Search Results (13)

A method for obtaining integer sequences is presented by defining simple rules for the evolution of a discrete dynamical system. This paper demonstrates that various Catalan-like recurrences of known integer sequences can be obtained from a single stochastic process defined by simple rules. The resulting exact equations that describe the stationary state of the process are derived using combinatorial analysis. A specific reduction of the process is applied, and the solvability of the reduced system of equations is demonstrated. Then, a procedure for providing appropriate parameters for a given sequence is formulated. The general method is illustrated with examples of Catalan, Motzkin, Schröder, and A064641 integer sequences. We also point out that by appropriately changing the parameters of the system, one can smoothly transition between distributions related to Motzkin numbers and shifted Catalan numbers. Full article

In this paper, we solve three Diophantine equations: $2^x\pm {\left(2^{k}p\right)}^y=z^2$ and $-2^x+{\left(2^{k}3\right)}^y=z^2$ with $k\ge 0$ and prime $p\equiv \pm 3$$(mod8)$. We obtain all the non-negative integer solutions by using elementary methods and the database of elliptic curves in “The L-functions and modular forms database” (LMFDB). Full article

Here, we describe the bicomplex $\left(p,q\right)$-Fibonacci numbers and the bicomplex $\left(p,q\right)$-Fibonacci quaternions based on these numbers to show that bicomplex numbers are not defined the same as bicomplex quaternions. Then, we give some of their equations, including the Binet formula, generating function, Catalan, Cassini, and d’Ocagne’s identities, and summation formulas for both. We also create a matrix for bicomplex $\left(p,q\right)$-Fibonacci quaternions, and we obtain the determinant of a special matrix that gives the terms of that quaternion. With this study, we get a general form of the second-order bicomplex number sequences and the second-order bicomplex quaternions. In addition, we show that these two concepts, defined as the same in many studies, are different. Full article

Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of $\{1,2,\dots ,N\}$, are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system. Full article

Temperature and viscosity are essential parameters in medicine, environmental science, smart materials, and biology. However, few fluorescent sensor publications mention the direct relationship between temperature and viscosity. Three anthracene carboxyimide-based fluorescent molecular rotors, 1DiAC∙Cl, 2DiAC∙Cl, and 9DiAC∙Cl, were designed and synthesized. Their photophysical properties were studied in various solvents, such as N, N-dimethylacetamide, N, N-dimethylformamide, 1-propanol, ethanol, dimethyl sulfoxide, methanol, and water. Solvent polarizability resulted in a solvatochromism effect for all three rotors and their absorption and emission spectra were analyzed via the Lippert–Mataga equation and multilinear analysis using Kamlet–Taft and Catalán parameters. The rotors exhibited red-shifted absorption and emission bands in solution on account of differences in their torsion angle. The three rotors demonstrated strong fluorescence in a high-viscosity environment due to restricted intramolecular rotation. Investigations carried out under varying ratios of water to glycerol were explored to probe the viscosity-based changes in their optical properties. A good linear correlation between the logarithms of fluorescence intensity and solution viscosity for two rotors, namely 2DiAC∙Cl and 9DiAC∙Cl, was observed as the percentage of glycerol increased. Excellent exponential regression between the viscosity-related temperature and emission intensity was observed for all three investigated rotors. Full article

This work aims to explore the usefulness of graphic design in awareness campaigns promoting sustainable tourist destinations and to identify their contribution to the success of the campaigns in terms of their generating increased protection of the natural and socioeconomic resources of the destination. The study applies semiotics to the field of social marketing to build a conceptual model that relates the graphic design of a campaign to public environmental awareness, and to the destination’s preservation. In order to test the conceptual model, the campaign “Que la montagne est belle!” of the “Parc Naturel Régional des Pyrénées catalanes” in the French Pyrenees is taken as a case study for analysis, as it aims to preserve the park’s natural environment and pastoral activities. The data are analysed using the partial least squares structural equation modelling technique (PLS-SEM), and the results are studied for different segments of the sample. The findings show that the graphic design semiotics influence public environmental awareness and destination preservation by generating in the audience a sensitive, emotional, and cognitive reaction towards the campaign. This innovative framework on graphic design can be adapted to other branding or marketing campaigns to improve destination images. Full article

In this work, the experimental solubility of ethyl candesartan in the selected solvents within the temperature ranging from 278.15 to 318.15 K was studied. It can be easily found that the solubility of ethyl candesartan increases with the rising temperature in all solvents. The maximum solubility value was obtained in N,N-dimethylformamide (DMF, 7.91 × 10⁻²), followed by cyclohexanone (2.810 × 10⁻²), 1,4-dioxanone (2.69 × 10⁻²), acetone (7.04 × 10⁻³), ethyl acetate (4.20 × 10⁻³), n-propanol (3.69 × 10⁻³), isobutanol (3.38 × 10⁻³), methanol (3.17 × 10⁻³), n-butanol (3.03 × 10⁻³), ethanol (2.83 × 10⁻³), isopropanol (2.69 × 10⁻³), and acetonitrile (1.15 × 10⁻²) at the temperature of 318.15 K. Similar results of solubility sequence from large to small were also obtained in other temperatures. The X-ray diffraction analysis illustrates that the crystalline forms of all samples were consistent, and no crystalline transformation occurred during the dissolution process. In aprotic solvents, except for individual solvents, the solubility data decreases with the decreasing values of hydrogen bond basicity (β) and dipolarity/polarizability (π*). The largest average relative deviation (ARD) data in the modified Apelblat equation is 1.9% and observed in isopropanol; the maximum data in λh equation is 4.3% and found in n-butanol. The results of statistical analysis show that the modified Apelblat equation is the more suitable correlation of experimental data for ethyl candesartan in selected mono solvents at all investigated temperatures. In addition, different parameters were used to quantify the solute–solvent interactions that occurred in the dissolution process including Abraham solvation parameters (AP_i), Hansen solubility parameters (HP_i), and Catalan parameters (CP_i). Full article

Surface tension is among the most important factors in chemical and pharmaceutical processes. Modeling the surface tension of solvents at different temperatures helps to optimize the type of solvent and temperature. The surface tension of solvents at different temperatures with their solvation parameters was used in this study to develop a model based on the van’t Hoff equation by multiple linear regression. Abraham solvation parameters, Hansen solubility parameters, and Catalan parameters are among the most discriminating descriptors. The overall MPD of the model was 3.48%, with a minimum and maximum MPD of 0.04% and 11.62%, respectively. The model proposed in this study could be useful for predicting the surface tension of mono-solvents at different temperatures. Full article

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) $T_1,T_2,\dots ,T_n$. In this case, for $1\le j\le n-1$, two consecutive tiles $T_j$ and $T_{j+1}$ share exactly one edge, either the edge at the east (west) of $T_j$ ($T_{j+1}$) or the edge at the north (south) of $T_j$ ($T_{j+1}$). Finding the number of perfect matchings associated with a given snake graph is one of the most remarkable problems regarding these graphs. It is worth noting that such a number of perfect matchings allows a bijection between the set of snake graphs and the positive continued fractions. Furthermore, perfect matchings of snake graphs have also been used to find closed formulas for cluster variables of some cluster algebras and solutions of the Markov equation, which is a well-known Diophantine equation. Recent results prove that snake graphs give rise to some string modules over some path algebras, connecting snake graph research with the theory of representation of algebras. This paper uses this interaction to define Brauer configuration algebras induced by schemes associated with some multisets called polygons. Such schemes are named Brauer configurations. In this work, polygons are given by some admissible words, which, after appropriate transformations, permit us to define sets of binary trees called groves. Admissible words generate codes whose energy values are given by snake graphs. Such energy values can be estimated by using Catalan numbers. We include in this paper Python routines to compute admissible words (i.e., codewords), energy values of the generated codes, Catalan numbers and dimensions of the obtained Brauer configuration algebras. Full article

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation, which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants, such as Catalan’s constant C and $\pi$. Full article

In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given $y=f\left(x\right)$, we provide the power series that defines its inverse $x=h_f\left(y\right)$. We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss–Catalan numbers. Full article

We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established. Full article