arXiv subjects
Publications and source records attributed to Eugen J. Ionascu.
We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and analyzed.
We derive integral inequalities governing drainage time in convex solids, inspired by Torricelli's Law, and introduce the Torricelli number as a shape invariant. We use these considerations to construct a class of solids that can be used in building asymmetrical clepsydrae.
We find a nontrivial upper bound on the average value of the function M(n) which associates to every positive integer n the minimal Hamming weight of a multiple of n. Some new results about the equation M(n)=M(n') are given.
In this paper, we study a class of functions defined recursively on the set of natural numbers in terms of the greatest common divisor algorithm of two numbers and requiring a minimality condition. These functions are permutations, products of infinitely many cycles that depend on certain breaks in the natural numbers involving the primes, and some special products of primes with a density of approximately $29.4\%$. We show that these functions split into only two equivalence classes (modulo the natural equivalence relation of eventually identical maps): one is the class of the identity map and the other is generated by a map whose discrete derivative is almost periodic with ``periods" the primorial numbers.
In this paper, we present an algorithm which allows us to search for all the bisections for the binomial coefficients $\{\binom{n}{k} \}_{k=0,...,n}$ and include a table with the results for all $n\le 154$. Connections with previous work on this topic is included. We conjecture that the probability of having only trivial solutions is $5/6$. \end{abstract}
In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point $O$. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point $O$. The formulae provide another way to approach the Sylvester's Four-Point Problem as we show in the last section. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: $\frac{2}{27}+20\frac{\ln 2}{81}$. We compute this probability in the case of a regular polygon and its center of mass for the point $O$. Other families of regions are studied. For the family of Limaçons $r=a+\cos t$, $a>1$, and $O$ the origin of the polar coordinates, the probability is $\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3π^2}$.
In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n <= 51.
In this paper we are constructing integer lattice squares, cubes or hypercubes in $\mathbb R^d$ with $d\in \{2,3,4\}$. For squares and cubes we find a complete description of their Ehrhart polynomial. For hypercubes, we compute one of the coefficients and show that there exists a simple linear relation between the other two unknown coefficients. This allows as to formulate a conjecture of what the Ehrhart polynomial is in this case.
In this paper we provide a new parametrization for the diophantine equation $A^2+B^2+C^2=3D^2$ and give a series of corollaries. We discuss some connections with Lagrange's four-square theorem. As applications, we find new parameterizations of equilateral triangles and regular tetrahedrons having integer coordinates in three dimensions.
In this paper we make a Gaussian integer version of the Erdős-Straus conjecture and we solve the Erdős-Straus diophantine equation over the rings of integers of norm-Euclidean quadratic fields.
We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations $a^2+b^2+c^2=3d^2$. In $\mathbb R^4$, the characterization is essentially based on two different solutions of the same equation.
We use the idea of the broken stick problem (which goes back to Poincare) and calculate the corresponding probabilities for the cases in which the three broken part are: the medians in a triangle, the altitudes, radii of excircles, angle bisectors, distances from I or O to the vertices, respectively sides, and some other three elements in a triangle which determine (more or less uniquely) the triangle. For each case we also look at the probability that the triangle that is (more or less uniquely) defined by the elements, being acute and compare to that of being obtuse.
Representations of primes by simple quadratic forms, such as $\pm a^2\pm qb^2$, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gauss and many others. We are interested in a comprehensive list of such results, for $q\le 20$. Some of the results can be established with elementary methods and we put them at work on some instances. We are introducing new relationships between various representations.
We study the problem of half-domination sets of vertices in vertex transitive infinite graphs generated by regular or semi-regular tessellations of the plane. In some cases, the results obtained are sharp and in the rest, we show upper bounds for the average densities of vertices in half-domination sets.
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and those for regular lattice octahedrons. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of the polytope and it depends heavily on the value $d$ and its divisors, where $d=\sqrt{\frac{a^2+b^2+c^2}{3}}$ and $(a,b,c)$ ($\gcd(a,b,c)=1$) is a vector with integer coordinates normal to the plane containing the triangle.
In this note we introduce three problems related to the topic of finite Hausdorff moments. Generally speaking, given the first n+1 (n in N or n=0) moments, alpha(0), alpha(1),..., alpha(n), of a real-valued continuously differentiable function f defined on [0,1], what can be said about the size of the image of df/dx? We make the questions more precise and we give answers in the cases of three or fewer moments and in some cases for four moments. In the general situation of n+1 moments, we show that the range of the derivative should contain the convex hull of a set of n numbers calculated in terms of the Bernstein polynomials, x^k(1-x)^{n+1-k}, k=1,2,...,n, which turn out to involve expressions just in terms of the given moments alpha(i), i=0,1,2,...n. In the end we make some conjectures about what may be true in terms of the sharpness of the interval range mentioned before.
In this note we present solutions to two problems which appeared in the American Mathematical Monthly. Although the problems seem to be of different nature when it comes to the hypothesis we show that they can be proved using essentially the same technique. In the end, we introduce a new problem which actually implies both.