The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each...

A polynomial Pythagorean–hodograph (PH) curve r(t) = (x1(t), . . . , xn(t)) in R n is characterized by the property that its derivative components satisfy the Pythagorean condition x′2 1 (t)+ · · ·+x ′2 n (t) = σ 2(t) for some polynomial σ(t), ensuring that the arc length s(t) = ∫ σ(t) dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R2 and...

The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if A,B,C are relatively prime integers such that A...

We propose a method based on finite mixture models for classifying a set of observations into number of different categories. In order to demonstrate the method, we show how the component densities for the mixture model can be derived by using the maximum entropy method in conjunction with conservation of Pythagorean means. Several examples of distributions belonging to the Pythagorean family a...

We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, scalar product, orthogonality, linear functional are derivable from statistical description micro-events Hilbertian sum squares $|\mathfrak{a}_1|^2+|\mathfrak{a}_2|^2+\cdots$. latter leads (non-axiomatically) to standard writing Born formula $\mathtt{f} = |\langle\psi|\varphi\rangle...

Let n be a number. We say that n is square if and only if: (Def. 3) There exists m such that n = m2. Let us note that every number which is square is also natural. Let n be a natural number. Note that n2 is square. Let us observe that there exists a natural number which is even and square. Let us observe that there exists a natural number which is odd and square. Let us mention that there exist...

The paper considers the need to develop some applied pedagogical issues of general theory modeling, as well psychological and problems solved by modeling in learning process, particular, formation concepts, interpretation sign mathematical models solution educational graphic problems, where does not act an additional method current verbal logical method. gives examples visualization planimetry,...

This paper summarizes basic properties of PPTs and shows that each PPT belongs to one of six different classes. Mapping an ordered sequence of PPTs into a corresponding sequence of these six classes makes it possible to use them in cryptography. We pose problems whose solution would facilitate such cryptographic application. Introduction A Pythagorean triple (a, b, c) consists of positive integ...