Search Results (18)

We investigate whether the geometric parameters of a three-dimensional domain can be recovered from the Dirichlet spectrum of the Laplacian. As a controlled benchmark, we consider rectangular boxes, about which the eigenvalues are explicitly known and the Weyl coefficients can be computed in closed form. Exploiting the short-time asymptotics of the heat trace, we extract the leading Weyl coefficients from finite spectral data and show how they encode volume, surface area, and the third spectral Weyl term. These coefficients uniquely determine the side lengths of the box via an explicit cubic reconstruction formula. Numerical experiments based on several thousand eigenvalues demonstrate that the method is stable, accurate, and robust with respect to spectral truncation. The box setting thus provides a stringent validation of the proposed inverse spectral methodology and serves as a foundation for its extension to smooth curved domains, such as triaxial ellipsoids, where explicit spectral formulas are no longer available. Full article

The Zeta-Minimizer Theorem establishes that the Riemann zeta function $\zeta \left(s\right)$ and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space $\left(X,\mu ,G\right)$ equipped with helical operators. Three fundamental axioms—strict concave entropy maximization (Axiom 1), spectral Gibbs minima with non-vanishing ground states (Axiom 2), and irreducible bounded oscillations with flux conservation (Axiom 3)—allow for the selection of the non-proper Archimedean conical helix as the sole topology satisfying all constraints. Primes emerge as indivisible minimal cycles in the associated representation graph $\Gamma$ (via Hilbert irreducibility and Maschke’s theorem), while the Euler product is recovered through the spectral Dirichlet mapping of the helical eigenvalues. The partial zeta product, $Z\left(s\right)=\prod _j{\displaystyle \frac{1}{1-p_j^{-s}}},s\in \mathbb{R}\left\{0\right\},$ constitutes the exact grand partition function of any finite subsystem. Numerical inversion of this product directly recovers the mixture frequency $s$ from any experimental compressibility factor $Z_{\mathrm{m}\mathrm{i}\mathrm{x}}$. Mole fractions $x_i\left(s\right)$, interaction parameters $\Delta \left(x_i\right)$, and the Lyapunov spectrum $\lambda \left(x_i\right)$ then follow deductively via the helical transfer matrix and the closed-form linear ODE for $\Delta$. Occupation numbers $N\left(x_i\right)$ attain sharp maxima precisely at Fibonacci ratios $F_r/F_{r+1}$, leading to the molecular prime-ID rule. For twelve representative purely binary (irreducible) systems spanning atomic noble gases, simple diatomics, polar molecules, and an aromatic ring, the residuals satisfy $|Z\left(s\right)-Z_{\mathrm{m}\mathrm{i}\mathrm{x}}|<1.5\times {10}^{-8}$. The resulting $\lambda \left(x_i\right)$ curves accurately reproduce critical points, liquid ranges, and thermodynamic anomalies with zero adjustable parameters. The Riemann Hypothesis follows rigorously as a theorem: the unique fixed point of the duality functor $s\mapsto 1-s$ that preserves the orthogonality condition $\sum {\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }^k=1$ is $\mathrm{R}\mathrm{e}\left(s\right)=1/2$, enforced by Axiom 1 concavity and Axiom 3 irreducibility. The framework is fully deductive and parameter-free and extends naturally to arbitrary mixtures and multiplicities through the helical representation graph. It provides a variational unification of analytic number theory, spectral geometry, thermodynamic phase behavior, and the Riemann Hypothesis from first principles. Full article

This study aims to analyze online job postings using machine learning-based, semantic approaches and to identify the expertise roles and competencies required for big data professions. The methodology of this study employs latent Dirichlet allocation (LDA), a probabilistic topic modeling technique, to reveal hidden semantic structures within a corpus of big data job postings. As a result of our analysis, we have identified seven expertise roles, six proficiency areas, and 32 competencies (knowledge, skills, and abilities) necessary for big data professions. These positions include “developer”, “engineer”, “architect”, “analyst”, “manager”, “administrator”, and “consultant”. The six essential proficiency areas for big data are “big data knowledge”, “developer skills”, “big data analytics”, “cloud services”, “soft skills”, and “technical background”. Furthermore, the top five skills emerged as “big data processing”, “big data tools”, “communication skills”, “remote development”, and “big data architecture”. The findings of our study indicated that the competencies required for big data careers cover a broad spectrum, including technical, analytical, developer, and soft skills. Our findings provide a competency map for big data professions, detailing the roles and skills required. It is anticipated that the findings will assist big data professionals in assessing and enhancing their competencies, businesses in meeting their big data labor force needs, and academies in customizing their big data training programs to meet industry requirements. Full article

The ‘dynamo problem’ requires that the origin of the primarily dipole geomagnetic field be found. The source of the geomagnetic field lies within the outer core of the Earth, which contains a turbulent magnetofluid whose motion is described by the equations of magnetohydrodynamics (MHD). A mathematical model can be based on the approximate but essential features of the problem, i.e., a rotating spherical shell containing an incompressible turbulent magnetofluid that is either ideal or real but maintained in an equilibrium state. Galerkin methods use orthogonal function expansions to represent dynamical fields, with each orthogonal function individually satisfying imposed boundary conditions. These Galerkin methods transform the problem from a few partial differential equations in physical space into a huge number of coupled, non-linear ordinary differential equations in the phase space of expansion coefficients, creating a dynamical system. In the ideal case, using Dirichlet boundary conditions, equilibrium statistical mechanics has provided a solution to the problem. As has been presented elsewhere, the solution also has relevance to the non-ideal case. Here, we examine and compare Galerkin methods imposing Neumann or mixed boundary conditions, in addition to Dirichlet conditions. Any of these Galerkin methods produce a dynamical system representing MHD turbulence and the application of equilibrium statistical mechanics in the ideal case gives solutions of the dynamo problem that differ only slightly in their individual sets of wavenumbers. One set of boundary conditions, Neumann on the outer and Dirichlet on the inner surface, might seem appropriate for modeling the outer core as it allows for a non-zero radial component of the internal, turbulent magnetic field to emerge and form the geomagnetic field. However, this does not provide the necessary transition of a turbulent MHD energy spectrum to match that of the surface geomagnetic field. Instead, we conclude that the model with Dirichlet conditions on both the outer and the inner surfaces is the most appropriate because it provides for a correct transition of the magnetic field, through an electrically conducting mantle, from the Earth’s outer core to its surface, solving the dynamo problem. In addition, we consider how a Galerkin model velocity field can satisfy no-slip conditions on solid boundaries and conclude that some slight, kinetically driven compressibility must exist, and we show how this can be accomplished. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article

In the rapidly advancing field of large language model (LLM) research, platforms like Stack Overflow offer invaluable insights into the developer community’s perceptions, challenges, and interactions. This research aims to analyze LLM research and development trends within the professional community. Through the rigorous analysis of Stack Overflow, employing a comprehensive dataset spanning several years, the study identifies the prevailing technologies and frameworks underlining the dominance of models and platforms such as Transformer and Hugging Face. Furthermore, a thematic exploration using Latent Dirichlet Allocation unravels a spectrum of LLM discussion topics. As a result of the analysis, twenty keywords were derived, and a total of five key dimensions, “OpenAI Ecosystem and Challenges”, “LLM Training with Frameworks”, “APIs, File Handling and App Development”, “Programming Constructs and LLM Integration”, and “Data Processing and LLM Functionalities”, were identified through intertopic distance mapping. This research underscores the notable prevalence of specific Tags and technologies within the LLM discourse, particularly highlighting the influential roles of Transformer models and frameworks like Hugging Face. This dominance not only reflects the preferences and inclinations of the developer community but also illuminates the primary tools and technologies they leverage in the continually evolving field of LLMs. Full article

Radar emitter identification (REI) aims to extract the fingerprint of an emitter and determine the individual to which it belongs. Although many methods have used deep neural networks (DNNs) for an end-to-end REI, most of them only focus on a single view of signals, such as spectrogram, bi-spectrum, signal waveforms, and so on. When the electromagnetic environment varies, the performance of DNN will be significantly degraded. In this paper, a multi-view adaptive fusion network (MAFN) is proposed by simultaneously exploring the signal waveform and ambiguity function (AF). First, the original waveform and ambiguity function of the radar signals are used separately for feature extraction. Then, a multi-scale feature-level fusion module is constructed for the fusion of multi-view features from waveforms and AF, via the Atrous Spatial Pyramid Pooling (ASPP) structure. Next, the class probability is modeled as Dirichlet distribution to perform adaptive decision-level fusion via evidence theory. Extensive experiments are conducted on two datasets, and the results show that the proposed MAFN can achieve accurate classification of radar emitters and is more robust than its counterparts. Full article

Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed. Full article

In this paper, we study quantum relativistic features of a scalar field around the Rindler–Schwarzschild wormhole. First, we introduce this new class of spacetime, investigating some energy conditions and verifying their violation in a region nearby the wormhole throat, which means that the object must have an exotic energy in order to prevent its collapse. Then, we study the behavior of the massless scalar field in this spacetime and compute the effective potential by means of tortoise coordinates. We show that such a potential is attractive close to the throat and that it is traversable via quantum tunneling by massive particles with sufficiently low energies. The solution of the Klein–Gordon equation is obtained subsequently, showing that the energy spectrum of the field is subject to a constraint, which induces a decreasing oscillatory behavior. By imposing Dirichlet boundary conditions on a spherical shell in the neighborhood of the throat we can determine the particle energy levels, and we use this spectrum to calculate the quantum revival of the eigenstates. Finally, we compute the Casimir energy associated with the massless scalar field at zero temperature. We perform this calculation by means of the sum of the modes method. The zero-point energy is regularized using the Epstein–Hurwitz zeta-function. We also obtain an analytical expression for the Casimir force acting on the shell. Full article

A frequency spectrum segmentation methodology is proposed to extract the frequency response of circuits and systems with high resolution and low distortion over a wide frequency range. A high resolution is achieved by implementing a modified Dirichlet function (MDF) configured for multi-tone excitation signals. Low distortion is attained by limiting or avoiding spectral leakage and interference into the frequency spectrum of interest. The use of a window function allowed for further reduction in distortion by suppressing system-induced oscillations that can cause severe interference while acquiring signals. This proposed segmentation methodology with the MDF generates an interleaved frequency spectrum segment that can be used to measure the frequency response of the system and can be represented in a Bode and Nyquist plot. The ability to simulate and measure the frequency response of the circuit and system without expensive network analyzers provides good stability coverage for reliable fault detection and failure avoidance. The proposed methodology is validated with both simulation and hardware. Full article

We consider a nonlinear eigenvalue problem driven by the Dirichlet $(p,2)$-Laplacian. The parametric reaction is a Carathéodory function which exhibits $(p-1)$-sublinear growth as $x\to +\infty$ and as $x\to 0^{+}$. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as $\lambda >0$ varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to $(p,q)$-equations ($q\ne 2$). Full article

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval $(1/2,1]$. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators. Full article

We derive a correspondence between the Hawking radiation spectra emitted from general classes of Taub-NUT black holes with that induced by the relativistic motion of an accelerated Dirichlet boundary condition (i.e., a perfectly reflecting mirror) in (1+1)-dimensional flat spacetime. We demonstrate that the particle and energy spectra is thermal at late times and that particle production is suppressed by the NUT parameter. We also compute the radiation spectrum in the rotating, electrically charged (Kerr–Newman) Taub-NUT scenario, and the extremal case, showing, explicitly, how these parameters affect the outgoing particle and energy fluxes. Full article

Mutation signatures are defined as the distribution of specific mutations such as activity of AID/APOBEC family proteins. Previous studies have reported numerous signatures, using matrix factorization methods for mutation catalogs. Different mutation signatures are active in different tumor types; hence, signature activity varies greatly among tumor types and becomes sparse. Because of this, many previous methods require dividing mutation catalogs for each tumor type. Here, we propose parallelized latent Dirichlet allocation (PLDA), a novel Bayesian model to simultaneously predict mutation signatures with all mutation catalogs. PLDA is an extended model of latent Dirichlet allocation (LDA), which is one of the methods used for signature prediction. It has parallelized hyperparameters of Dirichlet distributions for LDA, and they represent the sparsity of signature activities for each tumor type, thus facilitating simultaneous analyses. First, we conducted a simulation experiment to compare PLDA with previous methods (including SigProfiler and SignatureAnalyzer) using artificial data and confirmed that PLDA could predict signature structures as accurately as previous methods without searching for the optimal hyperparameters. Next, we applied PLDA to PCAWG (Pan-Cancer Analysis of Whole Genomes) mutation catalogs and obtained a signature set different from the one predicted by SigProfiler. Further, we have shown that the mutation spectrum represented by the predicted signature with PLDA provides a novel interpretability through post-analyses. Full article

This paper presents a general procedure for a rate-type creep analysis (based on the use of the continuous retardation spectrum) which avoids the need of recalculating the Kelvin chain stiffness elements at each time step. In this procedure are incorporated three different creep constitutive relations, two recommended by national codes such as the ACI (North-American) and EC2 (European) building codes and one by the RILEM research association. The approximate expressions of the different creep functions with the corresponding Dirichlet series are generated using the continuous retardation spectrum approach based on the Post–Widder formula. The proposed rate-type formulation is implemented into a 3D finite element code and applied to study the long-term deflections of a prestressed concrete bridge built in Romania, which crosses a wide artificial channel that connects the Danube river to the port of Constanta in the Black Sea. Full article