Locally 2-fold symmetric manifolds are locally symmetric

A manifold is locally k-fold symmetric if for any point and any k-dimensional vector subspace tangent to this point, there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that k-dimensional vector subspace is minus the identity. We show that for \(k \ge 2\), Riemannian, pseudoriemannian, and Finslerian locally k-fold symmetric manifolds are locally symmetric.     

On Defining Sets of Full Designs with Block Size Three

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d _D of D there exists a minimal defining set d _F of F such that \({d_D = d_F\cap D}\). The unique simple design with parameters \({{\left(v,k, {v-2\choose k-2}\right)}}\) is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.     

A new class for large sets of almost Hamilton cycle decompositions

A k-cycle system of order v with index λ, denoted by CS(v, k, λ), is a collection A of k-cycles (blocks) of K _v such that each edge in K _v appears in exactly λ blocks of A. A large set of CS(v, k, λ)s is a partition of the set of all k-cycles of K _v into CS(v, k, λ)_s, and is denoted by LCS(v, k, λ). A (v ?1)-cycle in K _v is called almost Hamilton. The completion of the existence problem for LCS(v, v?1, λ) depends only on one case: all v ≥ 4 for λ = 2. In this paper, it is shown that there exists an LCS(v, v ? 1, 2) for all v ≡ 2 (mod 4), v ≥ 6.     

Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S<Superscript>3</Superscript> × S<Superscript>3</Superscript>

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S³ × S³ is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((?h)(v, v, v), Jv) = λ holds for all unit tangent vector v.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

The existence of <Emphasis Type="Italic">J</Emphasis>-holomorphic curves in almost Hermitian manifolds

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and \(f:\Sigma \rightarrow M\) be an injective immersion. We prove that if the \(L_p\) functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.     

Optimum super-simple mixed covering arrays of type <Emphasis Type="Italic">a</Emphasis><Superscript>1</Superscript><Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>−1</Superscript>

A mixed covering array (MCA) of type (_v 1, v ₂,..., v _k ), denoted by MCA_λ (N; t, k, (v ₁, v ₂,..., v _k )), is an N × k array with entries in the i-th column from a set _{V i} of v _i symbols and has the property that each N × t sub-array covers all the t-tuples at least λ times, where 1 ≤ i ≤ k. An MCA _λ (N; t, k, (_v 1, v ₂,..., v _k )) is said to be super-simple, if each of its N × (t + 1) sub-arrays contains each (t + 1)-tuple at most once. Recently, it was proved by Tang, Yin and the author that an optimum super-simple MCA of type (a, b, b,..., b) is equivalent to a mixed detecting array (DTA) of type (a, b, b,..., b) with optimum size. Such DTAs can be used to generate test suites to identify and determine the interaction faults between the factors in a component-based system. In this paper, some combinatorial constructions of optimum super-simple MCAs of type (a, b, b,..., b) are provided. By employing these constructions, some optimum super-simple MCAs are then obtained. In particular, the spectrum across which optimum super-simple MCA₂(2b ²; 2, 4, (a, b, b, b))′s exist, is completely determined, where 2 ≤ a ≤ b.     

On the Normal Jacobi Operator of <Emphasis Type="Italic">CR</Emphasis>-Hypersurfaces in Conformal Kenmotsu Space Forms

We study the CR-hypersurfaces of a conformal Kenmotsu space form with a ξ-parallel normal Jacobi operator. We also present an illustrative example of a three-dimensional conformal Kenmotsu manifold that is not Kenmotsu.     

Flag-transitive point-primitive automorphism groups of non-symmetric 2-(<Emphasis Type="Italic">v</Emphasis>, <Emphasis Type="Italic">k</Emphasis>, 3) designs

In this paper, we show that if \({\mathcal {D}}\) is a non-trivial non-symmetric 2-(v, k, 3) design admitting a flag-transitive point-primitive automorphism group G, then G must be an affine or almost simple group.     

Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

In the present paper we prove a criterion of Lip ^k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Fréchet space F. We establish that a manifold is Lip ^k -paracompact if and only if the model space F is paracompact and Lip ^k -normal. We prove a sufficient condition for existence of Lip ^k -partition of a unity on a manifold of class Lip ^k .     

Lower bounds on the signed total k-domination number of graphs

Let G be a graph with vertex set V(G). For any integer k ≥ 1, a signed total k-dominating function is a function f: V(G) → {?1, 1} satisfying ∑_x∈N(v)f(x) ≥ k for every v ∈ V(G), where N(v) is the neighborhood of v. The minimum of the values ∑_v∈V(G)f(v), taken over all signed total k-dominating functions f, is called the signed total k-domination number. In this note we present some new sharp lower bounds on the signed total k-domination number of a graph. Some of our results improve known bounds.     

Generalized competition index of primitive digraphs

For any positive integers k and m, the k-step m-competition graph C _m ^k (D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v₁, v₂, · · ·, v _m in D such that there are directed walks of length k from x to v _i and from y to v _i for all 1 ≤ i ≤ m. The m-competition index of a primitive digraph D is the smallest positive integer k such that C _m ^k (D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.     

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume.The related questions that will also be studied are the following: given a contractible k-dimensional sphere in M ⁿ , how “fast” can this sphere be contracted to a point, if π _i (M ⁿ )={0} for 1≤i<k. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible k-dimensional sphere in a (k-1)-connected manifold M ⁿ providing that M ⁿ is not k-connected?     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Lower Bounds for the Degree of a Branched Covering of a Manifold

The problem of finding new lower bounds for the degree of a branched covering of a manifold in terms of the cohomology rings of this manifold is considered. This problem is close to M. Gromov’s problem on the domination of manifolds, but it is more delicate. Any branched (finite-sheeted) covering of manifolds is a domination, but not vice versa (even up to homotopy). The theory and applications of the classical notion of the group transfer and of the notion of transfer for branched coverings are developed on the basis of the theory of n-homomorphisms of graded algebras.The main result is a lemma imposing conditions on a relationship between the multiplicative cohomology structures of the total space and the base of n-sheeted branched coverings of manifolds and, more generally, of Smith–Dold n-fold branched coverings. As a corollary, it is shown that the least degree n of a branched covering of the N-torus T ^N over the product of k 2-spheres and one (N ? 2k)-sphere for N ≥ 4k + 2 satisfies the inequality n ≥ N ? 2k, while the Berstein–Edmonds well-known 1978 estimate gives only n ≥ N/(k + 1).     

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps.     

Tensor convolutions and Hankel tensors

Let A be an mth order n-dimensional tensor, where m, n are some positive integers and N:= m(n?1). Then A is called a Hankel tensor associated with a vector v ∈ ?^N+1 if A_σ = v _k for each k = 0, 1,...,N whenever σ = (i₁,..., i_m) satisfies i₁ +· · ·+i_m = m+k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.