Narrow Results

Let ψ = Γ′/Γ, where Γ is the Gamma function, and let x ≥ 0 and 0 < a ≤ b. Then

We present new quantum lower bounds and upper bounds for several computational geometry problems. The bounds presented here improve on currently known results in a number of ways. We give asymptotically optimal bounds for one of the problems considered, and we provide, up to logarithmic factors, optimal bounds for a number of other problems and, in particular, we settle an open problem of Bahadur et al. Some of these new bounds are obtained using a general algorithm for finding a minimum pair over a given arbitrary order relation.

A node-based smoothed finite element method (NS-FEM) for solving solid mechanics problems using a mesh of general polygonal elements was recently proposed. In the NS-FEM, the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh, and a number of important properties have been found, such as the upper bound property and free from the volumetric locking. The examination was performed only for two-dimensional (2D) problems. In this paper, we (1) extend the NS-FEM to three-dimensional (3D) problems using tetrahedral elements (NS-FEM-T4), (2) reconfirm the upper bound and free from the volumetric locking properties for 3D problems, and (3) explore further other properties of NS-FEM for both 2D and 3D problems. In addition, our examinations will be thorough and performed fully using the error norms in both energy and displacement. The results in this work revealed that NS-FEM possesses two additional interesting properties that quite similar to the equilibrium FEM model such as: (1) super accuracy and super-convergence of stress solutions; (2) similar accuracy of displacement solutions compared to the standard FEM model.

In this research, a general solution of volume constancy differential equation is presented based on the equation of deformation field for a general process. As the Bezier method is suitable for construction of complex geometries, the solution is used in conjunction with the Bezier method to analyze the equal channel angular extrusion (ECAE) process of rectangular cross section. Thus, a generalized kinematically admissible velocity field is derived from the equation of deformation zone such that the compatibility of the surface representing the deformation zone is fulfilled. The effects of die angle, friction between the billet and die wall, and the angle of outer curved corner, on extrusion pressure are all considered in the analysis. It is found that extrusion pressure decreases with increasing both the die angle and the outer curved corner angle and with decreasing the friction coefficient. Also, the effect of die curvature on inhemogenity of strain is assessed. It is exhibited that increasing the angle of outer curved corner decreases the extrusion pressure and increases the inhomogeniety of strain field of deformation zone. A good agreement is found between the predicted and experimental results pertaining to two dies of different outer curved corner.

A finite set of generators for a free product of two groups of type F₃ with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

The compaction of a package of monosized spherical solid grains by rate-independent plasticity deformation is examined in this paper through the use of both yield design homogenization method and finite element simulation. Both modes of compaction, isostatic and closed die, are considered. In this study, the arrangement of powder consists of hexagonal array of identical spherical grains touching each other in its initial state. During the compaction process the response of the powder compacts is monitored in terms of behaviors of appropriate representative unit cells subject to axisymmetrical loading conditions. The kinematic approach of the yield design homogenization method has been used to determine external estimates of macroscopic strength criteria of powders at various stages of compaction. The obtained upper bound estimates are based on consideration of discontinuous incompressible velocity fields satisfying conditions of homogeneous strain rate. The shapes and sizes of the macroscopic yield surfaces are determined at various stages of compaction and it has been found that they depend upon the loading history as well as the relative density of the compact. Finite element simulations similar to those of Ogbonna N. and Fleck N. A. [1995] "Compaction of an array of spherical particles," Acta Metall. Mater.43(2), 603–620. have also been performed in order to (i) obtain the deformation modes as well as the evolution of the deformation mechanism of the powder compact during the whole process of compaction; (ii) derive the evolution of contact sizes between adjacent grains; (iii) examine the dependence of the macroscopic yield surface upon the degree of compaction, using the "yield probing technique" Gurson, A. L. and Yuan, D. W. [1995] A Material Model for a Ceramic Powder Based on Ultrasound, TRS Bend Bar, and Axisymmetric Triaxial Compression Test Results (ASME, New York), pp. 57–68, and (iv) validate, to some extent, the results provided by the kinematic approach.

Let D₁, D₂, D, k, λ be fixed integers such that D₁ ≥ 1, D₂ ≥ 1, gcd(D₁, D₂) = 1, D = D₁D₂ is not a square, ∣k∣ > 1, gcd(D, k) = 1 and λ = 1 or 4 according as 2 ∤ k or not. In this paper, we prove that every solution class S(l) of the equation D₁x²-D₂y² = λk^z, gcd(x, y) = 1, z > 0, has a unique positive integer solution satisfying and , where z runs over all integer solutions (x,y,z) of S(l),(u₁,v₁) is the fundamental solution of Pell's equation u² - Dv² = 1. This result corrects and improves some previous results given by M. H. Le.

A composite positive integer $n$ has the Lehmer property if $\phi (n)$ divides $n-1,$ where $\phi$ is an Euler totient function. In this paper, we shall prove that if $n$ has the Lehmer property, then $n\le 2^{2^K}-2^{2^{K-1}}$, where $K$ is the number of prime divisors of $n$. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set

Let v₂(n) denote the 2-adic valuation of any positive integer n. Recently, Farhi introduced a curious arithmetic function f defined for any positive integer n by . Farhi showed that the inequality with c = 4.01055487… holds for all positive integer n and conjectured that one can replace the upper bound cⁿ by 4ⁿ in this inequality. In this paper, we show two identities about the product and then use it to prove partially Farhi's conjecture. Finally, we propose a conjecture from which the truth of Farhi's conjecture can be deduced. In particular, we confirm the truth of our conjecture for all positive integers n up to 100000 by using Matlab 7.1.

The objective of this paper is to obtain an upper bound to the second Hankel determinant for the functions belonging to strongly starlike and convex functions of order α(0 < α ≤ 1). Further, we introduce a subclass of analytic functions and obtain the same coefficient inequality for the functions in this class, using Toeplitz determinants.

For nonzero polynomials $f$ and $g$ over a field $K$, let $L(f/g)$ be the depth (length) of the continued fraction expansion for $f/g$. An upper bound on $L(fX)$, for nonzero polynomial $f$ and rational function $X$ is obtained. Applying this result, an upper bound on the depth of a linear fractional transformation is also established.

An upper bound for sorting permutations with an operation estimates the diameter of the corresponding Cayley graph and an exact upper bound equals the diameter. Computing tight upper bounds for various operations is of theoretical and practical (e.g., interconnection networks, genetics) interest. Akers and Krishnamurthy gave a Ω(n! n²) time method that examines n! permutations to compute an upper bound, f(Γ), to sort any permutation with a given transposition tree T, where Γ is the Cayley graph corresponding to T. We compute two intuitive upper bounds γ and δ′ each in O(n²) time for the same, by working solely with the transposition tree. Recently, Ganesan computed β, an estimate of the exact upper bound for the same, in O(n²) time. Our upper bounds are tighter than f(Γ) and β, on average and in most of the cases. For a class of trees, we prove that the new upper bounds are tighter than β and f(Γ).

For a real number $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of a graph $G$ is defined to be $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),$ where $A(G)$ and $D(G)$ are the adjacency matrix and degree diagonal matrix of $G$, respectively. The $A_{\alpha }$-spectral radius of $G$, denoted by ${\rho }_{\alpha }(G)$, is the largest eigenvalue of $A_{\alpha }(G)$. In this paper, we consider the upper bound of the $A_{\alpha }$-spectral radius ${\rho }_{\alpha }(G)$, also we give some upper bounds for the second largest eigenvalue of $A_{\alpha }$-matrix.

A permutation over alphabet $\mathrm{\Sigma }=(1,2,3,\dots ,n)$ is a sequence over $\mathrm{\Sigma }$, where every element occurs exactly once. $S_n$ denotes symmetric group defined over $\mathrm{\Sigma }$. $I_n=(1,2,3,\dots ,n)\in S_n$ denotes the Identity permutation. $R_n\in S_n$ is the reverse permutation i.e., $R_n=(n,n-1,n-2,\dots ,2,1)$. An operation, that we call as an LE operation, has been defined which consists of exactly two generators: set-rotate that we call Rotate and pair-exchange that we call Exchange (OEIS). At least two generators are the required to generate $S_n$. Rotate rotates all elements to the left (moves leftmost element to the right end) and Exchange is the pair-wise exchange of the two leftmost elements. The optimum number of moves for transforming $R_n$ into $I_n$ with LE operation are known for $n\le 10$; as listed in OEIS with identity A048200. However, no general upper bound is known. The contributions of this article are: (a) a novel upper bound for the number of moves required to sort $R_n$ with LE has been derived; (b) the optimum number of moves to sort the next larger $R_n$ i.e., $R_{11}$ has been computed; (c) an algorithm conjectured to compute the optimum number of moves to sort a given $R_n$ has been designed.

Permutations are discrete structures that naturally model a genome where every gene occurs exactly once. In a permutation over the given alphabet $\mathrm{\Sigma }$, each symbol of $\mathrm{\Sigma }$ appears exactly once. A transposition operation on a given permutation $\pi$ exchanges two adjacent sublists of $\pi$. If one of these sublists is restricted to be a prefix then one obtains a prefix transposition. The symmetric group of permutations with $n$ symbols derived from the alphabet $\mathrm{\Sigma }=\{0,1,2,\dots ,(n-1)\}$ is denoted by $S_n$. The symmetric prefix transposition distance between ${\pi }^{\star }\in S_n$ and ${\pi }^{\#}\in S_n$ is the minimum number of prefix transpositions that are needed to transform ${\pi }^{\star }$ into ${\pi }^{\#}$. It is known that transforming an arbitrary ${\pi }^{\star }\in S_n$ into an arbitrary ${\pi }^{\#}\in S_n$ is equivalent to sorting some ${\pi }^{\prime }\in S_n$. Thus, upper bound for transforming any ${\pi }^{\star }\in S_n$ into any ${\pi }^{\#}\in S_n$ with prefix transpositions is simply the upper bound to sort any permutation $\pi \in S_n$. The current upper bound is $n-{log}_{(\frac{7}{2})}n$ for prefix transposition distance over $S_n$. In this paper, we improve the same to $n-{log}_{3}n$.

Based upon Ritchken (1985), Levy (1985), Lo (1987), Zhang (1994), Jackwerth and Rubinstein (1996), and others, this chapter discusses the alternative method to determine option bound in terms of the first two moments of distribution. This approach includes stochastic dominance method and linear programming method, then we discuss semi-parametric method and non-parametric method for option-bound determination. Finally, we incorporate both skewness and kurtosis explicitly through extending Zhang (1994) to provide bounds for the prices of the expected payoffs for options, given the first two moments and skewness and kurtosis.