Superconvergent interpolants for Gaussian collocation solutions of mixed order BVODE systems

<abstract><p>The high quality COLSYS/COLNEW collocation software package is widely used for the numerical solution of boundary value ODEs (BVODEs), often through interfaces to computing environments such as Scilab, R, and Python. The continuous collocation solution returned by the code is much more accurate at a set of mesh points that partition the problem domain than it is elsewhere; the mesh point values are said to be superconvergent. In order to improve the accuracy of the continuous solution approximation at non-mesh points, when the BVODE is expressed in first order system form, an approach based on continuous Runge-Kutta (CRK) methods has been used to obtain a superconvergent interpolant (SCI) across the problem domain. Based on this approach, recent work has seen the development of a new, more efficient version of COLSYS/COLNEW that returns an error controlled SCI.</p> <p>However, most systems of BVODEs include higher derivatives and a feature of COLSYS/COLNEW is that it can directly treat such mixed order BVODE systems, resulting in improved efficiency, continuity of the approximate solution, and user convenience. In this paper we generalize the approach mentioned above for first order systems to obtain SCIs for collocation solutions of mixed order BVODE systems. The main contribution of this paper is the derivation of generalizations of continuous Runge-Kutta-Nyström methods that form the basis for SCIs for this more general problem class. We provide numerical results that (ⅰ) show that the SCIs are much more accurate than the collocation solutions at non-mesh points, (ⅱ) verify the order of accuracy of these SCIs, and (ⅲ) show that the cost of utilizing the SCIs is a small fraction of the cost of computing the collocation solution upon which they are based.</p></abstract>

Duality, Generalized Global Symmetries and Jet Space Isometries

We revisit universal features of duality in linear and nonlinear relativistic scalar and Abelian 1-form theories with single or multiple fields, which exhibit ordinary or generalized global symmetries. We show that such global symmetries can be interpreted as generalized Killing isometries on a suitable, possibly graded, target space of fields or its jet space when the theory contains higher derivatives. This is realized via a generalized sigma model perspective motivated from the fact that higher spin particles can be Nambu–Goldstone bosons of spontaneously broken generalized global symmetries. We work out in detail the 2D examples of a compact scalar and the massless Heisenberg pion fireball model and the 4D examples of Maxwell, Born–Infeld, and ModMax electrodynamics. In all cases we identify the ’t Hooft anomaly that obstructs the simultaneous gauging of both global symmetries and confirm the anomaly matching under duality. These results readily generalize to higher gauge theories for p-forms. For multifield theories, we discuss the transformation of couplings under duality as two sets of Buscher rules for even or odd differential forms.

Higher Derivatives and Polynomials of the Standard Nield-Kuznetsov Function of the First Kind

In this fundamental work, higher derivatives of the standard Nield-Kuznetsov function of the first kind, and the polynomials arising from this function and Airy’s functions, are derived and discussed. This work provides background theoretical material and computational procedures for the arising polynomials and the higher derivatives of the recently introduced Nield-Kuznetsov function, which has filled a gap that existed in the literature since the nineteenth century. The ease by which the inhomogeneous Airy’s equation can now be solved is an advantage offered by the Nield-Kuznetsov functions. The current analysis might prove to be invaluable in the study of inhomogeneous Schrodinger, Tricomi, and Spark ordinary differential equations.

Method for long-term coherent-noncoherent signal accumulation with non-zero higher derivatives range to radar target

A method of long-term combined accumulation of the reflected signal is justified, which provides for its division into disjoint subsets, coherent accumulation in subsets using one of the fast algorithms and subsequent incoherent accumulation of the squares of the modules of the results of processing the subsets. A distinctive method’s feature is the use with incoherent accumulation of maxima of the squares of the moduli of the coherent processing results, that are selected from the range / radial velocity regions in accordance with a given hypothesis about the minimum and maximum values of the target radial velocity and the radial acceleration detection channel setting.The efficiency of the method was confirmed by simulation modeling. Using the theories of ordinal statistics and the method of moments, a method for calculating the probability of correct detection is developed. Estimates of processing losses are made in comparison with coherent and incoherent accumulation algorithms for a signal reflected from a point target, for the case when there is no range and frequency migration. Estimates for the required number of receiver channels are given.

Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.

Physics of nonlinear oscillations with nonlocal variables

In cases where physical processes cannot be described by linear equations, and nonlinear equations are difficult to solve mathematically, we have to use approximate solutions to such problems. One such example is the description of the Kapitsa pendulum, which is a pendulum with a vibrating suspension point. In contrast to the previously known methods of describing such a problem, in this paper we propose to use additional variables in the form of higher derivatives, which allows us to obtain corrections that give a more detailed contribution to the description of this problem.

Benign ghosts in higher-derivative systems

A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however: ghosts are benign. This happens, in particular, in exactly solvable higher-derivative theories, but exact solvability seems to be a sufficient but not a necessary condition for the benign nature of the ghosts. We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe represents then a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

Natural higher-derivatives generalization for the Klein–Gordon equation

We propose a natural family of higher-order partial differential equations generalizing the second-order Klein–Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing higher-derivative terms. The limit obtained by considering arbitrarily higher-order powers of the d’Alembertian operator leading to a formal infinite-order partial differential equation is discussed. The general model is constructed using the exponential of the d’Alembertian differential operator. The canonical energy–momentum tensor densities and field propagators are explicitly computed. We consider both homogeneous and non-homogeneous situations. The classical solutions are obtained for all cases.