Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

On the Largest Graph-Lagrangian of 3-Graphs with Fixed Number of Edges

The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the \({r}\) -graph with \(m\) edges formed by taking the first \(\textit{m}\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({r}\) has the largest Graph-Lagrangian of all \(r\) -graphs with \(m\) edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed \(3\) -graphs with \(m\) edges is at most the Graph-Lagrangian of the \(\mathrm 3 \) -graph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({3}\) .     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

The Ubiquity of the Orders of Fractional Semifields of Even Characteristic

To each non-square integer \(2^{2N+1}\ge 2^5\) there correspond semifields \(D\) of order of \(2^{2N+1}\) that contain \(\text{ GF}(4)\) . Hence there exist affine planes for each non-square order \(2^{2N+1}\ge 2^{5}\) that contain subaffine planes of order \(2^2\) . Moreover, there also exists semifields \(D_1\) and \(D_2\) , with \(|D_1|= |D_2| =|D|\) such that \(D_1\) is commutative and \(D_2\) is non-commutative but neither \(D_1\) nor \(D_2\) contains \(\text{ GF}(4)\) .     

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Large collections of curves pairwise intersecting exactly once

Let \(\Omega =(\omega _{j})_{j\in I}\) be a maximum size collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus \(g\) surface \(S_{g}\) , such that \(\omega _{i}\) and \(\omega _{j}\) intersect exactly once for \(i\ne j\) . We show that for \(g\ge 3\) , there exists atleast two such collections up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran. As a consequence, we show that the automorphism group of the systole graph for \(S_{g}, g\ge 3\) (whose vertices are isotopy classes of simple closed curves, and whose edges correspond to pairs of curve intersecting once) does not act transitively on maximal complete subgraphs.     

Regularity and uniqueness of a class of biharmonic map heat flows

We consider a class of weak solutions of the heat flow of biharmonic maps from \(\Omega \subset \mathbb{R }^n\) to the unit sphere \(\mathbb{S }^L\subset \mathbb{R }^{L+1}\) , that have small renormalized total energies locally at each interior point. For any such a weak solution, we prove the interior smoothness, and the properties of uniqueness, convexity of hessian energy, and unique limit at \(t=\infty \) . We verify that any weak solution \(u\) to the heat flow of biharmonic maps from \(\Omega \) to a compact Riemannian manifold \(N\) without boundary, with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12), has small renormalized total energy locally and hence enjoys both the interior smoothness and uniqueness property. Finally, if an initial data \(u_0\in W^{2,r}(\mathbb{R }^n, N)\) for some \(r>\frac{n}{2}\) , then we establish the local existence of heat flow of biharmonic maps \(u\) , with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12).     

Classification of finite Morse index solutions for Hénon type elliptic equation -\Delta u = |x|^{\alpha } u_{+}^{p}

In this paper we investigate the non-autonomous elliptic equations \(-\Delta u = |x|^{\alpha } u_{+}^{p}\) in \( \mathbb{R }^{N}\) and in \( \mathbb{R }_+^{N}\) with the Dirichlet boundary condition, with \(N \ge 2\) , \(p>1\) and \(\alpha >-2\) . We consider the weak solutions with finite Morse index and obtain some classification results.     

Refined methods for the identifiability of tensors

We prove that the general tensor of size \(2^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.9997\frac{2^n}{n+1}\) (the constant 1 being the optimal value). Similarly, the general tensor of size \(3^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.998\frac{3^n}{2n+1}\) (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.     

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

New M-ary sequences with low autocorrelation from interleaved technique

Let \(p\) and \(q\) be two odd primes with \(p=Mf+1\) and \(M\) is even. A new construction of \(M\) -ary sequences of period \(pq\) with low periodic autocorrelation is presented in this paper based on interleaving the \(M\) -ary power residue sequence of period \(p\) according to the quadratic residue with respect to \(q\) . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if \(M=2\) . For \(M=4\) , a new class of quaternary sequences of period \(pq\) with maximal nontrivial autocorrelation value being either \(\sqrt{5}\) or \(3\) is obtained. This achieves the best known results for such kind of quaternary sequences.     

A study of saturated tensor cone for symmetrizable Kac–Moody algebras

Let \(\mathfrak {g}\) be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra \(\mathfrak {h}\) and the Weyl group \(W\) . Let \(P_+\) be the set of dominant integral weights. For \(\lambda \in P_+\) , let \(L(\lambda )\) be the integrable, highest weight (irreducible) representation of \(\mathfrak {g}\) with highest weight \(\lambda \) . For a positive integer \(s\) , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of \(\Gamma _s\) in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by \(\Gamma _s\) . These inequalities are indexed by products in \(H^*(G^{\mathrm{min }}/B; \mathbb {Z})\) for \(B\) the standard Borel subgroup, where \(G^{\mathrm{min }}\) is the ‘minimal’ Kac-Moody group with Lie algebra \(\mathfrak {g}\) . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that \(\mathfrak {g}\) is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer \(d>0\) is a saturation factor for \(A^{(1)}_1\) and 4 is a saturation factor for \(A^{(2)}_2\) .     

Weak lower semicontinuity for polyconvex integrals in the limit case

We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps \(u:\Omega \subset \mathbb{R }^n\rightarrow \mathbb{R }^m\) in \(W^{1,m}\) , \(2\le m\le n\) , bounded in \(W^{1,m-1}\) and convergent in \(L^1\) under mild technical conditions but without any extra coercivity assumption on the integrand.     

Approximate Unitary Equivalence to Skew Symmetric Operators

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called skew symmetric if \(T\) can be represented as a skew symmetric matrix relative to some orthonormal basis for \(\mathcal {H}\) . In this paper, we study the approximation of skew symmetric operators and provide a \(C^*\) -algebra approach to skew symmetric operators. We classify up to approximate unitary equivalence those skew symmetric operators \(T\in \mathcal {B(H)}\) satisfying \(C^*(T)\cap \mathcal {K(H)}=\{0\}\) . This is used to characterize when a unilateral weighted shift with nonzero weights is approximately unitarily equivalent to a skew symmetric operator.     

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle \(T^* {\mathcal {M}}\) of a differentiable manifold \({\mathcal {M}}\) enjoys a canonical symplectic form \(\Omega ^*\) . If \(({\mathcal {M}},\mathrm{J} ,g,\omega )\) is a pseudo-Kähler or para-Kähler \(2n\) -dimensional manifold, we prove that the tangent bundle \(T{\mathcal {M}}\) also enjoys a natural pseudo-Kähler or para-Kähler structure \(({\tilde{\hbox {J}}},\tilde{g},\Omega )\) , where \(\Omega \) is the pull-back by \(g\) of \(\Omega ^*\) and \(\tilde{g}\) is a pseudo-Riemannian metric with neutral signature \((2n,2n)\) . We investigate the curvature properties of the pair \(({\tilde{\hbox {J}}},\tilde{g})\) and prove that: \(\tilde{g}\) is scalar-flat, is not Einstein unless \(g\) is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if \(g\) has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if \(n=1\) and \(g\) has constant curvature, or \(n>2\) and \(g\) is flat. We also check that (i) the holomorphic sectional curvature of \(({\tilde{\hbox {J}}},\tilde{g})\) is not constant unless \(g\) is flat, and (ii) in \(n=1\) case, that \(\tilde{g}\) is never anti-self-dual, unless conformally flat.