Related papers: Some application of difference equations in Crypto…
A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
The realm of this thesis is cryptographic protocol theory in the quantum world. We study the security of quantum and classical protocols against adversaries that are assumed to exploit quantum effects to their advantage. Security in the…
We present a generalized notion of degree for rotating solutions of planar systems. We prove a formula for the relation of such degree with the classical use of Brouwer's degree and obtain a twist theorem for the existence of periodic…
Using Mawhin's coincidence degree theory, we obtain some new continuation theorems which are designed to have as a natural application the study of the periodic problem for cyclic feedback type systems. We also discuss some examples of…
Quantum cryptography is reviewed, first using entanglement both for the intuition and for the experimental realizations. Next, the implementation is simplified in several steps until it becomes practical. At this point entanglement has…
We derive a method for finding Lie Symmetries for third-order difference equations. We use these symmetries to reduce the order of the difference equations and hence obtain the solutions of some third-order difference equations. We also…
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize…
Difference equations have many applications and play an important role in numerical analysis, probability, statistics, combinatorics, computer science, quantum consciousness, etc. We first prove that the partial differential equation is…
We investigate a variety of statistical properties associated with the number of distinct degrees that exist in a typical network for various classes of networks. For a single realization of a network with N nodes that is drawn from an…
The laws of physics play a crucial role in the security of quantum key distribution (QKD). This fact has often been misunderstood as if the security of QKD would be based only on the laws of physics. As the experts know well, things are…
We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem,…
We review the plethora of uncertainty relations that appear in quantum mechanics and their nuances. We present both foundational applications, e.g. in understanding and defining complementarity, and practical applications, e.g. in quantum…
In this paper we use the Bott residue formula in equivariant cohomology to show a formula for the algebraic degree in semidefinite programming.
Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.
The quantum differential equations can be regarded as examples of equations with certain universal properties which are of wider interest beyond quantum cohomology itself. We present this point of view as part of a framework which…
We introduce a class of strongly \'{e}tale difference algebras, whose role in the study of difference equations is analogous to the role of \'{e}tale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's…
We explore the Klein-Gordon equation in the framework of crypto-Hermitian quantum mechanics. Solutions to common problems with probability interpretation and indefinite inner product of the Klein-Gordon equation are proposed.
In this article two implementations of a symmetric finite difference algorithm for a first-order partial differential equation are discussed. The considered partial differential equation discribes the time evolution of the crack length…