Structure and crystallization of bulk amorphous Pd41Ni10Cu28P21 alloy

Bulk amorphous Pd41Ni10Cu28P21 alloy has been prepared by water quenching method. The system shows excellent glass forming ability (GFA) with a high value of reduced glass transition temperature Trg, 0.714. Structural analyses indicate that the Pd41Ni10Cu28P21 alloy has a dense packing structure closer to "frozen liquid" than that of amorphous Pd40Ni40P20 alloy. Experiments on crystallization reveal that several crystalline phases simultaneously precipitate in the early part of crystallization. Below 710 K, a metastable phase forms, and subsequently disappears at elevated temperatures. In addition, the influence of partial substitute of Cu for Ni on GFA has been discussed with regard to thermodynamics and kinetics.     

Liquid structure of undercooled Cu70Ni30 alloy

Experiments of X-ray diffraction for liquid Cu₇₀Ni₃₀ alloy above and below its liquidus (l 230°C) have been carried out. By the analysis of experimental results, it is discovered that difference between structures of liquid and undercooled Cu₇₀Ni₃₀ alloy is their cluster sizes. The correlation radius of cluster is 1.125 nm and the atom number of cluster is 403 at l 250—l 400°C, and they are 1.3 nm and 704 respectively at the undercooled liquid state (1 200°C). The structure of liquid Cu₇₀Ni₃₀ alloy is fcc short order and its solid structure, fcc, is kept from liquid fcc short order.     

Formation of nanocrystalline Fe73.5Cu1Nb3Si13.5B9 alloy under high pressure

The formation of nanocrystalline Fe_73.5 Cu₁Nb₃Si_13.5 B₉ alloy by annealing an amorphous Fe_73.5Cu₁Nb₃Si_13.5B₉ alloy at a temperature of 823 K under pressures in the range of 1–5 GPa is investigated by using X-ray diffraction, electron diffraction, and transmission electron microscopy. The high pressure experiments are carried out in belt-type pressure apparatus. Experimental results show that the initial crystalline phase in these annealed alloys is a-Fe solid solution (named a-Fe phase below), and high pressure has a great influence on the crystallization process of the a-Fe phase. The grain size of the a-Fe phase decreases with the increase of pressure (P). The volume fraction of the a-Fe phase increases with increasing the pressure as the pressure is below 2 GPa, and then decreases (Pδ2 GPa). The mechanism for the effects of the high pressure on the crystallization process of amorphous Fe_73.5Cu₁Nb₃Si_13.5B₉ alloy is discussed Project supported by the National Natural Science Foundation of China (Grant No. 19674070) and the Natural Science Foundation of Hebei Province.     

The kernel of the Rost invariant, Serre''s Conjecture II and the Hasse principle for quasi-split groups 3,6 D 4 , E 6 , E 7

 We prove that for a simple simply connected quasi-split group of type ^3,6 D ₄ ,E ₆ ,E ₇ defined over a perfect field F of characteristic ≠=2,3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cd F≤2 (resp. vcd F≤2) then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a (C ₂ )-field, in particular ℂ(x,y), we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of type ^3,6 D ₄ ,E ₆ ,E ₇ . Received: 27 March 2002 / Published online: 28 March 2003 The author gratefully acknowledge the support of TMR ERB FMRX CT-97-0107 and Forschungsinstitut für Mathematik, ETH in Zürich     

L 1→ L q Poincaré Inequalities for 0 < q < 1 Imply Representation Formulas

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B ₀⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L ¹→L ¹ Poincaré inequality of the following form: for all metric balls B⊂B ₀⊂S, implies a variant of representation formula of fractional integral type: for ρ-a.e. x∈B ₀, One of the main results of this paper shows that an L ¹ to L ^q Poincaré inequality for some 0 < q < 1, i.e., for all metric balls B⊂B ₀, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved. Received December 27, 2000, Accepted May 28, 2001     

Solidification characteristics of Pd40Ni40P20 alloy under microgravity condition

The microstructure and solute distribution of Pd₄₀Ni₄₀P₂₀ alloy solidified both on board a Chinese retrievable satellite (μg) and on the earth (1g) were studied. It was found that the dendritic primary phase formed under microgravity condition was finer and shorter. In the central area of the sample some asteroidal patterns of the primary phase were present in the microstructure. The primary spacing of the dendrites at the cooling rate of 0. 056 K/s was smaller than that measured in the ground-based experiments at the same cooling rate, but almost the same as that cooled at 0.67 K/s on the ground. With these experimental results, mass transport coefficients both in space and on the earth were evaluated.     

P4k?1-factorization of bipartite multigraphs

Let λK _m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P _v-factorization of λK _m,n is a set of edge-disjoint P _v -factors of λK _m,n which partition the set of edges of λK _m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P _v -factorization of λK _m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k−1-factorization of λK _m,n is (1) (2k − 1)m ⩽ 2kn, (2) (2k − 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer.     

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 Crossing Number of Cartesian Products of Complete Bipartite Graphs K 2, m with Paths P n

Most results on the crossing number of a graph focus on the special graphs, such as Cartesian products of small graphs with paths P_n, cycles C_n or stars S_n. In this paper, we extend the results to Cartesian products of complete bipartite graphs K_2,m with paths P_n for arbitrary m ≥ 2 and n ≥ 1. Supported by the NSFC (No. 10771062) and the program for New Century Excellent Talents in University.     

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).     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R ₁, R ₂, R ₃) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000). This kind of algorithms are useful for many problems. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Research supported by: (1) Research Council K.U.Leuven: GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. M. Ishteva is supported by a K.U.Leuven doctoral scholarship (OE/06/25, OE/07/17, OE/08/007), L. De Lathauwer is supported by “Impulsfinanciering Campus Kortrijk (2007–2012)(CIF1)” and STRT1/08/023.     

Isotonic Approximation in L1

Let (Ω, ,P) be a measurable space and a sub-σ-lattice of the σ-algebra . For XL₁(Ω, ,P) we denote by P X the set of conditional 1-mean (or best approximants) of X given L₁( ) (the set of all -measurable and integrable functions). In this paper, we obtain characterizations of the elements in P X, similar to those obtained by Landers and Rogge for conditional s-means with 1<s<∞. Moreover, using these characterizations we can extend the operator P to a bigger space L₀(Ω, ,P). When, in certain sense, _n goes to _∞, we will be able to prove theorems about convergence and we will obtain bounds for the maximal function sup_nP _nX. A sharper characterization of conditional 1-means for certain particular σ-lattice was proved in previous papers. In the last section of this paper we generalize those results to all totally ordered σ-lattices.     

Frustrated quantum two-dimensional J1-J2-J3 antiferromagnet in a spherically symmetric self-consistent approach

In the framework of a spherically symmetric self-consistent approach to two-time retarded spin-spin Green’s functions, we develop the theory of a two-dimensional frustrated J ₁ -J ₂ -J ₃ quantum S=1/2 antiferromagnet. We show that taking the damping of spin fluctuations into account is decisive in forming both the spin-liquid state and the state with long-range order. In particular, the existence of damping allows explaining the scaling behavior of the susceptibility χ(q, ω) of the CuO ₂ cuprate plane, the behavior of the spin spectrum in the two-plane case, and the occurrence of an incommensurable χ(q, ω) peak. In the case of the complete J₁-J ₂ -J ₃ model, in a single analytic approach, we find continuous transitions between three phases with long-range order (“checkerboard,” stripe, and helical (q, q) phases) through the spin-liquid state. We obtain good agreement with cluster computations for the J ₁ -J ₂ -J ₃ model and agreement with the neutron scattering data for the J ₁ -J ₂ model of cuprates.     

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have lim_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O₂{\mathcal {O}_2} such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have lim inf_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d.     

K2 of a Ring via the G-Construction

We interpret the Steinberg symbols x_i,j(a) as homotopies contracting the elementary matrices e_i,j(a), the latters being represented by certain arcs in a simplicial model of the K-theory. We further prove the Steinberg relations for these homotopies. This provides an explicit map from K₂ of a ring, defined classically as ker(St(R) → GL(R)), to π₂ of the G-construction assigned to R. Though the two groups are known to be isomorphic, a certain work is to be done to prove that this explicit map is an isomorphism. Mathematics Subject Classification 1991: Primary 19B99, 19D99; secondary 18E10, 18F25.     

On inferring the probability of misclassification by the linear discriminant function

Summary Let the two alternative populationsP ₁ andP ₂ from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP ₁ orP ₂ according as (μ(2)-μ(1))′Σ^-1 x≶(1/2)(μ(2)-μ(1))′Σ^-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP ₁ andP ₂ and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP ₂ by this rule is π₂₁=Φ(-δ/2+cδ^-1), where Φ(.) is the distribution function of anN(0, 1) and . (Since we are free to choose which population we shall callP ₂, it is not necessary to consider separately the probability of misclassifying an individual fromP ₁.) LetP ₂₁ denote the probability of misclassification of an individual fromP ₂ by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates andV and letP ₂₁ ^* be the probability of misclassification of an individual fromP ₂ when the classification rule is the one with minimum risk among those based on . The fiducial distributions of π₂₁,P ₂₁ andP ₂₁ ^* are determined. Point estimates and confidence intervals for π₂₁,P ₂₁ andP ₂₁ ^* are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere as well is the distribution of a linear combination of a chi and an independent normal variable.     

Symplectic involutions on deformations of K3[2]

Let X be a hyperk?hler manifold deformation equivalent to the Hilbert square of a K3 surface and let φ be an involution preserving the symplectic form. We prove that the fixed locus of φ consists of 28 isolated points and one K3 surface, and moreover that the anti-invariant lattice of the induced involution on H ₂(X, ℤ) is isomorphic to E ₈(−2). Finally we show that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.     

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.     

Projective and indecomposableS-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which contains_s S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP≅ ∐ _{jϕ I} Se _i ,e _i ϕE. In particular,P is a projective, indecomposable and unitary object if and only ifP ≅Se for somee ∈E. These generalize some results obtained by Knauer and Talwar. Research partially supported by a UGC (HK) (Grant No. 2160092).