The unitary representation theory of locally compact contraction groups and their semi-direct products with $\mathbb{Z}$ is studied. We put forward the problem of completely characterising such groups which are type I or CCR and this article provides a stepping stone towards a solution to this problem. In particular, we determine new examples of type I and non-type-I groups in this class, and we completely classify the irreducible unitary representations of the torsion-free groups, which are shown to be type I. When these groups are totally disconnected, they admit a faithful action by automorphisms on an infinite locally-finite regular tree; this work thus provides new examples of automorphism groups of regular trees with interesting representation theory, adding to recent work on this topic.

In this paper it is argued that Barad's Agential Realism, an approach to quantum mechanics originating in the philosophy of Niels Bohr, can be the basis of a 'theory of everything' consistent with a proposal of Wheeler that observer-participancy is the foundation of everything. On the one hand, agential realism can be grounded in models of self-organisation such as the hypercycles of Eigen, while on the other agential realism, by virtue of the 'discursive practices' that constitute one aspect of the theory, implies the possibility of the generation of physical phenomena through acts of specification originating at a more fundamental level. Included in phenomena that may be generated by such a mechanism are the origin and evolution of life, and human capacities such as mathematical and musical intuition.

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any "Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin's classic papers and are written for mathematicians.

In light of Gödel's undecidability results (incomplete theorems) for math, quantum indeterminism indicates that physics and the Universe may be indeterministic, incomplete, and open in nature, and therefore demand no single unification theory of everything. The Universe is dynamic and so are the underlying physical models and spacetime. As the 4-d spacetime evolves dimension by dimension in the early universe, consistent yet different models emerge one by one with different sets of particles and interactions. A new set of first principles are proposed for building such models with new understanding of supersymmetry, mirror symmetry, and the dynamic phase transition mechanism - spontaneous symmetry breaking. Under this framework, we demonstrate that different models with no theory of everything operate in a hierarchical yet consistent way at different phases or scenarios of the Universe. In particular, the arrow of time is naturally explained and the Standard Model of physics is elegantly extended to time zero of the Universe.

In this paper, we introduce an approach to the protein folding problem from the point of view of statistical physics. Protein folding is a stochastic process by which a polypeptide folds into its characteristic and functional 3D structure from random coil. The process involves an intricate interplay between global geometry and local structure, and each protein seems to present special problems. We introduce CSAW (conditioned self-avoiding walk), a model of protein folding that combines the features of self-avoiding walk (SAW) and the Monte Carlo method. In this model, the unfolded protein chain is treated as a random coil described by SAW. Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. Conceptually, the mathematical basis is a generalized Langevin equation. To illustrate the flexibility and capabilities of the model, we consider several examples, including helix formation, elastic properties, and the transition in the folding of myoglobin. From the CSAW simulation and physical arguments, we find a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. The elastic energy gives rise to scaling laws $R_{g}\sim N^ν$ in different regions with exponents $ν=3/5,3/7,2/5$, consistent with the observed unfolded stage, pre-globule, and molten globule, respectively. These results indicate that CSAW can serve as a theoretical laboratory to study universal principles in protein folding.

In this paper we propose a quadratic programming model that can be used for calculating the term structure of electricity prices while explicitly modeling startup costs of power plants. In contrast to other approaches presented in the literature, we incorporate the startup costs in a mathematically rigorous manner without relying on ad hoc heuristics. Moreover, we propose a tractable approach for estimating the startup costs of power plants based on their historical production. Through numerical simulations applied to the entire UK power grid, we demonstrate that the inclusion of startup costs is necessary for the modeling of electricity prices in realistic power systems. Numerical results show that startup costs make electricity prices very spiky. In the second part of the paper, we extend the initial model by including the grid operator who is responsible for managing the grid. Numerical simulations demonstrate that robust decision making of the grid operator can significantly decrease the number and severity of spikes in the electricity price and improve the reliability of the power grid.

Down regulation of mRNA translation is an important problem in various bio-medical domains ranging from developing effective medicines for tumors and for viral diseases to developing attenuated virus strains that can be used for vaccination. Here, we study the problem of down regulation of mRNA translation using a mathematical model called the ribosome flow model (RFM). In the RFM, the mRNA molecule is modeled as a chain of $n$ sites. The flow of ribosomes between consecutive sites is regulated by $n+1$ transition rates. Given a set of feasible transition rates, that models the outcome of all possible mutations, we consider the problem of maximally down regulating the translation rate by altering the rates within this set of feasible rates. Under certain conditions on the feasible set, we show that an optimal solution can be determined efficiently. We also rigorously analyze two special cases of the down regulation optimization problem. Our results suggest that one must focus on the position along the mRNA molecule where the transition rate has the strongest effect on the protein production rate. However, this rate is not necessarily the slowest transition rate along the mRNA molecule. We discuss some of the biological implications of these results.

In this paper we treat a cavity QED quantum computation. Namely, we consider a model of quantum computation based on n atoms of laser-cooled and trapped linearly in a cavity and realize it as the n atoms Tavis-Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schr{\" o}dinger equation of the model in the weak coupling regime to construct the controlled NOT gate in the case of n=2, and to construct the controlled-controlled NOT gate in the case of n=3 by making use of several resonance conditions and rotating wave approximation associated to them. We also present an idea to construct general quantum circuits. The approach is more sophisticated than that of the paper [K. Fujii, Higashida, Kato and Wada, Cavity QED and Quantum Computation in the Weak Coupling Regime, J. Opt. B : Quantum Semiclass. Opt. {\bf 6} (2004), 502]. Our method is not heuristic but completely mathematical, and the significant feature is based on a consistent use of Rabi oscillations.

We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code all relevant domain concepts. These concepts encode a large portion of the knowledge in any domain. Furthermore, researchers should write programs that produce proofs of the kind that human mathematicians write (and publish); this means proofs that might sometimes have mistakes; and this means making inferences that are sometimes invalid. This approach is meant to contrast with the historically dominant "bottom up" approach: coding fundamental types (typically sets), axioms and rules for (valid) inference, and building up from this foundation to the theorems of mathematical practice and to their outstanding questions. It is an important fact that the actual proofs that mathematicians publish in math journals do not look like the formalized proofs of Russell & Whitehead's Principia Mathematica (or modern computer systems like Lean that automate some of this formalization). We believe some "lack of rigor" (in mathematical practice) is human-like, and can and should be leveraged for ATP.

This article introduces the Ω counter, a frequency counter -- or a frequency-to-digital converter, in a different jargon -- based on the Linear Regression (LR) algorithm on time stamps. We discuss the noise of the electronics. We derive the statistical properties of the Ω counter on rigorous mathematical basis, including the weighted measure and the frequency response. We describe an implementation based on a SoC, under test in our laboratory, and we compare the Ω counter to the traditional Π and Λ counters. The LR exhibits optimum rejection of white phase noise, superior to that of the Π and Λ counters. White noise is the major practical problem of wideband digital electronics, both in the instrument internal circuits and in the fast processes which we may want to measure. The Ω counter finds a natural application in the measurement of the Parabolic Variance, described in the companion article arXiv:1506.00687 [physics.data-an].