• Volume 126, Issue 3

August 2016,   pages  289-460

• Perfect 2-colorings of the generalized Petersen graph

In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs $GP(n, 2)$, where $n \geq 5$. We also present some basic results for $GP(n, k)$, where $n \geq 5$ and $k \geq 3$.

• Partial sums of arithmetical functions with absolutely convergent Ramanujan expansions

For an arithmetical function $f$ with absolutely convergent Ramanujan expansion, we derive an asymptotic formula for the $\sum_{n\leq N}$ f(n)$with explicit error term. As a corollary we obtain new results about sum-of-divisors functions and Jordan’s totient functions. • Harder–Narasimhan filtration for rank 2 tensors and stable coverings We construct a Harder--Narasimhan filtration for rank 2 tensors, where there does not exist any such notion {/it a priori,} as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank 2 tensors over a smooth projective complex variety, does not depend on a certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder--Narasimhan filtration for rank 2 tensors as this unique filtration coming from GIT. Symmetric rank 2 tensors over smooth projective complex curves define curve coverings lying on a ruled surface, hence we can translate the stability condition to define stable coverings and characterize the Harder--Narasimhan filtration in terms of intersection theory. • Weak point property and sections of Picard bundles on a compactified Jacobian over a nodal curve We show that the compactified Jacobian (and its desingularization) of an integral nodal curve$Y$satisfies the weak point property and the Jacobian of$Y$satisfies the diagonal property. We compute some cohomologies of Picard bundles on the compactified Jacobian and its desingularization • Diamond lemma for the group graded quasi-algebras Let$G$be a group. We prove that every expression in a$G$-graded quasialgebra can be reduced to a unique irreducible form and the irreducible words form abasis for the quasi-algebra. The result obtained is applied to some interesting classes of group graded quasi-algebras like generalized octonions. • Dimension of the$c$-nilpotent multiplier of Lie algebras The purpose of this paper is to derive some inequalities for dimension of the$c$-nilpotent multiplier of finite dimensional Lie algebras and their factor Lie algebras. We further obtain an inequality between dimensions of$c$-nilpotent multiplier of Lie algebra$L$and tensor product of a central ideal by its abelianized factor Lie algebra • Weighted and vector-valued inequalities for one-sided maximal functions In this paper, we study weighted and vector-valued inequalities for one-sided maximal functions. In particular, we give a new proof of$l_r$-valued extension of weighted$L^p$-inequalities for one-sided maximal functions. In the process, we prove an analogue of the well-known Fefferman–Stein’s weighted lemma in the context of one-sided maximal functions. Further, we also study$l_r$-valued extension of the lemma. • Extremely strict ideals in Banach spaces Motivated by the notion of an ideal introduced by Godefroy {\it et al.} ({\it Studia Math.} {\bf 104} (1993) 13–59), in this article, we introduce and study the notion of an extremely strict ideal. For a Poulsen simplex$K$, we show that the space of affine continuous functions on$K$is an extremely strict ideal in the space of continuous functions on$K$. For injective tensor product spaces, we prove a cancelation theorem for extremely strict ideals. We also exhibit non-reflexive Banach spaces which are not strict ideals in their fourth dual. • On prime and semiprime rings with generalized derivations and non-commutative Banach algebras Let$R$be a prime ring of characteristic different from 2 and$m$a fixed positive integer. If$R$admits a generalized derivation associated with a nonzero deviation$d$such that$[F(x), d(y)]_m = [x, y]$for all$x$,$y$in some appropriate subset of$R$, then$R$is commutative. Moreover, we also examine the case$R$is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer--Werner theorem. • On arrangements of pseudohyperplanes To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman--Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which is homeomorphic to the complexified complement in the realizable case. Further, we generalize the classical theorem of Salvetti and show that this space has the homotopy type of a cell complex defined in terms of the oriented matroid. • Ramanujan’s identities, minimal surfaces and solitons Using Ramanujan’s identities and the Weierstrass--Enneper representation of minimal surfaces, and the analogue for Born--Infeld solitons, we derive further nontrivial identities •$L_p$weak convergence method on BSDEs with non-uniformly Lipschitz coefficients and its applications In this paper, by using$L_p$($1 \lt p \leq 2$) weak convergence method on backward stochastic differential equations (BSDEs) with non-uniformly Lipschitz coefficients, we obtain the limit theorem of$g$-supersolutions. As applications of this theorem, we study the decomposition theorem of$\epsilon_g$-supermartingale, the nonlinear decomposition theorem of Doob-Meyer’s type and so on. Furthermore, by using the decomposition theorem of$\epsilon_g$-supermartingale, we provide some useful characterizations of an$\epsilon^g$-evaluation by the generating function$g(t; ω; y; z)$without the assumption that$g$is continuous with respect to$t\$. Our results generalize the known results in Ph. Briand et al., Electronic Commun. Probab. {\bf 5} (2000) 101–117; L Jiang, Ann. Appl. Probab. {\bf 18} (2008) 245–258; S Peng, Probab. Theory Relat. Fields {\bf 113} (1999) 473–499; S Peng, Modelling derivatives pricing with their generating functions (2006) http://arxiv.org/abs/math/0605599 and E Rosazza Gianin, Insur. Math. Econ. {\bf 39} (2006) 19–34.

• Proceedings – Mathematical Sciences

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