Dual Partitions Research Articles

Achieving acoustic comfort in the library: a sound masking initiative

Purpose The purpose of this paper is to highlight the sources of noise generation in a library and suggest the implementation of a sound masking system to provide acoustic comfort, maintain speech privacy and create an environment more engaging for the users. Design/methodology/approach Analyzing the existing literature and exploring the existing practices as observed in different libraries, the study gives an overview of the sound masking initiatives in libraries. Findings With practical examples of libraries, the study demonstrates how a sound masking system has been implemented to invoke better acoustic design in the library. The expansion of various activities in the library and a gradual shift from individual attention to a collaborative approach necessitates a strong focus on the acoustic design architecture of the library. The study showcases how the libraries adopt sound masking with the introduction of acoustic panels, dual panel partitions, sound-absorbent false ceilings, sound insulation, sound isolation and noise-dampening measures, installing furniture with sound containment features, adopting vibration control mechanism, mounting of white noise machines, etc., keeping the aesthetic quotient of the library alive. Originality/value The study attempts to show the current practices of the adoption of the sound masking system in libraries and promotes collaborative reading with the creation of an acoustic design-influenced library environment to control noise and reverberation and provide a comfortable reading environment.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

Asymptotics of pure dimer coverings on rail yard graphs

Abstract We study the asymptotic limit of random pure dimer coverings on rail yard graphs when the mesh sizes of the graphs go to 0. Each pure dimer covering corresponds to a sequence of interlacing partitions starting with an empty partition and ending in an empty partition. Under the assumption that the probability of each dimer covering is proportional to the product of weights of present edges, we obtain the limit shape (law of large numbers) of the rescaled height functions and the convergence of the unrescaled height fluctuations to a diffeomorphic image of the Gaussian free field (Central Limit Theorem), answering a question in [7]. Applications include the limit shape and height fluctuations for pure steep tilings [9] and pyramid partitions [20; 36; 39; 38]. The technique to obtain these results is to analyze a class of Macdonald processes which involve dual partitions as well.

A family of quadratic finite volume method for solving the Stokes equation

This paper presents a family of stable quadratic finite volume methods for solving the Stokes equation over triangular meshes. We introduce a novel mapping, which connects trial spaces with test spaces of the finite volume methods, and impose a geometric condition on the triangular meshes of primary partitions to ensure the ellipticity of discrete elliptic bilinear forms in the finite volume methods. By using the abstract theories of generalized saddle-point problems, the stability (or inf-sup condition in general) of the proposed finite volume methods could be established. The resulting finite volume methods have the optimal H1-norm error estimate for velocity of the Stokes problem without constraints on dual meshes of dual partitions of the finite volume methods, and have the optimal L2-norm error estimate for velocity with a specially selected dual mesh of the dual partitions. In addition, by adopting another appropriate dual mesh of the dual partitions, the finite volume methods may be identified as the corresponding finite element methods with differences only in their right-hand items. As a result, the optimal-order error estimate could be obtained in the H1-norm for velocity without any restriction on mesh geometry of the primary partitions. Finally, we investigate numerically the imposed geometric conditions on meshes of the primary partitions for ensuring the ellipticity of the discrete elliptic bilinear forms and we validate our theoretical error estimates through some numerical experiments.

Yield and evapotranspiration characteristics of potato-legume intercropping simulated using a dual coefficient approach in a tropical highland

An accurate estimation of crop evapotranspiration and soil water balance under potato-legume intercropping is important for improving the crop water productivity of rainfed potato. This study quantified the evapotranspiration, yield, and soil water balance of potato (Solanum tuberosum L.) intercropped with lima bean (Phaseolus lunatus L.) or dolichos (Lablab purpureus L.) in comparison to the same crops under monocropping. The experiment was set up in three agroecological zones of Kenya: upper midland (with an altitude range of 1500–1653 m above sea level (masl)), lower highland (1892–1923 masl) and upper highland (2502–2594 masl). The dual crop coefficient approach was adopted with the SIMDualKc model to estimate crop evapotranspiration and compute the soil water balance. The dual crop coefficient partitions crop evapotranspiration into crop transpiration and soil evaporation by applying both soil evaporation and basal crop coefficient. The model was calibrated and validated using field data observed along four crop growth seasons. The basal crop coefficients, the ratios of soil evaporation to evapotranspiration, and that of transpiration to evapotranspiration were determined. The yields of different intercrops were converted into equivalent yield of potato based on price of the produce. Good agreement between the observed and simulated data for available soil water and crop evapotranspiration was found with modeling efficiency > 0.8. The residual mean square error was low and ranged from 0.01−0.08 m3 m-3 for available soil water and 0.03−0.09 mm d-1 for crop evapotranspiration. The transpiration to actual evapotranspiration ratio of potato-legume intercropping was 6−28% greater than that of the sole potato because legume intercrops fully covered the ground by mid-season, thus limiting the energy available for soil evaporation. Crop yields were significantly greater under intercropping as transpiration occurred near its potential rate, thus not limiting yields. These results support the dual crop coefficient method as an appropriate tool to estimate and accurately partition crop evapotranspiration under potato-legume intercropping.

Homological mirror symmetry for generalized Greene\u2013Plesser mirrors

We prove Kontsevich’s homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev–Borisov’s ‘dual reflexive Gorenstein cones’ construction. In particular we prove HMS for all Greene–Plesser mirror pairs (i.e., Calabi–Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi–Yau complete intersections arising from Borisov’s construction via dual nef partitions, and also for certain Calabi–Yau complete intersections which do not have a Calabi–Yau mirror, but instead are mirror to a Calabi–Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov’s ‘K3 category of a cubic fourfold’, which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi–Yau threefold.

Computations of permeability of large rock images by dual grid domain decomposition

Digital rock physics (DRP) is an eminent technology that facilitates and repeatable core analysis and multi-physics simulation of rock properties. One of the challenges in this field is the scalability of the problem size, whereby large micro-CT images over the order of 10003 voxels incur a high computational demand on performance. We estimate the of permeability in large digital samples of rocks imaged by micro-CT by using a fast and efficient Dual Grid Domain Decomposition technique based on the Schwarz Alternating Method (slow, low memory) with Algebraic Multigrid (AMG) solvers (fast, high memory) to solve on an otherwise unfeasible shared-memory machine. The comparisons and differences to other methods commonly used have been added in the introduction. The method applies a scalable parallel simulation algorithm to solve pressure and velocity fields using the Semi Analytical Pore Scale Finite Volume Solver (PFVS) within real 3D pore-scale micro-CT images. The domain is split into non-overlapping coarse partitions and also split into a set of dual coarse partitions of varying width. The governing equations are then solved iteratively between the partition sets by updating the pressure and flux at the relevant boundaries before each step. The method is validated and shown to converge to flux continuous conditions requisite of the governing equations. Permeability estimation is within 5–10% of the fine scale solution and significantly reduces memory limitations and computational time for solving problems in micro-CT images, allowing ordinary workstations to solve images over 10003 within the magnitude of 1–100 h of CPU time. The permeability of a 2520 × 2520 × 7250 sample was calculated with a workstation to within 9% error of LBM calculated with a supercomputer within a similar timeframe.

Blocked pressure based Transfer Path Analysis (TPA) method to diagnose airborne sound transfer through building partitions

Airborne sound transmission through building elements or the sound insulation of the building element is usually rated by its Sound Reduction Index (SRI) or the Sound Transmission Class (STC). SRI/STC quantifies the overall sound transfer but gives no information about how the transfer takes place and what are the contributions of different sound transfer paths involved. Such problems are fairly common to the vehicle acoustics industry and are generally tackled by TPA techniques. The paper formulates an in-situ Airborne TPA technique to quantify the contributions of different sound transfer paths to the transmitted pressure. The airborne source is characterized by its blocked pressure and its direct measurement is discussed. Results are presented for dual leaf partitions excited by an airborne source. The method has shown to be significantly faster than the Blocked force based TPA method which relies on inverse measurement methods. The accuracy of the method is closely related to the wavelength of incident airborne waves.

$L^2$ Error Estimates for High Order Finite Volume Methods on Triangular Meshes

We establish a unified framework for $L^2$ error analysis for high order Lagrange finite volume methods on triangular meshes. Orthogonal conditions are originally proposed to construct dual partitions on triangular meshes, such that the corresponding finite volume method (FVM) schemes hold optimal $L^2$ norm convergence order. Moreover, with the Aubin--Nitsche technique, we prove the optimal $L^2$ error estimate for high order FVM schemes on triangular meshes. Some numerical experiments are presented to demonstrate the proved result.

Face-to-face partition of 3D space with identical well-centered tetrahedra

The motivation for this paper comes from physical problems defined on bounded smooth domains Ω in 3D. Numerical schemes for these problems are usually defined on some polyhedral domains Ω h and if there is some additional compactness result available, then the method may converge even if Ω h → Ω only in the sense of compacts. Hence, we use the idea of meshing the whole space and defining the approximative domains as a subset of this partition. Numerical schemes for which quantities are defined on dual partitions usually require some additional quality. One of the used approaches is the concept of well-centeredness, in which the center of the circumsphere of any element lies inside that element. We show that the one-parameter family of Sommerville tetrahedral elements, whose copies and mirror images tile 3D, build a well-centered face-to-face mesh. Then, a shape-optimal value of the parameter is computed. For this value of the parameter, Sommerville tetrahedron is invariant with respect to reflection, i.e., 3D space is tiled by copies of a single tetrahedron.

Association schemes on general measure spaces and zero-dimensional Abelian groups

Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. At the same time, direct extensions of this concept to infinite sets encounter some problems even in the case of countable sets, for instance, countable discrete Abelian groups. In an attempt to resolve these difficulties, we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. However, constructing a complete theory in the general case faces a set of obstacles related to the properties of the adjacency algebras and associated projection operators. To develop a theory of association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. Using the language of spectrally dual partitions, we prove that such groups support the construction of general Abelian (translation) schemes and establish properties of their spectral parameters (eigenvalues).Addressing the existence question of spectrally dual partitions, we show that they arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of p-adic numbers. This enables us to construct large classes of examples of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers (structural constants). We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group.Next we focus on the connection between schemes on zero-dimensional groups and harmonic analysis. We show that the eigenvalues have a natural interpretation in terms of Littlewood–Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal function systems on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelet bases on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and (non-Archimedean) harmonic analysis.We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes.

Higher‐order finite volume element methods based on Barlow points for one‐dimensional elliptic and parabolic problems

The article is devoted to a kind of higher‐order finite volume element methods, where the dual partitions are constructed by Barlow points, for elliptic and parabolic problems in one space dimension. Techniques to derive the stability and to control the nonsymmetry are presented. Superconvergence and the optimal order errors in the H1‐ and L2‐norms are obtained. Numerical results illustrate the theoretical findings. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 977–994, 2015

A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation

This paper introduces a novel method for deducing high-order compact difference schemes for the two-dimensional (2D) Poisson equation. Like finite volume method, a dual partition is introduced. Combining Simpson integral formula and parabolic interpolation, a family of fourth-order and sixth-order compact difference schemes are obtained based on three different types of dual partitions. Moreover, several new fourth-order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth-order schemes which have been presented in others’ work. The outline for the nonlinear Poisson equation is also given. Numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference schemes.

A Novel Method to Deduce a High-Order Compact Difference Scheme for the Three-Dimensional Semilinear Convection-Diffusion Equation with Variable Coefficients

In this article, a new family of fourth-order compact difference schemes for the three-dimensional semilinear convection-diffusion equation with variable coefficients is presented. Like the finite-volume method, a dual partition is introduced. Combining with the Simpson integral formula and parabolic interpolation, fourth-order schemes are derived based on two different types of dual partitions. Moreover, a sixth-order finite-difference discretization strategy is developed, which is based on the fourth-order compact discretization and Richardson extrapolation technique. This extrapolation technique can achieve a sixth-order-accurate solution on fine grids directly, without the need for interpolation. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order schemes and extrapolation formulas.

Stability of the primal-dual partition in linear semi-infinite programming

We analyse the primal-dual states in linear semi-infinite programming (LSIP), where we consider the primal problem and the so called Haar's dual problem. Any linear programming problem and its dual can be classified as bounded, unbounded or inconsistent, giving rise to nine possible primal-dual states, which are reduced to six by the weak duality property. Recently, Goberna and Todorov have studied this partition and its stability in continuous LSIP in a series of papers [M.A. Goberna and M.I. Todorov, Primal, dual and primal-dual partitions in continuous linear semi-infinite programming, Optimization 56 (2007), pp. 617–628; M.A. Goberna and M.I. Todorov, Generic primal-dual solvability in continuous linear semi-infinite programming, Optimization 57 (2008), pp. 239–248]. In this article we consider the general case, with no continuity assumptions, discussing the maintenance of the primal-dual state of the problem by allowing small perturbations of the data. We characterize the stability of all of the six possible primal-dual states through necessary and sufficient conditions which depend on the data, and can be easily checked, showing some differences with the continuous case. These conditions involve the strong Slater constraint qualification, and some distinguished convex sets associated to the data.

A Family of Fourth-Order and Sixth-Order Compact Difference Schemes for the Three-Dimensional Poisson Equation

In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.

Generic primal-dual solvability in continuous linear semi-infinite programming†

In this article, we consider the space of all the linear semi-infinite programming (LSIP) problems with a given infinite compact Hausdorff index set, a given number of variables and continuous coefficients, endowed with the topology of the uniform convergence. These problems are classified as inconsistent, solvable with bounded optimal set, bounded (i.e. finite valued), but either unsolvable or having an unbounded optimal set, and unbounded (i.e. with infinite optimal value), giving rise to the so-called refined primal partition of the space of problems. The mentioned LSIP problems can be also classified with a similar criterion applied to the corresponding Haar's dual problems, which provides the refined dual partition of the space of problems. We characterize the interior of the elements of the refined primal and dual partitions as well as the interior of the intersections of the elements of both partitions (the so-called refined primal-dual partition). These characterizations allow to prove that most (primal or dual) bounded problems have simultaneously primal and dual non-empty bounded optimal set. Consequently, most bounded continuous LSIP problems are primal and dual solvable. †In celebration of Prof. Dr Hubertus Th. Jongen's 60th Birthday.

Primal, dual and primal-dual partitions in continuous linear optimization

We associate with each natural number n and each compact Hausdorff topological space T the space of linear optimization problems with n primal variables and index set T (for the constraints) equipped with the topology of the uniform convergence. We consider three different partitions of this metric space. The primal and the dual partitions are the result of classifying a given optimization problem and its dual as either inconsistent or bounded or unbounded, whereas the primal-dual partition is formed by the nonempty intersections of the elements of both partitions. The elements of the three partitions are neither open nor closed and their topological interiors are formed by those problems for which sufficiently small perturbations maintain the membership of the problem, i.e. the problems that are stable for the corresponding property. We prove that the stable problems are the same for the three partitions, concluding that most problems are stable in the three senses. This is done by completing the topological analysis of the primal-dual partition carried out in a previous paper of the authors.

The polytope of dual degree partitions

We determine the extreme points and facets of the convex hull of all dual degree partitions of simple graphs on n vertices. (This problem was raised in the Laplace Energy group of the Workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns held at the American Institute of Mathematics Research Conference Center on October 23–27, 2006 [R. Brualdi, Leslie Hogben, Brian Shader, AIM Workshop – Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns, Final Report: Mathematical Results, November 17, 2006].)

An upwinding cell-centered method with piecewise constant velocity over covolumes

We analyze as a covolume method an upwinding cell-centered method derived from a mixed formulation of the convection{diffusion equation. The covolume method uses primal and dual partitions of the problem domain to assign degrees of freedom and is a natural generalization of the well known MAC (Marker and Cell) method. The concentration is cell-centered (piecewise constant with respect to the primal partition), and the velocity is covolume- or co-cell centered (piecewise constant with respect to the dual partition). Convergence results are demonstrated in theL 2 norm. c 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 49{?? , 1999