Affine symplectic geometry I: Applications to geometric inequalities

We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ³. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic oscillator operator inL ²(M), whereM is a Hadamard manifold. This article is dedicated to my dear friend Julia Rashba     

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q ²), the extended lines of L cover a non-singular Hermitian surface H ? H(3, q ²) of PG(3, q ²). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q ² + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q ³ + 1)-spans of the Hermitian variety H(5, q ²).     

Disconnectedness of sublevel sets of some Riemannian functionals

LetM be a compact manifold of dimension greater than four. Denote byRiem(M) the space of Riemannian structures onM (i.e. of isometry classes of Riemannian metrics onM) endowed with the Gromov-Hausdorff metric. LetRiem (M) Riem(M) be its subset formed by all Riemannian structures such that vol()=1 andinj() , whereinj() denotes the injectivity radius of.We prove that for all sufficiently small positive the spaceRiem (M) is disconnected. Moreover, if is sufficiently small, thenRiem (M) is representable as the union of two non-empty subsetsA andB such that the Gromov-Hausdorff distance between any element ofA and any element ofB is greater than/9. We also prove a more general result with the following informal meaning: There exist two Riemannian structures of volume one and arbitrarily small injectivity radius onM such that any continuous path (and even any sequence of sufficiently small jumps) in the space of Riemannian structures of volume one onM connecting these Riemannian structures must pass through Riemannian structures of injectivity radius uncontrollably smaller than the injectivity radii of these two Riemannian structures.These results can be generalized for at least some four-dimensional manifolds. The technique used in this paper can also be used to prove the disconnectedness of many other subsets of the space of Riemannian structures onM formed by imposing various constraints on curvatures, volume, diameter, etc.This work was partially supported by the New York University Research Challenge Fund grant, by NSF grant DMS 9114456 and by the NSERC operating grant OGP0155879.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Some properties of non-compact complete Riemannian manifolds

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold M. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on M, for any q∈(0,∞), every non-negative Lq subharmonic function is constant under a natural decay condition on the Ricci curvature.     

Properties of convergence for the q-Meyer-König and Zeller operators

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t²,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))_n?1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C¹[0,1], we show that the rate of convergence for the Meyer-König and Zeller operators is o(qn) if and only if

     

Generalized Obata theorem and its applications on foliations

We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension q?2 and a bundle-like metric gM. Then (M,F) is transversally isometric to (Sq(1/c),G), where Sq(1/c) is the q-sphere of radius 1/c in (q+1)-dimensional Euclidean space and G is a discrete subgroup of the orthogonal group O(q), if and only if there exists a non-constant basic function f such that for all basic normal vector fields X, where c is a positive constant and ∇ is the connection on the normal bundle. By the generalized Obata theorem, we classify such manifolds which admit transversal non-isometric conformal fields.     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

On the Geometry of Hermitian Matrices of Order Three Over Finite Fields

Some geometry of Hermitian matrices of order three over GF(q²) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M₇³of PG(8,q ) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM₇³ turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q²) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q²) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q²) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q²) new mixed partitions of PG(8,q ) into caps and linear subspaces are given.     

Inequalities for scalar-valued linear operators that extend to their vector-valued analogues

A theorem of Marcinkiewicz and Zygmund asserts that a linear operator satisfying a strong type (L^r, L^q) inequality with norm M automatically extends to a vector-valued operator satisfying a strong type (L^r(l^v), L^q(l^v)) inequality with norm not exceeding C_{r, q}^(γ)M. In this paper, this theorem is proved in a more general context by replacing the L^q metric with a more general class of metrics. In doing so, the theorem of Marcinkiewicz and Zygmund is not only extended to more general contexts, but improvements of that theorem are also realized. In particular, our results show that operators satisfying weak type inequalities automatically extend to their vector-valued analogues; also the constant C_{r, q}^(γ) may be taken as one in the theorem of Marcinkiewicz and Zygmund whenever q ? r, and this includes most cases of interest.     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

Counting colored planar maps: Algebraicity results

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χM(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q≠0,4 is of the form 2+2cos(jπ/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial PM(q,ν), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutte?s remarkable solution of properly q-colored triangulations.     

On the chromatic number of q-Kneser graphs

We show that the q-Kneser graph qK _2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q ^k ?+?q ^k?1 for k?=?3 and for k < q log q ? q. We obtain detailed results on maximal cocliques for k = 3.     

Knots and topologically transitive flows on 3-manifolds

Suppose that ? is a nonsingular (fixed point free) flow on a smooth three-dimensional manifold M. Suppose the orbit though a point p∈M is dense in M. Let D be an imbedded disk in M containing p which is transverse to the flow. Suppose that q∈D is a point in the forward orbit of p. Under certain assumptions on M, which include the case M=S³, we prove that if q is sufficiently close to p then the orbit segment from p to q together with a compact segment in D from p to q forms a nontrivial prime knot in M.     

A generalized numerical range: the range of a constrained sesquilinear form

Let M_n denote the algebra of all nxn complex matrices. For a given q?C with ∣Q∣≤1, we define and denote the q-numerical range of A?M_n by

W_q (A)={x ^? Ay:x,y?C ⁿ , _{x ^? x?y ^? y=1,x ^? y=q} }

The q-numerical radius is then given by r_q (A)=sup{∣z∣:z?W _q (A)}. When q=1,W _q (A) and r _q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W _q (A) is the inner q-numerical radius defined by [rtilde] _q (A)=inf{∣z∣:z?W _q (A)}

In this paper, we describe some basic properties of W _q (A), extending known results on the classical numerical range. We also study the properties of r_q considered as a norm (seminorm if q=0) on M_n .Finally, we characterize those linear operators L on M_n that leave W_q ,r_q of [rtilde]_q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.     

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 2²ⁿ⁺¹ and either q ? 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F⁴(q), where q = 2 ⁿ and either q⁴ + 1 or q⁴ ? q² + 1 is a prime number.     

A geometric criterion for compactness of invariant subspaces

Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M. We show, under general conditions, that the Ω-invariant subspace A _Ω of a normed vector space ${A\hookrightarrow L^q(M)}$ A ? L q ( M ) is compactly embedded into L ^q (M) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity.     

A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).     

NP-Completeness of the linear complementarity problem

We consider the linear complementarity problem (q, M) for which the data are the integer column vectorq εR ⁿ and the integer square matrixM of ordern. GLCP is the decision problem: Does (q, M) have a solution? We show that GLCP is NP-complete in the strong sense.