Multisymplectic formulation of fluid dynamics using the inverse map

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.</p

On asymptotic nonlocal symmetry of nonlinear Schr\"odinger equations

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schr\"odinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schr\"odinger equations. An important class of solutions of the free Schr\"odinger equation with improved smoothing properties is obtained

The effective temperature

This review presents the effective temperature notion as defined from the deviations from the equilibrium fluctuation-dissipation theorem in out of equilibrium systems with slow dynamics. The thermodynamic meaning of this quantity is discussed in detail. Analytic, numeric and experimental measurements are surveyed. Open issues are mentioned.Comment: 58 page

Characteristics of Conservation Laws for Difference Equations

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether's Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether's Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka-Volterra equation

Automorphisms of real Lie algebras of dimension five or less

The Lie algebra version of the Krull-Schmidt Theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form

The artistry of Jesus

Graduate Thesis (M.A.)--Boston University, 193

The priesthood of Jesus as presented by the Epistle to the Hebrews

Thesis (Ph.D.)--Boston UniversityThe author of the Epistle to the Hebrews claims that Jesus is a priest. This is a unique and original claim, since no other New Testament document calls him a priest, and none of Jesus' contemporaries apparently thought of him in this manner. The question naturally arises whether the author's claim that Jesus is a priest is a legitimate one or not. What does he mean by calling Jesus a priest? How does he justify the use of the term in connection with Jesus? Does he sufficiently support his case with the material which he advances for evidence? Is it possible to establish his claim with this evidence and this line of argument? Are his claims and procedure valid? Other interpreters of the priesthood of Jesus as presented by this epistle have said that the author starts with Jesus as priest and concludes from this assumption that Jesus fulfils priestly functions. They have said, too, that the proof which the author presents to establish Jesus as a priest is adequate for this purpose and valid. They hold that he proves his case by the methods he uses. But their treatments are dominated by theological and speculative considerations which raise no question in criticism of the author's presenting Jesus as a priest or of the validity of such a claim. An examination of the nature of priesthood in general and of the Aaronic priesthood of the Old Testament in particular indicates that priesthood is concerned with the relations between human beings and the superhuman beings whom they know as a god, gods, or God, for the benefit of both. In a word, a priest is the mediator between men and God, offering men's sacrifices and revealing God's judgments and will. Priests strive to gain forgiveness for men's sins and to establish communion between men and God. To become a priest a candidate must meet very rigorous requirements of a technical and professional nature. He must be intensively trained in all of the ceremonial practices of priesthood and elaborately initiated into the ranks of priests. Priests belong to an order of priesthood and are associated with a sanctuary. The author of this epistle appeals to the Aaronic priesthood as the very highest expression of professional priesthood that he knows. If any priesthood could have been expected to fulfil successfully the functions of priesthood, this priesthood should have been expected to do so in his judgment, for it was instituted by God himself. He uses this priesthood as a medium of comparison for the priesthood of Jesus, and because of the familiarity of his readers with it, as a means of explaining to them the nature of Jesus' priesthood. He is highly appreciative of the Aaronic priesthood and exalts it in order to exalt even more the priesthood of Jesus which he claims is better than it. Despite his high regard for the Aaronic priesthood, the author believes that it has failed to achieve the real purposes of priesthood and the true ends of religion. Its two chief failures are failure to remove the obstacles, hindrances, and hurdles which separate men from God, particularly men's sins, and failure to gain access for men to God. From the emphasis which the author lays on these two failures, it follows that he believes that true and successful priesthood must fulfil these two functions. Priesthood must deal adequately with sins and other barriers in the sinner's approach to God, and must bring men to God. The author is aware of the technical, professional and ceremonial requirements of priesthood. He recognizes, too, that Jesus could never be a priest according to any such professional priestly order. He admits frankly that Jesus is not a priest after the order of Aaron. Since he needs to belong to some order of priesthood, he makes him a priest after the order of Melchizedek on the strength of a quotation from Psalm 110:4 which calls the Messiah such a priest. Since Christians hold Jesus to be the Messiah, he must therefore be a Melchizedek priest. The Melchizedek priesthood allows the author to say "by interpretation" that Jesus belongs to an order of priesthood, and that this order is superior to the order of Aaron because eternal, prior to that of Aaron, deferred to by it, and permanent. The author then presents Jesus as "better" than the angels who mediated the Aaronic covenant, because he was better equipped to do for men what a priesthood is supposed to do. He is "better" than Moses because he leads men into the "rest of God" whereas Moses failed to do this. He is "better" than the Aaronic priests because he successfully removes the barrier of men's sin from obstructing their progress to God, and brings men fully into the presence of God. The sanctuary in which Jesus ministers is the true, original and actual sanctuary of God in heaven of which the Aaronic sanctuary is but an earthly copy made with hands. In this heavenly sanctuary there is no separation between men and God for Jesus has removed the veil of separation that kept them apart. Into this sanctuary Jesus enters as the forerunner of men by virtue of his own blood offered in sacrifice. This sacrifice is adequate to cleanse men's consciences of "dead works" and to bring men to the mercy seat of the Eternal. "Drawing nigh to God" is the author's way of representing this supreme priestly service of Jesus. Jesus fulfils the real purposes of priesthood and he achieves for men the true ends of religion. In him they gain complete salvation. The author does not start with Jesus as priest, he ends with him as such. Having noted the priestly services which Jesus successfully performs, he concludes from this fact that Jesus is a priest, a superior priest, and the supreme priest. This is the proof which the author presents for the priesthood of Jesus. When it is examined, it is found to consist of scriptural proof-texting, allegorical imagining and ingenious analogy making. It is all confessional claim, and not objective proof, judged by any standards of technical, logical argumentation. He has worked out the ministry of Jesus as a priest in terms of analogy and typology. These are literary rather than logical devices, however, His material is testimony, confession, witness and affirmation. Using this material the author has not proved and cannot prove his case that Jesus is a priest in any objective, logical sense. It is not material that can be so proved. The author's real case for the priesthood of Jesus rests upon his own Christian experience and that of his readers; it is proof or evidence of a religious character. When he claims that the Aaronic priesthood has failed to remove the barriers of sin and to bring men to God, and that Jesus has succeeded in doing these things, he is saying that the Aaronic priesthood failed to ·perform these priestly services for him and his readers and that Jesus has succeeded in performing them. This experience, though not subject to logical substantiation, cannot be refuted, and must be allowed to be a valid representation of truth. The author's justification for calling Jesus a priest is found in the fact that he accomplished for sinful men what priesthood should accomplish and failed. His proof does not stand up logically, but truth is truth whether it can be proved logically or not. Proof is intended only to help others to see the truth and to confirm it, never to create it. Formal proof is not the only kind of legitimate argumentation. Testimony, confession, witness of others and allegory are also legitimate methods of helping men to appropriate the truth. The author's proof is not the proof of logic, and so is not objectively, formally valid, but it is proof and legitimate proof, nonetheless. It is subjectively valid. His final appeal is to the experience of his readers as the confirmation of his own experience. The author uses the idea of Jesus as priest as an allegorical and illustrative interpretation of Jesus. The priesthood of Jesus is a sort of symbol of the service he renders to men. Jesus as priest is a matter of interpretation not of accurate statement of historical fact. Jesus was not a priest, but with a certain understanding of what it means to be a priest, he may be thought of as one. He is a priest by right of personality and achievement, not by right of descent and technical qualifications. It has been shown, therefore, that neither the author's method nor his material conforms to the usual technical standards of argument and disputation. From the standpoint of strict, formal reasoning his proof is not valid. His claims are not justified. However, when the confessional and experiential material of the author is understood as the irrefutable conviction of men" who know beyond a doubt what Jesus has done for them, and who are willing to have the matter tested or confirmed by the experience of others, then it can be accepted as valid subjective proof that cannot be denied. On the basis of religious evidence and Christian experience Jesus can most properly be presented as a priest arid his claim for priesthood be validly supported

Moving frames: difference and differential-difference Lagrangians

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's Theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the Nonlinear Schr\"odinger Equation.Comment: 28 pages, 3 figure

Modelling turbulence in axisymmetric wakes: an application to wind turbine wakes

A novel fast-running model is developed to predict the three-dimensional (3D) distribution of turbulent kinetic energy (TKE) in axisymmetric wake flows. This is achieved by mathematically solving the partial differential equation of the TKE transport using the Green's function method. The developed solution reduces to a double integral that can be computed numerically for a wake prescribed by any arbitrary velocity profile. It is shown that the solution can be further simplified to a single integral for wakes with Gaussian-like velocity-deficit profiles. Model results are compared and validated against detailed 3D laser Doppler anemometry data measured within the wake flow of a model wind turbine in a laboratory environment. This shows a remarkably good agreement in both the magnitude and shape of the radial TKE profiles at the turbine hub height. The wind-tunnel data also provide insights into the evolution of important turbulent flow quantities such as turbulent viscosity, mixing length, and the TKE dissipation rate within a wind turbine wake

Transformations, symmetries and Noether theorems for differential-difference equations

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether’s theorem. We state and prove the differential-difference version of Noether’s second theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether’s two theorems. These results are applied to various equations from physics