Mathematics

The Collatz conjecture asserts that repeatedly iterating f(x)=(3x+1)/ 2 a(x) , where a(x) is the highest exponent for which 2 a(x) exactly divides 3x+1 , always lead to 1 for any odd positive integer x . Here, we present an arborescence graph constructed from iterations of g(x)=( 2 e(x) x−1)/3 , which is the inverse of f(x) and where x≢[0 ] 3 and e(x) is any positive integer satisfying 2 e(x) x−1≡[0 ] 3 , with [0 ] 3 denoting 0(mod3) . The integer patterns inferred from the resulting arborescence provide new insights into proving the validity of the conjecture.

The double orthogonal projection of the 4-space onto two mutually perpendicular 3-spaces is a method of visualization of four-dimensional objects in a three-dimensional space. We present an interactive animation of the stereographic projection of a hyperspherical hexahedron on a 3-sphere embedded in the 4-space. Described are synthetic constructions of stereographic images of a point, hyperspherical tetrahedron, and 2-sphere on a 3-sphere from their double orthogonal projections. Consequently, the double-orthogonal projection of a freehand curve on a 3-sphere is created inversely from its stereographic image. Furthermore, we show an application to a synthetic construction of a spherical inversion and visualizations of double orthogonal projections and stereographic images of Hopf tori on a 3-sphere generated from Clelia curves on a 2-sphere.

The following material was created with the idea of being used for an introductory fractional calculus course. A recapitulation of the history of fractional calculus is presented, as well as the different attempts at fractional derivatives that existed before current definitions. Properties of the gamma function, beta function and the Mittag-Leffler function are presented, which are fundamental pieces in the fractional calculus. The basic properties of Riemann-Liouville and Caputo fractional derivatives are presented, as well as their implementation to different functions. It also presents the Laplace transform of a fractional operator and an application to the fractional free fall problem.

Let k≥1 be a small fixed integer. The rational approximations ∣ ∣ p/q− π k ∣ ∣ >1/ q μ( π k ) of the irrational number π k are bounded away from zero. A general result for the irrationality exponent μ( π k ) will be proved here. The irrationality exponents for the even parameters 2k correspond to those for the even zeta constants ζ(2k) . The specific results and numerical data for a few cases k=2 and k=3 are also presented and explained.

The first estimate of the upper bound μ(π)≤42 of the irrationality measure of the number π was computed by Mahler in 1953, and more recently it was reduced to μ(π)≤7.6063 by Salikhov in 2008. Here, it is shown that π has the same irrationality measure μ(π)=μ(α)=2 as almost every irrational number α>0 .

We propose a notion of iterating functions f: X k →X in a way that represents recurrence relations of the form a n+k =f( a n , a n+1 ,..., a n+k−1 ) . We define a function as n -involutory when its n th iterate is the identity map, and discuss elementary group-theoretic properties of such functions along with their relation to cycles of their corresponding recurrence relations. Further, it is shown that a function f: X k →X that is 2-involutory in each of its k arguments (holding others fixed) is (k+1) -involutory.

Musical gestures connect the symbolic layer of the score to the physical layer of sound. I focus here on the mathematical theory of musical gestures, and I propose its generalization to include braids and knots. In this way, it is possible to extend the formalism to cover more case studies, especially regarding conducting gestures. Moreover, recent developments involving comparisons and similarities between gestures of orchestral musicians can be contextualized in the frame of braided monoidal categories. Because knots and braids can be applied to both music and biology (they apply to knotted proteins, for example), I end the article with a new musical rendition of DNA.

Using nonstandard analysis (NSA), the proof of the Laplace's formula is given. The usage of NSA reduces the intricacy of taking limit, and the crude line of the proof would be clearly seen, compared to the done with the rigorous classical calculus. We use very elementary tools of NSA.

The maximal number of cells in a polyomino with no four cells equally spaced on a straight line is determined to be 142. This is based on several partial results, each of which can be verified with computer assistance.

A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1 - F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1 PF2 = a^2. Jacob Bernoulli first described the lemniscate in 1694. The Fagnano discovered the double angle formula of the lemniscate(1718). The Euler extended the Fagnano's formula to a more general addition theorem(1751). The lemniscate function was subsequently proposed by Gauss around the year 1800. These insights are summarized by Jacobi as the theory of elliptic functions. A leaf function is an extended lemniscate function. Some formulas of leaf functions have been presented in previous papers; these included the addition theorem of this function and its application to nonlinear equations. In this paper, the geometrical properties of leaf functions at n = 2 and the geometric relation between the angle theta and lemniscate arc length l are presented using the lemniscate curve. The relationship between the leaf functions sleaf2(l) and cleaf2(l) is derived using the geometrical properties of the lemniscate, similarity of triangles, and the Pythagorean theorem. In the literature, the relation equation for sleaf2(l) and cleaf2(l) (or the lemniscate functions, sl(l) and cl(l)) has been derived analytically; however, it is not derived geometrically.