Topological Conditions Research Articles

Learned load balancing

Effectively balancing traffic in datacenter networks is a crucial operational goal. Most existing load balancing approaches are handcrafted to the structure of the network and/or network workloads. Thus, new load balancing strategies are required if the underlying network conditions change, e.g., due to hard or grey failures, network topology evolution, or workload shifts. While we can theoretically derive the optimal load balancing strategy by solving an optimization problem given certain traffic and topology conditions, these problems take too much time to solve and makes the derived solution stale to deploy. In this paper, we describe a load balancing scheme Learned Load Balancing (LLB), which is a general approach to finding an optimal load balancing strategy for a given network topology and workload, and is fast enough in practice to deploy the inferred strategies. LLB uses deep supervised learning techniques to learn how to handle different traffic patterns and topology changes, and adapts to any failures in the underlying network. LLB leverages emerging trends in network telemetry, programmable switching, and “smart” NICs. Our experiments show that LLB performs well under failures and can be expanded to more complex, multi-layered network topologies. We also prototype neural network inference on smartNICs to demonstrate the workability of LLB.

ENHANCING 3T RETINOTOPIC MAPS USING DIFFEOMORPHIC REGISTRATION.

Retinotopic mapping aims to uncover the relationship between visual stimuli on the retina and neural responses on the visual cortical surface. This study advances retinotopic mapping by applying diffeomorphic registration to the 3T NYU retinotopy dataset, encompassing analyze-PRF and mrVista data. Diffeomorphic Registration for Retinotopic Maps (DRRM) quantifies the diffeomorphic condition, ensuring accurate alignment of retinotopic maps without topological violations. Leveraging the Beltrami coefficient and topological condition, DRRM significantly enhances retinotopic map accuracy. Evaluation against existing methods demonstrates DRRM's superiority on various datasets, including 3T and 7T retinotopy data. The application of diffeomorphic registration improves the interpretability of low-quality retinotopic maps, holding promise for clinical applications.

Topological properties and skin effects modulated by different-density nonreciprocal couplings in the Su-Schrieffer-Heeger structure

We pay attention to the topological properties and skin effects in the one-dimensional Su-Schrieffer-Heeger (SSH) structure, by considering the presence of different-density non-reciprocal couplings. We find that when only one single non-reciprocal coupling term is introduced, e.g., in the middle of the SSH structure, edge states can be induced around the sites suffering from the non-reciprocal coupling. As one non-reciprocal coupling term is introduced in each two cells, the topological phase transition is modified. Besides, new topological edge states and skin effects appear at the ends of the SSH structure, and the edge states can be differentiated by their localities, localization degrees, and the imaginary part of energy. Furthermore, if two neighboring cells of the SSH structure are coupled by the non-reciprocal coupling, the topological edge states and skin effect more well-defined, accompanied by the further change of the topological phase transition condition. These results suggest that the non-reciprocal coupling plays a unique and important role in modulating the topological properties and the skin effects of the one-dimensional SSH structure.

Thermodynamics and kinetics of interconversion between platinum nanoparticles and cations in zeolites

In metal-containing zeolites, sintering and redispersion processes exercise control over the identity of metal site structures, but the thermodynamic and kinetic factors that influence these molecular-level processes are not completely understood. Here, we assess the ability of first principles informed free energy models (for supported and unsupported nanoparticles) and kinetic models integrating Ostwald ripening with atom trapping to describe the interconversion between Pt cations and nanoparticles encapsulated in chabazite (CHA) zeolites. Density functional theory-derived thermodynamic phase diagrams show that the interconversion between cations, favored in oxidizing environments, and particles, favored in reducing environments, is fully reversible within a wide range of their respective conditions (temperatures and pressures) for CHA and several other zeolite topologies. Kinetic Monte Carlo simulations of Pt redispersion are consistent with experimentally observed redispersion kinetics of encapsulated Pt nanoparticles in CHA zeolites, and model results suggest the zeolite host imparts additional stability for Pt nanoparticles. We envision our thermodynamic and kinetic models for Pt-CHA are also capable of describing nanoparticle and cation interconversions for other zeolite frameworks under similar conditions.

AFM-IR study of interfacial nanostructures in high-temperature dilute nanocomposites

The dilute nanocomposite strategy facilitates a promising route for high-temperature polymer dielectrics to achieve high dielectric constant and energy density, robust breakdown strength, and low loss. The dielectric enhancement of the dilute nanocomposites strongly ties to the topological conditions of the nanofiller–polymer interface. Here, atomic force microscopy-based infrared spectroscopy was employed to observe the interfacial region of a high-temperature poly(arylene ether urea) (PEEU) dilute nanocomposite directly. The experimental results showed that the nanoparticles induce morphological changes in PEEU, and the interface region extends over 150 nm away from the nanofiller surfaces. Such nano-morphology changes are highly indicative of generating a large enhancement of the dielectric performance in the PEEU dilute nanocomposite.

Big Mapping Class Groups and the Co-Hopfian Property

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

Discord in the voter model for complex networks.

Online social networks have become primary means of communication. As they often exhibit undesirable effects such as hostility, polarization, or echo chambers, it is crucial to develop analytical tools that help us better understand them. In this paper we are interested in the evolution of discord in social networks. Formally, we introduce a method to calculate the probability of discord between any two agents in the multistate voter model with and without zealots. Our work applies to any directed, weighted graph with any finite number of possible opinions, allows for various update rates across agents, and does not imply any approximation. Under certain topological conditions, the opinions are independent and the joint distribution can be decoupled. Otherwise, the evolution of discord probabilities is described by a linear system of ordinary differential equations. We prove the existence of a unique equilibrium solution, which can be computed via an iterative algorithm. The classical definition of active links density is generalized to take into account long-range, weighted interactions. We illustrate our findings on real-life and synthetic networks. In particular, we investigate the impact of clustering on discord and uncover a rich landscape of varied behaviors in polarized networks. This sheds lights on the evolution of discord between, and within, antagonistic communities.

Some results regarding the ideal structure of C∗$C^*$‐algebras of étale groupoids

AbstractWe prove a sandwiching lemma for inner‐exact locally compact Hausdorff étale groupoids. Our lemma says that every ideal of the reduced ‐algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced ‐algebra, and triples consisting of two nested open invariant sets and an ideal in the ‐algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced ‐algebras of inner‐exact locally compact Hausdorff étale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.

Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system spectral gaps uniform in the system size. To obtain this result, we extend the Bravyi–Hastings–Michalakis strategy so it can be applied to perturbations of the GNS Hamiltonian of the infinite-system ground state.

Generalized essential spectra involving the class of g-g-Riesz operators

Abstract In this paper, we explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X. The results are formulated in terms of some topological conditions made on X or on its dual X * {X^{*}} . In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided.

Incomplete preferences or incomplete information? On Rationalizability in games with private values

I propose a notion of Rationalizability, called Incomplete Preference Rationalizability, for games with incomplete preferences. Under an appropriate topological condition, the incomplete preference rationalizable set is non-empty and compact. I argue that incomplete orderings can be used to model incomplete information in strategic settings. Drawing on this connection, I show that in games with private values the sets of incomplete preference rationalizable actions, of belief-free rationalizable actions (Battigalli et al., 2011; Bergemann and Morris, 2017), and of interim correlated rationalizable actions (Dekel et al., 2007) of the universal type space coincide.

Cross-Layer Methods for Ad Hoc Networks—Review and Classification

The OSI model used to be a common network model for years. In the case of ad hoc networks with dynamic topology and difficult radio communications conditions, gradual departure is happening from the classical kind of OSI network model with a clear delineation of layers (physical, channel, network, transport, application) to the cross-layer approach. The layers of the network model in ad hoc networks strongly influence each other. Thus, the cross-layer approach can improve the performance of an ad hoc network by jointly developing protocols using interaction and collaborative optimization of multiple layers. The existing cross-layer methods classification is too complicated because it is based on the whole manifold of network model layer combinations, regardless of their importance. In this work, we review ad hoc network cross-layer methods, propose a new useful classification of cross-layer methods, and show future research directions in the development of ad hoc network cross-layer methods. The proposed classification can help to simplify the goal-oriented cross-layer protocol development.

Deformations of Calabi–Yau varieties with k-liminal singularities

Abstract The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least $4$ . For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with $1$ -liminal singularities (which are exactly the ordinary double points in dimension $3$ but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions.

Supra v-open set and some of its properties

The idea of supra v-open set is introduced, and several different properties of supra v-open set are proven. Furthermore, the notion of supra v-space has been presented and which is satisfying the conditions of supra topology. In addition, the relations between v-open set and supra v-open set have been shown. Finally, the concept supra v-base and some of its properties were studied.

A new view of spaces and their properties in the sense of non-Newtonian measure.

This study presents a novel approach to metric spaces through the lens of geometric calculus, redefining traditional structures with new operations and properties derived from non-Newtonian measures. Specifically, we develop and prove geometric versions of the Hölder and Minkowski inequalities, which provide foundational support for applying these spaces in analysis. Additionally, we establish key relationships between geometric and classical metric spaces, examining concepts such as openness, closedness, and separability within this geometric framework. By exploring topological characteristics and separability conditions in geometric metric spaces, this work enhances the understanding of metric spaces' structural properties, offering potential applications in fields that require flexible metric adaptations, such as data science, physics, and computational geometry. This framework's adaptability makes it relevant for scenarios where non-Euclidean or high-dimensional spaces are needed, allowing for versatile applications and extending classical metric concepts into broader analytical contexts.

Can ephapticity contribute to brain complexity?

The inquiry into the origin of brain complexity remains a pivotal question in neuroscience. While synaptic stimuli are acknowledged as significant, their efficacy often falls short in elucidating the extensive interconnections of the brain and nuanced levels of cognitive integration. Recent advances in neuroscience have brought the mechanisms underlying the generation of highly intricate dynamics, emergent patterns, and sophisticated oscillatory signals into question. Within this context, our study, in alignment with current research, postulates the hypothesis that ephaptic communication, in addition to synaptic mediation's, may emerge as a prime candidate for unraveling optimal brain complexity. Ephaptic communication, hitherto little studied, refers to direct interactions of the electric field between adjacent neurons, without the mediation of traditional synapses (electrical or chemical). We propose that these electric field couplings may provide an additional layer of connectivity that facilitates the formation of complex patterns and emergent dynamics in the brain. In this investigation, we conducted a comparative analysis between two types of networks utilizing the Quadratic Integrate-and-Fire Ephaptic model (QIF-E): (I) a small-world synaptic network (ephaptic-off) and (II) a mixed composite network comprising a small-world synaptic network with the addition of an ephaptic network (ephaptic-on). Utilizing the Multiscale Entropy methodology, we conducted an in-depth analysis of the responses generated by both network configurations, with complexity assessed by integrating across all temporal scales. Our findings demonstrate that ephaptic coupling enhances complexity under specific topological conditions, considering variables such as time, spatial scales, and synaptic intensity. These results offer fresh insights into the dynamics of communication within the nervous system and underscore the fundamental role of ephapticity in regulating complex brain functions.

Distributed GNE Seeking Strategy for Second-Integrator Multiplayer Systems Over Directed Topologies.

This article studies the generalized Nash equilibrium (GNE) seeking problem of second-integrator multiplayer systems. In particular, each player is endowed with an individual payoff function with respect to collective decision variables, and simultaneously, a coupling inequality constraint and a set constraint are imposed to each player. The players communicate with their local neighbors over a directed topology. To begin with, a distributed-observer-based seeking strategy is synthesized by leveraging a proper composite variable. It is first demonstrated using nonsmooth analysis that the established distributed observer enables each player to accurately estimate the decision variables of others in terms of a strongly connected topology condition. Upon this basis, all the decision variables are then shown to converge to the expected GNE asymptotically borrowing from convex theory. In addition, three extension results are also given under the built GNE seeking framework. First, under the postulation that the velocity information is unavailable, a velocity-free distributed GNE seeking strategy is synthesized for second-integrator systems by implementing a proper auxiliary dynamics. Second, we consider nonlinear Euler-Lagrange systems with unknown inertia parameters and synthesize an improved distributed GNE seeking strategy resorting to an adaptation technique. Third, we focus on integrator chain systems and synthesize a modified distributed GNE seeking strategy using a new composite variable based on a proper coordinate transformation. For three extension cases, we all show in detail the achievement of the GNE seeking objective. Finally, a practical example is simulated to confirm the built GNE seeking results.

Persistent homology for functionals

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.

Some rigidity results for compact initial data sets

In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases. In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main result by Galloway and Mendes [Comm. Anal. Geom. 26 (2018), pp. 63–83]. We also obtain some extensions of results by Eichmair, Galloway, and Mendes [Comm. Math. Phys. 386 (2021), pp. 253–268]. A number of examples are given in order to illustrate some of the results presented in this paper.