On the automatic derivation of a set of geometric formulae

Leta, b, andc be the three sides of a triangleABC, a _i ,b _i ,c _i anda _e ,b _e , c_e be the lengths of the three internal and external bisectors of the three anglesA, B, andC respectively. It is easy to express the bisectors as formulae of the sides. In this paper, we solve a problem proposed by H. Zassenhaus: for any three different bisectors in {a_i, b_i, c_i, a_e, b_e, c_e}, finding the relations between each side of the triangle and the three chosen bisectors. We also prove that given any general values for three different bisectors (internal or external) of a triangle, we can not draw the triangle using a ruler and a pair of compasses alone. The formulae mentioned above are derived automatically using a general method of mechanical formula derivation.This work was partially supported by a Grant from Chinese NSF and by the NSF Grant CCR-917870.     

On the Distribution of the Sums of Binomial Coefficients Modulo a Prime

 Let be the binomial coefficient modulo b (b prime), with if l is greater than c, and let be the sum of binomial coefficients modulo b, that is (mod b). We prove the following property: the for which the couples c, l verify and are uniformly distributed in the residue classes modulo b as n tends to infinity. The method, using the Perron-Frobenius theory, applies also to and gives a new proof of the well known result for the non-zero binomial coefficients modulo b. (Received 21 June 1999; in revised form 13 July 2000)     

A class of problems for which cyclic relaxation converges linearly

The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value of each of the variables to its optimum subject to the condition that the remaining variables are fixed. We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when applied to the minimization of functions of the form for a _i,j ,b _i ,c _i ∈ℝ_≥0 with max {min {b ₁,b ₂,…,b _n },min {c ₁,c ₂,…,c _n }}>0 over the n-dimensional interval [l ₁,u ₁]×[l ₂,u ₂]×⋅⋅⋅×[l _n ,u _n ] with 0<l _i <u _i for 1≤i≤n. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing problems that arise in chip design.     

Number of points on certain hyperelliptic curves defined over finite fields

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums _l(v), ψ_l(v), and ψ_2l(v), and the Jacobsthal–Whiteman sums and , over finite fields F_q such that . These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n1, the exact number of F_qⁿ-rational points on the projective hyperelliptic curves aY²Z^e−2=bX^e+cZ^e (abc≠0) (for e=l,2l), and aY²Z^l−1=X(bX^l+cZ^l) (abc≠0), defined over such finite fields F_q. As a consequence, we obtain the exact form of the ζ-functions for these three classes of curves defined over F_q, as rational functions in the variable t, for all distinct cases that arise for the coefficients a,b,c. Further, we determine the exact cases for the coefficients a,b,c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F_q.     

Discontinuity of Cappings in the Recursively Enumerable Degrees and Strongly Nonbranching Degrees

We construct an r. e. degree a which possesses a greatest a-minimal pair b₀, b₁, i.e., r. e. degrees b₀ and b₁ such that b₀, b₁ < a, b₀ ∩ b₁ = a, and, for any other pair c₀, c₁ with these properties, c₀ ≤ b_i and c₁ ≤ b_1-i for some i ≤ 1. By extending this result, we show that there are strongly nonbranching degrees which are not strongly noncappable. Finally, by introducing a new genericity concept for r. e. sets, we prove a jump theorem for the strongly nonbranching and strongly noncappable r. e. degrees. Mathematics Subject Classification : 03D25.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

Surface area inequalities for ellipsoids using Minkowski sums

Given the convex body E=E(a,b,c) bounded by the ellipsoid with principal axes of lengths 2a, 2b, and 2c, its surface area, S(a,b,c), is a non-elementary integral unless a=b=c, (E is a ball) or two values of a,b, and c are equal (E is a solid spheroid). This leads to upper and lower estimates for S(a,b,c) in terms of the surface areas of balls or spheroids. We derive many of the known inequalities and some new inequalities for the surface areas of ellipsoids using Minkowski sums of ellipsoids and Minkowski's mixed volumes.     

The Γ-limit of a finite-strain Cosserat model for asymptotically thin domains and a consequence for the Cosserat couple modulus μc

We study the behaviour of a geometrically exact 3D Cosserat continuum model for an asymptotically flat domain. Despite the inherent nonlinearity, the Γ-limit of a corresponding canonically rescaled problem on a domain with constant thickness can be explicitly calculated. This “membrane” limit exhibits no bending contributions scaling with h ³ (similar to classical approaches) but features a transverse shear resistance scaling with h for strictly positive Cosserat couple modulus μ_c > 0. This result is physically inacceptable for a zero-thickness “membrane” limit model. Therefore it is suggested that the physically consistent value of the Cosserat couple modulus μ_c is zero. In this case, however, the Γ-limit looses coercivity for the midsurface deformation in H ^1,2(ω , ℝ³). For numerical purposes then, a transverse shear resistance can be reintroduced, establishing coercivity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An improvement and an extension of the Elzinga & Hearn's algorithm to the 1-center problem in ℝn withl 2b-normswithl 2b-norms

Summary In this paper we deal with the 1-center problem in ℝⁿ when the distance is measured by anyl _2b-norm. This type of norm generalizes the Euclidean norm (l ₂-norm) and can be used to estimate road distances or travel times in Locational Analysis, and to measure dissimilarities between data in Cluster Analysis. The problem is to find the smallestb-ellipsoid containing a given finite setA of points in ℝⁿ, which generalizes the one of finding the smallest sphere containingA (1-center problem with thel ₂-norm). We show that this problem has a unique optimal solution. For thel ₂-norm, we use the Elzinga-Hearn algorithm. New starting rules are proposed to initialize and to improve the algorithm. On the other hand, the Elzinga-Hearn algorithm is extended to solve the problem withl _2b-norms. Computational results are given for six differentl _2b-norms, when these new starting rules are used in order to show which is the best starting rule. Problems of up to 5.000 points in ℝⁿ,n=2,4,6,8 and 10, are solved in a few seconds.     

Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let C_a,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of C_a,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph C_a,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.     

Hardy inequalities for weighted Dirac operator

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r ^−b for functions in \mathbbRⁿ{\mathbb{R}^n}. The exact Hardy constant c _b  = c _b (n) is found and generalized minimizers are given. The constant c _b vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in \mathbbR²{\mathbb{R}^2}. Analogous inequalities are proved in the case c _b  = 0 under constraints and, with error terms, for a bounded domain.     

The Minimum Covering l pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in ⁿ. A hypersphere is a set S(c,r)={x ⁿ : d(x,c) r}, where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l ₂ (x-c). We consider the extension of this problem when d(x, c) is given by any l _pb -norm, where 1<p and b=(b ₁,...,b _n ) with b _j >0, j=1,...,n, then S(c,r) is called an l _pb -hypersphere, in particular for p=2 and b _j =1, j=1,..., n, we obtain the l ₂-norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallest l _pb -hypersphere enclosing a large set of points in ⁿ.     

Finite Families of lp-Spaces and Multirectangular Characteristics

Under consideration are finite families of Banach spaces with unconditional basis. The problem of isomorphic classification of families is studied together with the related problem of uniqueness of an unconditional basis for a family. Complete solution of both problems for l ₁- and c ₀-families is given by using some geometric invariants combined with the corresponding results for l ₁ and c ₀.     

Distance-regular subgraphs in a distance-regular graph, II

Let Γ be a distance-regular graph with r = l(1, a₁, b₁), a₁ > 0 and c_2r+1 = 1. We show the existence of a collinearity graph of a Moore geometry of diameter r + 1 as a subgraph in Γ. In particular, we show that r = 1.     

Two-dimensional model of a strip detector with two tunnel junctions

The article describes a two-dimensional model of a strip X-ray detector with a superconducting absorbing strip and two tunnel junctions at its ends, The motion of nonequilibrium current carriers (quasiparticles) is considered in the framework of a diffusion equation with boundary conditions of third kind. The detector signals Q _l and Q _r and the spectral line shape are computed as a function of the strip size, quasiparticle capture efficiency in the tunnel junction, the quasiparticle loss parameter on the absorber lateral boundaries, and the quasiparticle recombination parameter R. Analytical solutions for the detector signals Q _l and Q _r are derived for the case R = 0 . Numerical calculations are generally used. When boundary and recombination losses are allowed for, the detector signals turn out to depend on the photon absorption coordinate in the transverse direction. This broadens and distorts the spectral line shape.     

Identification of temperature‐dependent thermophysical properties in a partial differential equation subject to extra final measurement data

We consider an inverse problem for estimating the two coefficient functions c and k in a parabolic type partial differential equation c(u)u_t = ?[k(u)u_x]/?x with the aid of the measurements of u at two different times. The first‐ and second‐order one‐step group preserving schemes are developed. Solving the resultant algebraic equations with a closed‐form, we can estimate the unknown temperature‐dependent thermal conductivity and heat capacity. The new methods possess threefold advantages: they do not require any a priori information on the functional forms of thermal conductivity and heat capacity; no initial guesses are required; and no iterations are required. Numerical examples are examined to show that the new approaches have high accuracy and efficiency, even there are rare measured data. When the measured temperatures are polluted by uniform or normal random noise, the estimated results are also good. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On the Hausdorff dimensions of certain attractors

For any a,b∈R let ϕ_a,b(x)=ax+b(x∈R). Suppose 0<a<1. Let C_a,b be the generalized a-Cantor set with generating iterated function systme {ϕ_a,0, ϕ_a,b; ϕ_a,l}. Then we prove the Hausdorff dimension of C_a,c² C_{a,c^2 } is \fracln(3 - ?5 - ln2lna\frac{{ln(3 - \sqrt 5 - ln2}}{{lna}} when 0<a≤2 cos 80°.     

A remark on the intersection arrays of distance regular graphs and the distance regular graphs of diameter d = 3i − 1 with b i = 1 and k > 2

Let be a distance regular graph with intersection array b ₀, b ₁,..., b _d–1; c ₁,..., c _d. It is shown that in some cases (c _i–1, a _i–1, b _i–1) = (c ₁, a ₁, b ₁)and (c _2i–1, a _2i–1, b _2i–1) imply k 2b _i + 1. As a corollary all distance regular graphs of diameter d = 3i – 1 with b _i = 1 and k > 2 are determined.     

On the use of AGE algorithm with a high accuracy Numerov type variable mesh discretization for 1D non-linear parabolic equations

In this paper, the use of N-AGE and Newton-N-AGE iterative methods on a variable mesh for the solution of one dimensional parabolic initial boundary value problems is considered. Using three spatial grid points, a two level implicit formula based on Numerov type discretization is discussed. The local truncation error of the method is of O(k²h_l^-1 +kh_l +h_l³)O({k^2h_l^{-1} +kh_l +h_l^3}), where h _l  > 0 and k > 0 are the step lengths in space and time directions, respectively. We use a special technique to handle singular parabolic equations. The advantage of using these algorithms is highlighted computationally.     

Mathematical model for crack arrest of a transversely cracked piezoelectromagnetic strip – Part I

A magnetic, electric and mechanical yield model is proposed for a cracked piezoelectromagnetic ceramic narrow strip. The strip is subjected to anti-plane mechanical and in-plane electric and magnetic loads, consequently the crack opens in self-similar fashion forming a magnetic, a saturation and a slide zone ahead each tip. These in turn are arrested by prescribing a magnetic, electric and mechanical load, respectively. Employing Fourier integral transform the problem reduces to the solution of three dual integral equations. The solution of dual integral equations is then expressed in terms of Fredholm integral equation of second kind. Expressions are derived for yield induction zone, slide-yield zone and saturation zone lengths, energy release rate. A case study is carried for BaTiO₃–CoFe₂O₄ and results are presented graphically. It is shown that proposed model is capable of crack opening arrest under small-scale-yielding.