Search Results (3,709)

Asymmetric quantum codes are useful for quantum channels in which phase and bit errors occur with different probabilities, since the two distances, $d_z$ and $d_x$, can be controlled separately. We develop a permutation-paired matrix-product construction for such codes over ${\mathbb{F}}_q$. The main task is to build classical code pairs $\mathcal{C},\mathcal{D}\subseteq {\mathbb{F}}_{q^2}^{kn}$ satisfying the Hermitian inclusion ${\mathcal{D}}^{{\perp }_{\mathrm{H}}}\subseteq \mathcal{C}$, while keeping explicit dimension and distance bounds. Let $A\in {\mathbb{F}}_{q^2}^{k\times k}$ be a non-singular-by-columns (NSC) matrix with $A{A}^{†}=D{P}_{\tau }$, where D is an invertible diagonal and $P_{\tau }$ corresponds to an involution $\tau$. For $\mathcal{C}=[C_1,\dots ,C_k]A$ and $\mathcal{D}=[D_1,\dots ,D_k]A$, we prove ${\mathcal{D}}^{{\perp }_{\mathrm{H}}}=[D_{\tau \left(1\right)}^{{\perp }_{\mathrm{H}}},\dots ,D_{\tau \left(k\right)}^{{\perp }_{\mathrm{H}}}]A$. Thus, the global inclusion ${\mathcal{D}}^{{\perp }_{\mathrm{H}}}\subseteq \mathcal{C}$ is equivalent to the shorter paired inclusions $D_{\tau \left(i\right)}^{{\perp }_{\mathrm{H}}}\subseteq C_i$. This yields asymmetric quantum codes with parameters ${\left[[kn,{\sum }_{i=1}^k(r_i+s_i)-kn,d_z/d_x]\right]}_q$, where the bounds for $d_z$ and $d_x$ follow from NSC matrix-product distance estimates. For nested maximum distance separable (MDS) constituents, the paired conditions reduce to $r_i+s_{\tau \left(i\right)}\ge n$, giving explicit infinite families. Concrete $\tau$-OD matrices and numerical examples show that nontrivial permutations can increase the quantum dimension while preserving prescribed lower bounds for $d_z$ and $d_x$. Full article

Several important functions, including the gamma function, as well as several infinite sums, admit integral representations involving the Hankel contour. In addition, the large t asymptotic analysis of several recently derived identities satisfied by the Riemann zeta function requires the computation of the asymptotic form of certain integrals which also involve the Hankel contour; these integrals depend on a real parameter, $\alpha$. A rigorous asymptotic technique is presented here for computing such integrals to all orders. For certain values of $\alpha$, the relevant formula, in addition to an asymptotic series of explicit terms, also contains a specific integral. It is shown that, remarkably, the leading behavior of this integral can be written in the form of the leading order of the Bleistein integral. The latter integral arises in the implementation of the classical steepest descent method in the case that the stationary point coincides with one of the boundary points of the integral under consideration. Full article

This paper analyzes a single-server queueing system with infinite capacity, where arrivals follow a Markovian arrival process and service and repair times are modeled by phase-type distributions. The service mechanism is two-tier: every customer undergoes a mandatory primary service, after which an optional secondary service is available upon request. When the system is empty, the server initiates a closedown process before taking successive multiple vacations; upon return, the server goes through a setup process before beginning service again. Service can be interrupted by random breakdowns in either mode, triggering a phase-type repair. Matrix-analytic methods are used for the steady-state analysis, yielding the stability condition, stationary probability vectors, busy period analysis and key performance measures. A cost analysis framework is also developed. Numerical experiments validate the analytical results and illustrate the practical applicability of the model. Full article

This paper considers a multi-period two-sided asymmetric information model with infinitely long-lived sellers and short-lived buyers. I assume that two exogenously given qualities are offered in the market. Each period, a consumer, who is uncertain about the quality of the offered product, observes her pairwise matched seller’s price and a noisy signal of quality that cannot be manipulated by the seller. Prices are fixed and it is common knowledge that consumers are not willing to pay a high price for the low-quality product. A matched seller with a low-quality good can choose to be either honest (by charging the lower market price) or dishonest (by charging the higher price). Sellers’ incentives to misrepresent quality depend on how current trade outcomes affect future access to consumer traffic. I show that the strength of the informational role of prices is non-decreasing in the intensity of competition for future consumer traffic in equilibrium and that consumers do not benefit from more intense competition. Full article

A methodology for determining two parameters was proposed: the effective depth of heat penetration, and the thickness of the surface layer accumulating a given amount of heat. Explicit formulas allowing estimation of the values of these parameters for a semi-infinite body heated by heat flux with a variable time profile of intensity were obtained. Ten time profiles corresponding to different types of braking were analysed. The obtained results can be used at the design stage to determine the temperature mode and then to select materials for the friction elements of a disc braking system. Full article

Even before the discovery of DNA, Claude Shannon developed a mathematical model of Mendelian inheritance. Unlike the widely recognized Hardy–Weinberg equilibrium, Shannon’s genetic algebra has received little scholarly attention. Here, we revisit Shannon’s algebra and develop two complementary extensions for modern population genetics. First, we formulate a finite population version of Shannon’s framework, moving beyond the idealized infinite population setting to propagate allelic and genotypic frequencies under realistic sampling conditions. Second, we combine Shannon’s algebra with an analytical genotype–phenotype mapping framework to characterize genotypic configurations compatible with observed phenotypic frequencies, using auxiliary variables to express the inherent degeneracy of the genotype–phenotype relationship. Together, these extensions provide a unified framework in which phenotypic observations constrain the underlying genetic structure, and these constraints are propagated through inheritance via Shannon’s algebra. The resulting analytical expressions for offspring phenotypic distributions apply to both complete and incomplete penetrance and extend naturally to multilocus systems. This work highlights Shannon’s algebra as a flexible analytical tool for two complementary problems: (i) forward propagation of genetic information in finite populations, and (ii) analytical description of phenotypic inheritance from inferred genotypic information. Full article

Dynamic ON–OFF server control is widely used to reduce energy consumption in data centers. We study ON–OFF control in a queueing system with an s-staggered setup policy, which limits the number of servers that can simultaneously enter the setup state. Despite its potential energy benefits, analytical results for this policy remain limited, particularly for large-scale systems with infinite buffers. This paper presents a generating-function-based analysis of the s-staggered setup queueing model and derives exact expressions for the stationary queue-length distribution. We enhance the conventional generating-function approach by reformulating the non-homogeneous part of the underlying Markov chain, thereby reducing the number of computational states and improving scalability. The proposed algorithm requires approximately half the computational effort of the conventional generating-function approach when s is small relative to system capacity. Numerical experiments demonstrate that the algorithm can efficiently handle large-scale systems and provides insights into the energy–performance trade-off. Full article

Based on the Infinite State Representation (ISR) of the Riemann–Liouville integral, the energy stored in a fractional-order integrator is revisited, together with the energy dissipated through Joule losses. Using an idealized ultracapacitor model based on the Constant Phase Element (CPE), i.e., a fractional-order capacitor, theoretical expressions for the stored and dissipated energies during current charging of the CPE are derived. Numerical simulation of the fractional integrator over a frequency interval {ω_min, ω_max} validates a realistic CPE model, in which low-frequency modes correspond to energy storage, while high-frequency modes account for self-discharge and the origin of dissipated energy. This theoretical study leads to the definition of a new ultracapacitor model composed of an internal resistor and the previous realistic CPE, whose frequency-distributed representation enables prediction of the internal state variables and, consequently, the State of Charge. Full article

In this research, based on Adomian decomposition method (ADM), we construct true fractional-order differential equations. Due to the boosting function brought by the sine function, the system can output infinite coexistence attractors on y–z planes. In particular, this grid effect becomes increasingly obvious as the fractional order increases. Based on this boosting grid idea, in combination with the fractal dynamics, we construct some fractal patterns, e.g., Koch snows. These fractal diagrams all present grid fractal shapes. And then, we design a grid image encryption algorithm. This algorithm is proven to have higher security. The combination of chaos and fractals explores a new research direction. It provides new ideas for research in related fields. Full article

Symmetry operations are usually studied within the frameworks of group theory, geometry, and operator algebra. In the present work, a Ramsey-theoretic approach to symmetry is developed. Symmetry operations are treated as operators serving as vertices of complete bi-colored graphs, called symmetry graphs (SGs). Two symmetry operators are connected by a maroon edge when they commute and by a teal edge when they do not commute. Thus, the commutation structure of a symmetry group is transformed into a combinatorial object suitable for Ramsey-theoretic analysis. The introduced coloring is generally non-transitive, leading naturally to nontrivial complete bi-colored graphs constrained simultaneously by group-theoretical and combinatorial principles. It is shown that every symmetry graph containing six vertices necessarily contains either a monochromatic commuting triangle or a monochromatic non-commuting triangle as a direct consequence of the classical Ramsey theorem  R(3,3) = 6. The framework is illustrated for the symmetry groups of the equilateral triangle, regular tetrahedron, crystallographic point groups, infinite Cairo pentagonal tilings, and the triangular Ising ferromagnet. Higher-order structures, including teal quadrangles, second-order graph symmetries, infinite monochromatic cliques, and Lie-algebraic constraints arising from the Jacobi identity, are discussed. The proposed framework establishes a new connection between symmetry theory, Ramsey theory, graph theory, crystallography, and operator algebra. Full article

The transmission problem has emerged as an important component in various application fields, including industry, agriculture, and commerce. Simultaneously, it is also one of the most common challenges encountered in daily activities such as eating, drinking, living, and transportation. We have observed that numerous theories have been proposed regarding nonlocal transmission problems. However, how to find critical points of nonconvex functionals remains one of the major difficulties when one employs variational methods to establish the existence of solutions for nonlocal problems. This paper is devoted to the study of a class of weighted nonlocal transport equations. In order to establish the existence of solutions, we utilize algebraic analysis and combinatorial approximation techniques to overcome the nonconvexity of the corresponding Euler–Lagrange functional, and then prove the existence of a positive solution with the help of the strong maximum principle. Finally, by combining algebraic analysis with existing results, we obtain the existence of infinite solutions. Full article

This paper develops a unified asymptotic theory for conditional single-index U-statistics and the associated conditional U-processes in the setting of locally stationary functional time series subject to missing-at-random response mechanisms. The proposed framework addresses, within a single nonparametric inferential architecture, three major sources of complexity in modern functional data analysis: infinite-dimensional covariates, smoothly time-varying stochastic dynamics, and incomplete response observations. The methodology is based on a class of kernel-type estimators combining temporal localization, functional single-index smoothing, and inverse-propensity correction. Temporal localization captures the gradual evolution of the underlying regression structure, the single-index projection provides an effective dimension-reduction mechanism for functional covariates, and the propensity adjustment restores the target conditional functional under the MAR sampling scheme. The principal contribution of the paper is the establishment of weak convergence, in a suitable space of bounded functions, for the resulting propensity-adjusted conditional U-process indexed by a general class of measurable kernels. Under absolute regularity conditions, local stationarity assumptions, small-ball probability requirements, entropy restrictions of VC type, and uniform consistency of the propensity-score estimator, the normalized process is shown to converge weakly to a tight centered Gaussian process. The limiting covariance structure explicitly reflects the interaction between temporal smoothing, functional concentration, dependence, and the random loss of responses. In parallel, uniform convergence rates are derived for the associated conditional single-index U-statistic estimators, thereby quantifying the respective contributions of smoothing bias, stochastic fluctuation, local-stationarity approximation error, and missingness-induced variance inflation. A substantial part of the analysis is devoted to the technical difficulties created by the simultaneous presence of dependence, nonstationarity, functional covariates, and incomplete observations. The proofs combine Hoeffding-type decompositions adapted to weighted incomplete data, blocking and coupling arguments for absolutely regular triangular arrays, refined entropy bounds for kernel-indexed function classes, and small-ball probability techniques for functional covariates. The MAR mechanism is incorporated via inverse-propensity weighting, and its effects on the effective sample size, asymptotic variance, and bias structure are made explicit. The theory also provides a rigorous foundation for bandwidth selection through blocked, propensity-adjusted cross-validation and clarifies its relation to the corresponding oracle risk. The proposed framework encompasses a broad class of statistical learning and inference problems involving pairwise or higher-order functionals of functional time series. In particular, it applies to conditional Kendall-type functionals, discrimination problems, metric learning with incomplete labels, and conditional independence testing under local stationarity. A simulation study illustrates the finite-sample behavior of the proposed estimators and supports the theoretical findings across varying regimes of temporal nonstationarity, serial dependence, functional concentration, and response missingness. Overall, the results provide a mathematically rigorous and methodologically flexible foundation for inference from evolving functional data when dependence, infinite dimensionality, and incomplete observation are present simultaneously. Full article

This research explores the existence of solutions for a class of random fractional differential equations characterized by bounded delay, specifically within the context of Fréchet spaces. Random fractional differential equations serve as powerful mathematical tools for modeling complex phenomena subjected to stochastic perturbations and hereditary effects. Despite their significance, establishing solution existence in infinite-dimensional spaces remains a challenging task. By integrating the properties of the noncompactness measures with a generalized Darbo fixed point approach, we establish new existence results for the associated Darboux-type problem under milder compactness conditions. To illustrate the practical utility of these analytical results and demonstrate the validity of our theoretical framework, a representative example is provided. Full article

Non-Hermitian photonic lattices offer unconventional control over light evolution owing to modal non-orthogonality and the resulting non-normal dynamical response. In this work, we show that a uniform passive waveguide lattice with dissipation confined to one or a few sites near an edge can exhibit an algebraic(nearly linear) decay of optical power—an absorption law forbidden in orthogonal (normal-mode) dissipative systems, where any superposition of eigenmodes yields purely multi-exponential attenuation. We demonstrate that algebraic absorption arises when the input excitation is appropriately tailored to exploit non-orthogonal modal interference, effectively channeling energy toward the dissipative boundary. In particular, under the condition of coherent perfect absorption (CPA) associated with a spectral singularity of the semi-infinite lattice, nearly complete light absorption accompanied by algebraic decay of the optical power can be achieved. Starting from the minimal configuration of a single lossy edge site, we derive compact analytical expressions for the dynamics and identify the conditions under which linear-like absorption emerges. We then extend the analysis to multiple edge-proximal lossy sites. Our results show that simple dissipative photonic lattices, when driven by suitably prepared input states, enable robust sculpting of absorption laws through non-normal dynamics, providing a new route to programmable attenuation. Full article

Natural lateral particle segregation commonly occurs during the hydraulic deposition of slurry and thickened tailings in surface tailings storage facilities (TSFs), producing spatial heterogeneity in physical, hydrogeotechnical, and mineralogical properties, as well as in the water table. In sulfide-rich tailings, such heterogeneity complicates the design of reclamation cover systems, which are themselves affected by it. This study investigates the impact of physical and rheological properties of hard-rock mine tailings slurries on their segregation under hydrodynamic conditions. It proposes a multiparametric equation for the segregation index (SI) based on Buckingham’s π theorem. For this purpose, six flume experiments were conducted using tailings with initial solid mass concentrations of 63%, 66%, and 69% at slopes of 0.5% and 1%. Results revealed strong exponential correlations (R² > 0.95) between SI and tailings’ physical properties (solid concentration, bulk density) as well as rheological parameters (Herschel–Bulkley yield stress and flow index, Cross infinite dynamic viscosity). The SI equation was developed using MATLAB R2025b nonlinear least-squares optimization with a trust-region reflective algorithm. Using an SI threshold of 0.05 to define non-segregating behavior, the proposed model can predict segregation tendencies as a function of tailings properties and slope conditions. Further laboratory and field investigations are needed to validate and generalize the criterion. Full article