Maximal multihomogeneity of algebraic hypersurface singularities

From the degree zero part of the logarithmic vector fields along analgebraic hypersurface singularity we identify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one–to–one correspondence to maximal tori in the linear jet of the embedded automorphism group. These results are motivated by Kyoji Saito’s characterization of quasihomogeneity for isolated hypersurface singularities [Saito in Invent. Math. 14, 123–142 (1971)] and extend previous work with Granger and Schulze [Compos. Math. 142(3), 765–778 (2006), Theorem 5.4] and of Hauser and Müller [Nagoya Math. J. 113, 181–186 (1989), Theorem 4].     

Extrinsic diameter of immersed flat tori in <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S ³. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S ³. In this paper, we give a new method for characterizing immersed flat tori in S ³ with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H ³ and R ³ regarded as wave fronts has been studied. We also give here a formulation of flat tori in S ³ as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π.     

Equivalences between categories of modules and categories of comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.     

Isometric Embeddings of Families of Special Lagrangian Submanifolds

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g _t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ _t , then (Y,g _t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T ²-symmetry. Mathematics Subject Classification (2000): 53-XX, 53C38.Communicated by N. Hitchin (Oxford)     

On relations of vector optimization results with C 1,1 data

In this article we prove that some of the sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Luc [Appl. Math., 51, 5-36 （2006）] generalize （strictly） those presented by Guerraggio, Luc [J. Optim. Theory Appl., 109, 615-629 （2001）]. While the former paper shows examples for which the conditions given there are effective but the ones from the latter paper fail, it does not prove that generally the conditions it proposes are stronger. In the present note we complete this comparison with the lacking proof.     

Weak convergence theorem for Lipschizian pseudocontraction semigroups in Banach spaces

The purpose of this paper is to study the weak convergence problems of the irnplicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [Nonlinear Anal., 68, 2977-2983 （2008）], Chen, et ah [J. Math. Anal. Appl., 314, 701 709 （2006）], Xu and Ori [Numer. Funct. Anal. Optim, 22, 767-773 （2001）] and Osilike [J. Math. Anal. Appl., 294, 73-81 （2004）]. Keywords     

On the Wilson criterion

The U(1)-gauge theory with the Villain action is considered in a cubic lattice approximation of three-and four-dimensional tori. As the lattice spacing tends to zero, the naturally defined correlation functions converge to the correlation functions of theR-gauge electrodynamics on three- and four-dimensional tori only for a special scaling, which depends on the correlation functions. Another scaling gives degenerate continuum limits. The Wilson criterion for the confinement of charged particles is fulfilled for theR-gauge electrodynamics on a torus. If the radius of the initial torus tends to infinity, then the correlation functions converge to the correlation functions of theR-gauge Euclidean electrodynamics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 67–73, January, 1999.     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).      

Minimax signal detection forl q -ellipsoids withl p -balls removed

The paper is close to the investigations of M. Ermakov [Theory Probab. Appl.,35, 667–679 (1990)] and Yu. Ingster [Zap. Nauchn. Semin. LOMI,184, 152–168 (1990)]. On the basis of the latter, the case q>p, p∈[0, 1], is considered in detail. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 126–136, 1993. Translated by the author.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

In this paper, we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large-deviation prob- abilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang （J. Appl. Prob., 41, 93-107, 2004）.     

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.     

CLT for linear random fields with stationary martingale-difference innovation

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order.     

Representations of small braid groups in Temperley-Lieb algebra

In this paper we consider the question of faithfulness of the Jones' representation of braid group Bn into the Temperley-Lieb algebra TLn. The obvious motivation to study this problem is that any non-trivial element in the kernel of this representation (for any n) would almost certainly yield a non-trivial knot with trivial Jones polynomial (see [S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot Theory Ramifications 11 (4) (2002) 493-505], we will explain it in more detail in Section 1). As one of the two main results we prove Theorem 1 in which we present a method to obtain non-trivial elements in the kernel of the representation of B₆ into TL9,2—to the authors' knowledge the first such examples in the second gradation of the Temperley-Lieb algebra. Theorem 2 which is a refinement of Theorem 1 may be used to produce smaller examples of the same kind. We also show briefly how some braids that are used in Section 4 to construct specific examples were generated with a computer program.     

On Convergence Results for Weak Efficiency in Vector Optimization Problems with Equilibrium Constraints

In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack, even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997).     

Mise en position optimale de tores par rapport à un flot d'Anosov

Let Φ′ be an Anosov flow on a (non atoroidal) 3-manifoldM. We say that an incompressible torusT embedded inM admits an optimal position with respect to Φ′ if it is isotopic to a torus transverse to Φ′ outside a finite number of periodic orbits contained inT (there's an additional condition we dont's mention here). The first remark is that such an optimal position is quasi unique, i.e., we prove that if two tori in optimal position are homotopics inM, then they are homotopics along the flow. Then we give some sufficient condition for a torus admiting an optimal position. Eventually, we show that if a finite collection of disjoint tori is such that each torus admits an optimal position, then these optimal positions can be chosen disjoints one from each other.      

Minimal codewords in Reed–Muller codes

Minimal codewords were introduced by Massey (Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp 276–279, 1993) for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3 · 2 ^m−r in binary Reed–Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami and Tokura (IEEE Trans Inf Theory 16:752–759, 1970) and Kasami et al. (Inf Control 30(4):380–395, 1976) on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov et al. (Discrete Appl Math 128(1), 65–74, 2003), and computer aided results of Borissov and Manev (Serdica Math J 30(2-3), 303–324, 2004). This paper is in fact an extended abstract. Full proofs can be found on the arXiv.     

Likelihood ratio tests for comparingk populations —The two-parameter nonregular models

Summary The null and nonnull distributions of the likelihood ratio statistics for testing the homogeneity ofk given populations, each associated with a nonregular density depending on two truncation parameters, are investigated. This generalizes to the two-parameter case the work of Hogg (1956,Ann. Math. Statist.,27, 529–532), Barr (1966,J. Amer. Statist. Assoc.,61, 856–864) and Khatri and Jaiswal (1969,Aust. J. Statist.,11, 79–84; 1969, 1971,Ann. Inst. Statist. Math.,21, 127–136;23, 199–210).