Family algebras and generalized exponents for polyvector representations of simple Lie algebras of type <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We give an explicit formula for the exterior powers ∧ ^k π ₁ of the defining representation π ₁ of the simple Lie algebra ?ο(2n + 1, ?). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ?). We also give a survey of main results on family algebras.     

Multiplicative representation of bilinear operators

We establish that each lattice bimorphism from the Cartesian product of two vector lattices into a universally complete vector lattice is representable as the product of two lattice homomorphisms defined on the factors. This fact makes it possible to reduce the problem to the linear case and obtain some results on representation of an order bounded disjointness preserving bilinear operator as a strongly disjoint sum of weighted shift or multiplicative operators.     

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation

Sobolev-type integral representations for functions defined on Carnot groups

We obtain some integral representations of the form f(x) = P(f) + K(?f) on the Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. These representations are employed to prove the weak Poincaré inequality.     

Remarks on Naimark's duality

We present an extension of a version of Naimark's dilation theorem which states that complete systems in a Hilbert space are projections of -linearly independent systems of elements of an ambient Hilbert space. This result is presented in the context of other known extensions of Naimark's theorem.

     

Maximum of the modulus of kernels in Gauss-Turán quadratures

We study the kernels in the remainder terms of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at , when the weight is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel attains its maximum on the real axis (positive real semi-axis) for each . It was stated as a conjecture in [Math. Comp. 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes in the corresponding Gauss-Turán quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each . Numerical examples are included.

     

Explicit rational solutions of Knizhnik-Zamolodchikov equation

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group _n . We assume that parameter ρ = ±1. In previous paper [5] we proved that the fundamental solution of the corresponding KZ-equation is rational. Now we construct this solution in the explicit form.      

Waste marble dust-filled sustainable polymer composite selection using a multi-criteria decision-making technique

This research works with the optimal design of marble dust-filled polymer composites using a multi-criteria decision-making (MCDM) technique. Polylactic acid (PLA) and recycled polyethylene terephthalate (rPET)-based composites containing 0, 5, 10, and 20 wt% of marble dust were developed and evaluated for various physicomechanical and wear properties. The results showed that the incorporation of marble dust improved the modulus and hardness of both PLA and rPET. Moreover, a marginal improvement in flexural strength was noted while the tensile and impact strength of the matrices were deteriorating due to marble dust addition. The outcomes of wear analysis demonstrated an improvement in wear resistance up until 10 wt% filler reinforcement, after which the incidence of dust particles peeling off from the matrix was observed, thereby reducing its efficiency. The best tensile modulus of 3.23 GPa, flexural modulus of 4.39 GPa, and hardness of 83.95 Shore D were obtained for 20 wt% marble dust-filled PLA composites. The lowest density of 1.24 g/cc and the highest tensile strength of 57.94 MPa were recorded for neat PLA, while the highest impact strength of 30.94 kJ/m² was recorded for neat rPET. The lowest wear of 0.01 g was obtained for the rPET containing 5 wt% marble dust content. The experimental results revealed that for the examined criteria, the order of composite preference is not the same. Therefore, the optimal composite was identified by adopting a preference selection index-based MCDM technique. The findings demonstrated that the 10 wt% marble dust-filled PLA composite appears to be the best solution with favorable physical, mechanical, and wear properties.     

Clifiord分析中双正则函数及调和函数在平面上的积分表示

本文研究了取值在Clifiord代数上双正则函数及调和函数在超复平面上积分表示的问题.利用构造核函数的方法,获得了双正则函数及调和函数在超复平面上的积分表示公式,这些结果推广了Clifiord分析中正则函数在超复平面上的积分表示公式.     

An equivariant index for proper actions III: The invariant and discrete series indices

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of ${\mathrm{Spin}}^{\mathrm{c}}$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete series representations of semisimple groups, where we assume the Riemannian metric to have a certain product form. The other is an index defined in terms of sections invariant under a group action. We obtain a relation with the analytic assembly map, quantisation commutes with reduction results, and Atiyah–Hirzebruch type vanishing theorems. The arguments are based on an explicit decomposition of ${\mathrm{Spin}}^{\mathrm{c}}$-Dirac operators with respect to a global slice for the action.