Showing papers in "Open Mathematics in 2012"
TL;DR: In this paper, the notion of instanton bundles on a Fano threefold of index 2 was introduced and an analogue of a monadic description and the curve of jumping lines was discussed.
Abstract: We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.
69 citations
TL;DR: In this paper, the authors make an overview of some existence results for nonlinear differential and integral equations, which were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type.
Abstract: The aim of this paper is to make an overview of some existence results for nonlinear differential and integral equations. Those results were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type.
62 citations
TL;DR: It is shown that the Newman-Watts ones built on top of them by adding lattice-induced clustering are not hyperbolic as the network size goes to infinity, compared to the asymptotic Erdős-Rényi random graphs.
Abstract: The geometry of complex networks is closely related with their structure and function. In this paper, we investigate the Gromov-hyperbolicity of the Newman‐Watts model of small-world networks. It is known that asymptotic Erd ˝ os‐Renyi random graphs are not hyperbolic. We show that the Newman‐Watts ones built on top of them by adding lattice-induced clustering are not hyperbolic as the network size goes to infinity. Numerical simulations are provided to illustrate the eects of various parameters on hyperbolicity in this model.
52 citations
TL;DR: In this article, a simplified approach to the analytical approximation of the transition density related to a general local volatility model is presented, which is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.
Abstract: We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.
47 citations
TL;DR: In this article, it was shown that the fixed locus of the induced involution on H2(X, ℤ) is isomorphic to E8(−2).
Abstract: Let X be a hyperkahler manifold deformation equivalent to the Hilbert square of a K3 surface and let φ be an involution preserving the symplectic form. We prove that the fixed locus of φ consists of 28 isolated points and one K3 surface, and moreover that the anti-invariant lattice of the induced involution on H2(X, ℤ) is isomorphic to E8(−2). Finally we show that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.
47 citations
TL;DR: In this paper, the authors characterized the connected hypergraphs that achieve equality in the Lai-Chang bound and in the Chvatal-McDiarmid bound.
Abstract: Let H be a hypergraph on n vertices and m edges with all edges of size at least four. The transversal number τ(H) of H is the minimum number of vertices that intersect every edge. Lai and Chang [An upper bound for the transversal numbers of 4-uniform hypergraphs, J. Combin. Theory Ser. B, 1990, 50(1), 129–133] proved that τ(H) ≤ 2(n+m)/9, while Chvatal and McDiarmid [Small transversals in hypergraphs, Combinatorica, 1992, 12(1), 19–26] proved that τ(H) ≤ (n + 2m)/6. In this paper, we characterize the connected hypergraphs that achieve equality in the Lai-Chang bound and in the Chvatal-McDiarmid bound.
45 citations
TL;DR: For any complex valued Lp-function b(x), 2 ≤ p < ∞, or L∞-function with the norm b↾L∞ < 1, the spectrum of a perturbed harmonic oscillator operator in L2(ℝ1) is discrete and eventually simple as mentioned in this paper.
Abstract: For any complex valued Lp-function b(x), 2 ≤ p < ∞, or L∞-function with the norm ‖b↾L∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d2/dx2 + x2 + b(x) in L2(ℝ1) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L2(ℝ).
40 citations
TL;DR: In this article, the sharp hyperbolicity constant of a geodesic metric space is defined, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}.
Abstract: If X is a geodesic metric space and x1; x2; x3 ∈ X, a geodesic triangle T = {x1; x2; x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.
37 citations
TL;DR: In this article, a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces was considered and weak solutions to the zero Dirichlet boundary value problem were shown.
Abstract: We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Holder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.
30 citations
TL;DR: This paper considers meshbased and meshfree generalizations of the finite element method and the use of smooth, discontinuous, singular and numerical enrichment functions.
Abstract: This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To this end, we consider meshbased and meshfree generalizations of the finite element method and the use of smooth, discontinuous, singular and numerical enrichment functions.
29 citations
TL;DR: In this paper, the curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum, is shown to characterise explicitly the local biholomorphic equivalence of such M3 ⊂ ℂ 2 to the Heisenberg sphere ℍ3, such M 3 being necessarily real analytic.
Abstract: We study effectively the Cartan geometry of Levi-nondegenerate C6-smooth hypersurfaces M3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.
TL;DR: In this paper, it was shown that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson's axioms, and Ξ is a class in F1H2(L, ℂ), the first Hodge filtration piece of the second cohomology of L.
Abstract: Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and \( \equiv :Gr\Lambda \to Sym \bullet _{\mathcal{O}_X } \mathcal{G}\) is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on \(\mathcal{G}\) and Σ is a class in F1H2(L, ℂ), the first Hodge filtration piece of the second cohomology of L.
TL;DR: The tractor formalism from conformal geometry to the study of smooth metric measure spaces was introduced in this article, which gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles.
Abstract: We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.
TL;DR: In this paper, a family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered and a second order of convergence and a preservation of the discrete energy for this approach are proved.
Abstract: A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the proposed family of schemes is about two times more accurate than the family of schemes studied in [Kolkovska N., Two families of finite difference schemes for multidimensional Boussinesq paradigm equation, In: Application of Mathematics in Technical and Natural Sciences, Sozopol, June 21–26, 2010, AIP Conf. Proc., 1301, American Institute of Physics, Melville, 2010, 395–403].
TL;DR: Several examples of gaps between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations.
Abstract: Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.
TL;DR: The Poincare-Bendixson Theorem and the development of the theory are presented in this article from the papers of Bendixson to modern results, with a discussion of the relation between the two theories.
Abstract: The Poincare-Bendixson Theorem and the development of the theory are presented — from the papers of Poincare and Bendixson to modern results.
TL;DR: An algebraic extension of Cartan's method of equivalence for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W) was developed in this article.
Abstract: We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.
TL;DR: In this paper, the authors characterized the diameter-2-critical graphs with no antihole of length four, i.e., the complements have no induced 4-cycle and the extremal graphs are complete bipartite graphs with equal size partite sets.
Abstract: A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n
2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.
TL;DR: In this paper, a monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally.
Abstract: We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.
TL;DR: For general extensions characterizations of H-depth are possible using the H-equivalence generalization of Morita theory as mentioned in this paper, and for QF extensions it is derived from the odd depth of the endomorphism ring extension.
Abstract: Let Cn(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if Cn+1(A,B) is isomorphic to a direct summand of a multiple of Cn(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B. In this paper the H-depth of semisimple subalgebra pairs is derived from the transpose inclusion matrix, and for QF extensions it is derived from the odd depth of the endomorphism ring extension. For general extensions characterizations of H-depth are possible using the H-equivalence generalization of Morita theory.
TL;DR: The singular homology and Cech cohomology groups of the Harmonic Archipelago were shown to be not homotopy equivalent to the Griffiths space in this article.
Abstract: We calculate the singular homology and Cech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.
TL;DR: In this article, the authors completely determine the harmonicity properties of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields.
Abstract: Let (M = G/H;g)denote a four-dimensional pseudo-Riemannian generalized symmetric space and g = m + h the corresponding decomposition of the Lie algebra g of G. We completely determine the harmonicity properties of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.
TL;DR: In this article, the authors investigated the structure of X-groups in which every subgroup is permutable or of finite rank, and they showed that every sub group of such a group is a permutable subgroup.
Abstract: In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.
TL;DR: The connection between functional inequalities and functional inequalities is investigated in this paper, where D is a convex subset of a linear space and ρ is an integrable nonnegative function with ∫01ρ(t) dt = 1.
Abstract: The connection between the functional inequalities
$$f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}} {2} + \alpha _J \left( {x - y} \right), x,y \in D,$$
and
$$\int_0^1 {f\left( {tx + \left( {1 - t} \right)y} \right)\rho \left( t \right)dt \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \alpha _{\rm H} \left( {x - y} \right),} x,y \in D,$$
is investigated, where D is a convex subset of a linear space, f: D → ℝ, αH;αJ: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01ρ(t) dt = 1.
TL;DR: In this paper, Brauer's theorem has been applied to stabilization of control systems, including the case when the system is non-controllable, in the context of the Jordan form of A and Rado's deflations.
Abstract: Given a square matrix A, a Brauer’s theorem [Brauer A, Limits for the characteristic roots of a matrix IV Applications to stochastic matrices, Duke Math J, 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues Older and newer results can be considered in the framework of the above theorem In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable Other applications presented are related to the Jordan form of A and Wielandt’s and Hotelling’s deflations An extension of the aforementioned Brauer’s result, Rado’s theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues The same results considered by blocks can be put into the block version framework of the above theorem
TL;DR: In this paper, the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group is computed.
Abstract: We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.
TL;DR: In this paper, the Frobenius direct image of line bundles on toric varieties has been used to construct a strongly exceptional collection of lines in the derived category D====== b¯¯¯¯ (X), where X is the blow up of a line bundle along a multilinear subspace.
Abstract: This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D
b
(X), where X is the blow up of ℙ
n−r
×ℙ
r
along a multilinear subspace ℙ
n−r−1×ℙ
r−1 of codimension 2 of ℙ
n−r
×ℙ
r
. As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.
TL;DR: In this article, the Brill-Noether filtration on moduli spaces of vector bundles on polarized K3 surfaces is studied and a formula for the expected dimension of Brill-noether subschemes is given.
Abstract: Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces.
TL;DR: In this article, Gelfand and Zakharevich considered the problem of local classification of isotypic Kronecker webs and for a given web, they constructed a canonical connection and computed the curvature of the connection in the case of webs of equal rank and corank.
Abstract: An isotypic Kronecker web is a family of corank m foliations \(\{ \mathcal{F}_t \} _{t \in \mathbb{R}P^1 } \) such that the curve of annihilators t ↦ (TxFt)⊥ ∈ Grm(Tx* M) is a rational normal curve in the Grassmannian Grm(Tx*M) at any point x ∈ M. For m = 1 we get Veronese webs introduced by I. Gelfand and I. Zakharevich [Gelfand I.M., Zakharevich I., Webs, Veronese curves, and bi-Hamiltonian systems, J. Funct. Anal., 1991, 99(1), 150–178]. In the present paper, we consider the problem of local classification of isotypic Kronecker webs and for a given web we construct a canonical connection. We compute the curvature of the connection in the case of webs of equal rank and corank. We also show the correspondence between Kronecker webs and systems of ODEs for which certain sets of differential invariants vanish. The equations are given up to contact transformations preserving independent variable. As a particular case, with m = 1 we obtain the correspondence between Veronese webs and ODEs.
TL;DR: In this article, the authors introduced the notion of tame instantons by excluding a kind of pathological monads and showed that the locus I fixme * * of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r+ 1)
Abstract: Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2 We study the moduli space I
n;r
of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2) We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I
*
of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1)