Family Of Isomorphisms Research Articles

Affine Hecke algebras and generalizations of quiver Hecke algebras of type B

AbstractWe define and study cyclotomic quotients of affine Hecke algebras of type B. We establish an isomorphism between direct sums of blocks of these algebras and a generalization, for type B, of cyclotomic quiver Hecke algebras, which are a family of graded algebras closely related to algebras introduced by Varagnolo and Vasserot. Inspired by the work of Brundan and Kleshchev, we first give a family of isomorphisms for the corresponding result in type A which includes their original isomorphism. We then select a particular isomorphism from this family and use it to prove our result.

Affine Hecke algebras of type D and generalisations of quiver Hecke algebras

We define and study cyclotomic quotients of affine Hecke algebras of type D. We establish an isomorphism between (direct sums of blocks of) these cyclotomic quotients and a generalisation of cyclotomic quiver Hecke algebras which are a family of Z-graded algebras closely related to algebras introduced by Shan, Varagnolo and Vasserot. To achieve this, we first complete the study of cyclotomic quotients of affine Hecke algebras of type B by considering the situation when a deformation parameter p squares to 1. We then relate the two generalisations of quiver Hecke algebras showing that the one for type D can be seen as fixed point subalgebras of their analogues for type B, and we carefully study how far this relation remains valid for cyclotomic quotients. This allows us to obtain the desired isomorphism. This isomorphism completes the family of isomorphisms relating affine Hecke algebras of classical types to (generalisations of) quiver Hecke algebras, originating in the famous result of Brundan and Kleshchev for the type A.

Design of non-isomorphic symmetric descendants of the Miura-ori

The Miura fold pattern, or the Miura-ori, is a flat-foldable origami pattern with various applications in engineering and architecture. In addition to free-form variations, scholars have proposed a number of symmetric derivatives for this classic fold pattern over recent years. In a previous work, the authors of this paper studied isomorphic variations on the Miura-ori which led to the development of an ‘isomorphic family’ for this fold pattern. In this paper, we study non-isomorphic variations on the Miura-ori in order to develop a ‘non-isomorphic family’ for this pattern. Again we start with the Miura-ori, but reduce the symmetry by migrating from the original symmetry group to its subgroups, which may also include the enlargement of its unit cell. We systematically design and classify the non-isomorphic symmetric descendants of the Miura-ori which are either globally planar, or globally curved, flat-foldable tessellations.

Overconvergent Eichler–Shimura isomorphisms

AbstractGiven a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

Isomorphisms and strictly singular operators in mixed Tsirelson spaces

We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces TM[(Sn,θn)n] satisfying some regularity conditions and present results on existence of strictly singular non-compact operators on subspaces of mixed Tsirelson spaces defined by the families (An)n and (Sn)n.

Notes on complex hyperbolic triangle groups

We first demonstrate a family of isomorphisms between complex hyperbolic triangle groups and outline a systematic approach classifying the groups. Then we describe conditions that determine the discreteness of certain groups, in particular we prove a slightly weaker version of a conjecture given by Schwartz. Finally we collect together a list of known discrete triangle groups and propose some good candidates for discrete groups.

A CHARACTERIZATION OF SECOND-ORDER DIFFERENTIAL OPERATORS ON FINITE ELEMENT SPACES

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

KSbO(Ge0.32Si0.68)O4, a KTP isomorph.

A structural model of potassium antimony germanate/silicate (0.32/0.68), KSbO(Ge(0.32)Si(0.68))O(4), has been determined at room temperature. KSbO(Ge(0.32)Si(0.68))O(4) belongs to the KTiOPO(4) (KTP) isomorphic family and is composed of SbO(6) octahedra (site symmetry -1 and 2) arranged in helical chains bridged by (Ge/Si)O(4) tetrahedra. Germanium and silicon have a similar distribution in the crystallographically independent tetrahedra (site symmetry 2). The structure contains large cavities occupied by the K atom. Two partially occupied potassium positions have been identified 1.273 (8) A apart, with an indication of a third potassium position between them. At room temperature, KSbO(Ge(0.32)Si(0.68))O(4) crystallizes in the paraelectric phase of space group Pnan. This phase is found at elevated temperatures for almost all KTiOPO(4) isomorphic compounds and KSbO(Ge(0.32)Si(0.68))O(4) is the second isomorph that is paraelectric at room temperature.

Growth and raman scattering of aluminium gallium orthophosphate piezoelectric crystals

A new series of non-ferroelectric crystals, the piezoelectric solid solution aluminium gallium orthophosphate Al 1-xGa xPO 4 (AGP, 0 < x < 1) system has been successfully developed by a hydrothermal method. The AGP crystal with x = 0.10 shows a phase transition temperature T α− β = 587 ± 3 °C with a cell parameters a = b = 0.49386(3) and c = 1.09563(2) nm. Raman spectra including directional dispersion in AGP are similar to those from α-AIPO4 and α-quartz crystals, indicating all of them belong to an isomorphic family, the short-range interaction being dominant in AGP. Comparison with α-AIPO4 crystals, there exists an apparent red shift of the mode frequency and a smaller TO/LO mode splitting of the E species in the AGP crystals.

Approach Spaces, Limit Tower Spaces, and Probabilistic Convergence Spaces

The category LTS of limit tower spaces is defined and shown to be isomorphic to the category CAP of convergence approach spaces. The full subcategory of LTS determined by the objects satisfying a diagonal axiom due to Cook and Fischer is shown to be isomorphic to the category AP of approach spaces. A family of isomorphisms is also obtained between LTS and certain full subcategories of the category PCS of probabilistic convergence spaces.

The incompleteness of R�nyi entropies

Since the advent of the notion of mixing character in statistical mechanics, it has been conjectured over and over again that the Renyi entropies provide a mixing isomorphic family, that is: given two probability distributions p and q, the mixing character of q exceeds that of p, m[q]> m[p], if and only if Iα(q)≥Iα(p) for any positive α. This conjecture is disproved by means of counterexamples.

The use of the Clebsch-Gordan reduction of the Kronecker square of the typical representation in symmetry problems of crystal physics. I. Theoretical foundations

The theoretical background of a universal method for finite groups of calculating polynomial invariants and covariants (sets of functions transforming by non-identity representations) in the components of any tensor is considered. This includes the determination of the transformation properties of tensors, which is a question of determining linear covariants in the tensor components. The concepts of typical representation and typical variables, characteristic for an abstract group G and for the isomorphic family of groups of linear transformations imaged by G, are introduced. It is shown that the multilinear covariants in typical variables yield a prescription for the construction of tensor and polynomial covariants common for the whole isomorphic group family.

Nonlinear Optical Properties of Isomorphic Family of Crystals with Yttrium‐Aluminium Borate (YAB) Structure

AbstractThe second harmonics generation of Nd‐laser radiation (λ = 1.06 microns) of the isomorphic family of crystals with yttrium‐aluminium borate (YAB) structure — RAl3(BO3)4 (R = Y, Sm, Eu, Gd, Dy, Er, Ho, Tu) — investigated using the powder method and the theoretical calculation of SHG–coefficients — are reported. The refraction indices (for λ = 5481 Å) and optical sign for all studied YAB‐type crystals are determined with the help of the immersion method. All studied crystals possess PM direction and oee‐interaction. The SH‐intensity is of KDP order of magnitude, with reasonable correlation of experimental and theoretical data. Theoretical analysis of the contributions of different ions to the integral d magnitude has shown identical signs for all ion species in YAB structure type.