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We introduce an atomistic classifier based on a combination of spectral graph theory and a Voronoi tessellation method. This classifier allows for the discrimination between structures from different minima of a potential energy surface, making it a useful tool for sorting through large datasets of atomic systems. We incorporate the classifier as a filtering method in the Global Optimization with First-principles Energy Expressions (GOFEE) algorithm. Here, it is used to filter out structures from exploited regions of the potential energy landscape, whereby the risk of stagnation during the searches is lowered. We demonstrate the usefulness of the classifier by solving the global optimization problem of two-dimensional pyroxene, three-dimensional olivine, Au12, and Lennard-Jones LJ55 and LJ75 nanoparticles. [ABSTRACT FROM AUTHOR]

In this paper, Eu3+ doped multi-component transparent aluminosilicate and oxyfluoride aluminosilicate glasses were fabricated by the melt-quenching procedure, and the glass-ceramics containing LiAlSi 2 O 6 microcrystals (LAS glass-ceramics) were produced by a two-step heat treatment process. The XRD, SEM, and EDS data demonstrate that spherical single-phase LiAlSi 2 O 6 microcrystals have precipitated from the glass matrices. The microhardness and flexural strength of the glass-ceramics vary in 6.27–6.66 GPa and 89.18–155.75 MPa, respectively. Phonon sideband analysis reveals that the phonon energies for both fluorine-containing and fluorine-free LAS glass-ceramics are approximately 971 cm−1, and the multiphonon relaxation rates (W mp / W 0) of the former are lower than those of the latter. Given its lower W mp / W 0 values, the fluorine-containing LAS glass-ceramic exhibits a higher emission intensity and a longer lifetime. Under 394 nm excitation, the fluorine-containing LAS glass-ceramic presents a warm and nearly pure red emission, characterized by a correlated color temperature of less than 3300 K and a color purity approaching 100 %. Time-resolved emission spectra support the photoluminescent stability of the Eu3+ ions. According to the Judd-Ofelt theory and spectral results, the fluorine-containing LAS glass-ceramic possesses lower Ω 2 / Ω 4 and asymmetry ratios, leading to a more symmetrical local environment for the Eu3+ ions. Benefiting from its enhanced red emission, the 5D 0 →7F 2 transition in the fluorine-containing LAS glass-ceramic exhibits a higher branching ratio (69.28 % > 50 %), emission cross-section (8.94 × 10−22 cm2), and gain bandwidth (12.12 × 10−28 cm3). Additionally, temperature-dependent photoluminescence spectra substantiate the excellent thermal stability of the Eu3+ doped fluorine-containing LAS glass-ceramic. Specifically, the intensity of the 5D 0 →7F 2 transition at 150 °C remains 80 % of the initial intensity. Therefore, the Eu3+ doped multi-component transparent oxyfluoride aluminosilicate glass-ceramic has a significant competitive advantage in the field of red lasers and trichromatic WLEDs. [ABSTRACT FROM AUTHOR]

In this paper, we study block Jacobi operators on ℤ$$ \mathrm{\mathbb{Z}} $$ with quasi‐periodic meromorphic potential. We prove the nonperturbative Anderson localization for such operators in the large coupling regime. [ABSTRACT FROM AUTHOR]

One hundred ninety years ago, Lamé introduced the celebrated ellipsoidal system, which is an orthogonal curvilinear system that incorporates any anisotropic characteristics of three‐dimensional space. Since then, the theory of ellipsoidal harmonics has been successfully developed, but almost nothing has been achieved for ellipsoidal waves. This is due to the fact that, in order to deal with the ellipsoidal harmonics, Lamé introduced some ingenious arguments that led to a complete spectral theory for the Kernel space of the Laplacian, but no such arguments have been proposed for the Kernel space of the d'Alembert operator. So, one possible approach to develop a method that will give us ellipsoidal wave functions is to try to generalize the Lamé arguments (to the extent that this is possible) to four‐dimensional ellipsoidal harmonics. This procedure will give us the hyperellipsoidal harmonics. Then we can replace the fourth coordinate by ict in order for the four‐dimensional Laplace operator to become the three‐dimensional wave operator of d'Alembert and the four‐dimensional harmonics to become three dimensional waves. The present work is the first stage of this project. It develops the geometry of a four‐dimensional ellipsoidal system and calculates the Laplace operator in this hyperellipsoidal system. [ABSTRACT FROM AUTHOR]

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph G π are a subset of its eigenvalues, i.e. σ (G π) ⊆ σ (G). A natural question is whether it is possible to recover the remaining eigenvalues σ (G) − σ (G π) in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition. [ABSTRACT FROM AUTHOR]

We prove the inequality λ k ≤ d k + d k − 1 for all the eigenvalues λ 1 ≤ λ 2 ≤ ⋯ ≤ λ n of the Kirchhoff matrix K of a finite simple graph or quiver with vertex degrees d 1 ≤ d 2 ≤ ⋯ ≤ d n and assuming d 0 = 0. Without multiple connections, the inequality λ k ≥ max (0 , d k − (n − k)) holds. A consequence in the finite simple graph or multi-graph case is that the pseudo determinant Det (K) counting the number of rooted spanning trees has an upper bound 2 n ∏ k = 1 n d k and that det (1 + K) counting the number of rooted spanning forests has an upper bound ∏ k = 1 n (1 + 2 d k). [ABSTRACT FROM AUTHOR]

We consider a non-self-adjoint H acting on a complex Hilbert space. We suppose that H is of the form H = H 0 + C W C where C is a bounded, positive definite and relatively compact with respect to H 0 , and W is bounded. We suppose that C (H 0 − z) − 1 C is uniformly bounded in z ∈ ℂ ∖ ℝ. We define the regularized wave operators associated to H and H 0 by W ± (H , H 0) : = s-lim t → ∞ e ± i t H r ∓ (H) Π p (H ⋆) ⊥ e ∓ i t H 0 where Π p (H ⋆) is the projection onto the direct sum of all the generalized eigenspaces associated to eigenvalues of H ⋆ and r ∓ is a rational function that regularizes the "incoming/outgoing spectral singularities" of H. We prove the existence and study the properties of the regularized wave operators. In particular, we show that they are asymptotically complete if H does not have any spectral singularity. [ABSTRACT FROM AUTHOR]

Community composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction–diffusion–advection predator–prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near‐complete global dynamics of the system. In instances where the functional response to predation conforms to Holling‐type I or II, we explore the uniqueness and stability of positive steady‐state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species. [ABSTRACT FROM AUTHOR]

In this article we investigate no-resonance conditions for quantum many body chaotic systems and random matrix models. No-resonance conditions are properties of the spectrum of a model, usually employed as a theoretical tool in the analysis of late time dynamics. The first order no-resonance condition holds when a spectrum is non-degenerate, while higher order no-resonance conditions imply sums of an equal number of energies are non-degenerate outside of permutations of the indices. This resonance condition is usually assumed to hold for quantum chaotic models. In this work we use several tests from random matrix theory to demonstrate that the statistics of sums of eigenvalues, that are of interest to due to the no-resonance conditions, have Poisson statistics, and lack level repulsion. This result is produced for both a quantum chaotic Hamiltonian as well as the Gaussian Unitary Ensemble and Gaussian Orthogonal Ensemble. This implies some models may have violations of the no-resonance condition or "near" violations. We finish the paper by generalizing important bounds in quantum equilibration theory to cases where the no-resonance conditions are violated, and to the case of random matrix models. [ABSTRACT FROM AUTHOR]

The exploration of large-scale unknown environments can benefit from the deployment of multiple robots for collaborative mapping. Each robot explores a section of the environment and communicates onboard pose estimates and maps to a central server to build an optimized global multi-robot map. Naturally, inconsistencies can arise between onboard and server estimates due to onboard odometry drift, failures, or degeneracies. The mapping server can correct and overcome such failure cases using computationally expensive operations such as inter-robot loop closure detection and multi-modal mapping. However, the individual robots do not benefit from the collaborative map if the mapping server provides no feedback. Although server updates from the multi-robot map can greatly alleviate the robotic mission strategically, most existing work lacks them, due to their associated computational and bandwidth-related costs. Motivated by this challenge, this paper proposes a novel collaborative mapping framework that enables global mapping consistency among robots and the mapping server. In particular, we propose graph spectral analysis, at different spatial scales, to detect structural differences between robot and server graphs, and to generate necessary constraints for the individual robot pose graphs. Our approach specifically finds the nodes that correspond to the drift's origin rather than the nodes where the error becomes too large. We thoroughly analyze and validate our proposed framework using several real-world multi-robot field deployments where we show improvements of the onboard system up to 90% and can recover the onboard estimation from localization failures and even from the degeneracies within its estimation. [ABSTRACT FROM AUTHOR]

Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2 × 2 canonical systems). We prove a number of Szegő-type theorems for generalized indefinite strings and related spectral problems (including Krein strings, canonical systems and Dirac operators). More specifically, for several classes of coefficients (that can be regarded as Hilbert–Schmidt perturbations of model problems), we provide a complete characterization of the corresponding set of spectral measures. In particular, our results also apply to the isospectral Lax operator for the conservative Camassa–Holm flow and allow us to establish existence of global weak solutions with various step-like initial conditions of low regularity via the inverse spectral transform. [ABSTRACT FROM AUTHOR]

The issue of opinion sharing and formation has received considerable attention in the academic literature, and a few models have been proposed to study this problem. However, existing models are limited to the interactions among nearest neighbors, with those second, third, and higher-order neighbors only considered indirectly, despite the fact that higher-order interactions occur frequently in real social networks. In this paper, we develop a new model for opinion dynamics by incorporating long-range interactions based on higher-order random walks that can explicitly tune the degree of influence of higher-order neighbor interactions. We prove that the model converges to a fixed opinion vector, which may differ greatly from those models without higher-order interactions. Since direct computation of the equilibrium opinion is computationally expensive, which involves the operations of huge-scale matrix multiplication and inversion, we design a theoretically convergence-guaranteed estimation algorithm that approximates the equilibrium opinion vector nearly linearly in both space and time with respect to the number of edges in the graph. We conduct extensive experiments on various social networks, demonstrating that the new algorithm is both highly efficient and effective. [ABSTRACT FROM AUTHOR]

Additive perturbations, specifically matrix Uvarov transformations for matrix orthogonal polynomials, are under consideration. Christoffel–Uvarov formulas are deduced for the perturbed biorthogonal families, along with their matrix norms. These formulations are expressed in terms of the spectral jets of the Christoffel–Darboux kernels. [ABSTRACT FROM AUTHOR]

"Which graphs are determined by their spectra (DS for short)?" is an important and challenging topic in spectral graph theory. Let G be an n -vertex graph with adjacency matrix A (G) and adjacency walk matrix W A (G) = [ e , A (G) e , ... , A n − 1 (G) e ]. We call G controllable if the rank of W A (G) is n over R and noncontrollable otherwise. In Wang (2017), the author gave a simple condition for a controllable graph to be determined by their generalized adjacency spectrum (DGAS for short). However, the result fails for noncontrollable graphs. In this paper, we consider the problem in the context of the generalized distance spectrum. Let G be a connected graph with distance matrix D. A graph G is determined by its generalized distance spectrum (DG D S for short) if any graph H sharing the same generalized distance spectrum with G must be isomorphic to G. The paper aims to extend the result in Wang (2017) from generalized adjacency spectrum to generalized distance spectrum. Besides, a simple arithmetic criterion is given to determine whether a connected graph is DG D S or not, which is applicable to some kind of noncontrollable graphs, e.g., regular graphs. [ABSTRACT FROM AUTHOR]

In this paper, a distributed algorithm for reaching average consensus is proposed for multi-agent systems with tree communication graph, when the edge weight distribution is unbalanced. First, the problem is introduced as a key topic of core algorithms for several modern scenarios. Then, the relative solution is proposed as a finite-time algorithm, which can be included in any application as a preliminary setup routine, and it is well-suited to be integrated with other adaptive setup routines, thus making the proposed solution useful in several practical applications. A special focus is devoted to the integration of the proposed method with a recent Laplacian eigenvalue allocation algorithm, and the implementation of the overall approach in a wireless sensor network framework. Finally, a worked example is provided, showing the significance of this approach for reaching a more precise average consensus in uncertain scenarios. [ABSTRACT FROM AUTHOR]

We study singular Sturm–Liouville operators of the form 1 r j (− d d x p j d d x + q j) , j = 0 , 1 , in L 2 ((a , b) ; r j) with endpoints a and b in the limit point case, where, in contrast to the usual assumptions, the weight functions r j have different signs near a and b. In this situation the associated maximal operators become self-adjoint with respect to indefinite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights r j are allowed. Special attention is paid to Kneser type results in the indefinite setting and to L 1 perturbations of periodic operators. [ABSTRACT FROM AUTHOR]

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph D having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that ρ (D) ≥ α j = ∑ p = 1 m (χ j + 1 (p)) 2 ∑ p = 1 m (χ j (p)) 2 , such that equality is attained iff D = G ↔ + { DE ∉ Cycle}, where each component of associated graph is a k-regular or (k 1 , k 2) semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of D , the sequence of upper bounds of energy of D , where the sequence decreases when e U ≤ α j and increases when e U > α j , are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences. [ABSTRACT FROM AUTHOR]

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account of the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022. [ABSTRACT FROM AUTHOR]

The classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC '94) for space-bounded computation uses a seed of length O (log n · log (n w / ε) + log d) to fool ordered branching programs of length n, width w, and alphabet size d to within error ε . A series of works have shown that the analysis of the INW generator can be improved for the class of permutation branching programs or the more general regular branching programs, improving the O (log 2 n) dependence on the length n to O (log n) or O ~ (log n) . However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length O (log (n w d / ε)) . In this paper, we prove that any "spectral analysis" of the INW generator requires seed length Ω log n · log log min { n , d } + log n · log w / ε + log d to fool ordered permutation branching programs of length n, width w, and alphabet size d to within error ε . By "spectral analysis" we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size d = 2 except for a gap between their O log n · log log n term and our Ω log n · log log min { n , d } term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width ( w = O (1) ) permutation branching programs of alphabet size d = 2 to within a constant factor. To fool permutation branching programs in the measure of spectral norm, we prove that any spectral analysis of the INW generator requires a seed of length Ω log n · log log n + log n · log (1 / ε) when the width is at least polynomial in n ( w = n Ω (1) ), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS 2021) to within a constant factor. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the eigenvalues and the integral of the corresponding eigenfunctions, associated to the Newtonian potential operator, defined in a bounded domain \Omega \subset \mathbb {R}^{d}, where d=2,3, in terms of the maximum radius of \Omega. We first provide these estimations in the particular case of a ball and a disc. Then we extend them to general shapes using a, derived, monotonicity property of the eigenvalues of the Newtonian operator. The derivation of the lower bounds is quite tedious for the 2D-Logarithmic potential operator. Such upper/lower bounds appear naturally while estimating the electric/acoustic fields propagating in \mathbb {R}^{d} in the presence of small scaled and highly heterogeneous particles. [ABSTRACT FROM AUTHOR]

We develop a complete obstruction theory for the Z 2 -index of a compact connected 4-dimensional manifold with free involution. This Z 2 -index, equal to the minimum integer n for which there exists an equivariant map with target the n-sphere with antipodal involution, is computed in two steps using cohomology with twisted coefficients. The key ingredient is a spectral sequence computing twisted cohomology of the orbit space of a free involution on odd complex projective spaces. We illustrate the main results with various examples including computation of the secondary obstruction. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the principal spectral theory and asymptotic behavior of the spectral bound for cooperative nonlocal dispersal systems, specifically focusing on the case where partial diffusion coefficients are zero, referred to as the partially degenerate case. We propose two sufficient conditions that ensure the existence of the principal eigenvalue in these partially degenerate systems. Additionally, we study the asymptotic behavior of the spectral bound for nonlocal dispersal operators with small and large diffusion coefficients, considering both non-degenerate and partially degenerate cases. Notably, we find a threshold-type result as the diffusion coefficients tend towards infinity in the partially degenerate case. Finally, we apply these findings to discuss the asymptotic behavior of the basic reproduction ratio in a viral diffusion model. [ABSTRACT FROM AUTHOR]

Pumped storage units serve as a crucial support for power systems to adapt to large-scale and high-proportion renewable energy sources by providing a stable and flexible energy supply. However, due to the coupling effects of electric power load demands and the complex multi-source factors within the water–mechanical–electrical system, the interrelationship between unit parameters becomes more intricate, posing significant threats to the operational reliability and health status of the units. The complexity of fault diagnosis is further aggravated by the intricate and varied nature of fault characteristics, as well as the challenges in signal extraction under conditions of strong electromagnetic interference and high noise levels. To address these issues, this paper proposes a novel data-model hybrid-driven strategy that analyzes vibration signals to achieve rapid and accurate fault diagnosis of the units. Firstly, the spectral kurtosis theory is employed to enhance the traditional empirical mode decomposition, achieving optimal decomposition and noise reduction effects for vibration signals. Secondly, the intrinsic mode functions (IMFs) obtained from the decomposition are reconstructed, and the entropy values of effective IMFs are calculated as fault feature vectors. Subsequently, the CNN-LSTM model is utilized for fault diagnosis. The effectiveness and feasibility of the proposed method are verified through actual operational data from pumped storage units in a specific region. Through analysis, the fault diagnosis accuracy of the method proposed in this paper can be maintained above 95%, demonstrating robustness in complex engineering environments and effectively ensuring the safe and stable operation of pumped storage units. [ABSTRACT FROM AUTHOR]

A general theory for solving electromagnetic diffraction problems with impenetrable/penetrable wedges immersed in/made of an arbitrary linear (bianistropic) medium is presented. This novel and general spectral theory handles complex scattering problems by using transverse equations for layered planar and angular structures, the characteristic Green function procedure, the Wiener–Hopf technique, and a new methodology for solving GWHEs. The technique has been proven effective for analyzing problems involving wedges immersed in isotropic media; in this study, we extend the theory to more general cases while providing all necessary mathematical tools and corresponding validations. We obtain generalized Wiener–Hopf equations (GWHEs) from spectral functional equations in angular regions filled by arbitrary linear media. The equations can be interpreted with a network formalism for a systematic view. We recall that spectral methods (such as the Sommerfeld–Malyuzhinets (SM) method, the Kontorovich–Lebedev (KL) transform method, and the Wiener–Hopf (WH) method) are well-consolidated, fundamental, and effective tools for the correct and precise analysis of electromagnetic diffraction problems constituted by abrupt discontinuities immersed in media with one propagation constant, although they are not immediately applicable to multiple-propagation-constant problems. To the best of our knowledge, the proposed mathematical technique is the first extension of spectral analysis to electromagnetic problems in the presence of angular regions filled by complex arbitrary linear media, thereby providing novel mathematical tools. Validation through fundamental examples is proposed. [ABSTRACT FROM AUTHOR]

The success of many machine learning (ML) methods depends crucially on having large amounts of labeled data. However, obtaining enough labeled data can be expensive, time-consuming, and subject to ethical constraints for many applications. One approach that has shown tremendous value in addressing this challenge is semi-supervised learning (SSL); this technique utilizes both labeled and unlabeled data during training, often with much less labeled data than unlabeled data, which is often relatively easy and inexpensive to obtain. In fact, SSL methods are particularly useful in applications where the cost of labeling data is especially expensive, such as medical analysis, natural language processing, or speech recognition. A subset of SSL methods that have achieved great success in various domains involves algorithms that integrate graph-based techniques. These procedures are popular due to the vast amount of information provided by the graphical framework. In this work, we propose an algebraic topology-based semi-supervised method called persistent Laplacian-enhanced graph MBO by integrating persistent spectral graph theory with the classical Merriman–Bence–Osher (MBO) scheme. Specifically, we use a filtration procedure to generate a sequence of chain complexes and associated families of simplicial complexes, from which we construct a family of persistent Laplacians. Overall, it is a very efficient procedure that requires much less labeled data to perform well compared to many ML techniques, and it can be adapted for both small and large datasets. We evaluate the performance of our method on classification, and the results indicate that the technique outperforms other existing semi-supervised algorithms. [ABSTRACT FROM AUTHOR]

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three there exists a cone such that the heat kernel (the continuous analog of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles. [ABSTRACT FROM AUTHOR]

Transfer matrix techniques are used to provide a new proof of Widom's results on the asymptotic spectral theory of finite block Toeplitz matrices. Furthermore, a rigorous treatment of the skin effect, spectral outliers, the generalized Brillouin zone and the bulk-boundary correspondence in such systems is given. This covers chiral Hamiltonians with topological eigenvalues close to zero, but no line-gap. [ABSTRACT FROM AUTHOR]

In this paper, we prove pure point spectrum for a large class of Schrödinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough. [ABSTRACT FROM AUTHOR]

We define k-dimensional digraphs and initiate a study of their spectral theory. The kdimensional digraphs can be viewed as generating graphs for small categories called k-graphs. Guided by geometric insight, we obtain several new series of k-graphs using cube complexes covered by Cartesian products of trees, for k ≥ 2. These k-graphs can not be presented as virtual products and constitute novel models of such small categories. The constructions yield rank-k Cuntz–Krieger algebras for all k≥2. We introduce Ramanujan k-graphs satisfying optimal spectral gap property and show explicitly how to construct the underlying k-digraphs. [ABSTRACT FROM AUTHOR]

In recent years, discrete notions of curvature have been defined and exploited to understand various geometric properties of graphs; especially regarding heat flow, and spectral properties. In this paper, we study various combinatorial properties implied by satisfying the Bakry–Émery curvature dimension inequality CD(∞,K) $CD(\infty ,K)$. In particular we derive a local discrepancy inequality, similar in spirit to the expander mixing lemma from spectral graph theory, which certifies a type of “local pseudo‐randomness” of the edge set of the graph, for graphs satisfying a curvature lower bound. In addition, several other consequences are derived regarding graph connectivity and cycle statistics of the graph. [ABSTRACT FROM AUTHOR]

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series. [ABSTRACT FROM AUTHOR]

We show that the spectral theorem – which we understand to be a statement that every self-adjoint matrix admits a certain type of canonical form under unitary similarity – admits analogues over other $ * $ ∗ -algebras distinct from the complex numbers. If these $ * $ ∗ -algebras contain nilpotents, then it is shown that there is a consistent way in which many classic matrix decompositions – such as the Singular Value Decomposition, the Takagi decomposition, the skew-Takagi decomposition, and the Jordan decomposition, among others – are immediate consequences of these. If producing the relevant canonical form of a self-adjoint matrix were a subroutine in some programming language, then the corresponding classic matrix decomposition would be a 1-line invocation with no additional steps. We also suggest that by employing operator overloading in a programming language, a numerical algorithm for computing a unitary diagonalization of a complex self-adjoint matrix would generalize immediately to solving problems like SVD or Takagi. While algebras without nilpotents (like the quaternions) allow for similar unifying behaviour, the classic matrix decompositions which they unify are never obtained as easily. In the process of doing this, we develop some spectral theory over Clifford algebras of the form $ Cl_{p,q,0}(\mathbb R) $ C l p , q , 0 (R) and $ Cl_{p,q,1}(\mathbb R) $ C l p , q , 1 (R) where the former is admittedly quite easy. We propose a broad conjecture about spectral theorems. [ABSTRACT FROM AUTHOR]

The spectral graph theory explores connections between combinatorial features of graphs and algebraic properties of associated matrices. In this paper, we introduce modified maximum degree matrix MM(ζ) of a simple graph ζ and obtain a bound for eigenvalues of MM(ζ). We also introduce modified maximum degree energy ...(ζ) of a graph ζ and obtain bounds for E...(ζ). [ABSTRACT FROM AUTHOR]

According to the Bogoliubov theory the low energy behaviour of the Bose gas at zero temperature can be described by non-interacting bosonic quasiparticles called phonons. In this work the damping rate of phonons at low momenta, the so-called Beliaev damping, is explained and computed with simple arguments involving the Fermi Golden Rule and Bogoliubov’s quasiparticles. [ABSTRACT FROM AUTHOR]

Chain graphs and threshold graphs play a very important role in Spectral Graph Theory, since the maximizers for the largest eigenvalue of the adjacency matrix (for graphs of fixed order and size, either connected or disconnected) belong to these classes (threshold graphs in the general case, and chain graphs in the bipartite case). Nesting in the neighborhood of vertices in these graphs has gained the attention of various researchers. Motivated by this structure, we generalize and define a new class of graphs named it as 'partial threshold graphs' and study the properties. In this article, we give bounds and expressions for the Wiener index and Hyper-Wiener index of a partial threshold graph. We extend the study further and give a set of integers, except which every other integer is the Wiener index of some partial threshold graph. The highlight of the article is an algorithm for the inverse Wiener index problem of partial threshold graphs. [ABSTRACT FROM AUTHOR]

We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of Γ -calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by Klartag and Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general φ 4 -measures and discuss the interpretation in terms of transport maps. [ABSTRACT FROM AUTHOR]

This paper proves new results on spectral and scattering theory for matrix-valued Schrödinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half-bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations. [ABSTRACT FROM AUTHOR]

The original Reinforcement Sensitivity Theory (oRST) proposes two systems of approach (BAS) and avoidance (BIS) motivation to underpin personality and behavior. The revised‐RST (rRST) model separates avoidance motivation into passive (BIS; anxiety) and active (FFFS; fear) systems. Prior research has attempted to map RST onto lateralized frontal asymmetry to provide a neurophysiological marker of RST. The main aim is to examine the relationships of the o/rRST scales with trait (baseline) and state (manipulated through experimental paradigms) frontal asymmetry. A systematic review was conducted, resulting in 158 studies designated to neuroimaging research. In total, 54 studies were included in this review using either frontal asymmetry or spectral power. The results were split into three main categories: resting frontal alpha asymmetry (N = 23), emotional induction and state‐related frontal alpha asymmetry (N = 20), and spectral analysis (N = 16). Findings indicated that BAS was associated with enhanced left frontal asymmetry at baseline and during state‐related paradigms. Findings for BIS were more inconsistent, especially at rest, suggesting that BIS, in particular, may require active engagement with the environment. Only 9 of the 54 papers included used the revised RST model, highlighting the need for more rRST research. In a systematic review of reinforcement sensitivity (RST) and frontal asymmetry, we observed an over‐reliance on the original RST model (BIS/BAS). This may contribute to inconsistent findings between BIS and resting frontal asymmetry, as the original RST conflates passive (anxiety) and active (fear) avoidance into one system. We argue for the shift towards the revised‐RST model. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (reflected) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution. [ABSTRACT FROM AUTHOR]

Inspired by the 1995 paper of Baake–Grimm–Pisani, we aim to explain the empirical observation that the distribution of Lee–Yang zeros corresponding to a one-dimensional Ising model appears to follow the gap labelling theorem. This follows by combining two main ingredients: first, the relation between the transfer matrix formalism for the 1D Ising model and an ostensibly unrelated matrix formalism generating the Szegő recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices. [ABSTRACT FROM AUTHOR]

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In this paper, the concept of dual framelets on manifolds and its characterization are introduced. The accuracy of the proposed dual framelets transform is determined by sparse representation on graphs. If any pair of the framelet system is associated with filter-bank transform, then compactly supported refinable functions can have vanishing moments at most one and framelet approximation is the order of at most two. An algorithm of decomposition and reconstruction for the dual framelets transform on graph is presented. A new method called dual framelets filter-bank transform (DFFT) is employed, which is faster than the existing method spectral graph wavelet transform (SGWT). The theoretical results along with algorithms for accurate and efficient computation of the DFFT on discrete data sets are provided. Subsequently, some numerical examples are provided to show the importance of DFFT over SGWT on graphs. [ABSTRACT FROM AUTHOR]

Diffusion is a ubiquitous process in real-world syetems. In many complex systems, ranging from neuronal networks to traffic in cities, diffusion is nonconservative (NC) in the sense that diffusive particles can be created/annihilated at the entities of the system. Here, we consider the important problem of identifying potential navigational bottlenecks in NC diffusion occurring in the networks representing skeletons of complex systems. We develop a first-principles approach based on an NC diffusion using the Lerman-Ghosh Laplacian on graphs. By solving analytically this NC diffusion equation at two different times, we get an index which characterizes the capacity of every vertex in a network to spread the diffusive particles across the network in a short time. Vertices having such capacity diminished are potential navigational bottlenecks in this kind of dynamics. We solve analytically the situations in which the vertices with the highest degree (hubs) are at different distances in the network, allowing us to understand the structural significance of the index. Using algebraic methods, we derive a Euclidean distance between vertices in the context of NC diffusion with potential navigational bottlenecks. We then apply these indices to study several real-world networks. First, we confronted our theoretical results with experimental data about traffic congestion in a city. Then, we illustrated the application of the new methodologies to the study of a neuronal system, an air transportation network and two urban street networks. [ABSTRACT FROM AUTHOR]

Invertible members of some of the most famous multiplicative quasilinear spaces are exactly equal to invertible elements of the linear subspace of such spaces. But with the old definition of invertible elements, we do not get anything extra. So, in the present paper, we develop the introductory algebra and introduce an extended model of invertible elements. This new concept plays a fundamental role in the definition of the generalized spectrum model. These new objects are constructed in such a way that can provide acceptable results. For example, we will be able to generalize the well-known theorem “Spectral mapping property for polynomials” with these new notions. [ABSTRACT FROM AUTHOR]

We give asymptotics for shifted convolutions of the form ∑ n < X σ 2 u (n , χ) σ 2 v (n + k , ψ) n u + v for nonzero complex numbers u , v and nontrivial Dirichlet characters χ , ψ. We use the technique of automorphic regularization to find the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable. The error term we obtain is in some cases smaller than what the method we use typically yields. [ABSTRACT FROM AUTHOR]

In this paper, we consider the principal spectral theory for the nonlocal dispersal eigenvalue problem (1) d ρ p [ ∫ Ω J ρ (x − y) u (y) d y − u (x) ] + κ ρ q [ ∫ Ω G ρ (x − y) u (y) d y − u (x) ] + a (x) u (x) = − λ u (x) , x ∈ Ω ¯ , where Ω ⊂ R N is a bounded smooth domain, J ρ (x) = ρ N J (ρ x) , G ρ (x) = ρ N G (ρ x) , the kernel functions J (x) and G (x) are nonnegative, J (x) is symmetric, the parameter ρ is positive, d , κ are positive, p , q are given constants, and a ∈ C (Ω ¯). We investigate the limiting behavior of the principal spectral point or generalized principal eigenvalue of (1) as ρ → ∞ and ρ → 0. When ρ ≫ 1 , the nonlocal dispersal operator u (⋅) ↦ d ρ p (∫ Ω J ρ (⋅ − y) u (y) d y − u (⋅)) + κ ρ q (∫ Ω G ρ (⋅ − y) u (y) d y − u (⋅)) + a (⋅) u (⋅) with ν 0 ≠ 0 behaves like the elliptic operator u (⋅) ↦ d c 0 ρ p − 2 Δ u − κ ρ q − 1 ν 0 ⋅ ∇ u + a (⋅) u (⋅) on Ω with Dirichlet boundary condition on ∂Ω, where c 0 = 1 2 N ∫ R N J (y) | y | 2 d y , ν 0 = 〈 ∫ R N G (x) x 1 d x , ∫ R N G (x) x 2 d x , ⋯ , ∫ R N G (x) x N d x 〉. The results obtained in the paper are novel when the dispersal kernel function G (x) is asymmetric. Moreover, when G is asymmetric with ν 0 ≠ 0 , and p ≥ 2 , q > p − 1 or p < 2 , q > 1 , it is seen that the limiting behavior of the generalized principal eigenvalue of (1) as ρ → ∞ is strikingly different from the limiting behavior of the principal eigenvalue of the elliptic operator u (⋅) ↦ d c 0 ρ p − 2 Δ u − κ ρ q − 1 ν 0 ⋅ ∇ u + a (⋅) u (⋅) on Ω with Dirichlet boundary condition on ∂Ω. The main results are used to study the asymptotic dynamics of nonlinear nonlocal dispersal problems. [ABSTRACT FROM AUTHOR]

Let G be a tree on n vertices and let L = D − A denote the Laplacian matrix on G. The second-smallest eigenvalue λ 2 (G) > 0 , also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that 'behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs. [ABSTRACT FROM AUTHOR]

Recent developments in the spectral theory of Bayesian Networks has led to a need for a developed theory of estimation and inference on the eigenvalues of the normalized precision matrix, Ω. In this paper, working under conditions where n → ∞ and p remains fixed, we provide multivariate normal asymptotic distributions of the sample eigenvalues of Ω under general conditions and under normal populations, a formula for second-order bias correction of these sample eigenvalues, and a Stein-type shrinkage estimator of the eigenvalues. Numerical simulations are performed which demonstrate under what generative conditions each estimation technique is most effective. When the largest eigenvalue of Ω is small the simulations show that the second order bias-corrected eigenvalue was considerably less biased than the sample eigenvalue, whereas the smallest eigenvalue was estimated with less bias using either the sample eigenvalue or the proposed shrinkage method. [ABSTRACT FROM AUTHOR]