On integrability in elementary functions of certain classes of nonconservative dynamical systems

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium; see [210, 211], where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. The complete integrability of those systems is related to symmetries of a latent type. Therefore, it is of interest to study sufficiently wide classes of dynamical systems having analogous latent symmetries. As is known, the concept of integrability is sufficiently broad and undeterminate in general. In its construction, it is necessary to take into account in what sense it is understood (it is meant that a certain criterion according to which one makes a conclusion that the structure of trajectories of the dynamical system considered is especially “attractive and simple”), in which function classes the first integrals are sought for, etc. (see also [1, 4, 14, 17, 20–22, 35, 40–42, 47, 83–85, 117, 120]).     

On a class of secant-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

On the convergence of Newton-type methods under mild differentiability conditions

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics

Before we dive in this essay into the accessibility stream of nowadays indicatory applications of octonions and quaternions to computer and other sciences and to quantum physics (see for example [50-53], [41], [33]) and to Clifford algebras (see for example [17,16], 18) let us focus for a while on the crucially relevant events for today’s revival on interest to nonassociativities while the role of associative quaternions in eight periodicity constructive classification of associative Clifford algebras is now a text-book knowledge.     

On characters and formal degrees of discrete series of affine Hecke algebras of classical types

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ^∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).     

Order-Compactifications of Totally Ordered Spaces: Revisited

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Colloq Math 17:35–39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13:56–65, 1975) and Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.     

A strong form of almost differentiability

We present a uniformization of Reeken’s macroscopic differentiability (see [5]), discuss its relations to uniform differentiability (see [6]) and classical continuous differentiability, prove the corresponding chain rule, Taylor’s theorem, mean value theorem, and inverse mapping theorem. An attempt to compare it with the observability (see [1, 4]) is made too.     

On the fast solution of Toeplitz-block linear systems arising in multivariate approximation theory

When constructing multivariate Padé approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Padé approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Padé approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Padé approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12, 25], or approaches that require some symbolic computations [6, 18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14, 19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.Research partly funded by FWO-Vlaanderen.     

Vector partition functions and index of transversally elliptic operators

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory.