Eugen Ionascu

Columbus State University | CSU · Department of Mathematics

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.

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...

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 that involve the primes and so...

Class Notes

We investigate the analog of the circle of Apollonious in spher- ical geometry. This can be viewed as the pre-image through the stereo- graphic projection of an algebraic curve of degree three. This curve con- sists of two connected components each being the “reflection” of the other through the center of the sphere. We give an equivalent equation...

We investigate the analog of the circle of Apollonious in spherical geometry. This can be viewed as the pre-image through the stereographic projection of an algebraic curve of degree three. This curve consists of two connected components each being the ``reflection" of the other through the center of the sphere. We give an equivalent equation for i...

In this paper, we present an algorithm which allows us to search for all the bisections for the binomial coefficients {[Formula presented]}k=0,…,n and include a table with the results for all n≤154. Connections with previous work on this topic are included. We conjecture that the probability of having only trivial solutions is 5∕6.

Finding all (non-trivial) choices of signs in sum_{i=0}^n (\pm) binomial(n,i) =0

In this paper, we continue the study of the problem of bisecting binomial coefficients. We present an algorithm which allowed us to search for nontrivial solutions bisecting the binomial coefficients $\{\binom{n}{k} \}_{k=0,...,n}$ well beyond $n=50$. More data is included and we provide the exact formulae for two more infinite sequences for which...

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...

In Euclidean geometry the circle of Apollonious is the locus of points in the plane from which two collinear adjacent segments are perceived as having the same length. In Hyperbolic geometry, the analog of this locus is an algebraic curve of degree four which can be bounded or "unbounded". We study this locus and give a simple description of this c...

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 o...

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 o...

In this paper we provide a new parametrization for the diophantine equation A² +B² +C² = 3D² 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 are constructing integer lattice squares, cubes or hypercubes in R ⁿ with n ∈ {2, 3, 4}. We nd a complete description of their Ehrhart polynomial. We characterize all the integer squares in R ⁴ , in terms of two Pythagorean quadruple representations of the form a ² + b ² + c ² = d ² , and then prove a parametrization in terms of t...

In this paper, we prove a counting result for the number of polynomials with integer coefficients bounded by a positive integer n and having all roots integers.

In this paper, we prove a counting result for the number of polynomials with integer coefficients bounded by a positive integer n and having all roots integers.

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 will explore the solutions to the diophantine equation in the Erd\H{o}s-Straus conjecture. For $n \geq 2$ we are discussing the relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$$ We will separate the types of solutions into two cases. In particular we will argu...

In this article, we consider the Erdős-Straus conjecture in a more general setting. For instance, one can look at the diophantine equation (Forumal presented) where n and a, b, c are Gaussian integers. We have considered this problem in the case of rings of integers of the norm-Euclidean quadratic fields. Without any other restrictions on a, b and...

We go through a series of results related to the k-signum equation \(\pm 1^k\pm 2^k\pm\cdots\pm n^k=0\). We are investigating the number S k (n) of possible writings and the asymptotic behavior of these numbers, as k is fixed and \(n\to \infty\). The results are presented in connections with the Erdös–Surányi sequences. Analytic methods and algebra...

In this paper we introduce several natural sequences related to poly-nomials of degree s having coefficients in {1, 2, ..., n} (n ∈ N) which factor completely over the integers. These sequences can be seen as generalizations of A006218. We provide precise methods for calculating the terms and investigate the asymptotic behavior of these sequences f...

Representations of primes by simple quadratic forms, such as ±a2±qb2, 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 � 20. Some of the results can be established with elementary methods and we exemplify that in some instances. We are introduci...

In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in ℤ3. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of the triangle. We give some equivalent formula in terms of a parametrization of these objects which allows one to constru...

First, we calculate the Ehrhart polynomial associated with an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedra and those for regular lattice octahedra. These relations allow one to reduce the calculation of these polynomials to o...

The main aim of this paper is to describe a procedure for calculating the number of cubes that have coordinates in the set {0,1,...,n}. For this purpose we continue and, at the same time, revise some of the work begun in a sequence of papers about equilateral triangles and regular tetrahedra all having integer coordinates for their vertices. We ada...

We characterize all three point sets in ℝ4 with integer coordinates, the pairs of which are the same Euclidean distance apart. In three dimensions, the problem is characterized in terms of solutions of the Diophantine equation a 2 + b 2 + c 2 = 3d 2. In ℝ4, our characterization is essentially based on two different solutions of the same equation. T...

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. W...

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...

In this paper we introduce theoretical arguments for constructing a procedure that allows one to find the number of all regular tetrahedra that have coordinates in the set {0, 1, ..., n}. The terms of this sequence are twice the values of the sequence A103158 in the Online Encyclopedia of Integer Sequences [16]. These results lead to the considerat...

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 $d=\sqrt{\frac{a^2+b^2+c^2}{3}}$...

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 prec...

TextExtending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates (“in Z3Z3”), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the same property. To summarize, we show first that there is no regular icosahedron/ dodecahedron in Z3Z3. On the other...

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.

For various examples of geometric probabilities problems we construct a simulation and experimentally check the theoretical result.

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 triangl...

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving binomial coefficients and Fibonacci type sequences. Comment: 7 pages

In this paper we describe a procedure for calculating the number of regular octahedrons that have vertices with coordinates in the set {0,1,...,n}. As a result, we introduce a new sequence in ``The Online Encyclopedia of Integer Sequences" (A178797) and list the first one hundred terms of it. We adapt the method appeared in [11] which was used to f...

In this paper we describe a procedure of calculating the number cubes that have coordinates in the set {0,1,...,n}. We adapt the code that appeared in [11] developed to calculate the number of regular tetrahedra with coordinates in the set {0,1,...,n}. The idea is based on the theoretical results obtained in [13]. We extend then the sequence A09892...

We are interested in constructing concrete independent events in purely atomic probability spaces with geometric distribution. Among other facts we prove that there are many uncountable sequences of independent events.

Paul Erdos conjectured that for every n in N, n>1, there exist a, b, c natural numbers, not necessarily distinct, so that 4/n=1/a+1/b+1/c (see \cite{rg}). In this paper we prove an extension of Mordell's theorem and formulate a conjecture which is stronger than Erdos' conjecture. Comment: 9 pages, no figures

In this note we describe a procedure of calculating the number all regular tetrahedra that have coordinates in the set {0,1,...,n}. We develop a few results that may help in finding good estimates for this sequence which is twice A103158 in the Online Encyclopedia of Integer Sequences. Comment: 21 pages, 6 figures, preprint

Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the same property. To summarize, we show first that there is no regular icosahedron/ dodecahedron in $\mathbb Z^3$....

TextIn this note we characterize all regular tetrahedra whose vertices in R3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates [R. Chandler, E.J. Ionascu, A characterization of all equilateral triangles in Z3, Integers 8 (2008), Article A19]. Previous work on t...

We study the half-dependent problem for the king graph K n . We give proofs to establish the values h(K n ) for n∈{1,2,3,4,5,6} and an upper bound for h(K n ) in general. These proofs are independent of computer assisted results. Also, we introduce a two-player game whose winning strategy is tightly related with the values h(K n ). This strategy is...

Define a(k,q) to be the smallest positive multiple of k such that the sum of its digits in base q is equal to k. The asymptotic behavior, lower and upper bound estimates of a(k,q) are investigated. A characterization of the minimality condition is also considered.

We describe a procedure of counting allequilateral triangles in the three dimensional space whosecoordinates are allowed only in the set {0, 1, ..., n }³. Thissequence is denoted here by ET ( n ) and it has the entry A102698in "The On-Line Encyclopedia of Integer Sequences". The procedure is implemented in Maple and its main idea is base...

We introduce a two-player game in which one and his/her opponent attempt to pack as many ``prisoners'' as possible on the squares of an n-by-n checkerboard; each prisoner has to be ``protected'' by at least as many guards as the number of the other prisoners adjacent. Initially, the board is covered entirely with guards. The players take turns adju...

We describe a procedure of counting all equilateral triangles in the three dimensional space whose coordinates are allowed only in the set {0, 1, ..., n}. This sequence is denoted here by ET(n) and it has the entry A102698 in "The On-Line Encyclopedia of Integer Sequences". The procedure is implemented in Maple and its main idea is based on the res...

In this note we characterize all regular tetrahedra whose vertices in R^3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates contained in previous work. Then we use this characterization to point out some corollaries. The number of such tetrahedra whose vertices...

In this short note we introduce two new sequences defined using the sum of digits in the representation of an integer in a certain base. A connection to Niven numbers is proposed and some results are proven.

This paper is a continuation of the work started by the second author in a series of papers [see J. Integer Seq. 10, No. 6, Article 07.6.7, 17 p., electronic only (2007; Zbl 1140.11322) and Acta Math. Univ. Comen., New Ser. 77, No. 1, 129–140 (2008; Zbl 1164.11016)]. We extend to the general case the characterization previously found for those equi...

We determine the maximum and minimum area of all the rectangle ABCD having three of the vertices on fixed circles. It turns out that the last vertex is on a fixed circle too. As a corollary we show that if m, n, p, q be arbitrary positive integers, and x = mn + pq, z = |mq − np|, y = |mn − pq| and t = mq + np, then taking circles with radii x, y,...

We describe a procedure of counting all equilateral triangles in the three dimensional space whose coordinates are allowed only in the set $\{0,1,...,n\}$. This sequence is denoted here by ET(n) and it has the entry A102698 in "The On-Line Encyclopedia of Integer Sequences". The procedure is implemented in Maple and its main idea is based on the re...

These are some lecture notes for the Calculus I course. It deals with fundamental limits first and the rules of differentiation for all the elementary functions. The proofs of the fundamental limits are based on the differential calculus developed in general and the definitions of exp(), ln(), sin(),cos(), etc.

In this paper we prove some results which imply two conjectures proposed by Janous on an extension to the p-th power-mean of the Erdös-Debrunner inequality relating the areas of the four sub-triangles formed by connecting three arbitrary points on the sides of a given triangle.

We show that the class of (dyadic) wavelet sets is in one-to-one correspondence to a special class of Lebesgue measurable isomorphisms of [0, 1) which we call wavelet induced maps. We then define two natural classes of maps WI1 and WI2 which, in order to simplify their construction, retain only part of the characterization properties of a wavelet i...

Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider...

In this paper, we study the function H(a,b), which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e., it has integral area. In particular, we prove that H(p,q)⩽5 if p and q are primes, and that H(a,b)=0 for a random choice of positive integers a and b.

We study k-dependence and half domination problems for king's graphs in dimension n (n>1). Various sharp bounds are provided and a few conjectures are formulated in the cases the estimates are not the best possible.

We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also show a similar characterization for the sides of a regular tetrahedron in $\Z^3$: such a...

In this paper, we study the function $H(a,b)$, which associates to every pair of positive integers $a$ and $b$ the number of positive integers $c$ such that the triangle of sides $a,b$ and $c$ is Heron, i.e., has integral area. In particular, we prove that $H(p,q)\le 5$ if $p$ and $q$ are primes, and that $H(a,b)=0$ for a random choice of positive...

We develop a technique to compute asymptotic expansions for recurrent sequences of the form an+1 = f(an), where f(x) = x - axα + bxβ + o(xβ) as x → 0, for some real numbers α, β, a, and b satisfying a > 0, 1 < α < β. We prove a result which summarizes the present stage of our investigation, generalizing the expansions in [Amer. Math Monthly, Proble...

We associate a von Neumann algebra with each pair of complete wandering vectors for a unitary system. When this algebra is nonatomic, there is a norm - continuous path of a simple nature connecting the original pair of wandering vectors. We apply this technique to wavelet theory and compute the above von Neumann algebra in some special cases. Resul...

We study rank-one perturbations of diagonal Hilbert space operators mainly from the standpoint of invariant subspace problem. In addition to proving some general properties of these operators, we identify the normal operators and contractions in this class. We show that two well known results about the eigenvalues of rank-one perturbations and one-...

A wandering set for a map ϕ is a set containing precisely one element from each orbit of ϕ. We study the existence of Borel wandering sets for piecewise linear isomorphisms. Such sets need not exist even when the parameters involved are rational, but they do exist if in addition all the slopes are powers of 2. For ϕ having at most one discontinuity...

In this note we give a characterization of subwavelet sets and show that any point x ε ℝ\0 has a neighborhood which is contained in a regularized wavelet set.

It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L2(ℝ) of MSF wavelets for which the Fourier transform has support con...

We consider systems of unitary operators on the complex Hilbert spaceL2(Rn) of the form U≔UDA,Tv1,…,Tvn≔{DmTl1v1…Tlnvn:m,l1,…,ln∈Z}, whereDAis the unitary operator corresponding to dilation by ann×nreal invertible matrixAandTv1,…,Tvnare the unitary operators corresponding to translations by the vectors in a basis {v1,…,vn} for Rn. Orthonormal wavel...

In this paper we generalize the following consequence of a well- known result of Nagy: if T and T 1 are power bounded operators, then T is a polynomially bounded operator.

The aim of the present paper is to define a joint spectrum for an arbitrary finite family of permutable paraclosed transformations, which extends the corresponding concept introduced by J. L. Taylor for commuting bounded operators. The authors prove that the projection property still holds in this context and characterize the commutation of the str...

This is a personal (unfinished) approach I have on the teaching of Calculus. It requires a lot of more work.