Related papers: Classifying Linear Canonical Relations
The canonical structure of theories whose Lagrangian contains higher powers of time derivatives is often obscured by the nonlinear relationship between the velocities and momenta. We use the Dirac formalism and define a generalized Legendre…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially…
In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order…
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a…
Canonical transformations are ubiquitous in Hamiltonian mechanics, since they not only describe the fundamental invariance of the theory under phase-space reparameterisations, but also generate the dynamics of the system. In the first part…
We consider (finitary, propositional) logics through the original use of Category Theory: the study of the "sociology of mathematical objects", aligning us with a recent, and growing, trend of study logics through its relations with other…
We describe an inventory of semantic relations that are expressed by prepositions. We define these relations by building on the word sense disambiguation task for prepositions and propose a mapping from preposition senses to the relation…
In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a…
The aim of this paper is to constructs Boehmian space, the linear canonical transform for Boehmians is define and to study its properties.
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…
Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the…
The linear representation hypothesis is the informal idea that semantic concepts are encoded as linear directions in the representation spaces of large language models (LLMs). Previous work has shown how to make this notion precise for…
We generalize the solution of linear recurrence relations from fields to central division algebras, adapting the standard tools of companion matrices and characteristic polynomials to the non-commutative setting. We then solve linear…
Wehrheim and Woodward have shown how to embed all the canonical relations between symplectic manifolds into a category in which the composition is the usual one when transversality and embedding assumptions are satisfied. A morphism in…
We treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics and establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states…
In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of linear recurrence relation and deduce certain preliminary results…
For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear…
We provide some constructions using Lagrangian cobordisms which improve known examples for some symplectic squeezing problems. Additionally, we prove a flexibility result that Lagrangian submanifolds which are Lagrangian isotopic are also…
We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra…