Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.     

Appendix: Classes of Local Systems of Hermitian Vector Spaces

For a local system V on a topological manifold S associated to a representation ρ: π1(S,s)→GL(n,C) of the fundamental group, we denote by ci(V) =ci(ρ) =βi γi∈H2i-1(S,C/Z) the class defined in [4,1]: βi∈H2i-1(S,R) (see [1, (2.20)]), γi∈H2i-1(S,R/Z) (see [4, §4]).     

Method of “transition into space of derivatives”: 40 years of evolution

In 1975, the “method of transition into space of derivatives” was proposed. It is an efficiently verifiable frequency criterion for the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to which it could be applied. In this work, we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transition into space of derivatives, the well-known generalized Poincaré–Bendixson principle proposed by R. A. Smith, and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. In the recent years, the author obtained frequency criteria for the existence of orbitally stable cycles in multiinput multioutput (MIMO) control systems and methods for the construction of multidimensional systems having a unique equilibrium and an arbitrarily prescribed number of orbitally stable cycles, which are described in the paper. The extension of the generalized Poincaré–Bendixson principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to solving S. Smale’s problem in the chemical kinetics theory of biological cells and also to finding hidden attractors of the generalized Chua system and the minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.     

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_{ {\mathcal {X}}}One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let T _XT_{ {\mathcal {X}}} be the 3×3 matrix having only two nonzero entries (T _X)₁₂=(T _X)₂₁=1(T_{ {\mathcal {X}}})_{12}=(T_{ {\mathcal {X}}})_{21}=1 and let T_\varLambda {\mathcal {T}}_{\varLambda } be the set of real, symmetric tridiagonal matrices with the same spectrum as T _XT_{ {\mathcal {X}}} . There exists a neighborhood U ì T_\varLambda \boldsymbol {{\mathcal {U}}}\subset {\mathcal {T}}_{\varLambda } of T _XT_{ {\mathcal {X}}} which is invariant under Wilkinson’s shift strategy with the following properties. For T₀ ? UT_{0}\in \boldsymbol {{\mathcal {U}}} , the sequence of iterates (T _k ) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry (T _k )₂₃. In fact, quadratic convergence occurs exactly when limT_k=T _X\lim T_{k}=T_{ {\mathcal {X}}} . Let X\boldsymbol {{\mathcal {X}}} be the union of such quadratically convergent sequences (T _k ): The set X\boldsymbol {{\mathcal {X}}} has Hausdorff dimension 1 and is a union of disjoint arcs X^s\boldsymbol {{\mathcal {X}}}^{\sigma} meeting at T _XT_{ {\mathcal {X}}} , where σ ranges over a Cantor set.     

Time-consistent mean–variance hedging of longevity risk: Effect of cointegration

This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.     

Calculus of Generalized Quasi-differentiable Functions I: Some Results on the Space of Pairs of Convex-Set Collections

A Riesz space K1 whose elements are pairs of convex-set collections is presented for the study on the calculus of generalized quasi-differentiable functions. The space K1 is constructed by introducing a well-defined equivalence relation among pairs of collections of convex sets. Some important properties on the norm and operations in K1 are given.     

Reactions of tetrahalogeno-o-benzoquinones part VI: Reactions of tetrachloro-o-benzoquinone with methyl naphthols—Formation of naphthoquinone methides

The oxidation of a few methyl naphthols with tetrachloro-o-benzoquinone (1) has been described. Reaction of 1-methyl-2-naphthol (4) with the quinone (1) was found to give the spirodimer, 1,1′,2,2′-tetrahydro-3H-benzo (f) chromen-3-spiro-1′-naphthalene-2′-one (6) and a yellow solid which has been assigned the structure of 5,6,7,8-tetrachloro-1,4-benzodioxan-2-spiro-1′-naphthalen-2′-one (7) on the basis of chemical and spectral data. Among the other naphthols studied, 4-methyl-2-t-butyl-1-naphthol afforded 5,6,7,8-tetrachloro-1,4-benzodioxan-2-spiro-1′-naphthalen-3′-t-butyl-4′-one (22) while 2-methyl-1-naphthol and 4-methyl-1-naphthol gave 2-methyl-4,4-(tetrachloro-o-phenylenedioxy) naphthalen-1(4H)-one(15) and 4-methyl-2,2-(tetrachloro-o-phenylenedioxy) naphthelen-1(2H)-one (19) respectively. The present study has shown that oxidation of suitably substituted naphthols with the quinone (1) gives rise to naphthoquinone methides.     

Erratum to: Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.     

Analysis of Iterated Modular Exponentiation: The Orbits of xα mod N

Let N and α be integers larger than 1. Define an orbit to be the collection of residues in generated byiteratively applying mod N to an element which eventually maps back to itself.An orbit's length is the number of distinct residues in the orbit. When N isa large bicomposite integer, such as is commonly used in many cryptographicapplications, and when certain prime factorizations related to N are known,all orbit lengths and the number of orbits of each possible length can beefficiently computed using the results presented. If the required integerfactorizations are only partially known, the risk that a randomly selectedperiodic element might produce an orbit shorter than some (typically large)divisor of can be bounded. The information needed to producesuch a bound is fully available when the prime factors of N are generatedusing the prime generation algorithm defined in Maurer maur. Resultspresented can assist in choosing wisely a modulus N for the Blum, Blum, andShub pseudo-random bit generator. If N is a bicomposite RSA modulus, theanalysis shows how to quantify the risk posed by an iterated encryptionattack.     

On structure of cluster algebras of geometric type Ⅰ:In view of sub-seeds and seed homomorphisms

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for(rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also,we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.     

Comparison of various concepts of a random attractor:¶ A case study

Several non-equivalent definitions of an attractor of a random dynamical system have been proposed in the literature. We consider a rather simple special case: random dynamical systems with state space [0, ￥) [0, \infty) which fix 0. We examine conditions under which the set {0} is an attractor for three different notions of an attractor. It turns out that even in this simple case the various concepts are quite different. The purpose of this note is to highlight these differences and thus provide a basis for discussion about the "correct" concept of a random attractor.     

Approximations of e and π: an exploration

ABSTRACT

Fractional approximations of e and π are discovered by searching for repetitions or partial repetitions of digit strings in their expansions in different number bases. The discovery of such fractional approximations is suggested for students and teachers as an entry point into mathematics research.     

Initiation of Cracks in Griffith’s Theory: An Argument of Continuity in Favor of Global Minimization

The initiation of a crack in a sound body is a real issue in the setting of Griffith’s theory of brittle fracture. If one uses the concept of critical energy release rate (Griffith’s criterion), it is in general impossible to initiate a crack. On the other hand, if we replace it by a least energy principle (Francfort–Marigo’s criterion), it becomes possible to predict the onset of cracking in any circumstance. However this latter criterion can appear too strong. We propose here to reinforce its interest by an argument of continuity. Specifically, we consider the issue of the initiation of a crack at a notch whose angle ω is considered as a parameter. The result predicted by the Griffith criterion is not continuous with respect to ω, since no initiation occurs when ω>0 while a crack initiates when ω=0. In contrast, the Francfort–Marigo’s criterion delivers a response which is continuous with respect to ω, even though the onset of cracking is necessarily brutal when ω>0. The theoretical analysis is illustrated by numerical computations.     

Study of the Kepler��s conjecture: the problem of the closest packing

In the first and second parts the problem of the Kepler??s conjecture is reduced from a problem with an infinite number of parameters to a problem with a finite number of parameters. In the third part is it is shown that the latter problem can be solved by a numerical verification using only a finite number of computations. However that finite number remains large, even if modern computers can do it. The method of analysis is classical: to each sphere of an arbitrary packing is associated a corresponding large enough ??associated domain??, disjoint from all other similar domains, in the hope of obtaining an interesting upper limit of the space occupation coefficient. Many methods of association have been tried in the past and the new method presented in this study is complex, but very well adapted to the problem of interest.     

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.     

An Open Problem: Accumulation of Nonreal Eigenvalues of Indefinite Sturm–Liouville Operators

In this note we conjecture that the eigenvalues of singular indefinite Sturm–Liouville operators accumulate to the real axis whenever the eigenvalues of the corresponding definite Sturm–Liouville operator accumulate to the bottom of the essential spectrum from below.     

Interaction of intelligent and autonomous systems – part I: qualitative structuring of interaction

In this part I of the study, the development and the background of a meta-modelling technique to structure complex environments of systems is illustrated. A special situation-operator-model (SOM) developed to model the human-machine-interaction (HMI) is understood as a structuring approach modelling the interaction of intelligent systems in general. Part I of this study is structured as follows: Section 1 reflects on the term modelling with respect to the task: how to model interaction of unknown interaction partners, Section 2 briefly reviews the basics of the previously published foundations of the work and the SOM. The developed situation-operator-metamodelling approach is used to illustrate the connections between control loops and algorithms in Section 3, including a systematic view of systems' interaction mechanisms. In part II (part II available in this issue, pp. 319–339) of the contribution, the practical applications of the introduced approach are shown.     

Schoenberg’s polynomial B-splines of odd degrees: A brief review of application

A brief overview of applications of Schoenberg’s polynomial B-splines of odd degrees in mathematical statistics, computational mathematics, and statistical radio engineering is provided. Exact formulas for the found Schoenberg B-spline of 15th degree are presented. High-quality approximations of smooth functions with an infinite Fourier transform by functions with a finite Fourier transform are found.