Asymptotic periodicity of trajectories of an interval

We consider dynamical systems generated by continuous mappings of an interval I into itself. We prove that the trajectory of an interval J ⊂ I is asymptotically periodic if and only if J contains an asymptotically periodic point.     

Totally Ordered Monoids Based on Triangular Norms

A totally ordered monoid, or tomonoid for short, is a monoid together with a translation-invariant (i.e., compatible) total order. We consider in this paper tomonoids fulfilling the following conditions: the multiplication is commutative; the monoidal identity is the top element; all nonempty suprema exist; and the multiplication distributes over arbitrary suprema. The real unit interval endowed with its natural order and a left-continuous t-norm is our motivating example. A t-norm is a binary operation used in fuzzy logic for the interpretation of the conjunction.

Given a tomonoid of the indicated type, we consider the chain of its quotients induced by filters. The intention is to understand the tomonoid as the result of a linear construction process, leading from the coarsest quotient, which is the one-element tomonoid, up to the finest quotient, which is the tomonoid itself. Consecutive elements of this chain correspond to extensions by Archimedean tomonoids. If in this case the congruence classes are order-isomorphic to real intervals, a systematic specification turns out to be possible.

In order to deal with tomonoids and their quotients in an effective and transparent way, we follow an approach with a geometrical flavor: we work with transformation monoids, using the Cayley representation theorem. Our main results are formulated in this framework. Finally, a number of examples from the area of t-norms are included for illustration.     

Basic Interval Orders

An interval order of length n has elements in correspondence with a collection of intervals in the linearly ordered set {1, 2, ..., n}. A basic interval order of length n has the property that removal of any element yields an order with length less than n. We construct and enumerate the set of basic length n interval orders.     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

A general theory of self-similarity

A little-known and highly economical characterization of the real interval [0,1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Δn may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations.This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.     

On Dual Gabor Frame Pairs Generated by Polynomials

We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.     

A Refinement of Weak Order Intervals into Distributive Lattices

In this paper we consider arbitrary intervals in the left weak order on the symmetric group S _n . We show that the Lehmer codes of permutations in an interval form a distributive lattice under the product order. Furthermore, the rank-generating function of this distributive lattice matches that of the weak order interval. We construct a poset such that its lattice of order ideals is isomorphic to the lattice of Lehmer codes of permutations in the given interval. We show that there are at least ${\left(\lfloor {\frac{n}{2}} \rfloor \right)!}$ permutations in S _n that form a rank-symmetric interval in the weak order.     

MixingC rmaps of the interval without maximal measure

We construct aC ^rtransformation of the interval (or the torus) which is topologically mixing but has no invariant measure of maximal entropy. Whereas the assumption ofC ^∞ ensures existence of maximal measures for an interval map, it shows we cannot weaken the smoothness assumption. We also compute the local entropy of the example.     

Analytic Multifunctions, Holomorphic Motions and Hausdorff Dimension in IFSs

We study iterated function systems of contractions which depend holomorphically on a complex parameter λ. We first restrict our attention to systems which consist of similarities that satisfy the OSC. In this setting, we prove that the Hausdorff dimension of the limit set J(λ) is a continuous, subharmonic function of λ. In the remainder of the paper, systems consisting of conformal contractions are considered. We give conditions under which J(λ) and A(λ) = describe a holomorphic motion, and construct an example that shows that this is not the case in general. We finally show that A(λ) is best described as an analytic multifunction of λ, a notion that generalizes that of holomorphic motion. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT). This research was supported by the FQRNT.     

Weierstraßscher Approximationssatz und Gleichverteilung

Using uniformly distributed sequences modulo 1 polynomials are formed which approximate continuous functions ofJ, whereJ is at first a compact interval inR ^s . The error of the approximation is estimated using the discrepancy of the sequences. Some cases of unbounded intervals are also studied. Furthermore trigonometric polynomials are considered which approximate periodic continuous functions inR ^s .

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Meinem Freund Prof. L. Schmetterer zum 60. Geburtstag gewidmet     

The absolute order on the hyperoctahedral group

The absolute order on the hyperoctahedral group B _n is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen–Macaulay. This method results in a new proof of Cohen–Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on B _n is shellable and an example of a non-Cohen–Macaulay interval in the absolute order on D ₄ is given. Finally, the closed intervals in the absolute order on B _n and D _n which are lattices are characterized and some of their important enumerative invariants are computed.     

A Jacobi eigenreduction algorithm for definite matrix pairs

Summary We propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ ²=I,J diagonal (neitherJ norA itself need be definite). Our Jacobi reduction works only on one matrix and usesJ-orthogonal elementary congruences which include both trigonometric and hyperbolic rotations and preserve the symmetry throughout the process. For the rotation parameters only the pivotal elements of the current matrix are needed which facilitates parallelization. We prove the global convergence of the method; the quadratic convergence was observed in all experiments. We apply our method in two situations: (i) eigenreducing a single real symmetric matrix and (ii) eigenreducing an overdamped quadratic matrix pencil. In both cases our method is preceded by a symmetric indefinite decomposition and performed in its one-sided variant on the thus obtained factors. Our method outdoes the standard methods like standard Jacobi orqr/ql in accuracy in spite of the use of hyperbolic transformations which are not orthogonal (a theoretical justification of this behaviour is made elsewhere). The accuracy advantage of our method can be particularly drastic if the eigenvalues are of different order. In addition, in working with quadratic pencils our method is shown to either converge or to detect non-overdampedness.     

Comonotone Adaptive Interpolating Splines

For any given data we propose the construction of an interpolating spline of class C ¹, which is either a quadratic polynomial or a linear/linear rational function between the knots, and preserves the monotonicity of the data on the sections of rational intervals. We prove the uniqueness and existence of this spline. Numerical tests show good approximation properties and flexibility due to the non-coincidence of the given data arguments and the spline knots which can be chosen freely.     

On -Hadamard matrices

In this paper, we use the Sylvester's approach to construct another Hadamard matrix, namely a J_m-Hadamard matrix, from a given one. Consequently, we can generate other 2^m-1 Hadamard matrices from the constructed J_m-Hadamard matrix. Finally, we also discuss the Kronecker product of an Hadamard matrix and a J_m-Hadamard matrix.     

NOTE The Johnson Graphs Satisfy a Distance Extension Property

G has property if whenever F and H are connected graphs with and |H|=|F|+1, and and are isometric embeddings, then there is an isometric embedding such that . It is easy to construct an infinite graph with for all k, and holds in almost all finite graphs. Prior to this work, it was not known whether there exist any finite graphs with . We show that the Johnson graphs J(n,3) satisfy whenever , and that J(6,3) is the smallest graph satisfying . We also construct finite graphs satisfying and local versions of the extension axioms studied in connection with the Rado universal graph. Received June 9, 1998     

Continuous spin models on annealed generalized random graphs

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution.First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case.We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in Dommers et al. (2016). On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.     

Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Normal form construction for nearly-integrable systems with dissipation

We consider a dissipative vector field which is represented by a nearly-integrable Hamiltonian flow to which a dissipative contribution is added. The vector field depends upon two parameters, namely the perturbing and dissipative parameters, and by a drift term. We study an ?-dimensional, time-dependent vector field, which is motivated by mathematical models in Celestial Mechanics. Assuming to start with non-resonant initial conditions, we provide the construction of the normal form up to an arbitrary order. To construct the normal form, a suitable choice of the drift parameter must be performed. The normal form allows also to provide an explicit expression of the frequency associated to the normalized coordinates. We also give an example in which we construct explicitly the normal form, we make a comparison with a numerical integration, and we determine the parameter values and the time interval of validity of the normal form.     

Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ? with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ? in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ? in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ?. Finally, we define partial quadratic functionals associated with ? and a (strong) regularity of ?, which we relate to the positivity and nonnegativity of ?.     

On Jc-contraction and related fixed-point problem with applications

We introduce J_c-contraction, which encompasses both F-contraction and JS-contraction. We investigate a fixed-point problem subject to J_c-contraction. The existence of the solutions to the linear matrix equation and system of fractional differential equations are discussed by the application of obtained results.