The cube of the sum formula

Algebra ◽  
2004 ◽  
pp. 39-39
Author(s):  
Israel M. Gelfand ◽  
Alexander Shen
Keyword(s):  
Sum Formula

scholarly journals Trigonometric Inference Providing Learning in Deep Neural Networks

2021 ◽  
Vol 11 (15) ◽  
pp. 6704
Author(s):  
Jingyong Cai ◽  
Masashi Takemoto ◽  
Yuming Qiu ◽  
Hironori Nakajo
Keyword(s):  
Machine Learning ◽  
Neural Networks ◽  
Deep Neural Networks ◽  
Activation Function ◽  
Trigonometric Approximation ◽  
Model Parameters ◽  
Training Algorithms ◽  
Activation Functions ◽  
Classical Training ◽  
Sum Formula

Despite being heavily used in the training of deep neural networks (DNNs), multipliers are resource-intensive and insufficient in many different scenarios. Previous discoveries have revealed the superiority when activation functions, such as the sigmoid, are calculated by shift-and-add operations, although they fail to remove multiplications in training altogether. In this paper, we propose an innovative approach that can convert all multiplications in the forward and backward inferences of DNNs into shift-and-add operations. Because the model parameters and backpropagated errors of a large DNN model are typically clustered around zero, these values can be approximated by their sine values. Multiplications between the weights and error signals are transferred to multiplications of their sine values, which are replaceable with simpler operations with the help of the product to sum formula. In addition, a rectified sine activation function is utilized for further converting layer inputs into sine values. In this way, the original multiplication-intensive operations can be computed through simple add-and-shift operations. This trigonometric approximation method provides an efficient training and inference alternative for devices with insufficient hardware multipliers. Experimental results demonstrate that this method is able to obtain a performance close to that of classical training algorithms. The approach we propose sheds new light on future hardware customization research for machine learning.

scholarly journals ON MULTIPLE ZETA VALUES OF EXTREMAL HEIGHT

2015 ◽  
Vol 93 (2) ◽  
pp. 186-193 ◽  
Author(s):  
MASANOBU KANEKO ◽  
MIKA SAKATA
Keyword(s):  
Explicit Formula ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Maximal Height ◽  
Zeta Values ◽  
Sum Formula

We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

scholarly journals A novel copper precursor for electron beam induced deposition

2018 ◽  
Vol 9 ◽  
pp. 1220-1227 ◽  
Author(s):  
Caspar Haverkamp ◽  
George Sarau ◽  
Mikhail N Polyakov ◽  
Ivo Utke ◽  
Marcos V Puydinger dos Santos ◽  
...  
Keyword(s):  
Electron Beam ◽  
Optical Transmission ◽  
Pure Copper ◽  
Dielectric Behavior ◽  
Copper Oxidation ◽  
Transmission Electron ◽  
Sum Formula ◽  
Electron Beam Induced Deposition ◽  
High Degree ◽  
Good Agreement

A fluorine free copper precursor, Cu(tbaoac)2 with the chemical sum formula CuC16O6H26 is introduced for focused electron beam induced deposition (FEBID). FEBID with 15 keV and 7 nA results in deposits with an atomic composition of Cu:O:C of approximately 1:1:2. Transmission electron microscopy proved that pure copper nanocrystals with sizes of up to around 15 nm were dispersed inside the carbonaceous matrix. Raman investigations revealed a high degree of amorphization of the carbonaceous matrix and showed hints for partial copper oxidation taking place selectively on the surfaces of the deposits. Optical transmission/reflection measurements of deposited pads showed a dielectric behavior of the material in the optical spectral range. The general behavior of the permittivity could be described by applying the Maxwell–Garnett mixing model to amorphous carbon and copper. The dielectric function measured from deposited pads was used to simulate the optical response of tip arrays fabricated out of the same precursor and showed good agreement with measurements. This paves the way for future plasmonic applications with copper-FEBID.

scholarly journals A connected sum formula for involutive Heegaard Floer homology

2017 ◽  
Vol 24 (2) ◽  
pp. 1183-1245 ◽  
Author(s):  
Kristen Hendricks ◽  
Ciprian Manolescu ◽  
Ian Zemke
Keyword(s):  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Connected Sum ◽  
Sum Formula ◽  
Heegaard Floer

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

Constant term evaluation and two kinds of congruence

2018 ◽  
Vol 14 (07) ◽  
pp. 2013-2022
Author(s):  
Qing-Hu Hou ◽  
Yu-Shan Wang
Keyword(s):  
Constant Term ◽  
Term Evaluation ◽  
Sum Formula

We consider two kinds of congruence proposed recently by Sun. The first one is of the form [Formula: see text], where [Formula: see text] is a prime, [Formula: see text] is a character and [Formula: see text] is a sequence of integers. By using a constant term expression for [Formula: see text], we derive congruence for two sequences [Formula: see text]. The second one involves the sum [Formula: see text] where [Formula: see text] By giving a constant term expression for [Formula: see text], we derive a congruence for [Formula: see text] modulo a prime [Formula: see text].

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