搜索结果：Algebra and Number Theory

Motivated by the interplay between quadratic algebras, noncommutative geometry, and operator theory, we introduce the notion of quadratic subproduct systems of Hilbert spaces. Specifically, we study the subproduct systems induced by a finite number of complex quadratic polynomials in noncommuting variables, and describe their Toeplitz and Cuntz-Pimsner algebras. Inspired by the theory of graded associative algebras, we define a free product operation in the category of subproduct systems and show that this corresponds to the reduced free product of the Toeplitz algebras. Finally, we obtain results about the K-theory of the Toeplitz and Cuntz-Pimsner algebras of a large class of quadratic subproduct systems.

River networks are important landscape features that have been extensively studied over many years. While seminal works have focused on characterizing the topological properties of river networks, the quantification of their spectral properties has received limited attention. In this study, through a graph-theoretic formulation of river network topology, we investigate the eigenvalue spectra of its connectivity matrix (i.e., adjacency matrix). First, we explain the observed range of zero eigenvalues on the spectra using the notion of multiplicity (i.e., algebraic and geometric multiplicity) for both undirected and directed river networks. Next, we investigate the physical meaning of the multiplicity of zero eigenvalues on the dynamics of the river network. We show that multiplicity of zero eigenvalues is sufficient to determine the minimum set of driver nodes on the river network. The ratio of the number of driver nodes vs total number of nodes is a measurement of controllability of the river network, which is essential for a comprehensive understanding of the system's dynamics under external forcing. Using both synthetic and natural river networks, we show that with increasing heterogeneity, quantified via Tokunaga c-value, the number of zero eigenvalues increases indicating that basins in humid climate require more number of driver nodes to control their network dynamics. Finally, we show that driver nodes tend to avoid critical nodes identified via pairwise connectivity. Our results indicate that the multiplicity of zero eigenvalues in the eigenvalue spectrum can serve as a valuable tool for understanding and quantifying the physical and dynamical properties of river networks, such as controllability and heterogeneity. Furthermore, our findings establish a clear connection between controllability metrics and the vulnerability of river networks.

The study of algebraic structures through graph-theoretic representations provides a powerful visual and combinatorial framework for analyzing ring-theoretic properties. The ideal-based non-zero divisor graph &#x2205; I ( Z n ) , constructed from the ring of integers modulo n with respect to a proper ideal I . This graph extends the classical zero-divisor graph framework and serves as a visual and structural invariant for analyzing ideal interactions in finite commutative rings. Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of &#x2205; I ( Z n ) . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function &#x3d5; ( n ) . The analysis distinguishes cases based on the parity and primality of n , as well as the generator of I . Topological indices, including the Zagreb and Randi&#x107; indices, are formulated to quantify structural complexity. We establish necessary and sufficient conditions for the connectivity of &#x2205; I ( Z n ) , proving it is connected for all n &#x2265; 10 and any non-zero proper ideal I . For prime n &#x2209; { 2 , 3 } , the graph is shown to be complete. General formulas are provided for calculating vertex degrees based on gcd ( x , d ) where I = < d > . Furthermore, the structure and computation cut-sets are characterized for Z p 2 and composite n = xy . Moreover, the domination number &#x3b3; ( &#x2205; I ( Z n ) )=1 and girth gr ( &#x2205; I ( Z n ) )=3 is established for n &#x2265; 10 . General expressions for Zagreb and Randi&#x107; indices are derived, directly linking graph invariants to n and d . The graph &#x2205; I ( Z n ) serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in Z n . These results establish systematic connections between ring-theoretic properties and graph parameters, enabling both qualitative and quantitative analysis through connectivity, degree distributions, cut-sets, and topological indices.

A scheme for the calculation of electron-attachment (EA) processes within the framework of unitary coupled-cluster (UCC) theory is presented. Analogous to the description of electron-detachment, the intermediate state representation (ISR) approach is used for the formulation and its relation to the algebraic-diagrammatic construction scheme is pointed out. Due to the UCC ansatz, the resulting equations cannot be given by closed-form expressions, but need to be approximated. Explicit working equations for two computational schemes referred to as EA-UCC2 and EA-UCC3 are given, providing electron-attachment energies and spectroscopic amplitudes of electron-attached states dominated by one-particle excitations correct through second and third order in perturbation theory, respectively. In the derivation, an expansion of the UCC transformed Hamiltonian involving Bernoulli numbers as expansion coefficients is employed. In a benchmark against full configuration interaction (FCI) results including 50 states of 21 different species, both neutral and charged, closed- and open-shell, the novel methods are characterized by mean absolute errors of 0.15 eV (EA-UCC2) and 0.10 eV (EA-UCC3). Furthermore, an approach for the computation of physical properties of electron-attached as well as electron-detached states within the UCC framework is presented. It also builds upon the ISR approach, featuring an expectation value-like formulation similar to that of the equation-of-motion coupled-cluster (EOM-CC) method or the ISR approach of the algebraic-diagrammatic construction (ADC) method. Explicit expressions for the expectation value of a general one-particle operator correct through second order in perturbation theory are given and shown to be equivalent to those of the second-order ADC/ISR procedure.

In this paper, we introduce a general framework for analyzing the numerical conditioning of minimal problems in multiple view geometry, using tools from computational algebra and Riemannian geometry. Special motivation comes from the fact that relative pose estimation, based on standard 5-point or 7-point Random Sample Consensus (RANSAC) algorithms, can fail even when no outliers are present and there is enough data to support a hypothesis. We argue that these cases arise due to the intrinsic instability of the 5- and 7-point minimal problems. We apply our framework to characterize the instabilities, both in terms of the world scenes that lead to infinite condition number, and directly in terms of ill-conditioned image data. The approach produces computational tests for assessing the condition number before solving the minimal problem. Lastly, synthetic and real data experiments suggest that RANSAC serves not only to remove outliers, but in practice it also selects for well-conditioned image data, which is consistent with our theory.

Starbursts are the light-intensity patterns seen when small bright sources are observed at low illumination levels, typically stars at night. Starburst patterns are formed because the eye's wave aberrations generate caustics at the retina. However, a fascinating yet unexplained fact about starbursts is that they usually exhibit p-fold symmetry. Moreover, the number of peaks, related to the symmetry perceived by the subject, is not always the same. The main aim of this study is to explain these visual optics phenomena. For this purpose, we provide a theoretical framework based on the geometric and algebraic properties of the wave aberration function expressed as a Zernike polynomial expansion. Specifically, we investigated the number and distribution of the fertile cusps of Gauss of the wave aberration function. We also established the connections between these points with the symmetries and the number of starburst peaks. We found that starbursts are likely generated by wave aberrations dominated by axially symmetric polynomials combined with a certain amount of non-axially symmetric ones. For instance, whereas a wave aberration with a dominant spherical aberration (Zernike polynomial [Formula: see text]) plus [Formula: see text] may induce a 3-peaks starburst with a 3-fold symmetry, a wave aberration combining [Formula: see text] and [Formula: see text] may induce a 4-fold symmetry starburst with four or eight peaks. In addition to providing a comprehensive explanation of starburst symmetries, our theory has other promising applications; for instance, we could infer some basic properties of an eye's wave aberration function from a measurement (subjective or objective) of the starburst pattern.

Despite the availability of various sequence analysis models, comparative genomic analysis remains a challenge in genomics, genetics, and phylogenetics. Commutative algebra, a fundamental tool in algebraic geometry and number theory, has rarely been used in data and biological sciences. In this study, we introduce commutative algebra k-mer learning (CAKL) as the first-ever nonlinear algebraic framework for analyzing genomic sequences. CAKL bridges between commutative algebra, algebraic topology, combinatorics, and machine learning to establish a new mathematical paradigm for comparative genomic analysis. We evaluate its effectiveness on three tasks-genetic variant identification, phylogenetic tree analysis, and viral genome classification-typically requiring alignment-based, alignment-free, and machine-learning approaches, respectively. Across eleven datasets, CAKL outperforms five state-of-the-art sequence analysis methods, particularly in viral classification, and maintains stable predictive accuracy as dataset size increases, underscoring its scalability and robustness. This work ushers in a new era in commutative algebraic data analysis and learning.

Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields Q(&#x3b6;p) for primes p&#x2261;1(mod4) via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields Q(&#x3b6;pn) with n>1. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vign&#xe9;ras' differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems.

Hypercomplex Neural Networks (HNNs) represent the next frontier in deep learning, building on the mathematical theory of quaternions, octonions, and higher-dimensional algebras to generalize conventional neural architectures. This review synthesizes cutting-edge methods with their theoretical bases, architectural advancements, and primary applications, tracing the development of hypercomplex mathematics and its implementation in computational models. We distil key advances in quaternion and octonion networks, highlighting their ability to provide compact representations and computational efficiency. Particular attention is given to the unique challenge of non-associativity in octonions-where the order in which numbers are multiplied affects the result-requiring careful design of network operations. The article also discusses training complexity, interpretability, and the lack of standardized frameworks, alongside comparative performance with real- and complex-valued networks. Future directions include scalable algorithm construction, lightweight architectures through tensor decompositions, and integration with quantum-inspired systems using higher-order algebras. By presenting a systematic synthesis of current literature and linking these advances to practical applications, this review aims to equip researchers and practitioners with a clear understanding of the strengths, limitations, and potential of HNNs for advancing multidimensional data modelling.

The rapid advances in artificial intelligence (AI) and machine learning have transformed computer technology. Now, computers can learn and adapt autonomously, make sound judgments, and create environments in which humans and AI work in harmony. Analytical frameworks that can process data with uncertainty, imprecision, and multiple perspectives are essential for the success of human-AI collaboration. In this connection, linguistic Z-number (LZN) evolution models play a crucial role in selecting tools for human-AI collaboration because they can better capture uncertainty and fuzziness. Given this, we extend multicriteria group decision-making (MCGDM) by incorporating LZNs. The structure of MCGDM under LZNs, the consensus-building process, estimation of experts' weights, and the ranking of options are the main challenges. In the experts' primary opinion-based MCGDM, we believe that expert participation and the opportunity to revise their opinions with original linguistic terms (LTs) would be more practical. However, in existing studies, experts have no scope to revise their opinions based on the original LTs. Most studies on consensus-building either do so automatically or advise experts to revise their opinions based on virtual LTs. But either automatically or virtually, LTs cannot be physically represented by words, which is a significant drawback. Given these facts, we propose an interactive, strategy-based consensus-building in which we advise experts to revise their opinions within the original LTs. To estimate experts' weights, we developed an integrated subjective and objective method. By simultaneously assigning subjective weights and opinion-based relative-relevance indices to the experts, we derive the adjusted subjective expert weights. The objective weights of experts are derived from the criterion-based entropy method. The ranking of options, we proposed the prospect and regret theory-based TOPSIS ranking approach to rank the options. Finally, a case study on the optimal selection of an AI tool for human-AI collaboration validates the proposed method. Through a comparative analysis with alternative methodologies, we demonstrate the feasibility, stability, and superiority of the proposed model.

Quantum low-density parity-check (QLDPC) codes offer a promising path to low-overhead fault-tolerant quantum computation but lack systematic strategies for exploration. In this Letter, we establish a topological framework for studying the bivariate-bicycle codes, a prominent class of QLDPC codes tailored for real-world quantum hardware. Our framework enables the investigation of these codes through universal properties of topological orders. In addition to efficient characterizations using Gr&#xf6;bner bases, we also introduce a novel algebraic-geometric approach based on the Bernstein-Khovanskii-Kushnirenko theorem. This approach allows us to analytically determine how the topological order varies with the generic choices of bivariate-bicycle codes under toric layouts. Novel phenomena are unveiled, including topological frustration, where ground-state degeneracy on a torus deviates from the total anyon number, and quasifractonic mobility, where anyon movement violates energy conservation. We demonstrate their intrinsic link to symmetry-enriched topological orders and derive an efficient method for generating finite-size codes. Furthermore, we extend the connection between anyons and logical operators using Koszul complex theory. Our Letter provides a rigorous theoretical basis for exploring the fault tolerance of QLDPC codes and deepens the interplay among topological order, quantum error correction, and advanced algebraic structures.

The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra's numerical criterion. In this paper, we relate L ( 1 , &#x3c0; , Ad &#x2218; ) to the congruence ideals for cohomological cuspidal automorphic representations &#x3c0; of GL n over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of L ( 1 , &#x3c0; , Ad &#x2218; ) . This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of GL n , its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.

Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical insight into these transitions. This paper introduces an algebraic framework for analyzing the stability landscapes of ecological models defined by systems of first-order autonomous ordinary differential equations with polynomial or rational rate functions. Using tools from real algebraic geometry, we characterize parameter regions associated with steady-state feasibility and stability via three key boundaries: singular, stability (Routh-Hurwitz), and coordinate boundaries. With these boundaries in mind, we employ routing functions to compute the connected components of parameter space in which the number and type of stable steady states remain constant, revealing the stability landscape of these ecological models. As case studies, we revisit the classical Levins-Culver competition-colonization model and a recent model of coral-bacteria symbioses. In the latter, our method uncovers complex stability regimes, including regions supporting limit cycles, that are inaccessible via traditional techniques. These results demonstrate the potential of our approach to inform ecological theory and intervention strategies in systems with nonlinear interactions and multiple stable states.

Arguments by Sorkin (Impossible measurements on quantum fields. In: Directions in general relativity: proceedings of the 1993 International Symposium, Maryland, vol&#xa0;2, pp 293-305, 1993) and Borsten et al. (Phys Rev D 104(2), 2021. 10.1103/PhysRevD.104.025012) establish that a natural extension of quantum measurement theory from non-relativistic quantum mechanics to relativistic quantum theory leads to the unacceptable consequence that expectation values in one region depend on which unitary operation is performed in a spacelike separated region. Sorkin [1] labels such scenarios 'impossible measurements'. We explicitly present these arguments as a no-go result with the logical form of a reductio argument and investigate the consequences for measurement in quantum field theory (QFT). Sorkin-type impossible measurement scenarios clearly illustrate the moral that Microcausality is not by itself sufficient to rule out superluminal signalling in relativistic quantum theories that use L&#xfc;ders' rule. We review three different approaches to formulating an account of measurement for QFT and analyze their responses to the 'impossible measurements' problem. Two of the approaches are: a measurement theory based on detector models proposed in Polo-G&#xf3;mez et al. (Phys Rev D, 2022. 10.1103/physrevd.105.065003) and a measurement framework for algebraic QFT proposed in Fewster and Verch (Commun Math Phys 378(2):851-889, 2020). Of particular interest for foundations of QFT is that they share common features that may hold general morals about how to represent measurement in QFT. These morals are about the role that dynamics plays in eliminating 'impossible measurements', the abandonment of the operational interpretation of local algebras A ( O ) as representing possible operations carried out in region O, and the interpretation of state update rules. Finally, we examine the form that the 'impossible measurements' problem takes in histories-based approaches and we discuss the remaining challenges.

An even Eisenstein integer is a multiple of an Eisenstein prime of the least norm. Otherwise, an Eisenstein integer is called odd. An Eisenstein integer that is not an integer multiple of another one is said to be primitive. Such integers can be used to construct signal constellations and complex-valued codes over Eisenstein integers via a carefully designed modulo function. In this work, we establish algebraic properties of even, odd, and primitive Eisenstein integers. We investigate conditions for the set of all units in a given quotient ring of Eisenstein integers to form a cyclic group. We perform set partitioning based on the multiplicative group of the set. This generalizes the known partitioning of size a prime number congruent to 1 modulo 3 based on the multiplicative group of the Eisenstein field in the literature.

Glass networks model systems of variables that interact via sharp switching. A body of theory has been developed over several decades that, in principle, allows rigorous proof of dynamical properties in high dimensions that is not normally feasible in nonlinear dynamical systems. Previous work has, however, used examples of dimensions no higher than 6 to illustrate the methods. Here, we show that the same tools can be applied in dimensions at least as high as 20. An important application of Glass networks is to a recently proposed design of a true random number generator that is based on an intrinsically chaotic electronic circuit. In order for analysis to be meaningful for the application, the dimension must be at least 20. Bifurcation diagrams show what appear to be periodic and chaotic bands. Here, we demonstrate that the analytic tools for Glass networks can be used to rigorously show where periodic orbits are lost and the types of bifurcations that occur there. The main tools are linear algebra and the stability theory of Poincar&#xe9; maps. All main steps can be automated, and we provide computer code. The methods reviewed here have the potential for many other applications involving sharply switching interactions, such as artificial neural networks.

The occupation number is a key observable for diagnosing thermalization, as it connects directly to standard statistical laws such as Fermi-Dirac, Bose-Einstein, and Boltzmann distributions. In the context of spin systems, it represents the population of the sublevels of the magnetization in the z direction. We use this quantity to probe the onset of thermalization in an isolated one-dimensional quantum spin-1 Ising model with transverse and longitudinal fields and in its classical counterpart. Thermalization is achieved when the long-time average of the occupation number converges to the microcanonical prediction as the chain length L increases, consistent with the emergence of ergodicity. However, the finite-size scaling analysis in the quantum model is challenged by the exponential growth of the Hilbert space with L. To overcome this limitation, we turn to the classical model, which enables access to much larger system sizes. By tracking the dynamics of individual spins on their three-dimensional Bloch spheres and employing tools from random matrix theory, we establish a quantitative criterion for classical ergodicity in interacting spin systems. We find that deviations from classical ergodicity decay algebraically with system size. This power-law scaling then provides a quantitative bound on the approach to thermal equilibrium in the quantum model.

A survey of the literature reveals notable discrepancies among the purported exact results for the spectra of stochastic gene expression models. For self-repressing gene circuits, previous studies ([Phys. Rev. Lett. 99, 108103 (2007)], [Phys. Rev. E 83,062902 (2011)], [J. Chem. Phys. 160, 074105 (2024)], and [bioRxiv 2025.02.05.635946 (2025)]) have provided different exact solutions for the eigenvalues of the generator matrix. In this work, we propose a unified Hilbert space framework for the spectral theory of stochastic gene expression. Based on this framework, we analytically derive the spectra for models of constitutive, bursty, and autoregulated gene expression. The eigenvalues and eigenvectors obtained are then used to construct an exact spectral representation of the time-dependent distribution of gene product numbers. The spectral gap between the zero eigenvalue and the first nonzero eigenvalue, which reflects the relaxation rate of the system towards its steady state, is then compared with the prediction of the deterministic model, and we find that deterministic modeling fails to capture the relaxation rate when autoregulation is strong. In particular, our results demonstrate that for infinite-dimensional operators such as in stochastic gene expression models, many conclusions in linear algebra do not apply, and one must rely on the modern theory of functional analysis.

A lossless, exact compaction of the time-evolved state of the quantum dynamical system of a perturbed anharmonic molecule is demonstrated using dynamical symmetries. The density matrix of the anharmonic molecule is a linear combination of these symmetries, and it remains so as a time-dependent perturbation is applied. Accurate, unitary-but-approximate, and thereby irreversible compaction is further shown using fewer symmetries, and the fidelity of this lossy compaction is quantified. Perturbations are typically linear in the operators of a Lie algebra. For a Hamiltonian that is also linear, one knows well how to reversibly compact the state of a dynamical system. However, anharmonic vibrations have a finite number of unequally spaced energy levels, and a good description of their spectra typically requires an algebraic-type Hamiltonian that is bilinear in the operators of a Lie algebra. For a bilinear Hamiltonian we show how a matrix-based approach allows us to compact both the populations and the coherences, either exactly reversibly or inexactly irreversibly, with fewer symmetries. A forced Morse oscillator is used as an explicit analytical and numerical example covering the entire range of dynamics from the sudden to the adiabatic limits.