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.
This paper proves an equality for the case of differentiable functions involving the conformable fractional integrals. By using this equality, we establish new midpoint-type inequalities via convex functions with the aid of the conformable fractional integrals. Some significant inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Moreover, we give example using graph in order to show that our main results are correct. Furthermore, our results generalize known results from the literature by special choices.
In this paper, we give an identity involving Riemann–Liouville fractional integrals and partially differentiable function of two variables. Then, we use the newly established identity and prove some new Newton’s type inequalities for differentiable co-ordinated convex functions. We also recapture some existing inequalities and obtain some new inequalities in the special cases of newly established results. Furthermore, we present an example with graphs to illustrate the obtained main result.
In this paper, we investigate the solution of fractional system of Riccati equations. This system is important since it appears in several applications in science such as control theory. We use the operational matrix method to solve this system. The block-pulse operational matrices will initially assist in the reduction of the nonlinear fractional order Riccati-differential problem into an algebraic system. Benefits of this approach include inexpensive setup costs for the equations without the use of projection techniques like Galerkin, collocation, etc. In addition, we prove the convergence of the approximate solution using operational matrix method to the exact solution. Finally, we present two examples to provide numerical evidences to the efficiency of numerical approach used in this paper. We notice that the error is close to zero. Also, the approximate solutions are convergent to the exact solutions for different values of fractional derivative. Moreover, the approximate solutions approach to the solution of system of first order when the derivative order approaches to one.
In this work, amply r-supplemented modules are defined and some properties of these modules are investigated. It is proved that every factor module and every homomorphic image of an amply r-supplemented module are amply r-supplemented. Let M be a pi-projective and r-supplemented module. Then M is amply r-supplemented. Let M be an R-module. If every submodule of M is r-supplemented, then M is amply r-supplemented. Let R be any ring. Then every R-module is r-supplemented if and only if every R-module is amply r-supplemented. Let R be any ring. Then RR is (amply) r-supplemented if and only if every finitely generated R-module is (amply) r-supplemented.
In the theory of univalent analytic functions, the properties and characteristics of con-vex functions and starlike functions are avidely studied such as the estimates.on. Schippers higher-order Schwarzian derivatives at zero. In this paper, we consider the bounds for Schippers higher-order Schwarzian derivatives of f(z) at z = 0 when the function f belongs to the class of a kind of combination of convex functions and starlike functions, which consists of analytic functions f in the open unit disk D with the normalized conditions given by f(0) =f '(0)-1=0 and satisfying the following inequality: R((1)/(zF'(z))(2)/(f(z) )+(1)/(2)(1+(zf''(z))/(f'(z)))>0 (z is an element of D).
A new generalized class of m-type convex functions is presented. Some Hermite-Hadamard and Ostrowski type inequalities via Riemann-Liouville fractional integrals are obtained for these functions. Some particular cases are discussed as applications of the findings acquired in this study.
Given a graph G = (V, E), a dominating set is a subset D C V such that every vertex in V \D is adjacent with at least one vertex in D. The domination number of G, traditionally denoted by gamma(G), is the minimum cardinality of a dominating set in G. For any natural number k, a set D is k-distance dominating-called kdd set for short-if every vertex not in D is at distance at most d from some vertex in D. We define the universal k-distance domination number as uk(G) := min{d : HD C V with |D| >= d,D is a kdd set in G}. In sharp contrast to most of the standard domination parameters, determining the universal k-distance domination number uk(G) turns out to be computable in polynomial time. We also characterize the graphs satisfying gamma(G) = u1(G), and investigate the behavior of the function uk on random graphs. It remains a challenging open problem to characterize the equality of k-distance domination and universal k-distance domination numbers for a general k. A further problem is to sharpen the estimates on the edge probability in the auxiliary graph H = Gk.
In this paper, we study BKN rings, which are rings that satisfy the property HomR(M,N) not equal 0 for all nonzero left (right) R-modules M and N. Our aim is to provide a comprehensive overview of BKN rings and their connections to other types of rings, such as retractable and completely coretractable rings, commutative rings, and their connections to injective R-modules and non-trivial torsion theories. We prove several results, including the left-right symmetry of the BKN property and its preservation under various ring operations. Furthermore, we explore the relationship between BKN rings and the tensor product of modules. Our study provides insight into the comprehensive understanding of BKN rings, providing information about the relation between BKN rings and the Tor, Ext, and ext functors.
Fractional fourth-order partial differential equations find various applications such as image denoising, electrostatics, and geometric modeling. In this paper, we propose a new supervised machine learning algorithm for such problems. To do so, we develop a simulation method based on least squares support vector regression (LS-SVR). A polynomial kernel is used to approximate the solution in the training process by considering an inverse viewpoint to the residual function. For minimizing the loss function, we use the Petrov-Galerkin method for the constraints in the proposed LS-SVR. We study the stability and convergence of the method by providing some numerical examples and illustrating the error behavior as the degree of the solution increases.
The authors investigate the asymptotic behavior of solutions of the fourth-order halflinear neutral differential equation (alpha(t) ((mu (t)+d(t)mu (a(t)))''')b)'+m(t)mu b(delta(t)) =0 (e) without assuming alpha '(t) >= 0. By using a linearization method and deriving some new monotonic properties of the nonoscillatory solutions, they analyze the oscillatory behavior of solutions of (e). They use two different techniques, namely, a comparison with second-order delay differential inequalities, and a generalization of very effective Koplatadze's method. They illustrate the improvements over known results by providing specific examples.
This paper focus on a type of conformable non-autonomous non-instantaneous impulsive equation. Firstly, we give a kind of non-autonomous conformable Cauchy matrix to present the solution of linear and nonlinear systems. Then, we investigate the asymptotic stability of linear homogeneous problem and the exponential stability of linear perturbed problem. Also, one presents the solution of nonlinear problem and verifies its Ulam-Hyers-Rassias stability.
In mathematics and the applied sciences, fractional calculus is a significant generalization and a very useful tool as it overcomes many limitations of classical analysis. More importantly, it is better to use the new hybrid fractional operator, which merges the proportional and Caputo operator. In this study, because of its numerous applications, we concentrate on the proportional Caputo-hybrid operator. First, we propose a new integral identity with the help of twice-differentiable mappings for the proportional Caputo-hybrid operator. Then, with the help of this newly derived identity, we establish several integral inequalities related to the Milne-type integral inequalities for proportional Caputo-hybrid operator. Also, we obtain various Milne-type inequalities for bounded mappings and mappings of bounded variation. Finally, we point out that the obtained results enhance and generalize some of the previous findings in the field of integral inequalities. These results are the first kind of such results in this direction.
In the present paper, we will study the existence of at least one weak solution for the nonlinear parabolic initial boundary value problem associated with the following equation of Kirchoff type partial derivative u /partial derivative t -M (integral(ohm)(B(x,t,del u)+ 1/theta |del u|(theta))dx) div (b(x,t,del u)+|del u|(theta-2)del u =f(x,t)-g(x,t,u,del u). By using the Topological degree theory for operators of the type L + S + C, where L is a maximal monotone map, S is bounded demicontinuous map of class (S+) and C be compact and belongs to Gamma(tau)(sigma) (i.e there exist tau,sigma >= 0 such that parallel to C-x parallel to <= tau parallel to x parallel to+sigma). Our focus of the study is centered on this problem in space L-theta(0, T,W (1,theta) (ohm)), where theta >= 2 and ohm is a bounded open domain in R-N.
Proinov contraction has appeared a several years ago as a new approach in the study of nonlinear contractions that unifies and extends several well-known classes of nonlinear contractions. What was not discussed is the relation of Proinov contraction and nonlinear contractions, especially Meir-Keeler contraction which also presents an important concept in Fixed point theory. The aim of this paper is to show that Proinov contraction is a Meir-Keeler contraction under some restrictions and that the obtained results under the scope of this contraction and its generalizations in this case are the direct consequence of the analogous results concerning Meir-Keeler contraction mappings. We will also present an example of a Meir-Keeler contraction that is not Proinov contraction and, additionally an example of Proinov contraction which is not a Boyd-Wong contraction, but is a Meir-Keeler contraction.
For residual functions, two regularity notions are known in the literature, one is weak regularity introduced by Blyth and Janowitz, and the other is a closely related condition introduced by Nicholson et al., which we call N-regularity. In our paper, we provide several characterizations of weakly regular residuated maps defined in complete lattices and of N-regular residuated maps in complemented modular lattices. The relations between the mentioned notions and some related conditions (such as Jordan conditions J1 and J2 on residuated maps) are also investigated. We provide some significant examples of weakly regular residuated maps and of N-regular residuated maps.
