Boundary values of a convergent sequence of J-contractive matrix-functions

In this note it is proved that if W_n(z) are J-contractive matrix-functions which are meromorphic in the disk ¦z¦<1 (j–w=">_n ^* (z)JW_n(z) W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)     

Generalized factorization for a class ofn×n matrix functions — Partial indices and explicit formulas

The generalized factorization of a class of continuous non-rationaln×n matrix-functions is studied. The partial indices are determined and, in the case of existence of a canonical factorization, explicit formulas for the factors are obtained.     

Diamond-free circle graphs are Helly circle

The diamond is the graph obtained from K₄ by deleting an edge. Circle graphs are the intersection graphs of chords in a circle. Such a circle model has the Helly property if every three pairwise intersecting chords intersect in a single point, and a graph is Helly circle if it has a circle model with the Helly property. We show that the Helly circle graphs are the diamond-free circle graphs, as conjectured by Durán. This characterization gives an efficient recognition algorithm for Helly circle graphs.     

On Regularization of γ-Generating Pairs

This paper gives an affirmative answer to one of D. Sarason′s questions concerning the exposed points in H¹. Namely, one can infer nothing special about a from the knowledge that (a, b) is a Nehari Pair. The proof essentially uses D. Z. Arov′s concept of A-regular and A-singular j-inner matrix-functions and is based on an analysis of V. Katsnelson′s results on their regularization.     

Triplets of mutually orthogonal circles associated with any ellipse

The author has shown (J. Geom. 94, 159–173, 2009) that, for any general point P on a given ellipse H, four concyclic notable points exist which determine a circle (denoted by Ω) orthogonal to Monge’s circle. Now, it is shown that a new set of notable concyclic points exists; such points determine a circle (denoted by Δ) orthogonal to both Monge’s circle and the circle Ω. Moreover, it is possible to introduce a new ellipse (denoted by H _Δ) concentric with the circle Δ, which is tangent to the ellipse H at P, shares the same circle Ω with the ellipse H and admits the circle Δ as its own Monge’s circle. Only elementary facts from trigonometry and analytic geometry are used.     

Geometric fit of a point set by generalized circles

In our paper we approximate a set of given points by a general circle. More precisely, given two norms k ₁ and k ₂ and a set of points in the plane, we consider the problem of locating and scaling the unit circle of norm k ₁ such that the sum of weighted distances between the circumference of the circle and the given points is minimized, where the distance is measured by a norm k ₂. We present results for the general case. In the case that k ₁ and k ₂ are both polyhedral norms, we are able to solve the problem by investigating a finite candidate set.     

Two new sets of ellipse-related concyclic points

Two new circles (denoted by Γ _I and Γ _E ) are shown to be associated with any ellipse. Their analogies with two circles described by Barlotti are described. Two further new circles—denoted by Ω and Γ—are shown to be associated with any general point P of the ellipse. Tight relationships link the circles Ω and Γ with the circle K (previously introduced by the present author), as well as with Monge’s orthoptic circle, with Barlotti’s circles and with the circles Γ _I and Γ _E . In particular, the circle Ω is orthogonal to Monge’s circle. A new special point of the ellipse (the point T) is described. New properties of Fagnano’s point are described.     

Factorization of meromorphic matrix functions

The paper considers a class of matrix-functions defined on some contour in the complex plane that have meromorphic continuations in the interior or exterior domain of that contour. These matrix-functions generally do not admit Wiener-Hopf standard factorization. The paper studies the problem of index factorization, which is a version of Wiener-Hopf factorization. Some criterions for index factorization and exact formulas for particular indices are found along with a constructive method which applies the factorization to solution of an explicit, finite system of linear algebraic equations.     

The Lie Algebra of Homeomorphisms of the Circle

We define and study an infinite-dimensional Lie algebrahomeo₊which is shown to be naturally associated to thetopologicalLie grouphomeo₊of all orientation-preserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmüller theory developed before as motivation to provide Fréchet coordinates on the homogeneous space given byhomeo₊modulo the group of real fractional linear transformations, whose corresponding vector fields on the circle we then extend by the usual Lie algebrasl₂of real traceless two-by-two matrices in order to definehomeo₊. Surprisingly,homeo₊turns out to be equal to the algebra of all vector fields on the circle which are “piecewisesl₂” in the obvious sense. It is evidently important to consider the relationship between our new Fréchet coordinates and the usual trigonometric functions on the circle, and we undertake here both natural infinitesimal calculations. We finally apply some further previous work in order to give sufficient conditions on the Fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory.     

Optimization of nested polygons

The optimization problem under consideration requires to find the largest regular polygon withk sides to be fitted into a regular polygon withk – 1 sides. If the sequence of these maximal polygons is started with an equilateral triangle, then the final nested polygon, a circle, possesses a radiusr=0.3414r ₃, wherer ₃ is the radius of the inscribed circle of the equilateral triangle. Lower bounds for the ratior/r ₃ are also obtained.     

Partial characterizations of circle graphs

A circle graph is the intersection graph of a family of chords on a circle. There is no known characterization of circle graphs by forbidden induced subgraphs that do not involve the notions of local equivalence or pivoting operations. We characterize circle graphs by a list of minimal forbidden induced subgraphs when the graph belongs to one of the following classes: linear domino graphs, P₄-tidy graphs, and tree-cographs. We also completely characterize by minimal forbidden induced subgraphs the class of unit Helly circle graphs, which are those circle graphs having a model whose chords have all the same length, are pairwise different, and satisfy the Helly property.     

A dynamic adaptive local search algorithm for the circular packing problem

This paper studies the circular packing problem (CPP) which consists of packing n non-identical circles C_i of known radius r_i, i ∈ N = {1, … , n}, into the smallest containing circle C. The objective is to determine the coordinates (x_i, y_i) of the center of C_i, i ∈ N, as well as the radius r and center (x, y) of C. This problem, which is a variant of the two-dimensional open dimension problem, is solved using a two-step, dynamic, adaptive, local search algorithm. At each iteration, the algorithm identifies the set of potential “best local positions” of a circle C_i, i ∈ N, given the positions of the previously packed circles, and determines for each of these positions the coordinates and radius of the smallest containing circle. The “best local position” minimizes the radius of the current containing circle. That is, every time an additional circle is packed, both the center and the radius of the containing circle are dynamically updated, and the smallest containing circle is known. The experimental results reflect the good performance of the algorithm.     

Über das Stekloffsche Eigenwertproblem: Isoperimetrische Ungleichungen für symmetrische Gebiete

Summary Isoperimetric inequalities ofPólya [6] for symmetric membranes are extended to the Stekloff problem. The given symmetric domainG _z is mapped conformally onto a circle; some (harmonic) eigenfunctions of the circle are transplanted ontoG _z ; application of Rayleigh's and Poincaré's principles to the transplanted functions gives upper bounds for a number of eigenvalues ofG _z which depends on the order of symmetry of the domain.     

The convergence behavior of q-continued fractions on the unit circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of q-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, Y _G , on the unit circle such that if y ? Y _G then G(y) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

A problem in extremal quasiconformal extensions

A constantK ₀ ^(m) (h) is introduced for every quasisymmetric mappingh of the unit circle and every integerm≥4 which contains the constantK ₀(h) (indicated by the change in module of the quadrilaterals with vertices on the circle) as a special case. A necessary and sufficient condition is established forK ₀ ^(m) (h) =K ₁(h). It is shown that there are infinitely many quasisymmetric mappings of the unit circle having the property thatK ₀ ^(m) (h)<K ₁(h), wherek ₁(h) is the maximal dilatation ofh.     

Twistor quantization of the space of half-differentiable vector functions on the circle revisited

We discuss the twistor quantization problem for the classical system (V _d ,A _d ), represented by the phase space V _d , identified with the Sobolev space H ₀^1/2 (S ¹,ℝ ^d ) of half-differentiable vector functions on the circle, and the algebra of observables A _d , identified with the semi-direct product of the Heisenberg algebra of V _d and the algebra Vect(S ₁) of tangent vector fields on the circle.     

Partial indices for Toeplitz-like operators

The concept of partial indices, which is usually associated with matrix-functions, is defined for operators A satisfying rank (AU-VA)< for some fixed U and V (Toeplitz-like operators). A survey of the properties of partial indices is presented, some examples, and the connection with Kronecker indices as well. An application concerning Toeplitz plus Hankel matrices is supplemented.     

Independence Distribution Preserving Covariance Structures for the Multivariate Linear Model

Consider the multivariate linear model for the random matrixY_n×pMN(XB, VΣ), whereBis the parameter matrix,Xis a model matrix, not necessarily of full rank, andVΣ is annp×nppositive-definite dispersion matrix. This paper presents sufficient conditions on the positive-definite matrixVsuch that the statistics for testingH₀: CB=0vsH_a: CB≠0have the same distribution as under the i.i.d. covariance structureIΣ.     

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if (μ₀,μ₁) is a coherent pair of measures on the unit circle, then μ₀ is a semi-classical measure. Moreover, we obtain that the linear functional associated with μ₁ is a specific rational transformation of the linear functional corresponding to μ₀. Some examples are given.