scholarly journals The refined symplectic sum formula for Gromov–Witten invariants

2020 ◽  
Vol 31 (04) ◽  
pp. 2050032
Author(s):  
Mohammad F. Tehrani ◽  
Aleksey Zinger
Keyword(s):  
Product Operation ◽  
Abelian Covers ◽  
Sum Formula

We describe the extent to which Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants (GW-invariants) sharpens the usual symplectic sum formula. The key product operation on the target spaces for the refined invariants is specified in terms of abelian covers of symplectic divisors, making it suitable for studying from a topological perspective. We give several qualitative applications of this refinement, which include vanishing results for GW-invariants.

scholarly journals On the rim tori refinement of relative Gromov–Witten invariants

2020 ◽  
pp. 2050051
Author(s):  
Mohammad F. Tehrani ◽  
Aleksey Zinger
Keyword(s):  
Separate Paper ◽  
Abelian Covers ◽  
Sum Formula

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula and give further qualitative applications.

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.

scholarly journals Acceleration of Inner-Pairing Product Operation for Secure Biometric Verification

Sensors ◽  
2021 ◽  
Vol 21 (8) ◽  
pp. 2859
Author(s):  
Seong-Yun Jeon ◽  
Mun-Kyu Lee
Keyword(s):  
Mobile Technologies ◽  
Smart Devices ◽  
Biometric Data ◽  
Product Operation ◽  
Biometric Verification ◽  
Naive Method ◽  
Biometric Template ◽  
Need To Evaluate ◽  
Time Required ◽  
Pairing Product

With the recent advances in mobile technologies, biometric verification is being adopted in many smart devices as a means for authenticating their owners. As biometric data leakage may cause stringent privacy issues, many proposals have been offered to guarantee the security of stored biometric data, i.e., biometric template. One of the most promising solutions is the use of a remote server that stores the template in an encrypted form and performs a biometric comparison on the ciphertext domain, using recently proposed functional encryption (FE) techniques. However, the drawback of this approach is that considerable computation is required for the inner-pairing product operation used for the decryption procedure of the underlying FE, which is performed in the authentication phase. In this paper, we propose an enhanced method to accelerate the inner-pairing product computation and apply it to expedite the decryption operation of FE and for faster remote biometric verification. The following two important observations are the basis for our improvement—one of the two arguments for the decryption operation does not frequently change over authentication sessions, and we only need to evaluate the product of multiple pairings, rather than individual pairings. From the results of our experiments, the proposed method reduces the time required to compute an inner-pairing product by 30.7%, compared to the previous best method. With this improvement, the time required for biometric verification is expected to decrease by up to 10.0%, compared to a naive method.

scholarly journals Horospheres on abelian covers

1998 ◽  
Vol 29 (1) ◽  
pp. 195-195
Author(s):  
Fran�ois Ledrappier
Keyword(s):  
Abelian Covers

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 Distribution of points on Abelian covers over finite fields

2018 ◽  
Vol 14 (05) ◽  
pp. 1375-1401 ◽  
Author(s):  
Patrick Meisner
Keyword(s):  
Finite Fields ◽  
Galois Extension ◽  
Random Variables ◽  
Families Of Curves ◽  
Abelian Covers ◽  
Distribution Of Points ◽  
Genus Formula

We determine in this paper the distribution of the number of points on the covers of [Formula: see text] such that [Formula: see text] is a Galois extension and [Formula: see text] is abelian when [Formula: see text] is fixed and the genus, [Formula: see text], tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over [Formula: see text]. In all cases, the distribution is given by a sum of [Formula: see text] random variables.

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