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We generalize Poisson-Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the homogeneous Poisson-Nijenhuis structures and their relations with Jacobi structures., Comment: 21 pages, LateX

We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-Patodi-Singer boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.

Consider a sequence F → which is nondecreasing and onto. Then the quantity defined by F ′ m = 1 F−1 F m (where as usual F−1 k denotes the set F = k ) can be interpreted as the slope of F at m. Accordingly we call the function F ′ → the discrete derivative of F . And developing further the analogy with usual differential equations we can see that the notion of integration of this differential equation makes sense: it corresponds to reconstructing the sequence F from the sequence τ F = F−1 k k=1 of its runs.

Zeta-regularized products are used to define determinants of operators in infinite dimensional spaces. This article provides a general theory of regularized products and determinants which delivers a better approach to their existence and explicit determination.

We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived properties of the new methods.

The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapping) set of the flow similar to those of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly non-integrable horocycle foliations. As a consequence, we get exponential decay of correlation for Holder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp, we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re( s ) h T - e, where h T is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number π(λ) of closed orbits of least period λ 0.

For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexified rotation number tends to a rational value non-tangentially to the real axis, thus generalizing previous results of the authors. The function conjugating the dynamics to rotations by ω possesses a limit which is explicitly computed and related to hyperelliptic functions in the case of nonlinear terms which are trigonometric polynomials. The case of the standard map is shown to be non-generic.

The quantum mechanical three-body problem with Coulomb interaction is formulated within the adiabatic representation method using the hyperspherical coordinates. The Kantorovich method of reducing the multidimensional problem to the one-dimensional one is used. A new method for computing variable coefficients (potential matrix elements of radial coupling) of a resulting system of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabatic functions obtained as solutions of an auxiliary parametric eigenvalue problem. In the method proposed, a new boundary parametric problem with respect to unknown derivatives of eigensolutions in the adiabatic variable (hyperradius) is formulated. An efficient, fast, and stable algorithm for solving the boundary problem with the same accuracy for the adiabatic eigenfunctions and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sphere that has an analytical solution. The accuracy, efficiency, and robustness of the algorithm are studied in detail. The method is also applied to the computation of the ground-state energy of the helium atom and negative hydrogen ion.

We study the Lie algebra of infinitesimal isometries of seven-dimensional simply connected manifolds with Killing spinors. We obtain some splitting theorems for the action of this algebra on the space of Killing spinors, and as a corollary we prove that there is no infinitesimal isometry of constant length on a seven-dimensional 3-Sasakian manifold (not isometric to a space form) except the linear combinations of the Sasakian vector fields.

Phase space tunneling and exponential decay of eigenfunctions in phase space are well known for operators with symbols which are analytic in some neighborhood of the real axis. This can be used to prove an adiabatic theorem of exponential order if one assumes the Hamiltonian to depend analytically on time. However to study compactly supported switching processes one has to weaken the analyticity assumptions. Here we examine nonanalytic symbols with Gevrey class regularity and show that we get an exponential decay of the corresponding eigenfunctions with respect to ℏ1/a as ℏ→0, where a>1. The loss of regularity causes a slower decay in ℏ. The analysis is done using the methods of Martinez and its generalization by Nakamura. An upper bound for the rate of decay is given.

Let P P be a principal O ( 2 m ) O(2m) bundle over a manifold M M of dimension 4 m 4m , and let p p be its m m -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class p p can be realised as the Pontrjagin form of some connection on P P ?

Two different realizations of the supersymmetric sine algebra (SSA) are given. We show that the quantum superalgebra slq(2/1) is derived from the SSA. We discuss the relevance of the latter result to the study of spin- 1 2 Bloch electron in a constant magnetic field. The relation between the deformation parameter q and the degeneracy of Landau levels is established.

The Cauchy problem for Whitham equations with monotonic analytic initial data is studied. If the initial data f(u) satisfies the condition f(2N+1)(u)

Recently quite a few papers appeared proving complex bounds, local connectivity of Julia sets, absence of invariant linefields and quasisymmetric rigidity for real polynomial maps. Let us first discuss the unimodal case. In 1990 Martens [Ma] proved that there are real bounds for a certain sequence of first return maps to suitable central intervals. In 1990 Sullivan showed that if the unimodal maps are real analytic, infinitely renormalizable and have bounded combinatorics, then their first return maps extend to the complex plane as polynomial-like maps, see [Su] and also [MS]. In 1994 the present authors proved that all real analytic unimodal maps allow such polynomial-like complex extensions for the ‘non-central’ first return maps. The proof even gave explicit numerical lower bounds for the moduli of certain annuli, [LS1]. For the quadratic case, different proofs were later provided by [GS1], [LY]. For the smooth case see [Ko]. In 1996 Sands, see [Sa], improved our numerical bounds for maps of the form g(x ) so that the Schwarzian derivative of g is negative. This polynomial-like structure is one of the main ingredients in many recent results on smooth unimodal maps. For example, it is heavily used in the proof

Nous étudions divers types d'extensions galoisiennes d'anneaux non commutatifs et considérons quelques exemples. Nous énonc cons un critère qui permet de décider si un automorphisme donné d'une extension galoisienne appartient ou non au groupe de Galois. Nous calculons la cohomologie non abélienne de Borovoi en degré 0, 1 et 2 du groupe de Galois à coefficients dans un module croisé associé à l'extension galoisienne.

The splitting of a Q -deformed boson, in the Q → q = e 2πi k limit, is discussed. The equivalence between a Q -fermion and an ordinary one is established. The properties of quantum algebras U Q ( sl ( m )), the quantum superalgebras osp Q ( 1 2m ) and U Q (sl( m 1 )) and the deformed Virasoro algebra when their deformation parameter Q goes to a root of unity, are investigated. These properties are shown to be related to fractional supersymmetry and k -fermionic spin.

The dynamical entropy of space translations is used in the variational calculus of the pressure. It is shown that for quantum spin systems on a lattice of an arbitrary dimension, the pressure obtained in this way coincides with the usual one.

We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves., 50 pages, 6 Postscript figures

By assuming the symmetry group and using the inverse method, we find an integrable system which is constructed from superposition of two hierarchies of integrable equations. We discuss S U (2) and S L (2, R ) cases in detail. Furthermore, we present a procedure to construct integrable equations by taking a coefficient in the hierarchies as function of x and t .

A simplified color image formation model is used to construct an algorithm for image reconstruction from CCD sensors samples. The proposed method involves two successive steps. The first is motivated by Cok's (1994) template matching technique, while the second step uses steerable inverse diffusion in color. Classical linear signal processing techniques tend to oversmooth the image and result in noticeable color artifacts along edges and sharp features. The question is how should the different color channels support each other to form the best possible reconstruction. Our answer is to let the edges support the color information, and the color channels support the edges, and thereby achieve better perceptual results than those that are bounded by the sampling theoretical limit.

We classify completely all quasi-complete intersections of monomial curves in K an infinite field, see Theorem 4.1 and Theorem 4.2. This completes the investigations started in [4].

A filling by holomorphic dics of the levels of a Morse function on a hypersurface in C2 near the critical point is constructed. On the critical level it is similar to a filling by Bishop's families. It is the uniform limit of such families filling the levels consisting of two components near their elliptic points. Moreover, it can be deformed into families of holomorphic discs filling the levels consisting of one component. For hypersurfaces of contact type near the critical point the regularity of the induced foliation of the pseudoconvex side is shown. This is used to construct smooth, regular extensions with Levi-flat levels of certain Morse functions on (smooth) boundaries of Stein surfaces.

Let D be the unit disk andEbe a compact subset of theH∞maximal ideal space,M(H∞). We show that any continuous functionfon an open neighborhoodU⊂M(H∞) ofEsuch thatf|U∩D∈H∞(U∩D) can be uniformly approximated onEby ratiosh/g, whereh,g∈H∞andgis zero free onE. We also characterize those setsEfor whichh/gcan be replaced byhfor everyf. Finally, we prove that any inner functionucan be written as a rational function of interpolating Blaschke products. It follows thatucan be uniformly approximated by ratios of interpolating Blaschke products on any compact setE⊂M(H∞) whereuis zero free.

The central result of this paper is the following generalization of a result of the author on products of Severi­Brauer varieties. Let G be a semi­simple linear algebraic group over a field k. Let V be a generalized flag variety under G. Then there exist finite extensions ki of k for 1 6 i6 m, elements αi in Brki and a natural exact sequence m M i=1 k ∗ Nki/k(.∪αi) −−−−−→ Ker H 3 (k, Q/Z(2))→H 3 (k(V), Q/Z(2)) � →CH 2 (V)tors→0.

We will give the definition and basic properties of nearly parallel G2(2)∗-structures on pseudo-Riemannian manifolds of signature (4,3). In particular we explain the equivalence of their existence with that of Killing spinor fields. Furthermore, we will give first examples of pseudo-Riemannian manifolds of signature (4,3) with Killing spinors.

The paper examines some relationships between minimal generating sets of prime ideals of height 2, which are quasi-complete intersections. By characterizing the ideals p(n1, n2, n3) of monominal curves in P k3 with μ(p(n1, n2, n3)) = 4, which are quasi-complete intersections, it is shown that our results are best possible.

We present new fast algorithms for solving the Toeplitz and the Toeplitz-plus-Hankel least squares problems. These algorithms are based on a new fast algorithm for solving the Cauchy-like least squares problem. We perform an error analysis and provide conditions under which these algorithms are numerically stable. We also develop implementation techniques that significantly reduce the execution time. While no previous fast algorithm is known to be numerically stable for very ill conditioned problems, our numerical results indicate that these new algorithms are efficient and numerically stable for problems ranging from well conditioned to very ill conditioned to numerically singular.

A nearly parallel G 2 -structure on a seven-dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G 2 -manifolds with large symmetry group and in particular all homogeneous nearly parallel G 2 -structures.

When phrased in terms of Hausdorff convergence, M. Gromov's almost flat manifold theorem states that if a compact manifold M admits a bounded curvature collapse to a point, then a finite cover of M is necessarily diffeomorphic to a nilmanifold. It is then tempting to ask whether the prescription of more general Hausdorff limits will still place some kind of homogeneity conditions or other severe restrictions on a manifold, or, in fact, whether a compact manifold can even be topologically characterized by the bounded curvature collapses it allows. In this paper, we study these questions in mainly the solvable category. Improving earlier works on this problem, and solving a conjecture of Fukaya, we first show that if a compact manifold M admits a bounded curvature collapse to a compact flat orbifold of arbitrary dimension, then a finite cover of M is diffeomorphic to a solvmanifold. As a partial converse to this result, we obtain the theorem that any compact infrasolvmanifold M admits a generalized Seifert fibration over a compact flat orbifold, and a sequence of locally homogeneous metrics such that M allows a bounded curvature collapse to this orbifold. Locally homogeneous collapsing metrics with bounded curvature and diameter are also constructed on any infrahomogeneous space which is modelled on a Lie goup whose radical is nontrivial. In the topological category, the above results already lead to a metrical characterization of infrasolvmanifolds in the spirit of Farrell and Hsiang. However, in the smooth setting, the situation is quite different, and for an analogous smooth characterization our first theorem would have to be appropiately sharpened. Indeed, we show that the classes of compact smooth manifolds that admit finite coverings by homogeneous spaces are in general strictly bigger than the corresponding classes of infrahomogeneous spaces. In the standard literature on the almost flat manifold theorem, this crucial distinction has nearly never been made. Finally, we investigate some stability and closedness properties of classes of infrasolvmanifolds under Hausdorff convergence, give an account of the history of the results presented here, and discuss some related open questions and conjectures.

Let T be the collection of all Hs(s > 2) diffeomorphisms ηφ of the cylindrical surface M ≔ S1 × [p, q], where ηφ rotates each “horizontal” circle S1 × z rigidly by an angle φ = φ(z) which is a real valued Hs function. LetM be given the flat metric g induced from the Euclidean metric of R3. We shall prove in this paper that (1) Topologically, T is a real, infinite-dimensional, smooth, path-connected and closed submanifold of Diffvol relative to the Hs topology. (2) Algebraically, T is a maximal Abelian subgroup of Diffvol, it is equal to its centralizer in Diff, and its Weyl group in Diffvol is Z2. (3) Geometrically, with respect to the g-induced right-invariant L2 metric 〈·, ·〉, T is a totally geodesic and flat Riemannian submanifold of Diffvol; we also identify its normal bundle.

The interplay between temperature and q -deformation in the phase transition properties of many-body systems is studied in the particular framework of the collective q -deformed fermionic Lipkin model. It is shown that in phase transitions occuring in many-fermion systems described by su (2) q -like models are strongly influenced by the q -deformation.

The massive Curci-Ferrari model in the nonlinear Delbourgo-Jarvis gauge can be renormalized analogously to Yang-Mills theories in linear gauges if besides the (anti-) Slavnov-Taylor and (anti-) Delduc-Sorella identies also the equations of motion for the auxillary field and the (anti-)ghost fields as well as the so-called local Lee identity is required.

Some special cases of the generalized hypergeometric function q +1 F q with rational numbers as parameters are given in tabular form. These results complement existing tables. Some analytical aspects are discussed, and a derivation is given for those cases which correct existing table entries or replace numerical values by analytic expressions.

Modern analysis of HEP data needs advanced statistical tools to separate signal from background. This is the first book which focuses on machine learning techniques. It will be of interest to almost every high energy physicist, and, due to its coverage, suitable for students. Modern analysis of HEP data needs advanced statistical tools to separate signal from background. This is the first book which focuses on machine learning techniques. It will be of interest to almost every high energy physicist, and, due to its coverage, suitable for students.

Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.

Off-diagonal vacuum and nonvacuum configurations in Einstein gravity can mimic physical effects of modified gravitational theories of $f(R,T,R_{\mu \nu}T^{\mu \nu})$ type. To prove this statement, exact and approximate solutions are constructed in the paper, which encode certain models of covariant Ho\vrava type gravity with dynamical Lorentz symmetry breaking. The corresponding FLRW cosmological dynamics with possible nonholonomic deformations and the reconstruction procedure of certain actions closely related with the standard $\Lambda$CDM universe are studied. Off-diagonal generalizations of de Sitter universes are constructed which are generated through nonlinear gravitational polarization of fundamental physical constants and which model interactions with non-constant exotic fluids and effective matter. The problem of possible matter instability for such off-diagonal deformations in (modified) gravity theories is briefly discussed. Certain off-diagonal vacuum and nonvacuum configurations in Einstein gravity can mimic physical effects of modified gravitational theories of $ f(R,T,R_{\mu \nu }T^{\mu \nu })$ type. We prove this statement by constructing exact and approximate solutions which encode certain models of covariant Hořava type gravity with dynamical Lorentz symmetry breaking. Off-diagonal generalizations of de Sitter and nonholonomic $\Lambda $ CDM universes are constructed which are generated through nonlinear gravitational polarization of fundamental physical constants and which model interactions with non-constant exotic fluids and effective matter. The problem of possible matter instability for such off-diagonal deformations in (modified) gravity theories is discussed. Certain off-diagonal vacuum and nonvacuum configurations in Einstein gravity can mimic physical effects of modified gravitational theories of $ f(R,T,R_{\mu\nu}T^{\mu\nu})$ type. We prove this statement by constructing exact and approximate solutions which encode certain models of covariant Ho\v rava type gravity with dynamical Lorentz symmetry breaking. Off-diagonal generalizations of de Sitter and nonolonomic $\Lambda$CDM universes are constructed which are generated through nonlinear gravitational polarization of fundamental physical constants and which model interactions with non-constant exotic fluids and effective matter. The problem of possible matter instability for such off-diagonal deformations in (modified) gravity theories is discussed.

The goal of this book is to expose the reader to the indispensable role that mathematics---often very abstract---plays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multi-dimensional Green's functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fiber bundles and their applications to differential geometry and gauge theories. This second edition is a substantial revision of the first one with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras and spinors, fiber bundles, and gauge theories. The spirit of the first edition, namely the balance between rigor and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the "unreasonable effectiveness of mathematics" in modern physics. Einstein has famously said, "The most incomprehensible thing about nature is that it is comprehensible." What he had in mind was reiterated in another one of his famous quotes concerning the question of how " ... mathematics, being after all a product of human thought, is so admirably appropriate to the objects of reality." It is a question that comes to everyone's mind when encountering the highly abstract mathematics required for a deep understanding of modern physics. It is the experience that Eugene Wigner so profoundly described as "the unreasonable effectiveness of mathematics in the natural sciences."

Motivated by multi-centered black hole solutions of Maxwell-Einstein theories of (super)gravity in D=4 space-time dimensions, we develop some general methods, that can be used to determine all homogeneous invariant polynomials on the irreducible (SL_h(p,R) x G4)-representation (p,R), where p denotes the number of centers, and SL_h(p,R) is the "horizontal" symmetry of the system, acting upon the indices labelling the centers. The black hole electric and magnetic charges sit in the symplectic representation R of the generalized electric-magnetic (U-)duality group G4. We start with an algebraic approach based on classical invariant theory, using Schur polynomials and the Cauchy formula. Then, we perform a geometric analysis, involving Grassmannians, Pluecker coordinates, and exploiting Bott's Theorem. We focus on non-degenerate groups G4 "of type E7" relevant for (super)gravities whose (vector multiplets') scalar manifold is a symmetric space. In the triality-symmetric stu model of N=2 supergravity, we explicitly construct a basis for the 10 linearly independent degree-12 invariant polynomials of 3-centered black holes., 1+29 pages, 6 Tables

The concept of temporal networks is an extension of complex networks as a modeling framework to include information on when interactions between nodes happen. Many studies of the last decade examine how the static network structure affect dynamic systems on the network. In this traditional approach the temporal aspects are pre-encoded in the dynamic system model. Temporal-network methods, on the other hand, lift the temporal information from the level of system dynamics to the mathematical representation of the contact network itself. This framework becomes particularly useful for cases where there is a lot of structure and heterogeneity both in the timings of interaction events and the network topology. The advantage compared to common static network approaches is the ability to design more accurate models in order to explain and predict large-scale dynamic phenomena (such as, e.g., epidemic outbreaks and other spreading phenomena). On the other hand, temporal network methods are mathematically and conceptually more challenging. This book is intended as a first introduction and state-of-the art overview of this rapidly emerging field.

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost K\"ahler - Ricci geometric evolution equations. The second goal is to study how fixed Lie algebroid, i.e. Ricci soliton, configurations can be constructed for Riemannian manifolds and/or (co) tangent bundles endowed with nonholonomic distributions modelling (generalized) Einstein or Finsler - Cartan spaces. Finally, there are provided some examples of generic off-diagonal solutions for Lie algebroid type Ricci solitons and (effective) Einstein and Lagrange-Finsler algebroids., Comment: This version is accepted by Mediterranian J. Math. and modified following editor/referee's requests. File latex2e 11pt generates 29 pages