Diamond Operator Research Articles

On the Bessel Heat Equation Related to the Bessel Diamond Operator

In this article, we study the equation $$\frac{\partial }{\partial t}u(x,t)=c^{2}\Diamond _{B}^{k}u(x,t)$$ with the initial condition u(x,0)=f(x) for x∈ℝn+. The operator ⋄Bk is named to be Bessel diamond operator iterated k-times and is defined by $$\Diamond _{B}^{k}=\bigl[(B_{x_{1}}+B_{x_{2}}+\cdots +B_{x_{p}})^{2}-(B_{x_{p+1}}+\cdots +B_{x_{p+q}})^{2}\bigr]^{k},$$ where k is a positive integer, p+q=n, \(B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+\frac{2v_{i}}{x_{i}}\frac{\partial }{\partial x_{i}},\) 2vi=2αi+1, \(\;\alpha _{i}>-\frac{1}{2}\) , xi>0, i=1,2,…,n, and n is the dimension of the ℝn+, u(x,t) is an unknown function of the form (x,t)=(x1,…,xn,t)∈ℝn+×(0,∞), f(x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)):102–143, 1951; Yildirim, Ph.D. Thesis, 1995; Yildirim and Sarikaya, J. Inst. Math. Comput. Sci. 14(3):217–224, 2001; Yildirim, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4):375–387, 2004; Sarikaya, Ph.D. Thesis, 2007; Sarikaya and Yildirim, Nonlinear Anal. 68:430–442, 2008, and Appl. Math. Comput. 189:910–917, 2007). We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel diamond heat kernel. Moreover, such Bessel diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincar\'e (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.

On the Green function of the (⊕+m 2) k operator

In this article, we study the Green function of the operator (⊕+m 2) k which is iterated k-times and is defined by where m is a positive real number and p+q=n is the dimension of the n-dimensional Euclidean space ℝ n , x=(x 1, x 2, …, x n )∈ℝ n and k is a nonnegative integer. At first, we study the elementary solution or Green function of the operator (⊕+m 2) k . Moreover, the operator (⊕+m 2) k can be related to the ultra-hyperbolic Klein–Gordon operator (□+m 2) k , the Helmholtz operator (▵+m 2) k and the diamond operator of the form (♦+m 2) k , and also we obtain the elementary solutions of such operators. We also apply such a Green function to obtain the solution of the equation (⊕+m 2) k U(x)=f(x), where f is a generalized function and U(x) is an unknown function for x∈ℝ n .

Taut Monads, Dynamic Logic and Determinism

Studied are Kleisli categories of monads of sets which satisfy two properties motivated by functional properties of collections. Such categories have box and diamond operators which follow the laws of (loop-free) dynamic logic. A theorem of Kozen states that the category of sets and relations is complete for loop-free dynamic logic. It is shown that the Kleisli category of the filter monad is likewise complete. A morphism α is deterministic if <α>Q⊂[α]Q for all Q. Each “output value”αx is associated with a filter which is forced to be an ultrafilter when α is deterministic and αx is defined. Early work in the theory of domains abstracted from the partially ordered set of partial functions between two sets, ordered by extension. A different abstraction, suited to conditional constructs rather than recursive fixed point equations, is the notion of a locally Boolean poset, and this is used to compare restriction categories with deterministic Kleisli categories. The laws of dynamic logic in terms of [α]Q and <α>Q for a single α hold in any topological space with [α]Q the interior operator and <α>Q the closure operator.

On the weak solutions of the compound Bessel ultra-hyperbolic equation

In this article, we have studied the compound Bessel ultra-hyperbolic equation of the form ∑ r = 0 m C r □ B r u ( x ) = f ( x ) , where □ B r is the Bessel ultra-hyperbolic type operator iterated r-times, f is a given generalized function, u is an unknown function, x ∈ R n + and C r is a constant. In this work we study the weak solution u( x) of above the equation which is of the form Bessel Diamond operator and moreover, such a solution is unique.

On the Bessel diamond and the nonlinear Bessel diamond operator related to the Bessel wave equation

In this article, we study the solution of the equation ♢ B k u ( x ) = f ( x ) where u ( x ) is an unknown generalized function and f is a generalized function, ♢ B k is the Bessel diamond operator iterated k times and is defined by ♢ B k = [ ( B x 1 + B x 2 + ⋯ + B x p ) 2 − ( B x p + 1 + ⋯ + B x p + q ) 2 ] k where p + q = n , B x i = ∂ 2 ∂ x i 2 + 2 v i x i ∂ ∂ x i , where 2 v i = 2 α i + 1 , α i > − 1 2 [B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Uspekhi Matematicheskikh Nauk (N.S.) 6 2 (42) (1951) 102–143 (in Russian)], x i > 0 , i = 1 , 2 , … , n , k , is a nonnegative integer and n is the dimension of the R n + . Firstly, it found that the solution u ( x ) depends on the conditions of p and q and moreover such a solution is related to the solution of the Laplace Bessel equation and the Bessel wave equation. Finally, we study the solution of the nonlinear equation ♢ B k u ( x ) = f ( x , Δ B k − 1 □ B k u ( x ) ) . It is found that the existence of the solution u ( x ) of such an equation depends on the condition of f and Δ B k − 1 □ B k u ( x ) and moreover such a solution u ( x ) related to the Bessel wave equation depends on the conditions of p , q and k .

Algebras of modal operators and partial correctness

Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complementarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style. This claim is supported by novel algebraic soundness and completeness proofs for Hoare logic and by connecting this formalism with an algebraic decision procedure.

The solutions of then-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution

In this article, the operator\(\diamondsuit _B^k \) is introduced and named as the Bessel diamond operator iteratedk times and is defined by $$\diamondsuit _B^k = [(B_{x_1 } + B_{x_2 } + \cdots + B_{x_p } )^2 - (B_{x_{p + 1} } + \cdots + B_{x_{p + q} } )^2 ]^k $$ \(p + q = n,B_{x_i } = \tfrac{{\partial ^2 }}{{\partial x_i^2 }} + \tfrac{{2v_i }}{{x_i }}\tfrac{\partial }{{\partial x_i }}\) where\(2v_i = 2\alpha _i + 1,\alpha _i > - \tfrac{1}{2}[8],x_i > 0\),i = 1, 2, ...,nk is a non-negative integer andn is the dimension of ℝn+. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator\(\diamondsuit _B^k \) is called the Bessel diamond kernel of Riesz. Then, we study the Fourier-Bessel transform of the elementary solution and also the Fourier-Bessel transform of their convolution.

On the product of the non-linear Diamond operators related to the elastic wave

In this paper we study the solution of the equation ♢c1k♢c2ku(x)=f(x,Δc1k−1□c1k♢c2ku(x)) where ♢c1k♢c2k is the product of the Diamond operators defined by♢c1k=1c14∑i=1p∂2∂xi22−∑j=p+1p+q∂2∂xj22kand♢c2k=1c24∑i=1p∂2∂xi22−∑j=p+1p+q∂2∂xj22k,where c1 and c2 are positive constants, k is a non-negative integer, p+q=n is the dimension of the n-dimensional Euclidean space Rn, x=(x1,…,xn)∈Rn, u(x) is an unknown and f(x,Δc1k−1□c1k♢c2k) is a given function. It is found that, the existence of the solution u(x) of such equation depending on the conditions of f and Δc1k−1□c1k♢c2ku(x). Moreover such solution u(x) related to the elastic wave equation depending on the conditions of p, q and k.

On the nonlinear Diamond operator related to the wave equation

In this paper, we study the solution of nonlinear equation ♢ ku(x)=f(x, Δ k−1□ ku(x)) where ♢ k is the Diamond operator iterated k times and is defined by ♢ k= ∑ i=1 p ∂ 2 ∂x i 2 2− ∑ j=p+1 p+q ∂ 2 ∂x j 2 2 k or the operator ♢ k can be expressed by ♢ k=□ k Δ k= Δ k□ k . The operators Δ k and □ k are Laplacian and the ultrahyperbolic operator iterated k times, defined by □ k= ∂ 2 ∂x 1 2 + ∂ 2 ∂x 2 2 +⋯+ ∂ 2 ∂x p 2 − ∂ 2 ∂x p+1 2 − ∂ 2 ∂x p+2 2 −⋯− ∂ 2 ∂x p+q 2 k and Δ k=( ∂ 2 ∂x 1 2 + ∂ 2 ∂x 2 2 +⋯+ ∂ 2 ∂x n 2 ) k,p+q=n is the dimension of the n-dimensional Euclidean space R n, x=(x 1,x 2,…,x n)∈ R n, k is a nonnegative integer, u( x) is an unknown and f is a given function. It is found that, the existence of the solution u( x) of such equation depending on the conditions of f and Δ k−1□ ku(x) and moreover such solution u( x) related to the wave equation depending on the conditions of p, q and k.

On the Solutions of the Causal And Anticausaln -Dimensional Diamond Operator

In this paper, we consider the solution of the equation \\diamondsuit^{k}(P\\pm i0) = \\sum_{r=0}^{m} C_{r}\\diamondsuit^{r}\\delta where \\diamondsuit^{k} is introduced and named as the Diamond operator iterated k -times and is defined by \\diamondsuit^{k} = \\left[\\left({\\partial^{2} \\over \\partial x_{1}^{2}}+ \\cdots + {\\partial^{2} \\over \\partial x_{p}^{2}}\\right)^{2} - \\left({\\partial^{2} \\over \\partial x_{p+1}^{2}} + \\cdots + {\\partial^{2} \\over \\partial x_{p+q}^{2}}\\right)^{2}\\right]^{k}. Let x = (x_{1},x_{2},\\ldots,x_{n}) be a point of the n -dimensional Euclidean space. Consider a non-degenerate quadratic form in n variables of the form P = P(x) = x_{1}^{2} + \\cdots + x_{p}^{2} - x_{p+1}^{2} - \\cdots - x_{p+q}^{2} , where p + q = n , C_{r} is a constant, \\delta is the delta distribution \\diamondsuit^{0}\\delta =\\delta and k = 0,1,\\ldots. The distributions (P\\pm i0)^{\\lambda} are defined by (P\\pm i0)^{\\lambda} = \\lim_{\\varepsilon \\rightarrow 0}\\ \\{P\\pm i\\varepsilon \\vert x\\vert^{2}\\}^{\\lambda} , where \\varepsilon \\gt 0, \\vert x\\vert^{2} = x_{1}^{2} + \\cdots + x_{n}^{2} , \\lambda \\in \\open C . The distributions (P\\pm i0)^{\\lambda} are an important contribution of Gelfand (cf.[1], p. 274). The distributions (P\\pm i0)^{\\lambda} are analytic in \\lambda everywhere except at \\lambda = -n/2 - k , k = 0,1,\\ldots\\, , where they have simple poles (cf. [1], p. 275). By causal (anticausal) distributions, we mean distributions where P = P(x) = x_{1}^{2} + \\cdots + x_{n-1}^{2} - x_{n}^{2} . The causal distributions are particularly important when n = 4 because they appear frequently in the quantum theory of field. In this note we obtain the solutions of the causal and anti"causal n -dimensional Diamond operator by following, line by line, the paper entitled "On the solutions of the n -"dimensional Diamond operator" by Amnuay Kananthai (cf. [2]).

On the diamond operator related to the wave equation

On the diamond operator related to the wave equation

Heyting Algebras with Operators

In this paper, we will give a general description of subdirectly irreducible Heyting algebras with operators under some weak conditions, which includes the finite case, the normal case and the case for Boolean algebras with diamond operator. This can be done by normalizing these operators. This answers the question posed in Wolter [4].

On axiomatic characterisations of crisp approximation operators

In rough set theory one can find the result that the upper approximation defines a closure operator on the power set of the universe fixed. In the paper presented, first, we have solved the problem under which additional assumptions a given closure operator can be represented by an upper approximation within the concepts of rough set theory. Second, we have solved this problem within modal logic for a diamond operator represented by an arbitrary binary relation on the universe. Finally, we have generalized these problems to lower approximations and modal box operators, to rough approximations of fuzzy sets (rough fuzzy sets), to fuzzy approximations of crisp sets (fuzzy rough sets), and fuzzy approximations of fuzzy sets (`fuzzy' diamond and `fuzzy' box).

On the convolutions of the diamond kernel of Marcel Riesz

In this paper, we consider the equation ♢ k u( x)= δ where ♢ k is introduced and named as the diamond operator iterated k-times and is defined by ♢ k=((∑ p i=1( ∂ 2/ ∂x 2 i)) 2−(∑ p+q j=p+1( ∂ 2/ ∂x 2 j)) 2) k, u(x) is a generalized function, x=( x 1, x 2,…, x n )∈ R n the n-dimensional Euclidean space, p+ q= n, k=0,1,2,3,… and δ is the Dirac-delta distribution. Now u( x) is the elementary solution of the operator ♢ k and is called the diamond kernel of Marcel Riesz. The main part of this work is studying the convolution of u( x).

On the Fourier transform of the Diamond Kernel of Marcel Riesz

In this paper, the operator ♢ k is introduced and named as the Diamond operator iterated k-times and is defined by ♢ k =[(∂ 2/∂ x 1 2+∂ 2/∂ x 2 2+⋯+∂ 2/∂ x p 2) 2−(∂ 2/∂ x p+1 2+∂ 2/∂ x p+2 2+⋯+∂ 2/∂ x p+ q 2) 2] k , where n is the dimension of the Euclidean space R n,k is a nonnegative integer and p+ q= n. The elementary solution of the operator ♢ k is called the Diamond Kernel of Marcel Riesz. In this work we study the Fourier transform of the elementary solution and also the Fourier transform of their convolutions.

On the solutions of the n-dimensional diamond operator

In this paper, we consider the solution of the equation ▪ where ▪ is introduced and named as the diamond operator interated k times and is defined by ▪ Now x = ( x 1, x 2,…, x n ) ϵ R n , the n-dimensional Euclidian space, p + q = n, c r is a constant, δ is the Dirac-delta distribution, and ▪ and k = 0, 1, 2, 3,…. It is found that the type of solutions of this equation, such as the ordinary functions, the tempered distributions and the singular distributions, depend on the relationship between the values of k and m.

Deterministic process logic is elementary

Process logic (PL) is a language for reasoning about the behavior of a program during a computation, while propositional dynamic logic (PDL) can only reason about the input-output states of a program. Nevertheless, it is shown that to each PL model M , there corresponds in a natural way a PDL model M t such that each path in M is represented by a state in M t . Moreover, to every PL formula p , there corresponds a PDL formula p t , whose length is linear in that of p , such that p is true of a path in M iff p t is true of the state which represents that path in M t . Then it is shown that p is satisfiable iff p t is satisfiable in a finite PDL model with special properties which is called a pseudomodel . The size of the pseudomodel is in general nonelementary but is bounded by both the depth of nesting of the suf operator and the alternation of the suf and diamond operators. However, for DPL, a deterministic version of PL, the pseudomodel has exponential size, giving a deterministic exponential time procedure for deciding DPL validity. These results suggest that it is the interaction between nondeterministic programs and the suf operator that makes the general decision problem for PL so difficult.