Calculation of the FeZnSi phase diagram between 773 and 1173 K

The phase diagram for the Fe-Zn-Si system has been calculated from thermodynamic data for the binary systems. The excess Gibbs energies of the solid and liquid solutions in the ternary system are described by: ΔG^xs = ΔG^xs₁₂ + ΔG^xs₁₃ + ΔG^xs₂₃ + ΔG^xs₁₂₃ with: ΔG^xs_ij = x_ix_j(A_ijx_i + B_ijx_j)ΔG^xs₁₂₃ = A₁₂₃x₁x₂x₃ The ternary interaction parameter A₁₂₃ is obtained from experimental characteristics of the ternary diagram: temperature dependance on the solubility of Zn in Fe₃Si and FeSi in Zn. The diagram Fe-Zn-Si calculated between 773 and 1173 K is used for explaining diffusion path observed during galvanization of the Fe-Si alloys.     

Oscillation and nonoscillation for second-order linear difference equations

New oscillation and nonoscillation theorems are obtained for the second-order linear difference equation Δ²x_n−1 + p_nx_n = 0, where if p_n^∞_n=1 is a real sequence with p_n ≥ 0.     

Binary common-ion alkali halide mixtures thermodynamic analysis of solid-liquid phase diagrams: II. Systems with complete solid miscibility application of the liqfit method

LIQFIT is an iterative procedure — with intermediate phase diagram calculations — in which the calculated liquidus is made to run through the experimental liquidus points. The computations yield a reliable solidus and in addition the Redlich-Kister coefficients of the ΔG^E function. The latter is the difference between the excess Gibbs energies of the liquid and solid mixtures. The calculations are based on the equal-G curve (EGC), i.e. the projection on the TX plane of the intersection of the GTX surfaces of the liquid and solid mixtures. The EGC is the link between the phase diagram and the ΔG^E function. Results are presented for 18 common-ion alkali halide systems which show complete miscibility in the solid state.     

Phase stability of carbon in FCC and BCC metals

The Metal-Carbon interaction parameter was calculated from the binary systems Cr-C, Mn-C, Ni-C, Co-C, V-C, and the ternary systems Cr-Fe-C, Mn-Fe-C, Ni-Fe-C, Co-Fe-C, V-Fe-C, V-Cr-C as was done for Fe-C. In all cases the excess Gibbs energy of mixing can be expressed as Gm,ex = W3(x−x³), where W is the interaction parameter, and x is the atom fraction of the metal. Along the periodic table from V to Ni W changes smoothly. The phase stability of carbon was calculated from the invariant points of the binary phase diagrams. The same interaction parameter was used in the solid and liquid phases. From the equilibria in the Fe-Cr-C system given by Jackson the stability was also calculated in Fe-Cr alloys.     

Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients

In this paper, we consider the following higher-order neutral delay difference equations with positive and negative coefficients: Δ^m(x_n + cx_n−k) + p_nx_n−r − q_nx_n−l = 0, n≥n₀, where c ϵ R, m ⩾ 1, k ⩾ 1, r, l ⩾ 0 are integers, and {p_n}^∞_n=n0 and {q_n}_n=n0^∞ are sequences of nonnegative real numbers. We obtain the global results (with respect to c) which are some sufficient conditions for the existences of nonoscillatory solutions.     

Phase equilibria modeling in Bi2O3–SrO–MnOx system

The selected pseudobinary, isothermal and isoactivity sections of the Bi₂O₃–SrO–MnO_x phase diagram are constructed from the calculated phase equilibria. The starting point for the thermodynamic analysis were the assessments of the respective pseudobinary systems Bi₂O₃–SrO, Bi₂O₃–MnO_x and SrO–MnO_x. The only quaternary solid phase–the perovskite solid solution–is described in terms of the point defect model [ Bi³⁺_1−xSr²⁺_x] [ Mn³⁺_1−x+2δ Mn⁴⁺_x−2δ] [O²⁻_1−δ/3 V a_δ/3]₃. The quaternary liquid is approximated by a phenomenological model using the liquid species Mn, MnO, MnO_1.5, SrO, and BiO_1.5 with only binary Redlich–Kister coefficients. The ab initio electronic structure calculations of cohesive energies are employed to evaluate the enthalpies of formation of some stoichiometric phases.     

An extended collocation method

Given an approximate solutionx _n of a linear operator equation obtained by a collocation method, an improved solutionx ^* _n+m is obtained fromx _n by an «extended collocation method» which consists in solving a further (m)-order linear system instead of an (n+m)-order one, diminishing the effects of rounding error in carrying out the calculations. For a suitable choice of the knot, the method may be recursively performed both by spline approximation and by algebraic and trigonometric polynomial approximation. A numerical example with a two point boundary value problem confirms the advantages of the extended method with respect to the direct one.     

Inversion Problems in theq-Hahn Tableau

If { P_n(x;q)}_nis a family of polynomials belonging to the q -Hahn tableau then each polynomial of this family can be written as P_n(x;q) = ∑_{m = 0}ⁿD_m(n)_m(x) where _m(x) stands for (x;q)_mor x^m. In this paper we solve the corresponding inversion problem, i.e. we find the explicit expression for the coefficients I_m(n) in the expansion _n(x) = ∑_{m = 0}ⁿI_m(n)P_m(x;q).     

Set-contractive mappings and difference equations in Banach spaces

In this paper, we consider the difference equation on an arbitrary Banach space (X, ∥·∥_x), Δ(q_nΔx_n + f_n(x_n) = 0, where {q_n} is a positive sequence and f_n is X-valued. We shall give conditions so that for a given x ϵ X, there exists a solution of this equation asymptotically equal to x.     

An optimal algorithm for constructing the weighted voronoi diagram in the plane

Let S denote a set of n points in the plane such that each point p has assigned a positive weight w(p) which expresses its capability to influence its neighbourhood. In this sense, the weighted distance of an arbitrary point x from p is given by d_e(x,p)/w(p) where d_e denotes the Euclidean distance function. The weighted Voronoi diagram for S is a subdivision of the plane such that each point p in S is associated with a region consisting of all points x in the plane for which p is a weighted nearest point of S.An algorithm which constructs the weighted Voronoi diagram for S in O(n²) time is outlined in this paper. The method is optimal as the diagram can consist of Θ(n²) faces, edges and vertices.     

Analysis of a simple factorization algorithm

The probability that the k^th largest prime factor of a number n is at most n^x is shown to approach a limit F_k(x) as n → ∞. Several interesting properties of F_k(x) are explored, and numerical tables are given. These results are applied to the analysis of an algorithm commonly used to find all prime factors of a given number. The average number of digits in the k^th largest prime factor of a random m-digit number is shown to be asymptotically equivalent to the average length of the k^th longest cycle in a permutation on m objects.     

A sweepline algorithm for Euclidean Voronoi diagram of circles

Presented in this paper is a sweepline algorithm to compute the Voronoi diagram of a set of circles in a two-dimensional Euclidean space. The radii of the circles are non-negative and not necessarily equal. It is allowed that circles intersect each other, and a circle contains others.The proposed algorithm constructs the correct Voronoi diagram as a sweepline moves on the plane from top to bottom. While moving on the plane, the sweepline stops only at certain event points where the topology changes occur for the Voronoi diagram being constructed.The worst-case time complexity of the proposed algorithm is O((n+m)log n), where n is the number of input circles, and m is the number of intersection points among circles. As m can be O(n²), the presented algorithm is optimal with O(n² log n) worst-case time complexity.     

Monotone Einschließung von Inversen positiver Elemente durch Verfahren vom Schulz-Typ

The conditional iterationx _n+1 =sup (x _n ,x _n +x _n (e?ax _n )),y _n?1 =inf (y _n ,x _n +y _n (e?ax _n )) generating sequences (x _n ) and (y _n ) is considered in partially ordered spaces. Under certain conditions it is shown, that the inversea ^?1 of a positive elementa≧0 is monotonously enclosed in the sensex _n ≦x _n+1 ≦a ^?1 ≦y _n+1 ≦y _n and that (x _n ) and (y _n ) converge toa ^?1 quadratically.     

A pipelined computer architecture for unified elementary function evaluation

A pipelined computer architecture for rapid consecutive evaluation of several elementary functions (x/y, √x, sin x, cos, x, e^x, ln x, …) using basic CORDIC algorithms is proposed. Continued products iterations of the form (1 + σ_im 2^?k) allow linking n-identical ALU structures to permit n different function evaluations. New algorithms for sin^?1, cos^?1, cot^?1, sinh^?1, cosh^?1 and x^v are developed. Lastly, a new functional efficiency is defined for pipeline architectures which compares favorably to iterative arrays.Index terms—Digital Arithmetic, Pipeline, Unified Elementary Functions, Iterative Algorithms, CORDIC     

On the subword complexity of m-free D0L languages

A word is called m-free (m ? 2) if it does not contain a subword of the form x^m where x is a nonempty word. A language is called m-free if it consists of m-free words only. The subword complexity of a language K, denoted π_K, is a function of positive integers which to each positive integer n assigns the number of different subwords of length n occuring in words of K. It is known that if a D0L language K is m-free for some m ? 2, then, for all n, π_K(n) ? qn log₂n for some positive integer q. We demonstrate that there exists a 3-free D0L language K on three letters such that, for all n ? n₀, π_K(n) ? rn log₂n for some positive real r and a positive integer n₀. We also demonstrate that if m ? 3 and K is an m-free D0L language on two letters, then, for all n, π_K(n) ? pn for some positive integer p.     

Cylindrical decomposition for systems transcendental in the first variable

We present a generalization of the Cylindrical Algebraic Decomposition (CAD) algorithm to systems of equations and inequalities in functions of the form p(x,f₁(x),…,f_m(x),y₁,…,y_n), where p∈Q[x,t₁,…,t_m,y₁,…,y_n] and f₁(x),…,f_m(x) are real univariate functions such that there exists a real root isolation algorithm for functions from the algebra Q[x,f₁(x),…,f_m(x)]. In particular, the algorithm applies when f₁(x),…,f_m(x) are real exp-log functions or tame elementary functions.     

A note on maximum independent sets in rectangle intersection graphs

Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209-218] gave a (1+1/k)-factor algorithm with an O(nlogn+n^2k−1) time bound for any integer constant k?1; we describe a similar algorithm running in only O(nlogn+nΔ^k−1) time, where Δ?n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443-470] gave a ⌈log_kn⌉-factor algorithm with an O(n^k+1) time bound for any integer constant k?2; we describe similar algorithms running in O(nlogn+nΔ^k−2) and n^O(k/logk) time.     

Computational complexity of spline interpolation

The procedure for composing the spline function of order m that interpolates n data is roughly divided into two stages: (1) constructing the matrix A_n that transforms the fl-spline coefficient vector c into the sample value vector s; and (2) calculating the vector c. Then the effects of boundary conditions and the locations of the sampling points and the knots on the number of computations for spline interpolation are evaluated. There are no boundary condition effects of the construction of A" for equispacing. Non-periodic boundary conditions reduce the number of computations needed to calculate c from O(m²n) to O(m²n/4).

In the construction of A_n, choosing equispaced sampling points reduces the number of computations from O(m²n) to O(m²) in the case of sampling points located at a constant interval between each pair of adjacent knots. In the calculation of c, equispaced sampling points reduce the number of computations from O(m²n) to O(m²n /2) in the case where m is even and the locations of the knots are identical with those of the sampling points.