Strongly Continuous Research Articles

Approximation algorithms for minimum power k backbone node r-connected subgraph problem in wireless sensor networks

The minimum power assignment problem in a Wireless Sensor Network (WSN) is to assign transmission range to the nodes of the network with specified connectivity constraints minimizing the total power. This problem is termed as Strong Minimum Energy Topology (SMET) problem. Fault tolerance addresses the issue of a node or link failure in a WSN. So, with an objective of achieving high connectivity, we consider SMET problem with an additional property, i.e., the resultant network must satisfy the property of two or high connectivity. Minimum Power [Formula: see text]-Connected Subgraph (MP[Formula: see text]CS) problem is to contrive an [Formula: see text]-connected network with minimum power. Minimum Power [Formula: see text]-Connected Subgraph (MP2CS) problem is proved to be NP -hard. Minimum Power [Formula: see text] Backbone node [Formula: see text]-Connected Subgraph (MP[Formula: see text]B[Formula: see text]CS) problem is a special case of MP[Formula: see text]CS problem, which seeks a power assignment satisfying [Formula: see text]-connectivity with [Formula: see text] backbone nodes. We study MP[Formula: see text]B[Formula: see text]CS problem for [Formula: see text] backbone nodes, [Formula: see text]-connectivity and derive the approximation ratios. We also propose an algorithm of approximation ratio [Formula: see text], for Minimum Power [Formula: see text] Backbone node 3-Connected Subgraph (MP[Formula: see text]B3CS) problem for [Formula: see text] that runs in [Formula: see text] time and generalize for the case [Formula: see text] for any [Formula: see text] and [Formula: see text].

The Stokes Limit in a Three-Dimensional Chemotaxis-Navier–Stokes System

We consider initial-boundary value problems for the $\kappa$-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&=\Delta n-\nabla\!\cdot(n\nabla c),\ &x\in\Omega,& t>0,\\ c_{t}&+&u\cdot\!\nabla c&=\Delta c-cn,\ &x\in\Omega,& t>0,\\ u_{t}&+&\kappa(u\cdot\nabla)u&=\Delta u+\nabla P+n\nabla\phi,\ &x\in\Omega,& t>0,\\ &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary and given potential function $\phi\in C^{1+\beta}(\overline{\Omega})$ for some $\beta>0$. It is known that for fixed $\kappa\in\mathbb{R}$ an associated initial-boundary value problem possesses at least one global weak solution $(n^{(\kappa)},c^{(\kappa)},u^{(\kappa)})$, which after some waiting time becomes a classical solution of the system. In this work we will show that upon letting $\kappa\to0$ the solutions $(n^{(\kappa)},c^{(\kappa)},u^{(\kappa)})$ converge towards a weak solution of the Stokes variant $(\kappa=0)$ 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.

The discrete Gaussian free field on a compact manifold

In this article we define the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice Zd in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology.

Topological stable rank of mathcal {E}'(mathbb {R})

The set mathcal {E}'(mathbb {R}) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of mathcal {E}'(mathbb {R}) is 2.

Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

The weak sigma-attractor for the semi-dissipative 2D Boussinesq system

In this paper, we obtain the attraction properties for the velocity variable of the 2D Boussinesq equations with viscosity and without heat diffusion in the sense of the strong topology of V V and prove that the weak sigma-attractor has a pancake-like structure, which answer partly some questions arising in Biswas et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), pp. 381–405] and enrich the structure of the weak sigma-attractor.

Syntgen: a system to generate temporal networks with user-specified topology

AbstractIn the last few years, the study of temporal networks has progressed markedly. The evolution of clusters of nodes (or communities) is one of the major focus of these studies. However, the time dimension increases complexity, introducing new constructs and requiring novel and enhanced algorithms. In spite of recent improvements, the relative scarcity of timestamped representations of empiric networks, with known ground truth, hinders algorithm validation. A few approaches have been proposed to generate synthetic temporal networks that conform to static topological specifications while in general adopting an ad hoc approach to temporal evolution. We believe there is still a need for a principled synthetic network generator that conforms to problem domain topological specifications from a static as well as temporal perspective. Here, we present such a system. The unique attributes of our system include accepting arbitrary node degree and cluster size distributions and temporal evolution under user control, while supporting tunable joint distribution and temporal correlation of node degrees. Theoretical contributions include the analysis of conditions for graphic sequences of inter- and intracluster node degrees and cluster sizes and the development of a heuristic to search for the cluster membership of nodes that minimizes the shared information distance between clusterings. Our work shows that this system is capable of generating networks under user controlled topology with up to thousands of nodes and hundreds of clusters with strong topology adherence. Much larger networks are possible with relaxed requirements. The generated networks support algorithm validation as well as problem domain analysis.

Electronic Structure of a Graphene-like Artificial Crystal of NdNiO3.

Artificial complex-oxide heterostructures containing ultrathin buried layers grown along the pseudocubic [111] direction have been predicted to host a plethora of exotic quantum states arising from the graphene-like lattice geometry and the interplay between strong electronic correlations and band topology. To date, however, electronic-structural investigations of such atomic layers remain an immense challenge due to the shortcomings of conventional surface-sensitive probes with typical information depths of a few angstroms. Here, we use a combination of bulk-sensitive soft X-ray angle-resolved photoelectron spectroscopy (SX-ARPES), hard X-ray photoelectron spectroscopy (HAXPES), and state-of-the-art first-principles calculations to demonstrate a direct and robust method for extracting momentum-resolved and angle-integrated valence-band electronic structure of an ultrathin buckled graphene-like layer of NdNiO3 confined between two 4-unit cell-thick layers of insulating LaAlO3. The momentum-resolved dispersion of the buried Ni d states near the Fermi level obtained via SX-ARPES is in excellent agreement with the first-principles calculations and establishes the realization of an antiferro-orbital order in this artificial lattice. The HAXPES measurements reveal the presence of a valence-band bandgap of 265 meV. Our findings open a promising avenue for designing and investigating quantum states of matter with exotic order and topology in a few buried layers.

A VIEW TO POWER SET: A TOPOLOGICAL DISCUSSION

A Power Set is not only a container of all family of subsets of a set and the set itself,but ,in topology,it is also a generator of all topologies on the defined set. So, there is a topological existence of power set, being the strongest topology ever defined on a set,there are some properties of it's topological existence.In this paper, such properties are being proved and concluded. The following theorems stated are on the basis of the topological properties and separated axioms,which by satisfying, moves to a conclusion that,not only a power set is just a topology on the given defined set,but also it can be considered as a “Universal Topology”or a “Universal Topological Space”,that is the container of all topological spaces. This paper gives a general understanding about what a power set is,topologically and gives us a new perceptive from a “power set”to a “topological power space”.

On the J -reflexive sequences

We call a sequence $\mathcal{T}=(T_n)_n$ of bounded operators on a Banach space $X$, $J$-reflexive if every bounded operator on $X$ that leaves invariant, the $J$-sets of $\mathcal{T}$ is contained in the closure of $\{I, T_1, T_2, ... \}$ in the strong operator topology. We discuss some properties of $J$-reflexive sequences. We also give and prove some sufficient conditions under which an operator sequence is $J$-reflexive. Some examples are considered. Indeed, weakly $J^{mix}$-reflexivity is also defined. Finally, we extend the $J$-reflexive property in terms of subsets.

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

Density in $SBD$ and approximation of fracture energies

We prove three density theorems, in the strong BD topology, for the three subspaces of SBD functions: SBD ; SBD^p_\infty , where the absolutely continuous part of the symmetric gradient is in L^p , with p > 1 ; SBD^p , whose functions are in SBD^p_\infty and the jump set has finite \mathcal H^{n-1} -measure. We compare them with existing results, discussing related approximation of fracture energies.

Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure

We study measures on a real separable Hilbert space E that are invariant with respect to both shifts by arbitrary vectors of the space and orthogonal transformations. In particular, our first concern is a finitely additive analog of the Lebesgue measure. We present such an analog; namely, we construct a nonnegative finitely additive measure that is invariant with respect to shifts and rotations and is defined on the minimal ring of subsets of E that contains all infinite-dimensional rectangles such that the products of their side lengths converge absolutely. We also define a Hilbert space $${\mathcal H}$$ of complex-valued functions on E that are square integrable with respect to a shift- and rotation-invariant measure. For random vectors whose distributions are given by families of Gaussian measures on E that form semigroups with respect to convolution, we define expectations of the corresponding shift operators. We establish that such expectations form a semigroup of self-adjoint contractions in $${\mathcal H}$$ that is not strongly continuous, and find invariant subspaces of strong continuity for this semigroup. We examine the structure of an arbitrary semigroup of self-adjoint contractions of the Hilbert space, which may not be strongly continuous. Finally, we show that the method of Feynman averaging of strongly continuous semigroups based on the notion of Chernoff equivalence of operator-valued functions is also applicable to discontinuous semigroups.

Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW⁎-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW⁎-triples with strongly WCG predual and describe the order of seminorms defining the strong⁎ topology. As a byproduct we improve a characterization of weakly compact subsets of a JBW⁎-triple predual, providing so a proof for a conjecture, open for almost eighteen years, on weakly compact operators from a JB⁎-triple into a complex Banach space.

Analogs of the Lebesgue Measure and Diffusion in a Hilbert Space

We study shift-invariant measures on a real separable Hilbert space E, which are also invariant with respect to orthogonal transforms. In this article a finitely additive analogue of the Lebesgue measure is constructed. It is a non-negative finitely additive shift-invariant measure on a special ring of subsets from a space E, which is invariant with respect to orthogonal transforms. The ring contains all infinite-dimensional rectangles, whose products of side lengths are absolutely convergent. We define a Hilbert space \( \mathcal{\mathscr{H}} \), whose elements are complex-valued functions. The functions are square-integrable by some shift-invariant measure, which is also invariant with respect to rotations. We define the expected values of shift operators over random vectors, whose distribution is given by a family of Gaussian measures on a space E. The measures form a semigroup with respect to convolution. We prove that the expected values form a semigroup of contracting self-adjoint operators on a space \( \mathcal{\mathscr{H}} \). The semigroup is not strongly continuous. We also find invariant subspaces, where the semigroup is continuous in the strong operator topology. We investigate the structure of arbitrary (not necessarily continuous) operator semigroups of contracting self-adjoint transforms on a Hilbert space. We show that the suggested in Orlov et al. (Izv. Math. 80(6), 1131–1158 2016) method of the Feynman averaging is applicable to discontinuous semigroups.

On the removal of weak compactness arguments in proof mining

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.

Modern Physics of the Condensed State: Strong Correlations and Quantum Topology

The topic of the review is the application of new ideas of unconventional quantum states to the physics of condensed matter, in particular of solid state, in the context of modern field theory. A comparison is made with classical papers on many-electron theory, including the formalism of many-electron operators. The essentially many-particle nature of the ground state, individual and collective excitations, and quantum fluctuations in the systems under consideration, as well as quantum phase transitions are discussed with an emphasis on topological aspects and with allowance for the effects of frustration. Variational approaches and representations of auxiliary particles, corresponding mean-field approximation and gauge field theory, confinement-deconfinement problem, violation of the Fermi-liquid picture, and exotic non-Fermi-liquid states are considered. An overview is given of the modern theory of entangled topological states, formation of spin liquid, strings and string networks.

Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms

Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.

Three-dimensional superconductors with hybrid higher-order topology

We consider three dimensional superconductors in class DIII with a four-fold rotation axis and inversion symmetry. It is shown that such systems can exhibit higher order topology with helical Majorana hinge modes. In the case of even-parity superconductors we show that higher order topological superconductors can be obtained by adding a small pairing with the appropriate $C_4$ symmetry implementation to a topological insulator. We also show that a hybrid case is possible, where Majorana surface cones resulting from non-trivial strong topology coexist with helical hinge modes. We propose a bulk invariant detecting this hybrid scenario, and numerically analyse a tight binding model exhibiting both Majorana cones and hinge modes.

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

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.