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].

In a recent paper (quant-ph/0506105), A S Gupta, M. Gupta and A. Pathak proposed a modified Grover algorithm that would exponentially accelerate the unsorted database search problem if the number of marked items is known. If this were true, it would represent a major fundamental breakthrough in computer science, mathematics, quantum information and other related branches of sciences. However the algorithm is not valid. We will explain it in this brief comment.

We provide an overview of the hybrid compositional distributional model of meaning, developed in Coecke et al. (arXiv:1003.4394v1 [cs.CL]), which is based on the categorical methods also applied to the analysis of information flow in quantum protocols. The mathematical setting stipulates that the meaning of a sentence is a linear function of the tensor products of the meanings of its words. We provide concrete constructions for this definition and present techniques to build vector spaces for meaning vectors of words, as well as that of sentences. The applicability of these methods is demonstrated via a toy vector space as well as real data from the British National Corpus and two disambiguation experiments.

The data describing a process of echo-image formation in bottlenose dolphin sonar perception were accumulated in our experimental explorations. These data were formalized mathematically and used in the computational model, comparative testing of which in echo-discrimination tasks revealed no less capabilities then those of bottlenose dolphins.

Q-variance (so-called) posits a statistical relationship $\mathbf{E}(σ^2 | z) = σ_0^2 + \tfrac{1}{2}z^2$ between an asset's volatility $σ^2$, as observed in a time interval $T$, and its (suitably scaled) return $z$ in the same interval. We here show that this relationship is {\em exactly equivalent} to to positing an Inverse Gamma probability distribution for $σ^2$ itself. We then show that such a distribution is exactly generated by a multiplicative Langevin process with an arbitrary, settable coherence time $τ_c$, so that very nearly the same Q-variance relationship will hold for all $T \ll τ_c$.

We address the two issues raised by Bayle, Vallisneri, Babak, and Petiteau (in their gr-qc document arXiv:2106.03976) about our matrix formulation of Time-Delay Interferometry (TDI) (arXiv:2105.02054) \cite{TDJ21}. In so doing we explain and quantify our concerns about the results derived by Vallisneri, Bayle, Babak and Petiteau \cite{Vallisneri2020} by applying their data processing technique (named TDI-$\infty$) to the two heterodyne measurements made by a two-arm space-based GW interferometer. First we show that the solutions identified by the TDI-$\infty$ algorithm derived by Vallisneri, Bayle, Babak and Petiteau \cite{Vallisneri2020} {\underbar {do}} depend on the boundary-conditions selected for the two-way Doppler data. We prove this by adopting the (non-physical) boundary conditions used by Vallisneri {\it et al.} and deriving the corresponding analytic expression for a laser-noise-canceling combination. We show it to be characterized by a number of Doppler measurement terms that grows with the observation time and works for any time-dependent time delays. We then prove that, for a constant-arm-length interferometer whose two-way light times are equal to twice and three-times the sampling time, the solutions identified by TDI-$\infty$ are linear combinations of the TDI variable $X$. In the second part of this document we address the concern expressed by Bayle {\it et al.} regarding our matrix formulation of TDI when the two-way light-times are constant but not equal to integer multiples of the sampling time. We mathematically prove the homomorphism between the delay operators and their matrix representation \cite{TDJ21} holds in general. By sequentially applying two order-$m$ Fractional-Delay (FD) Lagrange filters of delays $l_1$, $l_2$ we find its result to be equal to applying an order-$m$ FD Lagrange filter of delay $l_1 + l_2$.

The mathematical concept of q-deformations, in particular the one of qnumbers, is used to study the genetic code(s). After considering two kinds of q-numbers, for comparison, a phenomenological classification scheme of the genetic code together with its numerous minor variants is, first, established. Next, numbers describing the presence of additional amino acids, such as Selenocysteine or/and Pyrrolysine, are also produced. Finally, a minimal number of amino acids, which could fit the small number of them which are thought to have been involved, at the origin of life on Earth, is found. All together, these results constitute our final semi-phenomenological model.

In the framework of the crystal basis model of the genetic code, where each codon is assigned to an irreducible representation of $U_{q \to 0}(sl(2) \oplus sl(2))$, single base mutation matrices are introduced. The strength of the mutation is assumed to depend on the "distance" between the codons. Preliminary general predictions of the model are compared with experimental data, with a satisfactory agreement.