We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

Spiral fibers were considered to be an ideal toughening phase of ultra-high torsional release effect. In this work, ZrB2 (Z)–20 vol% SiC (S) spiral fiber (ZSsf) with controllable structure was prepared by a combination approach liquid rope effect and non-solvent-induced separation. Dominantly depended on the kinematic viscosity (η), dropping height (H), flow ...

For a finite function class we describe the large sample limit of the sequential Rademacher complexity in terms of the viscosity solution of a G-heat equation. In the language of Peng’s sublinear expectation theory, the same quantity equals to the expected value of the largest order statistics of a multidimensional G-normal random variable. We illustrate this result by deriving upper and lower ...

In this thesis, we investigate two ways of generalising the preprojective algebra. First, we introduce the multiplicative preprojective algebra, Λ(Q), which is a multiplicative analogue of the deformed preprojective algebra, introduced by Crawley-Boevey and Holland. The special case q = 1 is the undeformed multiplicative preprojective algebra, which is an analogue of the ordinary (undeformed) p...

We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

and Applied Analysis 3 where ∗ denotes the corresponding symmetric terms, Σ 11 = Q 1 + τQ 2 − PC − CP − 2L 1 U 1 − 2M 1 U 4 , Σ 22 = − (1 − μ)Q 1 − 2L 1 U 2 , Σ 33 = Q 3 + τQ 4 − 2U 1 , Σ 44 = − (1 − μ)Q 3 − 2U 2 , Σ 55 = U 3 K (]) − 2U 4 , L 1 = diag (l− 1 l + 1 , . . . , l − n l + n ) , L 2 = diag (l− 1 + l + 1 , . . . , l − n + l + n ) , M 1 = diag (m− 1 m + 1 , . . . , m − n m + n ) , M 2 =...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra Bn(−q , q) and the quantum algebra associated to the symplectic Lie algebra sp2m. In particular, we deduce that this Schur–Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang ([37]) in the symplectic case. As a byproduct, we show tha...

We introduce a new algebra Bl(z, q) depending on two nonzero complex parameters such that Bl(q n , q) at q = 1 coincides with the Brauer algebra Bl(n). We establish an analog of the Brauer–Schur–Weyl duality where the action of the new algebra commutes with the representation of the twisted deformation Uq(on) of the enveloping algebra U(on) in the tensor power of the vector representation.