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).     

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.     

Coning, Symmetry and Spherical Frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53?C70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561?C598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric $\mathbb{M}^{d}$ and the hyperbolic metric ? ^d . This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among $\mathbb{E}^{d}$ , cones in $\mathbb{E}^{d+1}$ , $\mathbb{S}^{d}$ , $\mathbb{M}^{d}$ , and ? ^d . We also consider the further extensions associated with the other Cayley?CKlein geometries overlaid on the shared underlying projective geometry.     

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.     

A Characterization of the Locally Finite Networks Admitting Non-Constant Harmonic Functions of Finite Energy

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen (J Comb Theory, Ser B 49:87?C102, 1990) and of Lyons and Peres (2011) with the sufficient criterion of Soardi (1991). We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos (J Lond Math Soc, 2010).     

Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

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.     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Tilting Theory and Functor Categories I. Classical Tilting

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (mod_Λ), with Λ a finite dimensional algebra.     

Algorithms for lattice games

This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games (Guo et al. Oberwolfach Rep 22: 23–26, 2009; Guo and Miller, Adv Appl Math 46:363–378, 2010). Given a rational strategy for a lattice game, a polynomial time algorithm is presented to decide (i) whether a given position is a winning position, and to find a move to a winning position, if not; and (ii) to decide whether two given positions are congruent, in the sense of misère quotient theory (Plambeck, Integers, 5:36, 2005; Plambeck and Siegel, J Combin Theory Ser A, 115: 593–622, 2008). The methods are based on the theory of short rational generating functions (Barvinok and Woods, J Am Math Soc, 16: 957–979, 2003).     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces

In the previous work (Zhang and Zhu in J Differ Geom, http://arxiv.org/pdf/1012.4233v3, 2012), the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen and Wang (Sci Sin (A) 37:1–14, 1994), Chen and Wang (Sci Sin (A) 40:384–394, 1997) and Bakry–Qian (Adv Math 155:98–153, 2000), from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li–Yau’s estimate for positve solutions of heat equations on Alexandrov spaces.     

Pseudorandom number generators based on random covers for finite groups

Random covers for finite groups have been introduced in Magliveras et?al. (J Cryptol 15:285–297, 2002), Lempken et?al. (J Cryptol 22:62–74, 2009), and Svaba and van Trung (J Math Cryptol 4:271–315, 2010) for constructing public key cryptosystems. In this article we describe a new approach for constructing pseudorandom number generators using random covers for large finite groups. We focus, in particular, on the class of elementary abelian 2-groups and study the randomness of binary sequences generated from these generators. We successfully carry out an extensive test of the generators by using the NIST Statistical Test Suite and the Diehard battery of tests. Moreover, the article presents argumentation showing that the generators are suitable for cryptographic applications. Finally, we include performance data of the generators and propose a method of using them in practice.     

Computing Final Polynomials and Final Syzygies Using Buchberger’s Gröbner Bases Method

Final polynomials and final syzygies provide an explicit representation of polynomial identities promised by Hilbert’s Nullstellensatz. Such representations have been studied independently by Bokowski [2,3,4] and Whiteley [23,24] to derive invariant algebraic proofs for statements in geometry. In the present paper we relate these methods to some recent developments in computational algebraic geometry. As the main new result we give an algorithm based on B. Buchberger’s Gröbner bases method for computing final polynomials and final syzygies over the complex numbers. Degree upper bound for final polynomials are derived from theorems of Lazard and Brownawell, and a topological criterion is proved for the existence of final syzygies. The second part of this paper is expository and discusses applications of our algorithm to real projective geometry, invariant theory and matrix theory.     

Burgers type equations, Gelfand’s problem and Schumpeterian dynamics

Burgers?? equations have been introduced to study different models of fluids (Bateman, 1915, Burgers, 1939, Hopf, 1950, Cole, 1951, Lighthill andWhitham, 1955, etc.). The difference-differential analogues of these equations have been proposed for Schumpeterian models of economic development (Iwai, 1984, Polterovich and Henkin, 1988, Belenky, 1990, Henkin and Polterovich, 1999, Tashlitskaya and Shananin, 2000, etc.). This paper gives a short survey of the results and conjectures on Burgers type equations, motivated both by fluid mechanics and by Schumpeterian dynamics. Proofs of some new results are given. This paper is an extension and an improvement of (Henkin, 2007, 2011).     

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\) . For \(d\le 7\) , all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\) ; see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\) , see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\) ; see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\) , up to projectivity. For \(d=11,13,17,19\) , this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).     

The weight distributions of a class of cyclic codes II

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.