Hyperbolic polygons of minimal perimeter with given angles

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of Schlenker (Trans Am Math Soc 359(5): 2155–2189, 2007).     

Improved algorithms for the multicut and multiflow problems in rooted trees

Costa et al. (Oper. Res. Lett. 31:21–27, 2003) presented a quadratic O(min (Kn,n ²)) greedy algorithm to solve the integer multicut and multiflow problems in a rooted tree. (n is the number of nodes of the tree, and K is the number of commodities). Their algorithm is a special case of the greedy type algorithm of Kolen (Location problems on trees and in the rectilinear plane. Ph.D. dissertation, 1982) to solve weighted covering and packing problems defined by general totally balanced (greedy) matrices. In this communication we improve the complexity bound in Costa et al. (Oper. Res. Lett. 31:21–27, 2003) and show that in the case of the integer multicut and multiflow problems in a rooted tree the greedy algorithm of Kolen can be implemented in subquadratic O(K+n+min (K,n)log n) time. The improvement is obtained by identifying additional properties of this model which lead to a subquadratic transformation to greedy form and using more sophisticated data structures.      

Computational experiments with a lazy version of a <Emphasis Type="Italic">K</Emphasis> quickest simple path ranking algorithm

The quickest path problem is related to the classical shortest path problem, but its objective function concerns the transmission time of a given amount of data throughout a path, which involves both cost and capacity. The K-quickest simple paths problem generalises the latter, by looking for a given number K of simple paths in non-decreasing order of transmission time. Two categories of algorithms are known for ranking simple paths according to the transmission time. One is the adaptation of deviation algorithms for ranking shortest simple paths (Pascoal et al. in Comput. Oper. Res. 32(3):509–520, 2005; Rosen et al. in Comput. Oper. Res. 18(6):571–584, 1991), and another is based on ranking shortest simple paths in a sequence of networks with fixed capacity lower bounds (Chen in Inf. Process. Lett. 50:89–92, 1994), and afterwards selecting the K quickest ones. After reviewing the quickest path and the K-quickest simple paths problems we describe a recent algorithm for ranking quickest simple paths (Pascoal et al. in Ann. Oper. Res. 147(1):5–21, 2006). This is a lazy version of Chen’s algorithm, able to interchange the calculation of new simple paths and the output of each k-quickest simple path. Finally, the described algorithm is computationally compared to its former version, as well as to deviation algorithms.      

On the spectral expansion of hyperbolic Eisenstein series

In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla and Millson (Invent Math 54:193–211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive, hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going beyond the results in Kudla and Millson (Invent Math 54:193–211, 1979) and Risager (Int Math Res Not 41:2125–2146, 2004), we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface by first proving that the hyperbolic Eisenstein series is in L ². Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion.     

Unavoidable Crossings in a Thinnest Plane Covering with Congruent Convex Disks

Two convex disks K and L in the plane are said to cross each other if the removal of their intersection causes each disk to fall into disjoint components. Almost all major theorems concerning the covering density of a convex disk were proved only for crossing-free coverings. This includes the classical theorem of L. Fejes Tóth (Acta Sci. Math. Szeged 12/A:62–67, 1950) that uses the maximum area hexagon inscribed in the disk to give a significant lower bound for the covering density of the disk. From the early seventies, all attempts of generalizing this theorem were based on the common belief that crossings in a plane covering by congruent convex disks, being counterproductive for producing low density, are always avoidable. Partial success was achieved not long ago, first for “fat” ellipses (A. Heppes in Discrete Comput. Geom. 29:477–481, 2003) and then for “fat” convex disks (G. Fejes Tóth in Discrete Comput. Geom. 34(1):129–141, 2005), where “fat” means of shape sufficiently close to a circle. A recently constructed example will be presented here, showing that, in general, all such attempts must fail. Three perpendiculars drawn from the center of a regular hexagon to its three nonadjacent sides partition the hexagon into three congruent pentagons. Obviously, the plane can be tiled by such pentagons. But a slight modification produces a (non-tiling) pentagon with an unexpected covering property: every thinnest covering of the plane by congruent copies of the modified pentagon must contain crossing pairs. The example has no bearing on the validity of Fejes Tóth’s bound in general, but it shows that any prospective proof must take into consideration the existence of unavoidable crossings.     

A Numerical Investigation on Configurational Distortions in Nematic Liquid Crystals

When subjected to magnetic or electric fields, nematic liquid crystals confined between two parallel glass plates and initially uniformly oriented may undergo homogeneous one-dimensional spatial distortions (Fréedericksz and Zolina, Trans. Faraday Soc. 29:919, 1933) or periodic distortions (Lonberg and Meyer, Phys. Rev. Lett. 55(7):718–721, 1985; and Srajer et al., Phys. Rev. Lett. 67(9):1102–1105, 1991). According to the experimental observations, periodic phases are stable configurations at intermediate intensity of the acting field, while homogeneous phases are stable at higher strengths.     

Equivalence of Equilibrium Problems and Least Element Problems

In this paper, we introduce the concept of feasible set for an equilibrium problem with a convex cone and generalize the notion of a Z-function for bifunctions. Under suitable assumptions, we derive some equivalence results of equilibrium problems, least element problems, and nonlinear programming problems. The results presented extend some results of [Riddell, R.C.: Equivalence of nonlinear complementarity problems and least element problems in Banach lattices. Math. Oper. Res. 6, 462–474 (1981)] to equilibrium problems. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Educational Science Foundation of Chongqing (KJ051307).     

Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

Estimating the Scale Parameter of a Lévy-stable Distribution via the Extreme Value Approach

The characteristic exponent α of a Lévy-stable law S _α (σ, β, μ) was thoroughly studied as the extreme value index of a heavy tailed distribution. For 1 < α < 2, Peng (Statist. Probab. Lett. 52: 255–264, 2001) has proposed, via the extreme value approach, an asymptotically normal estimator for the location parameter μ. In this paper, we derive by the same approach, an estimator for the scale parameter σ and we discuss its limiting behavior.      

The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145–210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger–Müller theorem. We use a formula proved by Brüning and Ma (GAFA 16:767–873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Lück, J Diff Geom 37:263–322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695–714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529–533, 2009).     

On the Consistency of the Quantum-Like Representation Algorithm for Hyperbolic Interference

Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]–[28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations. The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric, hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric interference was recently proved [29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning. For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H ^b|a and H ^a|b (in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary equivalent (in the sense of hyperbolic Hilbert space).     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 retrial queue with retrials due to server failures

Sherman and Kharoufeh (Oper. Res. Lett. 34:697–705, [2006]) considered an M/M/1 type queueing system with unreliable server and retrials. In this model it is assumed that if the server fails during service of a customer, the customer leaves the server, joins a retrial group and in random intervals repeats attempts to get service. We suggest an alternative method for analysis of the Markov process, which describes the functioning of the system, and find the joint distribution of the server state, the number of customers in the queue and the number of customers in the retrial group in steady state.      

Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

   Abstract. A discrete analogue of the holomorphic maps z ^γ and log(z) is studied. These maps are given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z ^γ and log(z) , are established.     

Revisiting Generalized Nash Games and Variational Inequalities

Generalized Nash games with shared constraints represent an extension of Nash games in which strategy sets are coupled across players through a shared or common constraint. The equilibrium conditions of such a game can be compactly stated as a quasi-variational inequality (QVI), an extension of the variational inequality (VI). In (Eur. J. Oper. Res. 54(1):81–94, 1991), Harker proved that for any QVI, under certain conditions, a solution to an appropriately defined VI solves the QVI. This is a particularly important result, given that VIs are generally far more tractable than QVIs. However Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007) suggested that the hypotheses of this result are difficult to satisfy in practice for QVIs arising from generalized Nash games with shared constraints. We investigate the applicability of Harker’s result for these games with the aim of formally establishing its reach. Specifically, we show that if Harker’s result is applied in a natural manner, its hypotheses are impossible to satisfy in most settings, thereby supporting the observations of Facchinei et al. But we also show that an indirect application of the result extends the realm of applicability of Harker’s result to all shared-constraint games. In particular, this avenue allows us to recover as a special case of Harker’s result, a result provided by Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007), in which it is shown that a suitably defined VI provides a solution to the QVI of a shared-constraint game.     

Strong Limit Theorems for Weighted Sums of Negatively Associated Random Variables

In this paper, we establish strong laws for weighted sums of identically distributed negatively associated random variables. Marcinkiewicz-Zygmund’s strong law of large numbers is extended to weighted sums of negatively associated random variables. Furthermore, we investigate various limit properties of Cesàro’s and Riesz’s sums of negatively associated random variables. Some of the results in the i.i.d. setting, such as those in Jajte (Ann. Probab. 31(1), 409–412, 2003), Bai and Cheng (Stat. Probab. Lett. 46, 105–112, 2000), Li et al. (J. Theor. Probab. 8, 49–76, 1995) and Gut (Probab. Theory Relat. Fields 97, 169–178, 1993) are also improved and extended to the negatively associated setting.      

Singular homologically area minimizing surfaces of codimension one in Riemannian manifolds

For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (N ^{n +1},g), with non-trivial n dimensional integer homology, such that for some Γ∈H _n (N,Z), the hypersurface (n-current) M, which minimizes area among all hypersurfaces representing Γ, has singularities. The singular set of M consists of two isolated points, and the tangent cone at these points can be prescribed as any strictly stable, strictly minimizing, regular cone. To my knowledge these are the first examples of codimension one homological minimizers with singularities. Oblatum: 3-I-1997 & 13-II-1998 / Published online: 18 September 1998     

Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

Cotangent cohomology of rational surface singularities

In this paper we show that the number of generators of the cotangent cohomology groups T _Y ⁿ , n≥2, is the same for all rational surface singularities Y of fixed multiplicity. For a large class of rational surface singularities, including quotient singularities, this number is also the dimension. For them we obtain an explicit formula for the Poincaré series P _Y (t)=∑dim T ⁿ _Y ·t ⁿ . In the special case of the cone over the rational normal curve we give the multigraded Poincaré series. Oblatum: 18-XI-1998 & 25-III-1999 / Published online: 6 July 1999     

On Implicit Multifunction Theorems

We give implicit multifunction results generalizing to multifunctions the Robinson’s implicit function theorem (Robinson, Math Oper Res 16(2):292–309, 1991). To this end, we use parametric error bounds estimates for a suitable function refining the one given in Azé and Corvellec (ESAIM Control Optim Calc Var 10:409–425, 2004). Sharp approximations of the implicit multifunctions are given extending the results of Nachi and Penot (Control Cybernet 35:871–901, 2005). Dedicated to Boris Mordukhovich in honour of his 60th birthday.