Probabilistic metric spaces and hysteresis systems

A phenomenological theory of simple hysteresis is constructed with the aid of certain concepts from the theory of probabilistic metric spaces. The predicted forms of the dependence of average energy loss per hysteresis cycle on the maximum excursion of the hysteresis coordinate agree well with experimental results.     

Infinitesimal calculus in metric spaces

We study the possibility of defining tangent vectors to a metric space at a given point and tangent maps to applications from a metric space into another metric space. Such infinitesimal concepts may help in analysing situations in which no obvious differentiable structure is at hand. Some examples are presented; our interest arises from hyperspaces in particular. Our approach is simple and relies on the selection of appropriate curves. Comparisons with other notions are briefly pointed out.     

Mixing transformations on metric spaces

It is proved that if a metric space is subjected to a mixing transformation, then there exists a positive numberx ₀ such that the probability that any arbitrary set of positive measure is asymptotically mapped into a set of diameter less thanx ₀ is zero. Physical implications of this result, in particular the interpretation of Poincaré recurrence, are discussed.     

Metrizability of spaces of homomorphisms between metric vector spaces

This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?     

Stochastic phase spaces,fuzzy sets,and statistical metric spaces

This paper is devoted to the study of the notion of the phase-space representation of quantum theory in both the nonrelativisitic and the relativisitic cases. Then, as a derived concept, the stochastic phase space is introduced and its connections with fuzzy set theory and probabilistic topological (in particular, metric) spaces are discussed.Supported by NSERC Grant No. A5206.On leave of absence from the Theoretical Physics Institute, University of Gdask, 80-952 Gdask, Poland.     

On finsler spaces with kropina metric

The curvature tensors and other important tensors of a Finsler space with Kropina metric are studied. We find conditions for such a space to be affinely connected in Berwald's sense.     

Self-similarity of complex networks and hidden metric spaces

We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.     

On quantum symmetries of compact metric spaces

An action of a compact quantum group on a compact metric space $(X,d)$ is (D)-isometric if the distance function is preserved by a diagonal action on $X\times X$. In this study, we show that an isometric action in this sense has the following additional property: the corresponding action on the algebra of continuous functions on $X$ by the convolution semigroup of probability measures on the quantum group contracts Lipschitz constants. In other words, it is isometric in another sense due to Li, Quaegebeur, and Sabbe, which partially answers a question posed by Goswami. We also introduce other possible notions of isometric quantum actions in terms of the Wasserstein $p$-distances between probability measures on $X$ for $p\ge 1$, which are used extensively in optimal transportation. Indeed, all of these definitions of quantum isometry belong to a hierarchy of implications, where the two described above lie at the extreme ends of the hierarchy. We conjecture that they are all equivalent.     

Realization of the Onsager lemma in metric spaces

The proof of the Onsager lemma based on the metric properties of the space is presented and discussed using the technique of mathematical induction. It is shown that the Onsager lemma is correct for arbitrary metric spaces.     

Generalized lorenz-mie theory and applications

Many optical sizing techniques rely on particle/laser interactions. The classical Lorenz-Mie theory describing sphere/plane wave interactions is therefore misleading when designing instruments and processing data when the particle size is not small enough with respect to beam diameters. In such cases the use of the generalized Lorenz-Mie theory is required. After summarizing essential features of the generalized Lorenz-Mie theory for sphere/arbitray wave interactions, this paper describes applications of the theory with some emphasis on the analysis of phase-Doppler anemometers.     

Generalized Noether theorems and applications

We generalize the first and second Noether theorems (Noether identities) to a constrained system in phase space. As an example, the conservation law deriving from Lagrange's formalism cannot be obtained fromH _E via the generalized first Noether theorem (GFNT); Dirac's conjecture regarding secondary first-class constraints (SFCC) is invalid in this example. A preliminary application of the generalized Noether identities (GNI) to nonrelativistic charged particles in an electromagnetic field shows that on the constrained hypersurface in phase space one obtains electric charge conservation. This conservation law is valid whether Dirac's conjecture holds true or not.     

Generalized Lorenz-Mie theory and applications

The basic formulas of generalized Lorenz-Mie theory are presented, and are applied to scattering of a focused Gaussian laser beam by a spherical particle. Various applications of focused beam scattering are also described, such as optimizing the rate at which morphology-dependent resonances are excited, laser trapping, particle manipulation, and the analysis of optical particle sizing instruments. Each of these applications requires either special positioning the beam with respect to the particle or illumination of only part of the particle by the beam.     

Generalized Weierstrass representation for surfaces in multi-dimensional Riemann spaces

Generalizations of the Weierstrass formulae to surface immersed into ⁴, ⁴ and into multidimensional Riemann spaces are proposed. Integrable deformations of surfaces in these spaces via the modified Veselov-Novikov equation are discussed.     

Generalized hydrodynamic equations and correlation functions for thermoviscoelastic fluids

For very viscous liquids a phenomenological theory of thermoviscoelasticity is formulated, in which the retarded reaction of thermal variables, which arises from structural relaxation, is taken into account. The theory describes the effect of the slowing down of the structural relaxation near a glass transition on the fluctuation spectra of density and entropy; in particular, the intensity of the slow relaxational component of the fluctuation spectra, which is frozen in the glass below the glass transition, is derived. Conditions for positive energy dissipation and symmetry relations are obtained in the framework of thermodynamic relaxation theory, and the memory functions occurring in the Mori-Zwanzig projection operator formalism are calculated.Dedicated to K. Dransfeld on the occasion of his 60th birthday     

Analysis of Young's fringes in speckle photography: Generalized square imaging aperture

Processing speckle photography data using Young's fringes method is described when a square imaging aperture is used. The direction of the lateral motion and hence the fringe orientation is assumed to be general. The cases considered are step motion (doubleexposure) and sinusoidal vibration in time-average manner.     

Quadrature fringes wide-field optical coherence tomography and its applications to biological tissues

We propose and demonstrate quadrature fringes wide-field optical coherence tomography (QF WF OCT) to expand an optical Hilbert transformation to two-dimensions. This OCT simultaneously measures two quadrature interference images using a single InGaAs CCD camera to obtain en face OCT images. The axial and lateral resolutions are measured at 29 μm in air and 70 μm limited by a pixel size of camera using a superluminescent diode with a wavelength of 1.3 μm as the light source; the system sensitivity is determined to be −90 dB. The area of the en face OCT images is 4.0 mm × 4.0 mm (160 × 160  pixels). The OCT images are measured axially with steps of 10 μm. The en face OCT images of a in vivo human fingertip and a in situ rat brain are three-dimensionally measured up to the depth of about 3 mm with some degradations of a lateral resolution.