Invariant Theory

The aspects of many particle systems as far as their entanglement is concerned is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.

There are certain invariants of heroic poetry, elements that define it essentially: the theme – a narrative about “extraordinary deeds”, famous, adventurous topics, in the style of “traditional heroic mythology” (Marino) – and the idealized typology so that the epic hero usually occurs like
a sum of the virtues appreciated by the community which assumes him. Very likely influenced by the Mesopotamian one from the beginning of the second millennia B.C., the Homeric epic exerts its influence further in time and cultural geography, over the Latin one from the end of the 1st century B.C. and, via the Aeneid, over the Germanic epic from the Middle Ages. Through Virgil, Homer’s  self -confessed  competitor,  there  could  be  clarified  an  issue  of  the  European  literary history: how did the transition from the “heroic song” of the Germanic tribes –“lost, as long as it was not recorded in writing” (Curtius) –, to the Anglo-Saxon heroic epic or the Medieval German one (mittelhochdeutsch) occur? The Aeneid
appears to be, according to Heusler and Curtius, the missing link between the epic tradition of Mediterranean Antiquity and the Germanic Medieval
heroic epic. (The latter, the product of a culture that, by its “barbarian” roots, had the inspiration of  and  ability  to  assimilate,  from  the  defeated  Rome,  the  classic  inheritance  of  Greek -Latin humanism).  The  literary  heredity  that  unites  Homer,  Virgil  and  the  Germanic  epic  on  the background  of  Mesopotamian  loans,  provides  yet  another  very  interesting  example  of (inter)cultural intermediation and continuity in the Eurasian space, within a historical interval of almost four millennia –from Gilgamesh’s legendary rule in Uruk and its literary echoes to the
Germanic heroic epic from the Middle Ages.

In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress-strain measures, which reduces for one special case to the so-called Green-Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensorvalued isotropic tensor functions. The proposed model is formulated in terms of generalized stress-strain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed.

The structure of the turbulent flow over a simplified automotive model, the Ahmed body ͑S. R. Ahmed and G. Ramm, SAE Paper No. 8403001, 1984͒ with a 25°slanted back face, is investigated using high-order large-eddy simulations ͑LESs͒ at Reynolds number Re= 768 000. The numerical approach is carried out with a multidomain spectral Chebyshev-Fourier solver and the bluff body is modeled with a pseudopenalization method. The LES capability is implemented thanks to a spectral vanishing viscosity ͑SVV͒ technique, with particular attention to the near wall region. Such a SVV-LES approach is extended for the first time to an industrial three-dimensional turbulent flow over a complex geometry. In order to better understand the interactions between flow separations and the dynamic behavior of the released vortex wake, a detailed analysis of the flow structures is provided. The topology of the flow is well captured showing a partial separation of the boundary layer over the slanted face and the occurrence of two strong contrarotating trailing vortices expanding farther in the wake. The interactions of these large vortices with smaller structures reminiscent of horseshoe vortices, within the shear layer over the slanted face, form large helical structures providing strong unsteady phenomena in the wake. Mean velocity fields and turbulence statistics show a global agreement with the reference experiments of Lienhart et al. ͑DGLR Fach Symposium der AG SRAB, Stuttgart University, 15-17 November 2000͒. In order to provide a deeper insight into the nature of turbulence, the flow is analyzed using power spectra and the invariant theory of turbulence of Lumley ͓Adv. Appl. Mech. 18, 123 ͑1978͔͒.

This article discusses basic approaches to transcribe foreign and borrowed words in Ukrainian, and Russian, Belarusian, and Ukrainian words in Latin script. It is argued that the adopted and foreign words should be rendered on different bases, namely by invariant transcription and transliteration. Also, current problems of implementation of the Ukrainian Latinics as an international graphical presentation of Ukrainian, are analyzed. The scholarly grounded simple-correspondent transliteration system for Belarusian, Russian, and Ukrainian, is given.

A new three-dimensional failure criteria for fibre-reinforced composite materials based on structural tensors is presented. The transverse and the longitudinal failure mechanisms are formulated considering different criteria: for the transverse failure mechanisms, the proposed three-dimensional invariant-based criterion is formulated directly from invariant theory. Regarding the criteria for longitudinal failure, tensile fracture in the fibre direction is predicted using a noninteracting maximum allowable strain criterion. Longitudinal compressive failure is predicted using a three-dimensional kinking model based on the invariant failure criteria formulated for transverse fracture. The proposed kinking model is able to account for the nonlinear shear response typically observed in fibre-reinforced polymers. For validation, the failure envelopes for several fibre-reinforced polymers under different stress states are generated and compared with the test data available in the literature. For more complex three-dimensional stress states, where the test data available shows large scatter or is not available at all, a computational micro-mechanics framework is used to validate the failure criteria. It is observed that, in the case of the IM7/8552 carbon fibre-reinforced polymer, the effect of the nonlinear shear behaviour in the failure loci is negligible. In general, the failure predictions were in good agreement with previous three-dimensional failure criteria and experimental data. The computational micro-mechanics framework is shown to be a very useful tool to validate failure criteria under complex three-dimensional stress states.

Feature detection and matching are used in image registration, object tracking, object retrieval etc. There are number of approaches used to detect and matching of features as SIFT (Scale Invariant Feature Transform), SURF (Speeded up Robust Feature), FAST, ORB etc. SIFT and SURF are most useful approaches to detect and matching of features because of it is invariant to scale, rotate, translation, illumination, and blur. In this paper, there is comparison between SIFT and SURF approaches are discussed. SURF is better than SIFT in rotation invariant, blur and warp transform. SIFT is better than SURF in different scale images. SURF is 3 times faster than SIFT because using of integral image and box filter. SIFT and SURF are good in illumination changes images. Keywords-SIFT (Scale Invariant Feature Transform), SURF (Speeded up Robust Feature), invariant, integral image, box filter

"In solving practical problems in science and engineering arises as a direct consequence differential equations that explains the dynamics of the phenomena.Finding exact solutions to this equations provides importan informationabout the behavior of physical systems. The Lie symmetry method allows tofind invariant solutions under certain groups of transformations for differentialequations.This method not very well known and used is of great importance inthe scientific community. By this approach it was possible to find several exactinvariant solutions for the Klein Gordon Equation uxx − utt = k(u). A particularcase, The Kolmogorov equation uxx − utt = k1u + k2un was considered.These equations appear in the study of relativistic and quantum physics. Thegeneral solutions found, could be used for future explorations on the study forother specific K(u) functions"

Is there a woman in mathematics? Of course there is. I will assure you of one important discovery of one of them, the brilliant Emmy (Amalie Emmy Noether, 1882-1935), which, because of the theorem, is called a kind of icon of algebra and physics, but whose significance will only grow, I hope for the theory of information I'm just doing.

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here.

Secret sharing is an important topic in cryptography and has applications in information security. We use self-dual codes to construct secret-sharing schemes. We use combinatorial properties and invariant theory to understand the access structure of these secret-sharing schemes. We describe two techniques to determine the access structure of the scheme, the first arising from design properties in codes and the second from the Jacobi weight enumerator, and invariant theory.

Most soft tissues possess an oriented architecture of collagen fiber bundles, conferring both anisotropy and nonlinearity to their elastic behavior. Transverse isotropy has often been assumed for a subset of these tissues that have a single macroscopicallyidentifiable preferred fiber direction. Micro-structural studies, however, suggest that, in some tissues, collagen fibers are approximately normally distributed about a mean preferred fiber direction. Structural constitutive equations that account for this dispersion of fibers have been shown to capture the mechanical complexity of these tissues quite well. Such descriptions, however, are computationally cumbersome for two-dimensional (2D) fiber distributions, let alone for fully three-dimensional (3D) fiber populations. In this paper, we develop a new constitutive law for such tissues, based on a novel invariant theory for dispersed transverse isotropy. The invariant theory is derived from a novel closed-form 'splay invariant' that can easily handle 3D fiber populations, and that only requires a single parameter in the 2D case. The model fits biaxial data for aortic valve tissue as accurately as the standard structural model. Modification of the fiber stress-strain law requires no reformulation of the constitutive tangent matrix, making the model flexible for different types of soft tissues. Most importantly, the model is computationally expedient in a finite-element analysis, demonstrated by modeling a bioprosthetic heart valve.

Extending differential-based operations to color images is hindered by the multi-channel nature of color images. The derivatives in different channels can point in opposite di- rections, hence cancellation might occur by simple addi- tion. The solution to this problem is given by the structure tensor for which opposing vectors reinforce each other. We review the set of existing tensor based features which are applied on luminance images and show how to expand them to the color domain. We combine feature detectors with photometric invariance theory to construct invariant features. Experiments show that color features perform better than luminance based features and that the addi- tional photometric information is useful to discriminate be- tween different physical causes of features.

Anti-symmetric tensor gauge theories are quantized via the Faddeev-Popov procedure leading to a BRS invariant theory without the need for second order ghosts. The quantum mechanical equivalence of these theories to non-linear sigma models is established using the spontaneously broken global symmetry currents. Upon gauging these symmetries the I-Iiggs mechanism is displayed, while for anomalous symmetries an analog of the Wess-Zumino action is constructed in terms of the anti-symmetric tensor gauge fields.

The theory of equilibrium relationships for non-equilibrium dependencies previously created for the batch reactor is further developed for the description of the Temporal Analysis of Products (TAP) experimental data. Corrections are necessary to account for the different diffusivity values affecting the gas transport over the inert zones in the reactor. These corrections can be performed in the Laplace domain, the Fourier domain, or, using special series of exponentially weighted integrals, directly in the time domain.

Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic $\neq2$. We also describe canonical forms for sequences of $2\times2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.

The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly-even codes over fields of characteristic 2, doubly-even codes over the integers modulo a power of 2, and self-dual codes over the noncommutative ring $\F_q + \F_q u$, where $u^2 = 0$..

Nagata gave a fundamental sucient condition on group actions on nitely generated commutative algebras for nite generation of the subalge- bra of invariants. In this paper we consider groups acting on noncommutative algebras over a eld of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corre- sponding relatively

Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchař model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of

Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchař model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

The spectral invariants theory presents an alternative approach for modeling canopy scattering in remote sensing applications. The theory is particularly appealing in the case of coniferous forests, which typically display grouped structures and require computationally intensive calculation to account for the geometric arrangement of their canopies. However, the validity of the spectral invariants theory should be tested with empirical data sets from different vegetation types. In this paper, we evaluate a method to retrieve two canopy spectral invariants, the recollision probability and the escape factor, for a coniferous forest using imaging spectroscopy data from multiangular CHRIS PROBA and NADIR-view AISA Eagle sensors. Our results indicated that in coniferous canopies the spectral invariants theory performs well in the near infrared spectral range. In the visible range, on the other hand, the spectral invariants theory may not be useful. Secondly, our study suggested that retrieval of the escape factor could be used as a new method to describe the BRDF of a canopy.

Let A = (A1, ..., An, ...) be a finite or infinite sequence of 2 × 2 matrices with entries in an integral domain. We show that, except for a very special case, A is (simultaneously) triangularizable if and only if all pairs (Aj, A k ) are triangularizable, for 1 ≤ j, k ≤ ∞. We also provide a simple numerical criterion for triangularization.

Four level quantum systems, known as quartits, and their relation to two- qubit systems are investigated group theoretically. Following the spirit of Klein's lectures on the icosahedron and their relation to Hopf sphere bra- tions, invariants of complex re ection groups occuring in the theory of qubits and quartits are displayed. Then, real gates over octits leading to the Weyl group of E8 and its invariants are derived. Even multilevel systems are of interest in the context of solid state nuclear magnetic resonance.

Self-dual codes over $\Z_2\times\Z_4$ are subgroups of $\Z_2^\alpha \times\Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exist a code $\C\subseteq \Z_2^\alpha \times\Z_4^\beta$ are established. Moreover, the construction of a $\add$-linear code for each type and possible pair $(\alpha,\beta)$ is given. Finally, the standard techniques of invariant theory are applied to describe the weight enumerators for each type.

In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which automatically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are presented. Furthermore, we propose an extension to orthotropic symmetry. : S 0 0 2 0 -7 6 8 3 ( 0 2 ) 0 0 4 5 8 -4

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, A/G, of the space A/G of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory.

We give a rigorous and mathematically well defined presentation of the Covariant and Gauge Invariant theory of scalar perturbations of a Friedmann-Lemaître-Robertson-Walker universe for Fourth Order Gravity, where the matter is described by a perfect fluid with a barotropic equation of state. The general perturbations equations are applied to a simple background solution of R n gravity. We obtain exact solutions of the perturbations equations for scales much bigger than the Hubble radius. These solutions have a number of interesting features. In particular, we find that for all values of n there is always a growing mode for the density contrast, even if the universe undergoes an accelerated expansion. Such a behaviour does not occur in standard General Relativity, where as soon as Dark Energy dominates, the density contrast experiences an unrelenting decay. This peculiarity is sufficiently novel to warrant further investigation of fourth order gravity models.

A new three-dimensional failure criteria for fibre-reinforced composite materials based on structural tensors is presented. The transverse and the longitudinal failure mechanisms are formulated considering different criteria: for the transverse failure mechanisms, the proposed three-dimensional invariant-based criterion is formulated directly from invariant theory. Regarding the criteria for longitudinal failure, tensile fracture in the fibre direction is predicted using a noninteracting maximum allowable strain criterion. Longitudinal compressive failure is predicted using a three-dimensional kinking model based on the invariant failure criteria formulated for transverse fracture. The proposed kinking model is able to account for the nonlinear shear response typically observed in fibre-reinforced polymers. For validation, the failure envelopes for several fibre-reinforced polymers under different stress states are generated and compared with the test data available in the literature. For more complex three-dimensional stress states, where the test data available shows large scatter or is not available at all, a computational micro-mechanics framework is used to validate the failure criteria. It is observed that, in the case of the IM7/8552 carbon fibre-reinforced polymer, the effect of the nonlinear shear behaviour in the failure loci is negligible. In general, the failure predictions were in good agreement with previous three-dimensional failure criteria and experimental data. The computational micro-mechanics framework is shown to be a very useful tool to validate failure criteria under complex three-dimensional stress states.

We present a general discussion of B-L violation in an SU(5) invariant theory, for a single generation of light fermions. We realize the simplest of such mechanism through the 15 Higgs. Its consequences for neutron oscillation are discussed.

A class of identities in the Grassmann Cayley algebra which yields a large number of geometric theorems on the incidence of subspaces of projective spaces was found by Hawrylycz (``Geometric Identities in Invariant Theory, '' Ph.D. thesis, Massachusetts, Institute of Technology, 1994). In this paper we establish a link between such identities in the Grassmann Cayley algebra and a class of inequalities in the class of linear lattices, i.e., the lattices of commuting equivalence relations. We prove that a subclass of identities found by Hawrylycz, namely, the Arguesian identities of order 2, can be systematically translated into inequalities holding in linear lattices. As a consequence, we obtain a family of geometric theorems on the incidence of subspaces that are characteristic-free and independent of dimensions. work been recognized, developed, and extended, first by Doubilet et al. [7], and then by many others including, but not limited to, Barnabei, Brini, Crapo, Hawrylycz, Huang, Kung, Sturmfels, White, and Whitely.

We prove that Weyl invariant theories of gravity possess a remarkable property which, under very general assumptions, explains the stability of flat spacetime. We show explicitly how conformal invariance is broken spontaneously by the vacuum expectation value of an unphysical scalar field; this process induces general relativity as an effective long distance limit. We show that all ghost degrees of freedom acquire gauge fixing dependent masses, strongly suggest,ing the unitarity of the theory.

Extending differential-based operations to color images is hindered by the multi-channel nature of color images. The derivatives in different channels can point in opposite directions, hence cancellation might occur by simple addition. The solution to this problem is given by the structure tensor for which opposing vectors reinforce each other.

The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly-even codes over fields of characteristic 2, doubly-even codes over Z/2 f Z, and self-dual codes over the noncommutative ring F q + F q u, where u 2 = 0.. AMS 2000 Classification: Primary 94B05, 13A50; Secondary: 94B60 2 Self-dual isotropic codes.

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectively invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.

For a prime number p, we construct a generating set for the ring of invariants for the p + 1 dimensional indecomposable modular representation of a cyclic group of order p 2 , and show that the Noether number for the representation is p 2 + p − 3. We then use the constructed invariants to explicitly describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case. In the appendix, we use our results to compute the Hilbert series for the corresponding ring of invariants together with some other related generating functions.

The results of the paper of Verlinde [hep-th/0008140], discussing the holographic principle in a radiation dominated universe, are extended when allowing the cosmic fluid to possess a bulk viscosity. This corresponds to a non-conformally invariant theory. The generalization of the Cardy-Verlinde entropy formula to the case of a viscous universe seems from a formal point of view to be possible, although we question on physical grounds some elements in this kind of theory, especially the manner in which the Casimir energy is evaluated. Our discussion suggests that for non-conformally invariant theories the holographic definition of Casimir energy should be modified.

We consider an extension of Massey's construction of secret sharing schemes using linear codes. We describe the access structure of the scheme and show its connection to the dual code. We use the g-fold joint weight enumerator and invariant theory to study the access structure.

We prove a characteristic free version of Weyl's theorem on polariza- tion. Our result is an exact analogue of Weyl's theorem, the dierence being that our statement is about separating invariants rather than gener- ating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, A/G, of the space A/G of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory.

This paper is about generalized Fourier descriptors, and their application to the research of invariants under group actions. A general methodology is developed, crucially related to Pontryagin's, Tannaka's, Chu's and Tatsuuma's dualities, from abstract harmonic analysis. Application to motion groups provides a general methodology for pattern recognition. This methodology generalizes the classical basic method of Fourier-invariants of contours of objects. In the paper, we use the results of this theory, inside a Support-Vector-Machine context, for 3D objectsrecognition. As usual in practice, we classify 3D objects starting from 2D information. However our method is rather general and could be applied directly to 3D data, in other contexts.

A novel geometric model of a noncommutative plane has been constructed. We demonstrate that it can be construed as a toy model for describing and explaining the basic features of physics in a noncommutative spacetime from a field theory perspective. The noncommutativity is induced internally through constraints and does not require external interactions. We show that the noncommutative space-time is to be interpreted as having an internal angular momentum throughout. Subsequently, the elementary excitationsi.e. point particles -living on this plane are endowed with a spin. This is explicitly demonstrated for the zero-momentum Fourier mode. The study of these excitations reveals in a natural way various stringy signatures of a noncommutative quantum theory, such as dipolar nature of the basic excitations [7] and momentum dependent shifts in the interaction point . The observation [9] that noncommutative and ordinary field theories are alternative descriptions of the same underlying theory, is corroborated here by showing that they are gauge equivalent.

The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum theory.

Conventional regular moment functions have been proposed as pattern sensitive features in image classi®cation and recognition applications. But conventional regular moments are only invariant to translation, rotation and equal scaling. It is shown that the conventional regular moment invariants remain no longer invariant when the image is scaled unequally in the x-and y-axis directions. We address this problem by presenting a technique to make the regular moment functions invariant to unequal scaling. However, the technique produces a set of features that are only invariant to translation, unequal/equal scaling and re¯ection. They are not invariant to rotation. To make them invariant to rotation, moments are calculated with respect to the principal axis of the image. To perform this, the exact angle of rotation must be known. But the method of using the second-order moments to determine this angle will also be inclusive of an undesired tilt angle. Therefore, in order to correctly determine the amount of rotation, the tilt angle which diers for dierent scaling factors in the x-and y-axis directions for the particular image must be obtained. In order to solve this problem, a neural network using the back-propagation learning algorithm is trained to estimate the tilt angle of the image and from this the amount of rotation for the image can be determined. Next, the new moments are derived and a Fuzzy ARTMAP network is used to classify these Information Sciences 000 000±000 .my (P. Raveendran). 0020-0255/00/$ -see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 -0 2 5 5 ( 0 0 ) 0 0 0 8 7 -6 U N C O R R E C T E D P R O O F images into their respective classes. Sets of experiments involving images rotated and scaled unequally in the x-and y-axis directions are carried out to demonstrate the validity of the proposed technique. Ó