###### The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

## Abstract

As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals.

It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.

In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

###### On fixed-point theorems in synthetic computability

## Abstract

Lawvere’s fixed point theorem captures the essence of diagonalization arguments. Cantor’s theorem, Gödel’s incompleteness theorem, and Tarski’s undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds. We study Lawvere’s fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov’s principle, and the Enumeration axiom, which states that there are countably many countable subsets of ℕ. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory. We show that every countably generated ω-chain complete pointed partial order (ωcppo) is countable, and that countably generated ωcppos are closed under countable products. Therefore, Lawvere’s fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated ωcppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains (ωcppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an ωcpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere’s fixed point theorem for multi-valued endomaps.

###### Introduction. A Personal Tribute to Peter Freyd and Bill Lawvere

, Buffalo 1988. [31] F. W. Lawvere, Cohesive Toposes and Cantor's \Lauter Einsen", Phil. Math. 3-2 (1994), 5-15. [32] F. W. Lawvere, Volterra's functionals and covariant cohesion of space, in: R. Betti, F. W. Lawvere (Eds), Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie J, 64 (2000), 201-214. [33] F. W. Lawvere, Comments on the Development of Topos Theory, in: J. P. Pier, (ed.), Development of Mathematics 1950-2000. Birkhäuser, Basel-Boston-Berlin, 2000, 715-734. [34] F. W Lawvere

###### A characterization of weighted Besov spaces in quantum calculus

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###### Multiplicity result for a stationary fractional reaction-diffusion equations

differential equations, Academic Press, New York, 1999. [19] Rabinowitz P., Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., No 65, 1986. [20] Rivero M., Trujillo J., Vázquez L. and Velasco M., Fractional dynamics of populations, Appl. Math. Comput, 218, 1089 - 1095(2011). [21] Sabatier J., Agrawal O. and Tenreiro Machado J., Advances in fractional calculus. Theoretical developments and applications in physics and engineering, Springer-Verlag, Berlin, 2007

###### On Marshall-Olkin Burr X family of distribution

Inference (2004). [29] M. M. Ristic , K. K. Jose and A. Joseph, A Marshall-Olkin gamma distribution and minification process. STARS: Int. Journal (Sciences) 1 (2) (2007), 107-117. [30] R Development Core Team R. A Language and Environment for Statistical Computing, R Foundation for StatisticalNComputing (2015), Vienna, Austria. [31] N. Santos, M. Bourguignon and L. M. Zea, A. D. C. Nascimento and G. M. Cordeiro, The Marshall-Olkin extended Weibull family of distributions. J. Stat. Dist, Applic. 1Art. (2014), 1