In quantum mechanics, physical states are represented by rays in Hilbert space $\mathcal{H}$, which is a vector imbued an inner product $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$, whose meaning arises as the overlap $\ensuremath{\langle}\ensuremath{\phi}|\ensuremath{\psi}\ensuremath{\rangle}$ for $|\ensuremath{\psi}\ensuremath{\rangle}$ p...

In this paper, for the first time, notion of multi complex numbers and number valued inner product is introduced in linear (vector) space. Starting from definition, some basic properties spaces are studied along with examples. Multi parallelogram law polarization identity established

The mechanism of the self-reaction of HCO radicals is investigated by using high-level quantum-chemical methods including M05-2X, CCSD, CCSD(T) and CRCC(2,3). Next, the rate coefficients for several product channels as a function of pressure and temperature are computed by employing statistical rate theories. Four important product channels are predicted to be CO + CO + H2, HCOH + OH, cis-(HCO)...

The notions of intuitionistic fuzzy h-ideal and normal intuitionistic fuzzy h-ideal in Γ-hemiring are introduced and some of the basic properties of these ideals are investigated. Cartesian product of intuitionistic fuzzy h-ideals is also defined. Finally a characterization of intuitionistic fuzzy h-ideals in terms of fuzzy relations is obtained. Keywords—Γ-hemiring, fuzzy h-ideal, normal, cart...

In this paper, we apply the concept of fuzzy set to ideals and closed ideals in B-algebras. The notion of a fuzzy closed ideal of a B-algebra is introduced, and some related properties are investigated. Also, the product of fuzzy B-algebra is investigated. Keywords—B-algebras, fuzzy sets, fuzzy ideals, homomorphism, fuzzy closed ideals, homomorphism, equivalence relation, level cut, product of ...

considering convergent product as an important manufacturing technology for digital products, we integrate functions and sub-functions using a comprehensive fuzzy mathematical optimization process. to form the convergent product, a web-based fuzzy network is considered in which a collection of base functions and sub-functions configure the nodes and each arc in the network is to be a link betwe...

Let X be a finite subset of Euclidean space R. We define for each x ∈ X, B(x) := {(x, y) | y ∈ X, x 6= y, (x, x) ≥ (y, y)} where (, ) denotes the standard inner product. X is called an inside s-inner product set if |B(x)| ≤ s for all x ∈ X. In this paper, we prove that the cardinalities of inside s-inner product sets have the Fisher type upper bound. An inside s-inner product set is said to be ...

with a symmetric and positive definite operator A defined on a Hilbert space V with dimV = N and its preconditioned version PCG. We derive the CG from the projection in A-inner product point of view. We shall use (·, ·) for the standard inner product and (·, ·)A for the inner product introduced by the SPD operator A. When we say ‘orthogonal’ vectors, we refer to the default (·, ·) inner product...