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About 1,106 papers for “Sublinear function”
Mingq an Wei · Journal of Mathematical Inequalities · 2023
In this paper, we establish the block decomposition for Herz spaces associated with ball Banach function spaces.By using this decomposition, we obtain the boundedness for a class of sublinear operators on Herz spaces associated with ball Banach function spaces.
Cited by 1Open access
Anna Kamińska, Damian Kubiak · Commentationes Mathematicae · 2016
We show that under some assumptions on the Musielak−Orlicz function generating a quasi-Banach Musielak−Orlicz function space, the Banach envelope of the weighted Cesaro−Musielak−Orlicz space generated by a certain positive sublinear operator is a weighted \(L_1\)-space.
Cited by 1
George A. Anastassiou · Studia Universitatis Babes-Bolyai Matematica · 2018
Here we study the approximation of functions by a big variety of Max- Product operators under conformable fractional di¤erentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under conformable fractional initial conditions. So our approach is quan- titative by producing inequalities with their right hand sides involving the modulus of continuity of a high order conformable fractional derivative of the function under approximation.
Cited by 3Open access
C. Montaño-García, I. Gamboa‐deBuen · Radiation Protection Dosimetry · 2006
The absorption spectra and glow curves of LiF:Mg,Ti exposed to high doses of 60Co gamma rays as well as the optical density curves and thermoluminescent (TL) response as a function of dose were measured. Absorption peaks around 442 nm (2.8 eV), 307 nm (4.0 eV) and 248 nm (5.0 eV) were observed. The optical density is a linear function of dose up to 2500 Gy and then sublinear for the 5.0 eV band, and for the 4.0 eV band it is sublinear with a filling constant of 2.1 x 10(-4) and 6.2 x 10(-4) Gy(-1). The dose response of the 2.8 eV band is linear from 2500 Gy to 10 kGy with a slope of 1.0 x 10(-5) Gy(-1). Once the absorption spectra were obtained, the glow curves and the TL response were obtained. In the dose range, 290-10,000 Gy, the TL response of peaks 5 and 7 was supralinear-sublinear.
Cited by 12
Wenbo Liu, Rafael Pass · 2021
Let MKtP[s] be the set of strings x such that Kt(x) ≤ s(|x|), where Kt(x) denotes the t-bounded Kolmogorov complexity of the truthtable described by x. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every ε>0, polynomial t(n) ≥ (1+ε)n, and every “nice” class F of super-polynomial functions, the following are equivalent: (i) the existence of some function T ∈ F such that T-hard one-way functions (OWF) exists (with non-uniform security); (ii) the existence of some function T ∈ F such that MKtP[T−1] is mildly average-case hard with respect to sublinear-time non-uniform algorithms (with running-time nδ for some 0<δ<1). For instance, existence of subexponentially-hard (resp. quasi-poly-nomially-hard) OWFs is equivalent to mild average-case hardness of MKtP[poly logn] (resp. MKtP[2O(√logn))]) w.r.t. sublinear-time non-uniform algorithms. We additionally note that if we want to deduce T-hard OWFs where security holds w.r.t. uniform T-time probabilistic attackers (i.e., uniformly-secure OWFs), it suffices to assume sublinear time hardness of MKtP w.r.t. uniform probabilistic sublinear-time attackers. We complement this result by proving lower bounds that come surprisingly close to what is required to unconditionally deduce the existence of (uniformly-secure) OWFs: MKtP[polylogn] is worst-case hard w.r.t. uniform probabilistic sublinear-time algorithms, and MKtP[n−logn] is mildly average-case hard for all O(t(n)/n3)-time deterministic algorithms.
Cited by 0
Shahar Fattal, Dana Ron · 2008
In this paper we study the problem of approximating the distance of a function to convexity. Namely, we are interested in randomized sublinear algorithms that approximate the Hamming distance between a given function and the closest convex function. We present an efficient constant factor approximation algorithm for functions f: [n] → ℜ. 1
Cited by 0
Bin Fu · arXiv (Cornell University) · 2016
We develop a randomized approximation algorithm for the classical maximum coverage problem, which given a list of sets $A_1,A_2,\cdots, A_m$ and integer parameter $k$, select $k$ sets $A_{i_1}, A_{i_2},\cdots, A_{i_k}$ for maximum union $A_{i_1}\cup A_{i_2}\cup\cdots\cup A_{i_k}$. In our algorithm, each input set $A_i$ is a black box that can provide its size $|A_i|$, generate a random element of $A_i$, and answer the membership query $(x\in A_i?)$ in $O(1)$ time. Our algorithm gives $(1-{1\over e})$-approximation for maximum coverage problem in $O(p(m))$ time, which is independent of the sizes of the input sets. No existing $O(p(m)n^{1-ε})$ time $(1-{1\over e})$-approximation algorithm for the maximum coverage has been found for any function $p(m)$ that only depends on the number of sets, where $n=\max(|A_1|,\cdots,| A_m|)$ (the largest size of input sets). The notion of partial sublinear time algorithm is introduced. For a computational problem with input size controlled by two parameters $n$ and $m$, a partial sublinear time algorithm for it runs in a $O(p(m)n^{1-ε})$ time or $O(q(n)m^{1-ε})$ time. The maximum coverage has a partial sublinear time $O(p(m))$ constant factor approximation. On the other hand, we show that the maximum coverage problem has no partial sublinear $O(q(n)m^{1-ε})$ time constant factor approximation algorithm. It separates the partial sublinear time computation from the conventional sublinear time computation by disproving the existence of sublinear time approximation algorithm for the maximum coverage problem.
Cited by 1Open access
Junmei Wang · Pure and Applied Mathematics Journal · 2019
In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of <i>A<sub>p</sub></i> weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of <i>BMO</i> functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and <i>BMO</i> functions are also considered. And the boundedness of commutator is obtained by the lemma of function <i>BMO</i>.
Cited by 0Open access
Katsuo Matsuoka, Javier Soria · Kyoto University Research Information Repository (Kyoto University) · 2008
Cited by 0Open access
Ching-An Cheng, Jonathan Lee, Ken Goldberg et al. · arXiv (Cornell University) · 2019
Online learning is a powerful tool for analyzing iterative algorithms.\nHowever, the classic adversarial setup sometimes fails to capture certain\nregularity in online problems in practice. Motivated by this, we establish a\nnew setup, called Continuous Online Learning (COL), where the gradient of\nonline loss function changes continuously across rounds with respect to the\nlearner's decisions. We show that COL covers and more appropriately describes\nmany interesting applications, from general equilibrium problems (EPs) to\noptimization in episodic MDPs. In particular, we show monotone EPs admits a\nreduction to achieving sublinear static regret in COL. Using this new setup, we\nrevisit the difficulty of sublinear dynamic regret. We prove a fundamental\nequivalence between achieving sublinear dynamic regret in COL and solving\ncertain EPs. With this insight, we offer conditions for efficient algorithms\nthat achieve sublinear dynamic regret, even when the losses are chosen\nadaptively without any a priori variation budget. Furthermore, we show for COL\na reduction from dynamic regret to both static regret and convergence in the\nassociated EP, allowing us to analyze the dynamic regret of many existing\nalgorithms.\n
Cited by 2Open access
Bos, Philip, Hussain, Mumtaz, Simmons, David · Figshare · 2023
Let ψ : R+ → R+ be a non-increasing function. A real number x is said to be ψ-Dirichlet improvable if the system |qx − p| < ψ(t) and |q| < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(ψ). In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.
Cited by 0Open access
Bos, Philip, Hussain, Mumtaz, Simmons, David · La Trobe University · 2023
Let ψ : R+ → R+ be a non-increasing function. A real number x is said to be ψ-Dirichlet improvable if the system |qx − p| < ψ(t) and |q| < t has a non-trivial integer solution for all large enough t. Denote the collection of such points by D(ψ). In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.
Cited by 0Open access
Gord Sinnamon · Proceedings of the American Mathematical Society · 2003
Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the nonincreasing rearrangement are presented.
Cited by 2Open access
Ching-An Cheng, Jonathan Lee, Ken Goldberg et al. · arXiv (Cornell University) · 2019
Online learning is a powerful tool for analyzing iterative algorithms. However, the classic adversarial setup sometimes fails to capture certain regularity in online problems in practice. Motivated by this, we establish a new setup, called Continuous Online Learning (COL), where the gradient of online loss function changes continuously across rounds with respect to the learner's decisions. We show that COL covers and more appropriately describes many interesting applications, from general equilibrium problems (EPs) to optimization in episodic MDPs. In particular, we show monotone EPs admits a reduction to achieving sublinear static regret in COL. Using this new setup, we revisit the difficulty of sublinear dynamic regret. We prove a fundamental equivalence between achieving sublinear dynamic regret in COL and solving certain EPs. With this insight, we offer conditions for efficient algorithms that achieve sublinear dynamic regret, even when the losses are chosen adaptively without any a priori variation budget. Furthermore, we show for COL a reduction from dynamic regret to both static regret and convergence in the associated EP, allowing us to analyze the dynamic regret of many existing algorithms.
Cited by 7Open access
Shuoshuo Zhang, Meng Luan, Dan Zhao · 2025
This paper investigates distributed online Nash equilibrium learning with privacy preservation in dynamic non-cooperative games. In this context, players optimize their actions under local constraints to track the time-varying equilibrium without access to future cost information while protecting their private strategy information. To this end, a novel privacy-preserving distributed online algorithm with two-point bandit feedback is proposed, which enables Nash equilibrium tracking using only local function evaluations. Moreover, theoretical analysis establishes sublinear dynamic regret under sublinear path variation. Finally, numerical simulations validate theoretical results.
Cited by 0
Ana Maria Amarillo Bertone, J.V. Gonçalves · Discrete and Continuous Dynamical Systems · 2000
This paper deals with existence and regularity of positive solutions of sublinear equations of the form $-\Delta u + b(x)u =\lambda f(u)$ in $\Omega$ where either $\Omega\in R^N$ is a bounded smooth domain in which case we consider the Dirichlet problem or $\Omega =R^N$, where we look for positive solutions, $b$ is not necessarily coercive or continuous and $f$ is a real function with sublinear growth which may have certain discontinuities. We explore the method of lower and upper solutions associated with some subdifferential calculus.
Cited by 2Open access
George A. Anastassiou · Arabian Journal of Mathematics · 2018
Here, we consider the approximation of functions by a large variety of max-product operators under conformable fractional differentiability and using convexity. These are positive sublinear operators. Our study relies on our general results about positive sublinear operators. We derive Jackson-type inequalities under conformable fractional initial conditions and convexity. So our approach is quantitative by obtaining inequalities where their right hand sides involve the modulus of continuity of a high-order conformable fractional derivative of the function under approximation. Due to the convexity assumptions, our inequalities are compact and elegant with small constants.
Cited by 6Open access
George A. Anastassiou · Filomat · 2021
Here we consider the ordinary and fractional approximation of functions by sublinear positive operators with applications to generalized convolution type operators expressed by sublinear integrals such as of Choquet and Shilkret ones. The fractional approximation is under fractional differentiability of Caputo, Canavati and Iterated-Caputo types. We produce Jackson type inequalities under basic initial conditions. So our way is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of ordinary and fractional derivatives of the function under approximation. We give also an application related to Picard singular integral operators.
Cited by 2Open access
C.I. Burbidge · Journal of Applied Physics · 2015
The basic form of luminescence dose response is investigated, with the aim of developing a single function to account for the appearance of linear, superlinear, sublinear, and supralinear behaviors and variations in saturation signal level and rate. A function is assembled based on the assumption of first order behavior in different major factors contributing to measured luminescence-dosimetric signals. Different versions of the function are developed for standardized and non-dose-normalized responses. Data generated using a two trap two recombination center model and experimental data for natural quartz are analyzed to compare results obtained using different signals, measurement protocols, pretreatment conditions, and radiation qualities. The function well describes a range of dose dependent behavior, including sublinear, superlinear, supralinear, and non-monotonic responses and relative response to α and β radiation, based on change in relative recombination and trapping probability affecting signals sourced from a single electron trap.
Cited by 8
Pallavoor Suresh, Ramesh Krishnan · OpenBU (Boston University) · 2020
We study sublinear-time algorithms for testing fundamental properties of functions and graphs. Sublinear-time algorithms run in time sublinear in the input size and are especially useful for processing massive datasets. We investigate variants of the property testing model, which was introduced by Rubinfeld and Sudan (SIAM J. Comput., 1996) and Goldreich, Goldwasser and Ron (J. ACM, 1998) to formally study sublinear-time algorithms. The focus of this thesis is on designing better property testers, deepening our understanding of their limitations, and proposing new angles for investigating sublinear-time algorithms. \n \nFirst, we study the problem of testing unateness of real-valued functions over high-dimensional domains. A function over a multi-dimensional domain is unate if, along each dimension, the function is either nonincreasing or nondecreasing. We give efficient unateness testers and prove that they are optimal. \n \nWe then focus on testing monotonicity of functions when the input functions are partially corrupted or partially erased. We prove that, for some settings of parameters, testing monotonicity of functions with corrupted or erased values is exponentially harder than testing monotonicity of functions with neither corruptions nor erasures. \n \nWe then investigate the parameters in terms of which the complexity of property testers should be expressed. The complexity of algorithms and, in particular, property testers is usually expressed in terms of the size of the input. We prove that certain parameters capture the complexity of testing problems better than the input size, providing compelling evidence that the input size may not always be the best parameter to express the complexity of sublinear-time algorithms. \n \nFinally, we turn our attention towards testing properties of graphs. We first provide an algorithm for testing connectedness of graphs and prove that it is optimal. We then define a new model for studying graphs with missing information and investigate the problems of testing connectedness and estimating the average degree of graphs in the new model.
Cited by 0Open access