In this paper, we begin by defining finite versions of the digamma and cotangent functions, and examine their Laurent or power series expansions at integer points within a certain range. By constructing contour integrals involving these finite digamma and finite cotangent functions and performing residue calculations, we derive parity results for a finite version of the double polylogarithm function. Simply taking a limit then yields the known parity formulas satisfied by cyclotomic double zeta values.

Skew Laurent series and general cyclic convolutional codes

Abstract Let R be a ring, ( S , ⪯ ) {(S,\preceq)} a strictly totally ordered monoid and suppose also ω : S → End ⁢ ( R ) {\omega:S\rightarrow\mathrm{End}(R)} is a monoid homomorphism. A skew generalized power series ring R ⁢ [ [ S , ω , ⪯ ] ] {R[[S,\omega,\preceq]]} consists of all functions from a monoid S to a coefficient ring R whose support contains neither infinite descending chains nor infinite anti-chains, equipped with point-wise addition and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R . Special cases of the skew generalized power series ring construction are the skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Malcev–Neumann series rings and generalized power series rings as well as the untwisted versions of all of these objects. In the present article, we study the so-called ( S , ω ) {(S,\omega)} -McCoy condition on R that is a generalization of the standard McCoy condition from polynomials to skew generalized power series, thereby generalizing some of the existing results in the literature relevant to the subject.

Abstract Systems of $$N=1, 2, \dots $$ N = 1 , 2 , ⋯ first-order hyperbolic conservation laws feature $$N$$ N undamped waves propagating at finite speeds. On their own hand, multi-step Finite Difference and lattice Boltzmann schemes with $$q=N+1, N+2, \dots $$ q = N + 1 , N + 2 , ⋯ unknowns involve $$N$$ N “physical” waves, which are aimed at being as closely-looking as possible to the ones of the PDEs, and $$q-N$$ q - N “numerical–spurious–parasitic” waves, which are subject to their own speed of propagation, and either damped or undamped. The whole picture is even more complicated in the discrete setting—as numerical schemes act as dispersive media, thus propagate different harmonics at different phase (and group) velocities. For compelling practical reasons, simulations must always be conducted on bounded domains, even when the target problem is unbounded in space. The importance of transparent boundary conditions, preventing artificial boundaries from acting as mirrors producing polluting ricochets, naturally follows. This work presents, building on Besse, Coulombel, and Noble [ESAIM: M2AN, 55 (2021)], a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions. Our boundary conditions are “perfectly” transparent, at least for 1D problems, as they absorb both physical and spurious waves regardless of their frequency. After presenting, in a simple framework, several approaches to handle the fact that $$q>N$$ q > N , we elect the so-called “scalar” approach (which despite its name, also works when $$N>1$$ N > 1 ) as method of choice for more involved problems. This method solely relies on computing the coefficients of the Laurent series at infinity of the roots of the dispersion relation of the bulk scheme. We insist on asymptotics for these coefficients in the spirit of analytic combinatorics. The reason is two-fold: asymptotics guide truncation of boundary conditions to make them depending on a fixed number of past time-steps, and make it clear—during the process of computing coefficients—whether intermediate quantities can be safely stored using floating-point arithmetic or not. Numerous numerical investigations in 1D and 2D with $$N= 1$$ N = 1 and 2 are carried out, and show the effectiveness of the proposed boundary conditions.

ABSTRACT Based on the complex variable method and the corresponding principle of viscoelasticity, viscoelastic solutions for the stress and displacement of a lined non‐circular tunnel subjected to in‐situ stresses and internal water pressure is derived. The basic equations for solving the analytic functions are established according to the stress boundary condition along the inner boundary of the lining and the stress and displacement continuity conditions along the rock‐lining interface. The analytic functions are expressed as Laurent series and the Laplace transformation is performed on the basic equation. Herein, the power series method is applied to obtain the linear equations which are expressed by the analytic function coefficients in the Laplace domain. The stress and displacement solutions of tunnel in Laplace domain can be addressed by solving the equations, and then the viscoelastic solutions are obtained through Laplace Inverse transformation. Subsequently, an example for the horseshoe‐shaped tunnel is performed. The example used the generalized Kelvin model to simulate the rheological properties of surrounding rock mass. The obtained solution is compared with the numerical solution. The influences of the lateral pressure coefficient and the internal water pressure on the stresses and displacements of lining are analyzed.

Analytical time-dependent dynamics of stochastic gene expression with sRNA-mRNA interactions.

Principal Bautin ideal of monodromic singularities with inverse integrating factors

The linear viscoelasticity problem for an infinite anisotropic plate with an elliptical elastic inclusion under ideal mechanical contact conditions is solved. To obtain the solution, the small parameter method is applied, where the variation of Poisson's ratios over time is chosen as the parameter, effectively reducing the time-dependent problem to a sequence of analogous boundary value problems in the theory of elasticity. The construction of the solution is based on the complex potentials apparatus, conformal mapping methods, and Laurent series expansions. Boundary conditions at the contact interface are satisfied using the generalized least squares method, ensuring high accuracy of the unknown constants at any given moment. Analytical expressions for bending moments and shear forces in the plate are derived, explicitly incorporating viscoelastic time operators. For the case where the elliptical inclusion degenerates into a straight elastic line, formulas for calculating moment intensity factors at its endpoints are provided. The proposed approach allows for a correct description of the evolution of singular moment behavior and an evaluation of the material properties' influence on their temporal variation. Numerical studies were conducted for materials with various relaxation properties and different relative inclusion stiffnesses. It is established that the most intensive redistribution of moments occurs during the initial stage of the viscoelastic process, after which the stress state of the plate approaches a stationary phase. It is proven that moment concentration depends non-linearly on inclusion stiffness, being minimal at intermediate stiffness values and increasing sharply for holes or perfectly rigid inclusions. Isotropic plates are treated as a special case of anisotropic ones, allowing the results to be extended to a wide range of problems in composite mechanics and long-term strength prediction.

In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the q-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a q-analogue of the Hurwitz zeta function around s=1. We establish the closed-form expressions for the first two coefficients in the Laurent series of the q-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to γ 0 ( q , x ) for q>1 and 0<x<1, where γ 0 ( q , x ) is the constant term which appears in the Laurent series expansion of q-Hurwitz zeta function around s=1. Furthermore, we put forth a conjecture about the linear independence of special values of γ 0 ( q , x ) along with 1 at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals.

Sums of Laurent series with bounded partial quotients

This paper presents an effective approach for lower-order (LO) modeling of an electric vehicle–integrated off-grid microgrid (OMG) system. The seventh-order system (SOS) of the OMG is reduced to a second-order model (SOM) while preserving the original system’s dynamic characteristics and ensuring computational efficiency. The Taylor series (TS) and Laurent series (LS) expansions are employed to simplify the complex system that plays a significant role in the reduction process. The expansion parameters of the higher-order system (HOS) of OMG and its lower-order model (LOM) are exploited to construct the fitness function. The proposed approach constructs three sub-objective functions based on TS and LS. These sub-objective functions are then combined into a single fitness function to obtain an improved LOM by enhancing the transient and steady-state responses with respect to the HOS of OMG. To minimize the error, the resultant fitness function is optimized using the brown bear optimization (BBO) algorithm. The optimization is performed under two key constraints: (i) ensuring zero steady-state error, and (ii) satisfying the Hurwitz stability criterion. To demonstrate the efficacy of the proposed LOM, it is compared with other LOMs obtained from different approximation techniques. The proposed LOM and other LOMs are graphically validated through step, impulse, Bode, Nichols, and Nyquist response comparisons with the HOS. Additionally, the performance error criteria (PEC), time-domain specifications (TDSs) and frequency domain specifications (FDSs) of the proposed LOM are compared with other LOMs using the HOS to establish the validation and applicability of the proposed method.

Multiple zeta values and coefficients of Laurent series expansion of Beta function

Quantum sensing near exceptional points (EPs) in non-Hermitian systems has shown promising sensitivity enhancements. However, practical applications are often hindered by structural complexity and strict parameter constraints. In this work, we introduce a simplified anti-parity-time (anti-PT) symmetric platform consisting of two independently cavities, which are indirectly coupled to each other by a shared dissipative environment. We demonstrate a significantly enhanced sensing response at the EPs compared to non-EP configurations. This improvement is attributed to the dominant second-order term in the Laurent series expansion of the eigenvalue response to external perturbations- a characteristic feature of higher-order singularities at EPs. This mechanism not only reinforces the foundation for sensitivity enhancement but also offers a structurally compact and robust strategy for quantum sensing. Our results underscore the potential of anti-PT symmetric systems in enabling high-precision sensing technologies and bridging non-Hermitian physics with scalable photonic device platforms.

Bi-infinite Riordan matrices: A matricial approach to multiplication and composition of formal Laurent series

The higher Euler–Kronecker constants of a number field [Formula: see text] are the coefficients appearing in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function about [Formula: see text]. These coefficients are mysterious and seem to contain a lot of arithmetic information. In this paper, we study these coefficients. We prove arithmetic formulas and bounds satisfied by them, generalizing certain results of Ihara.

This study derives novel exact traveling wave solutions for the nonlinear (1+1)-dimensional Chafee-Infante equation by synthesizing the generalized first integral method (GFIM) with Laurent polynomial expansions. As a fundamental reaction-diffusion model, the Chafee-Infante equation governs pattern formation in diverse systems—from biological to chemical and physical contexts—yet its strong nonlinearity poses persistent challenges to classical integration techniques such as the inverse scattering transform or Hirota's method. We transform the equation into an autonomous polynomial system and employ the division theorem to systematically identify its first integrals, thereby circumventing the need for auxiliary equations or ansatz-based heuristics. By introducing Laurent polynomial ansatzes of varying complexity—ranging from first-degree to higher-order expansions—we yield compact rational-exponential solutions that are both exact and computationally tractable. The validity of these solutions is confirmed through symbolic computation in Mathematica, while a detailed graphical analysis elucidates their behavior—from bounded, dissipative profiles to singular structures—across different parameter regimes, including the critical thresholds $ C = 0 $ and $ C = 2 $ where blow-up phenomena emerge. This work underscores the efficacy of merging Laurent series with algebraic methods, offering a powerful and generalized tool for extracting exact solutions from a broader class of intractable nonlinear partial differential equations (PDEs) arising in mathematical physics and applied mathematics.

Continued fractions of cubic Laurent series and their effective irrationality exponents

The paper solves the problem of bending a piezo plate in the form of a half-plane with holes and cracks by using the complex potentials of the theory of electro-magneto-elastic bending of thin plates. In this case, functions that are holomorphic outside the contours of the holes and cracks are decomposed into Laurent series, and functions that are holomorphic in the lower half-planes are expressed using Cauchy-type integrals in terms of functions conjugate to these functions. При таком подходе полученные суммарные функции точно удовлетворяют граничным условиям на прямолинейной границе полуплоскости, а для определения неизвестных коэффициентов рядов Лорана используются граничные условия на контурах отверстий и трещин, которые в работе удовлетворяются обобщенным методом наименьших квадратов, приводящим задачу к переопределенной системе линейных алгебраических уравнений, решаемой методом сингулярного разложения. With this approach, the resulting total functions precisely satisfy the boundary conditions on the rectilinear boundary of the half-plane. To determine the unknown coefficients of the Laurent series we use boundary conditions on the contours of the holes and cracks, which are affected by the generalized least squares method, leading the problem to an overridden system of linear algebraic equations solved by the singular value decomposition method. The numerical results of the electro-magneto-elastic state of a half-plane with a circular hole or crack, with a circular hole and an internal crack in the jumper, with a circular hole having an edge crack in the jumper are described. We establish regularities of changes in the electro-magneto-elastic state of the plate depending on its material and geometric characteristics of holes and cracks, their mutual location. It has been found that as the hole or crack approaches the rectilinear boundary, the values of the moments at the points of the bridge increase sharply and change insignificantly in other zones. A large concentration of moments is also observed at the points of the rectilinear boundary near the jumper. The values of these moments are especially high in the problem for a half-plane with a circular hole having an edge crack in the jumper. The values of bending moments are significantly affected by taking into account the piezo properties of the material, especially in areas of high concentrations of the bending moments, therefore in these cases it is forbidden to limit ourselves to solving the problem of elasticity theory of plate bending, and it is necessary to solve the problem of electro-magneto-elasticity. Keywords: thin piezo plate, half-plane, holes, cracks, complex potentials, Cauchy type integrals, generalized least squares method, concentration of bending moments, moment intensity factors.

Abstract In 2004, de Mathan and Teulié stated the ‐adic Littlewood conjecture (‐LC) in analogy with the classical Littlewood conjecture. Let be a finite field be an irreducible polynomial with coefficients in . This paper deals with the analogue of ‐LC over the ring of formal Laurent series over , known as the ‐adic Littlewood conjecture (‐LC). First, it is shown that any counterexample to ‐LC for the case induces a counterexample to ‐LC when is any irreducible polynomial. Since Adiceam, Nesharim and Lunnon (2021) disproved ‐LC when and when is a finite field with characteristic 3, one obtains a disproof of ‐LC over any such field in full generality (i.e., for any choice of irreducible polynomial ). The remainder of the paper is dedicated to proving two metric results on ‐LC with an additional monotonic growth function over an arbitrary finite field. The first — a Khintchine‐type theorem for ‐adic multiplicative approximation — enables one to determine the measure of the set of counterexamples to ‐LC for any choice of . The second complements this by showing that the Hausdorff dimension of the same set is maximal in the critical case where . These results are in agreement with the corresponding theory of multiplicative Diophantine approximation over the reals. Beyond the originality of the results, the main novelty of the work comes from the methodology used. Classically, Diophantine approximation employs methods from either Number Theory or Ergodic Theory. This paper provides a third option: combinatorics. Specifically, an extensive combinatorial theory is developed relating ‐LC to the properties of the so‐called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. In full generality, the paper creates a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely.

Abstract The concept of degree independence, in the real number case, introduced in our earlier work is extended to non-archimedean fields. A sufficient condition for such independence is proved. As applications, sufficient conditions for degree independence of (i) elements in the field of formal Laurent series represented by Ruban continued fractions and of (ii) p -adic numbers are derived.