Effect of Mg and C contents in MgCNi3, and structure and superconductivity of MgCNi3-x -Co x

The effect of Mg and C contents on T_C in MgCN₃, and structure and superconductivity of MgCNi_3-x Co _x were studied. It is found that the excess of Mg and C in initial material mixture is favorable to improvement inT _c and helps to obtain single-phase samples. For preparing MgCNi₃ superconductor, the optimum composition of starting materials is MgC_1.45Ni₃ with excess of Mg (20 wt.%) of the stoichiometric composition. In gCNi_3-x Co _x system, a continuous solid solution is formed, lattice parameter decreases slightly andT _c decreases obviously with increasingx. A suppression of superconductivity is observed due to the substitution of Co (Mn) for Ni. The suppression effect is smaller for the substitution of Co than that of Mn     

Some group actions on K(x 1 , x 2 , x 3 )

LetK be any field which may not be algebraically closed,K(x ₁ ,x ₂ ,x ₃ ) be the rational function field of three variables overK, and σ:K(x ₁ ,x ₂ ,x ₃ ) → K(x ₁ ,x ₂ ,x ₃ ) be aK-automorphism defined by wherea _i ,b _i ,c _i ,d _i ∈K anda _i d _i −b _i c _i ≠0. Let ,f _i (T)=T ² −(a _i +d _i )T+(a _i d _i −b _i c _i )∈K[T] be the “characteristic polynomial” of σ _i . Theorem:Assume that charK≠2.Then the fixed field K(x ₁ ,x ₂ ,x ₃ ) ^<σ> is not rational (=purely transcendental) over K if and only if (i) for each 1≤i≤3, f _i (T) is irreducible; (ii) the Galois group of f ₁ (T)f ₂ (T)f ₃ (T) over K is of order 8; and (iii) for each 1≤i≤3,ord (σ _[itn] )is an even integer.     

The magnetism and CMR effect in FeCr2-x GaxS4

The magnetism and CMR effects of FeCr_2-xGa_xS₄ materials without double-exchange interaction are studied in this paper. The experimental results show that the ferromagnetism of system is strengthened and T_c is raised because the substitution of Ga for Cr ions destroys the interaction among Cr spins in Cr sub-lattice. But the amplitude of PM-FM phase transition decreases gradually with increasing Ga content. The electron spin resonance (ESR) measurements indicate that at temperatures below T_c, the spins in both Fe sub-lattice and Cr sub-lattice are ferromagnetic, while the spins between two sub-lattices are in AFM arrangement. For undoped sample, the macro magnetic moment of Fe sub-lattice and that of Cr sub-lattice cancel out each other. After doping, the net ferromagnetism is introduced at temperatures below T_c; thus the peak temperature of MR curve increases and the value of MR decreases.     

On the law S ( S ( x , y ), T ( x , y )) = S ( x , y ) of fuzzy logic

Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.     

Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x)

We provide generalizations of two of Euler’s classical transformation formulas for the Gauss hypergeometric function extended to the case of the generalized hypergeometric function _r+2 F _r+1(x) when there are additional numeratorial and denominatorial parameters differing by unity. The method employed to deduce the latter is also implemented to obtain a Kummer-type transformation formula for _r+1 F _r+1 (x) that was recently derived in a different way.     

On the Diophantine System a^2 ＋ b^2 = c^3 and a^x ＋ b^y = c^z for b is an Odd Prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.     

The decomposition of Kv into K2×K5''''s

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25).     

On the Function w( x)=|{1= s= k : x= a s (mod n s)}|

For a finite system of arithmetic sequences the covering function is w(x) = |{1 s k : x a_s (mod n_s)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n₀ of w(x) and the moduli n₁, . . . , n_k in such a system A. Here are three central results: (a) For each r=0,1, . . .,n_k/(n₀,n_k)–1 there exists a Jc{1, . . . , k–1} such that . (b) If n₁ ···n_k–l <n_k–l+1 =···=n_k (0 < l < k), then for any positive integer r < n_k/n_k–l with r 0 (mod n_k/(n₀,n_k)), the binomial coefficient can be written as the sum of some (not necessarily distinct) prime divisors of n_k. (c) max_(xw(x) can be written in the form where m₁, . . .,m_k are positive integers.The research is supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

Effect of A-site cation substitution on magnetic transition temperature of (LaCa) MnO3

The ferromagnetic transition temperatures have been measured on two bathes of perovskite manganese oxides such as (La_1-x R _x )_2/3Ca_1/3MnO₃(R=Ce, Pr, Nd, etc.) and La_2/3(Ca_1-y Cd _y )_1/3MnO₃. It is found that the (La_1-x R _x )_2/3Ca_1/3MnO₃ samples have obviously different Curie temperatureT _C although their (r_A) (the average A-site cation radius) are almost identical, and theT _C in the system of (La_1-x R _x )_2/3Ca_1/3MnO₃(R=Ce, Pr, Nd, etc.) and La_2/3(Ca_1-y Cd _y )_1/3MnO₃. It is found that the (La_1-x R _x )_2/3Ca_1/3MnO₃ compounds increases with increasingy although (r_A) decreases slightly. It is suggested that the Curie temperature should be influenced by the average local A-site magnetic moment in the ferromagnetic perovskite manganite system. Project supported by the National Natural Science Foundation of China (Grant No. 19504012) and the Chinese Academy of Sciences.     

Rado Numbers fora(x+y)=bz

In the case of existence the smallest numberN=Ra_kis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Ra_k(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {x_n} generated bya(x_n+x_n+1)=bx_n+2 is discussed.     

行列式detVn-(x1,...,xn)的准确值及其应用

本文给出行列式Ｖ_n^-(x₁,...,x_n)的准确值，它是通常的Ｖａｎｄｅｒｍｏｄｅ行列式计算公式的推广，以及它在理论上的一些重要应用．     

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).     

Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

On the completeness of the system of function {e sin λnx}n = 1

In this paper, the biorthogonal system corresponding to the system {e^−αnx sin nx}_{n = 1}^∞ is represented in an appropriate form so that it is possible to obtain sufficiently good estimates of its norm. Then, by the stability of a completeness property we prove that the system of functions {e^−αλ_nx sin λ_nx}_{n = 1}^∞ is complete.     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

Trace maps over M3(K) associated with the substitutions

Let K be a field of characteristic zero and M ₃(K) the ring of matrices 3×3 over K. In this paper, we establish first the certain identities of traces of some algebras of Min3(K), then we define the trace mapping of 3 × 3 matrices associated with a substitution over a two-letter alphabet on some algebraic variety and study their properties.     

The magnetism and CMR effect in FeCr2-x GaxS4

The magnetism and CMR effects of FeCr_2-xGa_xS₄ materials without double-exchange interaction are studied in this paper. The experimental results show that the ferromagnetism of system is strengthened and T_c is raised because the substitution of Ga for Cr ions destroys the interaction among Cr spins in Cr sub-lattice. But the amplitude of PM-FM phase transition decreases gradually with increasing Ga content. The electron spin resonance (ESR) measurements indicate that at temperatures below T_c, the spins in both Fe sub-lattice and Cr sub-lattice are ferromagnetic, while the spins between two sub-lattices are in AFM arrangement. For undoped sample, the macro magnetic moment of Fe sub-lattice and that of Cr sub-lattice cancel out each other. After doping, the net ferromagnetism is introduced at temperatures below T_c; thus the peak temperature of MR curve increases and the value of MR decreases.     

Best Approximation of Functions like |x| exp(−A|x|)

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like?_λ, α(x)=|x|^λ exp(−A|x|^−α). In particular, it is shown thatE(?_λ, α, _n, L_p(−1, 1))∼n^{−(2λp+αp+2)/2p(1+α)}×exp(−(1+α⁻¹)(Aα)^1/(1+α) cos απ/2(1+α) n^α/(1+α)), whereE(?_λ, α, _n, L_p(−1, 1)) denotes best polynomial approximation of?_λ, αinL_p(−1, 1),λ∈,α∈(0, 2],A>0, 1?p?∞. The problem, concerning the exact order of decrease ofE(?_0, 2, _n, L_∞(−1, 1)), has been posed by S. N. Bernstein.     

Magnetic impurity effect on superconductivity in systems with comparable Fermi and Debye energies

We obtain the basic system of equations for a superconductivity theory of a system with a chaotically distributed paramagnetic substitution impurity in which the Migdal theorem is violated (the condition ω_D ≪E _F cannot be assumed). The electron-phonon and impurity diagrams and also the additional diagrams corresponding to intersections of the electron-phonon and electron-impurity lines are taken into account. In the weak electron-phonon coupling limit, we obtain an equation for the superconducting transition temperature T_C that differs from the corresponding equation for the usual superconductors by renormalizations of T_C0 and of the impurity scattering parameter, ρ. These quantities depend essentially on the Migdal parameter ω_D/E _F and on the transferred momentum q_c. We show that the decrease of T_C with the increase of the impurity concentration is slowed, as compared with usual superconductors, to an extent determined by m and q_c. We also evaluate the isotopic coefficient α, whose behavior as a function of the impurity concentration depends on m and q_c. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp 455–474, June, 1999