arXiv subjects
Publications and source records attributed to Bojan Banjac.
In this paper, we generalize Cristinel Mortici's results on Wilker-Cusa-Huygens inequalities using stratified families of functions and SimTheP - a system for automated proving of MTP inequalities.
In the paper, we refine and extend Cusa-Huygens inequality by simple functions. In particular, we determine sharp bounds for $\sin(x) /x$ of the form $(2+\cos(x))/3 -(2/3-2/π)Υ(x)$, where $Υ(x) >0$ for $x\in (0, π/2)$, $Υ(0)=0$ and $Υ(π/2)=1$, such that $\sin x/x$ and the proposed bounds coincide at $x=0$ and $x=π/2$. The hierarchy of the obtained bounds is discussed, along with graphical study. Also, alternative proofs of the main result are given.
In this paper are given some estimates of precision of the Huygens approximation $x \approx \frac{2}{3} \sin x + \frac{1}{3} \tan x,$ for right neighbourhood of zero, by determining some boundaries for the Huygens function $f(x) = \frac{2}{3} \sin x + \frac{1}{3} \tan x,$ for $x \in \left(0, \fracπ{2} \right)$, in forms of some polynomial and some rational functions.
In this work the Erdos-Mordell's inequality is examined for the case of a triangle $ABC$ in the taxicab plane geometry. It is shown that the Erdos-Mordell's inequality $R_A + R_B + R_C \, \geq \, w \, (r_a + r_b + r_c)$ holds for triangles with appropriate positions for its points $A$, $B$ and $C$, if $w = 3/2$.
In this paper we present some new upper bounds of the Cusa-Huygens and the Huygens approximations. Bounds are obtained in the forms of some polynomial and some rational functions.
In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.
In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of appropriate inequalities, as well as allows direct establishment of the dependence between (the exponent of) some functions that occur as bounds of the approximation and the interval in which the corresponding inequality holds true.
This study considers rational approximations of musical constant $β=\log_2(3/2)$, which defines perfect fifth. This constant has been the subject of the numerous studies, and this paper determines quality of rational approximations in regards to absolute error. We analysed convergents and secondary convergents (some of these are the best Huygens approximations). Especially, we determined quality of the secondary convergents which are not the best Huygens approximations - in this paper we called them non-convergents approximations. Some of the microtonal scales have been positioned and determined by using non-convergents approximation of music constant which defines perfect fifth.
This paper presents a proof of the following conjecture, stated by Nishizawa in [Appl. Math. Comput. 269, (2015), 146--154.]: for $\displaystyle 0 \! \left(\frac{2}π + \frac{π\!-\!2}{π^{3}}(π^{2}\!-\!4x^{2})\right)^{θ(x)}\! $ holds, where $\displaystyle θ(x) \! = \! -\frac{(48\!-\!24π\!+\!π^{3})x^{3} }{3(π\!-\!2)π^{3}}+\frac{π^{3}}{24(π\!-\!2)}.$
In this paper we prove two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865--873], using a method for proving inequalities of mixed trigonometric polynomial functions.
In this paper we present our experience in using visualization in mathematics education. The experience with our university courses: "Computer tools in matematics" and "Symbolic algebra" provides the basis for mathematics teacher education program http://vizuelizacija.etf.rs/. The program is intended for elementary and high school teachers. The education program deals with modern techniques of visualization by using technologies such as GeoGegebra, JAVA and HTML.
In this paper we present a method for extraction of arcs of the algebraic curves of the higher order. Method is applied on conics, Cartesian ovals, trifocal curves and generalized Weber's curve.
In this paper we present geometry of some curves in Taxicab metric. All curves of second order and trifocal ellipse in this metric are presented. Area and perimeter of some curves are also defined.
In this paper we consider historical genesis of trifocal curve as an optimal curve for solving the Fermat's problem (minimizing the sum of distance of one point to three given points in the plane). Trifocal curves are basic plane geometric forms which appear in location problems. We also analyze algebraic equation of these curves and some of their applications in architecture, urbanism and spatial planning. The area and perimeter of trifocal curves are calculated using a Java application. The Java applet is developed for determining numerical value for the Fermat-Torricelli-Weber point and optimal curve with three foci, when starting points are given on an urban map. We also present an application of trifocal curves through the analysis of one specific solution in South Stream gas pipeline project.
In this paper we present modern texture mapping techniques and several applications of polynomial texture mapping in cultural heritage programs. We also consider some well-known and some new methods for mathematical procedure that is involved in generation of polynomial texture maps.