Rational Series Research Articles

On groups whose cogrowth series is the diagonal of a rational series

We show that if a group contains [Formula: see text] as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalizes a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag–Solitar groups [Formula: see text].

Finitely ambiguous and finitely sequential weighted automata over fields

Expressive power of finitely ambiguous and finitely sequential weighted automata over fields is examined. Rational series realised by such automata can be classified into infinite hierarchies according to ambiguity and sequentiality degrees of realising automata – it is shown that both these hierarchies are strict already over unary alphabets in case the field under consideration is not locally finite. Moreover, the expressive power of finitely sequential, finitely ambiguous, and polynomially ambiguous weighted automata is compared over different fields, both for unary and arbitrary finite alphabets, drawing a complete picture of the relations between corresponding classes of formal power series.

Antioptimization of mass-in-mass acoustic metamaterials based on interval analysis

In this study, an integrated analytical and experimental study on a type of one-dimensional periodic acoustic metamaterial with mass-in-mass unit cells is presented. Then, this paper proposes an antioptimization strategy to study the effect of the design parameters of the local resonators of the mass-in-mass acoustic metamaterial model on the band gap behavior. In this antioptimization strategy, the interval model is used to deal with the inner mass, spring stiffness and damping coefficient of the local resonators, and an interval mass-in-mass acoustic metamaterial model is established. Both the rational series expansion-interval perturbation finite element method (RSE-IPFEM) and interval affine finite element method (IAFEM) are used to evaluate the lower and upper bounds of the dynamic response interval vector, based on which an interval vibration transmission analysis can be implemented. Meanwhile, Monte Carlo method and experiments are also presented to demonstrate the efficiency and accuracy of the RSE-IPFEM and IAFEM. Results show that the design parameters of the local resonators significantly affect the beginning frequency of the band gap. Compared with the traditional optimization algorithm, the calculation accuracy of the antioptimization algorithm based on RSE-IPFEM and IAFEM can meet the engineering requirements, and the calculation efficiency is also significantly improved. Since IAFEM is more effective than RSE-IPFEM in suppressing conservatism, IAFEM is a better antioptimization strategy for dealing with the interval mass-in-mass acoustic metamaterial model. Therefore, the proposed antioptimization scheme is expected to inspire future researchers to analyze regulation rules of the band gap following the same procedures.

Bideterministic weighted automata

A finite automaton is called bideterministic if it is both deterministic and codeterministic – that is, if it is deterministic and its transpose is deterministic as well. The study of such automata in a weighted setting is initiated. All trim bideterministic weighted automata over integral domains and over positive semirings are proved to be minimal. On the contrary, it is observed that this property does not hold over commutative rings in general: non-minimal trim bideterministic weighted automata do exist over all semirings that are not zero-divisor free, and over many such semirings, these automata might not even admit equivalents that are both minimal and bideterministic. The problem of determining whether a given rational series is realised by a bideterministic automaton is shown to be decidable over fields and over tropical semirings. An example of a positive semiring over which this problem becomes undecidable is given as well.

Towards a Theory of Domains for Harmonic Functions and its Symbolic Counterpart

In this paper, we begin by reviewing the calculus induced by the framework of [10]. In there, we extended Polylogarithm functions over a subalgebra of noncommutative rational power series, recognizable by finite state (multiplicity) automata over the alphabet \(X=\{x_0,x_1\}\). The stability of this calculus under shuffle products relies on the nuclearity of the target space [32]. We also concentrated on algebraic and analytic aspects of this extension allowing to index polylogarithms, at non positive multi-indices, by rational series and also allowing to regularize divergent polyzetas, at non positive multi-indices [10]. As a continuation of works in [10] and in order to understand the bridge between the extension of this “polylogarithmic calculus” and the world of harmonic sums, we propose a local theory, adapted to a full calculus on indices of Harmonic Sums based on the Taylor expansions, around zero, of polylogarithms with index \(x_1\) on the rightmost end. This theory is not only compatible with Stuffle products but also with the Analytic Model. In this respect, it provides a stable and fully algorithmic model for Harmonic calculus. Examples by computer are also provided.

On some decidability questions concerning supports of rational series

A well-known construction of a weighted automaton over the integers, assigning zero to precisely the nonempty words from the equality set of a given Post's correspondence problem instance, is extended to the case of weights taken from an arbitrary integral domain that is not locally finite. Undecidability of problems for rational series such as universality or rationality of support thus generalises to such domains as well. In spite of being fairly simple, these findings imply that the support rationality problem can be undecidable over rings of positive characteristic, answering an open question of D. Kirsten and K. Quaas. In addition, a more general undecidability result for rational series over other than locally finite integral domains is proved, which subsumes, e.g., the undecidability of support rationality or context-freeness.

Chudnovsky-Ramanujan type formulae for non-compact arithmetic triangle groups

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun.As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series.

ПРЕДСТАВЛЕНИЕ ПАРНОГО ПОТЕНЦИАЛА ВЗАИМОДЕЙСТВИЯ В ВИДЕ МНОГОМЕРНЫХ РАЦИОНАЛЬНЫХ РЯДОВ В ПЕРЕМЕННЫХ ЯКОБИ ДЛЯ ЗАДАЧ МНОГИХ ТЕЛ

Scalar power functions of the form x1 + + xN -v Î are in some cases found in physical problems and applications, especially in many-body problems with paired interactions. There are known decompositions for two vectors in three-dimensional space. In this paper, we consider analogous decompositions with any number of N arbitrary M-dimensional vectors in Euclidean space as a product of a multidimensional rational series with respect to spatial variables and hyperspheric functions on the unit sphere SM-1. Such an advantage of expansion arises in three-body problems when solving the Faddeev equation, where it is known that the main problem is the approximate choice of approximation of interaction potentials, in which the t-matrix scattering elements acquired a separable form.

Inhibition of ERK5 Elicits Cellular Senescence in Melanoma via the Cyclin-Dependent Kinase Inhibitor p21.

Melanoma is the deadliest skin cancer with a very poor prognosis in advanced stages. Although targeted and immune therapies have improved survival, not all patients benefit from these treatments. The mitogen-activated protein kinase ERK5 supports the growth of melanoma cells in vitro and in vivo. However, ERK5 inhibition results in cell-cycle arrest rather than appreciable apoptosis. To clarify the role of ERK5 in melanoma growth, we performed transcriptomic analyses following ERK5 knockdown in melanoma cells expressing BRAFV600E and found that cellular senescence was among the most affected processes. In melanoma cells expressing either wild-type or mutant (V600E) BRAF, both genetic and pharmacologic inhibition of ERK5 elicited cellular senescence, as observed by a marked increase in senescence-associated β-galactosidase activity and p21 expression. In addition, depletion of ERK5 from melanoma cells resulted in increased levels of CXCL1, CXCL8, and CCL20, proteins typically involved in the senescence-associated secretory phenotype. Knockdown of p21 suppressed the induction of cellular senescence by ERK5 blockade, pointing to p21 as a key mediator of this process. In vivo, ERK5 knockdown or inhibition with XMD8-92 in melanoma xenografts promoted cellular senescence. Based on these results, small-molecule compounds targeting ERK5 constitute a rational series of prosenescence drugs that may be exploited for melanoma treatment. SIGNIFICANCE: This study shows that targeting ERK5 induces p21-mediated cellular senescence in melanoma, identifying a prosenescence effect of ERK5 inhibitors that may be exploited for melanoma treatment.

Determinisability of unary weighted automata over the rational numbers

Weighted finite automata over the field of rational numbers and unary alphabets are considered. The notion of a characteristic polynomial is introduced for such automata as a means to provide a decidable necessary and sufficient condition, under which a unary weighted automaton admits a deterministic, i.e., sequential equivalent. The sequentiality problem for univariate rational series is thus proved to be decidable both over the rational numbers and over the integers, confirming a conjecture of S. Lombardy and J. Sakarovitch; its decidability over the nonnegative integers is observed as well. The decision algorithm proposed for these tasks is shown to run in polynomial time. A determinisation algorithm for determinisable unary weighted automata over the rational numbers is also described.

The Abel – Poisson means of conjugate Fourier – Chebyshev series and their approximation properties

Herein, the approximation properties of the Abel – Poisson means of rational conjugate Fourier series on the system of the Chebyshev–Markov algebraic fractions are studied, and the approximations of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] by this method are investigated. In the introduction, the results related to the study of the polynomial and rational approximations of conjugate functions are presented. The conjugate Fourier series on one system of the Chebyshev – Markov algebraic fractions is constructed. In the main part of the article, the integral representation of the approximations of conjugate functions on the segment [–1,1] by the method under study is established, the asymptotically exact upper bounds of deviations of conjugate Abel – Poisson means on classes of conjugate functions when the function satisfies the Lipschitz condition on the segment [–1,1] are found, and the approximations of the conjugate Abel – Poisson means of conjugate functions with density | x |s , s ∈(1, 2), on the segment [–1,1] are studied. Estimates of the approximations are obtained, and the asymptotic expression of the majorant of the approximations in the final part is found. The optimal value of the parameter at which the greatest rate of decreasing the majorant is provided is found. As a consequence of the obtained results, the problem of approximating the conjugate function with density | x |s , s ∈(1, 2), by the Abel – Poisson means of conjugate polynomial series on the system of Chebyshev polynomials of the first kind is studied in detail. Estimates of the approximations are established, as well as the asymptotic expression of the majorants of the approximations. This work is of both theoretical and applied nature. It can be used when reading special courses at mathematical faculties and for solving specific problems of computational mathematics.

Commutation of Shintani descent and Jordan decomposition

Let GF be a finite group of Lie type, where G is a reductive group defined over F¯q and F is a Frobenius root. Lusztig’s Jordan decomposition parametrises the irreducible characters in a rational series E(GF,(s)G∗F∗) where s∈G∗F∗ by the series E(CG∗(s)F∗,1). We conjecture that the Shintani twisting preserves the space of class functions generated by the union of the E(GF,(s′)G∗F∗) where (s′)G∗F∗ runs over the semi-simple classes of G∗F∗ geometrically conjugate to s; further, extending the Jordan decomposition by linearity to this space, we conjecture that there is a way to fix Jordan decomposition such that it maps the Shintani twisting to the Shintani twisting on disconnected groups defined by Deshpande, which acts on the linear span of ∐s′E(CG∗(s′)F∗,1). We show a non-trivial case of this conjecture, the case where G is of type An−1 with n prime.

Noncommutative rational P\xf3lya series

A (noncommutative) Pólya series over a field K is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of K^times . We show that rational Pólya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.

Harmonic sums from the Kummer theorem

The Kummer summation theorem for F12(−1)-series is extended with three integer parameters and then examined by means of power series expansions. Quite a variety of new alternating series involving harmonic-like numbers and squared central binomial coefficients are evaluated in closed form, by making use of coefficient-extraction methods that have recently been applied in the determination of explicit identities for infinite families of rational series for 1π involving harmonic numbers. The same approach is also applied to obtain new identities for integrals involving complete elliptic integrals with complex arguments.

Approximation of the function |sin x| s by the partial sums of the trigonmometric rational fourier series

In the present article, the approximation of the function |sin x| s by the partial sums of the rational trigonometric Fourier series is considered. An integral representation, uniform and point estimates for the above-mentioned approximation were obtained. Based on them, several special cases of the selection of poles were studied. In the case of the approximation by the partial sums of the polynomial trigonometric Fourier series, an asymptotic equality was found. A detailed study is made of a fixed number of geometrically different poles.

English

English

A Sensitivity-Based Approach for Reliability Analysis of Randomly Excited Structures With Interval Axial Stiffness

Abstract Reliability assessment of linear discretized structures with interval parameters subjected to stationary Gaussian multicorrelated random excitation is addressed. The interval reliability function for the extreme value stress process is evaluated under the Poisson assumption of independent up-crossing of a critical threshold. Within the interval framework, the range of stress-related quantities may be significantly overestimated as a consequence of the so-called dependency phenomenon, which arises due to the inability of the classical interval analysis to treat multiple occurrences of the same interval variables as dependent ones. To limit undesirable conservatism in the context of interval reliability analysis, a novel sensitivity-based procedure relying on a combination of the interval rational series expansion and the improved interval analysis via extra unitary interval is proposed. This procedure allows us to detect suitable combinations of the endpoints of the uncertain parameters which yield accurate estimates of the lower bound and upper bound of the interval reliability function for the extreme value stress process. Furthermore, sensitivity analysis enables to identify the most influential parameters on structural reliability. A numerical application is presented to demonstrate the accuracy and efficiency of the proposed method as well as its usefulness in view of decision-making in engineering practice.

A FACTORIZATION THEORY FOR SOME FREE FIELDS

Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of minimal linear representations . We establish a factorization theory by providing an alternative definition of left (and right) divisibility based on the rank of an element and show that it coincides with the " classical'' left (and right) divisibility for non-commutative polynomials. Additionally we present an approach to factorize elements, in particular rational formal power series, into their (generalized) atoms. The problem is reduced to solving a system of polynomial equations with commuting unknowns.

Lamplighter groups, bireversible automata, and rational series over finite rings

We realize lamplighter groups A\wr \mathbb Z , with A a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize A\wr \mathbb Z as a bireversible automaton group if and only if the 2-Sylow subgroup of A has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.

Highly stable superparamagnetic iron oxide nanoparticles as functional draw solutes for osmotically driven water transport

In this work, we develop and demonstrate highly stable organic-coated engineered superparamagnetic iron oxide nanoparticles (IONPs), which provide effective osmotic pressure without aggregation, reverse diffusion, or membrane blocking (by nanoparticles) for osmotically driven membrane systems, considering both forward osmosis (FO) and pressure-retarded osmosis (PRO). For this, we synthesized highly water stable, monodisperse 12 nm IONPs with a rational series of water stabilizing surface coatings, including sodium dodecyl sulfate (SDS), cetyltrimethylammonium bromide (CTAB), and polyethylene glycol (PEG). We then compared the library of surface functionalized IONPs as draw solutes for osmotic pressure-driven membrane processes. As synthesized, surface (organic) coatings are compact, thin, and can have very similar surface charge as the membrane itself, which results in effective osmotic pressure in forward osmosis (FO) mode configuration. To increase the osmotic pressure further, on a per mass basis, we synthesized and demonstrated novel hollow IONPs with identical surface coatings. Finally, water flux was further enhanced for stable particle systems using an oscillating magnetic field, thus physically altering concentration gradients, as a function of particle magnetic properties.