Papers by Robert Owczarek
Witten constructed topological invariants of manifolds by introducing actions depending only on s... more Witten constructed topological invariants of manifolds by introducing actions depending only on smooth structures of the manifold without using a metric. He focused on an example of Chern-Simons' theory on 3D manifolds. Sławianowski based his field theory on frame/coframe fields without a metric. A Witten-type topological invariant can be introduced in Sławianowski's framework, using an integral over the reper fields. Such invariant behaves differently for group manifolds than for all others. It additionally distinguishes semi-simple Lie group manifolds from other group manifolds. Further study of this new invariant is needed, including possible generalizations to other field theories.
Technical Physics, 1995
Some considerations of the role of topology of configuration spaces of electrons for superconduct... more Some considerations of the role of topology of configuration spaces of electrons for superconductivity are presented. Geometries leading to the same exotic spinors as those following from the ring geometry are shown. Possibilities of more complicated topologies of electrons configuration spaces are pointed out. Reality of these structures is discussed.
International Journal of Theoretical Physics, 1991
Some aspects of classical field-theoretic phenomenology of superfluid helium are presented.
Arxiv preprint arXiv:0802.4112, 2008
Natural languages are described in this paper in terms of networks of synonyms: a word is identif... more Natural languages are described in this paper in terms of networks of synonyms: a word is identified with a node, and synonyms are connected by undirected links. Our statistical analysis of the network of synonyms in Polish language showed it is scale-free; similar to what is known for English. The statistical properties of the networks are also similar. Thus, the statistical aspects of the networks are good candidates for culture independent elements of human language. We hypothesize that optimization for robustness and efficiency is responsible for this universality. Despite the statistical similarity, there is no one-to-one mapping between networks of these two languages. Although many hubs in Polish are translated into similarly highly connected hubs in English, there are also hubs specific to one of these languages only: a single word in one language is equivalent to many different and disconnected words in the other, in accordance with the Whorf hypothesis about language relativity. Identifying language-specific hubs is vitally important for automatic translation, and for understanding contextual, culturally related messages that are frequently missed or twisted in a naïve, literary translation.
Journal of Mathematical Physics, 2005
Poisson structure for vortex filaments (loops and arcs) in 2D ideal incompressible fluid is analy... more Poisson structure for vortex filaments (loops and arcs) in 2D ideal incompressible fluid is analyzed in detail. Canonical coordinates and momenta on coadjoint orbits of the areapreserving diffeomorphism group, associated with such vortices, are found. The quantum space of states in the simplest case of "bosonic" vortex loops is built within a geometric quantization approach to the description of a quantum fluid. Fock-like structure and non-local creation and annihilation operators of quant urn vortex filaments are introduced.
for LLMC on Geometry Zbigniew Oziewicz Universidad Nacional Autonoma de Mexico Facultad de Estudi... more for LLMC on Geometry Zbigniew Oziewicz Universidad Nacional Autonoma de Mexico Facultad de Estudios Superiores Cuautitlan [email protected] This is an abstract of a lecture at Conference on Pure Mathematics LLMC, May 14-15, 2019. For the Session Geometry organized by Dr. Jerzy Kocik. The philosophy of conceptual Geometry needs category theory. The historical Geometry was reborn by Grassmann and Clifford algebras, and presently Geometry can be identified with Heinrich Brandt’ groupoid category. This lecture will be devoted to motivate and explain above statements also from historical perspective. Four-element group with unique unit, versus four-element groupoid with two units illustrate the essential distinction of a Brand’s groupoid from familiar concept of a group category. The Center of the Algebra of Observers William S. Page The primitive elements of the finite algebra of observers[1] are idempotent operators that may be represented as projectors in the Minkowski geome...
arXiv: Mathematical Physics, 2001
Structure constants of the $su(N)$ ($N$ odd) Lie algebras converge when N goes to infinity to the... more Structure constants of the $su(N)$ ($N$ odd) Lie algebras converge when N goes to infinity to the structure constants of the Lie algebra {\it sdiff}$(T^2)$ of the group of area-preserving diffeomorphisms of a 2D torus. Thus Zeitlin and others hypothesized that solutions of the Euler equations associated with $su(N)$ algebras converge to solutions of the Euler equations of incompressible fluid dynamics on a 2D torus. In the paper we prove the hypothesis. Our numerical experiments show the Galerkin method applied to Euler equation of hydrodynamics is computationally more efficient in the range of time in which it is stable than that based on the SU(N) approximation. However, the latter is stable for much longer time. These numerical results agree with theoretical expectations.
Physica B: Condensed Matter, 1992
In this paper v~c construct il canonical Hanliltollian svMclll +.)l Cqtlations Lu,~l()fe|'l]il]g ... more In this paper v~c construct il canonical Hanliltollian svMclll +.)l Cqtlations Lu,~l()fe|'l]il]g the C];AsMC:.II molioll ol ,i \oltcx lilamcnt in an incompressible fluid in a three-dimensional conliguration. Subsequently wc quantize the system, dcri~c the frequency spectrum of energy and compute some thcrmodynamical palalllClCFS O| the qllalltUlll excitations, such as spccilic heal and the mean-square displacement of the zero-point motion.
Journal of Knot Theory and Its Ramifications
The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A gene... more The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.
Journal of Physics a Mathematical and General, May 28, 1999
In this paper spinor structures over flag manifolds of compact simple Lie groups are considered a... more In this paper spinor structures over flag manifolds of compact simple Lie groups are considered and constructed explicitly, using the general method of Dabrowski and Trautman. In this way the existence and uniqueness of these structures is established, in accordance with purely topological results of Freed. Application of the structures for further studies of fermionic excitations in theories with coadjoint orbits as phase spaces, including infinite-dimensional systems such as superfluid helium, is suggested.
Mod Phys Lett B, 1993
In this letter, studies of knotted vortex structures in superfluid helium are continued. A model ... more In this letter, studies of knotted vortex structures in superfluid helium are continued. A model of superfluid phase transition (λ-transition) is built in this framework. Similarities of this model to the two-dimensional Ising model are shown. Dependence of specific heat of superfluid helium on temperature near the λ point is explained.
Journal of Knot Theory and Its Ramifications, 2016
Quantum computing is a field of great interest, attracting, among others, the attention of many m... more Quantum computing is a field of great interest, attracting, among others, the attention of many mathematicians. Although not all quantum mechanics is needed to successfully engage in research on quantum computing, the somewhat superficial approach usually applied by non-physicists is, in the opinion of the author of the lectures, not feasible. The following notes from lectures given at the mathematics department of George Washington University are meant to be a partial remedy to the situation, offering a very brief and slightly unorthodox introduction to one-particle quantum mechanics, and even shorter discussion of passage to multi-particle quantum mechanics, as needed for quantum computing.
Journal of Technical Physics, 1996
The equation: ReF`(T,Z)ZF`(T,Z) = 1 for conformal maps f(t,z) is important in interfacial dynamic... more The equation: ReF`(T,Z)ZF`(T,Z) = 1 for conformal maps f(t,z) is important in interfacial dynamics. We extend the results by Gustafsson on existence and uniqueness of solutions of this equation from the case when f(t,z) is a rational function of z to the case when the spatial derivative f`(t,z) is rational.
A systematic, language-independent method of finding a minimal set of paths covering the code of ... more A systematic, language-independent method of finding a minimal set of paths covering the code of a sequential program is proposed for application in White Box testing. Execution of all paths from the set ensures also statement coverage. Execution fault marks problematic areas of the code. The method starts from a UML activity diagram of a program. The diagram is transformed into a directed graph: graph's nodes substitute decision and action points; graph's directed edges substitute action arrows. The number of independent paths equals easy-to-compute cyclomatic complexity of the graph. Association of a vector to each path creates a path vector space. Independence of the paths is equivalent to linear independence of the vectors. It is sufficient to test any base of the path space to complete the procedure. An effective algorithm for choosing the base paths is presented.
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Papers by Robert Owczarek