scholarly journals Correlation Angles and Inner Products: Application to a Problem from Physics

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Adam Towsley ◽  
Jonathan Pakianathan ◽  
David H. Douglass
Keyword(s):  
Vector Space ◽  
Dimensional Space ◽  
Random Variables ◽  
Climate Science ◽  
Inner Product ◽  
Correlation Studies ◽  
Formal Vector ◽  
Inner Products

Covariance is used as an inner product on a formal vector space built on random variables to define measures of correlation across a set of vectors in a -dimensional space. For , one has the diameter; for , one has an area. These concepts are directly applied to correlation studies in climate science.

scholarly journals All about the ⊥ with its applications in the linear statistical models

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Augustyn Markiewicz ◽  
Simo Puntanen
Keyword(s):  
Statistical Models ◽  
Inner Product ◽  
Real Matrix ◽  
Inner Products ◽  
Column Space ◽  
The Matrix ◽  
Linear Statistical Models ◽  
Extensive List

Abstract For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

scholarly journals On the semi-inner product in locally convex spaces

1997 ◽  
Vol 20 (2) ◽  
pp. 219-224
Author(s):  
Shih-Sen Chang ◽  
Yu-Qing Chen ◽  
Byung Soo Lee
Keyword(s):  
Inner Product ◽  
Locally Convex Spaces ◽  
Basic Properties ◽  
Inner Products ◽  
Locally Convex ◽  
Convex Spaces

The purpose of this paper is to introduce the concept of semi-inner products in locally convex spaces and to give some basic properties.

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

Rational Tensor Representations of Hom(V, V) and an Extension of an Inequality of I. Schur.

1972 ◽  
Vol 24 (4) ◽  
pp. 686-695 ◽  
Author(s):  
Marvin Marcus ◽  
William Robert Gordon
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Linear Operators ◽  
Inner Product ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Semi Group ◽  
Rational Representation ◽  
Dimensional Vector Space ◽  
Tensor Representations

Let V be an n-dimensional vector space over the complex numbers equipped with an inner product (x, y), and let (P, μ) be a symmetry class in the mth tensor product of V associated with a permutation group G and a character χ (see below). Then for each T ∊ Hom (V, V) the function φ which sends each m-tuple (v1, … , vm) of elements of V to the tensor μ(TV1, … , Tvm) is symmetric with respect to G and x, and so there is a unique linear map K(T) from P to P such that φ = K(T)μ.It is easily checked that K: Hom(V, V) → Hom(P, P) is a rational representation of the multiplicative semi-group in Hom(V, V): for any two linear operators S and T on VK(ST) = K(S)K(T).Moreover, if T is normal then, with respect to the inner product induced on P by the inner product on V (see below), K(T) is normal.

scholarly journals Implementation of a Hamming distance–like genomic quantum classifier using inner products on ibmqx2 and ibmq_16_melbourne

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Kunal Kathuria ◽  
Aakrosh Ratan ◽  
Michael McConnell ◽  
Stefan Bekiranov
Keyword(s):  
Hamming Distance ◽  
Training Sample ◽  
General Purpose ◽  
Inner Product ◽  
Matching Problems ◽  
Binary Operations ◽  
Inner Products ◽  
Training Class ◽  
Dot Product ◽  
Product Measures

Abstract Motivated by the problem of classifying individuals with a disease versus controls using a functional genomic attribute as input, we present relatively efficient general purpose inner product–based kernel classifiers to classify the test as a normal or disease sample. We encode each training sample as a string of 1 s (presence) and 0 s (absence) representing the attribute’s existence across ordered physical blocks of the subdivided genome. Having binary-valued features allows for highly efficient data encoding in the computational basis for classifiers relying on binary operations. Given that a natural distance between binary strings is Hamming distance, which shares properties with bit-string inner products, our two classifiers apply different inner product measures for classification. The active inner product (AIP) is a direct dot product–based classifier whereas the symmetric inner product (SIP) classifies upon scoring correspondingly matching genomic attributes. SIP is a strongly Hamming distance–based classifier generally applicable to binary attribute-matching problems whereas AIP has general applications as a simple dot product–based classifier. The classifiers implement an inner product between N = 2n dimension test and train vectors using n Fredkin gates while the training sets are respectively entangled with the class-label qubit, without use of an ancilla. Moreover, each training class can be composed of an arbitrary number m of samples that can be classically summed into one input string to effectively execute all test–train inner products simultaneously. Thus, our circuits require the same number of qubits for any number of training samples and are $O(\log {N})$ O ( log N ) in gate complexity after the states are prepared. Our classifiers were implemented on ibmqx2 (IBM-Q-team 2019b) and ibmq_16_melbourne (IBM-Q-team 2019a). The latter allowed encoding of 64 training features across the genome.

scholarly journals The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Khodaei
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Vector Space ◽  
Fixed Point Method ◽  
Inner Product ◽  
Product Spaces ◽  
Real Vector ◽  
Fixed Integer ◽  
Inner Product Spaces ◽  
Fuzzy Approximation

Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer holds for all The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation which is said to be a functional equation associated with inner product spaces.

scholarly journals A ‘Dysonization’ scheme for identifying quasi-particles using non-Hermitian quantum mechanics

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120056 ◽  
Author(s):  
Katherine Jones-Smith
Keyword(s):  
Quantum Mechanics ◽  
High Temperature ◽  
High Temperature Superconductivity ◽  
Spin Waves ◽  
Low Energy ◽  
Inner Product ◽  
Inner Products ◽  
Weakly Interacting

Dyson analysed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of the non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here, we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly ‘Hermitian’. Then we extend his scheme to doped anti-ferromagnets described by the t – J model, with hopes of shedding light on the physics of high-temperature superconductivity.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

OPERATOR ACCRETIVE KUAT PADA RUANG HILBERT

2018 ◽  
Vol 1 (1) ◽  
pp. 60
Author(s):  
Razis Aji Saputro ◽  
Susilo Hariyanto ◽  
Y.D. Sumanto
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Vector Space ◽  
Cauchy Sequence ◽  
Accretive Operator ◽  
Inner Product ◽  
Accretive Operators ◽  
The Real ◽  
Adjoint Operators

Pre-Hilbert space is a vector space equipped with an inner-product. Furthermore, if each Cauchy sequence in a pre-Hilbert space is convergent then it can be said complete and it called as Hilbert space. The accretive operator is a linear operator in a Hilbert space. Accretive operator is occurred if the real part of the corresponding inner product will be equal to zero or positive. Accretive operators are also associated with non-negative self-adjoint operators. Thus, an accretive operator is said to be strict if there is a positive number such that the real part of the inner product will be greater than or equal to that number times to the squared norm value of any vector in the corresponding Hilbert Space. In this paper, we prove that a strict accretive operator is an accretive operator.

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