Orthogonal Sum Research Articles

A new orthogonal sum in Random Permutation Set

Random Permutation Set is a newly proposed method for handling uncertainty, which considers the order of the elements in evidence. However, how to fuse RPSs efficiently is still an open issue. To solve this problem, the space composed of order code is defined. Then the mutual mappings between this space and permutation event space are also presented. Finally, the new orthogonal sum is proposed. Compared with the existing left orthogonal sum, the new orthogonal sum can obtain more accurate results with lower entropy and deal with extreme situations. Numerical examples are used to illustrate the superiority of the new orthogonal sum.

Transposed Poisson structures on the Lie algebra of upper triangular matrices

We describe transposed Poisson structures on the upper triangular matrix Lie algebra T_{n}(F) , n>1 , over a field F of characteristic zero. We prove that, for n>2 , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n=2 , there is one more class of transposed Poisson structures on T_{n}(F) . We also show that, up to isomorphism, the full matrix Lie algebra M_{n}(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

Random Permutation Set Reasoning.

In artificial intelligence, it is crucial for pattern recognition systems to process data with uncertain information, necessitating uncertainty reasoning approaches such as evidence theory. As an orderable extension of evidence theory, random permutation set (RPS) theory has received increasing attention. However, RPS theory lacks a suitable generation method for the element order of permutation mass function (PMF) and an efficient determination method for the fusion order of permutation orthogonal sum (POS). To solve these two issues, this paper proposes a reasoning model for RPS theory, called random permutation set reasoning (RPSR). RPSR consists of three techniques, including RPS generation method (RPSGM), RPSR rule of combination, and ordered probability transformation (OPT). Specifically, RPSGM can construct RPS based on Gaussian discriminant model and weight analysis; RPSR rule incorporates POS with reliability vector, which can combine RPS sources with reliability in fusion order; OPT is used to convert RPS into a probability distribution for the final decision. Besides, numerical examples are provided to illustrate the proposed RPSR. Moreover, the proposed RPSR is applied to classification problems. An RPSR-based classification algorithm (RPSRCA) and its hyperparameter tuning method are presented. The results demonstrate the efficiency and stability of RPSRCA compared to existing classifiers.

Об одной начально-краевой задаче, возникающей в динамике сжимаемой идеальной стратифицированной жидкости

In this paper, we investigate the problem on small motions of a compressible ideal stratified fluid in a bounded domain. The problem is studied on the base of approach connected with application of so-called operator matrices theory, as well as abstract differential operator equations. For this purpose, Hilbert spaces and some of their subspaces are introduced. The original initial-boundary value problem reduces to the Cauchy problem for a second-order differential operator equation in the orthogonal sum of some Hilbert spaces. Further, an equation with a closed operator is associated with the resulting equation. On this basis, sufficient conditions for the existence of a solution to the corresponding problem are found.

Conjunctive and disjunctive combination rules in random permutation set theory: A layer-2 belief structure perspective

In uncertainty reasoning, conjunctive and disjunctive combination rules are the core tools for information updates. As an extension of evidential reasoning, random permutation set reasoning models uncertain information based on ordered focal sets. Existing combination rules in random permutation set theory, orthogonal sums, do not satisfy the commutativity of one of the original distributions and overly obey its order information. In this paper, the random permutation set theory is interpreted as an refined extension of Dempster–Shafer theory, and a layer-2 belief structure is proposed to describe the permutation event space. Compared with the traditional belief structure, the proposed structure can model both symbolic and numerical uncertainty. Based on the above, the conjunctive and disjunctive combination rules in Dempster–Shafer theory are extended to random permutation set theory. Through properties analysis and simulation demonstration, we demonstrate that the proposed methods can not only resolve the counter-intuitive results of orthogonal sums, but make full use of order information in distributions as well. In addition, we also extend the product space operations and discounting methods based on the proposed methods, and give a general framework of multi-source information fusion under the random permutation set theory.

Research on the development model of fusion of poetry communication and literary creation based on information fusion technology under the perspective of national fusion

Abstract Under the perspective of national integration, the fusion development of poetry communication and literary creation is flourishing. Through analyzing the characteristics of contemporary poetry dissemination and creation, this paper proposes an innovative model for the fusion development of poetry dissemination and literary creation based on information fusion technology. For poetry dissemination of multi-source information data, random set theory is used for representation, information fusion is performed on the representation results of random sets based on D-S evidence theory, and uncertainty is measured by introducing trust functions and multiple basic probability distribution functions synthesized into one basic trust function through orthogonal sum according to synthesis rules. From the viewpoint of dissemination information sources, news and information websites, video websites, podcasts or audiobooks are called the most important information sources for poetry dissemination, with a combined weight of more than 95%. In the case of poetry literature creation under the integration development mode, the periodicity of the works has significantly improved, with an improvement interval of (18.3% and 27.6%). The dissemination degree also increased effectiveness, with a maximum of 30.7% and a minimum of 21.5%. The development model based on information fusion can open up a new path for the fusion development of contemporary poetry communication and literary creation.

Symmetric Renyi-Permutation divergence and conflict management for random permutation set

Recently, a theory called random permutation set (RPS) is proposed, which is the extension of Dempster–Shafer evidence theory. It defines a permutation event space (PES) to represent the order of events and uses the RPS to represent the support of permutation events. In addition, this theory fuses multiple RPSs by developing the left orthogonal sum (LOS) and right orthogonal sum (ROS) algorithms. However, our experiments have found that when two RPSs are highly conflicting, using the LOS or the ROS algorithm to fuse them will generate conflicting results. Therefore, this paper proposes the symmetric Renyi-Permutation (SRP) divergence of the RPS and a LOS-SRP algorithm for conflict management. Furthermore, existing RPS-based reliability assessment methods have the drawbacks of subjectivity and low accuracy, which prompts us to combine SRP divergence with the RPS and propose the RASRP algorithm for reliability assessment. Experimental results show that the RASRP algorithm performs better than the most advanced existing methods.

Matrix operations in Random Permutation Set

In D-S evidence theory, belief and commonality functions are well-defined and deeply researched. However, these concepts have not been extended to Random Permutation Set (RPS) because of the difficulties in giving a suitable definition of containment relationships of permutations. In this paper, we addressed this issue by defining both ordered and unordered containment relationships. Based on the defined containment relationships, the corresponding belief function and commonality functions for RPS are derived naturally. Besides, the plausibility function for RPS is also proposed following the definition in D-S evidence theory. Furthermore, the transformations between them and permutation mass functions are discussed and implemented by matrix converters. Thanks to the properties of commonality functions, the combination process in RPS, including left and right orthogonal sum, are both implemented by matrix operations. Moreover, based on matrix operations, the reverse transformation of combination, that is, the decomposition of fusion results given one information source is achieved for the first time. Numerical examples are listed to illustrate the calculation process. The proposed matrix operations provide a way to implement and accelerate information transformations, combinations and decompositions.

Graded Lie-Rinehart algebras

The present work introduces the class of graded Lie-Rinehart algebras as a natural generalization of graded Lie algebras. It is demonstrated that a tight G-graded Lie-Rinehart algebra L over a commutative and associative G-graded algebra A, where G is an abelian group, can be decomposed into the orthogonal direct sums L=⨁i∈IIi and A=⨁j∈JAj, where each Ii and Aj is a non-zero ideal of L and A, respectively. Additionally, both decompositions satisfy that for any i∈I, there exists a unique j∈J such that AjIi≠0 and that any Ii is a graded Lie-Rinehart algebra over Aj. In the case of maximal length, the aforementioned decompositions of L and A are through indecomposable (graded) ideals, and the (graded) simplicity of any Ii and any Aj are also characterized.

Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras, and almost complex product structures

We study complex product structures on quadratic vector spaces and on quadratic Lie algebras analyzing the Lagrangian and orthogonal splittings associated with them. We show that a Manin triple equipped with generalized metric G+B such that B is an O-operator with extension G of mass −1 can be turned into another Manin triple that admits also an orthogonal splitting in Lie ideals. Conversely, a quadratic Lie algebra orthogonal direct sum of a pair of anti-isomorphic Lie algebras, following similar steps as in the previous case, can be turned into a Manin triple admitting an orthogonal splitting into Lie ideals.

Separation of spherically and translationally covariant finite quantum spaces within the XXX model

A method of separation of subspaces with definite values of the total spin, its z-projection and quasimomentum, within the state space of the XXX model, i.e. the N-th tensor power of a single qubit, is presented. We construct a complete system of orthogonal projectors onto such subspaces by use of the basis of wavelets and chosen measuring operators based on the total spin. We also demonstrate, on examples of N=6 and 8 magnetic spin rings, that the constructed projectors can be further decomposed into the orthogonal sum of density matrices of eigenstates of the XXX model in the corresponding subspaces. This decomposition exhibits a Galois symmetry, stemming from roots of indecomposable factors of characteristic polynomial of the Heisenberg Hamiltonian. In this way, we propose an alternative for Bethe Ansatz for moderate lengths of spin chains, by a return to the original spectral problem, avoiding the highly nonlinear system of Bethe Ansatz equations. Such results are easy to adapt in quantum information processing.

A Mixing Property for the Action of SL(3,ℤ) × SL(3,ℤ) on the Stone–Čech Boundary of SL(3,ℤ)

Abstract By analogy with the construction of the Furstenberg boundary, the Stone–Čech boundary of $\textrm {SL}(3,\mathbb {Z})$ is a fibered space over products of projective matrices. The proximal behaviour on this space is exploited to show that the preimages of certain sequences have accumulation points that belong to specific regions, defined in terms of flags. We show that the $\textrm {SL}(3,\mathbb {Z})\times \textrm {SL}(3,\mathbb {Z})$-quasi-invariant Radon measures supported on these regions are tempered. Thus, every quasi-invariant Radon boundary measure for $\textrm {SL}(3,\mathbb {Z})$ is an orthogonal sum of a tempered measure and a measure having matrix coefficients belonging to a certain ideal $c^{\prime}_0 ((\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$, slightly larger than $c_0 ((\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$. Hence, the left–right representation of $C^*(\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$ in the Calkin algebra of $\textrm {SL}(3,\mathbb {Z})$ factors through $C^*_{c^{\prime}_0} (\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$ and the centralizer of every infinite subgroup of $\textrm {SL}(3,\mathbb {Z})$ is amenable.

Decomposition of Linear Operators on Pre-Euclidean Spaces by Means of Graphs

In this work, we study a linear operator f on a pre-Euclidean space V by using properties of a corresponding graph. Given a basis B of V, we present a decomposition of V as an orthogonal direct sum of certain linear subspaces {Ui}i∈I, each one admitting a basis inherited from B, in such way that f=∑i∈Ifi. Each fi is a linear operator satisfying certain conditions with respect to Ui. Considering this new hypothesis, we assure the existence of an isomorphism between the graphs of f relative to two different bases. We also study the minimality of V by using the graph of f relative to B.

Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$. It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of the symbol. On the mixed spectrum, these two types of the asymptotic evolution of $\exp(-iTt) f$ coexist. This classification is justified in the framework of scattering theory. We prove the existence of wave operators that relate the model operators with the Toeplitz operator $T$. The ranges of these wave operators are pairwise orthogonal, and their orthogonal sum exhausts the whole space, i.e., the set of these wave operators is asymptotically complete.

On Convergence of Support Operator Method Schemes for Differential Rotational Operations on Tetrahedral Meshes Applied to Magnetohydrodynamic Problems

The problem of constructing and justifying the discrete algorithms of the support operator method for numerical modeling of differential repeated rotational operations of vector analysis (curlcurl) in application to problems of magnetohydrodynamics is considered. Difference schemes of the support operator method on the unstructured meshes do not approximate equations in the local sense. Therefore, it is necessary to prove the convergence of these schemes to the exact solution, which is possible after analyzing the error structure of their approximation. For this analysis, a decomposition of the space of mesh vector functions into an orthogonal direct sum of subspaces of potential and vortex fields is introduced. Generalized centroid-tensor metric representations of repeated operations of tensor analysis (div, grad, and curl) are constructed. Representations have flux-circulation properties that are integrally consistent on spatial meshes of irregular structure. On smooth solutions of the model magnetostatic problem on a tetrahedral mesh with the first order of accuracy in the rms sense, the convergence of the constructed difference schemes is proved. The algorithms constructed in this work can be used to solve physical problems with discontinuous magnetic viscosity, dielectric permittivity, or thermal resistance of the medium.

On the optional and orthogonal decompositions of a class of semimartingales

We consider a set \mathcal Q of probability measures, which are absolutely continuous with respect to the physical probability measure \mathbb{P} and at least one is equivalent to \mathbb{P} . We investigate necessary and sufficient conditions on \mathcal Q , under which any \mathcal Q -supermartingale can be decomposed into the sum of a local \mathcal Q -martingale and a decreasing process. We also provide an orthogonal decomposition of square integrable semimartingale as the orthogonal sum of a local \mathcal Q -martingale and a square integrable semimartingale. As one application, we state the orthogonal decomposition in an appropriate sense of the polar set of \mathcal Q . We generalise then the results of a previous article (2021), from finite probability space and discrete time case to general continuous time case.

Reducing subspaces for rank-one perturbations of normal operators

We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator $T=D + u\otimes v$ (uniquely determined by such expression) is contained in a line, since in such a case $T$ has a reducing subspace if and only if $T$ is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators $T=D + u\otimes v$ with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of $T$ is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces $M$ of $T$ such that the restriction $T\mid _M$ is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.

The anisotropic part of a quadratic form over a number field

It is well known that every non-degenerate quadratic form admits a decomposition into an orthogonal sum of its anisotropic part and a hyperbolic form. This decomposition is unique up to isometry. In this paper we present an algorithm for constructing an anisotropic part of a given form with coefficients in an arbitrary number field.

Affine density, von Neumann dimension and a problem of Perelomov

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as ‘Nyquist rate’) determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the PSL(2,R) variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of ‘Nyquist rate’ is obtained as the trace of a certain localization operator. The proof consists of first connecting the problem to the theory of von Neumann algebras, by introducing a new class of projective representations of PSL(2,R) acting on non-analytic Bergman-type spaces. We then adapt to this setting a new method for computing von Neumann dimensions, due to Sir Vaughan Jones. Our solution contains necessary conditions in the form of a ‘Nyquist rate’ dividing frames from Riesz sequences of coherent states and sampling from interpolating sequences. They hold for an infinite sequence of spaces of polyanalytic functions containing the eigenspaces of the Maass operator and their orthogonal sums. Within mild boundaries, we show that our result is best possible, by characterizing our sequence of function spaces as the only invariant spaces under the non-analytic PSL(2,R)-representations.

Self-adjointness of the 2D Dirac Operator with Singular Interactions Supported on Star Graphs

We consider the two-dimensional Dirac operator with Lorentz-scalar \(\delta \)-shell interactions on each edge of a star graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum of half-line Dirac operators with off-diagonal Coulomb potentials. This decomposition reduces the computation of the deficiency indices to determining the number of eigenvalues of a one-dimensional spin–orbit operator in the interval \((-1/2,1/2)\). If the number of edges of the star graph is two or three, these deficiency indices can then be analytically determined for a range of parameters. For higher numbers of edges, it is possible to numerically calculate the deficiency indices. Among others, examples are given where the strength of the Lorentz-scalar interactions directly change the deficiency indices, while other parameters are all fixed and where the deficiency indices are (2, 2), neither of which have been observed in the literature to the best knowledge of the authors. For those Dirac operators which are not already self-adjoint and do not have 0 in the spectrum of the associated spin–orbit operator, the distinguished self-adjoint extension is also characterized.