Lie Group

Copyright © 2014 Sonja Čalamani and Dončo Dimovski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a given (3,j,)-metric d on a set M, j{1,2}, we define seven topologies on M induced by d, give some examples, examine the connection among them, and show that some other topologies induced by generalized metrics are special cases of the above ones.

In this paper we present a complete classification, up to isomorphism, of affine and projective fully commutative m + k m -groups, defined on subsets of the set of complex numbers C. The classification is via characteristic vectors and their associated complex curves (real surfaces).

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.

In this present study, we propose the concept of tricomplex-controlled metric spaces as a generalization of both controlled metric-type spaces and tricomplex metric-type spaces. In this work, we establish fixed point results using Banach, Kannan and Fisher-type contractions supported with nontrivial examples in the setting of the proposed space. We apply the derived result to find the analytical solution of an integral equation using the fixed point technique under the same metric.

Let $\Omega$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $\Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.

F ∈ Aut(C n ), is that det J(F ) ∈ C × . The Jacobian Conjecture speculates the validity of the inverse statement. Thus the Jacobian Conjecture is This is true for dimension n = 1 but it is wide open for dimension n ≥ 2. The original version of the conjecture was stated by Ott Keller in 1939 . Keller worked over the integers Z and with polynomial mappings F : Z n → Z n that satisfy the unimodular Jacobian condition det J(F ) = 1 or (-1). Since that time, much research has been done on the so-called Keller Problem, giving rise to great many beautiful ideas and theories. Notable results are the degree reduction theorems . These are based on K-theoretic principles that allow us, for example, to reduce the proof of the conjecture to the seemingly simple case of mappings of the form F = (X 1 , . . . , X n ) + H(X 1 , . . . , X n ), where (X 1 , . . . , X n ) is the identity mapping and H(X 1 , . . . , X n ) is a cubic homogeneous mapping. However, one should prove the Jacobian Conjecture for such cubic mappings in an arbitrary dimension. We also note that the conjecture is known to be true in any dimension and for mappings of degree deg F ≤ 2, . So it seems that "we are almost there". Yet the degree 3 case seems to be out of our reach at least for the present. Some experts in this field tend to believe that the general conjecture (n ≥ 2) might be false but that the two-dimensional case is possibly true. We mention here two pivotal two dimensional results. The first is a theorem of Moh , which asserts the validity of the conjecture for n = 2 and degree d = deg F ≤ 100 or so. This is a difficult result. The second result is the ingenious counterexample of Pinchuk [17] to the so-called Real Jacobian Conjecture ; namely, if F : R 2 → R 2 is a real polynomial mapping that satisfies the real Jacobian condition det J F (X, Y ) = 0 ∀ (X, Y ) ∈ R 2 , then F -1 exists. We note that in this case det J F need not be a non-zero constant, but that the conclusion is also weaker, namely, F -1 does not need to be a polynomial mapping. This natural real version of the original Jacobian Conjecture was open for sometime until in 1993 Pinchuk found a counterexample. His original clever construction gave rise to a degree 35 counterexample, but almost immediately it was reduced to a degree 25 counterexample. It is interesting to mention that the minimal degree of a counterexample is still not known, but we do have some lower bounds (d ≥ 7). Another approach to solving the Jacobian Conjecture is via a thorough analysis of the structure of the semigroup of all the normalized étale mappings on C n .This approach is outlined in the papers of Kambyashi [7,

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory has already a long history that essentially started with two remarkable papers of Tatsujirô Shimizu that were published in 1931. The elements φ(z) of the group are defined by the automorphic equation f (φ(z)) = f (z), were f (z) is entire. Tatsujirô Shimizu also refers to the functions of this group as those functions that are determined by f -1 • f . He proved many remarkable properties of those automorphic functions. He indicated how they induce a beautiful geometric structure on the complex plane. Those structures were termed by Tatsujirô Shimizu, the system of normal polygonal domains, and the more refined system of the fundamental domains of f (z). The last system if exists tiles up the complex plane with remarkable geometric tiles that are conformally mapped to one another by the automorphic functions. In the Ph.D thesis of the author, those tiles were also called the system of the maximal domains of f (z). One can not avoid noticing the many similarities between this automorphic group and its accompanying geometric structures and analytic properties, and the more tame discrete groups that appear in the theory of hyperbolic geometry and also the arithmetic groups in number theory. This paper pursues further the theory initiated by Tatsujirô Shimizu, towards understanding global properties of the automorphic group, rather than just understanding the properties of the individual automorphic functions. We hope to be able in sequel papers to generalize arithmetic and analytic tools such as the Selberg trace formula, to this new setting. Contents 1 Some background and the contribution of Tatsujirô Shimizu 4 14 Consequences to Aut(f ) that follow from the classical theory of entire functions 15 The relations between scattering theory and automorphic functions 16 Local groups 17 The sums of the k'th derivatives of all the elements of the automorphic group Aut(f ), for any f ∈ E of order 0 < ρ < 1 2 , k = 1, 2, 3, . . . 18 The circular density of the orbits of the automorphic group Aut(f ), for any f ∈ E of order 0 < ρ < 1 2 19 The Vieta formulas for Aut(f ), f ∈ E of order 0 ≤ ρ < 1 20 Embedding the automorphic group within a larger group 21 Relations between the construction of the direct system of the automorphic groups, and Weierstrass products 22 Continuity properties of the automorphic groups 23 Amenability of the automorphic group

F ∈ Aut(C n ), is that det J(F ) ∈ C × . The Jacobian Conjecture speculates the validity of the inverse statement. Thus the Jacobian Conjecture is This is true for dimension n = 1 but it is wide open for dimension n ≥ 2. The original version of the conjecture was stated by Ott Keller in 1939 . Keller worked over the integers Z and with polynomial mappings F : Z n → Z n that satisfy the unimodular Jacobian condition det J(F ) = 1 or (-1). Since that time, much research has been done on the so-called Keller Problem, giving rise to great many beautiful ideas and theories. Notable results are the degree reduction theorems . These are based on K-theoretic principles that allow us, for example, to reduce the proof of the conjecture to the seemingly simple case of mappings of the form F = (X 1 , . . . , X n ) + H(X 1 , . . . , X n ), where (X 1 , . . . , X n ) is the identity mapping and H(X 1 , . . . , X n ) is a cubic homogeneous mapping. However, one should prove the Jacobian Conjecture for such cubic mappings in an arbitrary dimension. We also note that the conjecture is known to be true in any dimension and for mappings of degree deg F ≤ 2, . So it seems that "we are almost there". Yet the degree 3 case seems to be out of our reach at least for the present. Some experts in this field tend to believe that the general conjecture (n ≥ 2) might be false but that the two-dimensional case is possibly true. We mention here two pivotal two dimensional results. The first is a theorem of Moh , which asserts the validity of the conjecture for n = 2 and degree d = deg F ≤ 100 or so. This is a difficult result. The second result is the ingenious counterexample of Pinchuk [17] to the so-called Real Jacobian Conjecture ; namely, if F : R 2 → R 2 is a real polynomial mapping that satisfies the real Jacobian condition det J F (X, Y ) = 0 ∀ (X, Y ) ∈ R 2 , then F -1 exists. We note that in this case det J F need not be a non-zero constant, but that the conclusion is also weaker, namely, F -1 does not need to be a polynomial mapping. This natural real version of the original Jacobian Conjecture was open for sometime until in 1993 Pinchuk found a counterexample. His original clever construction gave rise to a degree 35 counterexample, but almost immediately it was reduced to a degree 25 counterexample. It is interesting to mention that the minimal degree of a counterexample is still not known, but we do have some lower bounds (d ≥ 7). Another approach to solving the Jacobian Conjecture is via a thorough analysis of the structure of the semigroup of all the normalized étale mappings on C n .This approach is outlined in the papers of Kambyashi [7,

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface S 3 × S 1 . As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

This paper is an introduction to cosymplectic topology. Through it, we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold (M, ω, η) : (i)− we define and present the features of the space of almost cosymplectic vector fields (resp. cosymplectic vector fields); (ii)− we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is C 0−closed in the group Diff∞ (M) (a rigidity result), while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the C 0−limit ϕ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition based on Reeb’s vector field which guarantees that ϕ is a cosymplectic diffeomorphism is given (a ˛exibility condition), the cosymplectic analogues of the usual symplectic capacity-inequality theorem are derived and the cosymplectic analogu...

Let G be a compact connected Lie group. One may define R (G), the com- plex representation ring of G, as in [1], [9]. If H isasubgroup of maximalrank we may consider R (G) as a subring of R (H) , making R (H) an R (G)- module. An extension of the Weyl character formula found in [3] allows us to define an R (G)-module homomorphism from R (H) --. HomR (R (H), R (G)) in a very natural way. It is conjectured that if 1 (G) and 1 (H) have no 2-tor- sion then this map provides a duality isomorphism between R (H) and its dual. This result together with [8] will yield a new proof of the conjecture of Atiyah-Hirzebruch that : R(H) -, K (G/H) is onto [2] (see 9). In this paper the duality isomorphism is proven for G any classical group and H, a suitable subgroup of maximal rank. Furthermore R (H) is shown to be a free R (G)-module. Though this result already appears in [8] our proof will provide an explicit basis. This together with the duality isomorphism will provide a basis for the free abelian group K (G/H) (see 9). For those more familiar with the equivariant K-theory of [10], we know that R (H) .g (G/H) and R (G) g (point). The duality isomorphism is then a map from K(G/H) Hompoin (K(G/H), g(point)), and provides a "Poincare duality" result for this cohomology theory.

This paper introduces a global notion of slice regularity for functions on arbitrary real Clifford algebras Cl(p, q). Our framework is founded on the algebraic structure of power series with real coefficients, which implies a Wirtinger-type holomorphy across all "slice planes" C J generated by unit vectors J satisfying J 2 =-1 (elliptic) or J 2 = +1 (hyperbolic). This approach provides a robust framework for analysis in mixed-signature settings, where elliptic and hyperbolic structures necessarily coexist. The theory is rich enough to support transcendental functions, as we show by constructing slice-regular exponentials. We generate families of firstorder, Dirac-type operators associated with each slice, formalize the concept of a slice connection and its curvature, and clarify the relationship between our unified slice regularity and classical Feuter monogenicity, showing them to be distinct theories. Concrete examples in Cl(1, 1) and Cl(2, 1) illustrate the theory's applicability to physics models.

**Abstract**

*The Composite Inverse Cotangent Function: Theory, Scope, and Applications* by Rupesh Ranjan presents a comprehensive exploration of the composite inverse cotangent function, \(\cot^{-1}(f(x))\), and its iterated forms, \(h_n(x) = \cot^{-1}(\cot^{-1}(\cdots \cot^{-1}(x)))\), offering a rigorous mathematical framework for understanding its properties and applications. This book delves into the function’s analytical characteristics, including its bounded range \((0, \pi)\), smooth derivatives, and multi-valued complex behavior, establishing a foundation for its use across diverse fields. Through twelve meticulously structured chapters, the text covers foundational theory, functional equations, inverse problems, and applications in signal processing, control theory, computational geometry, machine learning, complex analysis, numerical methods, and dynamical systems. Advanced techniques, such as high-precision numerical evaluation, Lyapunov exponent analysis, and conformal mapping, are developed, with detailed case studies in audio processing, 5G signal design, robotic control, antenna design, and chaotic dynamics. Supported by computational examples in Python, MATLAB, and Julia, and enriched with exercises for theoretical and applied exploration, this book bridges pure mathematics with practical applications. It serves as an essential resource for graduate students, researchers, and professionals in mathematics, engineering, and computational sciences, illuminating the composite inverse cotangent function’s profound versatility and its potential to inspire future research in interdisciplinary domains.

This paper presents a formal model describing the conservation of temporal experience through information degradation as observed by a traveler receding from a luminous source (e.g., the Sun) at relativistic velocity. Despite the extreme time dilation predicted by special relativity, we show that perceptual cues, particularly the evolving visual field and signal attenuation, encode a continuous and structured experience of external time. This model uses special relativity, radiative transfer, and retarded time-to demonstrate that an observer perceives an ordered decay in image fidelity, angular geometry, and intensity consistent with an external duration. We conclude that perceived time is conserved in the visual evolution of the lightcone.

Given a compact space $X$ and a commutative Banach algebra $A$, the character spaces of $A$-valued function algebras on $X$ are investigated. The class of natural $A$-valued function algebras, those whose characters can be described by means of characters of $A$ and point evaluation homomorphisms, is introduced and studied. For an admissible Banach $A$-valued function algebra $\mathcal{A}$ on $X$, conditions under which the character space $M(\mathcal{A})$ is homeomorphic to $M(\mathfrak{A}) \times M(A)$ are presented, where $\mathfrak{A}=C(X) \cap \mathcal{A}$ is the subalgebra of $\mathcal{A}$ consisting of scalar-valued functions. An illustration of the results is given by some examples.

Let E be a Banach space and △ * be the closed unit ball of the dual space E * . For a compact set K in E, we prove that K is polynomially convex in E if and only if there exist a unital commutative Banach algebra A and a continuous function f : △ * → A such that (i) A is generated by f (△ * ), (ii) the character space of A is homeomorphic to K, and (iii) K = sp(f ) the joint spectrum of f . In case E = C (X), where X is a compact Hausdorff space, we will see that △ * can be replaced by X. 1

È(K, A) be the closure in C (K, A) of the restrictions P | K of polynomials P in È(E, A). It is proved that È(K,A) is an A-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product P N (K) ⊗ǫ A, where P N (K) is the uniform algebra on K generated by nuclear scalar-valued polynomials. The character space of È(K, A) is then identified with KN × M(A), where KN is the nuclear polynomially convex hull of K in E, and M(A) is the character space of A.

Given a compact space X, a commutative Banach algebra A, and an A-valued function algebra A on X, the notions of vector-valued spectrum of functions f ∈ A are discussed. The A-valued spectrum sp A (f ) of every f ∈ A is defined in such a way that f (X) ⊂ sp A (f ). Utilizing the A-characters introduced in (M. Abtahi, Vector-valued characters on vector-valued function algebras, arXiv:1509.09215 [math.FA]), it is proved that sp A (f ) = {Ψ(f ) : Ψ is an A-character of A }. For the so-called natural A-valued function algebras, such as C(X, A) and Lip(X, A), we see that sp A (f ) = f (X). When A = C, Banach A-valued function algebras reduce to Banach function algebras, A-characters reduce to characters, and A-valued spectrums reduce to usual spectrums.

The n-dimensional extension of the one dimensional Position-dependent mass (PDM) Lagrangians under the nonlocal point transformations by Mustafa [38] is introduced. The invariance of the n-dimensional PDM Euler-Lagrange equations is examined using two possible/different PDM Lagrangian settings. Under the nonlocal point transformation of Mustafa [38], we have shown that the PDM Euler-Lagrange invariance is only feasible for one particular PDM-Lagrangians settings. Namely, when each velocity component is deformed by some dimensionless scalar multiplier that renders the mass position-dependent. Two illustrative examples are used as reference Lagrangians for different PDM settings, the nonlinear n-dimensional PDM-oscillators and the nonlinear isotonic n-dimensional PDM-oscillators. Exact solvability is also indulged in the process.

The n-dimensional extension of the one dimensional Position-dependent mass (PDM) Lagrangians under the nonlocal point transformations by Mustafa [38] is introduced. The invariance of the n-dimensional PDM Euler-Lagrange equations is examined using two possible/different PDM Lagrangian settings. Under the nonlocal point transformation of Mustafa [38], we have shown that the PDM Euler-Lagrange invariance is only feasible for one particular PDM-Lagrangians settings. Namely, when each velocity component is deformed by some dimensionless scalar multiplier that renders the mass position-dependent. Two illustrative examples are used as reference Lagrangians for different PDM settings, the nonlinear n-dimensional PDM-oscillators and the nonlinear isotonic n-dimensional PDM-oscillators. Exact solvability is also indulged in the process.

In this paper, we prove a cyclic Lefschetz formula for foliations. To this end, we define a notion of equivariant cyclic cohomology and show that its expected pairing with equivariant K-theory is well defined. This enables to associate to any Haefliger's transverse invariant current on a compact foliated manifold, a Lefschetz formula for leafwise preserving isometries. Mathematical subject classification (1991). 19L47, 19M05, 19K56.

Let G = SU (2) and let ΩG denote the space of based loops in SU (2). We explicitly compute the R(G)-module structure of the topological equivariant K-theory K * G (ΩG) and in particular show that it is a direct product of copies of K * G (pt) ∼ = R(G). (We describe in detail the R(G)-algebra (i.e. product) structure of K * G (ΩG) in a companion paper.) Our proof uses the geometric methods for analyzing loop spaces introduced by Pressley and Segal (and further developed by Mitchell). However, Pressley and Segal do not explicitly compute equivariant K-theory and we also need further analysis of the spaces involved since we work in the equivariant setting. With this in mind, we have taken this opportunity to expand on the original exposition of Pressley-Segal in the hope that in doing so, both our results and theirs would be made accessible to a wider audience. CONTENTS 1. Introduction 1 2. Classical Results 4 3. The Grassmannian Gr z (K) and its subspaces 7 4. Description of filtration quotients as Thom spaces 12 5. The G-homotopy equivalence SGr ′ z bdd,r (K) → S Gr z α(psm),r (K) 14 6. Proof of the main theorem 23 References 24

We compute the symplectic volume of the symplectic reduced space of the product of N coadjoint orbits of a compact connected Lie group G. We compare our result with the result of Suzuki and Takakura , who study this in the case G = SU(3) starting from geometric quantization.

8 Equivariant Cohomology 33 8.1 Homotopy quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8.2 The Cartan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.3 Characteristic classes of bundles over BU(1) and BT . . . . . . . . . 35 8.4 Characteristic classes in terms of the Cartan model . . . . . . . . . . 37 8.5 Equivariant first Chern class of a prequantum line bundle . . . . . . . 38 8.6 Euler classes and equivariant Euler classes . . . . . . . . . . . . . . . 38 8.7 Localization formula for torus actions . . . . . . . . . . . . . . . . . . 39 8.8 Proof of localization theorem when M consists of isolated fixed points 40 8.9 Equivariant characteristic classes . . . . . . . . . . . . . . . . . . . . 43 8.10 The abelian localization theorem . . . . . . . . . . . . . . . . . . . . 45

Let G be a semisimple compact connected Lie group. An N -fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N -fold reduced product of G. Suzuki and Takakura gave a volume formula for the N -fold reduced product of SU(3) in [25] by using geometric quantization and the Riemann-Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of SU(3).

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S 1 on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat's convexity theorem. Namely, the images of Ω(G) and Ω(G) τ (fixed point set of τ ) under the T × S 1 moment map on Ω(G) are equal. The space Ω(G) τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in Z 2 of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [Bo-Sa] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [Ha-Ho-Pu].

For a compact, connected, simply-connected Lie group G , the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H 1 -Sobolev maps S 1 ! G: The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops .G/ is an example of a homogeneous space of LG and has a natural Hamiltonian T S 1 action, where T is the maximal torus of G . We study the moment map for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T -space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image . .G// is convex. We also show that for the energy functional E , which is the moment map for the S 1 rotation action, each non-empty preimage is connected. 53D20; 22E65

The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasi-Hamiltonian K-manifolds, where K is a simply connected compact Lie group. The imploded cross-section of the double K × K turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasi-Hamiltonian K-manifolds, such as the "spinning 2n-sphere" for K = SU(n). Secondly we construct a universal ("master") moduli space of parabolic bundles with structure group K over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of K and a torus-invariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the four-sphere for K = SU(2)), the master moduli space turns out to be a complex algebraic variety. CONTENTS 1. Introduction 1 2. Hamiltonian and quasi-Hamiltonian manifolds 3 3. Imploded cross-sections 6 4. The universal imploded cross-section 12 5. Implosion and weighted projective spaces 21 6. The master moduli space 25 Appendix A. The conjugation action 28 Appendix B. The spinning 2n-sphere 34 Notation Index 38 References 38

Equivariant cohomology was designed to allow the study of spaces which are the quotient of a manifold M by the action of a compact group G. This can be accomplished by studying the fixed point sets of subgroups of G, notably the maximal torus T . The Cartan model replaces the study of infinite-dimensional manifolds by families of differential forms on finite-dimensional G-manifolds parametrized by an element X in the Lie algebra of G with polynomial dependence on X . A version of de Rham cohomology can be developed for the Cartan model. The localization theorem of Atiyah-Bott and Berline-Vergne describes the evaluation of such an equivariantly closed differential form on the fundamental class of the manifold. In this chapter, we first define homotopy quotients in Sect. 9.2. We introduce the Cartan model in Sect. 9.3. We treat characteristic classes of bundles over classifying spaces in Sect. 9.4. We consider these characteristic classes in terms of the Cartan model in Sect. 9.5. We treat the equivariant first Chern class of a prequantum line bundle in Sect. 9.6. We treat Euler classes and equivariant Euler classes in Sect. 9.7 We treat the localization formula for torus actions in Sect. 9.8. Finally, we treat the abelian localization theorem of Atiyah-Bott and Berline-Vergne in Sect. 9.10. References for this chapter are Audin [1], Sect. 5 and Berline-Getzler-Vergne [2], Sect. 7. In this section, we first define classifying spaces, and then use them to define homotopy quotients.

Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra 𝔫 such that 𝔫 = 𝔞 ⊕ 𝔟 ⊕ 𝔷 ${\mathfrak {n=a\oplus b\oplus z}}$ , [ 𝔞 , 𝔟 ] ⊆ 𝔷 ${[ \mathfrak {a},\mathfrak {b}] \subseteq \mathfrak {z}}$ , the algebras 𝔞 , 𝔟 , 𝔷 ${\mathfrak {a},\mathfrak {b,z}}$ are abelian, 𝔞 = ℝ - span { X 1 , X 2 , ... , X d } ${\mathfrak {a}=\mathbb {R}\mathrm {\hbox{-}span}\lbrace X_{1},X_{2},\ldots ,X_{d}\rbrace }$ , and 𝔟 = ℝ - span { Y 1 , Y 2 , ... , Y d } ${\mathfrak {b}=\mathbb {R}\mathrm {\hbox{-}span}\lbrace Y_{1},Y_{2},\ldots ,Y_{d}\rbrace }$ . Also, we assume that det [ [ X i , Y j ] ] 1 ≤ i , j ≤ d ${\det [ [ X_{i} ,Y_{j}] ] _{1\le i,j\le d}}$ is a non-vanishing homogeneous polynomial in the unknowns Z 1 , ... , Z n - 2 d ${Z_{1},\ldots ,Z_{n-2d}}$ where { Z 1 , ... , Z n - 2 d } ${\lbrace Z_{1},\ldots ,Z_{n-2d}\rbrace }$ is a basis for the center of the Lie algebra. Using well-known facts from time-frequency analysis, we provide some precise suffi...

In this paper we are concerned with the problem, posed by R. R. Phelps, of describing the into isometries of the disk algebra. We show that, in a certain sense, every isometry can be approximated by convex combinations of isometries of the form /->k(f°φ). We also give some sufficient conditions for an isometry to be of the form /->k(f°φ).

In Carnot-Carathéodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

Despite all our efforts, the "3" of the ratio 1 : 3 remains mysterious. In this article it simply arises out of the structure constants for G 2 and appears in the construction of the embedding of so 3 × so 3 into g 2 (section 5 and Appendix B). Algebraically speaking, this '3' traces back to the 3 edges in g 2 's Dynkin diagram and the consequent relative positions of the long and short roots in the root diagram (see figure 2 below) for g 2 which the Dynkin diagram is encoding. Open problem. Find a geometric or dynamical interpretation for the "3" of the 3 : 1 ratio.

Posing Kepler's problem of motion around a fixed "sun" requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the "sun") provides the potential energy. Posing Kepler's three laws (with input from Galileo) requires symmetry conditions. The metric space must be homogeneous, isotropic, and admit dilations. Any Riemannian manifold enjoying these three symmetry properties is Euclidean. So if we want a semblance of Kepler's three laws to hold but also want to leave the Euclidean realm, we are forced out of the realm of Riemannian geometries. The Heisenberg group (a subRiemannian geometry) and lattices provide the simplest examples of metric spaces enjoying a semblance of all three of the Keplerian symmetries. We report success in posing, and solving, the Kepler problem on the Heisenberg group. We report failures in posing the Kepler problem on the rank two lattice and partial success in solving the problem on the integers. We pose a number of questions.

Dedicated to Alan Weinstein on his 60th Birthday * The authors thank the Brazilian funding agencies CNPq and FAPERJ: a CNPq research fellowship (JK), a CNPq postdoctoral fellowship at Berkeley (PMR), a FAPERJ visiting fellowship to Rio de Janeiro (KE). (JK) thanks the E. Schrödinger Institute, Vienna, for financial support during Alanfest and the Poisson Geometry Program, August 2003.

We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W (E 8 ) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W (E 8 ) by entangling gates, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of W (E 8 ) into the four-letter alternating group A 4 , obtained from a chain of maximal subgroups. Group A 4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

We describe the fine (group) gradings on the Heisenberg algebras, on the Heisenberg superalgebras and on the generalized Heisenberg algebras. We compute the Weyl groups of these gradings. Also the results obtained respect to Heisenberg superalgebras are applied to the study of Heisenberg Lie color algebras.

Let p denote an odd prime. The following three identities (transformation formulae) involving the Legendre symbol ί -j are known to be valid for any complex-valued function F defined on the integers, which is periodic with period p: We consider a general class of transformation formulae, which includes the above examples. Let p denote a fixed odd prime and let GF(p) denote the Galois field with p elements. If X denotes an indeterminate we let Corresponding to any element θ(X)e6[X] (often just written θeθ) we define Θ*(X) = DX 2 + AX + d , where For any element ψ(X) e Φ[X] (often just written φeΦ) its value at #eGF(;p) is just φ(x) = qx 2 + ra; 4se GF(p). For any element ΰ(X) eθ[X], θ(x) will be defined provided Ax 2 + Bx + C Φ 0 and its value is Throughout this paper whenever we write Σ the summation is takem X over all x e GF(p). If we write Σ' the summation is over all x e GF(p) X for which the summand is defined. Further we let c ^ denote the complex number field and we denote by ^" the set of all functions with domain GF(p) and range g^. The particular function χ e j^~ defined for any x e GF(p) by

We develop many-valued logic, including a generic abstract model theory, over a fully abstract syntax. We show that important many-valued logic model theories, such as traditional first-order many-valued logic and fuzzy multi-algebras, may be conservatively embedded into our abstract framework. Our development is technically based upon the so-called theory of institutions of Goguen and Burstall and may serve as a template for defining at hand many-valued logic model theories over various concrete syntaxes or, from another perspective, to combine many-valued logic with other logical systems. We also show that our generic many-valued logic abstract model theory enjoys a couple of important institutional model theory properties that support the development of deep model theory methods.

In this chapter we begin a study of the relationship between observer theory and quantum mechanics. The first section presents an overview of the characterization of quantum systems initiated by von Neumann, Weyl, Wigner, and Mackey. For this section we have relied heavily on the book by V.S. Varadarajan (1985). The second section discusses the appearance of vector bundles in this context. In the third section we explore possible connections between these vector bundles and linearizations of the specialized chain bundles of 9-5. Our approach is based on the idea that theories of measurement, which form the basis of quantum formalism, must have a large overlap with theories of perception. Quantum interpretations rest entirely on the interaction between observer and observed, and on the irreducible effects on both of them subsequent to such interaction. Conversely, it is reasonable to require of a theory of perception that it provide some illumination on the paradoxes that have dogged measurement theory to date. We must, however, make clear that in this chapter our intention is to provide neither a scholarly treatment of quantum measurement theory, nor a full and rigorous grounding of that theory in observer mechanics. Rather, we initiate a line of enquiry into their relationship, making a first attempt at setting up a language within which quantum measurement and perception-in-general may both be discussed. There are other stochastic-foundational formalisms which seek to ground quantum theory, such as those of Nelson (1985) and Prugovecki (1984). We here make no comparisons with these theories.

We develop the notion of g-angle between two subspaces of a normed space. In particular, we discuss the g-angle between a 1-dimensional subspace and a t-dimensional subspace for t ≥ 1 and the g-angle between a 2-dimensional subspace and a t-dimensional subspace for t ≥ 2. Moreover, we present an explicit formula for the g-angle between two subspaces of ℓ p spaces.

In the first half of this thesis the algebraic properties of a class of minimal, polynomial systems on IRn are considered. Of particular interest in the sequel are the results that (i) a tensor algebra generated by the observation space and strong accessibility algebra is equal to the Lie algebra of polynomial vector fields on IRn and (ii) the observation algebra of such a system is equal to the ring of polynomial functions on IRn. The former result is proved directly, but to establish the second we construct a canonical form for which the claim is trivial, the general case then following from the properties of the diffeomorphism relating the two realisations. It is also shown that, as a consequence of the structure of the observation space, any system in the class considered has a finite Volterra series solution, thereby showing that the canonical form developed is dual to that of Crouch. The second part of the work is devoted to the algebraic aspects of nonlinear filtering. The fu...