scholarly journals Dual Lattice of ℤ-module Lattice

2017 ◽  
Vol 25 (2) ◽  
pp. 157-169
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Gram Matrix ◽  
Reduction Algorithm ◽  
Dual Basis ◽  
Dual Lattice ◽  
Inner Products ◽  
Definition Of

Summary In this article, we formalize in Mizar [5] the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20], [10] and [19].

scholarly journals Embedded Lattice and Properties of Gram Matrix

2017 ◽  
Vol 25 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Inner Product ◽  
Gram Matrix ◽  
Reduction Algorithm ◽  
Definition Of

Summary In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm [16] and cryptographic systems with lattice [17].

scholarly journals A Remark on the Homogeneity of Isosceles Orthogonality

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Chan He ◽  
Dan Wang
Keyword(s):  
Isosceles Orthogonality ◽  
Inner Products ◽  
Definition Of ◽  
Homogeneity Of Isosceles Orthogonality

Inspired by the definition of homogeneous direction of isosceles orthogonality, we introduce the notion of almost homogeneous direction of isosceles orthogonality and show that, surprisingly, these two notions coincide. Several known characterizations of inner products are improved.

Optimization Analysis on Dynamic Reduction Algorithm

2018 ◽  
Vol 6 (5) ◽  
pp. 447-458
Author(s):  
Yizhou Chen ◽  
Jiayang Wang
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Reduction Process ◽  
New Method ◽  
Reduction Algorithm ◽  
Optimization Analysis ◽  
Sampling Process ◽  
Dynamic Methods ◽  
Definition Of

Abstract On the basis of rough set theory, the strengths of dynamic reduction are elaborated compared with traditional non-dynamic methods. A systematic concept of dynamic reduction from sampling process to the generation of the reduct set is presented. A new method of sampling is created to avoid the defects of being too subjective. And in order to deal with the over-sized time consuming problem in traditional dynamic reduction process, a quick algorithm is proposed within the constraint conditions. We have also proved that dynamic core possesses the essential characteristics of a reduction core on the basis of the formalized definition of the multi-layered dynamic core.

A Quick Attribution Reduction Algorithm Based on Incomplete Decision Table

2010 ◽  
Vol 171-172 ◽  
pp. 154-158 ◽  
Author(s):  
Wen Hao Shu ◽  
Zhang Yan Xu ◽  
Shen Ruan
Keyword(s):  
Efficient Algorithm ◽  
Time Complexity ◽  
Decision Table ◽  
Reduction Algorithm ◽  
Discernibility Matrix ◽  
Positive Region ◽  
Definition Of

At present, some scholars have provided the attribution reduction algorithms of incomplete decision table. The time complexity of many algorithms are .To cut down the time complexity of the algorithms for computing attribution reduction , the definition of discernibility matrix based on positive region and the corresponding definition of the attribution reduction are provided. At the same time, it is proved that the attribution reduction is equivalent to the attribution reduction based on the positive region. The discernibility matrix is simplified for not comparing the objects between .On this condition, a efficient algorithm for computing attribution reduction is designed with the simplified discernibility matrix, whose time complexity is .At last, an emulate example is used to illustrate the efficiency of the new algorithm.

A Reproducing Kernel Hilbert Space Framework for Spike Train Signal Processing

2009 ◽  
Vol 21 (2) ◽  
pp. 424-449 ◽  
Author(s):  
António R. C. Paiva ◽  
Il Park ◽  
José C. Príncipe
Keyword(s):  
Signal Processing ◽  
Spike Train ◽  
Point Processes ◽  
Clustering Algorithm ◽  
Reproducing Kernel ◽  
Spike Trains ◽  
Inner Product ◽  
Inner Products ◽  
Definition Of ◽  
Intensity Functions

This letter presents a general framework based on reproducing kernel Hilbert spaces (RKHS) to mathematically describe and manipulate spike trains. The main idea is the definition of inner products to allow spike train signal processing from basic principles while incorporating their statistical description as point processes. Moreover, because many inner products can be formulated, a particular definition can be crafted to best fit an application. These ideas are illustrated by the definition of a number of spike train inner products. To further elicit the advantages of the RKHS framework, a family of these inner products, the cross-intensity (CI) kernels, is analyzed in detail. This inner product family encapsulates the statistical description from the conditional intensity functions of spike trains. The problem of their estimation is also addressed. The simplest of the spike train kernels in this family provide an interesting perspective to others' work, as will be demonstrated in terms of spike train distance measures. Finally, as an application example, the RKHS framework is used to derive a clustering algorithm for spike trains from simple principles.

scholarly journals Divisible ℤ-modules

2016 ◽  
Vol 24 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Coefficient Ring ◽  
Torsion Free

Summary In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

scholarly journals Method for Minimizing Total Generalized Squared Correlation of Synchronous DS-CDMA Signature Sequence Sets in Multipath Channels

2017 ◽  
Vol 2 (1) ◽  
pp. 20
Author(s):  
Paul Cotae ◽  
Matt Aguirre
Keyword(s):  
Code Division Multiple Access ◽  
Colored Noise ◽  
Multipath Channels ◽  
Gram Matrix ◽  
Channel State ◽  
Signature Sequences ◽  
Signature Sequence ◽  
Definition Of ◽  
Code Division Multiple ◽  
Synchronous Cdma

We characterize the Total Generalized Squared Correlation (TGSC) for a given signature sequence set used in uplink synchronous code division multiple access (S-CDMA) when channel state information is known perfectly at bothtransmitter and receiver. We give a definition of the TGSC based on the eigenvalues of Gram matrix associated to signature sequences set for multipath channels in the presence of the colored noise. Total Squared Correlation (TSC) and Total Weighted Squared Correlation (TWSC) measures are particular cases of TGSC. We present a method for minimizing TGSC (TSC, TWSC) in multipath channels and in the presence of the colored noise. Numerical results for overloaded synchronous CDMA systems are presented in order to support our analysis.

scholarly journals Lattice of ℤ-module

2016 ◽  
Vol 24 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Coding Theory ◽  
Positive Definite ◽  
Bilinear Forms ◽  
Reduction Algorithm ◽  
Real Numbers ◽  
Scalar Products ◽  
Definition Of

Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

scholarly journals Torsion Part of ℤ-module

2015 ◽  
Vol 23 (4) ◽  
pp. 297-307 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Rational Numbers ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Torsion Part ◽  
Torsion Free

Summary In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

Attribute Reduction Algorithm of Incomplete Dicision Table Based on Conditional Entropy

2013 ◽  
Vol 380-384 ◽  
pp. 1505-1509
Author(s):  
Zhang Yan Xu ◽  
Wei Zhang ◽  
Yan Ying Fan
Keyword(s):  
Hot Spot ◽  
Cardinal Number ◽  
Attribute Reduction ◽  
Conditional Entropy ◽  
Reduction Algorithm ◽  
Discernibility Matrix ◽  
Worst Case ◽  
Advantages And Disadvantages ◽  
Definition Of ◽  
Test Result

The search of the attribute reduction algorithm of rough set in incomplete decision table is a research hot spot. Though analysis of the advantages and disadvantages of the existing attribute reduction algorithms,we put forward a definition of relative discernibility matrix base on the positive area. Then we compute the tolerance class with the the idea of cardinal number sorting method, giving a quick heuristic algorithm of attribute reduction with theconditional entropy and relative discernibility matrix, which of the time complexity is in the worst case. The test result shows that the algorithm can obtain an attribute reduction efficiently.

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