In mathematics the basic operations are addition, subtraction, multiplication and division, a mathematical operator is symbol that used to indicate what operation be realized, for example, +, −, ×, ¸, so on. this article, I present procedure by which it possible reduce any natural number -even or odd, with without decimals-, one of first four odd numbers: 1, 3, 5, 7. suggest >n< reduction...

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for ...

We transform the oscillator algebra with κ-deformed multiplication rule, proposed in [1, 2], into the oscillator algebra with κ-deformed flip operator and standard multiplication. We recall that the κ-multiplication of the κ-oscillators puts them off-shell. We study the explicit forms of modified mass-shell conditions in both formulations: with κ-multiplication and with κ-flip operation. On the...

The Hardy space H(D) can be viewed as a module over the polynomial ring C[z, w] with module action defined by multiplication of functions. The core operator is a bounded self-adjoint integral operator defined on submodules of H(D), and it gives rise to some interesting numerical invariants for the submodules. These invariants are difficult to compute or estimate in general. This paper computes ...

where the integral in (1.1) should be interpreted in the principal value sense when (ax + b)(ax + d) < 0. Note that for a = 0, Ha,b,d is not a convolution operator. When a = 1, H1,b,d = H(fχ[−d,−b]) is the Hilbert transform composed with multiplication by the characteristic function of the interval [−d,−b]. When a = 0, H0,b,d is a well understood operator. In the sequel, we will concentrate our...

The optimal robust disturbance rejection problem plays an important role in feedback control theory. Here its time-varying version is solved explicitly in terms of duality and operator theory. In particular, the optimum is shown to satisfy a time-varying allpass property. Moreover, optimal performance is given in terms of the norm of a bilinear form. The latter depends on a lower triangular pro...

For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local contribution to...

We study parabolic stochastic partial differential equations (SPDEs), driven by two types of operators: one linear closed operator generating a C0−semigroup and one linear bounded operator with Wick-type multiplication, all of them set in the infinite dimensional space framework of white noise analysis. We prove existence and uniqueness of solutions for this class of SPDEs. In particular, we al...

The problem of quantum time and evolution systems, where is not a parameter, considered. In our model, following some earlier works, represented by operator. this paper, similarly to the position operators in Schrödinger representation mechanics, operator multiplication-type It can be also an appropriate positive operator-valued measure (POVM) which together with 3D operators/measures provide o...

We present a numerical approximation scheme for the Tomita-Takesaki modular operator of local subalgebras in linear quantum fields, working at one-particle level. This is applied to subspaces double cones vacuum sector massive scalar free field (1 + 1)- and (3 1)-dimensional Minkowski spacetime, using discretization time-0 data position space. In case wedge region, one component generator well-...