Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules

We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.

A New Method for Calculating Energy of Matter

An approximate calculation of the spatial characteristics on finite range is required, so one quantitative continuum represents the accumulation of infinite great quantities is artificially divided it into smaller and camparable parts in which calculus operation can be applied .This operation is defined as Theorem 1 in which infinity is not involved, there is a camparable finity is constantly (forever) approaching and not reaching infinity, and only staying within a finite range. Theorem 1 can exist in this paper as a new mathematical basis for physics. Because the essence of all physical quantities is size comparison, and the size comparison relation of matter can only be space/time, so relation formula space/time is the only expression of the concept of matter, all physical quantities are applicable to this expression, each different physical quantity is a multi-dimensional representation of this expression. A new mass energy formula is aslo derived from this paper.

First-Order Random Coefficient Multinomial Autoregressive Model for Finite-Range Time Series of Counts

In view of the complexity and asymmetry of finite range multi-state integer-valued time series data, we propose a first-order random coefficient multinomial autoregressive model in this paper. Basic probabilistic and statistical properties of the model are discussed. Conditional least squares (CLS) and weighted conditional least squares (WCLS) estimators of the model parameters are derived, and their asymptotic properties are established. In simulation studies, we compare these two methods with the conditional maximum likelihood (CML) method to verify the proposed procedure. A real example is applied to illustrate the advantages of our model.

A systematic bias in fitting the surface-density profiles of interstellar filaments

The surface-density profiles of dense filaments, in particular those traced by dust emission, appear to be well fit with Plummer profiles, i.e. Σ(b) = ΣB + ΣO{1 + [b/wO]2}[1 − p]/2. Here ΣB is the background surface-density; ΣB + ΣO is the surface-density on the filament spine; b is the impact parameter of the line-of-sight relative to the filament spine; wO is the Plummer scale-length (which for fixed p is exactly proportional to the full-width at half-maximum, $w_{{\rm O}}=\rm {\small FWHM}/2\lbrace 2^{2/[p-1]}-1\rbrace ^{1/2}$); and p is the Plummer exponent (which reflects the slope of the surface-density profile away from the spine). In order to improve signal-to-noise it is standard practice to average the observed surface-densities along a section of the filament, or even along its whole length, before fitting the profile. We show that, if filaments do indeed have intrinsic Plummer profiles with exponent pINTRINSIC, but there is a range of wO values along the length of the filament (and secondarily a range of ΣB values), the value of the Plummer exponent, pFIT, estimated by fitting the averaged profile, may be significantly less than pINTRINSIC. The decrease, Δp = pINTRINSIC − pFIT, increases monotonically (i) with increasing pINTRINSIC; (ii) with increasing range of wO values; and (iii) if (but only if) there is a finite range of wO values, with increasing range of ΣB values. For typical filament parameters the decrease is insignificant if pINTRINSIC = 2 (0.05 ≲ Δp ≲ 0.10), but for pINTRINSIC = 3 it is larger (0.18 ≲ Δp ≲ 0.50), and for pINTRINSIC = 4 it is substantial (0.50 ≲ Δp ≲ 1.15). On its own this effect is probably insufficient to support a value of pINTRINSIC much greater than pFIT ≃ 2, but it could be important in combination with other effects.

A New Method for Calculating Energy of Matter

An approximate calculation of the spatial characteristics on finite range is required, so one quantitative continuum represents the accumulation of infinite great quantities is artificially divided it into smaller and camparable parts in which calculus operation can be applied .This operation is defined as Theorem 1 in which infinity is not involved, there is a camparable finity is constantly (forever) approaching and not reaching infinity, and only staying within a finite range. Theorem 1 can exist in this paper as a new mathematical basis for physics. Because the essence of all physical quantities is size comparison, and the size comparison relation of matter can only be space/time, so relation formula space/time is the only expression of the concept of matter, all physical quantities are applicable to this expression, each different physical quantity is a multi-dimensional representation of this expression. A new mass energy formula is aslo derived from this paper.

Can Hubble Tension Be Eased by Invoking a Finite Range for Gravity?

The estimation of the Hubble constant in the past few decades has increasingly become more accurate with the advance of new techniques. But its value seems to depend on the epoch at which the measurements are made. The Planck estimate of the Hubble constant from the observations of the CMBR in the early Universe is about 67.

Do equidistant energy levels necessitate a harmonic potential?

Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only yields energy levels in a finite range and with a nonzero uncertainty. Within this study we review isospectral potentials from the literature and investigate the applicability of the harmonic oscillator hypothesis to recent measurements. First, we describe experimental results from literature by a harmonic oscillator model, obtaining a realistic size and depth of the resulting quantum well. Second, we use the shift-operator approach to calculate anharmonic non-polynomial potentials producing (partly) equidistant spectra. We discuss different potential types and interpret the possible modeling applications. Finally, by applying nth order perturbation theory we show that exactly equidistant eigenenergies cannot be achieved by polynomial potentials, except by the harmonic oscillator potential. In summary, we aim to give an overview over which conclusions may be drawn from the experimental determination of energy levels and which may not.