Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Existence of super-simple OA_{\lambda }(3, 5, v)^{\prime }s

It was proved recently that a super-simple orthogonal array (SSOA) of strength \(t\) and index \(\lambda \ge 2\) is equivalent to a minimum detecting array (DTA). In computer software tests in component-based systems, such a DTA can be used to generate test suites that are capable of locating \(d=\lambda -1\) \(t\) -way interaction faults and detect whether there are more than \(d\) interaction faults. It is proved in this paper that an SSOA of strength \(t=3\) , index \(\lambda \ge 2\) and degree \(k=5\) , or an SSOA \(_{\lambda }(3,5,v)\) , exists if and only if \(\lambda \le v\) excepting possibly a handful of cases.     

Coefficients of symmetric square L-functions

Let λsym2f(n) be the n-th coefficient in the Dirichlet series of the symmetric square L-function associated with a holomorphic primitive cusp form f.We prove Ω± results for λsym2f(n) and evaluate the number of positive(resp.,negative) λsym2f(n) in some intervals.     

Forbidden configurations and Steiner designs

Let ${\mathcal{F}}$ be a (0, 1) matrix. A (0, 1) matrix ${\mathcal{M}}$ is said to have ${\mathcal{F}}$ as a configuration if there is a submatrix of ${\mathcal{M}}$ which is a row and column permutation of ${\mathcal{F}}$ . We say that a matrix ${\mathcal{M}}$ is simple if it has no repeated columns. For a given ${v \in \mathbb{N}}$ , we shall denote by forb ${(v, \mathcal{F})}$ the maximum number of columns in a simple (0, 1) matrix with v rows for which ${\mathcal{F}}$ does not occur as a configuration. We say that a matrix ${\mathcal{M}}$ is maximal for ${\mathcal{F}}$ if ${\mathcal{M}}$ has forb ${(v, \mathcal{F})}$ columns. In this paper we show that for certain natural choices of ${\mathcal{F}}$ , forb ${(v, \mathcal{F})\leq\frac{\binom{v}{t}}{t+1}}$ . In particular this gives an extremal characterization for Steiner t-designs as maximal (0, 1) matrices in terms of certain forbidden configurations.     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

Alexander duality in experimental designs

If $F$ is a full factorial design and $D$ is a fraction of $F$ , then for a given monomial ordering, the algebraic method gives a saturated polynomial basis for $D$ which can be used for regression. Consider now an algebraic basis for the complementary fraction of $D$ in $F$ , built under the same monomial ordering. We show that the basis for the complementary fraction is the Alexander dual of the first basis, constructed by shifting monomial exponents. For designs with two levels, the Alexander dual uses the traditional definition for simplicial complexes, while for designs with more than two levels, the dual is constructed with respect to the basis for the design $F$ . This yields various new constructions for designs, where the basis and linear aberration can easily be read from the duality.     

Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus

Let $\mathcal P _\lambda $ be a homogeneous Poisson point process of rate $\lambda $ in the Clifford torus $T^2\subset \mathbb E ^4$ . Let $(f_0, f_1, f_2, f_3)$ be the $f$ -vector of conv $\,\mathcal P _\lambda $ and let $\bar{v}$ be the mean valence of a vertex of the convex hull. Asymptotic expressions for $\mathsf E \, f_1$ , $\mathsf E \, f_2$ , $\mathsf E \, f_3$ and $\mathsf E \, \bar{v}$ as $\lambda \rightarrow \infty $ are proved in this paper.     

On the non-existence of maximal difference matrices of deficiency 1

A \(k\times u\lambda \) matrix \(M=[d_{ij}]\) with entries from a group \(U\) of order \(u\) is called a \((u,k,\lambda )\) -difference matrix over \(U\) if the list of quotients \(d_{i\ell }{d_{j\ell }}^{-1}, 1 \le \ell \le u\lambda ,\) contains each element of \(U\) exactly \(\lambda \) times for all \(i\ne j.\) Jungnickel has shown that \(k \le u\lambda \) and it is conjectured that the equality holds only if \(U\) is a \(p\) -group for a prime \(p.\) On the other hand, Winterhof has shown that some known results on the non-existence of \((u,u\lambda ,\lambda )\) -difference matrices are extended to \((u,u\lambda -1,\lambda )\) -difference matrices. This fact suggests us that there is a close connection between these two cases. In this article we show that any \((u,u\lambda -1,\lambda )\) -difference matrix over an abelian \(p\) -group can be extended to a \((u,u\lambda ,\lambda )\) -difference matrix.     

Optimal mass transport and symmetric representations of their cost functions

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\) . Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\) -convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\) , we show that for every probability measure \(\mu \) on \(X\) , there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \) , and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\) -almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\) , then \(S\) is necessarily a \(\mu \) -measure preserving involution (i.e., \(S^2=I\) ) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \) -almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.     

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.     

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.     

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Consider the nonlinear heat equation $$v_t -\Delta v=|v|^{p-1}v \qquad \qquad \qquad (NLH)$$ in the unit ball of \({\mathbb{R}^2}\) , with Dirichlet boundary condition. Let \({u_{p,\mathcal{K}}}\) be a radially symmetric, sign-changing stationary solution having a fixed number \({\mathcal{K}}\) of nodal regions. We prove that the solution of (NLH) with initial value \({\lambda u_{p,\mathcal{K}}}\) blows up in finite time if |λ ?1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of \({u_{p,\mathcal{K}}}\) and of the linearized operator \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) .     

Salem numbers in dynamics on Kähler threefolds and complex tori

Let \(X\) be a compact Kähler manifold of dimension \(k\!\le \! 4\) and \(f{:}X\!\rightarrow \! X\) a pseudo-automorphism. If the first dynamical degree \(\lambda _1(f)\) is a Salem number, we show that either \(\lambda _1(f)=\lambda _{k-1}(f)\) or \(\lambda _1(f)^2=\lambda _{k-2}(f)\) . In particular, if \({\dim }(X)=3\) then \(\lambda _1(f)=\lambda _2(f)\) . We use this to show that if \(X\) is a complex 3-torus and \(f\) is an automorphism of \(X\) with \(\lambda _1(f)>1\) , then \(f\) has a non-trivial equivariant holomorphic fibration if and only if \(\lambda _1(f)\) is a Salem number. If \(X\) is a complex 3-torus having an automorphism \(f\) with \(\lambda _1(f)=\lambda _2(f)>1\) but is not a Salem number, then the Picard number of \(X\) must be 0, 3 or 9, and all these cases can be realized.     

Remarks on the existence of solutions to some quasilinear elliptic problems involving the Hardy-Leray potential

We study the solvability of the quasilinear problem $$\begin{aligned} -\Delta _p u =\frac{u^q }{|x|^p}+g(\lambda , x, u) \quad u>0 \quad \text{ in}\;\Omega , \end{aligned}$$ with $u=0$ on $\partial \Omega $ , where $-\Delta _p(\cdot )$ is the $p$ -Laplacian operator, $q>0, 1<p<N$ and $\Omega $ a smooth bounded domain in $\mathbb R ^N$ . We consider the following cases:

1. $g(\lambda ,x,u)\equiv 0$ ;
2. $g(\lambda ,x,u)=\lambda f(x)u^r$ , with $\lambda >0$ and $f(x) \gneq 0$ belonging to $L^{\infty }(\Omega )$ and $0 \le r<p-1$ .

In the case $(i)$ , the existence of solutions depends on the location of the origin in the domain, on the geometry of the domain and on the exponent $q$ . On the other hand, in the case $(ii)$ , the existence of solutions only depends on the position of the origin and on the coefficient $\lambda $ , but does not depend either on the exponent $q$ or on the geometry of $\Omega $ .     

Geodesics,distance, and the CAT(0) property for the manifold of Riemannian metrics

Let $\mathcal F ^a_\lambda $ be the PBW degeneration of the flag varieties of type $A_{n-1}$ . These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$ . We prove that $\mathcal F ^a_\lambda $ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda $ of $\mathcal F ^a_\lambda $ . The varieties $R_\lambda $ can be viewed as towers of successive $\mathbb{P }^1$ -fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda $ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda $ . Using the Atiyah–Bott–Lefschetz formula for $R_\lambda $ , we compute the $q$ -characters of the highest weight $\mathfrak sl _n$ -modules.     

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \mathbb R ^d$ be a smooth convex set and let $\mathcal{P }_{\lambda }$ be a Poisson point process on $\mathbb R ^d$ of intensity ${\lambda }$ . The convex hull of $\mathcal{P }_{\lambda }\cap K$ is a random convex polytope $K_{\lambda }$ . As ${\lambda }\rightarrow \infty $ , we show that the variance of the number of $k$ -dimensional faces of $K_{\lambda }$ , when properly scaled, converges to a scalar multiple of the affine surface area of $K$ . Similar asymptotics hold for the variance of the number of $k$ -dimensional faces for the convex hull of a binomial process in $K$ .     

Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

By analysing some explicit examples we investigate the positivity and the non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator \(\varDelta +\lambda I\) as \(\lambda \) varies. It is known that the semigroup is positive if \(\lambda <\lambda _1\) , where \(\lambda _1\) is the principal eigenvalue of \(-\varDelta \) with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on \(\lambda >\lambda _1\) .     

Complexity of unconstrained L_2-L_p minimization

We consider the unconstrained $L_q$ - $L_p$ minimization: find a minimizer of $\Vert Ax-b\Vert ^q_q+\lambda \Vert x\Vert ^p_p$ for given $A \in R^{m\times n}$ , $b\in R^m$ and parameters $\lambda >0$ , $p\in [0, 1)$ and $q\ge 1$ . This problem has been studied extensively in many areas. Especially, for the case when $q=2$ , this problem is known as the $L_2-L_p$ minimization problem and has found its applications in variable selection problems and sparse least squares fitting for high dimensional data. Theoretical results show that the minimizers of the $L_q$ - $L_p$ problem have various attractive features due to the concavity and non-Lipschitzian property of the regularization function $\Vert \cdot \Vert ^p_p$ . In this paper, we show that the $L_q$ - $L_p$ minimization problem is strongly NP-hard for any $p\in [0,1)$ and $q\ge 1$ , including its smoothed version. On the other hand, we show that, by choosing parameters $(p,\lambda )$ carefully, a minimizer, global or local, will have certain desired sparsity. We believe that these results provide new theoretical insights to the studies and applications of the concave regularized optimization problems.