Non-extensive transverse momentum distribution for identified particles at SNN = 7.7, 11.5, 19.6, 27, 39GeV

The transverse momentum distribution of charged particles formed in Au–Au collisions at Beam Energy Scan (BES) ([Formula: see text][Formula: see text]GeV) is investigated. In addition, [Formula: see text] spectra of [Formula: see text] particle at [Formula: see text][Formula: see text]GeV were examined. Tsallis distribution is used to extract the temperature, volume and the entropic index from the experimental results at mid-rapidity and zero chemical potential. We measure some particle ratios like [Formula: see text] and [Formula: see text] which are puzzling horn in the experiment and in the thermal model. We conclude that the horn vanished when we used Tsallis distribution, but this does not confirm a solution to the puzzle, which is primarily visible in the experimental results.

Dual addition formulas: the case of continuous q-ultraspherical and q-Hermite polynomials

We settle the dual addition formula for continuous q-ultraspherical polynomials as an expansion in terms of special q-Racah polynomials for which the constant term is given by the linearization formula for the continuous q-ultraspherical polynomials. In a second proof we derive the dual addition formula from the Rahman–Verma addition formula for these polynomials by using the self-duality of the polynomials. We also consider the limit case of continuous q-Hermite polynomials.

Force chain structure in a rod-withdrawn granular layer

When a rod is vertically withdrawn from a granular layer, oblique force chains can be developed by effective shearing. In this study, the force-chain structure in a rod-withdrawn granular layer was experimentally investigated using a photoelastic technique. The rod is vertically withdrawn from a two-dimensional granular layer consisting of bidisperse photoelastic disks. During the withdrawal, the development process of force chains is visualized by the photoelastic effect. By systematic analysis of photoelastic images, force chain structures newly developed by the rod withdrawing are identified and analyzed. In particular, the relation between the rod-withdrawing force [Formula: see text], total force-chains force [Formula: see text], and their average orientation [Formula: see text] are discussed. We find that the oblique force chains are newly developed by withdrawing. The force-chain angle [Formula: see text] is almost constant (approximately [Formula: see text] from the horizontal), and the total force [Formula: see text] gradually increases by the withdrawal. In addition, [Formula: see text] shows a clear correlation with [Formula: see text].

Using Area Method in Secondary School Geometry

Areas are actively used in the solution of many geometrical problems. In this work,smart and laconic solutions are found for various problems by means of the area method. The well-known trigonometric addition formula is also proved. In the area method, the given formulas are divided into the parts, whose areas are then calculated using problem data.

Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

Time Dilation and the Equivalence of Inertial Frames

We have compared the data of three clocks A, B and D moving in relative uniform motion with relative speed/velocity between A and B set at 0.6c, relative speed/velocity between A and D set at 0.8c and relative speed between B and D set at (5c/13) = 0.3846c as per the velocity addition formula (0.8-0.6)/(1-0.8*0.6). We have compared the time readings of the clocks when they meet at three events. Event 1 meeting of A and B, Event 2 meeting of A and D, Event 3 meeting of B and D.

Time Dilation and the Equivalence of Inertial Frames

We have compared the data of three clocks A, B and D moving in relative uniform motion with relative speed/velocity between A and B set at 0.6c, relative speed/velocity between A and D set at 0.8c and relative speed between B and D set at (5c/13) = 0.3846c as per the velocity addition formula (0.8-0.6)/(1-0.8*0.6). We have compared the time readings of the clocks when they meet at three events. Event 1 meeting of A and B, Event 2 meeting of A and D, Event 3 meeting of B and D.

Ionic conductivity of monohydroxy alcohol liquids doped with known probe ions and analysis of the liquid structure

The specific values of the conductivity [Formula: see text] and its variation with temperature [Formula: see text] of 2-pentanol and 2-methyl-1-pentanol liquids doped with different concentrations of NaI (abbreviated as 2PEN-[Formula: see text]NaI and 2M1PEN-[Formula: see text]NaI, respectively) ([Formula: see text], 1% and 5%) are measured by the commercial equipment. The results show that whether NaI is doped or not, [Formula: see text] of 2PEN-[Formula: see text]NaI and 2M1PEN-[Formula: see text]NaI all have a conductivity peak in the range of 300–350 K different from that of typical glass-formers of small molecules. In addition, [Formula: see text] goes up with increasing [Formula: see text], and the temperature corresponding to [Formula: see text] maximum increases with rising NaI content. Moreover, there is a nonlinear behavior of [Formula: see text] with [Formula: see text], i.e., with rising [Formula: see text], [Formula: see text] decreases at low-temperatures, but increases at high-temperatures. Moreover, the liquid structure and its variation with [Formula: see text] are further analyzed based on the ionic conductivity.

Polynomial and power series ring extensions from sequences

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.