Two-variable Problem Research Articles

Single variable-based unified numerical model for plane elastic problems of stress analysis in Cartesian and polar coordinate systems

A unified numerical model for stress analysis of plane problems of structural mechanics in Cartesian and polar coordinate systems is presented. The traditional two-variable problem of stress analys...

Multi-Objective Optimization of Off-Grid Hybrid Renewable Energy Systems in Buildings with Prior Design-Variable Screening

This work presents an optimization strategy and the cost-optimal design of an off-grid building served by an energy system involving solar technologies, thermal and electrochemical storages. Independently from the multi-objective method (e.g., utility function) and algorithm used (e.g., genetic algorithms), the optimization of this kind of systems is typically characterized by a high-dimensional variables space, computational effort and results uncertainty (e.g., local minimum solutions). Instead of focusing on advanced optimization tools to handle the design problem, the dimension of the full problem has been reduced, only considering the design variables with a high “effect” on the objective functions. An off-grid accommodation building is presented as test case: the original six-variable design problem consisting of about 300,000 possible configurations is reduced to a two-variable problem, after the analysis of 870 Monte Carlo simulations. The new problem includes only 220 possible design alternatives with a clear benefit for the multi-objective optimization algorithm. The energy-economy Pareto frontiers obtained by the original and the reduced problems overlap, showing the validity of the proposed methodology. The No-RES (no renewable energy sources) primary energy consumption can be reduced up to almost 0 kWh/(m2yr) and the net present value (NPV) after 20 years can reach 70 k€ depending on the number of photovoltaic panels and electrochemical storage size. The reduction of the problem also allows for a plain analysis of the results and the drawing of handy decision charts to help the investor/designer in finding the best design according to the specific investment availability and target performances. The configurations on the Pareto frontier are characterized by a notable electrical overproduction and a ratio between the two main design variables that goes from 4 to 8 h. A sensitivity analysis to the unitary price of the electrochemical storage reveals the robustness of the sizing criterion.

A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences

In 1927, Artin conjectured that any integer other than −1 or a perfect square generates the multiplicative group (Z/pZ)× for infinitely many p. In 2000, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture. In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the first N terms of a non-degenerate binary recurrence sequence. We also prove a weaker version of the same theorem, and give three proofs that we consider to be of independent interest. The first proof uses a transcendence result of Stewart, the second uses a theorem of Bombieri and Schmidt on Thue equations and the third uses Mumford's gap principle for counting points on curves by their height. We finally prove a disjunction theorem, where we consider the set of primes satisfying either our two-variable condition or the original condition of Artin's conjecture. We give an unconditional lower bound for the number of such primes.

Computational complexity and the conjugacy problem

The conjugacy problem for a finitely generated group $G$ is the two-variable problem of deciding for an arbitrary pair $(u,v)$ of elements of $G$, whether or not $u$ is conjugate to $v$ in $G$. We construct examples of finitely generated, computably presented groups such that for every element $u_0$ of $G$, the problem of deciding if an arbitrary element is conjugate to $u_0$ is decidable in quadratic time but the worst-case complexity of the global conjugacy problem is arbitrary: it can be any c.e. Turing degree , can exactly mirror the Time Hierarchy Theorem, or can be $\mathcal{NP}$-complete. Our groups also have the property that the conjugacy problem is generically linear time: that is, there is a linear time partial algorithm for the conjugacy problem whose domain has density $1$, so hard instances are very rare. We also consider the complexity relationship of the "half-conjugacy" problem to the conjugacy problem. In the last section we discuss the extreme opposite situation: groups with algorithmically finite conjugation.

Theoretical model of the helium zone plate microscope

Neutral helium microscopy is a new technique currently under development. Its advantages are the low energy, charge neutrality, and inertness of the helium atoms, a potential large depth of field, and the fact that at thermal energies the helium atoms do not penetrate into any solid material. This opens the possibility, among others, for the creation of an instrument that can measure surface topology on the nanoscale, even on surfaces with high aspect ratios. One of the most promising designs for helium microscopy is the zone plate microscope. It consists of a supersonic expansion helium beam collimated by an aperture (skimmer) focused by a Fresnel zone plate onto a sample. The resolution is determined by the focal spot size, which depends on the size of the skimmer, the optics of the system, and the velocity spread of the beam through the chromatic aberrations of the zone plate. An important factor for the optics of the zone plate is the width of the outermost zone, corresponding to the smallest opening in the zone plate. The width of the outermost zone is fabrication limited to around 10 nm with present-day state-of-the-art technology. Due to the high ionization potential of neutral helium atoms, it is difficult to build efficient helium detectors. Therefore, it is crucial to optimize the microscope design to maximize the intensity for a given resolution and width of the outermost zone. Here we present an optimization model for the helium zone plate microscope. Assuming constant resolution and width of the outermost zone, we are able to reduce the problem to a two-variable problem (zone plate radius and object distance) and we show that for a given beam temperature and pressure, there is always a single intensity maximum. We compare our model with the highest-resolution zone plate focusing images published and show that the intensity can be increased seven times. Reducing the width of the outermost zone to 10 nm leads to an increase in intensity of more than 8000 times. Finally, we show that with present-day state-of-the-art detector technology (ionization efficiency $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$), a resolution of the order of 10 nm is possible. In order to make this quantification, we have assumed a Lambertian reflecting surface and calculated the beam spot size that gives a signal 100 cts/s within a solid angle of 0.02 sr, following an existing helium microscope design.

An analytical study of the endoreversible Curzon–Ahlborn cycle for a non-linear heat transfer law

Abstract In the present article, an endoreversible Curzon–Ahlborn engine is studied by considering a non-linear heat transfer law, particularly the Dulong–Petit heat transfer law, using the `componendo and dividendo' rule as well as a simple differentiation to obtain the Curzon–Ahlborn efficiency as proposed by Agrawal in 2009. This rule is actually a change of variable that simplifies a two-variable problem to a one-variable problem. From elemental calculus, we obtain an analytical expression of efficiency and the power output. The efficiency is given only in terms of the temperatures of the reservoirs, such as both Carnot and Curzon–Ahlborn cycles. We make a comparison between efficiencies measured in real power plants and theoretical values from analytical expressions obtained in this article and others found in literature from several other authors. This comparison shows that the theoretical values of efficiency are close to real efficiency, and in some cases, they are exactly the same. Therefore, we can say that the Agrawal method is good in calculating thermal engine efficiencies approximately.

A Clustering Method for Calculating Membrane Currents in Cardiac Electrical Models

Many studies into cardiac electrical rhythm and disturbances use computer modeling as a valuable tool for testing hypotheses. Computer modeling is often limited by tractability and the availability of computing resources. Some of these limitations can be overcome by efficient numerical schemes that solve the equations modeling cardiac electrical activation. This work presents a new numerical method, the piecewise phase space approximation (PPSA), for the time advancement of the coupled systems of ordinary differential equations (ODEs) that are used to describe cardiac cell membrane currents. Using novel metrics, data structures and approximations, state variables are discretely clustered in the phase space of the ODEs. Extreme points of each cluster are advanced through time by an appropriate solver and the remainder of the cluster points are reconstructed using a predictor function. This new method can contribute additional efficiency to many other established and emerging techniques for discretizing and solving cardiac electrical activation problems. The PPSA is assessed by a two-variable problem and a suite of one and two-dimensional cardiac electrical activation models of varying complexity. Less computational operations are required for comparable results with methods where all points are solved independently. Electrical activation in a detailed model of canine ventricles confirms these results. The performance of the PPSA will continue to improve with further investigation into alternative predictor functions, relaxation of phase space discretizations and parallelization of the algorithm.

Progressive Design Processes and Bounded Rational Designers

This paper presents a method for assessing the quality of a progressive design process by measuring the profitability of the product that the process generates. The proposed approach uses separations, a type of problem decomposition, to model progressive design processes. The subproblems in the separations correspond roughly to phases in the progressive design processes. The proposed method simulates the choices of a bounded rational designer for each subproblem using different search algorithms. This paper presents a simple two-variable problem to help describe the approach and then applies the approach to assess motor design processes. Different types and versions of these search processes are considered to determine if the results are robust to the decision-making model. The results indicate that well-designed progressive design processes are the best way to generate profitable product designs. Methods for assessing the quality of engineering design processes can be used to guide improvements to engineering design processes and generate more valuable products.

Transient dynamics of a rotating spherical liquid drop

The transient rotation of a liquid drop in an infinite gaseous medium is analyzed for two cases: viscous retardation from constant-speed rotation, and steady-torque start-up with development to the steady state. This situation arises when a levitated liquid drop is rotated with an acoustically applied torque. Subsequent changes in the torque cause transient effects. To understand such a system, the two basic problems of start-up and retardation are studied. The rotational Reynolds number is considered to be low enough so that nonlinear inertial effects may be neglected. The Laplace-transform method is used to deal with the time dependence. Since the fluid velocity has only the azimuthal direction (and the profile is independent of it), and the other angular dependence is factorable as sin θ, the solution turns out to be effectively a two-variable problem in r and t. Nevertheless, the finite mass of the spherical drop and its finite viscosity make it a mathematically challenging problem. Besides the full analytical solution, results are obtained in the limits of a solid sphere, and small time for the liquid drop. In all liquid-drop cases, the results are limited to the drop viscosity being higher than the surrounding region. For several common liquids in a gas, the flow field indicates nearly a solid-sphere like behavior, except at small times in the region near the interface. The deviations from the asymptotic center-point velocity are amplified for illustration.

A method for optimizing distributions of Nafion and Pt in cathode catalyst layers of PEM fuel cells

Optimal distributions of Nafion content and Pt loading across the entire thickness of cathode catalyst layers of polymer electrode membrane (PEM) fuel cells are obtained by maximizing the cell current density at a given potential. When optimization is performed with either the Nafion content or Pt loading as the single variable, it is found that the optimal distribution is a linearly increasing function from the gas diffusion layer side to the membrane side; when optimization is treated as a two-variable problem, i.e. comprising of both Nafion content distribution and Pt loading distribution, the optimal distribution of Nafion content is still a linearly increasing function but the optimal distribution of Pt loading is a convex increasing function.

The Interactional Model: An Alternative to the Direct Cause and Effect Construct for Mutually Causal Organizational Phenomena

It is time that we in organization sciences develop and implement a new mental model for cause and effect relationships. The dominant model in research dates at least to the 1700s and no longer serves the full purposes of the social science research problems of the 21st century. Traditionally, research is “essentially concerned with two-variable problems, linear causal trains, one cause and one effect, or with few variables at the most” (von Bertalanffy, 1968, p. 12). However, the literature is replete with examples of phenomena in which the traditional cause and effect construct does not allow for greater understanding and insight into the phenomena. Different conceptions of cause and effect relationships have been developed including producer/product relationships (Ackoff 1981), design causality (Argyris and Schon, 1996), and four classes of causal models (Schwartz and Ogilvy, 1979). Of interest here is the possibility of mutual causality, “the assumption that the relationship between two (or more) phenomena is heavily influenced by the presence of feedback loops that are instantaneous, or nearly so” (Dent, 1999). Maturana’s (1998, Maturana and Varela, 1987) work on a new epistemology and ontology provides a foundation for the alternative model of cause and effect proposed here. This interaction model includes the dynamics of the traditional X and Y, but adds the structure of X (A), the structure of Y (B), the environment (E), and time (T).

動物權益問題

動物權益問題

Dynamic bifurcation during high-rate planar extension of a thin rectangular block

Dynamic bifurcations in deformation from a uniform state of rapid extension are studied for a rectangular block. The block may be viewed as a segment of a thin shell which is one period of a deformation which is periodic along the circumference. The material is taken to be incompressible and characterized by an incrementally linear, rate-independent elastic-plastic constitutive law. The stress-state in the block prior to bifurcation is approximately uniform and the effect of bifurcation mode inertia is taken into account. The bifurcation problem is formulated as a two-variable problem in terms of a constant stress and a variable representing the rate of growth of a bifurcation mode, here called the localization speed. A qualitative lower limit on growth of a diffuse mode in terms of the localization speed and a limit on the background motion due to the requirement of subsonic deformation are suggested. Bifurcation stress-localization speed distributions are computed and important changes from the quasistatic bifurcation behavior responsible for the phenomenon of multiple necking are shown to exist. In particular, it is found that the long wavelength modes are suppressed because their rate of formation is too slow compared to the background deformation. The main focus is on diffuse bifurcations; however, other bifurcation phenomena are touched upon by way of incremental wave propagation arguments.

Methods for finding feasible points in constrained optimization

Methods for finding feasible points are needed in some numerical optimization algorithms to initiate the search for a local minimum. Other methods perform more efficiently if a nearly feasible starting point is used. In addition, a feasible solution may be acceptable as the final design in some practical applications. Therefore, constraint correction methods are quite useful. Such methods based on penalty functions and primal approach are described and discussed. To gain insight into the methods a two-variable problem is used to analyze the methods. Several structural design problems are used to study the numerical behavior of the methods and compare their performance. It is concluded that, in general, the primal methods are more efficient than the penalty function methods. Also, no method has been found that is guaranteed to find a feasible point for general constrained nonlinear problems starting from an arbitrary point. In case a method fails, random points should be used to restart the procedure to search for a feasible point.

Generating Linear and Linear-Quadratic Bilevel Programming Problems

This paper describes a technique for generating both linear and linear-quadratic bilevel programming problems. The method is based on combining a number of single-parameter two-variable problems to obtain a separable multivariable problem with a number of desirable properties. The separability is then disguised using a simple transformation. The proposed technique, which requires very little computational effort, allows the user control over, among other things, the number and type of minima and the data density.

A variant of Rayleigh's method applied to Timoshenko beams embedded in a Winkler-Pasternak medium

This paper deals with the determination of the fundamental frequencies of Timoshenko beams in a Winkler-Pasternak medium by means of the variant of Rayleigh's method which allows an optimization of the approximate modal functions through a non-integer exponential parameter. The case of a Timoshenko beam gives rise to a two-variable problem which constitutes an extension of the title method with respect to previous work. A static functional relationship between the two unknown functions is proposed. Numerical results are given for simply supported and clamped-clamped beams of uniform and variable cross section. Comparisons with available exact solutions were made, showing very good agreement.

One-Dimensional Models for Neutral Particle Transport in Ducts

The problem of monoenergetic neutral particle transport in an evacuated duct of arbitrary cross-sectional geometry with partial isotropic reflection at the duct walls is considered. This five-variable (three in space, two in angle) transport problem is approximated by a coupled set of N two-variable (one in space, one in angle) problems via a simple Galerkin (or variational) procedure. Numerical comparisons with exact results indicate that the N=2 approximation is remarkably accurate for predicting the wall absorption and particle fluxes at the duct ends.

A user oriented modular package for the solution of general field problems under time varying conditions

In this paper a modular package, named COMPELL, for the solution of two-variable time-varying problems of the elliptic type is presented. The package allows the solution of standard magnetic problems, also under quasi-stationary or rapidly time-varying conditions, in the cases with nonlinear, anisotropic and nonhomogeneous materials with eddy currents, but can also face different problems, taking into account for instance the presence of moving bodies or the reciprocal interaction between magnetic fields, thermal fields and so on. The procedures used for field solution are generalized finite differences, least square finite differences, first presented in this paper, or finite elements. In the paper examples of magnetic field solutions are given, in particular by showing the field in a Tokamak structure accounting for the Hall effect. The data deck and the obtained results are displayed and the main features of the package are discussed.

On Double Chebyshev Series Approximation

Previous article Next article On Double Chebyshev Series ApproximationN. K. BasuN. K. Basuhttps://doi.org/10.1137/0710045PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractMany of the standard practical techniques for Chebyshev polynomial approximation in one-variable problems have been extended to two-variable problems.[1] Ferdinand Freudenstein, Bi-variate, rectangular, optimum-interval interpolation, Math. Comp., 15 (1961), 288–291 MR0129532 0168.14903 CrossrefGoogle Scholar[2] C. W. Clenshaw, A note on the summation of Chebyshev series, Math. Tables Aids Comput., 9 (1955), 118–120 MR0071856 0065.05403 CrossrefGoogle Scholar[3] David Elliott, Error analysis of an algorithm for summing certain finite series, J. Austral. Math. Soc., 8 (1968), 213–221 MR0230452 0159.45104 CrossrefGoogle Scholar[4] David Elliott, The numerical solution of integral equations using Chebyshev polynomials, J. Austral. Math. Soc., 1 (1959/1960), 344–356 MR0128087 0096.10201 CrossrefGoogle Scholar[5] L. Fox and , E. T. Goodwin, The numerical solution of non-singular linear integral equations, Philos. Trans. Roy. Soc. London. Ser. A., 245 (1953), 501–534 MR0054355 0050.12902 CrossrefISIGoogle Scholar[6] Google Scholar[7] L. Fox and , I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968ix+205 MR0228149 0153.17502 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Analytical and numerical investigation of Poiseuille flow through semi-elliptic annulusPhysics of Fluids, Vol. 34, No. 8 | 1 Aug 2022 Cross Ref Study of the free surface Green function at zero forward speed in finite water depth using Newman’s methodApplied Ocean Research, Vol. 120 | 1 Mar 2022 Cross Ref Comparison of existing methods for the calculation of the infinite water depth free-surface Green function for the wave–structure interaction problemApplied Ocean Research, Vol. 81 | 1 Dec 2018 Cross Ref Nondimensionalized Univariate Equation Characterizing Optimal State Feedback Control for Collision AvoidanceIEEE Transactions on Intelligent Transportation Systems, Vol. 19, No. 10 | 1 Oct 2018 Cross Ref The automatic solution of partial differential equations using a global spectral methodJournal of Computational Physics, Vol. 299 | 1 Oct 2015 Cross Ref Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flowJournal of Differential Equations, Vol. 257, No. 8 | 1 Oct 2014 Cross Ref A new kind of double Chebyshev polynomial approximation on unbounded domainsBoundary Value Problems, Vol. 2013, No. 1 | 22 January 2013 Cross Ref On Approximate Solutions of Second-Order Linear Partial Differential EquationsApplied Mathematics, Vol. 03, No. 09 | 1 Jan 2012 Cross Ref A fitting formula for the falloff curves of unimolecular reactions, II: Tunneling effectsInternational Journal of Chemical Kinetics, Vol. 43, No. 1 | 19 October 2010 Cross Ref Direct numerical simulation of preferential particle concentration in decaying turbulence under the influence of magnetic fieldInternational Journal for Numerical Methods in Fluids, Vol. 419 | 1 Jan 2009 Cross Ref Large and small signal distortion analysis using modified Volterra seriesAnalog Integrated Circuits and Signal Processing, Vol. 54, No. 2 | 29 November 2007 Cross Ref Volterra Analysis Using Chebyshev Series2007 IEEE International Symposium on Circuits and Systems | 1 May 2007 Cross Ref Large and Small Signal Distortion Analysis Using Modified Volterra Series2006 NORCHIP | 1 Nov 2006 Cross Ref An efficient numerical approximation for the linear class of mixed integral equationsApplied Mathematics and Computation, Vol. 167, No. 2 | 1 Aug 2005 Cross Ref Chebyshev polynomial solutions of second-order linear partial differential equationsApplied Mathematics and Computation, Vol. 134, No. 1 | 1 Jan 2003 Cross Ref Chebyshev series solution of linear Fredholm integrodifferential equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 29, No. 4 | 1 Jul 1998 Cross Ref Fast polynomial multiplication and convolutions related to the discrete cosine transformLinear Algebra and its Applications, Vol. 252, No. 1-3 | 1 Feb 1997 Cross Ref Chebyshev series solutions of Fredholm integral equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 27, No. 5 | 9 July 2006 Cross Ref An algorithm for the evaluation of multidimensional (direct and inverse) discrete Chebyshev transformsJournal of Computational and Applied Mathematics, Vol. 8, No. 4 | 1 Dec 1982 Cross Ref Optimum acoustic design of free-running low speed propellersAircraft Systems and Technology Meeting | 16 August 1977 Cross Ref Clenshaw sums in several variablesConstructive Theory of Functions of Several Variables | 2 October 2006 Cross Ref Volume 10, Issue 3| 1973SIAM Journal on Numerical Analysis History Submitted:05 March 1971Published online:14 July 2006 InformationCopyright © 1973 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0710045Article page range:pp. 496-505ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

On the stability of discrete-time systems with hysteretic controllers

The problem of deriving a frequency-domain sufficient condition for the global asymptotic stability of a discrete-time system having hysteretic controllers is considered. It is suggested that the hysteretic characteristic be replaced by a function of the form f(y, ẏ) where y(t) is the input to the controller. This allows the problem to be reformulated to that of deriving a sufficient condition for the stability of a two-controller system where both the controllers are nonhysteretic. This latter problem is solved by embedding it into a two-variable problem. Numerical examples are presented which illustrate the results.