A note on conservative galaxies,Skolem systems,cyclic cycle decompositions,and Heffter arrays

The conservative number of a graph $G$ is the minimum positive integer $M$, such that $G$ admits an orientation and a labeling of its edges by distinct integers in $\{1,2,\dots ,M\}$, such that at each vertex of degree at least three, the sum of the labels on the in-coming edges is equal to the sum of the labels on the out-going edges. A graph is conservative if $M=\left|E\left(G\right)\right|$. It is worth noting that determining whether certain biregular graphs are conservative is equivalent to find integer Heffter arrays.In this work we show that the conservative number of a galaxy (a disjoint union of stars) of size $M$ is $M$ for $M\equiv 0$, $3(mod4)$, and $M+1$ otherwise. Consequently, given positive integers $m_1$, $m_2$, …, $m_n$ with $m_i\ge 3$ for $1\le i\le n$, we construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of infinitely many circulant graphs, generalizing a result of Bryant, Gavlas and Ling (2003). In particular, it allows us to construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of the complete graph $K_{2M+1}$, where $M={\sum }_{i=1}^n{m}_i$. Also, we prove necessary and sufficient conditions for the existence of a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of $K_{2M+2}?F$, where $F$ is a 1-factor. Furthermore, we give a sufficient condition for a subset of ${\mathbb{Z}}_v?\left\{0\right\}$ to be sequenceable.     

A generalization of Heffter arrays

In this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of ${\mathbb{Z}}_v$ of order $t$. A ${\mathrm{H}}_t\left(m,n;s,k\right)$ Heffter array over ${\mathbb{Z}}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in ${\mathbb{Z}}_v$ such that (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in {\mathbb{Z}}_v\backslash J$, either $x$ or $?x$ appears in the array; and (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e., with entries chosen in $\pm \left\{1,\text{…},?\frac{2nk+t}{2}?\right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by ${\mathrm{H}}_k\left(n;k\right)$. In particular, we prove that for $3\le k\le n$, with $k\ne 5$, there exists an integer ${\mathrm{H}}_k\left(n;k\right)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\left(\text{mod}4\right)$; (b) $k\equiv 2\left(\text{mod}4\right)$ and $n$ is even; (c) $k\equiv 0\left(\text{mod}4\right)$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.     

Tight Heffter Arrays Exist for all Possible Values

A tight Heffter array is an matrix with nonzero entries from such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from appears twice. We prove that exist if and only if both m and n are at least 3. If H has the property that all entries are integers of magnitude at most , every row and column sum is 0 over the integers, and H also satisfies ), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if . Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both are even.     

A family of pandiagonal bimagic squares based on orthogonal arrays

In this article we give a construction of pandiagonal bimagic squares by means of four‐dimensional bimagic rectangles, which can be obtained from orthogonal arrays with special properties. In particular, we show that there exists a normal pandiagonal bimagic square of order n⁴ for all positive integer n≥7 such that gcd(n, 30) = 1 , which gives an answer to problem 22 of Abe in [Discrete Math 127 (1994), 3–13]. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:427‐438, 2011     

Uniform two‐class regular partial Steiner triple systems

A 2‐class regular partial Steiner triple system is a partial Steiner triple system whose points can be partitioned into 2‐classes such that no triple is contained in either class and any two points belonging to the same class are contained in the same number of triples. It is uniform if the two classes have the same size. We provide necessary and sufficient conditions for the existence of uniform 2‐class regular partial Steiner triple systems.     

On the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

On a problem of Mariusz Meszka

We consider a problem due to Mariusz Meszka similar to the well-known conjecture of Marco Buratti. Does there exist a near-1-factor in the complete graph on _Zp$Z_p$, p$p$ is an odd prime, whose set of edge-lengths equals a given multiset L$L$? We establish several sufficient conditions for the answer to be yes.     

Resolvable even‐cycle systems with a 1‐rotational automorphism

In this article, necessary and sufficient conditions for the existence of a 1‐rotationally resolvable even‐cycle system of λK_v are given, which are eventually for the existence of a resolvable even‐cycle system of λK_v. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 394–407, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10058     

Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =A ^tX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.     

A remark on a theorem by Deligne

We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.

     

Plane Hydrodynamic flows with steady vorticity: a reduction theorem

Summary Besides steady plane flows and unsteady plane flows of constant and steady vorticity, there are only two simple types of plane hydrodynamic flows with steady vorticity. These two types of unsteady flows have steady streamlines that are parallel straight lines or concentric circles.

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Zusammenfassung Neben stationären und nichtstationären ebenen Strömungen konstanter und stationärer Wirbelstärke, gibt es nur zwei einfache Typen von ebenen hydrodynamischen Strömungen mit stationärer Wirbelstärke. Diese zwei Typen von nichtstätionaren Strömungen haben stationäre Stromlinien, welche aus parallelen Geraden oder aus konzentrischen Kreisen bestehen.

     

The spectrum of 1-rotational Steiner triple systems over a dicyclic group

The spectrum of values v for which a 1-rotational Steiner triple system of order v exists over a dicyclic group is determined.     

The critical case for a Berestycki-Lions theorem

We consider the existence of the ground states solutions to the following Schrdinger equation:△u+V(x)u=f(u),u∈H1(RN),where N 3 and f has critical growth.We generalize an earlier theorem due to Berestycki and Lions about the subcritical case to the current critical case.     

An extension of a construction of covering arrays

By Raaphorst et al, for a prime power $q$, covering arrays (CAs) with strength 3 and index 1, defined over the alphabet ${\mathbb{F}}_q$, were constructed using the output of linear feedback shift registers defined by cubic primitive polynomials in ${\mathbb{F}}_q\left[x\right]$. These arrays have $2q^3?1$ rows and $q^2+q+1$ columns. We generalize this construction to apply to all polynomials; provide a new proof that CAs are indeed produced; and analyze the parameters of the generated arrays. Besides arrays that match the parameters of those of Raaphorst et al, we obtain arrays matching some constructions that use Chateauneuf‐Kreher doubling; in both cases these are some of the best arrays currently known for certain parameters.     

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.     

Almost convergence and a core theorem for double sequences

The idea of almost convergence was introduced by Moricz and Rhoades [Math. Proc. Cambridge Philos. Soc. 104 (1988) 283-294] and they also characterized the four dimensional strong regular matrices. In this paper we define the M-core for double sequences and determine those four dimensional matrices which transform every bounded double sequence x=[x_jk] into one whose core is a subset of the M-core of x.     

Deformation of hypersurfaces preserving the Möbius metric and a reduction theorem

A hypersurface without umbilics in the (n+1)$(n+1)$-dimensional Euclidean space f:Mⁿ→Rⁿ⁺¹$f:M^n\to R^{n+1}$ is known to be determined by the Möbius metric g and the Möbius second fundamental form B   up to a Möbius transformation when n?3$n?3$. In this paper we consider Möbius rigidity for hypersurfaces and deformations of a hypersurface preserving the Möbius metric in the high dimensional case n?4$n?4$. When the highest multiplicity of principal curvatures is less than n−2$n-2$, the hypersurface is Möbius rigid. When the multiplicities of all principal curvatures are constant, deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a reduction theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future.