Homomorphisms of matrix algebras and constructions of Butson–Hadamard matrices

An $n\times n$ matrix $H$ is Butson–Hadamard if its entries are $k$th roots of unity and it satisfies $H{H}^1=n{I}_n$. Write $\mathrm{BH}(n,k)$ for the set of such matrices.Suppose that $k=p^{\alpha }{q}^{\beta }$ where $p$ and $q$ are primes and $\alpha \ge 1$. A recent result of Östergård and Paavola uses a matrix $H\in \mathrm{BH}(n,pk)$ to construct $H^{\prime }\in \mathrm{BH}(pn,k)$. We simplify the proof of this result and remove the restriction on the number of prime divisors of $k$. More precisely, we prove that if $k=mt$, and each prime divisor of $k$ divides $t$, then we can construct a matrix $H^{\prime }\in \mathrm{BH}(mn,t)$ from any $H\in \mathrm{BH}(n,k)$.     

Saturation of Berge hypergraphs

Given a graph $F$, a hypergraph is a Berge- $F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subhypergraph that is a Berge-$F$, but for any edge $e\in E\left(\overline{H}\right)$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars and matchings in $k$-uniform hypergraphs.     

On Classification of Modular Tensor Categories

We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S-matrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Sect. 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Sect. 6.     

Non-quadratic defect energy: A comparison of gradient plasticity simulations to discrete dislocation dynamics results

A small-deformation strain gradient plasticity (GP) model for single-crystals has been proposed in [1], including a grain boundary (GB) yield condition without hardening. It has been extended by a hardening term for the GBs after a comparison to discrete dislocation dynamics (DDD) results in [2]. Differences between the strain gradients of the GP results and the DDD results motivate the consideration of a non-quadratic defect energy [3] in the GP model. It is shown that the gradients in the GP model can be improved using an exponent different from two. Remaining discrepancies in the strain profiles, compared to the DDD results, are attributed to the neglect of the individual gradients of plastic slip and due to the lack of a mechanism for the misorientation-dependent elastic interactions of dislocations across GBs [4] in the GP model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

REVERSIBLE SKELETAL SUBSTITUTION REACTIONS INVOLVING GROUP 13 HETEROPHOSPHAZENES

An unusual, reversible skeletal substitution reaction involving boron- and aluminum-based heterophosphazenes is described.     

Variable Charge-Densities in Hydrophilic Weak-Base Ion-Exchange Membranes. V. Equilibrium Properties with Pure 1-1 Electrolytes. Swelling and Ion-Exclusion

Abstract

The equilibrium distribution of chloride or nitrate counter-ions of K or NH₄ co-ions and of water is determined experimentally at different degrees of ionization X[xbar] of hydrophylic weak-base aminated polyvinyl-alcohol membranes. These data are analyzed and explained in terms of thermodynamics interrelating the molality of fixed-charge densities and of counter-ions to that of the amount of water and of diffused salt in the membrane.

Three domains have to be considered: the polyelectrolyte domain with small external concentration compared to the internal net fixed charge concentration (notion of “net-charge” densities), the domain of homogeneous interstitial liquid of the concentrated electrolyte type with sufficiently concentrated salt solution, and the intermediate domain.     

Stability constants of metal complexes of macrocyclic ligands with pendant donor groups

Abstract

In our studies of the stability constants of metal complexes, we have investigated a number of macrocyclic ligands with pendant donor groups. The ligands are characterized by the fact that they have nitrogen donors in the macrocyclic ring and oxygen or sulfur donors in the pendant arms. These ligands represent seven different macrocycles, and by varying the pendant donor groups, ten different ligands are indicated. The affinities of these ligands for fifteen metal ions will be described. The Fe(III) complex of triazanonane with o-hydroxypyridyl or o-hydroxybenzyl pendant donor groups are the most stable ferric complexes ever reported. The In(III) complex of triazacyclononane with pendant mercaptoethyl donor groups, is exceptionally stable. Also, the Ca(II) complex of DOTA probably has the highest stability of any calcium(II) complex. These, and other comparisons will be made on the basis of the thermodynamic stability constant data for the ligands described.