On a Differential Constraint in Asymmetric Theories of the Mechanics of Growing Solids

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

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Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0075 ◽

2013 ◽

Vol 469 (2157) ◽

pp. 20130075 ◽

Cited By ~ 7

Author(s):

Graeme W. Milton

Keyword(s):

Boundary Conditions ◽

Convex Function ◽

Convex Functions ◽

Quadratic Functions ◽

Differential Constraint ◽

Outstanding Problem ◽

Valued Field ◽

Spatial Dimensions ◽

Derivatives Of ◽

Quasi Convexity

Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region Ω of certain special quadratic functions f ( E ), where E ( x ) derives from a potential U ( x ). With E =∇ U , it is known that such sharp inequalities can be obtained when f ( E ) is a quasi-convex function and when U satisfies affine boundary conditions (i.e. for some matrix D , U = D x on ∂ Ω ). Here, we allow for other boundary conditions and for fields E that involve derivatives of a variety orders of U . We define a notion of convexity that generalizes quasi-convexity. Q *-convex quadratic functions are introduced, characterized, and an algorithm is given for generating sharply Q *-convex functions. We emphasize that this also solves the outstanding problem of finding an algorithm for generating extremal quasi-convex quadratic functions. We also treat integrals over Ω of special quadratic functions g ( J ), where J ( x ) satisfies a differential constraint involving derivatives with, possibly, a variety of orders. The results generalize an example of Kang, and the author in three spatial dimensions where J ( x ) is a 3×3 matrix-valued field satisfying ∇⋅ J =0.

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Application of differential constraint method to exact solution of second-grade fluid

Applied Mathematics and Mechanics ◽

10.1007/s10483-009-0401-x ◽

2009 ◽

Vol 30 (4) ◽

pp. 403-412 ◽

Cited By ~ 2

Author(s):

Dao-xiang Zhang ◽

Su-xiao Feng ◽

Zhi-ming Lu ◽

Yu-lu Liu

Keyword(s):

Exact Solution ◽

Second Grade ◽

Second Grade Fluid ◽

Differential Constraint ◽

Grade Fluid ◽

Constraint Method

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Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2015.0501 ◽

2015 ◽

Vol 471 (2183) ◽

pp. 20150501 ◽

Cited By ~ 5

Author(s):

G. A. Athanassoulis ◽

I. C. Tsantili ◽

Z. G. Kapelonis

Keyword(s):

Differential Equations ◽

Evolution Equation ◽

Nonlinear Differential Equations ◽

Limit State ◽

Time Derivative ◽

Missing Information ◽

Differential Constraint ◽

Link Type ◽

Closure Scheme ◽

Long Time

Uncertainty quantification for dynamical systems under non-white excitation is a difficult problem encountered across many scientific and engineering disciplines. Difficulties originate from the lack of Markovian character of system responses. The response–excitation (RE) theory, recently introduced by Sapsis & Athanassoulis (Sapsis & Athanassoulis 2008 Probabilistic Eng. Mech. 23, 289–306 ( doi:10.1016/j.probengmech.2007.12.028 )) and further studied by Venturi et al. (Venturi et al. 2012 Proc. R. Soc. A 468, 759–783 ( doi:10.1098/rspa.2011.0186 )), is a new approach, based on a simple differential constraint which is exact but non-closed. The evolution equation obtained for the RE probability density function (pdf) has the form of a generalized Liouville equation, with the excitation time frozen in the time-derivative term. In this work, the missing information of the RE differential constraint is identified and a closure scheme is developed for the long-time, stationary, limit-state of scalar nonlinear random differential equations (RDEs) under coloured excitation. The closure scheme does not alter the RE evolution equation, but collects the missing information through the solution of local statistically linearized versions of the nonlinear RDE, and interposes it into the solution scheme. Numerical results are presented for two examples, and compared with Monte Carlo simulations.

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Rational Interpolation of Car Motions

Journal of Mechanisms and Robotics ◽

10.1115/1.4030298 ◽

2015 ◽

Vol 7 (3) ◽

Author(s):

J. M. Selig

Keyword(s):

Final Section ◽

Rational Interpolation ◽

Polynomial Functions ◽

Optimal Motion ◽

Kinematic Constraint ◽

Differential Constraint ◽

Kinematic Constraints ◽

Cayley Map ◽

Polynomial Curve ◽

Rigid Body Motions

This work introduces a general approach to the interpolation of the rigid-body motions of cars by rational motions. A key feature of the approach is that the motions produced automatically satisfy the kinematic constraints imposed by the car wheels, that is, cars cannot instantaneously translate sideways. This is achieved by using a Cayley map to project a polynomial curve in the Lie algebra se(2) to SE(2) the group of rigid displacements in the plane. The differential constraint on se(2), which expresses the kinematic constraint on the car, is easily solved for one coordinate if the other two are given, in this case as polynomial functions. In this way, families of motions obeying the constraint can be found. Several families are found here and examples of their use are shown. It is shown how rest-to-rest motions can be generated in this way and also how these motions can be joined so that the motion is continuous and differentiable across the join. A final section discusses the optimization of these motions. For some cost functions, the optimal motions are known but can be rather impractical to use. By optimizing over a family of motions which satisfy the boundary conditions for the motion, it is shown that rational motions can be found simply and are close to the overall optimal motion.

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The qualitative representation of physical systems

The Knowledge Engineering Review ◽

10.1017/s0269888900006159 ◽

1992 ◽

Vol 7 (1) ◽

pp. 55-77 ◽

Cited By ~ 15

Author(s):

Enrico Coiera

Keyword(s):

Process Theory ◽

Multiple Model ◽

Differential Constraint ◽

Physical Systems ◽

Reasoning Systems ◽

Qualitative Representation ◽

Growing Body ◽

Recent Developments ◽

Qualitative Process Theory ◽

Functional Representations

AbstractThe representation of physical systems using qualitative formalisms is examined in this review, with an emphasis on recent developments in the area. The push to develop reasoning systems incorporating deep knowledge originally focused on naive physical representations, but has now shifted to more formal ones based on qualitative mathematics. The qualitative differential constraint formalism used in systems like QSIM is examined, and current efforts to link this to competing representations like Qualitative Process Theory are noted. Inference and representation are intertwined, and the decision to represent notions like causality explicitly, or infer it from other properties, has shifted as the field has developed. The evolution of causal and functional representations is thus examined. Finally, a growing body of work that allows reasoning systems to utilize multiple representations of a system is identified. Dimensions along which multiple model hierarchies could be constructed are examined, including mode of behaviour, granularity, ontology, and representational depth.

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Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

Mathematics ◽

10.3390/math8010090 ◽

2020 ◽

Vol 8 (1) ◽

pp. 90 ◽

Cited By ~ 2

Author(s):

Andrei D. Polyanin

Keyword(s):

Exact Solutions ◽

Applied Mathematics ◽

Separation Of Variables ◽

Surface Condition ◽

Traveling Wave Solutions ◽

Mathematical Physics ◽

Variable Coefficients ◽

Nonlinear Pdes ◽

Diffusion Type ◽

Differential Constraint

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

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Variant-Coherence Gaussian Sources

Photonics ◽

10.3390/photonics8090403 ◽

2021 ◽

Vol 8 (9) ◽

pp. 403

Author(s):

Franco Gori ◽

Massimo Santarsiero

Keyword(s):

Spectral Density ◽

Source Area ◽

Spectral Densities ◽

Differential Constraint ◽

Modal Expansion ◽

Shape Invariant ◽

Degree Of Coherence ◽

Cross Spectral Density ◽

The Laplace Transform ◽

Space Variant

The celebrated Gaussian Schell model source with its shift-invariant degree of coherence may be the basis for devising sources with space-variant properties in the spirit of structured coherence. Starting from superpositions of Gaussian Schell model sources, we present two classes of genuine cross-spectral densities whose degree of coherence varies across the source area. The first class is based on the use of the Laplace transform while the second deals with cross-spectral densities that are shape-invariant upon paraxial propagation. For the latter, we present a set of shape-invariant cross-spectral densities for which the modal expansion can be explicitly found. We finally solve the problem of ascertain whether an assigned cross-spectral density is shape-invariant by checking if it satisfies a simple differential constraint.

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Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2017062 ◽

2018 ◽

Vol 24 (4) ◽

pp. 1605-1624

Author(s):

Adam Prosinski

Keyword(s):

Vector Fields ◽

Additional Assumption ◽

Integral Functional ◽

Variational Problems ◽

Constant Rank ◽

Differential Constraint ◽

First Order ◽

Characteristic Cone ◽

Lower Semi ◽

Continuity Assumption

This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed 𝓐-p quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone of 𝓐.

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