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Lower bounds for in-network computation of arbitrary functions

DISTRIBUTED COMPUTING 2021年第3期34卷 181-193页

作者： Gillani, Iqra Altaf Vyavahare, Pooja Bagchi, Amitabha IIT Delhi Dept Comp Sci & Engn New Delhi India IIT Tirupati Dept Elect Engn Tirupati Andhra Pradesh India

In this paper, we provide a family of bounds for the rate at which the functions of many inputs can be computed in-network on general topologies. Going beyond simple symmetric functions where the output is invariant to the permutation of the operands, e.g., average, parity, we describe an algorithm that is analyzed to provide throughput bounds (both lower and upper) for the general functions. In particular, we analyze our algorithm when the function to be computed is given as a binary tree schema. Our lower bounds depend on schema parameters like the number of operands and graph parameters like the second largest eigenvalue of the transition matrix of simple random walk on network graph, the maximum and minimum degree of any node in the network. The lower bounding technique that we have used is based on the network flows and can capture general multi-commodity flow settings. Our proposed algorithm uses the well-known simple random walk on a network as its basic primitive for routing. We show that the lower bound obtained on the rate of computation is tight for the complete network topology, the hypercube and the star topology. We also present an upper bound on the expected latency of any data operand in terms of the height of schema, well-studied random walk parameter like the hitting time, and the relative distance from the critical data rate. For the computation time of symmetric functions on the random geometric graph under the gossip model, our approach achieves an order-optimal (O) over tilde (n) time despite enforcing a binary tree schema for function computation. In general, Big-O notation represents an upper bound and e.g hides poly log n factors.

关键词： In-network computation arbitrary functions Random walks Stable rate of computation

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Symmetric triangle quadrature rules for arbitrary functions

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2020年第10期79卷 2885-2896页

作者： Freno, Brian A. Johnson, William A. Zinser, Brian F. Campione, Salvatore Sandia Natl Labs POB 5800 Albuquerque NM 87185 USA

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Symmetric quadrature rules Quadrature rules for singularities Triangle quadrature rules arbitrary functions

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Johannes von Kries's Range Conception, the Method of arbitrary functions, and Related Modern Approaches to Probability

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JOURNAL FOR GENERAL PHILOSOPHY OF SCIENCE 2016年第1期47卷 151-170页

作者： Rosenthal, Jacob Univ Konstanz Fachbereich Philosophie D-78457 Constance Germany

A conception of probability that can be traced back to Johannes von Kries is introduced: the "Spielraum" or range conception. Its close connection to the so-called method of arbitrary functions is highlighted. Possible interpretations of it are discussed, and likewise its scope and its relation to certain current interpretations of probability. Taken together, these approaches form a class of interpretations of probability in its own right, but also with its own problems. These, too, are introduced, discussed, and proposals in response to them are surveyed, some of which also go back to von Kries.

关键词： arbitrary functions Chance Probability Range conception von Kries

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Fully integrable one-dimensional nonlinear wave equation: Solution of a general initial value problem

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Partial Differential Equations in Applied Mathematics 2022年 5卷

作者： Trifonov, E.V. Institute of Automation and Control Processes FEB RAS Radio 5 Vladivostok 690041 Russian Federation

A new integrable case of the one-dimensional nonlinear wave equation was found. A general solution for this case depending on two arbitrary functions was derived. The functional form of the speed of sound can be used to model weak nonlinear waves in non-dispersive media. For the initial value problem the nonlinear generalization of the d'Alembert’s formula was obtained. © 2022 The Author(s)

关键词： arbitrary functions Exact solutions General solution Nonlinear waves Riemann invariants Wave equation

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New types of nondegenerate solitons for a (2+1)-dimensional coupled system

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COMMUNICATIONS IN THEORETICAL PHYSICS 2025年第9期77卷 095001-095001页

作者： Meng, Yong Riaz, Hafiz Wajahat Ahmed Lin, Ji Zhejiang Normal Univ Dept Phys Jinhua 321004 Zhejiang Peoples R China

In this paper, we investigate the (2+1)-dimensional three-component long-wave-short-wave resonance interaction system, which describes complex systems and nonlinear wave phenomena in physics. By employing the Hirota bilinear method, we derive the general nondegenerate N-soliton solution of the system, where each short-wave component contains N arbitrary functions of the independent variable y. The presence of these arbitrary functions in the analytical solution enables the construction of a wide range of nondegenerate soliton types. Finally, we illustrate the structural features of several novel nondegenerate solitons, including M-shaped, multiple double-hump, and sawtooth double-striped solitons, as well as interactions between nondegenerate solitons, such as dromion-like solitons and solitoffs, with the aid of figures.

关键词： (2+1)-dimensional coupled system general nondegenerate N-soliton solution arbitrary functions modulation waveform

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A New Treatise on the Elements of the Differential and Integral Calculus

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2017年

作者： i. f. quinby horatio nelson robinson

A New Treatise on the Elements of the Differential and Integral Calculus by Horatio Nelson Robinson, I. F. Quinby; published by Ivison, Blakeman, Taylor & Co., 1867

关键词： arbitrary functions calculus curve differential differential equation elements equation explicit function function functions independent variable independent variables integral partial differential partial fractions radius vector second member tangent treatise values vjooqic vjooqic differential

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Oregonstate :: Os3 :: Cc-69-05 Mimic Apr69

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2016年

Oregonstate :: Os3 :: Cc-69-05 Mimic Apr69 by published by

关键词： apr69 arbitrary functions cc-69-05 data data card data cards function function code functions independent variable logical logical control logical unit logical variable mimic mimic program oregonstate output program unit values variable variable names

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Equivalence transformations of the Euler-Bernoulli equation

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NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS 2012年第5期13卷 2172-2177页

作者： Ndogmo, J. C. Univ Witwatersrand Sch Math ZA-2050 Johannesburg South Africa

We give a determination of the equivalence group of the Euler-Bernoulli equation and of one of its generalizations, and thus derive some symmetry properties of this equation. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Equivalence group arbitrary functions Euler-Bernoulli equation Differential operators Symmetry group

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Computing Blindfolded: New Developments in Fully Homomorphic Encryption

Computing Blindfolded: New Developments in Fully Homomorphic...

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52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS)

作者： Vaikuntanathan, Vinod Univ Toronto Toronto ON M5S 1A1 Canada

ISBN: (纸本)9780769545714

A fully homomorphic encryption scheme enables computation of arbitrary functions on encrypted data. Fully homomorphic encryption has long been regarded as cryptography's prized "holy grail" - extremely useful yet rather elusive. Starting with the groundbreaking work of Gentry in 2009, the last three years have witnessed numerous constructions of fully homomorphic encryption involving novel mathematical techniques, and a number of exciting applications. We will take the reader through a journey of these developments and provide a glimpse of the exciting research directions that lie ahead.

关键词： Databases Encryption Polynomials Privacy Public key arbitrary functions computing blindfolded cryptography data encryption fully homomorphic encryption mathematical analysis mathematical techniques

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On error operators related to the arbitrary functions principle

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JOURNAL OF FUNCTIONAL ANALYSIS 2007年第1期251卷 325-345页

作者： Bouleau, Nicolas ParisTech Ecole Ponts F-75007 Paris France

The error on a real quantity Y due to the graduation of the measuring instrument may be asymptotically represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the "arbitrary functions principle" (Poincare, Hopf). We give extensions of this property to R-d and to the Wiener space for some approximations of the Brownian motion. This gives new approximations of the Ornstein-Uhlenbeck gradient. These results apply to the discretization of some stochastic differential equations encountered in mechanics. (c) 2007 Elsevier Inc. All rights reserved.

关键词： arbitrary functions dirichlet forms Euler scheme girsanov theorem mechanical system rajchman measure square field operator stable convergence Stochastic differential equation

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