On the Arithmetic of Endomorphism Ring End$$({Z_p} \times {Z_{{p^m}}})$$

For a prime p, let \({E_{P,{P^m}}} = \left\{ {\left( {\begin{array}{*{20}{c}}aamp;b \\ {{p^{m – 1}}c}amp;d \end{array}} \right)\left| {a,b,c \in {Z_p},d} \right. \in {Z_{{p^m}}}} \right\}\). We first establish a ring isomorphism from \({Z_{p,{p^m}}}\)onto \({E_{p,{p^m}}}\). Then we provide a way to compute -d and d^–1 by using arithmetic in Z_p and \({Z_{{p^m}}}\), and characterize the invertible elements of \({E_{p,{p^m}}}\). Moreover, we introduce the minimal polynomial for each element in \({E_{p,{p^m}}}\)and give its applications.     

Binding number,minimum degree and bipancyclism in bipartite graphs

Let G =(V_1,V_2,E) be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B(G),is defined to be n if G =K_n and min i∈{1,2}|N(S)|n min |N(S)|/|S|otherwise.We call G bipancyclic if it contains a cycle of every even length m for 4 ≤ m ≤ 2n.A theorem showed that if G is a balanced bipartite graph with 2n vertices,B(G) 3 / 2 and n 139,then G is bipancyclic.This paper generalizes the conclusion as follows:Let 0 c 3 / 2 and G be a 2-colmected balanced bipartite graph with 2n(n is large enough) vertices such that B(G) c and δ(G)(2-c)n/(3-c)+2/3.Then G is bipancyclic.     

Critical Chords of Convex Bodies of Constant Width

In this paper, we show that when Minkowski measure of asymmetry of convex body K of constant width is bigger than α (n-1), K has at least n+1 critical chords, where \(\alpha (n) = \frac{{n + \sqrt {2n(n + 1)} }}{{n + 2}}\).     

Research on a Class of Equations over Finite Fields

Let F_q stand for the finite field of odd characteristic p with q elements(q=p~n,n∈N)and F_q~* denote the set of all the nonzero elements of F_q.In this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over F_q:f(x _1,x _2,...,x _n)=(a_1 x_1 x_2~d+a_2 x_2 x_3~d...+a_(n-1)x_(n-1)x_n~d+a_n x_n x_1~d)~λ-bx_1~(d1)x_2~d2...x_n~(dn),with a_i,b∈F_q~*,n≥2,λ0 being positive integers,and d,d_i being nonnegative integers for 1≤i n.This technique can be applied to the polynomials of the form h_1~λ=h_2 with λ being positive integer and h_1,h_2∈F_q[x _1,x _2,...,x _n].It extends the results of the Markoff-Hurwitz-type equations.     

Ranks of repeated-root constacyclic code

By the discussion of division in \(F_{^{^{^{_2 m} } } } \left[ u \right]/\left\langle {u^4 } \right\rangle \), the minimal spanning set and the rank of a (1 + u + u ²) - constacyclic code with an arbitrary length N = 2 ^e n over \(\Phi \) are determined based on the factorization of (x ⁿ - 1) over \({F_{{2^m}}}\).     

A generalized funk’s section theorem for measures

If L is a star body in R~n whose central(n-i)-slices have the same(n-i)-dimensional measure μ_(n-i)(with appropriate density)as the central(n-i)-slices of an origin-symmetric star body K,then the corresponding n-dimensional measures μ_n of K and L satisfy μ_n(K)≤μ_n(L).This extends a generalized Funk’s section theorem for volumes to that for measures.     

Properties of a class of groups on <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>

In this paper, we obtain the factorization of direct production and order of group GL(n, Z _m ) in a simple method. Then we generalize some properties of GL(2, Z _p ) proposed by Huppert, and prove that the group \(GL\left( {2,{Z_{{z^\lambda }}}} \right)\) is solvable. We also prove that group GL(n, Z _p ) is solvable if and only if GL(n, Z _p ) is solvable, and list the generators of groups GL(n, Z _p ) and SL(n, Z _p ). At last, we prove that PSL(2, Z _p )(p > 3) and PSL(n, Z _p )(n > 3) are simple.     

The periodic solution for a kind odd order delay differential equation

In this paper, we use the coincidence degree theory to research the odd order delay differential equation a(t)χ(2n+1)(t)+b(t)χ(t)+g(t,-τ(t)))=p(t), and we obtain the sufficient conditions for the existence of periodic solutions, which improves and generalizes some related results in the literatures.     

Further results on generalized LCM matrices

Let S={x_1,x_2,...,x_n } be a set of n distinct positive integers and f be an arithmetic function.By(f[S])(resp.( f[S])),we denote the n*n matrix whose i,j entry is Σ[x_i,x_j]|l l∈S f(l) (resp.Σx∈Sf(x)-Σ x_i,|l l∈S f(l)-Σ x_j,|l l∈S f(l)+Σ[x_i,x_j]|l l∈S f(l)).In this paper,we first investigate the structures of the matrices ( f[S]) and( f[S]),then we give the formulae for the determinants of these matrices.These extend the results obtained by Bege in 2011.Finally,we give two examples to demonstrate the validity of our main results.     

Multi-pattern matching algorithm with wildcards based on bit-parallelism

Multi-pattern matching with wildcards is a problem of finding the occurrence of all patterns in a pattern set{p~1,---,p~k}in a given text t. If the percentage of wildcards in pattern set is not high,this problem can be solved using finite automata. We introduce a multi-pattern matching algorithm with a fixed number of wildcards to overcome the high percentage of the occurrence of wildcards in patterns. In our proposed method,patterns are matched as bit patterns using a sliding window approach. The window is a bit window that slides along the given text,matching against stored bit patterns. Matching process is executed using bit wise operations. The experimental results demonstrate that the percentage of wildcard occurrence does not affect the proposed algorithm's performance and the proposed algorithm is more efficient than the algorithms based on the fast Fourier transform. The proposed algorithm is simple to implement and runs efficiently in O(n+d(n/σ)(m/w))time,where n is text length,d is symbol distribution over k patterns,m is pattern length,and σ is alphabet size.     

Vertex disjoint cycles in intersection graphs of bases of matroids

The intersection graph of bases of a matroid M=(E, B) is a graph G=G~I(M) with vertex set V(G) and edge set E(G) such that V(G)=B(M) and E(G)={BB′:|B∩B′|≠0, B, B′∈B(M), where the same notation is used for the vertices of G and the bases of M. Suppose that|V(G~I(M))| =n and k_1+k_2+…+k_p=n, where k_i is an integer, i=1, 2,…, p. In this paper, we prove that there is a partition of V(G~I(M)) into p parts V_1 , V_2,…, V_p such that |V_i| =k_i and the subgraph H_i induced by V_i contains a k_i-cycle when k_i ≥3, H_i is isomorphic to K_2 when k_i =2 and H_i is a single point when k_i =1.     

Fault-tolerant cycles embedding in folded hypercubes

The generalized conditional fault-tolerant embedding is investigated, in which the n-dimensional folded hypercube networks(denoted by FQ_n) acts as the host graph, and the longest fault-free cycle represents the guest graph. Under the conditions looser than that of previous works, it is shown that FQ_n has a cycle with length at least 2~n-2︱F_v︱ when the number of faulty vertices and non-critical edges is at most 2n-4; where ︱F_v︱ is the number of faulty vertices. It provides further theoretical evidence for the fact that FQ_n has excellent node-fault-tolerance and edge-fault-tolerance when used as a topology of large scale computer networks.     

An algorithm for the spectral immunity of binary sequence with period 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

To resist the fast algebraic attack and fast selective discrete Fourier transform attacks, spectral immunity of a sequence or a Boolean function was proposed. At the same time, an algorithm to compute the spectral immunity of the binary sequence with odd period N was presented, here N is a factor of 2 ⁿ ? 1, where n is an integer. The case is more complicated when the period is even. In this paper, we compute linear complexity of every orthogonal sequence of a given sequence using Chan-Games algorithm and k - error linear complexity algorithm. Then, an algorithm for spectral immunity of binary sequence with period N = 2 ⁿ is obtained. Furthermore, the time complexity of this algorithm is proved to be O(n).     

Influence of growth conditions on the electrochemical synthesis of SnS thin films and their optical properties

Tin sulfide (SnS) thin films were prepared by electrodeposition onto fluorine-doped tin oxide (FTO) glass substrates using an aqueous solution containing SnCl₂ and Na₂S₂O₃ at various deposition potentials (E) and bath concentrations. The pH value and temperature of the solution were kept constant. The deposited films were characterized using X-ray diffraction (XRD), field-emission scanning electron microscopy (FESEM), photoluminescence (PL), and ultraviolet–visible (UV–Vis) spectroscopy. The FESEM images demonstrated that changes in the deposition potential (E) and solution concentration led to marked changes in the morphology of the deposited SnS films. Energy-dispersive X-ray analysis (EDXA) results showed that the Sn/S atomic ratio strongly depended on both the solution concentration and the deposition potential. To obtain an Sn/S atomic ratio approximately equal to 1, the optimal Sn2+/S₂O₃2- molar ratio and E parameter were 1/8 and -1.0 V, respectively. The XRD patterns showed that the synthesized SnS was obviously polycrystalline, with an orthorhombic structure. The effects of the variations of bath concentration and deposition potential on the band-gap energy (E_g) were studied using PL and UV–Vis experiments. The PL spectra of all the SnS films contained two peaks in the visible region and one peak in the infrared (IR) region. The UV–Vis spectra showed that the optical band-gap energy varies from 1.21 to 1.44 eV.     

Lüroth Expansion Digits and Maclaurin's Inequality

It is well known that for almost all real number x, the geometric mean of the first n digits d_i(x) in the Lüroth expansion of x converges to a number K_0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits d_i(x) approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f(n) steps away from either extreme. We prove sufficient conditions on f(n) to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f(n) steps away from geometric mean.     

Electrical conductivity optimization of the Na3AlF6-Al2O3-Sm2O3 molten salts system for Al-Sm intermediate binary alloy production

Metal Sm has been widely used in making Al-Sm magnet alloy materials. Conventional distillation technology to produce Sm has the disadvantages of low productivity, high costs, and pollution generation. The objective of this study was to develop a molten salt electrolyte system to produce Al-Sm alloy directly, with focus on the electrical conductivity and optimal operating conditions to minimize the energy consumption. The continuously varying cell constant (CVCC) technique was used to measure the conductivity for the Na₃AlF₆-AlF₃-LiF-MgF₂-Al₂O₃-Sm₂O₃ electrolysis medium in the temperature range from 905 to 1055℃. The temperature (t) and the addition of Al₂O₃ (W(Al₂O₃)), Sm₂O₃ (W(Sm₂O₃)), and a combination of Al₂O₃ and Sm₂O₃ into the basic fluoride system were examined with respect to their effects on the conductivity (κ) and activation energy. The experimental results showed that the molten electrolyte conductivity increases with increasing temperature (t) and decreases with the addition of Al₂O₃ or Sm₂O₃ or both. We concluded that the optimal operation conditions for Al-Sm intermediate alloy production in the Na₃AlF₆-AlF₃-LiF-MgF₂-Al₂O₃-Sm₂O₃ system are W(Al₂O₃) + W(Sm₂O₃)=3wt%, W(Al₂O₃):W(Sm₂O₃)=7:3, and a temperature of 965 to 995℃, which results in satisfactory conductivity, low fluoride evaporation losses, and low energy consumption.     

Calibration of the U 37 K′ index of long-chain alkenones with the in-situ water temperature in Lake Qinghai in the Tibetan Plateau

Long-chain alkenones (LCAs) can potentially be used as indicators to understand past variations in lacustrine environments.Previous research has suggested that the relationship between the temperature and the unsaturation index of LCAs should be calibrated individually,because of the possible variations in the alkenone-producing algal species in the lacustrine environment.In this work,we have calibrated U37K’ of water filter samples against the in-situ water temperature in Lake Qinghai,Tibetan Plateau.There are significant relationships between U37K’ and the water temperature,a non-linear relationship was derived.Because the U37K’ values did not respond sensitively at lower temperatures,we suggested that a quadratic regression (U37K’ =0.0011×T2-0.0201×T+0.1959,n=15,r2=0.74) was appropriate than linear regression to represent the relationship between the in-situ temperatures and U37K’.Meanwhile,the U37K correlation relationship was not more significant than U37K’ index in our study.Because of the C37:4 effects by salinity change,we suggest U37K is not as robust as the U37K’ index as a temperature proxy,at least for the salt lake in the Tibetan Plateau.The calibration of the U37K’ index in this work has provided a new understanding of historic climatic changes in the Tibetan Plateau.     

Dissolution characteristics of sericite in chalcopyrite bioleaching and its effect on copper extraction

The effects of sericite particle size, rotation speed, and leaching temperature on sericite dissolution and copper extraction in a chalcopyrite bioleaching system were examined. Finer particles, appropriate temperature and rotation speed for Acidithiobacillus ferrooxidans resulted in a higher Al3+ dissolution concentration. The Al3+ dissolution concentration reached its highest concentration of 38.66 mg/L after 48-d leaching when the sericite particle size, temperature, and rotation speed were -43 μm, 30℃, and 160 r/min, respectively. Meanwhile, the sericite particle size, rotation speed, and temperature can affect copper extraction. The copper extraction rate is higher when the sericite particle size is finer. An appropriately high temperature is favorable for copper leaching. The dissolution of sericite fitted the shrinking core model, 1-(2/3)α-(1-α)2/3=k₁t, which indicates that internal diffusion is the decision step controlling the overall reaction rate in the leaching process. Scanning electron microscopy analysis showed small precipitates covered on the surface of sericite after leaching, which increased the diffusion resistance of the leaching solution and dissolved ions.     

Reduction in secondary dendrite arm spacing in cast eutectic Al-Si piston alloys by cerium addition

The effects of Ce on the secondary dendrite arm spacing (SDAS) and mechanical behavior of Al-Si-Cu-Mg alloys were investigated. The reduction of SDAS at different Ce concentrations was evaluated in a directional solidification experiment via computer-aided cooling curve thermal analysis (CA CCTA). -The results showed that 0.1wt%-1.0wt% Ce addition resulted in a rapid solidification time, △T_S, and low solidification temperature, △T_S, whereas 0.1wt% Ce resulted in a fast solidification time, △ta-Al, of the α-Al phase. Furthermore, Ce addition refined the SDAS, which was reduced to approximately 36%. The mechanical properties of the alloys with and without Ce were investigated using tensile and hardness tests. The quality index (Q) and ultimate tensile strength of (UTS) Al-Si-Cu-Mg alloys significantly improved with the addition of 0.1wt% Ce. Moreover, the base alloy hardness was improved with increasing Ce concentration.     

<Emphasis Type="Italic">a</Emphasis>-Wythoff’s game with <Emphasis Type="Italic">n</Emphasis> players

Letting a = 1 in a- Wythoff’s game introduced by Fraenkel yields Wythoff’s game which is a well-known 2-player impartial combinatorial game introduced by Wythoff in 1907. A method of solving n-player impartial games was presented by Krawec in 2012. In this paper, we employ Krawec’s function to analyze n > 2 players a-Wythoff’s game and obtain game values for all a 1. The results obtained cover n-player Wythoff’s game, a special case 1-Wythoff’s game.