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The generalized arithmetic–geometric matrix of a simple connected graph G = (V , E) is defined to be the | V | × | V | matrix whose i j -entry is d i α + d j α 2 d i α d j α if v i v j ∈ E , and 0 otherwise, where α is an arbitrary real number and d i is the degree of v i ∈ V. This matrix is a general form of the arithmetic–geometric matrix (α = 1) and the extended adjacency matrix (α = 2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρ a g α of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρ a g α are explored, and some extremal results on ρ a g α are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρ a g α are determined. [ABSTRACT FROM AUTHOR]

Nuclear reactions are complex, with a large number of possible channels. Understanding how different channels contribute to a given reaction is investigated by perturbing the continuous spectrum. Tools are developed to investigate reaction mechanisms by identifying the contributions from each reaction channel. Cluster decomposition methods, along with the spectral theory of proper subsystem problems, is used to identify the part of the nuclear Hamiltonian responsible for scattering into each channel. The result is an expression of the nuclear Hamiltonian as a sum over all scattering channels of channel Hamiltonians. Each channel Hamiltonian is constructed from solutions of proper subsystem problems. Retaining any subset of channel Hamiltonians results in a truncated Hamiltonian where the scattering wave functions for the retained channels differ from the wave functions of the full Hamiltonian by N-body correlations. The scattering operator for the truncated Hamiltonian satisfies an optical theorem in the retained channels. Because different channel Hamiltonians do not commute, how they interact determines their contribution to the full dynamics. [ABSTRACT FROM AUTHOR]

A theoretical model of pulsed photothermal radiometry based on conduction-radiation theory is introduced for a two-layered medium with a first layer having optical and thermal properties different from those of the semi-infinite substrate. This geometry closely represents the optical and thermal properties of biotissues, a major intended application. The theory derives the spatial distribution of the frequency spectrum of the pulsed photothermal signal from the composite two-layer boundary-value problem and matches the spectral frequency domain results to the measured photothermal transients through an efficient inverse Fourier transformation algorithm, which involves the optical, thermal, and geometric parameters of the experimental system. This approach avoids the complicated and computationally expensive analytical Laplace transform approach usually adopted in similar studies and yields a complete conduction–radiation description of photothermal signals without simplifying, yet restrictive, approximations encountered in the literature. Numerical and experimental tests of the model using intralipid solution as layer 1 and semi-infinite solid samples such black rubber and anodized aluminum as layer 2 of the medium are described. The radiation heat transfer coefficients from the air–intralipid and intralipid–solid interfaces were introduced to complement the conductive blackbody-radiated infrared signal component at the detecting camera. The best-fitted parameters and heat transfer coefficients are compared to theoretical and experimental data, and the results of these tests were found to be consistent with the theoretical model. [ABSTRACT FROM AUTHOR]

The objective of this paper is to develop, verify, and experimentally validate a mesh-free spectral graph theory-based approach for rapid prediction of thermal history in metal parts made using the wire arc additive manufacturing (WAAM) process. Accurate and rapid prediction of the thermal history is a critical prerequisite for functional quality assurance of WAAM parts. In the spectral graph method, the WAAM part is represented as a set of discrete nodes encompassed by a network graph. The thermal history is obtained by solving the heat equation on the network graph. The spectral graph theory approach thus bypasses the cumbersome computational burden associated with mesh generation in the finite element method. To validate the spectral graph theory approach, experimental temperature data is acquired for multi-layer mild steel WAAM parts processed under six different combinations of shape and inter-layer dwell time conditions. Further, each experiment was replicated resulting in a total of 12 parts. The accuracy of the thermal trends predicted by the spectral graph theory approach was quantified in terms of the symmetric mean absolute percent error (SMAPE) and root mean squared error (RMSE, °C). The thermal history trends were predicted with SMAPE < 5% and RMSE < 11 °C relative to the experimental temperature measurements. The computation time to obtain the thermal history of each 21-layer part was approximately 3.5 to 4 h. The spectral graph theory predictions were further verified with a commercial finite element software package (Simufact-Welding). For a similar level of SMAPE and RMSE as the spectral graph theory approach, the FE-based Simufact predictions required over 7.5 h. The computational advantage of the spectral graph theory approach is particularly valuable for devising physics-based process parameter optimization and feedforward control strategies to mitigate flaw formation in WAAM parts. [ABSTRACT FROM AUTHOR]

In this paper, transmission eigenvalues of a Schrödinger equation have been studied by constructing a new inner product and using Weyl theory. Necessary conditions for these eigenvalues to be negative, real, and finite have been examined. This method has provided a new framework related to transmission eigenvalue problems and the investigation of their properties. The conclusions have been verified for the special case of the problem. [ABSTRACT FROM AUTHOR]

We study Dirichlet‐type problems for the simplest third‐order linear dispersive partial differential equations (PDE), often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the spatial operator. Our specific interest is to determine whether the peculiar phenomenon of revivals, also known as Talbot effect, is supported by these boundary conditions, which for third‐order problems are not reducible to periodic ones. We prove that this is the case only for a very special choice of the boundary conditions, for which a new type of weak cusp revival phenomenon has been recently discovered. We also give some new results on the functional class of the solution for other cases. [ABSTRACT FROM AUTHOR]

We provide a variant of Geršgorin's circle theorem, where the ℓ1$\ell ^1$‐estimates are swapped for ℓ2$\ell ^2$‐estimates, more suitable for the infinite‐dimensional Hilbert space setting. [ABSTRACT FROM AUTHOR]

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with "few eigenvalues" is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is "highly" non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions. [ABSTRACT FROM AUTHOR]

The use of graph theory for solving labyrinths and mazes is well known, understanding the possible paths as the connections between the nodes that represent the corners or bifurcations. This work presents a new idea: minimizing the length of the optimal path formulated as a topology optimization problem. The maze is mapped with finite elements, and then, through the eigenvalues of the Laplacian matrix of the graph, a constraint is imposed over the connectivity between the input and the output. Several 2D examples are provided to support this approach, allowing for unequivocally finding the shortest path in mazes with multiple connections between entrance and exit, resulting in an nonheuristic algorithm. [ABSTRACT FROM AUTHOR]

Thunderstorm downbursts are transient in nature and have been responsible for a variety of structural damages in recent years. Currently, the researchers have done several works on the characteristics of downburst wind speed. Nonetheless, rare attention has been placed on the structural aerodynamics characteristics subjected to downburst winds. Based on this, an experimental investigation is performed to reproduce downburst-like winds physically and to study the transient surface pressures (SPs) on a 5:1 rectangular cylinder (RC). The experiment is conducted within a multiple-fan active control wind tunnel (MFACWT) and mainly focuses on simulating the transient characteristics of downburst-like flow, including time-varying mean (TVM) wind speed and nonstationary wind fluctuation. The resulting SPs are measured to understand the influence of transient wind on the aerodynamic behavior of bluff bodies. The spatiotemporal characteristics of the SPs are analyzed using wavelet transform and Priestley's classic spectral theory. The results indicate that the transient nature of the downburst-like flow can be physically reproduced by a MFACWT. The instantaneous pressures of a RC are illustrated by both the turbulence parameters of the transient flow and the flow-separation characteristics. The pressure coefficients normalized by the TVM of the downburst-like winds remain constant, which provides a more appropriate way to estimate the transient gust loading in a quasi-steady manner. Interestingly, the phenomenon of the time-varying phase shift and time-varying correlation of chordwise SPs is observed when the turbulent velocity changes dramatically. In addition, the normalized surface pressure can be regarded as a stationary stochastic process, which provides a significant basis for further establishing the theoretical model of nonstationary gust-loading. [ABSTRACT FROM AUTHOR]

A central task of theoretical physics is to analyse spectral properties of quantum mechanical observables. In this endeavour, Mourre’s conjugate operator method emerged as an effective tool in the spectral theory of Schrödinger operators. This paper introduces a novel class of examples from relativistic quantum field theory that are amenable to Mourre’s method. By assuming Lorentz covariance and the spectrum condition, we derive a limiting absorption principle for the energy-momentum operators and provide new proofs of the absolute continuity of the energy-momentum spectra. Moreover, under the assumption of dilation covariance, we show that the spectrum of the relativistic mass operator is purely absolutely continuous in (0 , ∞) . [ABSTRACT FROM AUTHOR]

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that the only obstruction for having norm resolvent convergence is the existence of an eigenvalue of infinite multiplicity for the limiting operator. More precisely, we prove the convergence of the resolvents in operator norm once projected into the orthogonal of the corresponding eigenspace. [ABSTRACT FROM AUTHOR]

We consider a competition system with lethal boundary conditions in advective environments. To obtain the combined effects of diffusion and advection on the local and global dynamics of the competition system, we employ the principal spectral theory and the theory of monotone dynamical systems. We find that the function describing advective direction plays a significant role in the global dynamics of the competition system. [ABSTRACT FROM AUTHOR]

We propose the novel spectral properties of the Neumann Laplacian in a two-dimensional bounded domain perturbed by an infinite number of compact sets with zero Lebesgue measure, so-called curved holes. These holes consist of segments or parts of curves enclosed in small spheres such that the diameters of holes tend to zero as the number of holes approaches infinity. Specifically, we rigorously demonstrate that the spectrum of the Neumann Laplacian on the perturbed domain converges to that of the original operator on the domain without holes under specific geometric assumptions and an appropriate selection of hole sizes. Furthermore, we derive sophisticated estimates on the convergence rate in terms of operator norms and estimate the Hausdorff distance between the spectra of the Laplacians. [ABSTRACT FROM AUTHOR]

To study the consumption of heroin in a heterogeneous environment, we propose and analyze a spatiotemporal model with a distributed delay. Using the spectral theory, we determine the basic reproduction number R0$$ {R}_0 $$, which serves a threshold role. If R0<1$$ {R}_0<1 $$, then the addiction‐free steady state is globally asymptotically stable while if R0>1$$ {R}_0>1 $$, then there is at least one addictive steady state. Moreover, when R0>1$$ {R}_0>1 $$, if one of the dispersal coefficients is zero, then there is only one addictive steady state, and it is globally asymptotically stable; if both diffusions of susceptible and addicted individuals are present, we cannot identify the temporal behavior of solutions, and hence, we study the asymptotic profile of addictive steady states when one of the dispersal coefficients tend to zero. [ABSTRACT FROM AUTHOR]

We investigate a competitive diffusion–advection Lotka–Volterra model with more general nonlinear boundary condition. Based on some new ideas, techniques, and the theory of the principal spectral and monotone dynamical systems, we establish the influence of the following parameters on the dynamical behavior of system (1.2): advection rates α u and α v , interspecific competition intensities c u and c v , the resources functions r u and r v of the two competitive species, and nonlinear boundary functions g 1 and g 2 . The models of (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018) are particular cases of our results when g i ≡ c o n s t for i = 1 , 2 , and hence this paper extends some of the conclusions from (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018). [ABSTRACT FROM AUTHOR]

We provide a combinatorial and self-contained proof of a result following from G. Besson [Ann. Inst. Fourier, 30 (1980), pp. 109-128] and Y. Colin de Verdière [Ann. Sci. Ec. Norm. Supér., 20 (1987), pp. 599-615] that for all graphs G embedded on a surface S, the Colin de Verdière parameter μ(G) is upper bounded by 7-2χ(S). [ABSTRACT FROM AUTHOR]

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks, since the possible removal of M edges requires that one compute all the eigenvalues of M distinct matrices. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and has a computational complexity that scales as O(MN) for networks with N nodes. Focusing on a form of VNE that is associated with a transport operator e-βL, where L is a graph Laplacian matrix and β> 0 is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate, and long timescales β for diffusion. [ABSTRACT FROM AUTHOR]

The exact expressions for scattering matrix and transmission coefficient in open multi-cascade nanostructure are obtained. The theory of spectral characteristics of electron quasi-stationary states is developed in two approaches. For the nanostructure with GaAs wells and AlGaAs barriers, it is established that resonance bands appear in the N-cascade structure. Each one is formed by the energies of N states. Only the scattering matrix allows to unambiguously determine the resonance energies and resonance widths of all electron states in the multi-cascade structure. The approach of transmission coefficient, due to the collapse of resonances, gives the incomplete information about the spectral characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral and scattering theory on the Hamiltonians which govern one-body quantum systems in crossed constant electric and magnetic fields. As one of the spectral features of the Hamiltonians, we show the absence of bound states of theirs under some appropriate conditions on the potentials. Moreover, by using the limiting absorption principle for the Hamiltonians, we give a simplified proof of the asymptotic completeness in short-range scattering. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method that integrates a tailored spectral graph metric and economic aspects within a coupled bi-objective optimization framework for the simultaneous pipe sizing and partitioning of water distribution networks (WDNs). Besides minimum pressure head and topological WDN connectivity, the telescopic diameter distribution criterion is applied through a new algorithm in the sizing phase to help the optimization converge towards sound engineering solutions. The application of the proposed method to a synthetic layout and a real Italian case study proves the superiority of the solutions obtained with the novel method from hydraulic and managerial/operational viewpoints in comparison with those obtained with a traditional decoupled method (separate pipe sizing and partitioning) in terms of i) compliance with service pressure constraints and ii) ease of management following WDN partitioning into district metered areas (DMAs). [ABSTRACT FROM AUTHOR]

Wireless mesh networks (WMNs) are a type of wireless network that can be used for various applications, such as Internet access, disaster response, and military communication. These networks consist of mesh routers that can communicate with each other and form a mesh topology, allowing them to provide connectivity even if some of the routers fail or are out of range. In this study, we used spectral graph theory to analyze the performance of a WMN. For the analysis process, software was developed to calculate the topological characteristics of the graph representing the WMN. The correlation between the values of the parameters of spectral graph theory and the topological characteristics of the observed network is analyzed. First, an analysis of the influence of the change in signal strength, in the observedWMN, on the algebraic connectivity was performed, and then the change in the spectral radius was also analyzed. The analysis was performed using special software, which was developed for that purpose. [ABSTRACT FROM AUTHOR]

We define a Bessel type equation in multiplicative calculus by some algebraic structures. Eigenfunctions of constructed problem are obtained by power series solution technique. While making these solutions, multiplicative Bessel functions are used strongly. We get a generator for that problem and construct integration representations for multiplicative Bessel functions. Finally, some basic spectral properties of multiplicative Bessel problem are examined in detail. [ABSTRACT FROM AUTHOR]

The study of twisted bilayer graphene (TBG) is a hot topic in condensed matter physics with special focus on magic angles of twisting at which TBG acquires unusual properties. Mathematically, topologically non-trivial flat bands appear at those special angles. The chiral model of TBG pioneered by Tarnopolsky, Kruchkov, and Vishwanath (2019) has particularly nice mathematical properties and we survey, and in some cases, clarify, recent rigorous results which exploit them. [ABSTRACT FROM AUTHOR]

We analyze the spectrum of a periodic quantum graph of the Cairo lattice form. The used vertex coupling violates the time reversal invariance and its high-energy behavior depends on the vertex degree parity; in the considered example both odd and even parities are involved. The presence of the former implies that the spectrum is dominated by gaps. In addition, we discuss two modifications of the model in which this is not the case, the zero limit of the length parameter in the coupling, and the sign switch of the coupling matrix at the vertices of degree three; while different they both yield the same probability that a randomly chosen positive energy lies in the spectrum. [ABSTRACT FROM AUTHOR]

We consider the interior inverse problem associated with the global conservative multipeakon solution of the Camassa-Holm equation. Based on the inverse spectral theory on the half-line and the oscillation property of eigenfunctions, some (non)uniqueness results of the interior inverse problem are obtained. In addition, we give the trace formula, which connects the global conservative multipeakon solution with the corresponding eigenvalues and normalized eigenfunctions. [ABSTRACT FROM AUTHOR]

High-speed railways are “lifeline projects” that shoulder the heavy responsibility of transporting relief supplies and medical forces for the first time after earthquakes. To ensure the train’s safety after earthquakes, it is of great urgency to ascertain a post-earthquake speed threshold. To that end, a target model for seismic irregularity emerges as a key parameter. In this paper, a null phase assumption-based technique for the mutual conversion between evolutionary power spectral density and non-stationary signal was proposed. Taking a high-speed railway track-bridge system as the research object, the target model of seismic irregularities was constructed based on the proposed technique. The rationality of the target model of seismic irregularities was verified, and the construction parameter settings were discussed. Moreover, a simplified frequency-domain fitting method for the target model of seismic irregularities was proposed based on the spectral decomposition theory. According to the research findings, the null phase assumption-based technique is capable of performing interconversion between seismic irregularity and its evolutionary power spectral density with satisfactory accuracy. It is recommended to set the minimum number of seismic irregularities, spatial sampling interval, hop size, and length of window function as 50, 0.25m, 1, and 40 to 100, respectively. [ABSTRACT FROM AUTHOR]

Spectral clustering, as an algorithm based on graph theory and spectral theory, has shown excellent performance in the classification tasks of hyperspectral images in recent years. Although better results have been achieved, some challenges still exist. The inclusion of a priori information can increase the performance of spectral clustering algorithms; however, in practice, it is often unable to meet the demand for a priori information; at the same time, spectral clustering for the quality of the similarity matrix is demanding, and some of the current algorithms are to improve the similarity matrix from the aspect of data planning, and ultimately for the label feature matrices are diluted. In response to the above problems, this paper proposes a self-supervised spectral clustering with spectral embedding (SESSC). The algorithm obtains a low-dimensional representation of the data through spectral embedding, which can simplify clustering while preserving feature information; then uses the similarity matrix about the data and the guidance of the sample constraint information to obtain a new similarity matrix in order to further refine the structural graph, and the result can feed back and optimize the label feature matrix and the low-dimensional representation of the data. Additionally, we introduce a fractional theory in the update of the sample variable matrix, which assures the integrity and validity of the information in the update. Experimental results show that the proposed algorithm has better performance in hyperspectral image classification than existing spectral clustering algorithms. [ABSTRACT FROM AUTHOR]

The normalized distance Laplacian of a graph G is defined as D L (G) = T (G) − 1 / 2 (T (G) − D (G)) T (G) − 1 / 2 where D (G) is the matrix with pairwise distances between vertices and T (G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius. [ABSTRACT FROM AUTHOR]

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it in terms of entries of the resolvent of the matrix and its "half-tree" restrictions. This formula is closely related to the one-dimensional Thouless formula and associates a natural phase with points in the bands. This allows streamlined proofs of the gap labeling and Aomoto index theorems. We give a complete proof of gap labeling and sketch the proof of the Aomoto index theorem. We also prove a version of this formula for the Anderson model on trees. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵ 2 u = f (| x | , u , | ∇ u | , ▵ u) in an annular domain Ω = { x ∈ R N : r 1 < | x | < r 2 } ( N ≥ 2 ) with the boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : [ r 1 , r 2 ] × R × R + × R → R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ 1 of the Laplace operator − ▵ with boundary condition u | ∂ Ω = 0 , an existence result and a uniqueness result are obtained. The inequality conditions allow for f (r , ξ , ζ , η) to be a superlinear growth on ξ , ζ , η as | (ξ , ζ , η) | → ∞ . Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates. [ABSTRACT FROM AUTHOR]

Remote sensing images (RSIs) are widely used in various fields due to their versatility, accuracy, and capacity for earth observation. Direct application of RSIs to harvest optimal results is generally difficult, especially for weak information features in the images. Thus, extracting the weak information in RSIs is reasonable to promote further applications. However, the current techniques for weak information extraction mainly focus on spectral features in hyperspectral images (HSIs), and a universal weak information extraction technology for RSI is lacking. Therefore, this study focused on mining the weak information from RSIs and proposed the deep multi-order spatial–spectral residual feature extractor (DMSRE). The DMSRE considers the global information and three-dimensional cube structures by combining low-rank representation, high-order residual quantization, and multi-granularity spectral segmentation theories. This extractor obtains spatial–spectral features from two derived sequences (deep spatial–spectral residual feature (DMSR) and deep spatial–spectral coding feature (DMSC)), and three RSI datasets (i.e., Chikusei, ZY1-02D, and Pasture datasets) were employed to validate the DMSRE method. Comparative results of the weak information extraction-based classifications (including DMSR and DMSC) and the raw image-based classifications showed the following: (i) the DMSRs can improve the classification accuracy of individual classes in fine classification applications (e.g., Asphalt class in the Chikusei dataset, from 89.12% to 95.99%); (ii) the DMSC improved the overall accuracy in rough classification applications (from 92.07% to 92.78%); and (iii) the DMSC improved the overall accuracy in RGB classification applications (from 63.25% to 63.6%), whereas DMSR improved the classification accuracy of individual classes on the RGB image (e.g., Plantain classes in the Pasture dataset, from 32.49% to 39.86%). This study demonstrates the practicality and capability of the DMSRE method to promote target recognition on RSIs and presents an alternative technique for weak information mining on RSIs, indicating the potential to extend weak information-based applications of RSIs. [ABSTRACT FROM AUTHOR]

This work presents a methodology to extract coherent structures from high-speed schlieren images of turbulent twin jets which are more physically interpretable than those obtained with currently existing techniques. Recently, Prasad and Gaitonde (J Fluid Mech 940:1–11, 2022) introduced an approach which employs the momentum potential theory of Doak (J Sound Vib 131(1):67–90, 1989) to compute potential (acoustic and thermal) energy fluctuations from the schlieren images by solving a Poisson equation, and combines it with spectral proper orthogonal decomposition (SPOD) to educe coherent structures from the momentum potential field instead of the original schlieren field. While the latter field is dominated by a broad range of vortical fluctuations in the turbulent mixing region of unheated high-speed jets, the momentum potential field is governed by fluctuations which are intimately related to acoustic emission, and its spatial structure in the frequency domain is very organized. The proposed methodology in this paper improves the technique of Prasad and Gaitonde (J Fluid Mech 940:1–11, 2022) in three new ways. First, the solution of the Poisson equation is carried out in the frequency-wavenumber domain instead of the time-space domain, which simplifies and integrates the solution of the Poisson equation within the SPOD framework based on momentum potential fluctuations. Second, the issue of solving the Poisson equation on a finite domain with ad hoc boundary conditions is explicitly addressed, identifying and removing those unphysical harmonic components introduced in the solution process. Third, the solution of the SPOD problem in terms of momentum potential fluctuations is used to reconstruct schlieren SPOD fields associated with each mode, allowing the visualization of the obtained coherent structures also in terms of the density gradient. The method is applied here to schlieren images of a twin-jet configuration with a small jet separation at two supersonic operation conditions: a perfectly-expanded and an overexpanded one. The SPOD modes based on momentum potential fluctuations retain the wavepacket structure including the direct Mach-wave radiation, together with upstream- and downstream-traveling acoustic waves, similar to SPOD modes based on the schlieren images. However, for the same dataset, they result in a lower-rank decomposition than schlieren-based SPOD and provide an effective separation of twin-jet fluctuations into independent toroidal and flapping oscillations that are recovered as different SPOD modes. These coherent structures are more consistent with twin-jet wavepacket models available in the literature than those originally obtained with direct schlieren-based SPOD, facilitating their interpretation and comparison against theoretical analyses. [ABSTRACT FROM AUTHOR]

Positive semidefinite kernels on spheres cross time are on demand in spatial statistics, machine learning and numerical analysis, motivated by applications in the climate, environmental and earth sciences. Most of the recent literature has focused on kernels that are isotropic on the sphere and stationary in time, but this assumption is often overly limited in quantifying spatial dependence properly. A broader class of kernels can be obtained by trading isotropy for axial symmetry, i.e., stationarity with respect to longitudes only. This paper challenges theoretical aspects by providing a spectral representation that uniquely characterizes space-time positive semidefinite kernels that are axially symmetric on the sphere and stationary in time. We also address the problem of partial Fourier inversion and show that it is feasible under mild technical conditions. The second part of the paper focuses on constructive methods to build nonseparable kernels having axial symmetry on the sphere and stationarity over time. [ABSTRACT FROM AUTHOR]

This article focuses on studying the existence of solutions for the following indefinite Choquard equation in R N : where N ≥ 2 . Here, α is a positive constant satisfying 0 < α < N , and I α represents the Riesz potential of order α . The potential V(x) is nonperiodic and changes sign, leading to inf σ (- Δ + V) < 0 , where σ (- Δ + V) denotes the spectrum of the operator - Δ + V , which the problem become indefinite. The exponent p is chosen from the range given by 0.1 N - 2 N + α < 1 p < 1 2 < N N + α. In this work, we establish the existence of nontrivial solutions by employing significant outcomes from spectral theory, along with a version of the linking theorem presented by [13], and the interaction between translated solutions of the problem at infinity. [ABSTRACT FROM AUTHOR]

Several spectral radii formulas for infinite bounded non-negative matrices in max algebra are obtained. We also prove some Perron–Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are similar to results on Frobenius normal form of $ n \times n $ n × n matrices. Some continuity results are also established. [ABSTRACT FROM AUTHOR]

Characterizing the thermodynamics of turbulent plasmas is key to decoding observable signatures from astrophysical systems. In magnetohydrodynamic (MHD) turbulence, non-linear interactions between counter-propagating Alfvén waves cascade energy to smaller spatial scales where dissipation heats the protons and electrons. When the thermal pressure far exceeds the magnetic pressure, linear theory predicts a spectral gap at perpendicular scales near the proton gyroradius where Alfvén waves become non-propagating. For simple models of an MHD turbulent cascade that assume only local non-linear interactions, the cascade halts at this gap, preventing energy from reaching smaller scales where electron dissipation dominates, leading to an overestimate of the proton heating rate. In this work, we demonstrate that non-local contributions to the cascade, specifically large-scale shearing and small-scale diffusion, can bridge the non-propagating gap, allowing the cascade to continue to smaller scales. We provide an updated functional form for the proton-to-electron heating ratio accounting for this non-local energy transfer by evaluating a non-local weakened cascade model over a range of temperature and pressure ratios. In plasmas where the thermal pressure dominates the magnetic pressure, we observe that the proton heating is moderated compared to the significant enhancement predicted by local models. [ABSTRACT FROM AUTHOR]

This paper delves into three research directions, leveraging the Lovász ϑ-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that n≥14, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on n vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász ϑ-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the ϑ-function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the ϑ-function variant by Schrijver does not possess the property of the Lovász ϑ-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the principal spectral theory of time-periodic cooperative systems with nonlocal dispersal and Neumann boundary condition. First we present a sufficient condition for the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we establish the monotonicity of principal eigenvalues with respect to the frequency and investigate the limiting properties of principal eigenvalues as the frequency tends to zero or infinity. We also study the effects of dispersal rates and dispersal ranges on the principal eigenvalues, and the difficulty is that principal eigenvalues of time-periodic cooperative systems with Neumann boundary conditions are not monotone with respect to the domain. Finally, we apply our theory to a man-environment-man epidemic model and consider the impacts of dispersal rates, frequency, and dispersal ranges on the basic reproduction number and positive time-periodic solutions. [ABSTRACT FROM AUTHOR]

Given a graph G $G$, we let s+(G) ${s}^{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of G $G$, and we similarly define s−(G) ${s}^{-}(G)$. We prove that χf(G)≥1+maxs+(G)s−(G),s−(G)s+(G) ${\chi }_{f}(G)\ge 1+\max \left\{\frac{{s}^{+}(G)}{{s}^{-}(G)},\frac{{s}^{-}(G)}{{s}^{+}(G)}\right\}$ and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chromatic number χ(G) $\chi (G)$. We in fact show a stronger result wherein we give a bound using the eigenvalues of G $G$ and H $H$ whenever G $G$ has a homomorphism to an edge‐transitive graph H $H$. Our proof utilizes ideas motivated by association schemes. [ABSTRACT FROM AUTHOR]

In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate dynamical analysis and optimal control for human immunodeficiency virus (HIV) transmission model with age-structure. Local asymptotic stability of disease-free equilibrium is proved by applying operator theory and spectral boundary principle. And the existence of endemic equilibrium is obtained by using fixed-point theorem and monotone iterative procedure. Optimal control of an age-structure HIV model is conducted by using prevention and control education and anti-HIV treatment as control strategies. Taking a suitable objective function is to prove existences of optimal control variables which are characterized by sensitivity system and adjoint system associated with optimal state system. Finally, numerical simulations with appropriate parameters are conducted to show that comprehensive publicity and education on prevention and control for susceptible individuals and prompt treatment of HIV-infected individuals can be better to reduce HIV impact. [ABSTRACT FROM AUTHOR]

Recent spectral graph sparsification research aims to construct ultra-sparse subgraphs for preserving the original graph spectral (structural) properties, such as the first few Laplacian eigenvalues and eigenvectors, which has led to the development of a variety of nearly-linear time numerical and graph algorithms. However, there is very limited progress for spectral sparsification of directed graphs. In this work, we prove the existence of nearly-linear-sized spectral sparsifiers for directed graphs under certain conditions. Furthermore, we introduce a practically-efficient spectral algorithm (diGRASS) for sparsifying real-world, large-scale directed graphs leveraging spectral matrix perturbation analysis. The proposed method has been evaluated using a variety of directed graphs obtained from real-world applications, showing promising results for solving directed graph Laplacians, spectral partitioning of directed graphs, and approximately computing (personalized) PageRank vectors. [ABSTRACT FROM AUTHOR]