Frac N2 Research Articles

Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$where $\Omega =\mathbb {R}^N$ or $\mathbb {R}^N\setminus \Omega$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (Dρ), or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (Dρ) is fulfilled. There are no assumptions on the size of $\mathbb {R}^N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}^N$.

Transversal Ck-factors in subgraphs of the balanced blow-up of Ck

AbstractFor a subgraph $G$ of the blow-up of a graph $F$ , we let $\delta ^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$ . Johansson proved that if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$ and $\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$ , then $G$ contains $n$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$ , then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. A similar conjecture was also made by Fischer and the case $k=3$ was proved for large $n$ by Magyar and Martin.In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of $G$ to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.

Efficient Assignment of Identities in Anonymous Populations

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size $n$ of the population is embedded in the transition function. Our labeling protocols are silent, i.e., eventually each agent reaches its final state and remains in it forever, as well as safe, i.e., never update the label assigned to any single agent. We first present a fast, silent and safe labeling protocol for which the required number of interactions is asymptotically optimal, i.e., $O(n \log n/\epsilon)$ w.h.p. It uses $(2+\epsilon)n+O(n^c)$ states, for any $c 1-\frac 1n$, uses $\ge n+\sqrt {n-1} -1$ states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Furthermore, we show that for any silent and safe labeling protocol utilizing $n+t<2n$ states the expected number of interactions required to achieve a valid labeling is $\ge \frac{n^2}{t+1}$.

The binary digits of n+t

The binary sum-of-digits function $s(n)$ counts the number of $\mathtt 1$s in the binary expansion of a nonnegative integer $n$. T.~W.~Cusick defined the asymptotic density \[ c_t=\lim_{N\rightarrow \infty} \frac 1N \lvert\{0\leq n 1/2$ for all $t$. We prove that there exists an explicit constant $M_0$ such that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M_0$ blocks of contiguous $\mathtt 1$s. Moreover, we sharpen a result by Emme and Hubert (2018), proving that the difference $s(n+t)-s(n)$ behaves according to a Gaussian distribution, up to an error tending to $0$ as the number of blocks in the binary expansion of $t$ grows.

Bayesian Algorithms for Decentralized Stochastic Bandits

We study a decentralized cooperative multi-agent multi-armed bandit (MAB) problem with K arms and N agents connected over a network. In this model, each arm's reward distribution is the same for every agent, and rewards are drawn independently across agents and over time steps. At each iteration, agents independently choose an arm to play and exchange at most poly(K) real-valued messages with their neighbors. Existing lower bound on the average per-agent regret over the network shows that cooperation with other agents can achieve a reduction of O(\frac 1N) over when playing in isolation. Motivated by this, we study a message-passing algorithm that can be combined with existing Bayesian MAB algorithms. Using this, we propose a decentralized Thompson Sampling (TS) algorithm and a decentralized Bayes-UCB algorithm. Under decentralized TS for bounded rewards, we establish a problem-dependent upper bound on the average per-agent regret in terms of the number of agents in the network and the network topology. For Bernoulli rewards, our upper bound asymptotically matches the lower bound. We empirically show that the proposed decentralized TS algorithm incurs significantly lesser per-agent regret than previously proposed algorithms. Furthermore, we show that the proposed decentralized TS can be extended to general bandit problems, where posterior distribution cannot be computed in closed form. We combine our decentralized TS algorithm with Variational Inference to apply our proposed algorithm to complex realistic reward distributions in a computationally efficient manner. We implement our proposed decentralized TS under gossip protocol and over time-varying networks, where each communication link has a fixed probability of failure and show that it incurs logarithmic regret.

A new criterion of physical measures for partially hyperbolic diffeomorphisms

We show that, for any C 1 C^1 partially hyperbolic diffeomorphism, there is a full volume subset such that any Cesàro limit of any point in this subset satisfies the Pesin formula for partial entropy. This result has several important applications. First, we show that, for any C 1 + C^{1+} partially hyperbolic diffeomorphism with one dimensional center, there is a full volume subset such that either every point in this set belongs to the basin of a physical measure with nonvanishing center exponent or the center exponent of any limit of the sequence 1 n ∑ i = 0 n − 1 δ f i ( x ) \frac 1n\sum _{i=0}^{n-1}\delta _{f^i(x)} is vanishing. We also prove that, for any diffeomorphism with mostly contracting center, it admits a C 1 C^1 -neighborhood such that every diffeomorphism in a C 1 C^1 residual subset of this open set admits finitely many physical measures whose basins have full volume.

Blow-up of Smooth Solutions to the Cauchy Problem for the Viscous Two-Phase Model

Let $\mathcal{C}(\mathcal{R},n,p,\Lambda,D,V_0)$ be the class of compact $n$-dimensional Riemannian manifolds with finite diameter $\leq D$, non-collapsing volume $\geq V_0$ and $L^p$-bounded $\mathcal{R}$-curvature condition $\|\mathcal{R}\|_{L^p}\leq \Lambda$ for some $p>\frac n2$. Let $(M,g_0)$ be a compact Riemannian manifold and $\mathcal{C}(M,g_0)$ the class of manifolds $(M,g)$ conformal to $(M,g_0)$. In this paper we use $\varepsilon$-regularity to show a rigidity result in the conformal class $\mathcal{C}(S^n,g_0)$ of standard sphere under $L^p$-scalar rigidity condition. Then we use harmonic coordinate to show $C^{\alpha}$-compactness of the class $\mathcal{C}(K,n,p,\Lambda,D,V_0)$ with additional positive Yamabe constant condition, where $K$ is the sectional curvature, and this result will imply a generalization of Mumford's lemma. Combining these methods together we give a geometric proof of $C^{\alpha}$-compactness of the class $\mathcal{C}(K,n,p,\Lambda,D,V_0)\cap \mathcal{C}(M,g_0)$. By using Weyl tensor and a blow down argument, we can replace the sectional curvature condition by Ricci curvature and get our main result that the class $\mathcal{C}(Ric,n,p,\Lambda,D,V_0)\cap \mathcal{C}(M,g_0)$ has $C^{\alpha}$-compactness.

Nonconventional random matrix products

Let $\xi_1,\xi_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi_{q_1(n)},\xi_{q_2(n)},...,\xi_{q_\ell(n)})$ where $0\leq q_1<q_2<...<q_\ell$ are increasing functions taking on integer values on integers. We study the asymptotic behavior as $N\to\infty$ of the singular values of the random matrix product $\Pi_N=X_N\cdots X_2X_1$ and show, in particular, that (under certain conditions) $\frac 1N\log\|\Pi_N\|$ converges with probability one as $N\to\infty$. We also obtain similar results for such products when $\xi_i$ form a Markov chain. The essential difference from the usual setting appears since the sequence $(X_n)$ is long-range dependent and nonstationary.

Some L p Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M n , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m goes to zero uniformly at infinity if for $p\geq \frac n2$ , the L p -norm of R m is finite. Moreover, If R is positive, then (M n , g) is compact. As applications, we prove that (M n , g) is isometric to a spherical space form if for $p\geq \frac n2$ , R is positive and the L p -norm of R m is pinched in [0, C 1), where C 1 is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).

Rigidity theorem for integral pinched shrinking Ricci solitons

We prove that an $n$-dimensional, $n\geq4$, compact gradient shrinking Ricci soliton satisfying a $L^{\frac n2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^n$, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).

A non-uniform distribution property of most orbits, in case the $3x+1$ conjecture is true

Let $T(n)=\left\{\begin{array}{ll}3n+1&(n\hbox{ odd})\frac n2&(n\hbox{ even})\end{array}\right.$ ($n\in\mathbb Z$). We call orbit of the integer $n$, the set $$ \mathcal O_n:=\{m\in\mathbb Z\;:\;\exists k\ge0, m=T^k(n)\} $$ and we put $c_i(n):=\#\{m\in\mathcal O_n\;:\;m\equiv i\hbox{ mod.}18\}$. Let $W$ be the set of the integers whose orbit contains $1$ and is, in the following sense, about well distributed modulo $18$ between the six elements of the set $I:=\{1,5,7,11,13,17\}$ (the elements of \{1,\dots,18\} that are odd and not divisible by $3$). More precisely: $$ W:=\Big\{n\in\mathbb N\;:\;\exists k\ge0, T^k(n)=1\hbox{ and }\forall i\in I, \frac{c_i(n)}{\sum_{i\in I}c_i(n)}\le\frac16+0.0215\Big\}. $$ We prove that $W$ has density $0$ in $\mathbb N$. Consequently, if the $3x+1$ conjecture is true, most of the positive integers $n$ satisfy $$ \frac{\max_{i\in I}c_i(n)}{\sum_{i\in I}c_i(n)}>\frac16+0.0215. $$

Global Existence of Small Equivariant Wave Maps on Rotationally Symmetric Manifolds

We introduce a class of rotationally invariant manifolds, which we call \emph{admissible}, on which the wave flow satisfies smoothing and Strichartz estimates. We deduce the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity $H^{\frac n2}$. The class of admissible manifolds includes in particular asymptotically flat manifolds and perturbations of real hyperbolic spaces $\mathbb{H}^{n}$ for $n\ge3$.

On the three dimensional minimal model program in positive characteristic

Let f : ( X , B ) → Z f:(X,B)\to Z be a threefold extremal dlt flipping contraction defined over an algebraically closed field of characteristic p &gt; 5 p&gt;5 , such that the coefficients of { B } \{ B\} are in the standard set { 1 − 1 n | n ∈ N } \{ 1-\frac 1n|n\in \mathbb N\} , then the flip of f f exists. As a consequence, we prove the existence of minimal models for any projective Q {\mathbb Q} -factorial terminal variety X X with pseudo-effective canonical divisor K X K_X .

Existence of Local Solutions of Nonlinear Wave Equations in $n$-Dimensional Space

In this paper, we investigate the local existence of solutions in $H^s$ for $n$-dimensional nonlinear wave equations with special nonlinear terms, such as $$u_{tt}-\Delta u=u^k|\nabla u|^l,\ \ x\in R^n,\ \ k\in Z^+,\ \ l\geq2.$$ where $\nabla u=(\frac{\partial u}{\partial x_1},\ \frac{\partial u}{\partial x_2},\ \cdots,\ \frac{\partial u}{\partial x_n})$. Meanwhile, we obtain that the regular index $s$ of Sobolev space $H^s$ satisfies $s>\max\{\frac{n+5}4;\ \frac n2+1-\frac1{l-1}\},\ n>3$.

Prescription de la multiplicité des valeurs propres du laplacien de Hodge–de Rham

On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum of the Hodge Laplacian acting on p -form for 1\leq p&lt;\frac n2 . In particular, we prescribe the multiplicity of the first eigenvalues.

Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms

Let Mn be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., \({\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}\), in an (n + p)-dimensional simply connected space form Nn+p(c) of constant curvature c, where |H| and |A|2 are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and \({\int_M|H|^n < \infty}\); (ii) n ≥ 5, c = −1 and \({|H| < 1-\frac{2}{\sqrt n}}\); (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L2 harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.

WEAK AND STRONG CONVERGENCE FOR SOME OF NONEXPANSIVE MAPPINGS

IIn this paper, we deal with a class of nonexpansive mappings with the property $D(\overline{co} F_{\frac 1n} (T),F(T))\to 0$, as $n\to \infty$, where $D$ is the Hausdorff metric. We show that nonexpansive mappings with compact domains enjoy this property and give some examples of this kind of mappings with noncompact domains in $l^\infty$. Then we prove a nonlinear ergodic theorem, and a convergence theorem of mann's type for this kind of mappings.

WEAK AND STRONG CONVERGENCE FOR SOME OF NONEXPANSIVE MAPPINGS

IIn this paper, we deal with a class of nonexpansive mappings with the property $D(\overline{co} F_{\frac 1n} (T),F(T))\to 0$, as $n\to \infty$, where $D$ is the Hausdorff metric. We show that nonexpansive mappings with compact domains enjoy this property and give some examples of this kind of mappings with noncompact domains in $l^\infty$. Then we prove a nonlinear ergodic theorem, and a convergence theorem of mann's type for this kind of mappings.

The structure of branched transportation networks

The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure μ + onto a measure μ − with the same mass. In contrast with the Monge–Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow s in a road or a tube or a wire, the transportation cost per unit length is supposed to be proportional to s α with 0 < α < 1. For the Monge–Kantorovich energy α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow $$s_1+s_2$$ is preferable to two individual roads s 1 and s 2 because $$(s_1+s_2)^\alpha < s_1^\alpha+s_2^\alpha$$ . Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When $$\alpha > 1-\frac 1N$$ such structures can irrigate a whole bounded open set of $${\mathbb{R}}^N$$ . The aim of this paper is to give a mathematical proof of several structure and regularity properties empirically observed in transportation networks. It is first proven that optimal transportation networks have a tree structure and can be monotonically approximated by finite graphs. An interior regularity result is then proven according to which an optimal network is a finite graph away from the irrigated measure. It is also proven that the branching number of optimal networks has everywhere a universal explicit bound. These results answer questions raised in two recent papers by Xia.

Minoration conforme du spectre du laplacien de Hodge-de Rham

Let Mn be an n-dimensional compact manifold, with n ≥ 3. For any conformal class C of riemannian metrics on M, we set \({\mu_k^c(M,C)=\inf_{g\in C}\mu_{\left[\frac n2\right],k}(M,g)\,{\rm Vol}(M,g)^{\frac2n}}\) , where μp,k(M,g) is the kth eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that \({0 < \mu_k^c(M,C)\leq \mu_k^c(S^n,[g_{\rm can}])\leq k^{\frac2n}\mu_1^c(S^n,[g_{\rm can}])}\) . We also prove that if g is a smooth metric such that \({\mu_{\left[\frac n2\right],1}(M,g)\,{\rm Vol}(M,g)^{\frac2n}=\mu_1^c(M,[g])}\) , and n = 0,2,3 mod 4, then there is a non-zero corresponding eigenform of degree \({\left[\frac{n-1}2\right]}\) with constant length. As a corollary, on a four-manifold with non vanishing Euler characteristic, there is no such smooth extremal metric.