Bożek, Bogusław
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Item type:Article, Access status: Open Access , Quaternionic Quantum Mechanics: the Particles, Their q-Potentials and Mathematical Electron Model(AGH University Press, 2026) Bożek, Bogusław; Danielewski, Marek; Sapa, LucjanIn this work we show the quaternionic quantum descriptions of physical processes from the Planck to macro scale. The results presented here are based on the concepts of the Cauchy continuum and the elementary cell at the Planck scale. The structurally symmetric quaternion relations and the postulate of the quaternion velocity have been important in the present development. The momentum of the expansion and compression $u̇_{0}(t, x)$ is the consequence of the scalar term $\sigma_{0}(t, x)$ in the quaternionic deformation potential. The quaternionic $G_{0}(m)(σ_{0} + \hat{\textstyle \phi} )$, vectorial $G_{0}(m) \hat{\textstyle \phi}$ and scalar $G_{0}(m)\sigma_{0}$ propagators are used to generate the second order PDE systems for the proton, electron and neutron. A mathematical model of an electron is formulated. It is described by the hyperbolic-elliptic partial differential system of quaternion equations with the initial-boundary conditions. The boundary conditions are generated by the quaternion energy flux that is found with the use of the Gauss theorem, the Cauchy–Riemann derivative and other mathematical formulas. The rigorous assessment of the second order PDE systems allows the proposal of two second order PDE systems for the $u$ and $d$ quarks from the up and down groups. It was verified that both the proton and the neutron obey experimental findings and are formed by three quarks. The proton and neutron are formed by the $d$-$u$-$u$ and $d$-$d$-$u$ complexes, respectively. The u and d quarks do not comply with the Cauchy equation of motion. The inconsistencies of the quarks’ PDE with the quaternion forms of the Cauchy equation of motion account for their short lifetime and the observed Quarks Chains. That is, they explain the Wilczek phenomenological paradox: Quarks are Born Free, but everywhere they are in Chains.Item type:Article, Access status: Open Access , A note on a family of quadrature formulas and some applications(2008) Bożek, Bogusław; Solak, Wiesław; Szydełko, ZbigniewIn this paper a construction of a one-parameter family of quadrature formulas is presented. This family contains the classical quadrature formulas: trapezoidal rule, mid-point rule and two-point Gauss rule. One can prove that for any continuous function there exists a parameter for which the value of quadrature formula is equal to the integral. Some applications of this family to the construction of cubature formulas, numerical solution of ordinary differential equations and integral equations are presented.Item type:Article, Access status: Open Access , Calculation of distribution of temperature in three-dimensional solid changing its shape during the process(2005) Bożek, Bogusław; Mączka, CzesławThe present paper suplements and continues [Bożek B., Filipek R., Holly K., Mączka C.: Distribution of temperature in three-dimensional solids. Opuscula Mathematica 20 (2000), 27-40]. Galerkin method for the Fourier–Kirchhoff equation in the case when $\Omega(t)$ – equation domain, dependending on time $t$, is constructed. For special case $\Omega(t) \subset \mathbb{R}^2$ the computer program for above method is written. Binaries and sources of this program are available on http://wms.mat.agh.edu.pl/~bozek.Item type:Article, Access status: Open Access , On some quadrature rules with Gregory end corrections(2009) Bożek, Bogusław; Solak, Wiesław; Szydełko, ZbigniewHow can one compute the sum of an infinite series $s := a_1 + a_2 + \ldots$? If the series converges fast, i.e., if the term $a_{n}$ tends to $0$ fast, then we can use the known bounds on this convergence to estimate the desired sum by a finite sum $a_1 + a_2 + \ldots + a_n$. However, the series often converges slowly. This is the case, e.g., for the series $a_n = n^{-t}$ that defines the Riemann zeta-function. In such cases, to compute $s$ with a reasonable accuracy, we need unrealistically large values $n$, and thus, a large amount of computation. Usually, the $n$-th term of the series can be obtained by applying a smooth function $f(x)$ to the value $n$: $a_n = f(n)$. In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum $R = f(n + 1) + f(n + 2) + \ldots$, we approximate this remainder by the corresponding integral $I$ of $f(x)$ (from $x = n + 1$ to infinity), and find good bounds on the difference $I - R$. First, we derive sixth order quadrature formulas for functions whose 6th derivative is either always positive or always negative and then we use these quadrature formulas to get good bounds on $I - R$, and thus good approximations for the sum $s$ of the infinite series. Several examples (including the Riemann zeta-function) show the efficiency of this new method. This paper continues the results from [W. Solak, Z. Szydełko, <i>Quadrature rules with Gregory-Laplace end corrections</i>, Journal of Computational and Applied Mathematics 36 (1991), 251–253] and [W. Solak, <i>A remark on power series estimation via boundary corrections with parameter</i>, Opuscula Mathematica 19 (1999), 75–80].