Asymptotic analysis of expectations of plane partition statistics

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around \(x=1\). The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.     

Dynamical analysis of an anemoscope in the phase plane

We study the dynamical aspects of an anemoscope indicating a wind direction by constructing a model in the discrete view of fluid. Due to the non-linearity of the rotational equation, the geometrical method is applied for the qualitative analysis. For a given wind direction, there are two fixed points of the parallel and anti-parallel configurations. Near the anti-parallel configuration of the stable fixed point, the anemoscope performs a damped harmonic oscillation. It can be noteworthy that there is always the damping term for both the configurations even in the absence of the non-aerodynamic frictional force from, for example, the axis of rotation. The argument using the index of a closed curve in the phase portrait seems to support the non-existence of periodic solution.     

Fringe analysis of plane trees related to cutting and pruning

Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.     

Hydromagnetic linear instability analysis of Giesekus fluids in plane Poiseuille flow

The effects of a fluid’s elasticity are investigated on the instability of plane Poiseuille flow on the presence of a transverse magnetic field. To determine the critical Reynolds number as a function of the Weissenberg number, a two-dimensional linear temporal stability analysis will be used assuming that the viscoelastic fluid obeys Giesekus model as its constitutive equation. Neglecting terms nonlinear in the perturbation quantities, an eigenvalue problem is obtained which is solved numerically by using the Chebyshev collocation method. Based on the results obtained in this work, fluid’s elasticity is predicted to have a stabilizing or destabilizing effect depending on the Weissenberg number being smaller or larger than one. Similarly, solvent viscosity and also the mobility factor are both found to have a stabilizing or destabilizing effect depending on their magnitude being smaller or larger than a critical value. In contrast, the effect of the magnetic field is predicted to be always stabilizing.     

Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t → ±∞, to two positive limits l _* and l ^*, and never goes below min(l _*, l ^*). Furthermore, the convergence to max(l _*, l ^*) is supposed to be not too fast (slower than a suitable exponential term).     

Elactic buckling of thin-walled beams with sandwich and double bends flanges

This of the paper are two thin–walled beams with sandwich and double bends flanges. Cross section of these beams is of C type. The beams are simply supported and subjected to a couple of moment – the pure bending. Geometric propeties (warping functions and inertia moments) of two sections with sandwich and double bendsflanges are separately described by dimensionless parameters. Values of critical loads for family of thin–walled beams are numerically determined on the ground of analytical solution. A comparative analysis for selected beams with the use of FEM is performed. Morover the values of critical loads for a family of thin–walled beams are experimentally researched in the Material Strength Laboratoey of the Poznan University of Technology. Finally, the results of the investigation for thin–walled beams are compared in paper. Results of the calculation are presented in tables and figures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis of the plane Couette flow for a model kinetic equation

The stability of the plane Couette flow is studied using the simplified Boltzmann equation (the BGK equation) in which the high modes in the space of velocities and coordinates are truncated. The solution to the Navier-Stokes equation with small additional terms depending on the Knudsen number is used as the stationary solution. We assume that the perturbations depend only on the coordinate that is orthogonal to the flow. The density perturbations are assumed to be nonzero. In this approximation, the problem is found to be unstable in the case of small Knudsen numbers.     

An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

Work-preserving emulations of shuffle-exchange networks: An analysis of the complex plane diagram

In this paper we show that for each n, the order-n shuffle-exchange network can be emulated by an n-node linear processor array or an n²-node mesh in a work-preserving manner. An emulation of a computation on a guest network is work-preserving on a host network , if the time-processor products are equal to within a constant factor. To achieve this result we demonstrate a uniform many-to-one embedding of the nodes of a shuffle-exchange network into a linear array. We then give a simple, deterministic routing algorithm on the linear array which schedules the communication of messages necessary to achieve the emulation within the required time bounds. The analysis of the algorithm relies on certain statistical properties of the complex plane diagram of the shuffle-exchange network.     

Phase plane analysis and classification of solutions of a mixed sublinear-superlinear elliptic problem

In this paper, some mixed sublinear-superlinear critical problem extending the famous problem of Brezis–Nirenberg are analysed. The existence of solutions is discussed. A phase plane analysis is performed in order to transform the problem into an ordinary differential equation. Finally, a full classification of radial solutions according to their behavior at the origin is performed for subcritical, critical and supercritical cases.     

A mathematical analysis of the elasto-plastic plane stress problem of a power-law material

In this paper a generic study on the plane stress problem ofa power-law material undergoing infinitesimal deformations iscarried out, and a general solution for the stress and strainfields is derived using a stress function method and analyticfunction theory. Hencky's deformation theory and von Mises'yield criterion are used, and a differential transformationis invoked in the analysis. As an example, the closed-form solutionof the pure bending problem of a thin beam of power-law materialis obtained by applying the general solution directly.     

Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints

The local dependence of static response and eigenvalues on the shape of plates and plane elastic solids is characterized. The so-called material derivative method is used. The shape sensitivity analysis includes, besides linear problems, nonlinear problems with unilateral conditions, e.g., the frictionless contact problem for an elastic body on a rigid foundation. The results on shape sensitivity analysis can be used to obtain expressions for variations of integral functionals that arise in structural optimization problems.The authors are indebted to Professor N. Olhoff and Dr. M. P. Bendsøe for stimulating discussions and valuable comments on design sensitivity analysis.     

Complexity analysis of the analytic center cutting plane method that uses multiple cuts

We analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle. The oracle is called at the analytic center of a polytope, which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle. If the center is not in the solution set, separating hyperplanes will be placed through the center to shrink the containing polytope. While the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration, the result remains open when multiple cuts are added. Moreover, adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods. In this paper, we show that the analytic center cutting plane algorithm, with multiple cuts added in each iteration, still is a fully polynomial approximation algorithm. The research of the author is supported by NSF grant DDM-9207347, an Iowa Business School Summer Grant, and a University of Iowa Obermann Fellowship.     

Elasto-plastic analysis for plane stress contact problems using a modified RBSM method

The rigid body-spring model proposed by Kawai (1977, 1978) (called RBSM) can be applied to the contact problem, but it leaves something to be desired as a method of elasto-plastic analysis.

Additionally, the results obtained from the RBSM do not agree well with the ones obtained by the FEM. Thus, a modified RBSM is proposed here which uses a more reasonable yielding criterion and a proper stress-strain matrix. A simple example is given, showing that the modified RBSM is effective in elasto-plastic analysis, producing nearly the same approximation as the FEM.

Using the modified RBSM and the original RBSM, the elasto-plastic behavior of the cantilever beam with the built-in edge taken as a contact problem is analyzed under the two contact conditions, the sticking friction and the frictionless lubrication. Comparing the results, it is clear that the modified RBSM provides reasonable results but the original RBSM should be used with care.