{(pb)^{j}s_j}: s_j \in S \right\},
\]
where \(S=pD\). We give conditions on \(\tau \in \mathbb Z\) under which the scaling set \(\tau \Lambda (pb, S)\) is also a spectrum of
\(\mu_{pb, D}\). These investigations link number theory and spectral measures.''
One can note that necessary or sufficient conditions under which a composite number is complete or incomplete, are studied. Connections between obtained results and known ones are described.
Reviewer: Symon Serbenyuk (Kyïv)Intrinsic differentiability and intrinsic regular surfaces in Carnot groups.https://www.zbmath.org/1456.352062021-04-16T16:22:00+00:00"Di Donato, Daniela"https://www.zbmath.org/authors/?q=ai:di-donato.danielaSummary: A Carnot group \(\mathbb{G}\) is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as \(\mathbb{C}^1\) surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize the results on \textit{L. Ambrosio} et al. [J. Geom. Anal. 16, No. 2, 187--232 (2006; Zbl 1085.49045)] valid in Heisenberg groups to the more general setting of Carnot groups.On fractional and fractal Einstein's field equations.https://www.zbmath.org/1456.830062021-04-16T16:22:00+00:00"El-Nabulsi, Rami Ahmad"https://www.zbmath.org/authors/?q=ai:el-nabulsi.rami-ahmad"Golmankhaneh, Alireza Khalili"https://www.zbmath.org/authors/?q=ai:golmankhaneh.alireza-khaliliWeak convergence and weak compactness in the space of integrable functions with respect to a vector measure.https://www.zbmath.org/1456.460272021-04-16T16:22:00+00:00"Swartz, Charles"https://www.zbmath.org/authors/?q=ai:swartz.charles-wLet \(\Sigma\) be a \(\sigma\)-algebra on a set \(S\) and \(m:\Sigma\rightarrow X\) be a Banach space-valued \(\sigma\)-additive measure. A function \(f:S\rightarrow \mathbb{R}\) is \(m\)-integrable if \(f\) is \(x'm\)-integrable for every \(x'\in X'\) and for every \(A\in\Sigma\) there exists \(x_A\in X\) such that \(x'(x_A)=\int_A fd(x'm)\) for \(x'\in X'\). The space \(L^1(m)\) of all \(m\)-integrable functions \(f:S\rightarrow \mathbb{R}\) is a Banach space with the norm \(||f||_1:= \sup \{\int |f|d|x'm|: x'\in X', \,||x'||\leq 1\}\). Using \textit{S. Okada}'s description [J. Math. Anal. Appl. 177, No. 2, 583--599 (1993; Zbl 0804.46049)] of the continuous dual of \(L^1(m)\), the author gives a necessary and sufficient condition for a sequence in \(L^1(m)\) to be weak Cauchy and then characterizes conditionally sequentially weakly compact subsets of \(L^1(m)\). Recall that a subset \(K\) of \(L^1(m)\) is conditionally sequentially weakly compact (sometimes also called weakly sequentially precompact) if every sequence in \(K\) has a subsequence which is weakly Cauchy.
Reviewer: Hans Weber (Udine)A domain-theoretic investigation of posets of sub-\(\sigma\)-algebras (extended abstract).https://www.zbmath.org/1456.060112021-04-16T16:22:00+00:00"Battenfeld, Ingo"https://www.zbmath.org/authors/?q=ai:battenfeld.ingoSummary: Given a measurable space \((X,\mathcal{M})\) there is a (Galois) connection between sub-\(\sigma\)-algebras of \(\mathcal{M}\) and equivalence relations on \(X\). On the other hand equivalence relations on \(X\) are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-\(\sigma\)-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question.
For the entire collection see [Zbl 1391.03010].Fourier quasicrystals with unit masses.https://www.zbmath.org/1456.420322021-04-16T16:22:00+00:00"Olevskii, Alexander"https://www.zbmath.org/authors/?q=ai:olevskii.alexander-moiseevich"Ulanovskii, Alexander"https://www.zbmath.org/authors/?q=ai:ulanovskii.alexander-mSummary: The sum of \(\delta\)-measures sitting at the points of a discrete set \(\Lambda\subset\mathbb{R}\) forms a Fourier quasicrystal if and only if \(\Lambda\) is the zero set of an exponential polynomial with imaginary frequencies.On fractional differentiation.https://www.zbmath.org/1456.260082021-04-16T16:22:00+00:00"Gladkov, S. O."https://www.zbmath.org/authors/?q=ai:gladkov.sergei-oktyabrinovich"Bogdanova, S. B."https://www.zbmath.org/authors/?q=ai:bogdanova.sofya-borisovnaSummary: Due to the operation of fractional differentiation introduced with the help of Fourier integral, the results of calculating fractional derivatives for certain types of functions are given. Using the numerical method of integration, the values of fractional derivatives for arbitrary dimensionality \(\varepsilon \), (where \(\varepsilon\) is any number greater than zero) are calculated. It is proved that for integer values of \(\varepsilon\) we obtain ordinary derivatives of the first, second and more high orders. As an example it was considered heat conduction equation of Fourier, where spatial derivation was realized with the use of fractional derivatives. Its solution is given by Fourier integral. Mmoreover, it was shown that integral went into the required results in special case of the whole \(\varepsilon\) obtained in \(n\)-dimensional case, where \(n = 1, 2\dots \), etc.Multifractal analysis for core-periphery structure of complex networks.https://www.zbmath.org/1456.051552021-04-16T16:22:00+00:00"Liu, Jin-Long"https://www.zbmath.org/authors/?q=ai:liu.jinlong"Yu, Zu-Guo"https://www.zbmath.org/authors/?q=ai:yu.zuguo"Anh, Vo"https://www.zbmath.org/authors/?q=ai:anh.vo-v|anh.vo-ngocMultiscale functional inequalities in probability: concentration properties.https://www.zbmath.org/1456.600532021-04-16T16:22:00+00:00"Duerinckx, Mitia"https://www.zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://www.zbmath.org/authors/?q=ai:gloria.antoineSummary: In a companion article we have introduced a notion of multiscale functional inequalities for functions \(X(A)\) of an ergodic stationary random field \(A\) on the ambient space \(\mathbb{R}^d\). These inequalities are multiscale weighted versions of standard Poincaré, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields \(A\) arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions \(X(A)\). This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto [\textit{A. Gloria} et al., Milan J. Math. 88, No. 1, 99--170 (2020; Zbl 1440.35064)], and to the corresponding quantitative stochastic homogenization results.Estimates for Taylor coefficients of Cauchy transforms of some Hausdorff measures. II.https://www.zbmath.org/1456.300672021-04-16T16:22:00+00:00"Li, Hong-Guang"https://www.zbmath.org/authors/?q=ai:li.hongguang"Dong, Xin-Han"https://www.zbmath.org/authors/?q=ai:dong.xinhan"Zhang, Peng-Fei"https://www.zbmath.org/authors/?q=ai:zhang.pengfeiSummary: Let \(q\) be even, \(F\) be the Cauchy transform of the self-similar measure \(\mu = \frac{1}{q} \sum_{j = 0}^{q - 1} \mu \circ S_j\) where \(S_j(z) = e^{2j \pi i / q} + \rho(z - e^{2 j \pi i / q})\) with \(\rho \in (0, 1)\), and \(K\) be the attractor of \(\{S_j\}_{j = 0}^{q - 1}\) and \(R_q = \text{dist}(0, K)\). As a follow-up of [the authors, ibid., Article ID 108653, (2021; Zbl 07306981)], we not only study the asymptotic formula for the Taylor coefficients \(\{ b_{qk - 1} \}_{k = 1}^\infty\) of \(F\) in \(|z| < R_q\), but also the chaotic behavior of \(F^{(n)}\) near the attractor \(K\).
For part I, see [Zbl 1456.30068].Estimates for Taylor coefficients of Cauchy transforms of some Hausdorff measures. I.https://www.zbmath.org/1456.300682021-04-16T16:22:00+00:00"Li, Hong-Ping"https://www.zbmath.org/authors/?q=ai:li.hongping"Dong, Xin-Han"https://www.zbmath.org/authors/?q=ai:dong.xinhan"Zhang, Peng-Fei"https://www.zbmath.org/authors/?q=ai:zhang.pengfei"Wu, Hai-Hua"https://www.zbmath.org/authors/?q=ai:wu.haihuaSummary: Suppose that \(q = 2m + 1 \geq 5\). Let \(F\) be the Cauchy transform of the self-similar measure \(\mu = \frac{1}{q} \sum_{j = 0}^{q - 1} \mu \circ S_j\) where \(S_j(z) = e^{2j \pi i / q} + \rho (z - e^{2j \pi i / q})\) with \(\rho \in (0, 1)\). Let \(K\) be the attractor of \(\{S_j\}_{j = 0}^{q - 1}\) and \(R_q = \text{dist}(0, K)\). The Laurent coefficients of \(F\) in \(| z | > 1\) was studied in [\textit{J.-P. Lund} et al., Exp. Math. 7, No. 3, 177--190 (1998; Zbl 0959.28006)] and [\textit{X.-H. Dong} and \textit{K.-S. Lau}, J. Funct. Anal. 202, No. 1, 67--97 (2003; Zbl 1032.28005)].
In this paper, we study the asymptotic formula of the Taylor coefficients \(\{ b_{qk - 1} \}_{k = 1}^\infty\) of \(F\) in \(|z| < R_q\). We prove that the supremum of \(\beta\) satisfying \(\{ ( q k )^\beta R_q^{qk} b_{qk - 1} \}_{k = 1}^\infty \in l^\infty\) is the Hausdorff dimension \(\alpha\) of \(K\), and that the accumulation points set of \(\{(qk)^\alpha R_q^{qk} b_{qk - 1}\}_{k = 1}^\infty\) is a union of some non-degenerate line segments.Morse spectra, homology measures, spaces of cycles and parametric packing problems.https://www.zbmath.org/1456.530352021-04-16T16:22:00+00:00"Gromov, Misha"https://www.zbmath.org/authors/?q=ai:gromov.mikhaelA motivating question for this long and densely-written paper is the following: For an ensemble of moving particles in a space, what happens if the effectively observable number of states is replaced by the number of effective/persistent degrees of freedom? ``We suggest in this paper several mathematical counterparts to the idea of persistent degrees of freedom and formulate specific questions, many of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in a ball. And often we present variable aspects of the same ideas in different sections of the paper.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)On spectral eigenvalue problem of a class of self-similar spectral measures with consecutive digits.https://www.zbmath.org/1456.420332021-04-16T16:22:00+00:00"Wang, Cong"https://www.zbmath.org/authors/?q=ai:wang.cong"Wu, Zhi-Yi"https://www.zbmath.org/authors/?q=ai:wu.zhi-yiSummary: Let \(\mu_{p,q}\) be a self-similar spectral measure with consecutive digits generated by an iterated function system \(\{f_i(x)=\frac{x}{p}+\frac{i}{q}\}_{i=0}^{q-1} \), where \(2\le q\in\mathbb{Z}\) and \(q|p\). It is known that for each \(w=w_1w_2\cdots \in \{-1,1\}^\infty :=\{i_1i_2\cdots :\text{ all }i_k \in \{-1,1\}\} \), the set
\[
\Lambda_w=\left\{\sum_{j=1}^na_j w_j p^{j-1}:a_j\in \{0,1,\ldots ,q-1\},n\geq 1\right\}
\]
is a spectrum of \(\mu_{p,q}\). In this paper, we study the possible real number \(t\) such that the set \(t\Lambda_w\) are also spectra of \(\mu_{p,q}\) for all \(w\in \{-1,1\}^\infty\).Scalings and fractals in information geometry: Ornstein-Uhlenbeck processes.https://www.zbmath.org/1456.601282021-04-16T16:22:00+00:00"Oxley, William"https://www.zbmath.org/authors/?q=ai:oxley.william"Kim, Eun-Jin"https://www.zbmath.org/authors/?q=ai:kim.eunjinHigher dimensional divergence for mapping class groups.https://www.zbmath.org/1456.200412021-04-16T16:22:00+00:00"Behrstock, Jason"https://www.zbmath.org/authors/?q=ai:behrstock.jason-a"Druţu, Cornelia"https://www.zbmath.org/authors/?q=ai:drutu.corneliaSummary: In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces. We show that these functions exhibit phase transitions at the quasi-flat rank (as measured by \(3 \cdot \text{ genus }+ \text{ number of punctures }- 3)\).Appendix to the paper ``On the Billingsley dimension of Birkhoff average in the countable symbolic space''.https://www.zbmath.org/1456.370052021-04-16T16:22:00+00:00"Selmi, Bilel"https://www.zbmath.org/authors/?q=ai:selmi.bilelSummary: This appendix gives a lower bound of the Billingsley-Hausdorff dimension of a level set related to Birkhoff average in the \textit{``non-compact''} symbolic space \(\mathbb{N}^{\mathbb{N}}\), defined by Gibbs measure.
This is an appendix to [\textit{N. Attia} and the author, C. R., Math., Acad. Sci. Paris 358, No. 3, 255--265 (2020; Zbl 1445.28005)].Fractal transformed doubly reflected Brownian motions.https://www.zbmath.org/1456.602092021-04-16T16:22:00+00:00"Ehnes, Tim"https://www.zbmath.org/authors/?q=ai:ehnes.tim"Freiberg, Uta"https://www.zbmath.org/authors/?q=ai:freiberg.uta-renataThe geometry of synchronization problems and learning group actions.https://www.zbmath.org/1456.051052021-04-16T16:22:00+00:00"Gao, Tingran"https://www.zbmath.org/authors/?q=ai:gao.tingran"Brodzki, Jacek"https://www.zbmath.org/authors/?q=ai:brodzki.jacek"Mukherjee, Sayan"https://www.zbmath.org/authors/?q=ai:mukherjee.sayanSummary: We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group \(G\) on connected graph \(\Gamma\) with a flat principal \(G\)-bundle over \(\Gamma\), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma\) into \(G\). We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal \(G\)-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham-Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions -- partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations -- and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.