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Given a real function $f$, the rate function for the large deviations of the diffusion process of drift $\nabla f$ given by the Freidlin-Wentzell theorem coincides with the time integral of the energy dissipation for the gradient flow associated with $f$. This paper is concerned with the stability in the hilbertian framework of this common action functional when $f$ varies. More precisely, we show that if $(f_h)_h$ is uniformly $\lambda$-convex for some $\lambda \in \mathbb{R}$ and converges towards $f$ in the sense of Mosco convergence, then the related functionals $\Gamma$-converge in the strong topology of curves.

The (third order in time) JMGT equation [Jordan (J Acoust Soc Am 124(4):2491–2491, 2008) and Cattaneo (C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3:83–101, 1948)] is a nonlinear (quasi-linear) partial differential equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second-order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell–Cattaneo’s law gives rise to the third-order in time derivative scaled by a small parameter $$\tau >0$$ , the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper, we provide an asymptotic analysis of the third-order model when $$\tau \rightarrow 0 $$ . It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher-order regularity. This addresses an open question raised in [20], where a related JMGT equation has been studied and weak star convergence of the solutions when $$\tau \rightarrow 0$$ has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the “smallness” of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

We consider reaction–diffusion systems with random terms that oscillate rapidly in space variables. Under the assumption that the random functions are ergodic and statistically homogeneous we prove that the random trajectory attractors of these systems tend to the deterministic trajectory attractors of the averaged reaction-diffusion system whose terms are the average of the corresponding terms of the original system. Special attention is given to the case when the convergence of random trajectory attractors holds in the strong topology.

In this paper, the concept of probabilisticg-metric space with degreel, which is a generalization of probabilisticG-metric space, is introduced. Then, by endowing strong topology, the definition ofl-dimensional asymptotic density of a subsetAof$\mathbb{N}^{l}$Nlis used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.

We introduce the notions of strongly $\lambda$-statistically pre-Cauchy and strongly Vall´ee-Poussin pre-Cauchy sequences in probabilistic metric spaces endowed with strong topology. And we show that these two new notions are equivalent. Strongly $\lambda$-statistically convergent sequences are strongly $\lambda$-statistically pre-Cauchy sequences, and we give an example to show that there is a sequence in a probabilistic metric space which is strongly $\lambda$-statistically pre-Cauchy but not strongly $\lambda$-statistically convergent.

It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $\varphi$ per unit area, then any spectral island $\sigma_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $\mathcal{I}_b=M \varphi$. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any $2d$ Bloch-Landau operator $H_b$ which also has a bounded $\mathbb{Z}^2$-periodic electric potential. Assume that $H_b$ has a spectral island $\sigma_b$ which remains isolated from the rest of the spectrum as long as $\varphi$ lies in a compact interval $[\varphi_1,\varphi_2]$. Then $\mathcal{I}_b=c_0+c_1\varphi$ on such intervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$. The integer $c_1$ is the Chern marker of the spectral projection onto the spectral island $\sigma_b$. This result also implies that the Fermi projection on $\sigma_b$, albeit continuous in $b$ in the strong topology, is nowhere continuous in the norm topology if either $c_1\ne0$ or $c_1=0$ and $\varphi$ is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields., Comment: 20 pages, no figures. Appendix C added. Final version accepted for publication in Journal of the European Mathematical Society

We study the continuity and the measurability of the solution to Schrodinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This is a continuation of our previous result where the space of all Borel probability measures was endowed with the strong topology. As an application, we construct a convex function of which the moment measure is a given probability measure, by the zero noise limit of a class of stochastic optimal transportation problems.

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self maps. Then representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and smooth representations. It is known that in the topological category, this correspondence can be reversed for certain topological groupoids. We extend this result to the smooth category under weaker assumptions on the groupoids., Comment: 33 pages, v2: Corrected numerous typos and some small mistakes, small rewrite to improve readability, main results remain unchanged

We study the continuity in the sense of the strong topology for the flux function υ → l(υ) = |υ|p(⋅) − 2υ acting from the Lebesgue–Orlicz space Lp(⋅)(Ω, ℝm) to the dual Lp ′ (⋅)(Ω, ℝm), where p′(⋅) is the Holder-conjugate exponent, under the assumption that p(·) is an L∞(Ω)-function such that 1 < α ≤ p(·) ≤ β < ∞. We obtain estimates for the convergence $$ {\left\Vert l\left({\upsilon}_n\right)-l\left(\upsilon \right)\right\Vert}_{p^{\prime}\left(\cdot \right)}\to 0 $$ with respect to the smallness order as ‖υn − υ‖p(⋅) → 0. The strong continuity of the energy functional $$ \underset{\varOmega }{\int }{\left|\upsilon \right|}^{p\left(\cdot \right)} dx $$ is a consequence of the strong continuity of the flux function.

In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.

We generalize Brazas' topology on the fundamental group to the whole universal path space X ˜ , i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action.

Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (τ,μ)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting.

The paper investigates the well-posedness and longtime dynamics of Boussinesq type equations with fractional damping: u t t + Δ 2 u + ( − Δ ) α u t − Δ f ( u ) = g ( x ) , with α ∈ ( 1 , 2 ) . The main results focus on the relations among the dissipative exponent α , the growth exponent p of nonlinearity f ( u ) and the well-posedness and the longtime dynamics of the equations. We find a new critical exponent p α ≡ N + 2 ( 2 α − 1 ) ( N − 2 ( 2 α − 1 ) ) + rather than p ∗ ≡ N + 2 N − 2 ( N ≥ 3 ) as known before and show that when 1 ≤ p p α : (i) The equations are like parabolic, that is, not only the IBVP of the equations are well-posedness, but also their weak solutions are of higher global regularity as t > 0 . (ii) The related solution semigroup has a global attractor A α in natural energy space, and also has an exponential attractor A e x p α in the sense of partially strong topology. In particular, when 1 ≤ p p α ′ ≡ N + 2 α ( N − 2 α ) + ( p α ) , the partially strong topology becomes the strong one. (iii) For any α 0 ∈ [ 1 , 2 ) , the family of global attractors A α is upper semicontinuous at the point α 0 .

The aim of this paper is to introduce a single structure which carries the subsets of X as well as the subsets of Y for studying the information about the ordered pair (A, B) of subsets of X and Y. Such a structure is called a binary structure from X to Y. Mathematically a binary structure from X to Y is defined as a set of ordered pairs (A, B) where AX and BY. The purpose of this paper is to introduce a new topology between two sets called a binary topology and investigate its basic properties where a binary topology from X to Y is a binary structure satisfying certain axioms that are analogous to the axioms of topology. MSC 2010: 54A05, 54A99.

In this paper, we study a replicator dynamics for asymmetric and symmetric evolutionary games when the strategy sets are metric spaces. In this case, the replicator dynamics evolves in a Banach space. We provide conditions under which this replicator dynamics can be approximated by finite-dimensional dynamical systems. These approximations are studied in the weak topology using the Kantorovich–Rubinstein metric, and also in the strong topology using the norm of total variation. Some numerical examples illustrate our results.

In a dual pair 〈E, F〉 one wants to define topologies on E associated with collections of suitable subsets of F. (This generalises the definition of the norm topology on the dual E′ of a Banach space E, in this case for the dual pair 〈E′, E〉.) Such a collection \(\mathcal M\) defines a ‘polar topology’ on E, where the corresponding neighbourhoods of zero in E are polars of the members of \(\mathcal M\). Examples of such topologies are the weak topology and the strong topology. In the first part of the chapter we define polars and investigate some of their properties.

We consider high order phase field functionals introduced in Borden et al. (2014) and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy. We take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in Sobolev spaces and in spaces of tensor product B -splines. In the latter convergence holds when the mesh size vanishes faster than the internal length of the phase-field model. On the theoretical level, this condition is natural since the size of the phase field layer, around the crack, itself scales like the internal length; on the numerical level, it should be satisfied by local h -refinement. Technically, convergence holds in the sense of Γ -convergence, with respect to the strong topology of L 1 , while the sharp crack energy is defined in G S B D 2 . The constraint on the phase field to take values in [ 0 , 1 ] is taken into account both in the Sobolev setting and in the iso-geometric setting; in the latter, it requires a special treatment since the projection operator on the space of tensor product B -splines is not Lagrangian (i.e., interpolatory).

Schweizer, Sklar and Thorp proved in 1960 that a Menger space $(G,D,T)$ under a continuous $t$-norm $T$, induce a natural topology $\tau$ wich is metrizable. We extend this result to any probabilistic metric space $(G,D,\star)$ provided that the triangle function $\star$ is continuous. We prove in this case, that the topological space $(G,\tau)$ is uniformly homeomorphic to a (deterministic) metric space $(G,\sigma_D)$ for some canonical metric $\sigma_D$ on $G$. As applications, we extend the fixed point theorem of Hicks to probabilistic metric spaces which are not necessarily Menger spaces and we prove a probabilistic Arzela-Ascoli type theorem.

We present a further generalization of the 𝒯AI -density topology introduced in [W. Wojdowski, A category analogue of the generalization of Lebesgue density topology, Tatra Mt. Math. Publ. 42 (2009), 11–25] as a generalization of the I-density topology. We construct an ascending sequence {𝒯 A I(n) } n ∈ℕ of density topologies which leads to the 𝒯 A I(ω) -density topology including all previous topologies. We examine several basic properties of the topologies.

To the best of our knowledge, till date there are only two directions along which fuzzy topology (general topology) can be compared, one is the generalization of fuzzy topology (such as fuzzy supra topology, generalized fuzzy topology etc.) and the other is the stronger form of fuzzy topology which is known as Alexandroff fuzzy topology. It means that a given topology is always linear. This paper aims to propose the third direction that is a new parallel form of fuzzy topology (or a non-linear topology) called fuzzy independent topology which is neither a generalization nor a stronger form of the given fuzzy topology though it is a unique natural offshoot of the given fuzzy topology but very rare in existence and that has been shown by defining fuzzy $$\gamma ^{*}$$ -open set in the sense of Dimitrije Andrijevic, by proving that fuzzy $$\gamma ^{*}$$ -open set and fuzzy open set are completely independent of each other though the collection of fuzzy $$\gamma ^{*}$$ -open sets are themselves a fuzzy topology therein. At the same time we claim that it is beyond the scope of general topology with the existing generalized open sets in literature and consequently we move one more step towards learning the difference between topology and fuzzy topology. To this end, we study the fundamental properties of the new structure. Also we study some of the basic properties and characterizations of fuzzy $$\gamma ^{*}$$ -open sets and few of their applications. The investigation enables us to present a new covering property of the given fuzzy topological space and the preservation theory is obtained. Finally to illustrate the advantage of the proposed concept, we compare the obtained results with some already existing ones.

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: \begin{document}$ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $\end{document} . In the case when the nonlinearities \begin{document}$ h(u_t) $\end{document} and \begin{document}$ g(u) $\end{document} are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

We study properties of the infimal topology τinf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τinf to have the Frechet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τμ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τinf to satisfy the first axiom of countability.

We consider initial-boundary value problems for the $��$-dependent family of chemotaxis-(Navier--)Stokes systems \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=��n-\nabla\!\cdot(n\nabla c),\ &x\in��,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=��c-cn,\ &x\in��,& t>0,\\ u_{t}&+&��(u\cdot\nabla)u&=��u+\nabla P+n\nabla��,\ &x\in��,& t>0,\\ &&\nabla\cdot u&=0,\ &x\in��,& t>0, \end{array}\right. \end{align*} in a bounded domain $��\subset\mathbb{R}^3$ with smooth boundary and given potential function $��\in C^{1+��}(\overline��)$ for some $��>0$. It is known that for fixed $��\in\mathbb{R}$ an associated initial-boundary value problem possesses at least one global weak solution $(n^{(��)},c^{(��)},u^{(��)})$, which after some waiting time becomes a classical solution of the system. In this work we will show that upon letting $��\to0$ the solutions $(n^{(��)},c^{(��)},u^{(��)})$ converge towards a weak solution of the Stokes variant $(��=0)$ of the systems above with respect to the strong topology in certain Lebesgue and Sobolev spaces. We thereby extend the recently obtained result on the Stokes limit process for classical solutions in the two-dimensional setting to the more intricate three-dimensional case., 34 pages

In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that incorporate the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology, or to specific invariant subspaces (obtained via the so-called Ott-Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. Besides, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here, we establish an explicit criterion for their spectral stability and we prove their local asymptotic stability in weak topology, for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We show in particular that no loss of generality results in assuming the OA ansatz. To our best knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria, in absence of dissipation., 29 pages, 1 figure

In this paper, we examine properties of the division topology on the set of positive integers introduced by Rizza in 1993. The division topology on [Formula: see text] with the division order is an example of [Formula: see text]-Alexandroff topology. We mainly concentrate on closures of arithmetic progressions and connected and compact sets. Moreover, we show that in the division topology on [Formula: see text], the continuity is equivalent to the Darboux property.

Some kind of abstract density topology in a topological Baire space is considered. The semiregularization of this type of topology on the real line in many cases is the coarsest topology for which real functions continuous with respect to the abstract density topology are continuous.

In this note it is proved that for a quasicontinuous lattice L, the lower topology ω ( L ) and the Scott topology σ ( L ) are duals for each other; and if L is a complete lattice such that σ ( L ) is continuous but not hypercontinuous (equivalently, L is not quasicontinuous), then ω ( L ) is not the dual of σ ( L ) and hence they are not duals for each other.

In two-dimensions, a fracture network consist of a system of branches and nodes that can be used to define both geometrical features, such as length and orientation, and the relationship between elements of the network – topology. Branch lengths are preferred to trace lengths as they can be uniquely defined, have less censoring and are more clustered around a mean value. Many important properties of networks are more related to topology than geometry. The proportions of isolated (I), abutting (Y) and crossing (X) nodes provide a basis for describing the topology that can be easily applied, even with limited access to the network as a whole. Node counting also provides an unbiased estimate of frequency and can be used in conjunction with fracture intensity to estimate the characteristic length and dimensionless intensity of the fractures. The nodes can be used to classify branches into three types – those with two I-nodes, one I-node and no I-nodes (or two connected nodes). The average number of connections per branch provides a measure of connectivity that is almost completely independent of the topology. We briefly discuss the extension of topological concepts to 3-dimensions.

In this paper, we define and study a new topology constructed from any given topology on a set, using irreducible sets. The manner in which this derived topology is obtained is inspired by how the Scott topology on a poset is constructed from its Alexandroff topology. This derived topology leads us to a weak notion of sobriety called k-bounded sobriety. We investigate the properties of this derived topology and k-bounded sober spaces. A by-product of our theory is a novel type of compactness, which involves crucially the Scott irreducible families of open sets. Some related applications on posets are also given.

Most optimization problems in the field of fluid mechanics can be classified as either topology or shape optimization. Although topology and shape have been considered mutually exclusive optimization methods since their inception, it is conceivable that they will find choicest solutions in tandem, with shape optimization refining a solution found by topology. However, linking the topology optimization problem to that of shape is not trivial and, to the authors' knowledge, has yet to be formally attempted. This paper pro- poses a novel transitional procedure that post-processes 2D adjoint topology solutions,fitting the interface between the solid and fluid topological domains to create a parameterized solution which can be used as either a CAD-compatible representation of the interface or a source for grid generation from which a shape optimization loop can be initialized. The interface to be fit can be extracted from any topological field with distinct fluid and solid domains, meaning that the proposed transition process is independent of the topology approach utilized. To conveniently describe the interface between the solid and fluid topological domains, the topology optimization process employed in this paper is ltered using the level set method. The interface is fit with non-uniform rational B-spline (NURBS) curves through application of sensitivitiesgarnered from the solution of an auxiliary inverse design problem which aims at reducing the difference between signed-distance fields generated about both the NURBS curve being optimized and the section of interface being fit. The geometry defined by the fit NURBS curves is then (optionally) used to build a boundary-fitted grid on which a shape optimization loop is performed. The parameterized result of the topology to shape transition process is compared to that of shape optimization in 2D cases with internal, incompressible fluid flows.

Let M be a smooth oriented connected n-dimensional manifold and let M be the space of pseudo-Riemannian metrics on M of a given signature (n+,n−),n++n−=n>1. A system of n metric invariants is attached to each metric in M, called the Ricci invariants, and by using the geometric properties of such invariants, the following result is proved: The subset O⊂M of metrics with no Killing vector fields other than the trivial one is open and dense with respect to the strong topology.

The present paper aims to address a long-standing and challenging problem in structural topology optimization: explicit feature control of the optimal topology. The basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure. Numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach.

The main observation of this paper is that some sequential weak compactness arguments in Hilbert space theory can be replaced by Heine/Borel compactness arguments (for the strong topology). Even though the latter form of compactness fails in (infinite-dimensional) Hilbert spaces, it nevertheless trivializes under the so-called bounded functional interpretation. As a consequence, the proof mining program of extracting computational bounds from ordinary proofs of mathematics can be applied to modified proofs which use these false Heine/Borel compactness arguments. Additionally, the bounded functional interpretation provides good logical guidance in formulating quantitative versions of analytical statements. We illustrate these claims with three minings. The bounded functional interpretation is here used for the first time in proof mining.

For a signature L with at least one constant symbol, an L-structure is called minimal if it has no proper substructures. Let be the set of isomorphism types of minimal L-structures. The elements of can be identified with ultrafilters of the Boolean algebra of quantifier-free L-sentences, and therefore one can define a Stone topology on . This topology on generalizes the topology of the space of n-marked groups. We introduce a natural ultrametric on , and show that the Stone topology on coincides with the topology of the ultrametric space iff the ultrametric space is compact iff L is locally finite (that is, L contains finitely many n-ary symbols for any ). As one of the applications of compactness of the Stone topology on , we prove compactness of certain classes of metric spaces in the Gromov-Hausdorff topology. This slightly refines the known result based on Gromov's ideas that any uniformly totally bounded class of compact metric spaces is precompact.

We establish that if ℱ is a countable field of characteristic p, U is a unitary action of ℱl on a Hilbert space ℋ, and P is an essential idempotent ultrafilter on ℱ, then for every polynomial r: ℱ → ℱl with r(0) = 0 one has $$P - {\lim _g}{U_{r(g)}}f = {P_r}f{\rm{ weakly,}}$$ , where P r is the projection onto the closed subspace $${H_r} = \{ f \in H:{\{ {U_{r(g)}}f\} _{g \in F}}{\text{ is precompact in the strong topology\} }}{\text{.}}$$ . We then derive combinatorial consequences of this result, including results for sufficiently large finite fields.

We characterize the completely regular Lindelof spaces X such that C ( X ) with the finest splitting topology is Frechet, a topological group, or a k -space.

Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] the authors define a topology on $X$ by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction given in [arXiv:0707.1525] and, starting from a set $X$ and a collection of subsets $\mathcal{F}$ of $X$, we define by using ultrafilters a topology on $X$ in which $\mathcal F$ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several new examples of spectral spaces., 16 pages. To appear in Communications in Algebra

This paper will discuss about a new topology, obtained from a grill and a filter on the same set. The Characterizations and open base of the new topology are also aim of this paper. The generalized continuity is also a part of this paper.

In this article we present a method to reconstruct the interconnectedness of dynamically related stochastic processes, where the interactions are bi-directional and the underlying topology is a tree. Our approach is based on multivariate Wiener filtering which recovers spurious edges apart from the true edges in the topology reconstruction. The main contribution of this work is to show that all spurious links obtained using Wiener filtering can be eliminated if the underlying topology is a tree based on which we present a three stage network reconstruction procedure for trees. We illustrate the effectiveness of the method developed by applying it on a typical distribution system of the electric grid., Comment: 6 pages

The evolution of a mixture of two incompressible and (partially) immiscible fluids is here described by Ladyzhenskaya–Navier–Stokes type equations for the (average) fluid velocity coupled with a convective Cahn–Hilliard equation with a singular (e.g., logarithmic) potential. The former is endowed with no-slip boundary conditions, while the latter is subject to no-flux boundary conditions so that the total mass is conserved. Here we first prove the existence of a weak solution in three-dimensions and some regularity properties. Then we establish the existence of a weak trajectory attractor for a sufficiently general time-dependent external force. Finally, taking advantage of the validity of the energy identity, we show that the trajectory attractor actually attracts with respect to the strong topology.

Topology control has received much attention in stationary sensor networks by effectively minimizing energy consumption, reducing interference, and shortening end to-end delay. The transience of mobile nodes in Vehicular Ad hoc A network (VANETs) renders topology control a great challenge. the existing edge connected topology control algorithms need to determine the value priori, but moving speeds of nodes are unpredictable, and therefore, these algorithms are not practical in VANETs. In the proposed method if node failure occurs the described algorithm automatically determines the appropriate alternate node based on the efficient values for the required graph, using Local information while ensuring the required connectivity ratio of the whole network. The results show that the dynamic method can enhance the practicality and packet transmission in undisturbed channel with the strong topology control algorithms while guaranteeing the network connectivity.

Given random samples drawn i.i.d. from a probability measure $\mathbb{P}$ (defined on say, $\mathbb{R}^d$), it is well-known that the empirical estimator is an optimal estimator of $\mathbb{P}$ in weak topology but not even a consistent estimator of its density (if it exists) in the strong topology (induced by the total variation distance). On the other hand, various popular density estimators such as kernel and wavelet density estimators are optimal in the strong topology in the sense of achieving the minimax rate over all estimators for a Sobolev ball of densities. Recently, it has been shown in a series of papers by Gin\'{e} and Nickl that these density estimators on $\mathbb{R}$ that are optimal in strong topology are also optimal in $\|\cdot\|_{\mathcal{F}}$ for certain choices of $\mathcal{F}$ such that $\|\cdot\|_{\mathcal{F}}$ metrizes the weak topology, where $\|\mathbb{P}\|_{\mathcal{F}}:=\sup\{\int f\,\mathrm{d}\mathbb{P}: f\in\mathcal{F}\}$. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing $\mathcal{F}$ to be a unit ball in a reproducing kernel Hilbert space (say $\mathcal{F}_H$ defined over $\mathbb{R}^d$), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that $\|\cdot\|_{\mathcal{F}_H}$ metrizes the weak topology and the kernel density estimator (with $L^1$ optimal bandwidth) estimates $\mathbb{P}$ at dimension independent optimal rate of $n^{-1/2}$ in $\|\cdot\|_{\mathcal{F}_H}$ along with providing a uniform central limit theorem for the kernel density estimator., Comment: Published at http://dx.doi.org/10.3150/15-BEJ713 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We consider in this article the system of (pure) gravity water waves in any dimension and in fluid domains with general bottoms. The unique solvability of the problem was established by Alazard-Burq-Zuily [Invent. Math, 198 (2014), no. 1, 71--163] at a low regularity level where the initial surface is $C^{\frac{3}{2}+}$ in terms of Sobolev embeddings, which allows the existence of free surfaces with unbounded curvature. Our result states that the solutions obtained above depend continuously on initial data in the strong topology where they are constructed. This completes a well-posedness result in the sense of Hadamard., Explanation of the strategy expanded, some typos fixed. To appear in Journal of Hyperbolic Differential Equations

This paper is dealing the abstract density topologies in the family of Lebesgue measurable sets generated by an operator similar the density lower operator defined in the family of a measurable sets. In the theory of density topologies on the real line in the family of Lebesgue measurable sets the density topology is introduced usually by an operator defined on the family of Lebesgue measurable sets which has the desired properties. In that way we get the classical density topology, ψdensity topology, f -density topology, 〈s〉-density topology (see [8], [6], [5], [3], [7], [1], [2]). Let us assume that R is the set of reals, ` is the Lebesgue measure on R, L is the family of Lebesgue measurable sets on R and L is the family of `-null set. By Tnat and Td we will denote the natural topology and the classical density topology on R respectively. The fact that `(A4B) = 0 for any set A,B ∈ L will be denoetd by A ∼ B. We shall say that operator Φ : L → L is a lower density operator if the following conditions are satisfied: ∗2000 Mathematics Subject Classification: 54A05, 28A05, 11B05.

We proved that the union of rational curves is dense on a very general K3 surface and the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology., Comment: 23 pages. To appear in Mathematische Annalen

Using the theory of liftings, we give simple new proofs of the characterizations of the relatively weakly compact subsets and weak Cauchy sequences of L ∞ ( E ) . Also a different proof, of a deep result of J. Diestel, W. Ruess, W. Schachermayer about weak compactness in L 1 ( E ) , is given.