Alessandro Sanzeni (9389509), Mark H. Histed (9389512), Nicolas Brunel (82387) · Figshare · 2020
<p>Network response in strongly coupled networks (left) features linear regions separated by abrupt transitions (red circles). In the weak coupling limit, the network transfer function becomes identical to the single cell transfer function. It is supralinear at low inputs, but then becomes sublinear at high inputs.</p>
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Qi Ding · Analysis & PDE · 2025
Let be a complete Riemannian manifold of nonnegative Ricci curvature.We prove a Liouville-type theorem: every smooth solution u to the minimal hypersurface equation on is a constant provided u has sublinear growth for its negative part.Here, the sublinear growth condition is sharp.Our proof relies on a gradient estimate for minimal graphs over with small linear growth of the negative parts of graphic functions via iteration.which is a nonlinear partial differential equation describing the minimal graph M D f.x; u.x// 2 j x 2 g over .The smooth solution u to (1-1) is the height function of the minimal graph M in . Therefore, we call u a minimal graphic function on .When is Euclidean space n , equation (1-1) has been studied successfully by many mathematicians.Bombieri, De Giorgi, and Miranda [Bombieri et al. 1969b] (see also [Gilbarg and Trudinger 1983]) proved interior gradient estimates for solutions to the minimal hypersurface equation on n , where the 2-dimensional case had already been obtained in [Finn 1954].As a corollary, they immediately got a Liouville-type theorem in [Bombieri et al. 1969b] as follows.Theorem 1.1.If a minimal graphic function u on n satisfies sublinear growth for its negative part, i.e., lim supx!1 maxf u.x/; 0g jxj D 0;(1-2) then u is a constant.The condition (1-2) is sharp since any affine function is a minimal graphic function on n .When the minimal graphic function u on n has the uniformly bounded gradient, Moser [1961] proved u is affine
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Yanfu Qin, Kaihong Lu, Hongxia Wang · IEEE Transactions on Network Science and Engineering · 2024
In this paper, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents. Each agent's objective function is time-varying and nonconvex and the agents exchange information with their neighbors through a time-varying directed graph. Particularly, here the graph is not assumed to be balanced. To handle this problem, an online distributed algorithm is proposed based on the projection-free strategy and the push-sum protocol. The algorithm's performance is measured using dynamic regrets whose offline benchmark is to find a stationary point of the objective function at each time. Under mild assumptions on the graph and objective functions, we prove that if the deviation of the objective function sequence is sublinear with the square root of the time horizon, and the deviation of the objective function gradient sequence is sublinear with the time horizon, then the dynamic regret increases sublinearly. Finally, simulation experiments are presented to verify the effectiveness of the theoretical results.
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Junping Shi, Ratnasingham Shivaji · SHILAP Revista de lepidopterología · 2016
The uniqueness of positive solution of a semilinear cooperative Hamiltonian elliptic system with two equations is proved for the case of sublinear and superlinear nonlinearities. Implicit function theorem, bifurcation theory, and ordinary differential equation techniques are used in the proof.
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Alexandru Kristály, Vicenţiu D. Rădulescu · Studia Mathematica · 2009
Let $(M,g)$ be a compact Riemannian manifold without boundary, with $\dim M\geq 3,$ and $f:\mathbb R \to \mathbb R$ a continuous function which is {sublinear} at infinity. By various variational approaches, existence of multiple solutions of the eigenva
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Thabo Samakhoana, Benjamin Grimmer · ArXiv.org · 2025
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for any target level of smoothness. Optimal smoothings restricting to either inner or outer approximations also follow from our theory. Finally, we apply our theory to provide insights into smoothing amenable functions given by compositions with sublinear functions and generic convex sets by expressing them as conic sections.
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Ido Grayevsky, Gabriel Pallier · ArXiv.org · 2024
We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. We employ our product theorem to distinguish up to quasiisometry certain families of solvable groups which share the same dimension, cone-dimension and Dehn function; actually we do this by distinguishing them up to sublinear bilipschitz equivalence, which is slightly stronger. As an application, we recover the fact, recently obtained by Bourdon and Rémy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.
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Ying Wai Lee · arXiv (Cornell University) · 2026
The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in which the distinct-digit count grows at a positive linear rate or at a prescribed sublinear rate are determined. The resulting dimension laws exhibit a sharp phase transition: imposing any positive linear rate forces the dimension to collapse to a value determined solely by the tail index, whereas under a broad class of sublinear growth rates, the exceptional sets retain full Hausdorff dimension.
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Shoham Letzter · Cambridge University Press eBooks · 2024
In this survey we aim to give a comprehensive overview of results using sublinear expanders. The term sublinear expanders refers to a variety of definitions of expanders, which typically are defined to be graphs G such that every not-too-small and not-too-large set of vertices U has neighbourhood of size at least α|U|, where α is a function of n and |U|. This is in contrast with linear expanders, where α is typically a constant. We will briefly describe proof ideas of some of the results mentioned here, as well as related open problems.
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Ying Wai Lee · ArXiv.org · 2026
The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in which the distinct-digit count grows at a positive linear rate or at a prescribed sublinear rate are determined. The resulting dimension laws exhibit a sharp phase transition: imposing any positive linear rate forces the dimension to collapse to a value determined solely by the tail index, whereas under a broad class of sublinear growth rates, the exceptional sets retain full Hausdorff dimension.
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Jincheng Zhang · Zenodo (CERN European Organization for Nuclear Research) · 2026
This paper proposes a novel optimization framework that organically combines the constraints of Dual Semi-Definite Programming (Dual SDP) with the sublinear rate of convergence (O(1/t)) method, resulting in a convergent and efficient algorithm for large-scale problems. By constructing dual relaxations for the primal problem, introducing a dynamic step-size adjustment strategy, and combining this with sublinear convergence rate analysis, we construct a new iterative optimization equation: x_{t+1} = x_t - α_t ∇x L_d(x_t, λ_t), λ{t+1} = Π_Λ [ λ_t + β_t ∇_λ L_d(x_t, λ_t) ], where L_d(x, λ) represents the dual Lagrangian function, α_t and β_t are adaptive step sizes dependent on t, and Π_Λ represents the projection operation on λ. Theoretical analysis shows that this framework converges to the ε-optimal solution in nonsmooth semidefinite constrained problems in O(1/ε) iterations, while also considering computational complexity. Experimental results demonstrate that this method outperforms the combination of the traditional dual SDP and standard sublinear methods on multinomial semidefinite optimization problems.
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Jincheng Zhang · Zenodo (CERN European Organization for Nuclear Research) · 2026
This paper proposes a novel optimization framework that organically combines the constraints of Dual Semi-Definite Programming (Dual SDP) with the sublinear rate of convergence (O(1/t)) method, resulting in a convergent and efficient algorithm for large-scale problems. By constructing dual relaxations for the primal problem, introducing a dynamic step-size adjustment strategy, and combining this with sublinear convergence rate analysis, we construct a new iterative optimization equation: x_{t+1} = x_t - α_t ∇x L_d(x_t, λ_t), λ{t+1} = Π_Λ [ λ_t + β_t ∇_λ L_d(x_t, λ_t) ], where L_d(x, λ) represents the dual Lagrangian function, α_t and β_t are adaptive step sizes dependent on t, and Π_Λ represents the projection operation on λ. Theoretical analysis shows that this framework converges to the ε-optimal solution in nonsmooth semidefinite constrained problems in O(1/ε) iterations, while also considering computational complexity. Experimental results demonstrate that this method outperforms the combination of the traditional dual SDP and standard sublinear methods on multinomial semidefinite optimization problems.
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Sadygov Misraddin Allahverdi oglu · Zenodo (CERN European Organization for Nuclear Research) · 2025
In this paper we study n-convex functions. We mainly study four types of n-convex functions: lower semicontinuous n-convex functions, simple n-convex, n(A)-sublinear and n-sublinear functions. We also consider nconvex functions that are the pointwise upper bound of some non-empty family of n-affine, n(A)-affine, n-linear and n(A)-linear functions, respectively. It is shown that n-convex functions are closely related to convex functions defined on the tensor product of spaces. In this paper we study n-subdifferentiability of n-convex functions, simple n-convex, n-sublinear functions and study a number of their properties. In this paper we also study the properties of n-conjugate functions. In this paper we study the n-subdifferential of the maximum function. Using convex hulls of n-convex functions on the tensor product of spaces, criteria for n-subdifferentiability of n-convex functions are obtained. A characterization of n-convex functions and n-convex sets is considered.
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Alberto Seeger · HAL (Le Centre pour la Communication Scientifique Directe) · 2014
Up to orthogonal transformation, a solid closed convex cone K in the Euclidean space Rn+1 is the epigraph of a nonnegative sublinear function f: Rn → R. This work explores the link between the geometric properties of K and the analytic properties of f.
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Ram Surat Chauhan, Debdas Ghosh, Jaroslav Ramı́k et al. · Research Square · 2021
Abstract This paper is devoted to the study of gH-Clarke derivative for interval-valued functions. To develop the properties of gH-Clarke derivative, the concepts of limit superior, limit inferior, and sublinear interval-valued functions are studied in the sequel. It is proved that the upper gH-Clarke derivative of a gH-Lipschitz continuous interval- valued function (IVF) always exists. Further, it is found that for a convex and gH-Lipschitz IVF, the upper gH-Clarke derivative coincides with the gH-directional derivative. It is observed that the upper gH-Clarke derivative is a sublinear IVF. Several numerical examples are provided to support the study.
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Waed Muhsin, Osama Moaaz, Sameh Askar et al. · Symmetry · 2023
In this work, new oscillation criteria are established for a second-order differential equation with several sublinear neutral terms and in the canonical case. To determine the oscillation conditions, we followed the Riccati approach and also compared the studied equation with a first-order delay equation. Obtaining the oscillation conditions required deducing some new relationships linking the solution to the corresponding function as well as its derivatives. The paper addresses some interesting analytical points in the study of the oscillation of equations with several sublinear neutral terms. These new findings complement some well-known findings in the literature. Furthermore, an example is provided to show the importance of the results.
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Ran Canetti, Yuval Ishai, Ravi Kumar et al. · 2001
Motivated by the application of private statistical analysis of large databases, we consider the problem of selective private function evaluation (SPFE). In this problem, a client interacts with one or more servers holding copies of a database x = x1, … , xn in order to compute f(xi1, … , xim), for some function f and indices i = i1, … , im chosen by the client. Ideally, the client must learn nothing more about the database than f(xi, … , xim), and the servers should learn nothing.Generic solutions for this problem, based on standard techniques for secure function evaluation, incur communication complexity that is at least linear in n, making them prohibitive for large databases even when f in relatively simple and m is small. We present various approaches for constructing sublinear-communication SPFE protocols, both for the general problem and for special cases of interest. Our solutions not only offer sublinear communication complexity, but are also practical in many scenarios.
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Romain D. Cazé, Alexandra Tran-Van-Minh, Boris Gutkin et al. · Scientific Reports · 2024
Theory predicts that nonlinear summation of synaptic potentials within dendrites allows neurons to perform linearly non-separable computations (LNSCs). Using Boolean analysis approaches, we predicted that both supralinear and sublinear synaptic summation could allow single neurons to implement a type of LNSC, the feature binding problem (FBP), which does not require inhibition contrary to the exclusive-or function (XOR). Notably, sublinear dendritic operations enable LNSCs when scattered synaptic activation generates increased somatic spike output. However, experimental demonstrations of scatter-sensitive neuronal computations have not yet been described. Using glutamate uncaging onto cerebellar molecular layer interneurons, we show that scattered synaptic-like activation of dendrites evoked larger compound EPSPs than clustered synaptic activation, generating a higher output spiking probability. Moreover, we also demonstrate that single interneurons can indeed implement the FBP. Using a biophysical model to explore the conditions in which a neuron might be expected to implement the FBP, we establish that sublinear summation is necessary but not sufficient. Other parameters such as the relative sublinearity, the EPSP size, depolarization amplitude relative to action potential threshold, and voltage fluctuations all influence whether the FBP can be performed. Since sublinear synaptic summation is a property of passive dendrites, we expect that many different neuron types can implement LNSCs.
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Amitabh Basu, Giacomo Zambelli · 2009
We show that, given a closed convex set K containing the origin in its interior, the support function of the set {y ∈ K ∗ | ∃x ∈ K such that 〈x, y 〉 = 1} is the pointwise smallest among all sublinear functions σ such that K = {x |σ(x) ≤ 1}. 1
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G. N. Chhatria, Said R. Grace, John R. Graef · Journal of the Egyptian Mathematical Society · 2021
Abstract The authors present necessary and sufficient conditions for the oscillation of a class of second order non-linear neutral dynamic equations with non-positive neutral coefficients by using Krasnosel’skii’s fixed point theorem on time scales. The nonlinear function may be strongly sublinear or strongly superlinear.
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