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Abstract This comment concerns the doubtful new auxiliary equation method (Khater method) included in the above paper. We prove by simple calculation that this method is wrong. The exact solutions of the proposed nonlinear equation obtained in the above paper are all wrong. Highlights • The new auxiliary equation method is called Khater method. • We prove by simple calculation that Khater method is wrong. • We show that the exact solutions obtained in the above paper are all wrong. [ABSTRACT FROM AUTHOR]

The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert. [ABSTRACT FROM AUTHOR]

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

The paper is dedicated to analysing the role of "denominate numbers" in textbooks for Polish schools, whose author or co-author was Stefan Banach. Banach used the concepts of "number" and "denominate number" in his textbooks. The ways of introducing these concepts, as well as the associated manner of understanding numeration, are analysed in the paper. The methods of manipulating "denominate numbers" through various procedures are also discussed. The analysis is carried out in the context of the Polish ministerial ordinances related to teaching mathematics and the content of other textbooks, especially those used by Banach as a secondary school student. [ABSTRACT FROM AUTHOR]

Leibniz's manuscripts on perspective geometry remained unpublished and unknown until very recently. Among them, Scientia perspectiva stands out as the most complex and the most original. In this paper, we offer a thorough analysis of this manuscript, showing how Leibniz moves from perspective concepts fairly common at that time to a completely new idea of the practice that could have affected its entire history. This new science represents not only Leibniz's unique contribution to the development of perspective but also casts a new light on his own notion of space and geometry and their philosophical grounding. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

Stefan Kempisty was a Polish mathematician, working on the theory of real functions, set theory, integrals, interval functions and the theory of surface area. In 1919 he defended his Ph.D. thesis, On semi-continuous functions , at the Jagiellonian University in Cracow under the supervision of Kazimierz Żorawski. In December 1924 he did his habilitation at the University of Warsaw and continued his work at the Stefan Batory University in Vilnius. Kempisty published over forty scientific papers, three textbooks and one monograph. Kempisty's name in mathematics appears in connection with the definition of quasi-continuous functions, different kinds of continuity of functions of several variables, the classification of Baire, Young and Sierpiński functions, interval functions, and Denjoy or Burkill integrals. This paper is prepared for a wide range of readers. It is an abridged version of the article written in Polish by the same authors (cf. Jóźwik et al., 2017), where can be found more detailed information. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in S n (123) and S n (132). We first study the distribution of consecutive pattern γ -matches in S n (123) and S n (132) for each length 3 consecutive pattern γ. Then we extend our methods to study the joint distributions of multiple consecutive patterns. Some more general cases are discussed in this paper as well. [ABSTRACT FROM AUTHOR]

Abstract A variation of direct velocity feedback, often referred to as skyhook damping, is discussed in this paper. Skyhook damping cannot be regarded as collocated control method since only the action force component is collocated with the velocity sensor mounted onto the receiving part of the structure. The reaction control force component reacting off the source part of the structure does not have a collocated sensor. Depending on the characteristics of the passive structure under control, the feedback loop may be quite insensitive to the effects produced by the non-collocated reaction control force component, and maintain stability properties that are otherwise characteristic only for collocated control. Moreover, there exist additional effects related to the response of structures activated by the application of skyhook damping. It is shown in this paper that the structure subjected to such active control, although exhibiting stable response and linear input-output relationships, no longer complies with the reciprocity principle. The absence of reciprocity is interesting given the recent efforts in developing metamaterial cloaks, where one of the critical issues is how to design material structures or systems that demonstrate non-reciprocal behaviour. [ABSTRACT FROM AUTHOR]

Text By Zeckendorf's theorem each positive integer is uniquely written as a sum of distinct non-adjacent terms of the Fibonacci sequence. This representability remains true for so called the Nth order Fibonacci sequence , and for a further generalization to linear recurrences with positive coefficients. In this paper we consider sequences { G n } that have the same linear recurrence relations as the N th order Fibonacci sequence but has different initial values, and investigate the number of positive integers up to X that are written as a sum of distinct terms of G n . We also introduce a converse of Zeckendorf's theorem that does not require the increasing condition. Our method extends to general linear recurrences, and a generalization is introduced in this paper. Video For a video summary of this paper, please visit https://youtu.be/vSwSJ_sppns . [ABSTRACT FROM AUTHOR]

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by Élie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925–1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions. [ABSTRACT FROM AUTHOR]

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

Text Two infinite sequences A and B of nonnegative integers are called additive complements , if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β , then B is not an additive complement of S . In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S . Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 . [ABSTRACT FROM AUTHOR]

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand [19] established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the global wellposedness and large time behavior results of strong solutions for the compressible Oldroyd-B model derived by Barrett, Lu, Süli (Commun. Math. Sci., 15, 1265–1323, 2017). Exploiting the harmonic analysis tools (especially Littlewood-Paley theory), we first study the global well-posedness of the model with small initial data in spaces with low regularity. Then, under some suitable conditions involving only the low frequency of the initial data, we also obtain the optimal decay rates of the solutions. Compared with the result by Wang and Wen (Math. Models Methods Appl. Sci., 30, 139–179, 2020), the polymer number density is allowed to vanish and the stress tensor is near zero equilibrium. [ABSTRACT FROM AUTHOR]

In the paper, we construct compact embedded convex λ -hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. As the special case of our result, we solve Sun's problem (Int. Math. Res. Not. (2021) 11818–11844). In this sense, one can not expect to have Alexandrov type theorem for λ -hypersurfaces. [ABSTRACT FROM AUTHOR]

This paper outlines the results of a study aimed at charting the spread and development of the mathematical sciences in Sardinia over the period from 1720, the year in which the island was awarded to the Savoy dynasty following the Treaty of The Hague, to 1848, the year in which King Charles Albert of Sardinia placed Sardinia on a level political and administrative footing with the Kingdom's mainland territory. In particular, the paper analyses the role played in the transmission and spread of mathematics by the Savoy educational institutions. [ABSTRACT FROM AUTHOR]

This paper tackles the history of tactics, a field of investigation at the crossroads of algebra, combinatorics and recreational mathematics. Tactics was only taken up by mathematicians now and then between the 1850s and the 1900s, and its emergence was a process of mathematization of questions linked to the notions of “order” and “position”. To understand the long-term history of this field of investigation—one that became neither a theory nor a discipline—the paper analyzes the different historical configurations in which tactics took on its scientific meaning. It thus investigates how, under the banner of tactics, a continuity could be claimed by mathematicians that were, finally, working in very different scientific and historical context. [ABSTRACT FROM AUTHOR]

The Italian mathematician Antonio Bordoni is mainly known for his adherence to the Lagrangian approach to the foundations of calculus and for his role in creating an important school of mathematics. In this paper, I consider his less known work on the application of probability to design exams and analyze their outcomes. Within this framework, he obtained in 1837, as Mondésir and Poisson, the result that would lead Catalan to formulate his "new principle" of probability (Jongmans and Seneta, 1994). Moreover, in 1843, Bordoni also gave an early complete proof of the finite rule of succession. [ABSTRACT FROM AUTHOR]

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field analogue of the classical result proved by A. Weil in his 1965 Acta Math. paper. We also give a convergence criterion for the theta integral by using Harder's reduction theory over function fields. [ABSTRACT FROM AUTHOR]

In recent years there has been a focus on two types of scheme for dynamical substructure testing: the hybrid scheme (HS) and the dynamically substructured system (DSS) scheme. Although these approaches are fundamentally different in the manner of compensation/control, the comparative performance of the schemes has not yet been studied in simulated or real experimental tests. This paper studies the performance of these schemes via tests on a simple base-isolated structure with a natural rubber bearing. In dynamical tests, a guarantee of stability for the experimental system is essential in order to achieve a safe and reliable result. Pure time delays can be present in such experimental systems, primarily due to discrete-time computational elements, which will promote instability. This paper studies the stability of HS and DSS test systems, with a prime focus on the effect of pure time delays. In particular the schemes were applied to the substructure testing of a rubber isolation bearing, where the overall system included a digital computation time delay of τ =6 ms resulting from a series of control, data acquisition and signal filter elements. Neither delay compensation nor adaptive/self-tuning methods were included in these tests, in order to establish a fair and objective comparison between the basic forms of the two schemes. It was found that DSS stability was preserved in the presence of the 6 ms delay, whereas HS was rendered unstable. This increase in robustness was due to a linear substructure controller (LSC) used in the DSS, which was based upon a knowledge of the dynamics of the transfer system and substructures. In addition, DSS exhibited significant robustness to further artificial increases in the pure time delay, eventually yielding to instability at a value of τ =17 ms. Analysis of the two schemes was found to be effective at predicting accurate stability conditions in advance of actual experimentation. [ABSTRACT FROM AUTHOR]

Text The aim of this paper is to present a method to calculate the rate of growth of divergent series. In the case of the harmonic series, we study an alternate of other classical methods presented by L. Comtet, S.M. Zeyman, W. DeTemple and S.-H. Wang. Moreover, the method presented in this paper is suitable for obtaining similar results on other divergent series, as we will show in the final part of our investigation. Video For a video summary of this paper, please visit http://youtu.be/SFR7zY4pvG4 . [ABSTRACT FROM AUTHOR]

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations as viscosity vanishes with a convergence rate ϵ 1 / 6 | ln ⁡ ϵ |. In this paper, the vanishing viscosity limit of 2D non-isentropic compressible Navier-Stokes equations is studied. In contrast to the work [17] , the convergence rate for the 2D full compressible Navier-Stokes equations is improved to ϵ 2 / 7 | ln ⁡ ϵ | 2 by choosing a different scaling argument and performing more detailed energy estimates. [ABSTRACT FROM AUTHOR]

Though the central role of Gösta Mittag-Leffler in the promotion of specialized, research-oriented mathematics at Stockholms Högskola is widely acknowledged, the specific social and technical means by which he sought to cultivate a fledgling research community there during the early- to mid-1880s have received little attention. In particular, a detailed study of the relationship of his own research activity to that of his first Swedish students is absent from the existing literature. Through the juxtaposition of their research activities and unpublished correspondence, this paper explores Mittag-Leffler's active and deliberate efforts to engage his students Ivar Bendixson and Edvard Phragmén in open problems within his own research agenda, support them through his institutional connections, and instill within them norms concerning research ideologies, practices of communication and criticism, and frameworks for shared knowledge. It also illuminates the extent to which his teachings took root in at least one student to emerge from his program, who would perpetuate the mathematical practices set in place by his teacher and set forth on the international stage to promote his newly-acquired system of values. [ABSTRACT FROM AUTHOR]

This paper is devoted to the conception of mathematical objects and methods according to d'Alembert. We first recall his vision of the place of mathematics in the knowledge of nature, then the internal hierarchy of the various fields of this science, based on their degree of abstraction from sensations (§1 and 2). Then we come to the ideas of definitions, primitive ideas , simple ideas , and their generation as well as their generalization (§3 and 4). Then, having looked at what he means by quantities, numbers, quantities, as well as his conception of the objects and rules of algebra as abstract ideas by generalization (§5), we approach the question of the reality of mathematical objects with the example of the irrational (§6). The following paragraphs of the text are devoted to the difficulties encountered in various fields and the way d'Alembert tries to solve them: algebra and negative quantities (§7); principles of geometry (§8); the notion of limit as the basis of infinitesimal calculus (§9). His reflections, even if unfinished, were not without posterity (§10). [ABSTRACT FROM AUTHOR]

In the paper 'A probabilistic analysis of a discrete-time evolution in recombination' [4] the evolution of the recombination transformation Ξ = ∑ δ ρ δ ⨂ J ∈ δ μ J was described by a Markov chain (Y n) on a set of partitions, which converges to the finest partition. Our main results were the description of the geometric decay rate to the limit and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit. All these results continue to be true, but the Markov chain (Y n) that was claimed to satisfy Ξ n = E (⨂ J ∈ Y n μ J) required to be modified. This is done in this Corrigendum. [ABSTRACT FROM AUTHOR]

• Variational Laplace (VL) is a scheme for Bayesian modelling. • VL is widely used in neuroimaging, in particular DCM. • This paper provides a tutorial explanation of the math and algorithms. • New standalone code is provided to enable re-implementation. • The supplementary materials provide worked derivations. This article details a scheme for approximate Bayesian inference, which has underpinned thousands of neuroimaging studies since its introduction 15 years ago. Variational Laplace (VL) provides a generic approach to fitting linear or non-linear models, which may be static or dynamic, returning a posterior probability density over the model parameters and an approximation of log model evidence, which enables Bayesian model comparison. VL applies variational Bayesian inference in conjunction with quadratic or Laplace approximations of the evidence lower bound (free energy). Importantly, update equations do not need to be derived for each model under consideration, providing a general method for fitting a broad class of models. This primer is intended for experimenters and modellers who may wish to fit models to data using variational Bayesian methods, without assuming previous experience of variational Bayes or machine learning. Accompanying code demonstrates how to fit different kinds of model using the reference implementation of the VL scheme in the open-source Statistical Parametric Mapping (SPM) software package. In addition, we provide a standalone software function that does not require SPM, in order to ease translation to other fields, together with detailed pseudocode. Finally, the supplementary materials provide worked derivations of the key equations. [ABSTRACT FROM AUTHOR]

Abstract This paper provides new equivalence theorems for regular chains and regular differential chains, which are generalizations of Ritt's characteristic sets. These theorems focus on regularity properties of elements of residue class rings defined by these chains, which are revealed by resultant computations. New corollaries to these theorems have quite simple formulations. [ABSTRACT FROM AUTHOR]

Abstract The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max ⁡ , +) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent. [ABSTRACT FROM AUTHOR]

• Dynamic gestures are associated with constructing valid mathematical proofs. • Repetitive speech is associated with constructing valid mathematical proofs. • Syntactic complexity is associated with constructing valid mathematical proofs. • Logical inference is associated with constructing valid mathematical proofs. • Dynamic gestures and logical inference uniquely predict valid proof construction. Grounded and embodied theories of cognition suggest that both language and the body play crucial roles in grounding higher-order thought. This paper investigates how particular forms of speech and gesture function together to support abstract thought in mathematical proof construction. We use computerized text analysis software to evaluate how speech patterns support valid proof construction for two different tasks, and we use gesture analysis to investigate how dynamic gestures —those gestures that depict and transform mathematical objects—further support proof practices above and beyond speech patterns. We also evaluate the degree to which speech and gesture convey distinct information about mathematical reasoning during proving. Dynamic gestures and speech indicating logical inference support valid proof construction, and both dynamic gestures and speech uniquely predict variance in valid proof construction. Thus, dynamic gestures and speech each make separate and important contributions to the formulation of mathematical arguments, and both modalities can convey elements of students' understanding to teachers and researchers. [ABSTRACT FROM AUTHOR]

Mathematical writers, above all, Euclid, tend to present their theorems as decontextualized, abstract propositions, which has become the standard modus of textual presentation in theoretical mathematics. Mathematical commentators, however, provide their readers with personal names and historical facts in order to elucidate problems, provide contexts of discovery, or construct doxographies, among other things. Modern readers have used such information for the construction of histories of science. When we look at these passages, however, we see that personal names and information about mathematicians of the past can serve quite a range of different objectives, such as the strategic self-positioning of the commentator vis-à-vis the past or present of mathematics, the education of the reader, mathematical or moral, the construction of the history of the field, etc. Not only does the commentator present a persona of himself to the reader, he can also turn colleagues and predecessors into personae. This paper attempts to elucidate the practice, by offering four examples of such plays of and with personae , in Pappus, Eutocius, al-Nayrīzī, and Proclus. [ABSTRACT FROM AUTHOR]

Abstract The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship | a d − b c | = 1 for a , b , c , d integers as it arises in the Farey tessellation of the hyperbolic plane H 2 and the array of Ford circles in the upper-half of the real plane R 2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H 2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition | α γ − β δ | = 1 for α , β , γ , δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H 3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers. [ABSTRACT FROM AUTHOR]

Abstract Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when | P | = 2. However, despite various attempts, characterising this number in terms of agreement forests for | P | > 2 remains elusive. In this paper, we characterise the minimum hybridisation number for when P is of arbitrary size and consists of not necessarily binary trees. Building on our previous work on cherry-picking sequences, we first establish a new characterisation to compute the minimum hybridisation number in the space of tree-child networks. Subsequently, we show how this characterisation extends to the space of all rooted phylogenetic networks. Moreover, we establish a particular hardness result that gives new insight into some of the limitations of agreement forests. [ABSTRACT FROM AUTHOR]

Abstract Let S = GR (p e , m) be a Galois ring of characteristic p e and cardinality p e m. An additive code over S of length n is a subgroup of S n under addition. In this paper, we study additive codes over S. We introduce a correspondence between linear codes over Z p e and additive codes over S and we describe additive codes over S by the structure of linear codes over Z p e . In particular, we find the generator matrix and the number of additive codes over S , and we determine some classes of MDR additive codes over S. Among other results, permutation equivalent additive codes and decomposable additive codes are described. Also we prove MacWilliams identity and Delsarte theorem for additive codes over S. [ABSTRACT FROM AUTHOR]

Abstract In this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial δ -uniform permutations is obtained, for any δ , as well as an upper bound for differentially two-valued functions. Concerning quadratic functions, significant relations between nonlinearity and differential uniformity are exhibited. In particular, we show that the quadratic differentially 4-uniform permutations should be differentially two-valued and possess the best known nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract Let R be a noncommutative ring with unity. The commuting graph of R , denoted by Γ (R) , is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if x y = y x. Let F be a finite field and n ≥ 2. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if Γ (R) ≅ Γ (M n (F)) , then R ≅ M n (F). In this paper, we prove the conjecture whenever n is of the form 2 k 3 l with k ≠ 0. [ABSTRACT FROM AUTHOR]

Abstract Let q be a power of a prime and F q be a finite field with q elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form c x − x s + x q s , and investigate the relationship between two types of widely discussed permutation polynomials. Based on this relation, many classes of permutation polynomials having the form (x q k − x + δ) s + c x without restriction on δ are derived. [ABSTRACT FROM AUTHOR]

Abstract In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these polynomials. [ABSTRACT FROM AUTHOR]

Abstract Howie and Duncan observed that a word in a free product with length at least two, which is not a proper power and involves no letter of order two can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about a one-relator product of groups with such word as the relator. In this paper, we show that similar results hold in a more general setting where we allow a certain number of elements of order two. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n (a , b). We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers. [ABSTRACT FROM AUTHOR]

Abstract During the 20th century, impossibility theorems have become an important part of mathematics. Arrow's impossibility theorem (1950) stands out as one of the first impossibility theorems outside of pure mathematics. It states that it is impossible to design a welfare function (or a voting method) that satisfies some rather innocent looking requirements. Arrow's theorem became the starting point of social choice theory that has had a great impact on welfare economics. This paper will analyze the history of Arrow's impossibility theorem in its mathematical and economic contexts. It will be argued that Arrow made a radical change of the mathematical model of welfare economics by connecting it to the theory of voting and that this change was preconditioned by his deep knowledge of the modern axiomatic approach to mathematics and logic. [ABSTRACT FROM AUTHOR]

Abstract Weak fault detection, as a key step in the condition-based maintenance, is a significant but difficult issue because the fault signals are usually submerged in strong background noise. Contrary to traditional denoising and filtering methods, vibrational resonance (VR), as well as stochastic resonance (SR), is an effective way to detect weak signals by utilizing high-frequency interferences or random noise on purpose. In this paper, we investigate the application of VR to weak bearing fault detection. In order to enhance the detection performance, we construct an array of bistable systems based on VR by injecting different high-frequency sinusoidal interferences. Considering the frequency of fault signal which is usually greater than 1 Hz in practice, the frequency-shifted and rescaling transform method is adopted. Levenberg-Marquardt algorithm is utilized to optimize the system parameters, which is different from the most existing evolutionary algorithms. The proposed VR-based method is validated by simulation data, bearing data with single implanted fault and bearing data with multiple naturally-developed faults. The experimental results show that, compared with bistable SR system, this method by using an array of bistable systems based on VR is more practical to enhance the detection performance of bearing weak faults. Highlights • Propose a noise-utilization method for weak bearing fault detection. • Construct an array of bistable systems based on vibrational resonance. • Different high-frequency sinusoidal interferences are injected into the array. • Multiple faults can be detected simultaneously. • Method effectiveness and efficiency are validated by comparative analysis. [ABSTRACT FROM AUTHOR]

Abstract With the increasing use of composite material and external energy dissipation device in civil structures, the classical Rayleigh damping becomes an inappropriate assumption in structural damage identification. In this paper, a parametric sensitivity-based approach is developed to identify the structural damage and damping defect of non-classically damped shear-type building simultaneously. As the preliminary step, the complex eigenvalues and eigenvectors of damaged structure are obtained by a novel identification technique, which decouples the first-order modal analysis equation under physical coordinate so that the complex modal parameters of each mode can be identified independently. Thereafter, by analysing the sensitivities of complex eigenparameter, non-linear least square method is applied to identify the locations and magnitudes of damage with respect to stiffness reduction and damping defect. The proposed identification approach is illustrated by identifying the damage in a numerical five-storey shear building. The identification results show that the proposed approach can locate and quantify the stiffness reductions satisfactorily even when measurement noise is taken into account. However, in order to quantify the damping defects consistently, their locations need to be known as a priori. Highlights • A novel complex eigenparameter identification approach under state-space coordinate. • Damage identification on non-proportionally damped shear-type building. • Identify the stiffness reduction and damping defect simultaneously. • Identify the design parameters which locate and quantify the damages. • Numerical validation on a five-storey shear building by considering measurement noise. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we define the linear complexity for multidimensional sequences over finite fields, generalizing the one-dimensional case. We give some lower and upper bounds, valid with large probability, for the linear complexity and k -error linear complexity of multidimensional periodic sequences. [ABSTRACT FROM AUTHOR]

In this paper we deal with a problem of Pethő related to existence of a quartic algebraic integer α for which β = 4 α 4 α 4 − 1 − α α − 1 is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many α 's such that the corresponding β is quadratic. Moreover, we present a description of quartic numbers α such that the corresponding β is a quadratic real number. [ABSTRACT FROM AUTHOR]

Energy efficiency in irrigation distribution networks can be estimated by following the method proposed by Abadia et al (2008) [Biosystems Engineering, 101, 21–27], in which energy efficiency is quantified in terms of global energy efficiency (GEE), a parameter that is a compound of the pumping energy efficiency (PEE) and the supply energy efficiency (SEE). PEE is related to pump functioning, while SEE is related to the design and management of the distribution system. Calculation of SEE by this method is limited to irrigation networks that have a single water intake at an elevation below that of the irrigated area and a single supplied sector. Results from the method are confusing for networks with more than one water supply point or irrigation sectors supplied by intakes located at elevations higher than that of the irrigated area. This paper proposes a modification of the earlier method that is appropriate to any type of layout and configuration of the irrigation network, with one or several water intake points and supplied sectors. The new model is applied to four different types of collective pressurised irrigation network, and the results are compared with those of the previous model. The influence of different measures taken to improve the SEE and determine the variables that affect this efficiency was analysed. The results show that adequate SEE values can be obtained with the proposed model for all the situations analysed, and that it is possible to quantify the improvements in SEE with the model. Highlights • Knowing the energy efficiency of a network can improve design and energy management. • Previous Supply Energy Efficiency (SEE) definition unfit for some network designs. • New method proposed and applied to four distribution networks. • New method is coherent and appropriate for all networks designs. • New SEE definition allows for identification of badly designed or managed networks. [ABSTRACT FROM AUTHOR]