Showing papers in "Mathematika in 2002"

On the reverse Lp–busemann–petty centroid inequality

Abstract: The volume of the L p -centroid body of a convex body K ⊂ R d is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the L p -centroid body and related to classical open problems like the slicing problem. Some variants of the L p -Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

Numerical Treatment of Delay Differential Equations by Runge-Kutta Method Using Hermite Interpolation

Abstract: Embedded Diagonally Implicit Runge-Kutta methods of different orders are used for the treatment of delay differential equations The delay argument is approximated using an appropriate Hermite Interpolation The numerical results based on these methods are compared and the Q-stability region of the methods are presented

Non-linear boundary value problems on the semi-infinite interval: an upper and lower solution approach

Abstract: Existence criteria are presented for non-linear boundary value problems on the half line. In particular, the theory includes a problem in the theory of colloids and a problem arising in the unsteady flow of a gas through a semi-infinite porous medium.

Local inversion for differentiable functions and the Darboux property

Abstract: The main goal of this paper is to prove that the classical theorem of local inversion for functions extends in finite dimension to everywhere differentiable functions. As usual, a theorem of implicit functions can be deduced from this “Local Inversion Theorem”. The deepest part of the local inversion theorem consists of showing that a differentiable function with non-vanishing Jacobian determinant is locally one-to-one. In turn, this fact allows one to extend the Darboux property of derivative functions on ℝ (the range of the derivative is an interval) to the Jacobian function D f of a differentiable function, under the condition that this Jacobian function does not vanish. It is also proved that these results are no longer true in infinite dimension. These results should be known in whole or part, but references to a complete proof could not be found.

On shifted products which are powers

Abstract: Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj + 1 is a square for 1 ≤ i < j ≤ 4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai’s are rational numbers. It is conjectured that there do not exist five positive integers such that their pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is the set of positive integers of the form x with x and k positive integers and k > 1. How large can a set of positive integers A be if aa + 1 is in V whenever a and a are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

Rational homology of spaces of complex monic polynomials with multiple roots

Abstract: This paper treats rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces X λ,μ . A combinatorial description of the cell structure of X λ,μ is presented using the language of marked forests. As applications are obtained a new proof of a theorem of Arnold and a counterexample to a conjecture of Sundaram and Welker, along with a few other smaller results.

On minimal pairs and residually transcendental extensions of valuations

Abstract: In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure λ of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to λ is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

The longest segment in the complement of a packing

Abstract: Let K be a compact convex body in R not contained in a hyperplane, and denote the norm whose unit ball is 1 2 (K − K) by ‖ · ‖K . Given a translative packing of K, we are interested in how long segments (with respect to ‖ · ‖K) lie in the complement of the interiors of the translates. The main result of this note is showing the existence of a translative packing such that the length of the longest segments avoiding it is only exponential in the dimension n (see below). We start here with a lower bound showing that this bound is close to optimal for balls. We show that any packing of the unit Euclidean ball B avoids a segment of length exponential in n. It is a rather interesting question to find how long segments necessarily exist that avoid any packing of any convex, open body in R . Our lower bound proof does not work for bodies allowing dense packings. Let | · | denote the n–dimensional Lebesgue measure. Let us consider any packing of B, and denote the area and the packing density of the unit ball by κn and δ(B ), respectively. Choose a unit segment s, and denote the projection of B into some hyperplane orthogonal to s by B, and set

Lipschitz quotient mappings with good ratio of constants

Abstract: It is shown that Lipschitz quotient mappings between finite dimensional spaces behave nicely (e.g., are bijective in the case of equal dimensions) if the Lipschitz and co-Lipschitz constants are close to each other. For Lipschitz quotient mappings of the plane, a bound for the cardinality of the pre-image of a point in terms of the ratio of the constants is obtained.

Local Nonsimilarity Solution for Vertical Free Convection Boundary Layers

Abstract: Suatu analisis dalam mengkaji ciri-ciri aliran dan pemindahan haba bagi masalah lapisan sempadan olakan bebas berlamina terhadap plat menegak di dalam bendalir likat dibincangkan dengan menggunakan kaedah ketakserupaan setempat. Kes apabila suhu permukaan berubah secara eksponen dipertimbangkan dalam mengilustrasikan kaedah tersebut. Dalam kaedah ini, sebutan ketakserupaan yang wujud pada persamaan momentum dan tenaga dikekalkan dan pemansuhan sebutan hanya dilakukan pada persamaan bantu yang diterbitkan. Sistem persaman yang diperolehi diselesaikan secara berangka menggunakan kaedah Keller-box. Kadar pemindahan haba dan tegasan ricih dinding yang diperolehi menunjukkan keputusan yang memuaskan. Kesan parameter $\xi$ dan nombor Prandtl terhadap taburan halaju dan suhu juga ditunjukkan. Katakunci: Olakan bebas; lapisan sempadan; kaedah ketakserupaan setempat; kaedah Keller-box. Laminar free convection boundary layer over a vertical flat plate with an exponential variation of surface temperature in viscous fluids is analyzed using the local nonsimilarity method. The present approach takes into consideration the nonsimilarity terms appearing in the momentum and energy equations, which have been unaccounted for previously, in for example, the similarity and the local similarity methods. The governing equations are solved numerically using the Keller-box method, an efficient implicit finite difference scheme. Numerical results are presented in the form of heat transfer rates, local wall shear stress, velocity and temperature profiles. The heat transfer rates and local wall shear stress obtained show good agreement with available nonsimilar solutions. The effects of various values of transformed stream-wise coordinate $\xi$ and Prandtl numbers on velocity and temperature profiles are also presented. Keywords: Free convection heat transfer; boundary layers; local nonsimilarity method; Keller-box

Absolute curvature measures for unions of sets with positive reach

Abstract: Absolute curvature measures for locally finite unions of sets with positive reach are introduced, extending the definition of Zahle [13] by taking into account the absolute value of the index function. It is shown that this definition differs from that of Matheron [5] and Schneider [12]. An intersection formula of Crofton type for absolute curvature measures is proved. The role of absolute curvature measures in geometric statistics is illustrated by an example.

On extensions generated by roots of lifting polynomials

Abstract: Let υ be a Henselian valuation of any rank of a field K and υ its unique prolongation to a fixed algebraic closure K of K having value group G. For any subfield L of K, let R(L) denote the residue field of the valuation obtained by restricting υ to L. Using the canonical homomorphism from the valuation ring of υ onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, υ)-minimal pair (α, δ) belonging to K × G. As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, υ)-minimal pair (α, δ). If F, F 1 are two such lifted polynomials with coefficients in K having roots θ, θ 1 , respectively, then it is proved in the present paper that υ(K(θ)) = υ(K(θ 1 )), R(K(θ)) = R(K(θ 1 )); in case (K, υ) is a tame field, it is shown that K(θ) and K(θ 1 ) are indeed K-isomorphic.

Epidermiological Approach to Predator-Prey Interaction

Abstract: An epidemiological approch is used to model predator-prey interaction. The concept is borrowed from SIS model of disease transmission. In the predator-prey interaction, predation is viewed as a "disease" of the predator transmitted by the prey. In this approch, the predator population is partitioned into hungry and satisfied subpopulation. The stability of the steady states from the equations is analyzed. Keywords: Predator-prey interaction; Model SIS Pendekatan penyakit berjangkit digunakan untuk memodelkan intearksi mangsa-pemangsa. Konsep ini dipinjam daripada Model SIS jangkitan penyakit. Dalam interaksi mangsa-pemangsa, pemangsa dianggap sebagai "penyakit" pemangsa yang dijangkiti daripada mangsa. Pendekatan ini membahagikan populasi pemangsa kepada subpopulasi lapar dan kenyang. Kestabilan titik-titik kesimbangan daripada persamaan dianalisa. Katakunci: Intearksi mangsa-pemangsa; Model SIS

Nonparametric Kernel Estimation of Annual Maximum Stream Flow Quantiles

Abstract: Kaedah Nonparametric Kernel dicadangkan dan dinilaikan perlaksanaannya dalam menganggar kuantil aliran tahunan maksimum. Penganggar bagi bandwidth dianggarkan menggunakan teknik optimal dan `cross-validation'. Hasil keputusan menggunakan data sebenar yang terhad dari Malaysia, menunjukkan bahawa penganggar kuantil berdasarkan model nonparametric menggunakan kedua-dua teknik ini menghasilkan nilai punca min ralat kuasa dua dan punca min ralat mutlak yang kecil. Berdasarkan ujian pekali korelasi menunjukkan bahawa pendekatan model nonparametric adalah tepat, seragam dan ianya boleh dijadikan sebagai kaedah alternatif bagi model parametric dalam analisis frekuensi banjir. Katakunci: Bandwidth; `cross-validation'; kernel; pekali korelasi. A nonparametric kernel methods is proposed and evaluated performance for estimating annual maximum stream flow quantiles. The bandwidth of the estimator is estimated by using an optimal technique and a cross-validation technique. Results obtained from a limited amount of real data from Malaysia show that quantiles estimated by nonparametric method using these techniques have small root mean square error and root mean absolute error. Based on correlation coefficient test shown that the nonparametric model approach is accurate, uniform and flexible alternatives to parametric models for flood frequency analysis. Keywords: Bandwidth; cross-validation; kernel; correlation coefficient.

Comparisons of the LH Moments and the L Moments

Abstract: Kertas ini membincangkan perbandingan antara kaedah LH momen dengan kaedah L momen. LH momen adalah L-momen umum yang berasaskan kepada gabungan linear bagi statistik tertib tinggi digunakan untuk dipadankan dengan ciri-ciri hujung taburan dan data ekstrim. Analisis terhadap data menggunakan penganggar LH momen bagi hujung atas taburan didapati LH lebih baik berbanding penganggar L momen. Perbandingan bagi gambarajah LH momen dan gambarajah L momen ke atas data juga menunjukkan bahawa taburan GEV dapat dipadankan dengan baik dengan nisbah LH momen berbanding dengan nisbah L momen. Katakunci: LH Moments; GEV; PWM; Gabungan Linear; Statistik Peringkat-Tinggi This paper discusses comparisons of the LH moments method with L moments method. LH moments, a generalization of L moments, based on linear combinations of higher-order statistics was introduced for charactering the upper part of distributions and larger events in data by Wang (1997). Analysis of observed data shows that using LH moments estimates of the upper part of distribution events are expected to be more reasonable than the L moments estimates. A comparison of the LH moment diagram and the L moment diagram of the data also shows that the GEV distribution describes the LH moment ratios better than the L moments. Keywords: LH moments; GEV; PWM; Linear Combination; Higher-Order Statistics

An analogue of Van der Corput's A^5-process for exponential sums

Abstract: A diophantine system is studied from which is deduced an analogue of van der Corput's A 5 -process in order to bound analytic exponential sums of the form Σ 1≤m≤M e(f(m)). The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a ninth derivative test for exponential sums which is essential for giving new exponent pairs in [3].

The Hooley-Huxley contour method for problems in number fields II: factorization and divisibility

Abstract: Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above isfor 0

Penganggaran Parameter bagi Model BL$(p,0,1,1)$ dengan Pendekatan Bayesian

Abstract: Model bilinear merupakan salah satu model tak linear bagi data siri masa dan diwakili oleh BL$(p,q,r,s)$. ARMA merupakan kes khas apabila $r$ dan $s$ mengambil nilai sifar. Model ini dipercayai sesuai digunakan ke atas data hidrologi dan meteorologi. Beberapa kaedah telah disarankan untuk menganggar kesemua parameter bagi model ini. Di dalam kertas kerja ini, kaedah bayesian digunakan untuk membuat penganggaran parameter. Walau bagaimanapun, ia memerlukan maklumat berkenaan dengan reja terdahulu. Oleh itu, kaedah kuasa dua ralat terkecil tak linear digunakan. Set data sunspot digunakan sebagai ilustrasi. Katakunci: Bilinear; Bayesian; siri masa tak linear; Taburan prior tak tentu Jeffrey Bilinear model is one of the nonlinear models for time series data, which is denoted by BL$(p, q, r, s)$. The ARMA model is a special case of Bilinear model when the value $r$ and $s$ are zero. This model is believed to suit best for hydrology and meteorology data. A few estimation methods have been suggested to estimate the parameters. In this paper, the bayesian approach is used. But the residual must be known first. For that the Least Square methods is chosen. The sunspot data is used as an illustration. Keywords: Bilinear; Bayesian; nonlinear time series; Jeffreys' prior distribution

Numerical Experiments on Eigenvalues of Weakly Singular Integral Equations Using Product Simpson's Rule

Abstract: This paper discusses the use of Product Simpson's rule to solve the integral equation eigenvalue problem $\lambda f(x) = \int_{-1}^1k(|x - y|)f(y)dy$ where $k(t) = \ln|t|$ or $k(t) = t^{-1}, 0 < < 1,\lambda, f$ and are unknowns which we wish to obtain. The function $f(y)$ in the integral above is replaced by an interpolating function $L^f_n(y) = \sum_{i=0}^n f(x_i)\phi_i(y),$ where $\phi(y)$ are Simpson interpolating elements and $x_0, x_1,...,x_n$ are the interpolating points and they are chosen to be the appropriate non-uniform mesh points in $[-1, 1].$ The product integration formula $\int_{-1}^1 k(y)f(y)dy\approx \sum_{i=0}^n w_if(x_i)$ is used, where the weights wi are chosen such that the formula is exact when $f(y)$ is replaced by $L^f_n(y)$ and $k(y)$ as given above. The five eigenvalues with largest moduli of the two kernels $K(x, y) = \ln|x-y|$ and $K(x, y) = |x- y|^{-n}, 0 < \alpha < 1$ are given. Keywords: igenvalue; product integration; singular kernel; integral eequation.

Quick asymptotic upper bounds for lattice kissing numbers

Abstract: General upper bounds for lattice kissing numbers are derived using Hurwitz zeta functions and new inequalities for Mellin transforms.

Uniqueness theorems for convex bodies in non-Euclidean spaces

Abstract: The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

Alternative Approach to Deterministic Inventory Problem

Abstract: Dalam kertas ini kita akan menentukan polisi terbaik bagi masalah penambahan perolehan inventori apabila kadar permintaannya tentu berubah dengan masa secara selanjar untuk sesuatu tempoh yang terhad. Kita tentukan bila dan jumlah kuantitinya untuk setiap kitaran. Kita gunakan prosedur genetik algoritma yang berasaskan kepada prinsipal kewujudan Darwin dan model hamparan di dalam Microsoft Excel Solver. Keputusan berangka dari beberapa contoh menunjukkan kedua-kedua prosedur mampu memberikan penyelesaian yang optimum. Katakunci: Permintaan berubah dengan masa; Genetik algoritma; Model hamparan dan Inventori. This paper addresses the problem of determining the best policy for an inventory replenishment with continuous deterministic time-varying demand over a finite planning horizon. We determine when to replenish and how much for each batch. We propose a genetic algorithm procedure which is based on Darwin's survival of the fittest principle and a method using spreadsheet modelling in Microsoft Excel Solver. Numerical results from our examples showed that both procedures produced optimal solutions reported in the literature. Keywords: Time-Varying Demand; Genetic Algorithm; Spreadsheet Modelling and Inventory.

Multiplicities of eigenvalues of a vector-valued Sturm-Liouville problem

Abstract: This paper concerns the spectrum of the r -dimensional Sturm–Liouville equation y ″ + ( λI − Q ( x )) y = 0 with the Dirichlet boundary conditions, where Q is an r × r symmetric matrix. It is proved that, under certain conditions on Q , this problem can only have a finite number of eigenvalues with multiplity r . Further discussion is given for the multiplicities of eigenvalues when Q is an r × r Jacobian matrix.

Reka Bentuk Algoritma Blok Baru Stensil 9-titik dalam Masalah Sempadan

Abstract: Dalam kertas ini dikemukakan kaedah lelaran kumpulan 4 dan 9 titik yang baru diterbitkan daripada stensil 9-titik putaran [4] dengan cara pendekatan yang sama seperti Yousef [7]. Ujikaji berangka yang dijalankan menunjukkan kaedah-kaedah kumpulan baru ini sememangnya lebih cepat menumpu berbanding dengan skema lelaran titiknya, selaras dengan analisis kompleksitinya. Katakunci: Kaedah lelaran kumpulan; persamaan eliptik; stensil 9-titik putaran . In this paper, new 4 and 9 point group iterative methods derived from thr rotated nine-point stencil [4] similar to the approach of Yousef [7] is presented. Numerical experiments carried out confirmed that these new group methods converge faster than its poinywise counterpart which coincides with its complexity analysis Keywords: Group Iterative Method; elliptic equation; rotated 9-point stencil.

The universal quaternary quadratic form with the maximal discriminant

Abstract: A positive definite integral quadratic form over rational integers is said to be universal, if it represents all positive integers. The universal quaternary quadratic form is determined with the maximal discriminant, which is 1073/4.

Approximate Solution of Forced Korteweg-de Vries Equation

Abstract: Beberapa keputusan tentang penjanaan soliton paksaan oleh persamaan paksaan Korteweg-de Vries (fKdV) telah dibincangkan dalam kertas kerja ini Sistem persamaan seperti ini telah hilang sifat simetri kumpulannya akibat dari gangguan atau paksaan ke atasnya Kaedah teori kumpulan tidak lagi mampu memberikan penyelesaian secara analitik kerana tidak wujud lagi ketakterhinggaan banyaknya hukum keabadian Dengan itu kaedah penyelesaian secara penghampiran dan berangka sahaja yang mampu menyelesaikannya Dalam kertas kerja ini kita akan tunjukkan bagaimana penyelesaian secara penghampiran mampu menyelesaikan persamaan fKdV dan seterusnya menjana soliton paksaan seragam Penyelesaian hampir telah diterbitkan secara terperinci dan beberapa profil bagi fKdV seperti kedalaman zon tertekan; $h_d$, amplitud; $a_s$, laju; $s$ dan tempoh; $T_s$ penjanaan soliton paksaan seragam telah diberikan Katakunci: Soliton paksaan; soliton seragam; perlanggaran soliton dan persamaan paksaan Korteweg-de Vries Several findings on forced solitons generated by the forced Korteweg-de Vries equation (fKdV) are discussed in this paper This equation has lost group symmetries d ue to the forcing term The traditional group-theoretical approach can no longer generate analytic solution of solitons, because there are no infinitely many conservation laws Approximate solution and numerical simulation seem to be the only way to solve fKdV equations In this paper we show how approximate scheme can be used to solve the fKdV equation and generate uniform forced solitons A detail derivation of the approximate solution was provided and various profiles of fKdV such as the depth of depression zone; $h_d$, amplitude; $a_s$, speed; $s$ and the period; $T_s$ of generation of forced uniform solitons was given Keywords: Forced soliton; uniform soliton; soliton collision and forced Korteweg de-Vries equation

Penggunaan Kaedah Analisis Regresi Teguh untuk Menentukan Kecekapan Relatif

Abstract: Kajian ini membincangkan penggunaan analisis regresi teguh menerusi kaedah kuasa dua trim terkecil (LTS) untuk menilai kecekapan relatif unit-unit sesebuah organisasi dalam keadaan wujudnya data pencilan dan pengaruh faktor luaran yang tidak homogen. Kecekapan unit atau DMU dalam kajian ini adalah kecekapan pakar di hospital yang mengambil kira kos ke atas servis sebagai input dan jumlah pesakit kes berat dan ringan yang keluar dari hospital sebagai output. Hasil kajian menunjukkan penganggaran kecekapan unit menerusi kaedah LTS adalah lebih stabil. Katakunci: Analisis Regresi Teguh; Kaedah Kuasa Dua Trim Terkecil; Pengukuran Kecekapan. This study discusses the use of robust regression analysis using the least trimmed of squares method to evaluate the relative efficiency of decision making units in organizations in the case of the existance of outliers and the effect of nonhomogeneus outside factor. Efficiency unit evaluated in this study is the efficiency of hospital physicians where the service cost is the input and the number of high severity discharges and low severity discharges of patients as the outputs. The results show that the estimation of the unit efficiency using LTS method is more stable. Keywords: Robust Regression Analysis; Least Trimmed of Squares Method; Efficiency Evaluation.

Irreducible and end type group actions on Λ-trees

Abstract: Synopsis. Let Λ be an ordered abelian group. It is shown that groups in a certain class can have no non-trivial action of end type on a Λ-tree. A similar result is obtained for irreducible actions.

Characterizations of a two dimensional Euclidean ring among near-rings

Abstract: All those multiplications on the two-dimensional Euclidean group are determined such that the resulting non-associative topological near-ring has (1, 0) for a left identity and has the additional property that every element of the near-ring is a right divisor of zero. This result, together with several previous results, is then used to show that any one of several common algebraic properties is sufficient to characterize one particular two-dimensional Euclidean ring within the class of all two dimensional Euclidean near-rings. Specifically, it is proved that, if N is a topological near-ring with a left identity whose additive group is the two-dimensional Euclidean group, then the following assertions are equivalent: (1) the left identity is not a right identity, (2) N contains a non-zero left annihilator, (3) every element of N is a right divisor of zero, (4) Nw ¬= N for all w ∈ N, (5) N is isomorphic to the topological ring whose additive group is the two dimensional Euclidean group and whose multiplication is given by (v 1 , v 2 )(w 1 , w 2 ) = (v 1 w 1 , v 1 w 2 ).