In this paper we show that each factorization structure M on a small category X , satisfying certain conditions, yields a presheaf M on X and a morphism of presheaves m :. 鈭掆啋 M. We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on X (b) subobjects of M (c) morphisms from M to (d) weak Lawvere-Tierney topologies (e) weak Grothendieck topologies (f) closure operators on Sets X op .
In this article the notion of quasi right factorization structure in a category X is given. The main result is a one to one correspondence between certain classes of quasi right factorization structures and 2-reflective subobjects of a predefined object in Lax(PrOrd X op ). Also a characterization of quasi right factorization structures in terms of images is given. As an application, the closure operators are discussed and it is shown that quasi closed members of certain collections are quasi right factorization structures. Finally several examples are furnished.
In this paper, we develop compositional vector-based semantics of positive transitive sentences using quantum natural language processing (Q-NLP) to compare the parametrized quantum circuits of two synonymous simple sentences in English and Persian. We propose a protocol based on quantum long short-term memory (Q-LSTM) for Q-NLP to perform various tasks in general but specifically for translating a sentence from English to Persian. Then, we generalize our method to use quantum circuits of sentences as an input for the Q-LSTM cell. This enables us to translate sentences in different languages.Our work paves the way toward representing quantum neural machine translation, which may demonstrate quadratic speedup and converge faster or reaches a better accuracy over classical methods.
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