Physical Mathematics

Understanding the nature of time requires more than philosophy and geometry. Past, present, and future are defined through event sequences and state transitions. Communication and the construction of geometry are key elements for constructing time. Special and general relativity bring together the effects of communication and geometry in understanding time. Quantum theory and particle-to-force relationships underscore the need for precise definitions of time. Time and information flow are inextricably entangled. Significant challenges lie ahead for physics, time, and information science.

Photodynamics refers to the dynamics of particles, like the photon, that have a rest mass and travel at the speed of light. Recently we derived an energy-momentum theorem which provides the dynamics of such particles. In this paper we derive the same theorem from a slightly different point of view which better illustrates the connection between kinematics and dynamics. We use the photon, and its associated photokinetics to reveal how its rest mass is created. We reveal the same process at work in Quantum Mechanics. We then discuss the process of Photogenesis which we believe, at the fundamental level, is used to create matter, and argue that it has universal applicability.

A generalized Chua’s circuit with a single parametric nonlinearity is introduced in this letter. The circuit is obtained by replacing the Chua’s diode of the classical Chua’s circuit with a parametric active diode pair. The obtained circuit can be described by three differential equations. Preliminary dynamic properties of the circuit are categorized with respect to its parameters and it is found that the circuit has two nonzero stable node-foci leading to complex coexisting behaviours. A plethora of coexisting symmetric and asymmetric attractors are depicted. More importantly, the symmetry of the circuit can be monitored with a single parameter k (i.e. a single control resistor). Also, multistability in the symmetry boundary is discussed by monitoring the single bifurcation parameter k. Finally, Pspice circuit implementation results are consistent with the complex dynamic behaviours observed during numerical analysis.

This paper investigates the topological properties on the structure generated by the addition modulus of the Aunu permutation pattern 𝑤𝑖. The pattern 𝑤𝑖 has the generating function defined as:

𝜔𝑖=(1 (1+𝑖)𝑚𝑜𝑑𝑝 (1+2𝑖)𝑚𝑜𝑑𝑝 (1+3𝑖)𝑚𝑜𝑑𝑝………(1+((𝑝−1)𝑖)𝑚𝑜𝑑𝑝)

For 𝑖=1,2,..𝑝−1.

The numbers are arrangement on the structure 𝑋={1,….,𝑃} for primes p ≥ 5.

It has been established in this paper that the generated set is a topology on 𝑋, it is not a convex set and finally, not a 𝜎-algebra.

In this paper, p 2 Z Z p p + - @ for n ≥ 3, where p=2n-1 and Zp-1 p =e (e is the multiplicative identity). The diagonal elements of Z+ 2n+1 can generate Z2k-2 2n-1; k=2, 3,… n-1 by simple algorithm.