Recent zbMATH articles in MSC 28https://zbmath.org/atom/cc/282024-09-27T17:47:02.548271ZWerkzeugFractal networks: topology, dimension, and complexityhttps://zbmath.org/1541.051642024-09-27T17:47:02.548271Z"Bunimovich, L."https://zbmath.org/authors/?q=ai:bunimovich.leonid-a"Skums, P."https://zbmath.org/authors/?q=ai:skums.p-v(no abstract)Additive and geometric transversality of fractal sets in the integershttps://zbmath.org/1541.110172024-09-27T17:47:02.548271Z"Glasscock, Daniel"https://zbmath.org/authors/?q=ai:glasscock.daniel"Moreira, Joel"https://zbmath.org/authors/?q=ai:moreira.joel"Richter, Florian K."https://zbmath.org/authors/?q=ai:richter.florian-karlIn analogy to \(\times r\)-invariant sets on \(\mathbb{S}^1\) the authors introduce a new class of fractal sets in \(\mathbb{Z}\) and introduce notions of dimension of such sets. They study the additive and geometric independence between two such sets. Particular they find:
\begin{itemize}
\item A classification of all subsets of \(\mathbb{N}\) that are simultaneously \(\times r\) and \(\times s\) invariant.
\item The dimensions of the intersection \(A\cap B\) and the sumset \(A+B\) of \(\times r\) and \(\times s\) invariant sets \(A\) and \(B\) when \(r\) and \(s\) are multiplicaively independent.
\item The dimension of iterated sumsets \(A+\dots+A\) for any \(\times r\) invariant set \(A\).
\end{itemize}
It is interesting to see that concepts develop for the continuum could here by transferred to a discrete setting.
Reviewer: Jörg Neunhäuserer (Goslar)On dimension of the space of derivations on commutative regular algebrashttps://zbmath.org/1541.160132024-09-27T17:47:02.548271Z"Ayupov, Shavkat"https://zbmath.org/authors/?q=ai:ayupov.sh-a"Kudaybergenov, Karimbergen"https://zbmath.org/authors/?q=ai:kudaybergenov.karimbergen-k"Karimov, Khakimbek"https://zbmath.org/authors/?q=ai:karimov.khakimbekThe dimension of the space of derivations on a commutative von Neumann regular subalgebra of a measure space is determined. Specifically, \((\Omega,\Sigma,\mu)\) is a measure space with a finite countably additive measure, \(S(\Omega)\) is the algebra of \(\mathbb{F}\)-valued measurable functions on \((\Omega,\Sigma,\mu)\) (for \(\mathbb{F} = \mathbb{R}\) or \(\mathbb{C}\)), and \(\mathcal{A}\) is a von Neumann regular subalgebra of \(S(\Omega)\). \textit{A. F. Ber} et al. [Extr. Math. 21, No. 2, 107--147 (2006; Zbl 1129.46056)] and \textit{A. G. Kusraev} [Sib. Mat. Zh. 47, No. 1, 97--107 (2006; Zbl 1113.46043); translation in Sib. Math. J. 47, No. 1, 77--85 (2006)] proved that \(S(\Omega)\) admits nonzero derivations if and only if \(\nabla(S(\Omega))\), the Boolean algebra of idempotents in \(S(\Omega)\), is not atomic. Also, \textit{A. F. Ber} [Mat. Tr. 13, No. 1, 3--14 (2010; Zbl 1249.13020); translation in Sib. Adv. Math. 21, No. 3, 161--169 (2011)] showed that \(\dim_{\mathbb{F}} \operatorname{Der} (S([0,1]))\) is uncountable, being at least \(\dim_{\mathbb{F}} S([0,1])^I\) for an uncountable set \(I\).
In the present paper, both \(S(\Omega)\) and \(\mathcal{A}\) are assumed to be homogeneous in the sense that the Boolean algebras \(\nabla(S(\Omega))\) and \(\nabla(\mathcal{A})\) are homogeneous. Moreover, the transcendence degree of \(\mathcal{A}\) over \(\mathbb{F}\), as introduced by the authors [Positivity 26, No. 1, Paper No. 11, 23 p. (2022; Zbl 1494.46055)], is assumed to be infinite, as is the weight \(\tau(\nabla(\mathcal{A}))\), that is, the least cardinality of a set generating \(\nabla(\mathcal{A})\). The authors prove that \(\dim_{\mathbb{F}} \operatorname{Der}(\mathcal{A}) = \tau(\nabla(\mathcal{A}))^{\operatorname{trdeg}(\mathcal{A})}\).
Reviewer: Kenneth R. Goodearl (Santa Barbara)A geometric based connection between fractional calculus and fractal functionshttps://zbmath.org/1541.260252024-09-27T17:47:02.548271Z"Liang, Yong Shun"https://zbmath.org/authors/?q=ai:liang.yongshun"Su, Wei Yi"https://zbmath.org/authors/?q=ai:su.weiyiSummary: Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist. After further discussion, fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Hölder continuous functions. Investigation about other fractional calculus, such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary. This work is helpful to reveal the mechanism of fractional calculus on continuous functions. At the same time, it provides some theoretical basis for the rationality of the definition of fractional calculus. This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.Influence of nonlinearity to box-counting dimensions of spiral orbits for two-dimensional differential systemshttps://zbmath.org/1541.340502024-09-27T17:47:02.548271Z"Onitsuka, Masakazu"https://zbmath.org/authors/?q=ai:onitsuka.masakazu"Tanaka, Satoshi"https://zbmath.org/authors/?q=ai:tanaka.satoshiSummary: This study focuses on spiral orbits approaching the origin of two-dimensional nonautonomous nonlinear systems. The main result explains that the fractal dimensions (box-counting dimensions) of spiral orbits of the systems on the phase plane can be derived from the power of the nonlinear term. The examples given show that when the coefficients are power functions, the balance between their power and the power of the nonlinear term determines the fractal dimension. In addition, some numerical simulations are also included.On Gibbs measures and topological solitons of exterior equivariant wave mapshttps://zbmath.org/1541.353132024-09-27T17:47:02.548271Z"Bringmann, Bjoern"https://zbmath.org/authors/?q=ai:bringmann.bjornSummary: We consider \(k\)-equivariant wave maps from the exterior spatial domain \(\mathbb{R}^3 \setminus B(0,1)\) into the target \(\mathbb{S}^3\). This model has infinitely many topological solitons \(\mathcal{Q}_{n,k}\), which are indexed by their topological degree \(n \in \mathbb{Z}\). For each \(n \in \mathbb{Z}\) and \(k \geq 1\), we prove the existence and invariance of a Gibbs measure supported on the homotopy class of \(\mathcal{Q}_{n,k}\). As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives.Dynamic boundary conditions for time dependent fractional operators on extension domainshttps://zbmath.org/1541.355362024-09-27T17:47:02.548271Z"Creo, Simone"https://zbmath.org/authors/?q=ai:creo.simone"Lancia, Maria Rosaria"https://zbmath.org/authors/?q=ai:lancia.maria-rosariaSummary: We consider a parabolic semilinear non-autonomous problem \((\tilde{P})\) for a fractional time dependent operator \(\mathcal{B}^{s,t}_{\Omega}\) with Wentzell-type boundary conditions in a possibly non-smooth domain \(\Omega\subset\mathbb{R}^N\). We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem \((P)\) via an evolution family \(U(t,\tau)\). We then prove that the mild solution of the abstract proble \((P)\) actually solves problem \((\tilde{P})\) via a generalized fractional Green formula.Generic extensions of ergodic systemshttps://zbmath.org/1541.370022024-09-27T17:47:02.548271Z"Ryzhikov, Valerii V."https://zbmath.org/authors/?q=ai:ryzhikov.valery-vSummary: The paper is devoted to problems concerning the generic properties of extensions of dynamical systems with invariant measures. It is proved that generic extensions preserve the singularity of the spectrum, the mixing property and some other asymptotic properties. It is discovered that the preservation of algebraic properties generally depends on statistical properties of the base. It is established that the \(P\)-entropy of a generic extension is infinite. This fact yields a new proof of the result due to \textit{T. Austin} et al. [Ergodic Theory Dyn. Syst. 43, No. 10, 3216--3230 (2023; Zbl 1531.37008)] on the nondominance of deterministic actions. Generic measurable families of automorphisms of a probability space are considered. It is shown that the asymptotic behaviour of representatives of a generic family is characterized by a combination of dynamic conformism and dynamic individualism.Metric mean dimension via preimage structureshttps://zbmath.org/1541.370132024-09-27T17:47:02.548271Z"Liu, Chunlin"https://zbmath.org/authors/?q=ai:liu.chunlin"Rodrigues, Fagner B."https://zbmath.org/authors/?q=ai:rodrigues.fagner-bLet \((X, d)\) be a compact metric space and \(f : X\to X\) be a continuous map. \textit{M. Hurley} [Ergodic Theory Dyn. Syst. 15, No. 3, 557--568 (1995; Zbl 0833.54021)] introduced the topological preimage entropy of \(f\) as follows
\[
h_m(f)=\lim_{\varepsilon \rightarrow 0} h_m(f,\varepsilon ) =\underset{n\rightarrow \infty }{\lim \sup }\frac{1}{n}\log \underset{x\in X}{\sup }s(n,\varepsilon ,f^{-n}(x)), \] where \(s(n,\varepsilon,f^{-n}(x))\) is the maximal cardinality of all \((n,\varepsilon)\)-separated subsets of \(f^{-n}(x)\) with respect to \(f\). Another potential definition of preimage entropy, proposed by \textit{W.-C. Cheng} and \textit{S. E. Newhouse} [Ergodic Theory Dyn. Syst. 25, No. 4, 1091--1113 (2005; Zbl 1098.37012)], is the following
\[
h_{pre}(f)=\lim_{\varepsilon \rightarrow 0} h_{pre}(f,\varepsilon)=\underset{n\rightarrow \infty }{\lim \sup }\frac{1}{n}\log\underset{x\in X,k\geq n}{\sup }s(n,\varepsilon ,f^{-k}(x)).
\]
In this paper, inspired by earlier definitions of preimage entropy, the authors propose two concepts of upper and lower metric mean dimension based on preimage structures. The authors investigate the measure-theoretic preimage entropy and the preimage metric mean dimension. Inspired by previous studies, they estimate new upper and lower bounds depending on whether the function \(f\) is continuous and surjective. Several intermediate results are given to prove the main theorems.
Reviewer: Hasan Akin (Şanlıurfa)Multifractal analysis of local polynomial entropieshttps://zbmath.org/1541.370142024-09-27T17:47:02.548271Z"Liu, Lei"https://zbmath.org/authors/?q=ai:liu.lei.12"Zhao, Cao"https://zbmath.org/authors/?q=ai:zhao.caoSummary: In this paper, we introduce the Bowen polynomial entropy and study the multifractal spectrum of the local polynomial entropies for arbitrary Borel probability measures.\(q\)-invariance entropy for control systemshttps://zbmath.org/1541.370152024-09-27T17:47:02.548271Z"Zhong, Xingfu"https://zbmath.org/authors/?q=ai:zhong.xingfu"Chen, Zhijing"https://zbmath.org/authors/?q=ai:chen.zhijingSummary: We introduce the notions of \(q\)-invariance entropy and \(q\)-metric invariance entropy for control systems by Carathéodory-Pesin structures. We obtain some basic properties of these entropies and some relations between \(q\)-invariance entropies and invariance pressures. Furthermore, we discuss relations between \(q\)-metric invariance entropies and local metric invariance entropies.Multifractal analysis of measures arising from random substitutionshttps://zbmath.org/1541.370562024-09-27T17:47:02.548271Z"Mitchell, Andrew"https://zbmath.org/authors/?q=ai:mitchell.andrew.1|mitchell.andrew|mitchell.andrew-j"Rutar, Alex"https://zbmath.org/authors/?q=ai:rutar.alexIn this excellent paper, the authors derive symbolic expressions for the \(L^q \)-spectrum of frequency measures associated with random substitutions under certain weak assumptions. With an additional assumption of recognisability, they establish a closed-form analytic expression for the \( L^q \)-spectrum and a variational formula, which together imply the multifractal formalism. Recognisable random substitutions display unique properties not observed in other well-understood classes of measures for the multifractal formalism: often, the unique frequency measure of maximal entropy is not a Gibbs measure with respect to the zero potential, and the corresponding subshift is not sofic. These techniques and results offer significant new insights into the geometry and dynamics of the respective measures.
Let \(\mu\) be a Borel probability measure in a compact metric space. The \( L^q \)-spectrum of \(\mu\) is defined by
\[
\tau _{\mu }(q)=\underset{r\rightarrow 0}{\lim \inf }\frac{\log \sup \sum _i\mu \left(B(x_i,r)\right){}^q}{\log r},
\]
where the supremum is taken over 2\(r\)-separated subsets \(\{x_i\}_i\) of the support of \(\mu\). The authors introduce the definitions of the local dimension of \(\mu\) at \(x\) and the multifractal spectrum of \(\mu\). Given a random substitution \(\vartheta_\mathbf{P}\), the authors define the Perron-Frobenius eigenvalue \(\lambda\) and the corresponding right eigenvector \(\mathbb{R}\) of the substitution matrix associated with \(\vartheta_\mathbf{P}\). In their basic results, the authors prove that for every \(q\geq 0\), the inflation word \( L^q \)-spectrum of \(\vartheta_\mathbf{P}\) and the \( L^q \)-spectrum of \(\mu_\mathbf{P}\) correspond to each other. The authors describe several fundamental characteristics of the \(L^q \)-spectrum of the measure \( \mu \), and investigate a random substitution that is primitive, compatible, and recognizable, along with its associated frequency measure \(\mu_\mathbf{P}\). Many examples, counterexamples and applications are given in the last section.
Reviewer: Hasan Akin (Şanlıurfa)Pseudo-Fubini entire functions on the plane in the sense of Riemannhttps://zbmath.org/1541.460082024-09-27T17:47:02.548271Z"Bernal-González, Luis"https://zbmath.org/authors/?q=ai:bernal-gonzalez.luis"Fernández-Sánchez, Juan"https://zbmath.org/authors/?q=ai:fernandez-sanchez.juanSummary: We prove the existence of a maximal dimensional vector space of real functions on the real plane all of whose nonzero members are bounded, entire, non-Lebesgue-integrable, and satisfy the equality of the two iterated integrals given in the conclusion of the Fubini theorem, with the additional property that these iterated integrals exist in the Riemann sense, but not in the Lebesgue one. Moreover, this vector space is dense in the space of smooth functions on the plane under its natural topology.A trace inequality for solenoidal chargeshttps://zbmath.org/1541.460182024-09-27T17:47:02.548271Z"Raiţă, Bogdan"https://zbmath.org/authors/?q=ai:raita.bogdan"Spector, Daniel"https://zbmath.org/authors/?q=ai:spector.daniel-e"Stolyarov, Dmitriy"https://zbmath.org/authors/?q=ai:stolyarov.dmitry-mSummary: We prove that for \(\alpha \in (d - 1,d)\), one has the trace inequality
\[
\int_{\mathbb{R}^d} |I_{\alpha} F| d\nu \leq C |F|(\mathbb{R}^d)\|\nu\|_{\mathcal{M}^{d-\alpha}(\mathbb{R}^d)}
\]
for all solenoidal vector measures \(F\), i.e., \(F\in M_b(\mathbb{R}^d;\mathbb{R}^d)\) and div \(F = 0\). Here \(I_\alpha\) denotes the Riesz potential of order \(\alpha\) and \(\mathcal M^{d-\alpha }(\mathbb{R}^d)\) the Morrey space of \((d - \alpha )\)-dimensional measures on \(\mathbb{R}^d \).A biorthogonal approach to the infinite dimensional fractional Poisson measurehttps://zbmath.org/1541.460252024-09-27T17:47:02.548271Z"Bendong, Jerome B."https://zbmath.org/authors/?q=ai:bendong.jerome-b"Menchavez, Sheila M."https://zbmath.org/authors/?q=ai:menchavez.sheila-m"da Silva, José Luís"https://zbmath.org/authors/?q=ai:da-silva.jose-luisSummary: In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure \(\pi_{\sigma}^{\beta}\), \(0<\beta\leq 1\), on the dual of Schwartz test function space \(\mathcal{D}^{\prime}\). The Hilbert space \(L^2 (\pi_{\sigma}^{\beta})\) of complex-valued functions is described in terms of a system of generalized Appell polynomials \(\mathbb{P}^{\sigma, \beta, \alpha}\) associated to the measure \(\pi_{\sigma}^{\beta}\). The kernels \(C_n^{\sigma, \beta}(\cdot)\), \(n\in \mathbb{N}_0\), of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system \(\mathbb{P}^{\sigma, \beta, \alpha}\), there is a generalized dual Appell system \(\mathbb{Q}^{\sigma, \beta, \alpha}\) that is biorthogonal to \(\mathbb{P}^{\sigma, \beta, \alpha}\). The test and generalized function spaces associated to the measure \(\pi_{\sigma}^{\beta}\) are completely characterized using an integral transform as entire functions.Dimension-free estimates on distances between subsets of volume \(\varepsilon\) inside a unit-volume bodyhttps://zbmath.org/1541.520132024-09-27T17:47:02.548271Z"Ismailov, Abdulamin"https://zbmath.org/authors/?q=ai:ismailov.abdulamin"Kanel-Belov, Alexei"https://zbmath.org/authors/?q=ai:kanel-belov.alexei"Ivlev, Fyodor"https://zbmath.org/authors/?q=ai:ivlev.fyodorGiven a family of bounded convex bodies in \(\mathbb{R}^n\) with unit volume, consider the supremum of possible Euclidean distances between two subsets of fixed volume \(\varepsilon\in(0,1/2)\). This quantity is denoted by \(d_n(\varepsilon)\). The authors derive asymptotics for \(d_n(\varepsilon)\) when \(n\to\infty\) for various families of convex bodies. For example, for the family of unit-volume Euclidean balls, the authors show that
\[
\lim_{n\to\infty}d_n(\varepsilon)=-\frac{2}{\sqrt{e}}\Phi^{-1}(\varepsilon)\,,
\]
where \(\Phi(a)=\int_{-\infty}^ae^{-\pi x^2}\text{d}x\). Similarly, for the family of unit cubes,
\[
-2\sqrt{\frac{\pi}{6}}\Phi^{-1}(\varepsilon)\leq\liminf_{n\to\infty}d_n(\varepsilon)\leq\limsup_{n\to\infty}d_(\varepsilon)\leq-2\Phi^{-1}(\varepsilon)\,,
\]
and for the family of unit-volume simplices,
\[
-\frac{\sqrt{2}}{e}\ln(2\varepsilon)\leq\liminf_{n\to\infty}d_n(\varepsilon)\leq\limsup_{n\to\infty}d_(\varepsilon)\leq-c\ln(\varepsilon),
\]
for some constant \(c>0\) independent of \(n\) and \(\varepsilon\). Some analogous results are also presented for unit-volume \(\ell_p\) balls for \(p\geq1\), and for the Manhattan distance in place of the Euclidean distance. Proofs make use of isoperimetric inequalities, and links are drawn with concentration of measure.
Reviewer: Fraser Daly (Edinburgh)Group action and \(L^2\)-norm estimates of geometric problemshttps://zbmath.org/1541.520232024-09-27T17:47:02.548271Z"Pham, Thang"https://zbmath.org/authors/?q=ai:pham.thang-van|pham-van-thang.\textit{M. Bennett} et al. [Forum Math. 29, No. 1, 91--110 (2017; Zbl 1429.52020)] applied a group action approach to the study of Erdős-Falconer-type problems in vector spaces over finite fields and used it to obtain non-trivial exponents for the distribution of simplices. The paper under review gives a number of new applications of their powerful framework, on the product and quotient of distance sets, the \(L^2\) norm of the direction set and the \(L^2\) norm of scales in difference sets. The proofs employ Fourier analysis.
Reviewer: László A. Székely (Columbia)Wellposedness and controllability results of stochastic integrodifferential equations with noninstantaneous impulses and Rosenblatt processhttps://zbmath.org/1541.600462024-09-27T17:47:02.548271Z"Kasinathan, Ravikumar"https://zbmath.org/authors/?q=ai:kasinathan.ravikumar"Kasinathan, Ramkumar"https://zbmath.org/authors/?q=ai:kasinathan.ramkumar"Sandrasekaran, Varshini"https://zbmath.org/authors/?q=ai:sandrasekaran.varshini"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose(no abstract)On representations of the Helmholtz Green's functionhttps://zbmath.org/1541.810382024-09-27T17:47:02.548271Z"Beylkin, Gregory"https://zbmath.org/authors/?q=ai:beylkin.gregorySummary: We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number \(k\). The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in \(\mathcal{O} (k^d \log k)\) operations. The non-oscillatory component has a multiresolution representation via a linear combination of Gaussians and is applied efficiently in space.
Since the Helmholtz Green's function can be viewed as a point source, this partitioning can be interpreted as a splitting into propagating and evanescent components. We show that the non-oscillatory component is significant only in the vicinity of the source at distances \(\mathcal{O} (c_1 k^{- 1} + c_2 k^{- 1} \log_{10} k)\), for some constants \(c_1, c_2\), whereas the propagating component can be observed at large distances.Semiclassical approximation of functional integrals containing the centrifugal potentialhttps://zbmath.org/1541.810622024-09-27T17:47:02.548271Z"Malyutin, Viktor Borisovich"https://zbmath.org/authors/?q=ai:malyutin.viktor-borisovich"Nurzhanov, Berdakh Orynbaevich"https://zbmath.org/authors/?q=ai:nurzhanov.berdakh-orynbaevichSummary: In this paper, we consider the class of functional integrals with respect to the conditional Wiener measure, which is important for applications. These integrals are written using the action functional containing terms corresponding to kinetic and potential energies. For the considered class of integrals an approach to semiclassical approximation is developed. This approach is based on the decomposition of the action with respect to the classical trajectory. In the expansion of the action, only terms with degrees zero and two are used. A numerical analysis of the accuracy of the semiclassical approximation for functional integrals containing the centrifugal potential is done.Configurational Entropy and the \(\mathcal{N}^\ast(1440)\) Roper resonance in QCDhttps://zbmath.org/1541.812092024-09-27T17:47:02.548271Z"Karapetyan, G."https://zbmath.org/authors/?q=ai:karapetyan.garnik-a|karapetyan.gevork-yaSummary: The electroexcitation of the \(\mathcal{N}^\ast(1440)\) Roper resonance, which defines the first radially excited state of the nucleon, is examined within the soft-wall AdS/QCD model. Such excited Fock states are characterized by the leading three-quark component, which determines the main properties of Roper resonance. The differential configurational entropy (DCE) was used in the nuclear interaction with a gauge vector field for \(\mathcal{N}^\ast(1440)\) transition. Comparing the main results with the recent data of the CLAS Collaboration at JLab shows a good agreement on the accuracy of the computed data.Quantifying the distortion by spin-orbit and spin-spin coupling in molecular clusters using molecular quantum similarityhttps://zbmath.org/1541.812232024-09-27T17:47:02.548271Z"Morales-Bayuelo, Alejandro"https://zbmath.org/authors/?q=ai:morales-bayuelo.alejandroSummary: The manuscript discusses the concepts of spin-orbit and spin-spin coupling in atomic physics and Molecular Quantum Similarity (MQS) in molecular clusters. spin-orbit and spin-spin coupling arises from the interaction between an electron's spin and its motion around the nucleus and electron-electron interaction and plays a crucial role in determining energy levels and spectral lines in atoms with heavy nuclei. On the other hand, MQS is a computational approach to compare the electronic density distributions in different molecular systems. In this order of ideas, the study aims to answer questions about electronic and structural differences caused by the spin-orbit and spin-spin coupling from the initial geometry [Steradians (SR) geometry] using the MQS framework. The MQS is based on the Molecular Quantum Similarity Measure (MQSM) using different positive operators such as Dirac delta and Coulomb operators to quantify the similarity between molecular systems. The paper presents tables with MQSM indices and Euclidean distances for different molecular clusters using initial geometry vs. geometry involved spin-orbit and spin-spin coupling. The scalar, spin-orbit and spin-spin relativistic coupling were incorporated using Amsterdam Density Functional code. The results show significant coupling of spin-orbit and spin-spin coupling on the similarity measures between different molecules. The manuscript suggests that understanding the relationship between spin-orbit and spin-spin coupling and quantum similarity could lead to deeper insights into electronic interactions in complex molecular systems and has potential applications in quantum mechanics and molecular physics.Density of states for the Anderson model on nested fractalshttps://zbmath.org/1541.820082024-09-27T17:47:02.548271Z"Balsam, Hubert"https://zbmath.org/authors/?q=ai:balsam.hubert"Kaleta, Kamil"https://zbmath.org/authors/?q=ai:kaleta.kamil"Olszewski, Mariusz"https://zbmath.org/authors/?q=ai:olszewski.mariusz"Pietruska-Pałuba, Katarzyna"https://zbmath.org/authors/?q=ai:pietruska-paluba.katarzynaLet \(\mathfrak{K}^{\infty}\) be a planar unbounded simple nested fractal with the Good Labeling Property and let \( \mathfrak{L}\) be the associated Laplacian. The authors propose and study the random Anderson operator \(H^\omega = H_0 + V^\omega\), acting in \(L^2(\mathfrak{K}^{\infty},\mu )\) with the kinetic term \(H_0\) takes the form \(\phi(-\mathfrak{L}),\) for a sufficiently regular operator monotone function \(\phi,\) and \(V^\omega\) is the operator of multiplication by a function that is called a fractal alloy-type potential. The main goal of this paper is to establish the existence and then to study asymptotic properties of the integrated density of states of the operator \(H^\omega.\)
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Fractal and fractional SIS model for syphilis datahttps://zbmath.org/1541.920892024-09-27T17:47:02.548271Z"Gabrick, Enrique C."https://zbmath.org/authors/?q=ai:gabrick.enrique-c"Sayari, Elaheh"https://zbmath.org/authors/?q=ai:sayari.elaheh"Souza, Diogo L. M."https://zbmath.org/authors/?q=ai:souza.diogo-l-m"Borges, Fernando S."https://zbmath.org/authors/?q=ai:borges.fernando-s"Trobia, José"https://zbmath.org/authors/?q=ai:trobia.jose"Lenzi, Ervin K."https://zbmath.org/authors/?q=ai:kaminski-lenzi.ervin"Batista, Antonio M."https://zbmath.org/authors/?q=ai:batista.antonio-marcos(no abstract)Inequalities for entropies and dimensionshttps://zbmath.org/1541.940322024-09-27T17:47:02.548271Z"Shen, Alexander"https://zbmath.org/authors/?q=ai:shen.alexanderSummary: We show that linear inequalities for entropies have a natural geometric interpretation in terms of Hausdorff and packing dimensions, using the point-to-set principle and known results about inequalities for complexities, entropies and the sizes of subgroups.
For the entire collection see [Zbl 1528.68022].