Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Biharmonic PNMC submanifolds in spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$ . Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in $\mathbb{S}^{n}$ , their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in $\mathbb{S}^{n}$ is of 1-type or 2-type if and only if it has constant mean curvature f=1 or f∈(0,1), respectively; and (ii) there are no proper-biharmonic 3-type submanifolds with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$ .     

Computable representations for convex hulls of low-dimensional quadratic forms

Let ${\mathcal{C}}$ be the convex hull of points ${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$ . Representing or approximating ${\mathcal{C}}$ is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and ${\mathcal{F}}$ is a simplex, then ${\mathcal{C}}$ has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and ${\mathcal{F}}$ is a box, then ${\mathcal{C}}$ has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ when ${\mathcal{F}\subset\Re^2}$ is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ . When n = 3 and ${\mathcal{F}}$ is a box, we show that a representation for ${\mathcal{C}}$ can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.     

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H ² over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H ² with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T _z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason.     

On the distinctness of modular reductions of primitive sequences over Z/(232−1)

This paper studies the distinctness of modular reductions of primitive sequences over ${\mathbf{Z}/(2^{32}-1)}$ . Let f(x) be a primitive polynomial of degree n over ${\mathbf{Z}/(2^{32}-1)}$ and H a positive integer with a prime factor coprime with 2³²?1. Under the assumption that every element in ${\mathbf{Z}/(2^{32}-1)}$ occurs in a primitive sequence of order n over ${\mathbf{Z}/(2^{32}-1)}$ , it is proved that for two primitive sequences ${\underline{a}=(a(t))_{t\geq 0}}$ and ${\underline{b}=(b(t))_{t\geq 0}}$ generated by f(x) over ${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$ if and only if ${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$ for all t ≥ 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

Parabolicity of maximal surfaces in Lorentzian product spaces

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form ${M^2 \times \mathbb {R}_1}$ , where M ² is a connected Riemannian surface with non-negative Gaussian curvature and ${M^2 \times \mathbb {R}_1}$ is endowed with the Lorentzian product metric ${{\langle , \rangle}={\langle , \rangle}_M-dt^2}$ . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain ${\Omega \subseteq M}$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi–Bernstein result for entire maximal graphs in ${M^2 \times \mathbb {R}_1}$ .     

On the L p -distortion of finite quotients of amenable groups

We study the L ^p -distortion of finite quotients of amenable groups. In particular, for every ${2\leq p < \infty}$ , we prove that the ? ^p -distortions of the groups ${C_2\wr C_n}$ and ${C_{2^n}\rtimes C_n}$ are in ${\Theta((\log n)^{1/p}),}$ and that the ? ^p -distortion of ${C_n^2 \rtimes_A \mathbf{Z}}$ , where A is the matrix ${{\left({\small\begin{array}{cc}2 & 1 \\ 1 & 1 \end{array}} \right)}}$ is in ${\Theta((\log \log n)^{1/p}).}$      

Wintgen Ideal Surfaces in Four-dimensional Neutral Indefinite Space Form {R^4_2(c)}

For an oriented space-like surface M in a four-dimensional indefinite space form ${R^4_2(c)}$ , there is a Wintgen type inequality; namely, the Gauss curvature K, the normal curvature K ^D and mean curvature vector H of M in ${R^4_2(c)}$ satisfy the general inequality: ${K+K^D \geq \langle H,H \rangle+c}$ . An oriented space-like surface in ${R^4_2(c)}$ is called Wintgen ideal if it satisfies the equality case of the inequality identically. In this paper, we study Wintgen ideal surfaces in ${R^4_2(c)}$ . In particular, we classify Wintgen ideal surfaces in ${R^4_2(c)}$ with constant Gauss and normal curvatures. We also completely classify Wintgen ideal surfaces in ${\mathbb E^4_2}$ satisfying |K| = |K ^D | identically.     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

We prove a global implicit function theorem. In particular we show that any Lipschitz map ${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$ (with n-dim. image) can be precomposed with a bi-Lipschitz map ${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$ such that ${f \circ \bar{g}}$ will satisfy, when we restrict to a large portion of the domain ${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$ , that ${f \circ \bar{g}}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map ${\bar{g}}$ distorts ${\mathbb{R}^{n+m}}$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C ¹ map from [0, 1]³ onto [0, 1]² with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function ${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$ gives rise to a large (in an appropriate sense) subset ${E \subset [0,1]^{n}}$ such that ${f|_E}$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of ${\mathbb{R}^{n}}$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in ${\mathbb{R}^{D}}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in ${\mathbb{R}^{D}}$ . This was previously known only for D ≥ 2n?+?1 by a result of David and Semmes.     

Some relations on the ordering of trees by minimal energies between subclasses of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. A tree is said to be non-starlike if it has at least two vertices with degree more than 2. A caterpillar is a tree in which a removal of all pendent vertices makes a path. Let $\mathcal{T}_{n,d}$ , $\mathbb{T}_{n,p}$ be the set of all trees of order n with diameter d, p pendent vertices respectively. In this paper, we investigate the relations on the ordering of trees and non-starlike trees by minimal energies between $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ . We first show that the first two trees (non-starlike trees, resp.) with minimal energies in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same for 3≤d≤n?2 (3≤d≤n?3, resp.). Then we obtain that the trees with third-minimal energy in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same when n≥11, 3≤d≤n?2 and d≠8; and the tree with third-minimal energy in $\mathcal{T}_{n,8}$ is the caterpillar with third-minimal energy in $\mathbb{T}_{n,n-7}$ for n≥11.     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

A Bernstein Property of a Class of Fourth Order Complex Partial Differential Equations

Let ${f:\Omega \rightarrow \mathbb{R}}$ be a smooth function on a domain   ${\Omega \subset \mathbb{C}^n}$ with its Hessian matrix ${\left( \frac{\partial^2 f}{\partial z^i \partial\bar{z}^j}\right)}$ positive Hermitian. In this paper, we investigate a class of partial differential equations $$\Delta \ln \det (f_{i\bar{j}}) = \beta \;\| \text{grad} \ln \det (f_{i\bar{j}}) \|^2, $$ where Δ and ${\| \cdot \|}$ are the Laplacian and tensor norm, respectively, with respect to the metric ${G = \sum f_{i\bar{j}} \,dz^i \otimes d\bar{z}^j}$ , and β > 1 is some real constant depending on the dimension n. We prove that the above PDEs have a Bernstein property when the metric G is complete, provided that ${\det (f_{i\bar{j}})}$ and the Ricci curvature are bounded.     

Some new classes of Kadison-Singer lattices in Hilbert spaces

Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).     

Nielsen coincidence,fixed point and root theories of n-valued maps

Let ${(\phi, \psi)}$ be an (m, n)-valued pair of maps ${\phi, \psi : X \multimap Y}$ , where ${\phi}$ is an m-valued map and ${\psi}$ is n-valued, on connected finite polyhedra. A point ${x \in X}$ is a coincidence point of ${\phi}$ and ${\psi}$ if ${\phi(x) \cap \psi(x) \neq \emptyset}$ . We define a Nielsen coincidence number ${N(\phi : \psi)}$ which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to ${(\phi, \psi)}$ . We calculate ${N(\phi : \psi)}$ for all (m, n)-valued pairs of maps of the circle and show that ${N(\phi : \psi)}$ is a sharp lower bound in that setting. Specifically, if ${\phi}$ is of degree a and ${\psi}$ of degree b, then ${N(\phi : \psi) = \frac{|an - bm|}{\langle m, n \rangle}}$ , where ${\langle m, n \rangle}$ is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for n-valued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.