Inner Product Research Articles

English

Let V be a unitary space. For an arbitrary subgroup G of the full symmetric group Sm and an arbitrary irreducible unitary representation Λ of G, we study the generalized symmetry class of tensors over V associated with G and Λ. Some important properties of this vector space are investigated.

Study of information entropy for involved quantum models in complex Cayley-Klein space

In the language of the complex formalism, we study the information entropy of a particle on the motion groups from a family of the unitary Cayley-Klein space with constant curvature κ. Hence, in making use of the constant curvature, all the results here presented will be simultaneously valid for the 2D coset space SUκ(2)/U(1) in forms of 2D sphere , hyperbolic plane and Euclidean plane . In addition to physical complex coordinates , their corresponding components are expressed in a 2D complex formalism in terms of the constant curvature in Cayley-Klein space. This process enables us to derive information entropies located in the circular well on two spaces, which are the basis for achieving the relationship between Shannon entropy and Fisher information and the inequalities among them. In particular, we notice that the particle has freely behavior in a spherical state , while it behaves as a constraint particle in the situation of the hyperbolic plane .

Order topology on orthocomplemented posets of linear subspaces of a pre-Hilbert space

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair $$(S,{\mathscr {L}})$$ , where S is a pre-Hilbert space and $${\mathscr {L}}$$ is an orthocomplemented poset of orthogonally closed linear subspaces of S, closed w.r.t. finite-dimensional perturbations (i.e., if $$M\in {\mathscr {L}}$$ and F is a finite-dimensional linear subspace of S, then $$M+F\in {\mathscr {L}}$$ ). We study the order topology $$\tau _o({\mathscr {L}})$$ on $${\mathscr {L}}$$ and show that completeness of S can by characterized by the separation properties of the topological space $$({\mathscr {L}},\tau _o({\mathscr {L}}))$$ . It will be seen that the remarkable lack of a proper probability theory on pre-Hilbert space logics—for an incomplete S—comes out elementarily from this topological characterization.

Educational Tourism Potentials of Argapura Village, Bogor Regency towards Pongkor UNESCO Global Geopark

Argapura is one of the village areas in Cigudeg Subdistrict which has potential unique characteristics for a tourist destination, namely Geopark. Argapura Village is one of the Pongkor National Geopark development villages covering 15 sub-districts in Bogor Regency. Pongkor National Geopark is a unitary space with geodiversity, culturediversity and biodiversity which aims to preserve and protect the wealth of natural resources. The geopark is located very close to tourism activities, but the existing tour has not specifically touched the educational aspect (educational tourism). Educational tourism is a tourism packaging program that is loaded with educational values in each of its activities. Argapura Village needs to be developed educational tourism aimed at all visitors. The objectives of this study were to identify and analyze the potential of educational tourism objects in Argapura Village, and to arrange educational tourism program in Argapura Village. Data collection through literature studies, field surveys and interviews with stakeholders. Data analysis using tables and described based on the criteria of educational values and geopark elements. The results of study showed that Argapura Village has two main potential attractions/educational tourism objects, namely Angklung Gubrag and Gudawang Caves. Angklung Gubrag is an educational tourism attraction that has high cultural, historical and religious values. Gudawang Cave is an educational tourism object in the form of karst ecosystem. In addition to the both of tourism objects, there are several other potential tourism objects and characteristics of the Geopark. The Argapura Village educational tourism was compiled includes two main programs, namely the half-day and the full-day visiting programs.

Planning and design of underground space construction during longwall mining in coal mines

ABSTRACT A large area of underground spaces form after the extraction of underground mineral resources, in which coal mining leads to billions of cubic meters of underground spaces every year. In China, there are 2,469 coal mines exist within 50 km (straight line distance) of the cities, and the annual volume of the goaf in the coal mines is 442 million m3. However, most of these spaces are filled caved rock fragmentation and can hardly be retained and used. To fill this research gap, an innovative scientific conception is proposed in this paper. High-water and quick-setting materials (HWQSMs) are used to build backfill bodies rapidly in mined-out spaces during longwall mining to retain a large amount of underground spaces. Three different methods of building cubic or cylindrical backfill bodies in the mining goaf are introduced. An integration craft and the potential backfilling materials for the construction of the underground space are described in detail. Technical and economical rationality of the proposed conception is evaluated. Further research prospects and the problems that need to be overcome to achieve this goal are discussed. The results shows that compressive strength reaches 10.5 MPa when the HWQSM specimen has a water to HWQSM ratio of 1.5 and the specimen sustains an obvious residual load-bearing capacity. Hence the backfill body will not fail instantaneously in the process of roof supporting. The average unitary underground space value is estimated to be 2,838.89 Yuan per m3, which has great economic value.

Weighted Moore–Penrose inverses of products and differences of weighted projections on indefinite inner-product spaces

An adjointable operator acting on a Hilbert $$C^*$$ -module is called a weight if it is self-adjoint and invertible. An indefinite inner-product space as well as weighted projections can be induced by a weight. Some new formulas are provided for weighted Moore–Penrose inverses associated to products and differences of weighted projections. As a result, some characterizations of Moore–Penrose inverses associated to projections are generalized to the weighted case.

Operators on positive semidefinite inner product spaces

Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form 〈⋅,⋅〉. If a linear operator A:U→U is bounded (i.e., ‖Au‖⩽c‖u‖ for some c∈R and all u∈U), then the subspace U0:={u∈U|〈u,u〉=0} is invariant, and so A defines the linear operators A0:U0→U0 and A1:U/U0→U/U0.Let A be an indecomposable bounded operator on U such that 0≠U0≠U. Let λ be an eigenvalue of A0. We prove that the algebraic multiplicity of λ in A1 is not less than the geometric multiplicity of λ in A0, and the geometric multiplicity of λ in A1 is not less than the number of Jordan blocks Jt(λ) of each fixed size t×t in the Jordan canonical form of A0.We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U.For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.

PROTECTION OF THE SEA REGION FROM OIL POLLUTION BY TANKER SHIP

In the context of national law, Law Number 32 Year 2009 concerning Environmental Protection and Management (UUPPLH) provides an understanding of the environment as a unitary space with all objects, forces, and living things, including humans and their behavior, which affect nature itself, continuity of life, and the welfare of humans and other living things. Environmental protection has general principles, namely: Sovereignty over natural resources and the responsibility not to cause damage to the environment of other states or areas beyond national jurisdiction; Principle of preventive action; Cooperation; Sustainable development. Specifically the mandate of the 1945 Constitution is spelled out in the weighing section of Law Number 32 of 2009 concerning Protection of Environmental Management (UUPPLH) which states that a good and healthy environment is the human right of every Indonesian citizen as mandated in Article 28H of the Law 1945 Constitution of the Republic of Indonesia. This law states that environmental protection and management are systematic and integrated efforts undertaken to preserve environmental functions and prevent environmental pollution and / or damage which includes planning, utilization, control, maintenance, supervision and law enforcement.

Birkhoff Orthogonality and Different Particular Cases of Carlsson's Orthogonality on Normed Linear Spaces

Let x, y ∈ X, where X is an inner-product space. We say x is orthogonal to y if ⟨x, y⟩ = 0. When we move to general normed spaces there are many possibilities of extending the notion of orthogonality. Since 1934, different types of orthogonality relations in normed spaces have been introduced and studied. In this study, we enlist some properties of Birkhoff's orthogonality and Carlsson's orthogonality along with it we introduce two new particular cases of Carlsson's orthogonality and check some properties of othogonality in relation to these particular cases in normed spaces.

A Classification Algorithm Based on Complex Number Feature

In this study, a classification algorithm based on complex number feature is proposed. Specifically, the SVM framework is reformulated, so each example would be classified in the unitary space. The cost function is redefined by considering the maximum margin of real and imaginary units of the complex number feature at the same time. The cost function is based on the expectation of the hinge loss, and its derivatives can be calculated in closed forms. Using a stochastic gradient descent (SGD) algorithm, this method allows for efficient implementation. For complex number feature, the example uncertainty is modeled by a sample preprocessing method based on within-class Euclidean distance Gaussian distribution sample (DGS). In addition, a complex number feature selection method based on improved hybrid discrimination analysis (HDA) is proposed by considering the correlation between real and imaginary units of complex number feature. The proposed classification algorithm is tested on synthetic data and three publicly available and popular datasets, namely, MNIST, WDBC, and Voc2012. Experimental results verify the effectiveness of the proposed method. The codes are available: https://github.com/luckysomebody/paper-code.

Analysis of a Composition Operator’s Eigenvalue Equation on Unitary Spaces by the Krein-Rutman Theorem

The Krein-Rutman theorem is vital in partial differential equations that are non-linear and provides evidence of the presence of several significant eigenvalues useful in topological degree calculations, stability analysis, and bifurcation theory. Schr?der’s equation which has been used extensively in studies of turbulence is an equation with a single independent variable suitable for encoding self-similarity. The concept of Hilbert spaces has been an inner product space frequently used due to its convenience in countless dimensional vector analysis. This paper is aimed at proving a number of solutions through the Krein-Rutman theorem in unitary spaces especially in Hilbert spaces. It has been certainly observed that the whole Krein-Rutman theorem system has a fairly stable scope, and has strong regular features, and many non-linear elliptic operators need the most ethical principles to satisfy the comparison policy.

ABOUT THE SYSTEMS WITH FULL SPARK

Frames of a finite-dimensional Euclidean and unitary spaces composed of discrete Fourier transform matrices are considered. The relationship of phaseless reconstruction systems with the alternative completeness property is presented. In the complex case, alternative completeness is only a necessary condition for phaseless reconstruction. A system of vectors is constructed such that each of its subsystems with a volume equal to the dimension of space is linearly independent. These systems are called systems with full spark. In particular, such systems are optimal for phase retrieval.

Equations of motion for noncompact symplectic dynamical symmetry

When the high dimension of the quantum Hilbert space creates a serious challenge for conventional numerical diagonalisation of the Hamiltonian matrix, the equations of motion (EOM) method may solve efficiently the eigenvalue problem in some instances. EOM assumes the existence of a spectrum generating Lie algebra (SGA) for which the Hamiltonian operator is a polynomial (typically a quadratic) in the algebra’s basis operators. The method computes values of low-energy excitation energies and corresponding algebra matrix elements by solving a coupled set of commutation relations. To test EOM’s effectiveness, the paper applies it to SU(1, 1) Sp(1, ), whose unitary discrete series representation spaces are infinite dimensional. The paper investigates two (1, 1) Hamiltonians. The first Hamiltonian has a phase transition that challenges brute force diagonalisation. The second Hamiltonian is the one-dimensional quartic potential. For the quartic case, the algebraic mean field Hamiltonian, an approximation to the SGA Hamiltonian, provides a qualitative solution and a useful starting point for an EOM computation. EOM yields fast and accurate results on a laptop computer for both Hamiltonians.

Codebooks from generalized bent [formula omitted]-valued quadratic forms

Codebooks with small inner-product correlation have applications in unitary space–time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Levenshtein bound. This paper presents a class of generalized bent Z4-valued quadratic forms, which contains functions proposed by Heng and Yue (2017). Using these generalized bent Z4-valued quadratic forms, we construct optimal codebooks achieving the Levenshtein bound. These codebooks have parameters (22m+2m,2m) and alphabet size 6.

The Mellin Transform and the Plancherel Theorem for the Discrete Heisenberg Group

In the classical representation theory of locally compact groups, there are well-known constructions of a unitary dual space of irreducible representations, the Fourier transform, and the Plancherel theorem. In this paper, we present analogs of these constructions for the discrete Heisenberg group and its irreducible infinite-dimensional representations in a vector space without topology.

On a Certain Space of Centrally Symmetric Polyhedra Which is a Reproducing Kernel Hilbert Space

In this paper a certain unitary space will be constructed. The elements of this space can be interpreted as a special type of three dimensional (n-dimensional) polyhedra. Among them one may find, besides the cube, also such elegant mathematical objects as for example the truncated cuboctahedron or the truncated icosidodecahedron. This unitary space, after completion, will turn out to be a reproducing kernel Hilbert space.

High-Accuracy and Fault Tolerant Stochastic Inner Product Design

In this brief, we present a novel inner product (IP) design for stochastic computing (SC). SC is an emerging computing technique, that encodes a number in the probability of observing a one in a random bit stream. This leads to reduced hardware costs and high error tolerance. The proposed IP design is based on a two-line bipolar encoding format and applies sequential processing of the input in a central accumulation unit. Sequential processing significantly increases the computation accuracy, since it allows for preliminary cancelation of carry bits. Moreover, the central accumulation unit gives a much better scalability compared to conventional adder tree approaches. We show that the proposed IP design outperforms a state-of-the-art design in terms of hardware costs for high accuracy requirements and fault tolerance.

Deterministically generated negative selection algorithm for damage detection in civil engineering systems

In the framework of Structural Health Monitoring, vibration-based methods are commonly used to assess the condition of a structural system, being the dynamic properties sensitive to damage-induced changes. Within this context, negative selection, a bio-inspired classification algorithm, can be exploited to distinguish anomalous from normal behaviours by comparing the monitored system features with a set of detectors appropriately trained to spot any possible anomaly inside the unitary feature space. Such method results particularly convenient due to its easy implementation, low computational cost and capability to carry out the classification based on a training set of data belonging only to a healthy-state condition. This circumstance is extremely common in real civil engineering applications where no knowledge might exist about different structural conditions over time. In this paper, a negative-selection algorithm with a non-random strategy for detector generation is developed and tested on a numerical case study, namely a model simulating the I-40 Bridge over the Rio Grande in Albuquerque, New Mexico (USA). The work carried out proves that the algorithm is suitable for the purpose of damage detection (a binary classification problem) and, by introducing the anomaly score as a qualitative measure of the level of damage, provides a sound analysis of the method multiclass classification skills, aiming at the quantification of the damage.

Some Refinements of Ostrowski’s Inequality and an Extension to a 2-Inner Product Space

The purpose of this paper is to prove certain refinements of Ostrowski’s inequality in an inner product space. We study extensions of Ostrowski type inequalities in a 2-inner product space. Finally, some applications which are related to the Chebyshev function and the Grüss inequality are presented.

Fast faulted line identification method for wide‐area backup protection with PMU optimal placement strategy

This study presents a novel faulted line identification method that is applicable in power systems with only sparse synchronised measurements, and at the same time, with significantly reduced computational burden so that it can be a competitive choice for wide-area backup protection schemes. By revealing the behaviour pattern of bus voltages from the perspective of high-dimensional unitary space, the method is enabled to avoid unnecessary computation and thus to remarkably improve its efficiency. Analysis in high-dimensional unitary space also gives a direct and feasible guideline for phasor measurement unit (PMU) placement, from which a PMU optimal placement strategy is developed. Performed on the IEEE benchmark systems of 39 and 118 buses, the proposed method gives encouraging results.