The canonical representation of multiplication operation on triangular fuzzy numbers

The representation of multiplication operation on fuzzy numbers is very useful and important in the fuzzy system such as the fuzzy decision making. In this paper, we propose a new arithmetical principle and a new arithmetical method for the arithmetical operations on fuzzy numbers. The new arithmetical principle is the L⁻¹-R⁻¹ inverse function arithmetic principle. Based on the L⁻¹-R⁻¹ inverse function arithmetic principle, it is easy to interpret the multiplication operation with the membership functions of fuzzy numbers. The new arithmetical method is the graded multiple integrals representation method. Based on the graded multiple integrals representation method, it is easy to compute the canonical representation of multiplication operation on fuzzy numbers. Finally, the canonical representation is applied to a numerical example of fuzzy decision.     

Trivializing a central simple algebra of degree 4 over the rational numbers

We give an algorithm for finding an isomorphism of a given central simple algebra of degree 4 over the rationals and the full matrix algebra, provided it exists. It reduces the task to classical problems in number theory, for which there are already known algorithms.     

A more precise rounding algorithm for rational numbers

We adjoin a more precise companion to the classical mediant rounding algorithm for rational numbers.     

Constructive belief and rational representation

It is commonplace in artificial intelligence to divide an agent's explicit beliefs into two parts: the beliefs explicitly represented or manifest in memory, and the implicitly represented or constructive beliefs that are repeatedly reconstructed when needed rather than memorized. Many theories of knowledge view the relation between manifest and constructive beliefs as a logical relation, with the manifest beliefs representing the constructive beliefs through a logic of belief. This view, however, limits the ability of a theory to treat incomplete or inconsistent sets of beliefs in useful ways. We argue that a more illuminating view is that belief is the result of rational representation. In this theory, the agent obtains its constructive beliefs by using its manifest beliefs and preferences to rationally (in the sense of decision theory) choose the most useful conclusions indicated by the manifest beliefs.     

A simple optimal representation for balanced parentheses

We consider succinct, or highly space-efficient, representations of a (static) string consisting of n   pairs of balanced parentheses, which support natural operations such as finding the matching parenthesis for a given parenthesis, or finding the pair of parentheses that most tightly enclose a given pair. This problem was considered by Jacobson [Space-efficient static trees and graphs, in: Proc. of the 30th FOCS, 1989, pp. 549–554] and Munro and Raman [Succinct representation of balanced parentheses and static trees, SIAM J. Comput. 31 (2001) 762–776] who gave O(n)$\mathrm{O}(n)$-bit and 2n+o(n)$2n+\mathrm{o}(n)$-bit representations, respectively, that supported the above operations in O(1)$\mathrm{O}(1)$ time on the RAM model of computation. This data structure is a fundamental tool in succinct representations, and has applications in representing suffix trees, ordinal trees, planar graphs and permutations.     

保护隐私的有理数科学计算

安全多方计算作为密码学的基本组成部分,是各种密码协议的基础,是国际密码学界的研究热点。近年来,许多学者研究了各种各样的安全多方计算问题,包括保密的信息比较、保密的集合问题和保密的计算几何等,并提出相应的解决方案。而在许多实际应用场景中,安全多方计算问题需要应用有理数进行描述,因此研究有理数域上的安全多方计算问题具有重要的理论与实际意义。但现有的安全多方计算问题的研究成果大多数局限于整数范围,且研究的数据主要是单维度数据。关于有理数域上多维度数据安全多方计算问题的研究较少且无法推广应用。基于有理数的分数表示形式,设计了新的编码方案（有理数编码方案和有理向量编码方案）,可将有理数域上任意维数的数据进行编码,为研究有理数域上其他安全多方计算问题提供了新的解决思路。以该编码方案和单向哈希函数为基础,分别设计了有理数相等、有理向量相等和集合问题的保密判定协议。所设计的协议仅采用基本算术运算和单向哈希函数进行计算,不需要使用公钥加密算法,使得协议的计算效率较高;且协议对研究问题中的数据范围没有限制,适用范围更广。进一步应用模拟范例严格证明了协议在半诚实模型下的安全性;并通过理论分析和模拟实验验证了...     

Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.     

Visibility representation of plane graphs via canonical ordering tree

In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [Rectilinear planar layouts and bipolar orientations of planar graphs, Discrete Comput. Geom. 1 (1986) 343], Tamassia and Tollis [An unified approach to visibility representations of planar graphs, Discrete Comput. Geom. 1 (1986) 321] independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of the representation. In this paper, we prove that any plane graph G has a VR with height bounded by . This improves the previously known bound . We also construct a plane graph G with n vertices where any VR of G requires a size of . Our result provides an answer to Kant's open question about whether there exists a plane graph G such that all of its VR require width greater that cn, where c>1 [G. Kant, A more compact visibility representation, Internat. J. Comput. Geom. Appl. 7 (1997) 197].     

URR: Universal representation of real numbers

A new internal representation is proposed for real numbers that has been named URR for Universal Representation of Real Numbers. This approach is based on a bisection method which is applied to real number intervals. With this method, the point of division increases or decreases double-exponentially in global area. The main characteristics of the method are as follows. First, overflow/underflow does not, in practice, occur. Secondly, since data format does not depend on the length but on the value of the data, transformation operation is virtually not needed between systems of long and short data. Finally, only 1 bit of resolution is lost compared with the fixed point form. This new method is thus the most suitable uniform interface with the internal form, not only for computers but also for digital systems which deal with real numbers or physical (scalar) values.     

Object recognition by spectral feature derived from canonical shape representation

In this paper, we introduce a new spectral shape feature that can be used in content-based object recognition. We explain a new canonical string representation for a polygonal shape approximation from which the proposed spectral feature is derived. This spectral feature is a composition of Fourier coefficients of the shape function that is derived from the canonical representation. We applied the proposed feature in classification of lung nodules by means of our hierarchical learning scheme proposed in another study. The results show that the spectral feature is promising for lung nodule recognition.     

Fast modular multiplication based on complement representation and canonical recoding

Modular multiplication is the fundamental operation in implementing circuits for cryptosystem, as the process of encrypting and decrypting a message requires modular exponentiation that can be decomposed into multiplications. In this paper, a proposed multiplication method utilizes the complement recoding method and canonical recoding technique. By performing the complement representation method and canonical recoding technique, the number of partial products can be further reduced. Based on these techniques, an algorithm with efficient multiplication method is proposed. For multiplication operation, in average case, the proposed algorithm can reduce the number of k-bit additions from 1/4k+(log (k)/k)+5/2 to 1/6k+(log (k)/k)+5/2, where k is the bit length of the multiplicand and multiplier. Besides, if we perform the proposed technique to compute common-multiplicand multiplication, the computational complexity can reduce the number of k-bit additions from 1/2k+2×(log (k)/k)+5 to 1/3k+2×(log (k)/k)+5. We can, therefore, efficiently speed up the overall computing performance of the multiplication operation.     

Hamming weight pyramid - A new insight into canonical signed digit representation and its applications

Signed-power-of-two terms are widely used in design automation algorithms for digital filter synthesis and optimization, linear transformation and other multiple constant multiplication problems. In these applications, the computation efficiency or solution quality tends to degrade with the number of nonzero digits in the signed digit representation of the a priori fixed coefficients. This paper provides a new perspective to interpret the hamming weights of fixed-point coefficients represented in signed-power-of-two terms with minimal number of nonzero digits, called the minimal signed digit (MSD) representation. A new hamming weight pyramid (HWP) is proposed to succinctly compress the information about the distribution of the hamming weights of canonical signed digit (CSD) representation in a visually appealing manner for analysis and synthesis. CSD is a unique and popularly used subset of the general MSD representation. Many interesting properties of CSD are uncovered in this regularly structured HWP. These properties are exploited to develop a novel and elegant algorithm for the direct conversion of decimal number to CSD representation. We also show that the HWP can also be employed to overcome the limit imposed on the word length of the coefficients for the reduced adder graph (RAG) algorithm and filter coefficient synthesis.