scholarly journals Some Grüss Type Inequalities for Fréchet Differentiable Mappings

2020 ◽  
Vol 44 (4) ◽  
pp. 571-579
Author(s):  
T. TEIMOURI-AZADBAKHT ◽  
A. G GHAZANFARI
Keyword(s):  
Open Neighborhood ◽  
Continuous Functions ◽  
Inner Product ◽  
C Algebra ◽  
Inner Products ◽  
Fréchet Differentiable ◽  
Grüss Type Inequalities ◽  
Differentiable Mappings

Let X be a Hilbert C∗-module on C∗-algebra A and p ∈ A. We denote by Dp(A,X) the set of all continuous functions f : A → X, which are Fréchet differentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A,X), and using them some Grüss type inequalities in semi-inner product C∗-module Dp(A,X) and Dp(A,Xn) are established.

scholarly journals BESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES

2007 ◽  
Vol 50 (1) ◽  
pp. 23-36 ◽  
Author(s):  
Senka Banić ◽  
Dijana Ilišević ◽  
Sanja Varošanec
Keyword(s):  
Inner Product ◽  
C Algebra ◽  
Grüss Type Inequalities

AbstractIn this paper we give Bessel- and Grüss-type inequalities in an inner product module over a proper $H^*$-algebra or a $C^*$-algebra.

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.

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 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.

scholarly journals Sphere and projective space of a C*-algebra with a faithful state

2019 ◽  
Vol 52 (1) ◽  
pp. 410-427
Author(s):  
Andrea C. Antunez
Keyword(s):  
Initial Data ◽  
Projective Space ◽  
Banach Spaces ◽  
Lie Group ◽  
Unit Sphere ◽  
Homogeneous Spaces ◽  
Inner Product ◽  
C Algebra ◽  
Faithful State ◽  
Banach Lie Group

AbstractLet 𝒜 be a unital C*-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊ϕ = {x ∈ 𝒜 : ϕ(x*x) = 1} and the projective space ℙϕ = 𝕊ϕ/𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰ϕ(𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙϕ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

A Priority-Based Weighted Inner Products Matching Coarsening Algorithm on Multilevel Hypergraph Partitioning

2013 ◽  
Vol 717 ◽  
pp. 466-474
Author(s):  
Yao Yuan Zeng ◽  
Wen Tao Zhao ◽  
Zheng Hua Wang
Keyword(s):  
Combinatorial Optimization ◽  
Primary Component ◽  
Experimental Results ◽  
Inner Product ◽  
Priority Rules ◽  
Hypergraph Partitioning ◽  
Inner Products ◽  
Matching Mechanism ◽  
Benchmark Suite ◽  
A Priority

Multilevel hypergraph partitioning is an significant and extensively researched problem in combinatorial optimization. Nevertheless, as the primary component of multilevel hypergraph partitioning, coarsening phase has not yet attracted sufficient attention. Meanwhile, the performance of coarsening algorithm is not very satisfying. In this paper, we present a new coarsening algorithm based on multilevel framework to reduce the number of vertices more rapidly. The main contribution is introducing the matching mechanism of weighted inner product and establishing two priority rules of vertices. Finally, the effectiveness of our coarsening algorithm was indicated by experimental results compared with those produced by the combination of different sort algorithms and hMETIS in most of the ISPD98 benchmark suite.

scholarly journals Representation and coding of signal geometry

2017 ◽  
Vol 6 (4) ◽  
pp. 349-388 ◽  
Author(s):  
Petros T Boufounos ◽  
Shantanu Rane ◽  
Hassan Mansour
Keyword(s):  
Coding Theory ◽  
Rate Distortion ◽  
Inner Product ◽  
Sufficient Information ◽  
Specific Information ◽  
Rate Distortion Theory ◽  
Nonlinear Approximations ◽  
Inner Products ◽  
Linear And Nonlinear ◽  
Broad Framework

Abstract Approaches to signal representation and coding theory have traditionally focused on how to best represent signals using parsimonious representations that incur the lowest possible distortion. Classical examples include linear and nonlinear approximations, sparse representations and rate-distortion theory. Very often, however, the goal of processing is to extract specific information from the signal, and the distortion should be measured on the extracted information. The corresponding representation should, therefore, represent that information as parsimoniously as possible, without necessarily accurately representing the signal itself. In this article, we examine the problem of encoding signals such that sufficient information is preserved about their pairwise distances and their inner products. For that goal, we consider randomized embeddings as an encoding mechanism and provide a framework to analyze their performance. We also demonstrate that it is possible to design the embedding such that it represents different ranges of distances with different precision. These embeddings also allow the computation of kernel inner products with control on their inner product-preserving properties. Our results provide a broad framework to design and analyze embeddings and generalize existing results in this area, such as random Fourier kernels and universal embeddings.

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.

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