Subsonic irrotational flows in a two-dimensional finitely long curved nozzle

This is a continuation of our previous paper Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf), where we have characterized a set of physical boundary conditions that ensures the existence and uniqueness of subsonic irrotational flow in a flat nozzle. In this paper, we will investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angle of inclination of nozzle walls play the same role as the end pressure for the stabilization of subsonic flows. In other words, the L ² and L ^∞ bounds of the derivative of these two quantities cannot be too large, similar as we have indicated in Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf) for the end pressure. The curvatures of the nozzle walls will also play an important role in the stability of the subsonic flow.     

A bound for the number of different basic solutions generated by the simplex method

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems (LP) having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the problem is primal nondegenerate, it becomes a bound for the number of iterations. The result includes strong polynomiality for Markov Decision Problem by Ye (http://www.stanford.edu/~yyye/simplexmdp1.pdf, 2010) and utilize its analysis. We also apply our result to an LP whose constraint matrix is totally unimodular and a constant vector b of constraints is integral.     

Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several examples are given illustrating the results.     

Period Doubling in the Rössler System—A Computer Assisted Proof

Using rigorous numerical methods, we validate a part of the bifurcation diagram for a Poincaré map of the Rössler system (Rössler in Phys. Lett. A 57(5):397–398, 1976)—the existence of two period-doubling bifurcations and the existence of a branch of period two points connecting them. Our approach is based on the Lyapunov–Schmidt reduction and uses the C ^r -Lohner algorithm (Wilczak and Zgliczyński, available at http://www.ii.uj.edu.pl/~wilczak) to obtain rigorous bounds for the Rössler system.     

Direct and Indirect Approaches to Blockmodeling of Valued Networks in Terms of Regular Equivalence

The aim of this article is to compare methods for detecting and/or measuring regular equivalence in valued networks. The methods compared can be divided into direct and indirect methods. The indirect methods considered here are all variants of the REGE algorithm (White, 2005 White , D. R. ( 2005 ). REGGE (web page) . Retrieved May 12, 2005, from http://eclectic.ss.uci.edu/~drwhite/REGGE/  [Google Scholar]). All direct methods have roots in Batagelj, Doreian, and Ferligoj (1992 Batagelj , V. , Doreian , P. , &; Ferligoj , A. ( 1992 ). An optimizational approach to regular equivalence . Social Networks , 14 , 121 – 135 .[Crossref] , [Google Scholar]); however they have been adapted for valued networks (?iberna, 2006b ?iberna , A. ( 2006b ). Generalized Blockmodeling of Valued Networks . Social Networks(in press) , Retrieved May 30, 2006, from http://dx.doi.org/10.1016/j.socnet.2006.04.002  [Google Scholar]). All approaches are compared on an empirical valued social network. Variants of the REGE algorithm are also compared on a few artificial valued networks in order to clarify their functionality. This is not necessary for direct approaches, which have clearly defined ideal blocks. In discussion, advantages and disadvantages of both indirect and direct approaches are discussed.     

A remark on Feuerbach hyperbolas

Recently many authors have studied properties of triangles and the theory of perspective triangles in the Euclidean plane (see Kimberling et al. J Geom Graph 14:1–14, 2010; Kimberling et al. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html, 2012; Moses and Kimberling J Geom Graph 13:15–24, 2009; Moses and Kimberling Forum Geom 11:83–93, 2011; Odehnal Elem Math 61:74–80, 2006; Odehnal Forum Geom 10:35–40, 2010; Odehnal J Geom Graph 15: 45–67, 2011). The aim of this paper is to present a new approach to the construction of points on the Feuerbach hyperbola. Surprisingly, these points can be obtained as centers of perspectivity of a triangle ABC and a certain one-parametric set of triangles A′B′C′. The presented construction is based on partitions of the triangle’s sides and—in a way—dual to the construction of points on the Kiepert hyperbola. It can also be generalized to spherical triangles. The proofs are based on an affine property of triangles, which amazingly can also be used for the proof of the spherical theorem.     

Spatial power indices with applications on real voting data from the Chamber of Deputies of the Czech Parliament

Participants in parliamentary voting—usually political parties—are evaluated with a value that assigns them predicted power with respect to their strength by so called power indices. However, in real world, political parties’ representatives act not strictly as predicted in theory. One possibility how the representatives differ from theory is the way they form coalitions; the coalitions can be announced (for example official governmental coalition parties in multiparty parliaments) or hidden. To incorporate the coalition forming influence, Bilal et al. (http://aei.pitt.edu/2052/1/001591_1.pdf, 2001) proposed to consider additional weights to possible coalitions into power indices. This article applies the concept of additional weights to calculate an ex post power distribution using Shapley–Shubik power index together with Banzhaf power index on real voting data, namely the data from the Chamber of Deputies of the Czech Parliament with the emphasis on the State Budget voting issues during 2006–2010 parliamentary period.     

The second Dirichlet coefficient starts out negative

Classical modular forms of small weight and low level are likely to have a negative second Fourier coefficient. Similarly, the labeling scheme for elliptic curves tends to give smaller labels to the higher-rank curves. These observations are easily made when browsing the L-functions and Modular Forms Database, available at http://www.LMFDB.org/. An explanation lies in the L-functions associated to these objects.     

On the Brauer-Feit bound for abelian defect groups

We improve the Brauer-Feit bound on the number of irreducible characters in a $p$ -block for abelian defect groups by making use of Halasi and Podoski (Every coprime linear group admits a base of size two. http://arxiv.org/abs/1212.0199v1, [7]) and Kessar and Malle (Ann Math 178(2):321–384, [11]). We also prove Brauer’s $k(B)$ -Conjecture for 2-blocks with abelian defect groups of rank at most 5 and 3-blocks and 5-blocks with abelian defect groups of rank at most 3.     

Penrose Type Inequalities for Asymptotically Hyperbolic Graphs

In this paper, we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space ${\mathbb{H}^n}$ . The graphs are considered as unbounded hypersurfaces of ${\mathbb{H}^{n+1}}$ which carry the induced metric and have an interior boundary. For such manifolds, the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence, we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition, this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam’s article (The graph cases of the Riemannian positive mass and Penrose inequalities in all dimensions. http://arxiv.org/abs/1010.4256, 2010) concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu (The equality case of the penrose inequality for asymptotically flat graphs. http://arxiv.org/abs/1205.2061, 2012), we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.     

Finite number of fibre products of Kummer covers and curves with many points over finite fields

We study fibre products of a finite number of Kummer covers of the projective line over finite fields. We determine the number of rational points of the fibre product over a rational point of the projective line, which improves the results of Özbudak and Temür (Appl Algebra Eng Commun Comput 18:433–443, 2007) substantially. We also construct explicit examples of fibre products of Kummer covers with many rational points, including a record and two new entries for the current table (http://www.manypoints.org, 2011).     

The concept of hyperellipticity on surfaces with nodes

A surface with nodes X is hyperelliptic if there exists an involution such that the genus of X/〈h〉 is 0. We prove that this definition is equivalent, as in the category of surfaces without nodes, to the existence of a degree 2 morphism satisfying an additional condition where the genus of Y is 0. Other question is if the hyperelliptic involution is unique or not. We shall prove that the hyperelliptic involution is unique in the case of stable Riemann surfaces but is not unique in the case of Klein surfaces with nodes. Finally, we shall prove that a complex double of a hyperelliptic Klein surface with nodes could not be hyperelliptic.     

The Goulden—Jackson cluster method: extensions,applications and implementations

Th powerful (and so far under-utilized) Goulden—Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of ‘dirty words’, is tutorialized and extended in various directions. The authors' Maple implementations, contained in several Maple packages available from this paper's website www.math.temple.edu/zeilberg/gj.html, ar described and explained.     

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes (Sel. Math. 1(3):411–457, 1995). In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Consani and Marcolli (Adv. Math. 214(2):761–831, 2007), and using a classification result of Borger and de Smit (arXiv:1105.4662) for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreover the uniqueness of the arithmetic model is shown by Sergey Neshveyev in an appendix. In the second part of the paper we introduce a base-change functor for a class of algebraic endomotives and construct in this way an algebraic refinement of a functor from the category of number fields to the category of Bost-Connes systems, constructed recently by Laca, Neshveyev and Trifkovic (arXiv:1010.4766).     

The number of binary cube-free words of length up to 47 and their numerical analysis

An analog of the work of Noonan and Zeilberger on square-free ternary words given for the case of cube-free binary words. As in their work, the Goulden-Jackso Cluster Method is used to derive a rigorous upper bound, as well as a non-rigorot estimate, for the limit of thenth roots of the terms. The Maple implementatin of the work is available from this paper's website www.math.temple. edu/`anne/cut free.html.     

Some remarks on the inverse semigroup associated to the Cuntz-Li algebras

In this article, we prove that the inverse semigroup associated to the Cuntz-Li relations is strongly 0-E unitary and is an F ^?-inverse semigroup. We also identify the universal group of the inverse semigroup. This gives a conceptual explanation for the result obtained in S. Sundar (arXiv:1201.4620v1, 2012).     

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p constant of singular integral operators on classical weighted Lebesgue spaces $L^{p}_{w}$ in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill.?J.?Math. 52:653?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 2002). Our main result is stated in Theorem?1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.     

Cell decomposition of some unitary group Rapoport–Zink spaces

In this paper we study the \(p\) -adic analytic geometry of the basic unitary group Rapoport–Zink spaces \(\mathcal {M}_K\) with signature \((1,n-1)\) . Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special fiber \(\mathcal {M}_{red}\) defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain \(\mathcal {D}_K\) in \(\mathcal {M}_K\) for the action of \(G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)\) , the product of the associated \(p\) -adic reductive groups, and prove that \(\mathcal {M}_K\) admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720, 2013).