In this short note, we prove a version of the Johnson-Lindenstrauss flattening Lemma for point sets taking values in discrete subgroups. More precisely, given d , λ 0 , N 0 ∈ ℕ and 𝜖 ∈ ( 0 , 1 2 ) suitably chosen, we show there exists a natural number k = k ( d , 𝜖 ) = O ( 1 𝜖 2 log d ) , such that for every sufficiently large scaling factor λ ∈ ℕ and any point set 𝒟 ⊂ λ λ 0 ℤ d ∩ B ( 0 , λ N 0 ) with cardinality d , there exists an embedding F : 𝒟 → 1 λ 0 ℤ k , with distortion at most ( 1 + 𝜖 + 𝜖 λ λ 0 ) .
The so-called Sasaki projection was introduced by U. Sasaki on the lattice L(H) of closed linear subspaces of a Hilbert space H as a projection of L(H) onto a certain sublattice of L(H). Since L(H) is an orthomodular lattice, the Sasaki projection and its dual can serve as the logical connectives conjunction and implication within the logic of quantum mechanics. It was shown by the authors in a previous paper that these operations form a so-called adjoint pair. The natural question arises if this result can be extended also to lattices with a unary operation which need not be orthomodular or to other algebras with two binary and one unary operation. To show that this is possible is the aim of the present paper. We determine a variety of lattices with a unary operation where the Sasaki operations form an adjoint pair and we continue with so-called λ-lattices and certain classes of semirings. We show that the Sasaki operations have a deeper sense than originally assumed by their author and can be applied also outside the lattices of closed linear subspaces of a Hilbert space.
In this paper, concepts of (cofinitely) (D & lowast;)-modules which are a proper generalization of concept of circle plus delta-supplemented modules are studied. We say that M is a (D & lowast; 12 )-module if for 12 every submodule A of M, there exists a direct summand B of M and an epimorphism f : B-M such that ker ( f) << delta B. The module M is called cofinitely (D & lowast;)-module if for every cofiniteA 12 submodule A of M, there exists a direct summand B of M and an epimorphism f : B-MA such that ker ( f ) << delta B. In this paper, various properties of these modules are given. In addition, a new characterization of delta-semiperfect rings is given using cofinitely (D & lowast; )-modules. 12
Quantum calculus, also known as q-calculus, extends classical calculus with a deformation parameter q. The aim of this paper is to present new integral identities of q-trapezoidal and q-midpoint types for first-order q-differentiable functions. These identities are fundamental for establishing novel q-trapezoidal and q-midpoint type integral inequalities that can be applied to functions characterized by their first quantum derivatives and their absolute values being (11,12)-convex functions. To achieve this goal, the paper employs various mathematical tools, including the q-power mean inequality and q-Holder's inequality. These tools are essential for deriving and demonstrating the proposed inequalities, which play a crucial role in understanding and characterizing the behavior of functions within the framework of quantum calculus.
In this paper, we are interested in the solvability in closed-form of the following threedimensional system of nonlinear difference equations xn+1 = yn+1 = zn+1 = a1xn-kyn-k , axn-k + byn-k + czn-k a2yn-kzn-k , axn-k + byn-k + czn-k a3xn-kzn-k , axn-k + byn-k + czn-k where n, k is an element of N0, the parameters a1, a2, a3, a, b, c are real numbers, and the initial values x-k, ..., x0, y-k, . ..,y0, z-k, . .., z0, are non-zero real numbers. Firstly, we establish some preliminaries results for the general case, then we solve in a closed form, via some change of variables, our system in the two particular cases k = 0 and k = 1.
