Existence of positive periodic solutions for second-order functional differential equations

In this paper, we show the existence of positive $T$ -periodic solutions of second-order functional differential equations $u^{\prime \prime }(t)-\rho ^2u(t)+\lambda g(t)f(u(t-\tau (t)))=0,\ \ t\in \mathbb R , $ where $\rho >0$ is a constant, $g\in C(\mathbb R ,[0,\infty ))$ , $\tau \in C(\mathbb R ,\mathbb R )$ are $T$ -periodic functions, $f\in C([0,\infty ),[0,\infty ))$ and $\lambda $ is a positive parameter. Our approach based on global bifurcation theorem.     

The distribution of the zeros of the Goss zeta-function for A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)

Let $F$ be a global function field over a finite constant field and $\infty $ a place of $F$ . The ring $A$ of functions regular away from $\infty $ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta $ -function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty $ of $F$ at $\infty $ . He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$ . In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in  $F_\infty $ . In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$ —this $A$ has class number $1$ , it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty $ is $\mathbb{F }\,\!{}_2$ -rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty $ is a rational place of $F$ , we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus  $2$ .     

The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let ${\mathcal{S} = \{T(s): 0 \leq s < \infty\}}$ be a nonexpansive semigroup on E such that ${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$ , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and ${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$ . When the sequences of real numbers {α _n } and {t _n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to ${\tilde{x}}$ , where ${\tilde{x}}$ is the unique solution in ${Fix(\mathcal{S})}$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.     

Quasirandom permutations are characterized by 4-point densities

For permutations ${\pi}$ and ${\tau}$ of lengths ${|\pi|\le|\tau|}$ , let ${t(\pi,\tau)}$ be the probability that the restriction of ${\tau}$ to a random ${|\pi|}$ -point set is (order) isomorphic to ${\pi}$ . We show that every sequence ${\{\tau_j\}}$ of permutations such that ${|\tau_j|\to\infty}$ and ${t(\pi,\tau_j)\to 1/4!}$ for every 4-point permutation ${\pi}$ is quasirandom (that is, ${t(\pi,\tau_j)\to 1/|\pi|!}$ for every ${\pi}$ ). This answers a question posed by Graham.     

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.     

The classification of (m, \rho)-quasi-Einstein manifolds

If there exist a smooth function f on $(M^n, g)$ and three real constants $m,\rho ,\lambda $ ( $0<m\le \infty $ ) such that $$\begin{aligned} R_{ij}+f_{ij}-\frac{1}{m}f_if_j=(\rho R+\lambda ) g_{ij}, \end{aligned}$$ we call $(M^n,g)$ a $(m,\rho )$ -quasi-Einstein manifold. Here $R_{ij}$ is the Ricci curvature and R is the scalar curvature of the metric g, respectively. This is a special case of the so-called generalized quasi-Einstein manifold which was a natural generalization of gradient Ricci solitons associated with the Hamilton’s Ricci flow. In this paper, we first obtain some rigidity results for compact $(m,\rho )$ -quasi-Einstein manifolds. Then, we give some classifications under the assumption that the Bach tensor of $(M^n,g)$ is flat.     

On the existence of Evans potentials

It is shown that, for every noncompact parabolic Riemannian manifold $X$ and every nonpolar compact $K$ in  $X$ , there exists a positive harmonic function on $X\setminus K$ which tends to $\infty $ at infinity. (This is trivial for $\mathbb{R }$ , easy for  $\mathbb{R }^2$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space  $X$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set  $K$ , there is a symmetric (positive) Green function for $X\setminus K$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $\left[0,\infty \right)\times \{0\}, \left[0,\infty \right)\times \{1\}$ , and the line segments $\{n\}\times [0,1], n=0,1,2,\dots $ ).     

On the zeros of Meixner polynomials

We investigate the zeros of a family of hypergeometric polynomials $M_n(x;\beta ,c)=(\beta )_n\,{}_2F_1(-n,-x;\beta ;1-\frac{1}{c})$ , $n\in \mathbb N ,$ known as Meixner polynomials, that are orthogonal on $(0,\infty )$ with respect to a discrete measure for $\beta >0$ and $0<c<1.$ When $\beta =-N$ , $N\in \mathbb N $ and $c=\frac{p}{p-1}$ , the polynomials $K_n(x;p,N)=(-N)_n\,{}_2F_1(-n,-x;-N;\frac{1}{p})$ , $n=0,1,\ldots , N$ , $0<p<1$ are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials $M_n(x;\beta ,c)$ , $c<0$ and $n<1-\beta $ , the quasi-orthogonal polynomials $M_n(x;\beta ,c)$ , $-k<\beta <-k+1$ , $k=1,\ldots ,n-1$ and $0<c<1$ or $c>1,$ as well as the polynomials $K_{n}(x;p,N)$ with non-Hermitian orthogonality for $0<p<1$ and $n=N+1,N+2,\ldots $ . We also show that the polynomials $M_n(x;\beta ,c)$ , $\beta \in \mathbb R $ are real-rooted when $c\rightarrow 0$ .     

Parabolic power concavity and parabolic boundary value problems

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .     

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [0,\infty [

In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $\alpha ,\beta \rightarrow \infty $ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces ${H}_d(\mathbb F )$ with growing dimensions $d$ over the fields $\mathbb F =\mathbb R ,\ \mathbb C $ or the quaternions $\mathbb H $ .     

Optimal relationships between L^p-norms for the Hardy operator and its dual

We obtain sharp two-sided inequalities between $L^p$ -norms $(1<p<\infty )$ of functions $\textit{Hf}$ and $H^*f$ , where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty ).$ In an equivalent form, it gives sharp constants in the two-sided relationships between $L^p$ -norms of functions $H\varphi -\varphi $ and $\varphi $ , where $\varphi $ is a nonnegative nonincreasing function on $(0,+\infty )$ with $\varphi (+\infty )=0.$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $p=2k \,\,(k\in \mathbb N )$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $p\ge 2$ , and gives a sharp version of this result for $1<p<2$ .     

Convergence of the eigenvalues of the p-Laplace operator as p goes to 1

Let $(\lambda ^k_p)_k$ be the usual sequence of min-max eigenvalues for the $p$ -Laplace operator with $p\in (1,\infty )$ and let $(\lambda ^k_1)_k$ be the corresponding sequence of eigenvalues of the 1-Laplace operator. For bounded $\Omega \subseteq \mathbb{R }^n$ with Lipschitz boundary the convergence $\lambda ^k_p\rightarrow \lambda ^k_1$ as $p\rightarrow 1$ is shown for all $k\in \mathbb{N }$ . The proof uses an approximation of $BV(\Omega )$ -functions by $C_0^\infty (\Omega )$ -functions in the sense of strict convergence on $\mathbb{R }^n$ .     

Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), ${t\in [0,\infty)}$ , which has unbounded curvature for all ${t\in [0,\infty)}$ .     

The Saito–Kurokawa lifting and Darmon points

Let $E_{/_\mathbb{Q }}$ be an elliptic curve of conductor $Np$ with $p\not \mid N$ and let $f$ be its associated newform of weight $2$ . Denote by $f_\infty $ the $p$ -adic Hida family passing though $f$ , and by $F_\infty $ its $\varLambda $ -adic Saito–Kurokawa lift. The $p$ -adic family $F_\infty $ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde{A}_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$ . We relate explicitly certain global points on $E$ (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their $p$ -adic derivatives, evaluated at weight $k=2$ .     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

Lax embeddings of the Hermitian unital

In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital ${\mathcal{U}}$ of ${\mathsf{PG}(2,\mathbb{L}), \mathbb{L}}$ a quadratic extension of the field ${\mathbb{K}}$ and ${|\mathbb{K}| \geq 3}$ , in a ${\mathsf{PG}(d,\mathbb{F})}$ , with ${\mathbb{F}}$ any field and d ≥ 7, such that disjoint blocks span disjoint subspaces, is the standard Veronesean embedding in a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ (and d = 7) or it consists of the projection from a point ${p \in \mathcal{U}}$ of ${\mathcal{U}{\setminus} \{p\}}$ from a subgeometry ${\mathsf{PG}(7,\mathbb{K}^{\prime})}$ of ${\mathsf{PG}(7,\mathbb{F})}$ into a hyperplane ${\mathsf{PG}(6,\mathbb{K}^{\prime})}$ . In order to do so, when ${|\mathbb{K}| >3 }$ we strongly use the linear representation of the affine part of ${\mathcal{U}}$ (the line at infinity being secant) as the affine part of the generalized quadrangle ${\mathsf{Q}(4,\mathbb{K})}$ (the solid at infinity being non-singular); when ${|\mathbb{K}| =3}$ , we use the connection of ${\mathcal{U}}$ with the generalized hexagon of order 2.