Sublinear Growth Research Articles

Dirichlet problem in one-dimensional billiard space with velocity dependent right-hand side

The paper brings multiplicity results for a Dirichlet problem in one-dimensional billiard space with right-hand side depending on the velocity of the ball, i.e. a problem in the formx'' = f(t, x, x') if x(t) ∈ int K, x'(t+) = -x'(t-) if x(t) ∈ ∂K, x(0) = A, x(T) = B,where T > 0, K = [0, R], R > 0, f is a Carathéodory function on [0, T] × K × ℝ, A, B ∈ int K. Sufficient conditions ensuring the existence of at least two solutions having prescribed number of impacts with the boundary of the billiard table K are obtained. In particular, if the right-hand side has at most sublinear growth in the last variable, there exist infinitely many solutions of the problem.

Unpacking sublinear growth: diversity, stability and coexistence

How can so many species coexist in natural ecosystems remains a fundamental question in ecology. Classical models suggest that competition for space and resources should maintain the number of coexisting species far below the staggering diversity commonly found in nature. To overcome this paradox, theoretical studies have long highlighted a number of mechanisms that can favour species coexistence, from the distribution of interaction strengths between species to the shape of population growth functions. In particular, a family of mathematical models finds that, when sublinear population growth (SG) rates are coupled with competition between species, species diversity can stabilize community dynamics. This could suggest that SG may explain the stable coexistence of many species in natural ecosystems. Here we clarify why SG models do not solve the paradox of species coexistence. This is because, in the SG model, coexistence emerges from an unrealistic property, in which population per capita growth rates tend to infinity at low abundance, preventing species from ever going extinct due to competitive exclusion. Infinite growth at low abundance can be regularized by assuming a minimal abundance threshold, below which a species goes extinct or follows non‐infinite growth curves. When this is done, the SG model recovers the classical result: increasing the diversity of the species pool leads to competitive exclusion and species extinctions.

An online dual consensus algorithm for distributed resource allocation over networks

We address the problem of online resource allocation in a distributed environment where requests arrive dynamically over time at different agents in the network. As each request arrives, the receiving agent must make an immediate decision that incurs a cost and consumes a certain amount of resources. The requests are drawn independently from unknown distributions that are different for each agent. First, we present an Online Consensus Alternating Direction Method of Multipliers (OC-ADMM) algorithm for the dual counterpart of the online distributed resource allocation problem, focusing on the dual variables. Then, we propose an Online Dual Consensus ADMM (ODC-ADMM) algorithm for the primal problem to derive the primal variables from the dual update process in the OC-ADMM algorithm. The ODC-ADMM algorithm exhibits sublinear growth in both regret and expected constraint violation with respect to the time horizon. Furthermore, extensive numerical results on both synthetic and real-world data confirm its effectiveness.

On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature

On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature

Analysis of harmonic functions under lower bounds of N-weighted Ricci curvature with ε-range

The behavior of harmonic functions on Riemannian manifolds under lower bounds of the Ricci curvature has been studied from both analytic and geometric viewpoints. For example, some Liouville type theorems are obtained under lower bounds of the Ricci curvature. Recently, those results are generalized under lower bounds of the N-weighted Ricci curvature RicψN with N∈[n,∞]. In this paper, we present a Liouville type theorem for harmonic functions of sublinear growth and a gradient estimate of harmonic functions on weighted Riemannian manifolds under weaker lower bounds of RicψN with N<0. We also prove an Lp-Liouville theorem under lower bounds of RicψN with N∈[n,∞] in a different way from previous studies. Our results are obtained as a consequence of an argument under lower bounds of RicψN with ε-range, which is a unification of constant and variable curvature bounds. Among various methods considered for the analysis of harmonic functions, this paper focuses on methods using the Moser's iteration procedure.

On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature

We prove that an open noncollapsed manifold with nonnegative Ricci curvature and linear volume growth always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets for given large volumes in the above setting. We also find that under our assumptions, the diameters of the level sets of any Busemann function are uniformly bounded as opposed to a classical result stating that they can have sublinear growth when the end is collapsing. Moreover, some equivalent characterizations of linear volume growth are given. Finally, we construct an example to show that for manifolds in our setting, although their limit spaces at infinity are always cylinders, the cross sections can be nonhomeomorphic.

Decay rate of discrete-time homogeneous delayed systems with cone invariance

This paper is dedicated to analysing the decay rate of discrete-time homogeneous delayed systems that exhibit order preservation within a proper cone. First, a sufficient criterion ensuring the cone invariance of the system is introduced. By exploring the monotonicity of system trajectories, a condition that is both sufficient and necessary for guaranteeing the asymptotic stability of the system is presented. Importantly, under a weak assumption for the growth rate of time-varying delays, a universal framework is provided for characterising the decay rate of the system. Furthermore, typical examples of decay rate analysis are presented when delays are bounded or experience linear, sublinear, or logarithmic growth rates. Eventually, the rationality of the results is illustrated through a numerical simulation.

Positive Radial Symmetric Solutions of Nonlinear Biharmonic Equations in an Annulus

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵2u=f(u,▵u) on an annular domain Ω in RN with the Navier boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:R+×R−→R+ is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f(ξ,η) to have superlinear or sublinear growth on ξ,η as |(ξ,η)|→0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones.

Multi-Armed Bandit-Based User Network Node Selection.

In the scenario of an integrated space-air-ground emergency communication network, users encounter the challenge of rapidly identifying the optimal network node amidst the uncertainty and stochastic fluctuations of network states. This study introduces a Multi-Armed Bandit (MAB) model and proposes an optimization algorithm leveraging dynamic variance sampling (DVS). The algorithm posits that the prior distribution of each node's network state conforms to a normal distribution, and by constructing the distribution's expected value and variance, it maximizes the utilization of sample data, thereby maintaining an equilibrium between data exploitation and the exploration of the unknown. Theoretical substantiation is provided to illustrate that the Bayesian regret associated with the algorithm exhibits sublinear growth. Empirical simulations corroborate that the algorithm in question outperforms traditional ε-greedy, Upper Confidence Bound (UCB), and Thompson sampling algorithms in terms of higher cumulative rewards, diminished total regret, accelerated convergence rates, and enhanced system throughput.

Positive solutions for a coupled systems involving the fractional p(x)- Laplacian operator in unbounded domains

In this paper, we study the existence of at least one weak solution for a class of nonlocal elliptic systems involving the fractional p(x)-Laplacian. These operators are utilized to solve a system defined within unbounded domain. The right sides of systems treated are closely related to a type of gradient C1-functional which satisfy a sublinear growth conditions. Under some additional assumptions on the nonlinearities and leveraging the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we can use a variational approach to establish the existence result.

Normalized homoclinic solutions of discrete nonlocal double phase problems

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text]([Formula: see text] or [Formula: see text], [Formula: see text] or [Formula: see text]) is the discrete fractional [Formula: see text]-Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

Acyclic and tree unique colourings in random graphs

In this paper we study acyclic colouring in the random subgraph G of the complete graph Kn on n vertices where each edge is present with probability p, independent of the other edges. We show that unlike the ordinary chromatic number, the acyclic chromatic number exhibits a phase transition from sub-linear to linear growth as the edge probability increases, even in the sparse regime and obtain estimates for the critical exponent. We show that allowing for a small relaxation in the acyclic colouring condition allows for sublinear growth for all values of the edge probability exponent. We also study a generalization of unique colourings where we impose the condition that at least one vertex in any copy of a given tree has a unique colour and obtain bounds for the minimum number of colours needed for such a colouring.

Diversity begets stability: Sublinear growth and competitive coexistence across ecosystems.

The worldwide loss of species diversity brings urgency to understanding how diverse ecosystems maintain stability. Whereas early ecological ideas and classic observations suggested that stability increases with diversity, ecological theory makes the opposite prediction, leading to the long-standing "diversity-stability debate." Here, we show that this puzzle can be resolved if growth scales as a sublinear power law with biomass (exponent <1), exhibiting a form of population self-regulation analogous to models of individual ontogeny. We show that competitive interactions among populations with sublinear growth do not lead to exclusion, as occurs with logistic growth, but instead promote stability at higher diversity. Our model realigns theory with classic observations and predicts large-scale macroecological patterns. However, it makes an unsettling prediction: Biodiversity loss may accelerate the destabilization of ecosystems.

Observation of Subdiffusive Dynamic Scaling in a Driven and Disordered Bose Gas.

We explore the dynamics of a tuneable box-trapped Bose gas under strong periodic forcing in the presence of weak disorder. In absence of interparticle interactions, the interplay of the drive and disorder results in an isotropic nonthermal momentum distribution that shows subdiffusive dynamic scaling, with sublinear energy growth and the universal scaling function captured well by a compressed exponential. We explain that this subdiffusion in momentum space can naturally be understood as a random walk in energy space. We also experimentally show that for increasing interaction strength, the gas behavior smoothly crosses over to wave turbulence characterized by a power-law momentum distribution, which opens new possibilities for systematic studies of the interplay of disorder and interactions in driven quantum systems.

Quasilinear Lane–Emden type systems with sub-natural growth terms

Global pointwise estimates are obtained for quasilinear Lane–Emden-type systems involving measures in the “sublinear growth” rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff’s potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.

Asymptotic decay of solutions for sublinear fractional Choquard equations

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))f(u)onRNwhere s∈(0,1), N≥2, α∈(0,N), μ>0, Iα denotes the Riesz potential and F(t)=∫0tf(τ)dτ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than ∼1|x|N+2s. The result is new even for homogeneous functions f(u)=|u|r−2u, r∈[N+αN,2), and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case s=1 in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new “s-sublinear” threshold that we detect on the growth of f. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.

INTaaS: Provisioning In-band Network Telemetry as a service via online learning

Provisioning In-band Network Telemetry as a service for INT-based and INT follow-up applications in an online manner suffers multiple challenges, including control decisions of different INT path algorithms, resources provisioning coupled over time, and the unforeseeable INT query workloads. To overcome these challenges, this study formulates an online non-linear time-varying integer programming problem that maximizes the overall quality of service through algorithm selection and INT query workloads distribution. Specifically, an online learning algorithm is proposed to make the fractional decisions and it uses a primal–dual mechanism by simultaneously minimizing a convex problem and maximizing a concave problem, based on the previously observed inputs. Furthermore, we design a randomized rounding strategy to convert these fractional decisions to integers with an expectation guarantee. We implement our system prototype based on INTCollector upon real devices, i.e., Barefoot Wedge100BF and Inspur Rack. Our rigorous theoretical analysis shows that our INTaaS achieves sub-linear growth on dynamic regret for the optimal loss and incurs sub-linear growth on dynamic fit for the long-term constraint violations. Finally, extensive evaluations on real-world data indicate that INTaaS exhibits up-lift performance up to 40% over other state-of-the-art algorithms.

Robustness of attractors in thin domains for a nonlocal reaction-diffusion equation with sublinear growth term

Robustness of attractors in thin domains for a nonlocal reaction-diffusion equation with sublinear growth term

Obsolescence effects in second language phonological networks

Phonological networks are representations of word forms and their phonological relationships with other words in a given language lexicon. A principle underlying the growth (or evolution) of those networks is preferential attachment, or the “rich-gets-richer” mechanisms, according to which words with many phonological neighbors (or links) are the main beneficiaries of future growth opportunities. Due to their limited number of words, language lexica constitute node-constrained networks where growth cannot keep increasing in a linear way; hence, preferential attachment is likely mitigated by certain factors. The present study investigated obsolescence effects (i.e., a word’s finite timespan of being active in terms of growth) in an evolving phonological network of English as a second language. It was found that phonological neighborhoods are constructed by one large initial lexical spurt, followed by sublinear growth spurts that eventually lead to very limited growth in later lexical spurts during network evolution. First-language-given neighborhood densities are rarely reached even by the most advanced language learners. An analysis of the strength of phonological relationships between phonological word forms revealed a tendency to incorporate phonetically more distant phonological neighbors at earlier acquisition stages. Overall, the findings suggest an obsolescence effect in growth that favors younger words. Implications for the second-language lexicon include leveraged learning mechanisms and learning bouts focused on a smaller range of phonological segments, and involve questions concerning lexical processing in aging networks.

Liouville theorem for [formula omitted]-harmonic maps under non-negative [formula omitted]-Ricci curvature for non-positive [formula omitted

Let V be a C1-vector field on an n-dimensional complete Riemannian manifold (M,g). We prove a Liouville theorem for V-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into Cartan–Hadamard manifolds, which extends Cheng’s Liouville theorem proved in Cheng (1980) for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan–Hadamard manifolds. We also prove a Liouville theorem for V-harmonic maps from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt et al. (1980) and Choi (1982). Our probabilistic proof of Liouville theorem for several growth V-harmonic maps into Hadamard manifolds enhances an incomplete argument in Stafford (1990). Our results extend the results due to Chen et al. (2012) and Qiu (2017) in the case of m=+∞ on the Liouville theorem for bounded V-harmonic maps from complete Riemannian manifolds with non-negative (∞,V)-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V-harmonic maps and the recurrence property of ΔV-diffusion processes on manifolds. Our results are new even in the case V=∇f for f∈C2(M).