0000000000652383
AUTHOR
Marco Pavone
A C-free approach to linear ODEs with constant coefficients
The purpose of this paper is to present an alternative way of establishing the general solution of the equation x''(t)+x(t)=0 without relying on complex numbers. Our method can be extended to the corresponding non-homogeneous equation and, more generally, to higher-order equations.
An algebraic representation of Steiner triple systems of order 13
Abstract In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V = GF ( 5 ) 13 , with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V , which can also be characterized as the intersections of B with thirteen linear hyperplanes of V .
Additive Steiner triple systems
A Steiner triple system is additive if it can be embedded in a commutative group in such a way that the sum of the three points in any given block is zero. In this paper we show that a Steiner triple system is additive if and only if it is the point-line design of either a projective space PG(d,2) over GF(2) or an affine space AG(d,3) over GF(3), for d ≥ 1. Our proof is based on algebraic arguments and on the combinatorial characterization of finite projective geometries in terms of Veblen points.
The overland flow equation for constant rainfall excess: an evaluation of runoff volume and time to equilibrium
In this paper we study the runoff volume V(t) per unit area generated by the overland flow in the time interval [0,t], under a stationary rainfall r and initially dry conditions. For all positive values of the rating exponent m we express V(t) in closed form in terms of the solution q(t) of the overland flow equation. We define a simpler, approximated value of V(t) and show that, for m≥1, the error is smaller than a quantity of the form c e^{-pt}. Finally, for 1≤m≤3, we find an explicit upper bound for the time to equilibrium t_e, by showing that q(t) differs from the equilibrium outflow r by a quantity of the form c e^{-pt}.
On the solutions of the differential overland flow equation
In this paper we study the overland flow equation for an arbitrary positive value of the rating exponent m. We write the general solution of the equation and generalize the series solution given in [1] and [2]. Finally, we show how the five solutions presented in [5] are actually a special case of a general formula valid for any rational m≥1.
1-(v,3,r) designs and the equation x+y+z=0 in finite abelian groups
Let (G, +) be a finite abelian group with more than three elements and let B_3 be the family of all the unordered triples {x,y,z} of distinct elements of G such that x+y+z=0. We show that (G, B_3) is a 1-(v,3,r) design if and only if G is either an elementary abelian 3-group or the direct sum of Z/2Z with an elementary abelian 3-group. We also characterize the groups containing at least an element that does not belong to any triple in B_3
On the 2-(25, 5, λ) design of zero-sum 5-sets in the Galois field GF(25)
In this paper we consider the incidence structure ${\mathcal{D}}=({\mathcal{F}},{\mathcal{B}}_5^{0}),$ where ${\mathcal{F}}$ is the Galois field with $25$ elements, and ${\mathcal{B}}_5^{0}$ is the family of all $5$-subsets of $\mathcal F$ whose elements sum up to zero. It is known that ${\mathcal{D}}$ is a $2$-$(25,5,71)$ design. Here we provide two alternative, direct proofs of this result and, moreover, we prove that ${\mathcal{D}}$ is not a $3$-design. Furthermore, if ${\mathcal{B}}_5^{x}$ denotes the family of all $5$-subsets of $\mathcal F$ whose elements sum up to a given element $x \in \mathcal F,$ we also provide an alternative, direct proof that $({\mathcal{F}},{\mathcal{B}}_5^{x}…
On the additivity of block designs
We show that symmetric block designs $${\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})$$D=(P,B) can be embedded in a suitable commutative group $${\mathfrak {G}}_{\mathcal {D}}$$GD in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of $${\mathrm {PG}}(d,2)$$PG(d,2) and $${\mathrm {AG}}(d,3)$$AG(d,3). In both cases, the blocks can be characterized as the only k-subsets of $$\mathcal {P}$$P whose elements sum to zero. It follows that the group of automorphisms of any such design $$\mathcal {D}$$D is the group of automorphisms of $${\mathfrak {G}}_\mathcal {D}$$GD that leave $$\mathcal {P}$$P in…
On the fixed ends of hyperbolic translations of infinite graphs
Let X be an infinite, connected, locally finite and vertex-transitive graph with infinitely many ends and let G be a subgroup of Aut(X) which acts transitively on X. In this note we provide a necessary and sufficient condition for the existence of a hyperbolic translation g in G with fixed ends in two prescribed open subsets of the space of ends of X. We also give an explicit combinatorial construction of the hyperbolic translation g in the special case where X is a (right) Cayley graph of a (non-abelian) free group of finite type G.
On the hyperbolic limit points of groups acting on hyperbolic spaces
We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg +↦g −, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG.
Binary Hamming codes and Boolean designs
AbstractIn this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we…
On the subset sum problem for finite fields
Abstract Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G, by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper we give an alternative proof of the two formulas. Our argument is purely combinatorial, as in the original proof by Li and Wan, but follows a different and somehow more “natural” approach. We also indicate some new connections with coding theory and combinatorial designs.
Kirkman's tetrahedron and the fifteen schoolgirl problem
We give a visual construction of two solutions to Kirkman's fifteen schoolgirl problem by combining the fifteen simplicial elements of a tetrahedron. Furthermore, we show that the two solutions are nonisomorphic by introducing a new combinatorial algorithm. It turns out that the two solutions are precisely the two nonisomorphic arrangements of the 35 projective lines of PG(3,2) into seven classes of five mutually skew lines. Finally, we show that the two solutions are interchanged by the canonical duality of the projective space.
Permutations of zero-sumsets in a finite vector space
Abstract In this paper, we consider a finite-dimensional vector space 𝒫 {{\mathcal{P}}} over the Galois field GF ( p ) {\operatorname{GF}(p)} , with p being an odd prime, and the family ℬ k x {{\mathcal{B}}_{k}^{x}} of all k-sets of elements of 𝒫 {\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for x = 0 {x=0} , of the permutations of 𝒫 {\mathcal{P}} inducing permutations of ℬ k 0 {{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space 𝒫 {\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space 𝒫 {\mathcal{P}} if p divides k. The same question is answered also in the case where …
Additivity of affine designs
We show that any affine block design $$\mathcal{D}=(\mathcal{P},\mathcal{B})$$ is a subset of a suitable commutative group $${\mathfrak {G}}_\mathcal{D},$$ with the property that a k-subset of $$\mathcal{P}$$ is a block of $$\mathcal{D}$$ if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design $$\mathcal{D}$$ is the group of automorphisms of $${\mathfrak {G}}_\mathcal{D}$$ that leave $$\mathcal P$$ invariant. Whenever k is a prime p, $${\mathfrak {G}}_\mathcal{D}$$ is an elementary abelian p-group.
On the weight distribution of perfect binary codes
In this paper, we give a new proof of the closed-form formula for the weight distribution of a perfect binary single-error-correcting code.