Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.     

An interval version of separation by semispaces in max-min convexity

In this paper, we study separation of a closed box from a max-min convex set by max-min semispaces. This can be regarded as an interval extension of the known separation results. We give a constructive proof of the separation in the case when the box satisfies a certain condition, and we show that the separation is never possible when the condition is not satisfied. We also study the separation of two max-min convex sets by a box and by a box and a semispace.     

Commutative matrix subalgebras and length function

It is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion. The examples are given showing that the converse does not hold. To establish this result, we prove several fundamental properties of the length function.     

The circumpolygon with maximal area

The convex circumpolygon with maximal area to a given convex polygon can be determined by means of dynamic programming. The effort of this method increases cubically with respect to the number of sides. It is further shown that the optimal circum-polygon can be constructed with ruler and circle. The applied version of dynamic programming can be also used for solving Steiner's problem of the inpoiygon with minimal circumference but it demands a higher effort than Phú's method.     

A note on an error estimate for least squares approximation

An asymptotic expansion is obtained which provides upper and lower bounds for the error of the bestL ₂ polynomial approximation of degreen forx ⁿ⁺¹ on [–1, 1]. Because the expansion proceeds in only even powers of the reciprocal of the large variable, and the error made by truncating the expansion is numerically less than, and has the same sign as the first neglected term, very good bounds can be obtained. Via a result of Phillips, these results can be extended fromx ⁿ⁺¹ to anyfC ⁿ⁺¹[–1, 1], provided upper and lower bounds for the modulus off ⁽ⁿ⁺¹⁾ are available.     

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.     

Fractal Penrose tilings I. Construction and matching rules

Summary Quasiperiodic tilings of kite-and-dart type, widely used as models for quasicrystals with decagonal symmetry, are constructed by means of somewhat artificial matching rules for the tiles. The proof of aperiodicity uses a self-similarity property, or inflation procedure, which requires drawing auxiliary lines. We introduce a modification of the kite-and-dart tilings which comes very naturally with both properties: the tiles are strictly self-similar, and their fractal boundaries provide perfect matching rules.     

Theq-gamma function forx<0

F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!) _q =1(1+q)(1+q+q ²)...(1+q+q ²+...+q ^n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q ^x –1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of _q asq changes and showed that asq1^–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of _q (x) forx<0 and allq>0. In addition to some basic properties of _q , we will study the behavior of the sequence {x _n (q)} of critical points asn orq changes.     

Solution générale sur un groupe abélien de I'équation fonctionnellef(x * f(y)) = f(f(x) * y)

Sans résumé

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Survival in a quasi-death process

We consider a Markov chain in continuous time with one absorbing state and a finite set S of transient states. When S is irreducible the limiting distribution of the chain as t→∞, conditional on survival up to time t, is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which S may be reducible, and show that it remains valid if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on S has geometric (but not, necessarily, algebraic) multiplicity one. The result is then applied to pure death processes and, more generally, to quasi-death processes. We also show that the result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of S satisfy an accessibility constraint. A key role in the analysis is played by some classic results on M-matrices.     

Hyponormal and strongly hyponormal matrices in inner product spaces

Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on earlier works on normal matrices, the notions of hyponormal and strongly hyponormal matrices are introduced. A full characterization of such matrices is given and it is shown how those matrices are related to different concepts of normal matrices in degenerate inner product spaces. Finally, the existence of invariant semidefinite subspaces for strongly hyponormal matrices is discussed.     

Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

For a given convex body K in with C ² boundary, let P ^c _n be the circumscribed polytope of minimal volume with at most n edges, and let P ⁱ _n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P ^c _n and P ⁱ _n are asymptotically regular triangles and squares, respectively, in a suitable sense. Supported by OTKA grants 043520 and 049301, and by the EU Marie Curie grants Discconvgeo, Budalggeo and PHD. Authors’ addresses: Károly J. B?r?czky, Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest H–1364, Hungary, and Department of Geometry, Roland E?tv?s University, Pázmány Péter sétány 1/C, Budapest 1117, Hungary; Salvador S. Gomis, Department of Mathematical Analysis, University of Alicante, 03080 Alicante, Spain; Péter Tick, Gyűrű utca 24, Budapest H–1039, Hungary     

Generalized sine equations,III

Summary We solve the equationf(x + y)f(x – y) = P(f(x), f(y)) under various conditions on the unknown functionsf, P.