Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Doubles of Quasi-Quantum Groups

Abstract: In [Dr1] Drinfeld showed that any finite dimensional Hopf algebra \(\) extends to a quasitriangular Hopf algebra \(\), the quantum double of \(\). Based on the construction of a so-called diagonal crossed product developed by the authors in [HN], we generalize this result to the case of quasi-Hopf algebras \(\). As for ordinary Hopf algebras, as a vector space the “quasi-quantum double”\(\) is isomorphic to \(\), where \(\) denotes the dual of \(\). We give explicit formulas for the product, the coproduct, the R-matrix and the antipode on \(\) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi-Hopf algebra. In particular \(\) becomes an associative algebra containing \(\) as a quasi-Hopf subalgebra. On the other hand, \(\) is not a subalgebra of \(\) unless the coproduct on \(\) is strictly coassociative. It is shown that the category \(\) of finite dimensional representations of \(\) coincides with what has been called the double category of \(\)-modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi-quantum doubles in terms of a Tannaka–Krein-like reconstruction procedure. The whole construction is shown to generalize to weak quasi-Hopf algebras with \(\) now being linearly isomorphic to a subspace of \(\).

Function-Hiding Inner Product Encryption Is Practical

Abstract: In a functional encryption scheme, secret keys are associated with functions and ciphertexts are associated with messages. Given a secret key for a function f, and a ciphertext for a message x, a decryptor learns f(x) and nothing else about x. Inner product encryption is a special case of functional encryption where both secret keys and ciphertext are associated with vectors. The combination of a secret key for a vector \({\mathbf {x}}\) and a ciphertext for a vector \(\mathbf {y}\) reveal \(\langle {\mathbf {x}}, \mathbf {y}\rangle \) and nothing more about \(\mathbf {y}\). An inner product encryption scheme is function-hiding if the keys and ciphertexts reveal no additional information about both \({\mathbf {x}}\) and \(\mathbf {y}\) beyond their inner product.

Hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making

Abstract: Hamacher product is a t-norm and Hamacher sum is a t-conorm. They are good alternatives to algebraic product and algebraic sum, respectively. Nevertheless, it seems that most of the existing hesitant fuzzy aggregation operators are based on the algebraic operations. In this paper, we utilize Hamacher operations to develop some hesitant fuzzy aggregation operators. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy multiple attribute decision making problems. Finally, a practical example for evaluating the customer satisfaction of e-commerce websites is given to verify the developed approach and to demonstrate its practicality and effectiveness.