Alexander Balinsky

Cardiff University | CU · School of Mathematics

UDC 517.9 We analyze the cohomology structure of the fundamental two-form deformation related to a modified Monge–Ampère type on the complex Kähler manifold P 2 ( ℂ ) . Based on the Levi-Civita connection and the related vector-field deformation of the fundamental two-form, we construct a hierarchy of bilinear symmetric forms on the tangent bundle...

We present a review of differential-geometric and Lie-algebraic approaches to the investigation of a broad class of nonlinear integrable differential systems of “heavenly” type associated with Hamiltonian flows on the spaces conjugate to the loop Lie algebras of vector fields on the tori. These flows are generated by the corresponding orbits of the...

The Special Issue “Symmetry of Hamiltonian Systems: Classical and Quantum Aspects” is addressed to mathematical physicists wanting to find some fresh views on results and perspectives in symmetry analysis of a wide class of Hamiltonian systems featuring their many applications in modern classical and quantum theory [...]

УДК 517.9 Наведено огляд диференціально-геометричних і Лі-алгебраїчних підходів до вивчення широкого класу нелінійних інтегровних диференціальних систем „небесного'' типу, асоційованих із гамільтоновими потоками на спряжених просторах до петельних алгебр Лі векторних полів на торах. Ці потоки породжуються відповідними орбітами коприєднаної дії пете...

УДК 517.9Наведено огляд диференцiально-геометричних i Лi-алгебраїчних пiдходiв до вивчення широкого класу нелiнiйних iнтегровних диференцiальних систем „небесного” типу, асоцiйованих iз гамiльтоновими потоками на спряжених просторах до петельних алгебр Лi векторних полiв на торах. Цi потоки породжуються вiдповiдними орбiтами коприєднаної дiї петель...

We propose some new matrix models for clonal network dynamics that are typical for simulating various biological clonal-type networks and study their dynamics to stable states. We present a detailed description and deduce the corresponding matrix equations governing the dynamics of immune systems on the basis of the general gradient-type principles...

We review some analytic, measure-theoretic and topological techniques for studying ergodicity and entropy of discrete dynamical systems, with a focus on Boole-type transformations and their generalizations. In particular, we present a new proof of the ergodicity of the 1-dimensional Boole map and prove that a certain 2-dimensional generalization is...

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directl...

The aim of this paper is to develop an algebraically feasible approach to solutions of the oriented associativity equations. Our approach was based on a modification of the Adler–Kostant–Symes integrability scheme and applied to the co-adjoint orbits of the diffeomorphism loop group of the circle. A new two-parametric hierarchy of commuting to each...

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the...

We review a modern differential geometric description of the fluid isotropic motion and featuring it the diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. There is analyzed the adiabatic liquid dynamics, within which, following the general approach, th...

There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-...

We review main differential-algebraic structures \ lying in background of \ analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and \ Leibniz type algebraic structures are derived, a new nonassociativ...

BACKGROUND Clinical trials are an important step in introducing new interventions into clinical practice by generating data on their safety and efficacy. Clinical trials need to ensure that participants are similar to be able to attribute any findings to the interventions studied and not some other factors. Therefore, each clinical trial defines el...

There are reviewed modern investigations devoted to studying nonlinear dispersiveless heavenly type integrable evolutions systems on functional spaces within the modern differential-geometric and algebraic tools. Main accent is done on the loop diffeomorphism group vector fi elds on the complexifi ed torus and the related Lie-algebraic structures,...

Background: Clinical trials are an important step in introducing new interventions into clinical practice by generating data on their safety and efficacy. Clinical trials need to ensure that participants are similar so that the findings can be attributed to the interventions studied and not some other factors. Therefore, each clinical trial defines...

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebrai...

Automatic document segmentation gets more and more attention in the natural language processing field. The problem is defined as text division into lexically coherent fragments. In fact, most of realistic documents are not homogeneous, so extracting underlying structure might increase performance of various algorithms in problems like topic recogni...

We present some results from a joint project between HP Labs, Cardiff University and Dyfed Powys Police on predictive policing. Applications of the various techniques from recommender systems and text mining to the problem of crime patterns recognition are demonstrated. Our main idea is to consider crime records for different regions and time perio...

Keyword and feature extraction is a fundamental problem in text data mining and document processing. A majority of document processing applications directly depend on the quality and speed of keyword extraction algorithms. In this article, an approach, introduced in [1], to rapid change detection in data streams and documents is developed and analy...

Keyword and feature extraction is a fundamental problem in data mining and document processing. A majority of applications directly depend on the quality and speed of keyword and feature extraction pre-processing results. In the current paper we present novel algorithms for feature extraction and change detection in unstructured data, primarily in...

The Hardy and Sobolev inequalities are of fundamental importance in many branches of mathematical analysis and mathematical physics, and have been intensively studied since their discovery. A rich theory has been developed with the original inequalities on \((0,\infty )\) extended and refined in many ways, and an extensive literature on them now ex...

In lectures delivered at New York University in 1953, and published posthumously in the proceedings [128] of the International Congress of Mathematicians held in Amsterdam in 1954, Rellich proved the following inequality which bears his name: for n ≠ 2 $$\displaystyle{ \int _{\mathbb{R}^{n}}\vert \Delta u(\mathbf{x})\vert ^{2}d\mathbf{x} \geq \frac...

In classical mechanics the motion of charged particles depends only on electric and magnetic fields E, B which are uniquely described by Maxwell’s equations: $$\displaystyle{\nabla \cdot \mathbf{E} = 4\pi \rho,}$$ $$\displaystyle{\nabla \cdot \mathbf{B} = 0,}$$ $$\displaystyle{\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t},}$$...

From the Hardy and Sobolev inequalities $$\displaystyle{\|\nabla u\|_{p,\Omega }^{p} \geq C_{ H}\|u/\delta \|_{p,\Omega }^{p},\ \ \ \|\nabla u\|_{ p,\Omega }^{p} \geq C_{ S}\|u\|_{p^{{\ast}},\Omega }^{p},\ \ \ u \in D_{ 0}^{1,p}(\Omega ),}$$ where \(\delta (\mathbf{x}) =\mathrm{ dist}(\mathbf{x},\partial \Omega ),C_{H},C_{S}\) are the optimal const...

Let \(\Omega \) be a domain (an open, connected set) in \(\mathbb{R}^{n}\) with non-empty boundary, \(1 < p < \infty \), and denote by δ(x) the distance from a point \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega,\) i.e., $$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mat...

Let \(\Omega \) be an open subset of \(\mathbb{R}^{n},\ n \geq 2,\) with non-empty boundary, and set $$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}}$$ for the distance of \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega \). Our main objectiv...

Automatic text segmentation, which is the task of breaking a text into topically-consistent segments, is a fundamental problem in Natural Language Processing, Document Classification and Information Retrieval. Text segmentation can significantly improve the performance of various text mining algorithms, by splitting heterogeneous documents into hom...

Example embodiments relate to keyword determination based on a weight of meaningfulness. In example embodiments, a computing device may determine a number of occurrences of a word in a particular document and may then determine a weight of meaningfulness for the word based on the number of occurrences. The computing device may then add the word to...

Unusual behaviour detection and information extraction in streams of short documents and files (emails, news, tweets, log files, messages, etc.) are important problems in security applications. In [1], [2], a new approach to rapid change detection and automatic summarization of large documents was introduced. This approach is based on a theory of s...

The extraction of keywords and features is a fundamental problem in text data mining. Document processing applications directly depend on the quality and speed of the identification of salient terms and phrases. Applications as disparate as automatic document classification, information visualization, filtering and security policy enforcement all r...

In this paper we describe the possibility of constructing the well-known small world topology for an ordinary document, based on the actual document structure. Sentences in such a graph are represented by nodes, which are connected if and only if the corresponding sentences are neighbors or share at least one common keyword. This graph is built usi...

Automatic text summarization is an important and challenging problem. Over the years, the amount of text available electronically has grown exponentially. This growth has created a huge demand for automatic methods and tools for text summarization. We can think of automatic summarization as a type of information compression. To achieve such compres...

Statistics of natural images have shown to be non-Gaussian quantities displaying high kurtosis, heavy tails and sharp central cusps, i.e. sparse distributions. In this paper we extend on these results to show that the non-linear filter response distributions of natural colour images are also sparse. We then incorporate the statistics of natural ima...

Dirac-Sobolev and Dirac-Hardy inequalities in $L^1$ are established in which the $L^p$ spaces which feature in the classical Sobolev and Hardy inequalities are replaced by weak $L^p$ spaces. Counter examples to the analogues of the classical inequalities are shown to be provided by zero modes for appropriate Pauli operators constructed by Loss and...

A Hardy inequality of the form \[\int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, \] for all $f \in C_0^{\infty}({\tilde{\Omega}})$, is considered for $p\in (1,\infty)$, where ${\tilde{\Omega}}$ can...

Non-linear filter responses of natural colour images have been shown to display non-Gaussian heavy tailed distributions which we call sparse. These filters operate in the YUV colour space on the chroma channel U (and V) using weighting functions obtained from the gray image Y. In this paper we utilise this knowledge for denoising the chroma channel...

Statistical analysis of natural luminance and colour images have shown to display non-Gaussian distributions. We review a recent result on the non-linear sparse filter response distribution observed in, and illustrate its application to the problem of colorizing natural gray images. We further utilise elements of both image statistics and colorizat...

Keyword extraction is a fundamental problem in text data mining and document processing. A large number of document processing applications directly depend on the quality and speed of keyword extraction algorithms. In this article, a novel approach to rapid change detection in data stream. and documents is developed. It is based on ideas from image...

We present some recent results on Hardy type inequalities in ℝ n , on open subset and for magnetic Dirichlet forms.

Natural images in the colour space YUV have been observed to have a non-Gaussian, heavy tailed distribution (called `sparse') when the filter ¿(U)(r) = U(r) - _s¿N(r)¿ w(Y)rsU(s), is applied to the chromacity channel U (and equivalently to V), where w is a weighting function constructed from the intensity component Y. In this paper we...

Over the last decade, there has been considerable interest and progress in determining the spectral properties of various operators that take relativistic effects into account, with important implications for mathematics and physics. Difficulties are encountered in many-particle problems due to the lack of semiboundedness of the Dirac operator, and...

In this article we establish a connection between semi-supervised learning and compressive sampling. We show that sparsity and compressibility of the learning function can be obtained from heavy-tailed distributions of filter responses or coefficients in spectral decompositions. In many cases the NP-hard problems of finding sparsest solutions can b...

We observe a non-Gaussian heavy tailed distribution for the non-linear filter \labelFilter g(U)(r) = U(\bold r) - å s Î N(r) w(Y)r s U(s), \label{Filter} \gamma(U)(\mathbf r) = U(\bold r) - \sum_{ \mathbf s \in N(\mathbf r)} w{(Y)_{\mathbf r \mathbf s}} U(\mathbf s), (1) applied to the chromacity channel ’U’ (and equivalently to ’V’) on indivi...

Hardy–Sobolev–type inequalities associated with the operator L := x · are established, using an improvement to the Sobolev embedding theo-rem obtained by M. Ledoux. The analysis involves the determination of the operator semigroup {e −tL * L } t>0 .

The paper analyses the decay of any zero modes that might exist for a massless Dirac operator $H:= \ba \cdot (1/i) \bgrad + Q, $ where $Q$ is $4 \times 4$-matrix-valued and of order $O(|\x|^{-1})$ at infinity. The approach is based on inversion with respect to the unit sphere in $\R^3$ and establishing embedding theorems for Dirac-Sobolev spaces of...

We use the spectra of Dirac type operators on the sphere $S^n$ to produce sharp $L^2$ inequalities on the sphere. These operators include the Dirac operator on $S^n$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inver...

The main result includes features of a Hardy-type inequality and an inequality of either Sobolev or Gagliardo-Nirenberg type. It is inspired by the method of proof of a recent improved Sobolev inequality derived by M. Ledoux which brings out the connection between Sobolev embeddings and heat kernel bounds. Here Ledoux's technique is applied to the...

Visual data mining (VDM) is an emerging research area of Data Mining and Visual Analytics gaining a deep visual understanding of data. A border between patterns can be recognizable visually, but its analytical form can be quite complex and difficult to discover. VDM methods have shown benefits in many areas, but these methods often fail in visualiz...

This report relates to the field of image restorations and features extracting from noisy and blurred images. Since their introduction in a classical paper by Rudin, Osher and Fatemi (2), Total Variation (TV) minimising models have become o ne of the most popular and successful tools for image restorations. Whilst invariance under affine transforma...

We present a Kato-type inequality for bounded domain Omega \subset R^n, n>1. Comment: 17 pages

In this article we present Sobolev-type inequalities for the localization of pseudo-relativistic energy.

We obtain lower bounds for the magnetic Dirichlet form in dimensions d greater than or equal to 2. For d = 2 the results generalize a well known lower bound by the magnetic field strength: we replace the actual magnetic field B by an non-vanishing effective field which decays outside the support of B as dist( x, supp B)(-2). In the case d greater t...

The paper discusses the existence of zero modes of Pauli and Weyl-Dirac operators and some mathematical and physical implications. A formal treatment in terms of quaternions is given, motivated by previous works, which clarifies the construction of examples of magnetic fields which give rise to zero modes. Recent results of the authors which prove...

The aim of this article is to extend the Laptev-Weidl inequalities to the case of Aharonov-Bohm magnetic potentials with multiple singularities.

The Brown–Ravenhall operator describes an electron under a Coulomb force and subject to relativity and is defined in terms of the associated Dirac operator and the projection onto the positive spectral subspace of the free Dirac operator. For a specific optimal charge range it is known to be positive. The paper investigates the following properties...

It is proved that the existence of zero modes of Weyl–Dirac operators is a rare phenomenon. An estimate of the multiplicity is given in term of the magnetic potential.

It is proved that for V+ = max(V, 0) in the subspace L 1 (R + ; L 1 (S 1 ); rdr) of L 1 (R 2 ), there is a Cwikel-Lieb-Rosenblum type inequality for the number of negative eigenvalues of the operator 1 i ~ r+ ~ A 2 V in L 2 (R 2 ) when ~ A is an Aharonov-Bohm magnetic potential with non-integer flux. It is shown that L 1 (R + ; L 1 (S 1 ); rdr) can...

In a previous article, the first two authors have proved that the existence of zero modes of Pauli operators is a rare phenomenon; inter alia, it is proved that for |B|L3/2(3), the set of magnetic fields B which do not yield zero modes contains an open dense subset of [L3/2(3)]3. Here the analysis is taken further, and it is shown that Sobolev, H...

In a previous article, the first two authors have proved that the existence of zero modes of Pauli operators is a rare phenomenon; inter alia, it is proved that for [iopmath latex="$|{\bi B}| \in L^{3/2}({\Bbb R}^3)$"] |B|L3/2(3) [/iopmath] , the set of magnetic fields [iopmath latex="${\bi B}$"] B [/iopmath] which do not yield zero modes contains...

. It is proved that the existence of zero modes of Weyl-Dirac operators is a rare phenomenon. An estimate of the multiplicity is given in term of the magnetic potential. 1. Introduction In [1] it was proved that the set of magnetic elds which give rise to zero modes of the associated Pauli operators in R 3 is rather sparse. Specically the following...

It is proved that the form domain of the magnetic Schrödinger operator SA in L 2 (R 2 ) with an Aharonov-Bohm magnetic field is continuously embedded in L 1 (R + ; rdr) L 2 (S 1 ). An implication of this is that, when V 2 L 1 (R + ; rdr) L 1 (S 1 ), SA and SA + V have the same form domain and coincident essential spectrum, namely [0, 1)

Two results are proved for nul P-A, the dimension of the kernel of the Pauli operator P-A = {sigma . ( 1/t del + (A) over right arrow}(2) in [L-2(P-3)/(2): (i) for /(B) over right arrow/ is an element of L-3 (2)(R-3), where (B) over right arrow = curl (A) over right arrow is the magnetic field, nul Pt-A = 0 except for a finite number of values of t...

ON THE BROWN-RAVENHALL RELATIVISTIC HAMILTONIAN AND THE STABILITY OF MATTER ON THE BROWN-RAVENHALL RELATIVISTIC HAMILTONIAN AND THE STABILITY OF MATTER Abstract . This is a survey of recent work on the stability of relativistic matter with and without magnetic field , mainly for the case when the no- pair Hamiltonian of Brown and Ravenhall is tak...

We study the asymptotic behavior, as Planck's constant $\hbar\to 0$, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field $\mu\mathbf{B}(x)$ and an electric field. The magnetic field strength $\mu$ is allowed to tend to infinity as $\hbar\to 0$. Two main types of results are established: in th...

In appropriate units, the Brown-Ravenhall Hamiltonian for a system of 1 electron relativistic molecules with K fixed nuclei having charge and position Zk, Rk, k=1,2, ¼,Kk=1,2, \ldots,K, is of the form \bB1,K = L+ ( D0 + aVc) L+ \bB_{1,K}= \Lambda_+ \bigl( D_0 + \alpha V_c\bigr) \Lambda_+ , where &#118+ is the projection onto the positive spectral...

In appropriate units, the no-pair Hamiltonian for a system of one-electron relativistic molecules with K fixed nuclei, having charge and position Z k ,R k ,k=1,2,⋯,K, is of the form B 1,K =Λ + (D 0 +αV c )Λ + , where Λ + is the projection onto the positive spectral subspace of the free Dirac operator D 0 and V c =-∑ k=1 K αZ k /|x-R k |+∑ k<l,k,l=1...

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than max(2 \frac12 \frac{1}{2} )mc2, where is the fine structure constant, for all values of the nuclear charge Z below the critical value Zc:...

The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative struc...

Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie groups are canonically equipped with a Poisson Li...

New algebraic structure on the orbits of dressing transformations of the quasitriangular Poisson Lie groups is provided. This give the topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation. Some applications to the three-dimensional topological quantum field theor...

Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of...

Topological interpretation of the link invariants associated with the Weinstein--Xu classical solutions of the quantum Yang-Baxter equation are provided.

We present the method for finding of the nonlinear Poisson-Lie groups structures on the vector spaces and for their quantization. For arbitrary central extension of Lie algebra explicit formulas of quantization are proposed.