A complete solution to the infinite Oberwolfach problem

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. We will also consider the same problem in the more general contest of graphs $F$ that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when $F$ is locally finite. Moreover, we characterize the infinite subgraphs $L$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(L)$

Frame difference families and resolvable balanced incomplete block designs

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $(pq+1,p+1,1)$-RBIBDs over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ are derived, and the existence of $(125q+1,6,1)$-RBIBDs is discussed. We construct $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976,5720,5776,10200,14176,24480\}$, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.Comment: arXiv admin note: text overlap with arXiv:1702.0750

New 22-designs from strong difference families

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to $2$-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of $(\mathbb{F}_{p}\times \mathbb{F}_{q},\mathbb{F}_{p}\times \{0\},k,\lambda)$-DFs for $k\in\{p,p+1\}$. We improve Buratti's existence results for $2$-$(13q,13,\lambda)$ designs and $2$-$(17q,17,\lambda)$ designs, and establish the existence of seven new $2$-$(v,k,\lambda)$ designs for $(v,k,\lambda)\in\{(694,7,2),(1576,8,1),(2025,9,1),(765,9,2),(1845,9,2),(459,9,4)$, $(783,9,4)\}$.Comment: Version 1 is named "Improved cyclotomic conditions leading to new 2-designs: the use of strong difference families". Major revision according to the referees' comment

On the Bandwidth of High-Impedance Frequency Selective Surfaces

In this letter, the bandwidth of high-impedance surfaces (HISs) is discussed by an equivalent circuit approach. Even if these surfaces have been employed for almost 10 years, it is sometimes unclear how to choose the shape of the frequency selective surface (FSS) on the top of the grounded slab in order to achieve the largest possible bandwidth. Here, we will show that the conventional approach describing the HIS as a parallel connection between the inductance given by the grounded dielectric substrate and the capacitance of the FSS may induce inaccurate results in the determination of the operating bandwidth of the structure. Indeed, in order to derive a more complete model and to provide a more accurate estimate of the operating bandwidth, it is also necessary to introduce the series inductance of the FSS.We will present the explicit expression for defining the bandwidth of a HIS, and we will show that the reduction of the FSS inductance results in the best choice for achieving wide operating bandwidth in correspondence with a given frequency.Comment: 4 pages, 6 figure

A Circuit-based Model for the Interpretation of Perfect Metamaterial Absorbers

A popular absorbing structure, often referred to as Perfect Metamaterial Absorber, comprising metallic periodic pattern over a thin low-loss grounded substrate is studied by resorting to an efficient transmission line model. This approach allows the derivation of simple and reliable closed formulas describing the absorption mechanism of the subwavelength structure. The analytic form of the real part of the input impedance is explicitly derived in order to explain why moderate losses of the substrate is sufficient to achieve matching with free space, that is, perfect absorption. The effect of the constituent parameters for tuning the working frequency and tailoring the absorption bandwidth is addressed. It is also shown that the choice of highly capacitive coupled elements allows obtaining the largest possible bandwidth whereas a highly frequency selective design is achieved with low capacitive elements like a cross array. Finally, the angular stability of the absorbing structure is investigated.Comment: Accepted for publication on IEEE Transactions on Antennas and Propagatio

A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures