A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces

We consider the two-player, complete information game of Cops and Robber played on undirected, finite, reflexive graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. Let c(g) be the supremum over all cop numbers of graphs embeddable in a closed orientable surface of genus g, and likewise ${\tilde c(g)}$ for non-orientable surfaces. It is known (Andreae, 1986) that, for a fixed surface, the maximum over all cop numbers of graphs embeddable in this surface is finite. More precisely, Quilliot (1985) showed that c(g) ≤ 2g + 3, and Schröder (2001) sharpened this to ${c(g)\le \frac32g + 3}$ . In his paper, Andreae gave the bound ${\tilde c(g) \in O(g)}$ with a weak constant, and posed the question whether a stronger bound can be obtained. Nowakowski & Schröder (1997) obtained ${\tilde c(g) \le 2g+1}$ . In this short note, we show ${\tilde c(g) \leq c(g-1)}$ , for any g ≥ 1. As a corollary, using Schröder’s results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3, the maximum cop number of graphs embeddable in the Klein Bottle is at most 4, ${\tilde c(3) \le 5}$ , and ${\tilde c(g) \le \frac32g + 3/2}$ for all other g.     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

C∞ functions in infinite dimension and linear partial differential difference equations with constant coefficients

This work is closed to [2] where a dense linear subspace \(\mathbb{E}\) (E) of the space ?(E) of the Silva C ^∞ functions on E is defined; the dual of \(\mathbb{E}\) (E) is described via the Fourier transform by a Paley-Wiener-Schwartz theorem which is formulated exactly in the same way as in the finite dimensional case. Here we prove existence and approximation result for solutions of linear partial differential difference equations in \(\mathbb{E}\) (E) with constant coefficients. We also obtain a Hahn-Banach type extension theorem for some C^∞ functions defined on a closed subspace of a DFN space, which is analogous to a Boland’s result in the holomorphic case [1].     

Ideal Divisibility Monoids

Inspired by the monograph of Larsen/McCarthy, [26], in [10] and [11] the author started a series of articles concerning abstract multiplicative ideal theory along the problem lines of [26]. In this paper we turn to multiplicative lattices having the left Priifer property, that is to m-lattices satisfying the implication a¹ + … + a_n ? B ? a₁ +… + a_n ¦? B or even the multiplication property A ? B ? A ¦B, respectively. Clearly, studying such structures includes studying substructures of d-semigroups.     

Geometric rigidity for analytic estimates of Müller–?verák

In the paper Müller–?verák (J Differ Geom 42(2):229–258, 1995) conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature ${\int_{\Sigma} |A|^2 < 8 \pi}$ in dimension three were embedded and conformal to the plane with one end. Here, using techniques from Kuwert–Li (W ^2,2-conformal immersions of a closed Riemann surface into R ⁿ . arXiv:1007.3967v2 [math.DG], 2010), we will show that if the total curvature ${ \int_{\Sigma}|A|^2\leq8\pi}$ , then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper’s minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if n?≥ 3 and ${ \int_{\Sigma} | A|^{2}\leq 16 \pi }$ then Σ is embedded or Σ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss–Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks (Topology 22(2):203–221, 1983). Using this theorem, we then prove an inversion formula for the Willmore energy.     

Equivalence of Two Gyrogroup Structures on Unit Balls

In special relativity all possible velocities form an open ball, R_c ³, of radius c in R³ with a binary operation, ⊕, given by Einstein’s velocity composition law. The concept of gyrogroups arose in the study of the algebraic structure underlying the resulting groupoid, (R_c ³, ⊕), giving rise to a generalization of the usual abelian group theory. Two different gyrogroup structures were defined on the unit ball of the Euclidean space resulting in the so-called standard and nonstandard Lorentzian groups [14]. The aim of this paper is (i) to define gyrogroup isomorphism and hence (ii) to show that the two structures are isomorphic in the gyrogroup sense; and (iii) to show that the resulting two Lorentzian groups are isomorphic, in the ordinary group sense, as pointed out by Urbantke [23]. Our results also include determining the group of all continuous automorphisms of the unit ball, when the latter is considered as a gyrogroup.     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

Algebraische Unabhängigkeit der Werte gewisser Lückenreihen in nicht-archimedisch bewerteten Körpern

In 1844 Liouville proved the transcendence of α = ∑_h≥1 10^{?h h!} over Q. The number α can be considered as the value of the gap power series ∧(x) =∑_h≥1 at tne point 1/10 Since then, the above result has been generalized in this direction by different authors by applying improved “Liouville-estimates”. For instance, in 1973 Cijsouw and Tijdeman [2] showed that a gap series with algebraic coefficients takes on transcendental values (over Q) at non-zero algebraic points under some conditions on the growth of the coefficients and the gaps. In 1988 Bundschuh [1] resp. Zhu [9] proved the algebraic independence (over Q) of the values of several gap series at different algebraic points. In particular this result includes the algebraic independence of A(α₁),…, α(α_s) for non-zero algebraic numbers α₁,…, α_s of distinct absolute values less than 1. Moreover in [1] a set of continuum-many algebraically independent numbers was constructed. In 1978 Geijsel [4] obtained a result analogous to that of Cijsouw and Tijdeman underlying a non-archimedian valued function field over a finite field, and in 1983 Sieburg [7] was concerned with the algebraic independence of “Liouville-series” in non-archimedian valued fields of characteristic zero. In this paper some of the results of [1] resp. [9] will be transfered to the situation of some non-archimedian valued fields. If the characteristic of the field is prime, we have to require stronger conditions as in the “classical case”. An example shows that in this case the numbers A(c*i),..., A(aa) need not to be algebraically independent. But a set of continuum-many algebraically independent numbers still exists. In characteristic zero, results of the same kind will be obtained like in the “classical case”.     

The von Kármán theory for incompressible elastic shells

We rigorously derive the von Kármán shell theory for incompressible materials, starting from the 3D nonlinear elasticity. In case of thin plates, the Euler-Lagrange equations of the limiting energy functional reduce to the incompressible version of the classical von Kármán equations, obtained formally in the limit of Poisson’s ratio ν → 1/2. More generally, the midsurface of the shell to which our analysis applies, is only assumed to have the following approximation property: ${\mathcal C^3}$ first order infinitesimal isometries are dense in the space of all W ^2,2 infinitesimal isometries. The class of surfaces with this property includes: subsets of ${\mathbb R^2}$ , convex surfaces, developable surfaces and rotationally invariant surfaces. Our analysis relies on the methods and extends the results of Conti and Dolzmann (Calc Var PDE 34:531–551, 2009, Lewicka et al. (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX:253–295, 2010, Friesecke et al. (Comm. Pure. Appl. Math. 55, no. 2, 1461–1506, 2002).     

The Mellin Transform and Spectral Properties of Toric Varieties

In this paper we apply the results of [W] on the twisted Mellin transform to problems in toric geometry. In particular, we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, z ^k , $ k \in \mathbb{Z}^{d} $ , in Bargmann space and prove an “upstairs” version of the spectral density theorem of [BGU]. We also obtain for the z ^k ’s, “upstairs” versions of the results of [STZ] on distribution laws for eigenstates on toric varieties.     

Further remarks on the non-degeneracy condition

The structure of the set of positive solutions of the semilinear elliptic boundary value problem depends on a certain non-degeneracy condition, which was proved by K.J. Brown [2] and T. Ouyang and J. Shi [12], with a shorter proof given later by P. Korman [8]. In this note we present a more general result, communicated to us by L. Nirenberg [13]. We also discuss the extensions in cases when the domain D is in R ², and it is either symmetric or convex.     

Syntactic Rings

If the state set and the input set of an automaton are Ω-groups then near-rings are useful in the study of automata (see [5]). These near-rings, called syntactic near-rings, consist of mappings from the state set Q of the automaton into itself. If, as is often the case, Q bears the structure of a module, then the zerosymmetric part N₀(A) of syntactic near-rings is a commutative ring with identity. If N₀(A) is a syntactic ring then its ideals are useful for determining reachability in automata (see [1] or [2]). In this paper we investigate syntactic rings.     

Rigidity around Poisson submanifolds

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to compute the Poisson moduli space of the sphere in the dual of a compact semisimple Lie algebra [17].     

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).     

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I: rank-one actions on the centre

The spectrum of a Gelfand pair of the form ${(K\ltimes N,K)}$ , where N is a nilpotent group, can be embedded in a Euclidean space ${{\mathbb R}^d}$ . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on ${{\mathbb R}^d}$ has been proved already when N is a Heisenberg group and in the case where N?=?N _3,2 is the free two-step nilpotent Lie group with three generators, with K?=?SO₃ (Astengo et?al. in J Funct Anal 251:772–791, 2007; Astengo et?al. in J Funct Anal 256:1565–1587, 2009; Fischer and Ricci in Ann Inst Fourier Gren 59:2143–2168, 2009). We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra ${{\mathfrak n}}$ of N for all Gelfand pairs ${(K\ltimes N,K)}$ in Vinberg’s list (Vinberg in Trans Moscow Math Soc 64:47–80, 2003; Yakimova in Transform Groups 11:305–335, 2006).     

A note on propositional proof complexity of some Ramsey-type statements

A Ramsey statement denoted ${n \longrightarrow (k)^2_2}$ says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n ^k ) and with terms of size ${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$ . Let r _k be the minimal n for which the statement holds. We prove that RAM(r _k , k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák??s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4 ^k , k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.     

Lipscomb’s L(A) Space Fractalized in l p (A)

In this paper, by using some of our new results concerning the shift space for an infinite IFS (see A. Mihail and R. Miculescu, The shift space for an infinite iterated function system, Math. Rep. Bucur. 11 (2009), 21?C32), we show that, for an infinite set A, the embedded version of the Lipscomb space L(A) in l ^p (A), ${p \in [1,\infty)}$ , with the metric induced from l ^p (A), denoted by ${\omega_p^A}$ , is the attractor of an infinite iterated function system comprising affine transformations of l ^p (A). In this way we provide a generalization of the positive answer that we gave to an open problem of J.C. Perry (see Lipscomb??s universal space is the attractor of an infinite iterated function system, Proc. Amer. Math. Soc. 124 (1996), 2479?C2489) in one of our previous works (see R. Miculescu and A. Mihail, Lipscomb space ?? ^A is the attractor of an infinite IFS containing affine transformations of l ²(A), Proc. Amer. Math. Soc. 136 (2008), 587?C592). Moreover, as a byproduct, we provide a generalization of Corollary 15 from Perry??s paper by proving that ${\omega_p^A}$ is a closed subset of l ^p (A).     

Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.     

Finite simple 3′-groups are cyclic or Suzuki groups

In this note we prove that all finite simple 3′-groups are cyclic of prime order or Suzuki groups. This is well known in the sense that it is mentioned frequently in the literature, often referring to unpublished work of Thompson. Recently an explicit proof was given by Aschbacher [3], as a corollary of the classification of ${\mathcal{S}_3}$ -free fusion systems. We argue differently, following Glauberman’s comment in the preface to the second printing of his booklet [8]. We use a result by Stellmacher (see [12]), and instead of quoting Goldschmidt’s result in its full strength, we give explicit arguments along his ideas in [10] for our special case of 3′-groups.     

A fourier-type transform on translation-invariant valuations on convex sets

Let V be a finite-dimensional real vector space. Let V al ^sm (V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism $$ \mathbb{F}_V :Val^{sm} (V)\tilde \to Val^{sm} (V^* ) \otimes Dens(V) $$ such that $ \mathbb{F}_V $ commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.