Twofold triple systems with cyclic 2‐intersecting Gray codes

Given a combinatorial design with block set , the block‐intersection graph (BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . The i‐block‐intersection graph (i‐BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . In this paper, several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian 2‐BIGs and result in larger TTSs that also have Hamiltonian 2‐BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian 2‐BIGs (equivalently TTSs with cyclic 2‐intersecting Gray codes) as well as the complete spectrum for TTSs with 2‐BIGs that have Hamilton paths (i.e. for TTSs with 2‐intersecting Gray codes). In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel classes in arbitrary TTSs, and then construct larger TTSs with both cyclic 2‐intersecting Gray codes and parallel classes.     

Pedagogical alternatives for triple integrals: moving towards more inclusive and personalized learning

This paper is based on the presumption that teaching multiple ways to solve the same problem has academic and social value. In particular, we argue that such a multifaceted approach to pedagogy moves towards an environment of more inclusive and personalized learning. From a mathematics education perspective, our discussion is framed around pedagogical approaches to triple integrals seen in a standard multivariable calculus curriculum. We present some critical perspectives regarding the dominant and long-standing approach to the teaching of triple integrals currently seen in hegemonic calculus textbooks; and we illustrate the need for more diverse pedagogical methods. Finally, we take a constructive position by introducing a new and alternate pedagogical approach to solve some of the classical problems involving triple integrals from the literature through a simple application of integration by parts. This pedagogical alternative for triple integrals is designed to question the dominant one-size-fits-all approach of rearranging the order of integration and the privileging of graphical methods; and to enable a shift towards a more inclusive, enhanced and personalized learning experience.     

On tortkara triple systems

The commutator [a,b] = ab?ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T(a,b,c,d) which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T(a,b,c,d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product [a,b,c] = [[a,b],c] in a free Zinbiel algebra and use computer algebra to construct a relation TT(a,b,c,d,e) which implies every relation satisfied by [a,b,c] in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by [a,b,c] which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity n≥9.     

THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN H^＊-ALGEBRA^＊

We study the Banach-Lie group Ltaut(A) of Lie triple automorphisms of a complex associative H*-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder(A), are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut(A)0 = Aut(A) implying Ltder(A) = Der(A). In the finite-dimensional case Ltaut(A)0 is a direct product of Aut(A) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

Non-vanishing functions and Toeplitz operators on tube-type domains

We prove an index theorem for Toeplitz operators on irreducible tube-type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.     

二阶微分方程多点边值问题的三重正解

We consider the second-order differential equation u(t) + q(t)f(t,u(t),u (t)) = 0,0 t 1,subject to three-point boundry condition u(0) = 0,u(1) = a 0 u(ξ 0 ),or to m-point boundary conditionu (0) = m2 i=1 b i u (ξ i ),u(1) = m2 i=1 a i u(ξ i ).We show the existence of at least three positive solutions of the above multi-point boundary-value problem by applying a new fixed-point theorem introduced by Avery and Peterson.     

Revealing Triple‐Shape Memory Effect by Polymer Bilayers

Bilayer polymers that consist of two epoxy dual‐shape memory polymers of well‐separated glass transition temperatures have been synthesized. These bilayer epoxy samples exhibit a triple‐shape memory effect (TSME) with shape fixities tailorable by changing the ratio between the two layers. The triple‐shape fixities of the bilayer epoxy polymers can be explained by the balance of stress between the two layers. Based on this work, it is believed that the following three molecular design criterions should be considered in designing triple‐shape memory polymers with optimum TSME: 1) well‐separated thermal transitions, 2) a strong interface, and 3) an appropriate balance of moduli and relative ratios between the layers (or microphases).

     

On the gonality of stable curves

In this paper we use admissible covers to investigate the gonality of a stable curve C over $\mathbb{C}$. If C is irreducible, we compare its gonality to that of its normalization. If C is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension‐1 subscheme of the boundary of the target Hurwitz scheme.