Narrow Results

The frequency-hopping sequence set with low-hit-zone (LHZ FHS set) is useful for eliminating multiple-access interference in quasi-synchronous frequency-hopping multiple access systems. In this paper, basing on Cyclotomy, the discrete logarithm function and the Chinese remainder theorem, we construct two classes of LHZ FHS sets which have new parameters not covered in the literature. It is proved that proposed LHZ FHS sets are optimal respectively with respect to the Peng-Fan-Lee bounds.

In this paper, we give an equivalent form of Dickson’s conjecture. Based on this equivalent form, we propose a computational problem and obtain several interesting results.

In this paper, we consider the problem of how to design a minimal perfect hashing function which is suitable for the Mandarin Phonetic Symbols system. Our main idea is inspired by Chang’s letter-oriented minimal perfect hashing scheme. By using our hashing function, 1303 Mandarin phonetic symbol transcriptions will be hashed to 1303 locations in the way of one-to-one correspondence.

Field programmable gate array (FPGA)-based digital signal processing has been widely used in multimedia applications. By combining distributed arithmetic (DA) and residue number system (RNS) in such designs, efficient area, speed and power efficiency can be achieved. In this paper, we propose novel techniques for the design and FPGA implementation of DA-RNS finite impulse response (FIR) filters. By introducing a novel low-cost moduli set and its selection method, efficient modulo arithmetic units inside the subfilters are designed. Then, a new residue-to-binary conversion algorithm, a so-called modified DA Chinese remainder theorem, is derived to reduce the modulo operations and provide an efficient residue-to-binary converter suitable to FPGA implementation. Based on these proposed techniques, a seventh-order DA-RNS FIR filter is designed, implemented and tested by using Xilinx FPGA tools. The implementation results show that the proposed filter design consumes only 77% of the power that the existing filter^12,13 requires, while maintaining the same speed (throughput).

Residue number systems are attractive due to their basic arithmetic operations, such as addition and multiplication, are carry-free; this facilitates low-power, high-speed implementations. But converting from residue to conventional notation is generally problematic. In this paper, we propose a high-speed architecture for reverse conversion; this is based on Base Extension and the Chinese Remainder Theorem.

The efficiency of residue number system depends on the reverse converter due to several modulo operations like addition, subtraction and multiplication. In this paper, a design of new four moduli set $\{2^{n-1},2^n-1,2^n+1,2^{2n+1}-1\}$, reverse converter is presented. The moduli set have moduli with length ranging from ($n-1$) to ($2n+1$)-bits. The reverse conversion for moduli set $\{2^{n-1},2^n-1,2^n+1\}$ has been optimized in existing state of art. Thus, proposed converter is based on two new moduli set $\{2^{n-1}(2^{2n}-1),2^{2n+1}-1\}$ and utilizes the mixed radix conversion. This converter is memoryless, and occupies least area. The proposed converter is based on carry save adder (CSA) and modulo adder enabling more speed and less hardware complexity for dynamic range of $5n$-bit, offering good area-delay product.

Let $R$ be a commutative ring with identity. In this paper, a Cohen-type theorem for $w$-Artinian modules is given, i.e. a $w$-cofinitely generated $R$-module $M$ is $w$-Artinian if and only if ${(M/{\mathrm{\text{ann}}}_M(P))}_w$ is $w$-cofinitely generated for every prime $w$-ideal $P$ of $R$. As a by-product of the proof, we also obtain a detailed representation of elements of a $w$-module and the $w$-theoretic version of the Chinese remainder theorem for both modules and rings.

The Chinese remainder theorem (CRT), which appeared in ancient China, is widely used in many modern computer applications. This paper presents the CRT implementation by using the interval-index characteristic and minimum redundancy residue code. The proposed algorithm does not use large modulo addition operations and provides low computational complexity compared to conventional non-redundant RNS. The efficiency factors of using the minimally redundant RNS increase with the number $k$ of non-redundant moduli, asymptotically approaching the threshold $k/2$. The new approach presented here will have a significant impact on many non-modular operations in RNS arithmetic, which currently use the CRT.

For a nilpotent group $G$ without $\pi$-torsion, and $x$, $y\in G$, if $x^n=y^n$ for a $\pi$-number $n$, then $x=y$; if $x^m{y}^n=y^n{x}^m$ for $\pi$-numbers $m$, $n$, then $xy=yx$. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let $m$ be a given positive integer and $A$ a complex square matrix satisfying that (i) all eigenvalues of $A$ are nonnegative, and (ii) $\mathrm{rank}{A}^2=\mathrm{rank}A$; then $A$ has a unique $m$-th root $X$ with $\mathrm{rank}{X}^2=\mathrm{rank}X$, all eigenvalues of $X$ are nonnegative, and moreover there is a polynomial $f(\lambda )$ with $X=f(A)$. In addition, let $A$ and $B$ be complex $n\times n$ matrices with all eigenvalues nonnegative, and $\mathrm{rank}{A}^2=\mathrm{rank}A$, $\mathrm{rank}{B}^2=\mathrm{rank}B$; then (i) $A=B$ when $A^r=B^r$ for some positive integer $r$, and (ii) $AB=BA$ when $A^s{B}^t=B^t{A}^s$ for two positive integers $s$ and $t$.

In this work, we investigate the structure of the digraph associated with the square mapping on the ring of Gaussian integers by using the exponent of the unit group modulo γ. The formula for the fixed points of is established. Some connections of the lengths of cycles with the exponent of the unit group modulo γ are presented. Furthermore, we study the maximum distance from the cycle on each component.

In this article, we will give a new proof of the reciprocity law for Dedekind sums, as well as a proof of the transformation formula for the Dedekind $\eta$-function using the Chinese Remainder Theorem.

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$,

In this paper, we consider the problem of how to design a minimal perfect hashing function which is suitable for the Mandarin Phonetic Symbols system. Our main idea is inspired by Chang’s letter-oriented minimal perfect hashing scheme. By using our hashing function, 1303 Mandarin phonetic symbol transcriptions will be hashed to 1303 locations in the way of one-to-one correspondence.