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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0742.00056).]
This paper is a survey work on the recent development of studies on Igusa's local zeta functions and related topics. It is closely related to the number of solutions of congruences \(\bmod p^ m\), and to exponential sums \(\bmod p^ m\). For a \(p\)-adic field \(K\), we denote by \(R\) the ring of integers of \(K\) and set \(q\) the cardinal of the residue field. Let \(f(x)\) be a polynomial on \(K^ n\) and let \(\chi\) be a character of \(R^ \times\). We define Igusa's local zeta function associated to \(f(x)\) by
\[
Z_ \Phi(s,\chi)=Z_ \Phi(s,\chi,K,f):=\int_{K^ n}\Phi(x)(acf(x))\;| f(x)|^ s\;| dx|
\]
where \(\Phi(x)\) is a Schwartz-Bruhat function and \(| dx|\) is the Haar measure on \(K^ n\) normalized that \(R^ n\) has measure 1. It is proved that \(Z_ \Phi(s,\chi)\) is convergent if the real part \(\text{Re}(s)\) is sufficiently large and is a rational function in \(q^{-s}\).
In this survey, after stating some basic properties of this zeta function, the author gives the relation between the monodromy and the \(b\)-function of \(f(x)^ s\), and some functional equations involved in them. The latter half of this article is devoted to the explanation of some special topics such as prehomogeneous vector spaces, the integration on \(p\)-adic subanalytic sets and so on. survey; Igusa's local zeta functions; number of solutions of congruences; exponential sums; \(p\)-adic field; Schwartz-Bruhat function; Haar measure; monodromy; \(b\)-function; functional equations; prehomogeneous vector spaces; integration on \(p\)-adic subanalytic sets J. Denef, Report on Igusa's local zeta function. Séminaire Bourbaki, vol. 1990/1991. Astérisque 201-203 (1991). Exp. No. 741, 359-386 (1992) Zeta functions and \(L\)-functions, Research exposition (monographs, survey articles) pertaining to number theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Homogeneous spaces and generalizations, Local ground fields in algebraic geometry Report on Igusa's local zeta function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider topologically non-trivial Higgs \(G\)-bundles over Riemann surfaces \(\Sigma _{g }\) with marked points and the corresponding Hitchin systems. We show that if \(G\) is not simply-connected, then there exists a finite number of different sectors of the Higgs bundles endowed with the Hitchin Hamiltonians. They correspond to different characteristic classes of the underlying bundles defined as elements of \(H^{2}\big(\Sigma_g, \mathcal{Z}(G)\big)\), (\({\mathcal{Z}(G)}\) is the center of \(G\)). We define the conformal version \(CG\) of \(G\) -- an analog of \(\mathrm{GL}(N)\) for \(\mathrm{SL}(N)\), and relate the characteristic classes with degrees of \(CG\)-bundles. We describe explicitly bundles in the genus one case (\(g = 1\)). If \(\Sigma_{1}\) has one marked point and the bundles are trivial, then the Hitchin systems coincide with Calogero-Moser (CM) systems. For nontrivial bundles we call the corresponding systems modified Calogero-Moser (MCM) systems. Their phase space has the same dimension as the phase space of the CM systems with spin variables, but a smaller number of particles and a greater number of spin variables. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, and define the phase spaces and the Poisson structure using dynamical \(r\)-matrices. The latter are the completion of the classification list of Etingof-Varchenko corresponding to the trivial bundles. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of 't Hooft matrices for
\[
\mathrm{sl}(N)
\]
. We find that the MCM systems contain the standard CM subsystems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the stable holomorphic bundles with non-trivial characteristic classes. Riemann surfaces with marked points; Higgs \(G\)-bundles; Hitchin systems Levin, A., Olshanetsky, M., Smirnov, A., Zotov, A.: Characteristic classes and Hitchin systems. General construction. Commun. Math. Phys. \textbf{316}(1), 1-44 (2012). arXiv:1006.0702 Compact Riemann surfaces and uniformization, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Holomorphic bundles and generalizations Characteristic classes and Hitchin systems. General construction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result is an effective estimation of the best exponent \(\rho \in \lbrack 0,1)\) in the Lojasiewicz gradient inequality
\[
\left\vert \nabla f(x)\right\vert \geq C\left\vert f(x)\right\vert ^{\rho }
\]
in a neighbourhood of \(a\in U\subset \mathbb{R}^{n}\) for some constant \(C>0,\) for a Nash function \(f:U\rightarrow \mathbb{R},\) where \(f(a)=0.\) Namely, if \(d\) is the degree of a nonzero polynomial \(P\) such that \( P(x,f(x))=0,\) \(x\in U,\) then the above inequality holds for
\[
\rho =1-\frac{1}{2(2d-1)^{3n+1}}.
\]
This is a generalization of the \textit{P. Solernó} [Appl. Algebra Eng. Commun. Comput. 2, No. 1, 1--14 (1991; Zbl 0754.14035)] and \textit{D. D'Acunto} and \textit{K. Kurdyka} [Ann. Pol. Math. 87, 51--61 (2005; Zbl 1093.32011)] type estimation for polynomials. As a corollary the authors obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities. semialgebraic function; Nash function; Lojasiewicz gradient inequality; Lojasiewicz exponent Nash functions and manifolds, Affine geometry Effective Lojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider a wide class of models of plane algebraic curves, so-called \((n, s)\)-curves. The case \((2, 3)\) is the classical Weierstrass model of an elliptic curve. On the basis of the theory of multivariate sigma functions, for every pair of coprime \(n\) and \(s\) we obtain an effective description of the Lie algebra of derivations of the field of fiberwise Abelian functions defined on the total space of the bundle whose base is the parameter space of the family of nondegenerate \((n, s)\)-curves and whose fibers are the Jacobi varieties of these curves. The essence of the method is demonstrated by the example of Weierstrass elliptic functions. Details are given for the case of a family of genus 2 curves. sigma function; differentiation with respect to parameters; universal bundle of Jacobi varieties Bukhshtaber, V. M. and Leykin, D. V., Solution of the problem of differentiation of Abelian functions over parameters for families of {\((n,s)\)}-curves, Functional Analysis and its Applications, 42, 4, 268-278, (2008) Theta functions and curves; Schottky problem, Analytic theory of abelian varieties; abelian integrals and differentials, Plane and space curves Solution of the problem of differentiation of abelian functions over parameters for families of \((n, s)\)-curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper reviews spectral properties of a class of singular Schrödinger operators with the interaction supported by manifolds or complexes of codimension 1; in particular, the relation of these properties to the geometric setting of the problem is discussed. We describe how these operators can be approximated by operators of other classes and how such approximations can be used. Furthermore, we present asymptotic expansions of the eigenvalues in terms of the parameters characterizing the coupling strength and geometric deformations. We also give an example illustrating the influence of a magnetic field of the Aharonov-Bohm type and briefly describe results on singular perturbations of Dirac operators. singular Schrödinger operators; codimension 1 manifolds; spectral properties; asymptotic expansions; Dirac operators Selfadjoint operator theory in quantum theory, including spectral analysis, Operations with distributions and generalized functions, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices, Linear operator approximation theory, Deformations of singularities, Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory, General topics in linear spectral theory for PDEs Leaky quantum structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the authors study the moduli space of polarized abelian varieties, and they construct it as Heisenberg invariant part of a Hilbert scheme. In the particular case of \((1,7)\)-polarized abelian surfaces, they obtain only a birational model of the moduli space. But they obtain a birational parametrization of the moduli space defined over \(\mathbb{Q}\). They prove that the moduli space \(X(1,7)\) of \((1,7)\)-polarized abelian surfaces with canonical level structure is birational to the Fano 3-fold \(V_{22}\) of polar hexagons of the Klein quartic \(\overline{X}(7)\). In particular, they obtain that \(X(1,7)\) is rational with birational map to \(\mathbb{P}^{3}\) defined over \(\mathbb{Q}\). Furthermore they give the equations of the \((1,7)\)-very ample polarized abelian surfaces embedded in \(\mathbb{P}^{6}\). abelian surface; moduli space; syzygies; Heisenberg group; Hilbert scheme; Fano threefold Manolache, N.; Schreyer, F.-O., Moduli of \((1, 7)\)-polarized abelian surfaces via syzygies, Math. nachr., 226, 177-203, (2001), MR 1839408 Algebraic moduli of abelian varieties, classification, \(3\)-folds, Syzygies, resolutions, complexes and commutative rings Moduli of \((1,7)\)-polarized abelian surfaces via syzygies | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author in [``Nonabelian Jacobian of smooth projective varieties'', Science China, Vol. 56, No. 1, 1--42 (2013)] is a survey of the work done in [J. Differ. Geom. 74, No. 3, 425--505 (2006; Zbl 1106.14030)] as well as in [{I. Reider}, ``Nonabelian Jacobian of smooth projective surfaces and representation theory'', \url{arXiv:1103.4749}]. Central in this work is the construction of a nonabelian Jacobian \(J(X;L,d)\) of a smooth projective surface \(X\), a scheme over the Hilbert scheme \(X^{[d]}\) of subschemes of length \(d\) in \(X\), with a morphism to the stack of torsion free sheaves of rank 2 on \(X\) with determinant \(\mathcal{O}_X(L)\) and second Chern class \(d\). Among the various constructions covered in this survey, the existence of a sheaf of reductive Lie algebras on \(J(X;L,d)\) takes center stage. It originates from a well-chosen filtration on \(X^{[d]}\). This sheaf, which is the object of the second part of the paper, paves the way for the use of representation theoretic methods in the study of projective surfaces. There is some interesting work covered in the survey, though what makes it alluring are the possible applications. The seasoned algebraic geometer will no doubt appreciate the expansive coverage done in this work; worthy of further investigation are the connections with quantum gravity, homological mirror symmetry, the geometric Langlands program as well as quiver representations/Gromov-Witten invariants. Jacobian; Hilbert scheme; vector bundle; sheaf of reductive Lie algebras; Fano toric varieties; period maps; stratifications; Hodge-like structures; relative Higgs structures; perverse sheaves; Langlands program Reid, I, Nonabelian Jacobian of smooth projective surfaces -- a survey, Sci China Math, 56, 1-42, (2013) Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of orders, lattices, algebras over commutative rings, Variation of Hodge structures (algebro-geometric aspects), Geometric Langlands program (algebro-geometric aspects), Jacobians, Prym varieties Nonabelian Jacobian of smooth projective surfaces -- a survey | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(N(B)\) denote the number of rational points on the variety
\[
X^3_0= X_1 X_2 X_3
\]
lying off the lines with \(X_0= 0\), and for which the height
\[
h^*(x_0, x_1, x_2, x_3)= \max|x_i|
\]
is at most \(B\). (Here we take the \(x_i\) to be coprime integers.) A number of authors have proved Manin's conjecture for this variety, showing that
\[
N(B)\sim cB(\log B)^6.
\]
In particular \textit{R. de la Bretèche} [Astérisque 251, 51--77 (1998; Zbl 0969.14014)] has shown that \(N(B)= Bf(\log B)+ O(B^{7/8})\) for an explicit polynomial \(f\) of degree 6. It follows that the height zeta-function \(Z(s; h^*)= \sum_P h^*(P)^{-s}\) (where \(P\) runs over the same points as for \(N(B)\)) has an analyitc continuation for \(\sigma> 7/8\).
It is natural to ask whether there are alternative height functions which are ``canonical'', and for which the height zeta-function has even better properties. The author finds a height which is a sum of local heights, each of which is a specialization of a positive definite bilinear form under the natural group law. The resulting zeta-function has an analytic continuation to \(\sigma> 0\) with a natural boundary on the imaginary axis. Manin conjecture; height zeta-function; canonical height; toric variety Rational points, Heights, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Toric varieties, Newton polyhedra, Okounkov bodies A canonical height on \(X^3_0=X_1 X_2 X_3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We develop a ``universal'' support theory for derived categories of constructible (analytic or étale) sheaves, holonomic \(\mathcal{D}\)-modules, mixed Hodge modules and others. As applications we classify such objects up to the tensor triangulated structure and discuss the question of monoidal topological reconstruction of algebraic varieties. constructible sheaves; holonomic \(\mathcal{D}\)-modules; mixed Hodge modules; motivic sheaves; constructible systems; support datum; tensor-triangular geometry; smashing spectrum; classification; reconstruction Étale and other Grothendieck topologies and (co)homologies, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Motivic cohomology; motivic homotopy theory, Derived categories, triangulated categories, Monoidal categories, symmetric monoidal categories Supports for constructible systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0752.00052.]
Igusa's local zeta function \(Z_ K(s)\) is an integral of the complex power of a polynomial with respect to the \(p\)-adic measure; \(Z_ K(s)=\int_{O^ n_ K}| f(x)|^ s_ K | dx|\). In this definition, \(K\) is taken to be a \(p\)-adic completion of a number field \(k\), \(O_ K\) is the ring of integers in \(K\), \(\pi_ K O_ K\) is the ideal of non-units in \(O_ K\), and \(q_ K\) is the cardinality of the residue field \(O_ K/\pi_ K O_ K\). Furthermore, \(| dx|\) is the Haar measure on \(K^ n\) normalized such that \(| \pi_ K|_ K=q^{-1}_ K\) and the measure of \((O^ n_ K)=1\). It is proved that \(Z_ K(s)\) is a rational function of \(q^{-s}_ K\). In this paper, the author gives some algebraic identities to compute \(Z_ K(s)\). Igusa's local zeta function; rational function; algebraic identities Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Quaternion and other division algebras: arithmetic, zeta functions Algebraic identities useful in the computation of Igusa local zeta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author shows that cotangent bundles of moduli spaces of vector bundles over a Riemann surface are algebraically completely integrable Hamiltonian systems. More precisely, let G be a complex semisimple Lie group, let N be the moduli space of stable G-bundles with prescribed topological invariants on a compact Riemann surface and let n be the dimension of N. The cotangent space to N at the point represented by a G- bundle P is \(H^ 0(M;ad(P\otimes K))\) where ad(P) is the bundle associated to P via the adjoint representation of G on its Lie algebra g. Thus a choice of basis \(p_ 1,...,p_ k\) for the ring of invariant polynomials on g induces a holomorphic map \(\phi: T*N\to \oplus H^ 0(M;K^{d_ i})\) where \(d_ i\) is the degree of \(p_ i\). The components of \(\phi\) are n functionally independent Poisson-commuting functions on T*N, and when G is a classical group the generic fibre of \(\phi\) is an open set in an abelian variety on which the Hamiltonian vector fields defined by the components of \(\phi\) are linear. This is what it means to say that T*N is an algebraically completely integrable Hamiltonian system. The abelian varieties occurring are either Jacobian or Prym varieties of curves covering M. cotangent bundles of moduli spaces of vector bundles over a Riemann surface; completely integrable Hamiltonian systems; adjoint representation; Jacobian; Prym varieties N. Hitchin, \textit{Stable bundles and integrable systems}. Duke Math. J. 54 (1987), no. 1, 91--114. Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Stable bundles and integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by \textit{P. Candelas, X. de la Ossa} and \textit{F. Rodriguez Villegas} [Calabi-Yau manifolds over finite fields. I. \url{arXiv:hep-th/0012233}, II in: Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 38, 121--157 (2003; Zbl 1100.14032)]. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link. quintic threefold; hypergeometric curves; zeta function factorization Arithmetic mirror symmetry, Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Calabi-Yau manifolds (algebro-geometric aspects) On the zeta function of a family of quintics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This short note studies actions of Frobenius groups on curves. By definition, a Frobenius group is a finite group \(G\) that has a proper non-trivial subgroup \(H\) such that \(H\cap H^g=\{1\}\) for all \(g\in G\setminus H\). Both \(H\) and \(N=(G\setminus \bigcup_{g\in G}H^g)\cup\{1\}\) are uniquely determined subgroups of \(G\), called Frobenius complement and Frobenius kernel.
The main result is Corollary 3.1 (which in the paper itself is referred to as `Theorem 3.1'): Let \(G\) be a Frobenius group and \(X\) a smooth projective curve over an algebraically closed field \(k\). Let \(G\) act faithfully as a group of automorphisms on \(X\) such that \(X/G\) is of genus zero, or, in other words, fix a \(G\)-Galois cover \(X\rightarrow\mathbb{P}^1\). Then the genus \(g(X)\) of \(X\) can be expressed as
\[
g(X) = g(X/N) + g(X/H)|H|.
\]
From this result, the author then deduces classifications of Frobenius group actions on curves in certain cases. The special case \(g(X/H)=0\) of the main result was proven already in [\textit{J. Flynn}, Near-exceptionality over finite fields, PhD thesis, Berkeley, (2001)] and [\textit{R. M. Guralnick}, Rational functions with monodromy group a Frobenius group, preprint, (2000)]. In this special case it follows that \(g(X)\leq1\).
The proof of the main result applies tools from representation theory to the action of \(G\) on the Tate module of \(X\). In the final section, the author considers a more general class of group actions on curves and again deduces some genus estimates. Frobenius group; monodromy group; coverings of curves; algebraic function field Coverings of curves, fundamental group, Algebraic functions and function fields in algebraic geometry, Separable extensions, Galois theory, Finite automorphism groups of algebraic, geometric, or combinatorial structures Frobenius groups as monodromy groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We discuss ways in which momentum operators can be introduced on an oriented metric graph. A necessary condition appears to the balanced property, or a matching between the numbers of incoming and outgoing edges; we show that a graph without an orientation, locally finite and at most countably infinite, can made balanced oriented if and only if the degree of each vertex is even. On such graphs we construct families of momentum operators; we analyze their spectra and associated unitary groups. We also show that the unique continuation principle does not hold here. differential geometry; algebraic geometry Exner, P.: Momentum operators on graphs. arXiv:1205.5941v2 Graphs and linear algebra (matrices, eigenvalues, etc.), Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices, Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Momentum operators on graphs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The object of this paper is to describe a simple method for proving that certain groups are residually torsion-free nilpotent, to describe some new parafree groups and to raise some new problems in honour of the memory of Wilhelm Magnus. \({\mathcal D}\)-groups; residually torsion-free nilpotent groups; one-relator groups Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Musings on Magnus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A model of nondeterministic finite automaton over (finite) partial orders is introduced. Roughly speaking: the automaton scans local neighbourhoods in a given partial order (an acyclic digraph) and assigns nondeterministically states to the points of this partial order. It is shown that a set of (finite and labelled) partial orders is recognizable by such a finite automaton iff it is definable in existential monadic second-order logic. If the partial orders are words or if they are label trees, then recognizable sets are definable in unrestricted monadic second-order logic. Some special forms are discussed, e.g. deterministic automata; logical and algorithmic properties are analyzed, e.g. closure under complement, decidability of the nonemptiness problem. The mentioned problems are studied for different classes of partial orders, trees, Mazurkiewicz traces and rectangular grids. Finally, directions for further research are offered. recognizable set; automaton; monadic second-order logic Thomas W. Elements of an automata theory over partial orders. InProc. Workshop on Partial Order Methods in Verification, (Peled D, ed.), DIMACS Ser. in Discr. Math. and TCS, 1997, 29: 25--40. Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Elements of an automata theory over partial orders | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this survey article the authors describe the background of the Mackey topology on groups. They start with the results of Mackey and Arens who proved the existence of the Mackey topology for locally convex vector spaces [\textit{G. W. Mackey}, Trans. Am. Math. Soc. 60, 519--537 (1946; Zbl 0061.24302)] and [\textit{R. Arens}, Duke Math. J. 14, 787--794 (1947; Zbl 0030.03403)]. Afterwards they discuss results for topological vector spaces which are not locally convex and for vector groups and proceed to results of \textit{N. T. Varopoulos} [Proc. Camb. Philos. Soc. 60, 465--516 (1964; Zbl 0161.11103)] who investigated analogous questions for locally precompact groups. Finally, they present results of the initial paper on the Mackey topology for locally quasi-convex groups [\textit{M. J. Chasco} et al., Stud. Math. 132, No. 3, 257--284 (1999; Zbl 0930.46006)] and give an overview about the progress in this area. Mackey topology; locally convex space; locally quasi--convex group; vector group; dually embedded subgroup; precompact group; pre-locally compact group; compatible topology E. Martín-Peinador and V. Tarieladze, Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110 (2016), no. 2, 667-679. Structure of general topological groups, Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler's equations preserving independently scattered random measures. differential geometry; algebraic geometry Linear composition operators, Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Infinitesimal invariance of completely random measures for 2D Euler equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Sei \(g\) die Liealgebra der Liegruppe \(G\) und \(g^*\) ihr Dualraum. Dann existiert eine zur adjungierten Darstellung duale (im allgemeinen nicht unitäre) Darstellung von \(G\) auf \(g^*\), koadjungierte Darstellung genannt.
Für den Bahnenraum \(g^*/G\) dieser Operation gilt nun im Falle einer zusammenhängenden, einfach zusammenhängenden nilpotenten Gruppe \(G\), dass er mit dem unitären Dual, d.h. der Menge aller unitären irreduziblen Darstellungen modulo Äquivalenz, übereinstimmt.
In der vorliegenden Vorlesungsausarbeitung erläutert der Autor die Erweiterung seiner ``Bahnmethode'' auf allgemeinere Gruppen, insbesondere auflösbare, kompakte und halbeinfache. Dabei werden auch die sich ergebenden Probleme aufgezeigt, da etwa die komplementäre Serie auf diese Weise nicht erhalten wird.
Der Hintergrund des (nach Vogan wundersamen) Zusammenhangs zwischen den Bahnen in \(g^*\) und dem unitären Dual ist physikalischen Ursprungs. orbit method; geometric quantization; coadjoint representation; nilpotent Lie group; semisimple Lie group A. Kirillov, ``The orbit method, I: Geometric quantization'' in Representation Theory of Groups and Algebras , Contemp. Math. 145 , Amer. Math. Soc., Providence, 1993, 1--32. Lie algebras of Lie groups, Representations of Lie algebras and Lie superalgebras, analytic theory, Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters, Finite-dimensional groups and algebras motivated by physics and their representations The orbit method. I: Geometric quantization | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article computes explicitly the vertical and horizontal monodromies of Hirzebruch-Jung surfaces. The actions are computed in terms of a common eigenbasis. Using a formula of Steenbrink the monodromies are applied to deduce Sp, the Spectrum of the Yorndin series associated to the Hirzebruch-Jung surface. Structure of families (Picard-Lefschetz, monodromy, etc.), Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Vertical monodromy and spectrum of a Yomdin series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For Part I see the preceding review.]
Im zweiten Teil der Vorlesungsausarbeitung über die Bahnmethode geht es um deren Anwendung auf unendlich-dimensionale Liegruppen.
Nach Diskussion wichtiger Beispiele werden viele interessante offene Probleme erörtert und einige Ergebnisse der Moskauer Schule des Autors aus den Jahren 1970-1990 angeführt.
Bemerkenswert ist, dass sich bei der Untersuchung des unendlich-dimensionalen homogenen Raumes
\[
M=G/H,\quad G=\text{Diff}_+S^1,\quad H=\text{Rot }S^1
\]
ein Zusammenhang mit der Bieberbach-Vermutung (Theorem von de Branges) und die Hoffnung auf einen ``geometrischen'' Beweis ergibt. orbit method; infinite-dimensional Lie groups; Lie algebras; Bieberbach conjecture A.A. Kirillov, ''The orbit method, II: Infinite-dimensional Lie groups and Lie algebras'', Contemporary Mathematics145 (1993), p. 33-33 Lie algebras of Lie groups, Infinite-dimensional Lie (super)algebras, Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters, Infinite-dimensional Lie groups and their Lie algebras: general properties, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations The orbit method. II: Infinite-dimensional Lie groups and Lie algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The error bounds of the rectangular, trapezoidal and Simpson's rules which are commonly used in approximating the integral of a function \(f(x)\) over an interval \([a, b]\) were estimated. The error bounds of the second, and third generating functions of the Gauss-Legendre quadrature rules were also estimated in this paper. It was shown that for an \(f(t)\) whose smoothness is increasing, the accuracy of the fourth, sixth and eighth error bound of the second, and third generating functions of the Gauss-Legendre quadrature rule does not increase. It was also shown that the accuracy of the fourth error bound of the Simpson's 1/3 and 3/8 rules does not increase. generating functions; Newton-Cotes formulas; quadrature rules; numerical integration Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Error bounds for numerical integration of functions of lower smoothness and Gauss-Legendre quadrature rule | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A well-known problem of \textit{B. Grünbaum} [``Partitions of mass-distributions and of convex bodies by hyperplanes,'' Pac. J. Math. 10, 1257--1261 (1960; Zbl 0101.14603)] asks whether for every continuous mass distribution (measure) \( d\mu = f\, dm\) on \( \mathbb{R}^n\) there exist \( n\) hyperplanes dividing \( \mathbb{R}^n\) into \(2^n\) parts of equal measure. It is known that the answer is positive in dimension \(n=3\) (see \textit{H. Hadwiger} [``Simultane Vierteilung zweier Körper,''Arch. Math. 17, 274--278 (1966; Zbl 0137.41501)]) and negative for \( n\geq 5\) (see \textit{D. Avis} [``Non-partitionable point sets,'' Inf. Process. Lett. 19, 125--129 (1985; Zbl 0564.51007)] and \textit{E. Ramos} [``Equipartition of mass distributions by hyperplanes,'' Discrete Comput. Geom. 15, No.\,2, 147--167(1996; Zbl 0843.68120)]). We give a partial solution to Grünbaum's problem in the critical dimension \( n=4\) by proving that each measure \( \mu\) in \( \mathbb{R}^4\) admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional affine subspace \( L\) of \( \mathbb{R}^4\). Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on Koschorke's exact singularity sequence (1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4; see G. C. Tootill (1956) and D. E. Knuth (2001). Geometric combinatorics; partitions of masses; gray codes Živaljević, Rade T., Equipartitions of measures in \(\mathbb{R}^4\), Trans. Amer. Math. Soc., 360, 1, 153-169, (2008) Computer graphics; computational geometry (digital and algorithmic aspects), Computing methodologies for image processing, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Equipartitions of measures in \(\mathbb{R}^4\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The homotopy perturbation method proposed by \textit{J.-H. He} [Int. J. Mod. Phys. B 20, No.~10, 1141--1199 (2006; Zbl 1102.34039)] is adopted for solving multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations. The numerical results of the present method are compared with the exact solution of an artificial multi-dimensional nonlinear coupled system of parabolic and hyperbolic model to show the efficiency of the method. Moreover, comparison is made between the results obtained by the present method and that obtained by the Adomian decomposition method. It is found that the present method works extremely well, very efficient, simple and convenient. comparison of methods; homotopy perturbation method; multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations; numerical results; Adomian decomposition method Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs, Nonlinear parabolic equations, Second-order nonlinear hyperbolic equations, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Homotopy perturbation method for multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any \(p\in[1,\infty]\), denote by \(p'\) the dual exponent of \(p\); namely, \(1/p+1/p'=1\). In this paper, the authors proved that, for any \(p\in(1,\infty)\), there exists a pair of weights \((u,v)\) such that the Hardy-Littlewood maximal operator \(M\) is bounded from the weighted Lebesgue space \(L^p(v)\) to \(L^p(u)\) and from \(L^{p'}(u^{1-p'})\) to \(L^{p'}(v^{1-p'})\), respectively, but the sparse operator is not bounded from \(L^p(v)\) to \(L^p(u)\). In particular, for any \(p\in(1,\infty)\), there exists a weight \(w\) such that \(M\) is bounded on \(L^p(w)\), but the sparse operator is not bounded on \(L^p(w)\). two weight inequalities; Hardy-Littlewood maximal function; dyadic sparse operators Maximal functions, Littlewood-Paley theory, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Muckenhoupt-Wheeden conjectures for sparse operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Mathematically rigorous approaches to functional integrals of two dimensional Yang-Mills gauge theory and three-dimensional Chern-Simons theory are described. Sengupta, A. N.: Yang-Mills in two dimensions and Chern-Simons in three, AMS/IP studies in advanced mathematics 50, 307-316 (2011) Yang-Mills and other gauge theories in quantum field theory, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Yang-Mills in two dimensions and Chern-Simons in three | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies geometrical properties of a unit vector field on a Riemannian 2-manifold. The field is considered as an embedding of the manifold into its tangent sphere bundle endowed with the Sasaki metric. A number of results are proved. For example, it is shown that a unit vector field \(\xi \) generating a totally geodesic submanifold in \(T_1M^ 2\), where \(M^ 2\) is a Riemannian manifold of constant Gauss curvature \(K\), exists if and only if \(K=0\) or \(K=1\). In both cases, the form of \(\xi \) is described in detail. The properties of the submanifold \(\xi (M^ 2)\subset T_1(M^ 2)\) are studied, in particular it is shown that it has non-positive extrinsic curvature.
The cases in which it has constant Gauss curvature are described. A family \(\xi _\omega \) of vector fields on the hyperbolic 2-plane \(L^ 2\) of curvature \(-c^ 2\) which generate foliations on \(T_1L^ 2\) with leaves of constant intrinsic curvature \(-c^ 2\) and of constant extrinsic curvature \(-c^ 2/4\) are found. The basic tool is the description of the second fundamental form of the submanifold \(\xi (M)\subset T_1(M)\) from which an expression for the Gaussian curvature of \(\xi (M^ 2)\) is derived. unit vector field; Riemannian 2-manifold; tangent sphere bundle; Sasaki metric; sectional curvature; totally geodesic submanifolds Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Global Riemannian geometry, including pinching On the intrinsic geometry of a unit vector field. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The articles of this volume will be reviewed individually. Weiping Li, Loretta Bartolini, Jesse Johnson, Feng Luo, Robert Myers, and J. Hyam Rubinstein , Topology and geometry in dimension three, Contemporary Mathematics, vol. 560, American Mathematical Society, Providence, RI, 2011. Triangulations, invariants, and geometric structures; Papers from the Jacofest Conference in honor of William Jaco's 70th birthday held at Oklahoma State University, Stillwater, OK, June 4 -- 6, 2010. Proceedings, conferences, collections, etc. pertaining to manifolds and cell complexes, Proceedings, conferences, collections, etc. pertaining to group theory, General low-dimensional topology, Topology of general 3-manifolds, Modular representations and characters, Geometric group theory, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Proceedings of conferences of miscellaneous specific interest, Festschriften Topology and geometry in dimension three. Triangulations, invariants, and geometric structures. Conference in honor of William Jaco's 70th birthday, Stillwater, OK, USA, June 4--6, 2010 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of \(s\) polynomials in \(R[X_1,\dots,X_k]\) whose degrees are at most \(d\) is bounded by
\[
\frac{(2d)^k}{k!} s^k + O(s^{k-1}) .
\]
This improves the best upper bound known previously which was
\[
\frac 12\, \frac{(8d)^k}{k!} s^k + O(s^{k-1}).
\]
The new bound matches asymptotically the lower bound obtained for families of polynomials each of which is a product of generic polynomials of degree one. semi-algebraically connected components Basu, S., Pollack, R., Roy, M.-F.: An asymptotically tight bound on the number of connected components of realizable sign conditions. Combinatorica 29, 523--546 (2009) Semialgebraic sets and related spaces, Topology of real algebraic varieties, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use the theory of large deviations on function spaces to extend Erdős and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths \(\{W_{N,n}\}\) defined by
\[
W_{N,n} (t,\omega)={S_{n+[h(N)t]} (\omega)-S_ n (\omega) \over h(N)}+{h(N)t-[h(N)t] \over h(N)} X_{n+[h(N)t]+1} (\omega),
\]
where \(S_ n={1 \over n} \sum^ n_ 1X_ i\), \(\{X_ i:i \geq 1\}\) is an i.i.d. sequence of random variables, and \(h(N)=[c \log N]\) is relatively compact; the limit set is given by the set \([x:I^*(x) \leq 1/c]\) where \(I^*(x)=\int^ 1_ 0 I(x'(t)) dt\) and \(I\) is Cramér's rate function. large deviations; Erdős and Rényi's law of large numbers; limit set; Cramér's rate function G.R. Sanchis, A functional limit theorem for Erdos and Rényi's law of large numbers. Probab. Theory Relat. Fields 98, 1--5 (1994) Functional limit theorems; invariance principles, Large deviations, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions A functional limit theorem for Erdős and Rényi's law of large numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this research paper is threefold: first, to introduce new constructions in the category of real semigroups (RS): locally constant functions on a topological space with values in a RS, directed inductive limits and RS-sums, an analog of a coproduct for RSs. Secondly, to obtain some understanding of the space of RS-characters of an infinite product of RSs and, lastly, to establish the preservation of \textbf{finite} products and arbitrary directed inductive limits by the natural functor from the category of commutative unitary rings into that of RSs. The basic references for real semigroups, whose notation and terminology we adopt, are the papers cited under Dickmann-Petrovich (see References). We are grateful to the referee for the careful reading of the text and the valuable comments and corrections that improved the presentation. real semigroups; geometrical theory Algebraic theory of quadratic forms; Witt groups and rings, Semialgebraic sets and related spaces, Rings and algebras of continuous, differentiable or analytic functions, Algebraic properties of function spaces in general topology Constructions in the category of real semigroups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any \(\ell >0\), we present an algorithm which takes as input a semi-algebraic set, \(S\), defined by \(P_{1}\leq 0,\ldots, P_{s}\leq 0\), where each \(P_{i}\in R [ X_{1},\ldots, X_{k}]\) has \(\text{degree}\leq 2\), and computes the top \(\ell\) Betti numbers of \(S, b_{k - 1}(S),\ldots, b_{k - \ell}(S)\), in polynomial time. The complexity of the algorithm, stated more precisely, is \(\sum_{i=0}^{\ell+2}{\binom si}k^{2^{o(\min(\ell,s))}}\). For fixed \(\ell\), the complexity of the algorithm can be expressed as \(s^{\ell+2}+k^{2^{O(\ell)}}\), which is polynomial in the input parameters \(s\) and \(k\). To our knowledge this is the first polynomial time algorithm for computing nontrivial topological invariants of semialgebraic sets in \(R^{k}\) defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed \(s\), we obtain, by letting \(\ell = k\), an algorithm for computing all the Betti numbers of \(S\) whose complexity is \(k^{2^{O(S)}}\).
In the erratum, an amended version of Corollary 1.3. is given. Section 4.3. has been deleted, and substantial changes are made to Sections 6 and 7. Betti numbers; Quadratic inequalities; Semialgebraic sets S. Basu. Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time. \textit{Found. Comput. Math}., 8(1):45-80, 2008. Semialgebraic sets and related spaces, Topology of real algebraic varieties, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Computing the top Betti numbers of semialgebraic sets defined by quadratic inequalities in polynomial time | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper sufficient conditions for the existence of a solution for the inclusion problem \(x^*\in \operatorname{conv}(A(x^*))\) are given. Some consequences are obtained. Unfortunately, the paper contains some typos/misprints that make it difficult to understand. set-valued mapping; KKM mapping; topological vector space; quasi-equilibrium problem; intersection theorem Fixed-point theorems on manifolds, Fixed-point theorems, Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Set-valued and function-space-valued mappings on manifolds A new form of Kakutani fixed point theorem and intersection theorem with applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Concerns ibid. 98, No. 1, 1-5 (1994; Zbl 0794.60018). large deviations; Erdős and Rényi's law of large numbers; limit set; Carmér's rate function G.R. Sanchis, A functional limit theorem for Erdos and Rényi's law of large numbers - Addendum. Probab. Theory Relat. Fields 99, 475 (1994) Functional limit theorems; invariance principles, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Large deviations A functional limit theorem for Erdős and Rényi's law of large numbers (Addendum) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we introduce a method for resolving singularities of Galois closure covers for 5-fold covers between smooth surfaces. Applying this method, we determine types of singular fibers of a family of Galois closure curves for plane sextic curves. Consequently,we obtain an explicit construction of smooth projective minimal surfaces of general type with positive indices obtained as families of Galois closure curves of smooth plane sextic curves. \(\mathcal{S}_5\)-covers; canonical resolution; surfaces of general type with positive indices; Galois closure curves; Galois points Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Surfaces of general type Galois closure covers for 5-fold covers between smooth surfaces and its application | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The quadratic stable range property is discussed. The ring \(A\) has quadratic stable range \(1\) (\(\text{qsr}(A) = 1\)) if every primitive quadratic form over \(A\) represents a unit. The property is motivated by the ring of holomorphic functions on a connected noncompact Riemann surface. The authors prove the following results:
(1) if \(\text{qsr} (A)=1\) then the stable range of \(A\) equals \(1\) and \(\text{Pic} (A)=1\).
(2) \(\text{qsr} (A)=1\) iff \(\text{Pic}(T)=1\) for every quadratic \(A\)-algebra \(T\).
They also classify quadratic forms over Bezout domains of characteristic not 2 satisfying a very strong approximation property (defined in the paper). This classification applies to the ring of holomorphic functions mentioned above. quadratic stable range; Picard group; quadratic form; holomorphic function; Riemann surface; Bezout domain D. R. Estes and R. M. Guralnick, ''A stable range for quadratic forms over commutative rings,'' J. Pure Appl. Algebra, 120, No. 3, 255--280 (1997). Quadratic forms over local rings and fields, Quadratic forms over global rings and fields, Quadratic and bilinear forms, inner products, Picard groups, Rings and algebras of continuous, differentiable or analytic functions A stable range for quadratic forms over commutative rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) and \(Y\) be smooth projective varieties over \(\mathbb{C}\) of dimension \(n \geq 2\) and \(p : Y \to X\) be a finite abelian cover, i.e. a proper finite map with a faithful action of a finite abelian group \(G\). The aim of this article is to compare the topological fundamental groups \(\pi_1 (X)\) and \(\pi_1 (Y)\). We can define a geometrical datum which characterizes the cover \(p : Y \to X\):
To any irreducible component \(D_j\) of the branch locus \(D\) of \(p\) we can associate a cyclic subgroup \(G_j\) of \(G\), the inertia subgroup, and a faithful representation \(\psi_j\) of \(G_j\);
the action of \(G\) on \(p_*({\mathcal O}_Y)\) induces a splitting: \(p_* ({\mathcal O}_Y) = \bigoplus_{\chi\in G^*} L^{- 1}_\chi\), where \(G^*\) is the group of characters of \(G\) and where \(L_\chi^{- 1}\) is a line bundle on \(X\) such that \(G\) acts on \(L_\chi^{- 1}\) via the character \(\chi\).
We say that the \(G\)-cover \(p : Y \to X\) is totally ramified if the inertia subgroups of all the components of \(D\) generate \(G\). We consider the totally ramified abelian cover \(p : Y \to X\), such that the components \(D_1, \ldots, D_k\) of the branch locus \(D\) are ample and flexible. Then: The natural map \(p_* : \pi_1 (Y) \to \pi_1 (X)\) is surjective, the kernel \(K\) is a finite abelian group and \(0 \to K \to \pi_1 (Y) @>p_*>> \pi_1 (X) \to 1\) is a central extension.
Let \(\pi : \widetilde X \to X\) be the universal covering of \(X\) and \(\widetilde D = \pi^{- 1} (D)\), then \(\widetilde D_j = \pi^{- 1} (D_j)\) is connected for every \(j\); then \(K \simeq \ker (\bigoplus G_j \to G)/ \text{Im} (\sigma \circ \rho) \subset (\bigoplus G_j)/ \text{Im} (\sigma \circ \rho)\), where \(\rho\) is the restriction map \(H_c^{2n - 2} (\widetilde X) \to H_c^{2n - 2} (\widetilde D)\) and \(\sigma\) is the map \(H_c^{2n - 2} (\widetilde D) \simeq \bigoplus \mathbb{Z} \widetilde D_j \to \bigoplus G_j\). The cohomology class of the central extension can be computed in terms of the Chern classes of the \(D_j\)'s and the \(L_\chi\)'s.
Let \(c(p) \in H^2 (\pi_1 (X), K) \subset H^2 (X,K)\) be the cohomology class of the extension and \(i_* (c(p)) \in H^2 (X, \widetilde G)\) its image induced by the inclusion \(K \subset \widetilde G = (\bigoplus G_j)/ \text{Im} (\sigma \circ \rho)\), then we have: \(i_* (c(p)) = \Phi_* ([D_1], \ldots, [D_k])\), where \(\Phi\) is the map \(\mathbb{Z}^k \to \widetilde G\).
The authors give some applications of this result. They give a class of examples of nonhomeomorphic covers with the same branch divisors \(D_j\), inertia subgroups \(G_j\) and Galois groups \(G\). In fact the cohomology class \(c(p)\) of the extension of the fundamental groups depends on the choice of the solution \(\{L_\chi\}\) of the characteristic relations of the cover \(p\). Covers corresponding to different solutions may give different extensions and varieties \(Y\) with different fundamental group. finite abelian cover; cohomology class; extension of the fundamental groups Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Projective techniques in algebraic geometry A note on the topology of a totally ramified abelian cover | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We suggest a general method of computation of the homology of certain smooth covers \(\hat{\mathcal{M}}_{g, 1}(\mathbb{C})\) of moduli spaces \(\mathcal{M}_{g, 1}(\mathbb{C})\) of pointed curves of genus \(g\). Namely, we consider moduli spaces of algebraic curves with level \(m\) structures. The method is based on the lifting of the Strebel-Penner stratification of \(\mathcal{M}_{g, 1}(\mathbb{C})\). We apply this method for \(g \leq 2\) and obtain Betti numbers; these results are consistent with \textit{R. C. Penner} [Commun. Math. Phys. 113, 299--339 (1987; Zbl 0642.32012); J. Differ. Geom. 27, No. 1, 35--53 (1988; Zbl 0608.30046)] and \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] results on Euler characteristics. moduli spaces of algebraic curves; Betti numbers; dessins d'enfants Dunin-Barkowski, P.; Popolitov, A.; Shabat, G.; Sleptsov, A., On the homology of certain smooth covers of moduli spaces of algebraic curves, Differential Geom. Appl., 40, 86-102, (2015) Families, moduli of curves (algebraic), Dessins d'enfants theory, Coverings of curves, fundamental group, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Stratifications in topological manifolds On the homology of certain smooth covers of moduli spaces of algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors turn their attention to the Gaussian map of a double cover. They make a cohomological study of the Gaussian map for double covers of smooth projective toric variety, and, in particular, for the case of Hirzebruch surfaces. In general, they focus on cohomological analyses for divisors, not on geometrical aspects. toric varieties; double cover; Gaussian map Duflot, J., Peters, P.: Gaussian maps for double covers of toric surfaces. Rocky Mt. J. Math. 42(5), 1471--1520 (2012) Toric varieties, Newton polyhedra, Okounkov bodies, Coverings in algebraic geometry Gaussian maps for double covers of toric surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies a morphism \(\pi: X\to S\) from a smooth complex algebraic surface to a smooth complex curve which is locally trivial from the differential viewpoint. He assumes that the fundamental groups of the base and the fibre are non-commutative, and that the universal covering \(\tilde X\) is not a bidisk. He indicates the proofs for the following results: The group \(\Aut \tilde X\) of all biholomorphic automorphisms is countable. The fundamental group \(\pi_1(X)\) has finite index in \(\Aut \tilde X\). There are only finitely many subgroups of \(\Aut \tilde X\) which act properly discontinuously and are isomorphic to \(\pi_1(X)\). hyperbolic curve; group of biholomorphic automorphisms; fundamental group Shabat, GB, Local reconstruction of complex algebraic surfaces from universal coverings, Funktsional. Anal. i Prilozhen., 17, 90-91, (1983) Coverings in algebraic geometry, Special surfaces, Complex Lie groups, group actions on complex spaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Low codimension problems in algebraic geometry, Group actions on varieties or schemes (quotients), Homotopy theory and fundamental groups in algebraic geometry Local construction of complex algebraic surfaces with respect to the universal covering | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\pi:Y\to X\) be a tamely ramified covering of Dedekind schemes over \(\mathbb{Z}[1/2]\), in which all the ramification indices \(e_ y\) are odd. We give a formula expressing the second Stiefel-Whitney class of the quadratic \(X\)-bundle
\[
E=\pi_ * O_ Y\left( \sum_{y\in Y} {{e_ y- 1} \over 2} y\right)
\]
in terms of other invariants of the covering, of Galois and geometric nature. It is a common generalisation of two formulas of \textit{J.-P. Serre} [Comment. Math. Helv. 59, 651-676 (1984; Zbl 0565.12014) and C. R. Acad. Sci., Paris, Sér. I 311, No. 9, 547-552 (1990; Zbl 0742.14030)], one in the case of a field extension and the other in the case of a ramified covering of Riemann surfaces. The method is to prove first the formula in the étale case (where it makes sense and holds for arbitrary schemes \(X\) and \(Y\) over \(\mathbb{Z}[1/2]\)), and then reduce to the étale case by ``swallowing'' the ramification by an auxiliary Kummer covering. ramified covering of Dedekind schemes; second Stiefel-Whitney class Hélène Esnault, Bruno Kahn, and Eckart Viehweg, Coverings with odd ramification and Stiefel-Whitney classes, J. Reine Angew. Math. 441 (1993), 145 -- 188. Characteristic classes and numbers in differential topology, Coverings in algebraic geometry Coverings with odd ramification and Stiefel-Whitney classes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0521.14009. semistable degenerate orthogonal bundles; semistable symplectic bundles; vector bundles on curve; ramified covering; moduli spaces; quadratic forms; symplectic forms Bhosle U, Degenerate symplectic and orthogonal bundles on \(\mathbb{P}\)1,Math. Ann. 267 (1984) 347--364 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), General binary quadratic forms, Coverings in algebraic geometry Degenerate symplectic and orthogonal bundles on \({\mathbb{P}}^ 1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author generalizes Catanese's isotropic subspace theorem to the non-compact case [see \textit{F. Catanese}, Invent. Math. 104, No. 2, 263-289 (1991; Zbl 0743.32025)]. In fact, the following logarithmic analogue is proved:
``Let \(X\) be a quasiprojective manifold and let \(\overline{X}\) be a smooth compactification of \(X\), such that \(Y:= \overline{X}-X\) is a divisor with normal crossings. Every maximal real isotropic subspace \(V\subset H^1(X,\mathbb{C})\) determines a logarithmic irrational pencil (i.e., a surjective holomorphic map \(f:X\to C\) with connected fibres from \(X\) to a quasiprojective curve \(C\) with first Betti number \(b_1(:=\dim H^1(C,\mathbb{C}))= \beta\geq 3)\). The genus \(g\) of a smooth compactification \(\overline{C}\) of \(C\) is
\[
\frac 12\dim (V\cap H^1(\overline{X}, \mathbb{C})).
\]
\(C\) is complete iff \(V\) is contained in \(H^1 (\overline{X}, \mathbb{C})\); in this case \(f^*(H^1(C,\mathbb{C}))= V\oplus \overline{V}\), and in particular \(\dim(V)= \text{genus} (C)\). If \(C\) is non complete then \(V= f^*(H^1(C,\mathbb{C}))\), \(\dim(V)= g+g^*\), where \(g^*\) is the logarithmic genus of \(C\). In this way one has established a bijective correspondence between the set of maximal real isotropic subspaces \(V\subset H^1(X,C)\) of dimension \(\beta\geq 3\), which are not contained in \(H^1(\overline{X}, \mathbb{C})\), and the set of strictly logarithmic irrational pencils \(f:X\to C\) with first Betti number \(b_1(C)= \beta\).''
Then, a generalization of a result of Gromov and Green-Lazarsfeld on fundamental groups of algebraic varieties with few relations to open algebraic manifolds is given. compactification; logarithmic irrational pencil; fundamental groups Catanese, F., Lönne, M., Perroni, F.: Genus stabilization for moduli of curves with symmetries. arXiv:1301.4409 (to appear in Algebraic Geometry) Coverings in algebraic geometry, Compactification of analytic spaces, Homotopy theory and fundamental groups in algebraic geometry Irrational pencils on non-compact algebraic manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that the notions of gerbe and stack introduced by Grothendieck occur naturally in the theory of coverings. To a \(G\)-covering \(\overline f\) of the field of moduli \(K\) is associated a \(K\)-gerbe \({\mathcal G}(\overline f)\) bound by the center \(Z(G)\) of \(G\). This gerbe is, in fact, the residual gerbe in the point Spec\(K\) of a more general algebraic stack defined over \(Z[\frac{1}{|G|}]\). We can use the diophantine approximations in the gerbs and stacks to get results of the type Hasse principle. Generalizations (algebraic spaces, stacks), Fine and coarse moduli spaces, Coverings in algebraic geometry Descent, stacks and Hurwitz gerbs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper, of which three proofs are given (two of them due to L. Moret-Bailly and N. M. Katz, respectively) is the following theorem: Suppose that \(X\) is a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(G\) be a finite \(p\)-group acting freely on \(X\) and \(Y=X/G\). Then \(\chi_ c(X,{\mathbb Q}_ p)=| G| \cdot \chi_ c(Y,{\mathbb Q}_ p).\) Here \(\chi_ c\) denote the Euler-Poincaré characteristic for cohomology with compact supports.
There are some consequences of which the following three should be mentioned:
(1) the formula of Deuring-Shafarevich for the \(p\)-ranks of \(X\) and \(Y\) for a finite morphism \(f: X\to Y\) of smooth projective curves over \(k\) [cf. e.g. \textit{M. L. Madan}, Manuscr. Math. 23, 91--102 (1977; Zbl 0369.12011)];
(2) the fact that \(\pi_ 1(X)\) has no p-torsion [in the case of the algebraic closure of a finite field this was proved by \textit{T. Katsura}, C. R. Acad. Sci., Paris, Sér. A 288, 45--47 (1979; Zbl 0429.14011)]; and
(3) if \(p=2\) and \(X\) is a singular Enriques surface, then its double-cover \(K3\)-surface is ordinary. étale \(p\)-covers; torsionless fundamental group; group acting on scheme; \(p\)-ranks of smooth projective curves; characteristic \(p\); Euler-Poincaré characteristic; singular Enriques surface Crew, Richard M., Etale \(p\)-covers in characteristic \(p\), Compositio Math., 52, 1, 31-45, (1984) Coverings in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Homotopy theory and fundamental groups in algebraic geometry, Finite ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients) Étale \(p\)-covers in characteristic \(p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) be a vector bundle over a smooth curve \(X\). In [\textit{I. Biswas} et al., Beitr. Algebra Geom. 60, No. 1, 137--156 (2019; Zbl 1420.14036)], it is shown that realisations of \(E\) as a direct image under an étale map are in bijection with torus subgroup-schemes of the bundle \(\mathrm{Ad} (E)\) of invertible endomorphisms of \(E\). In the present paper, the authors generalise this result to the parabolic setting. We quote the main results of the paper:
{Theorem 6.3.} Let \(E_*\) be a parabolic vector bundle on a connected smooth complex projective curve \(X\) with parabolic divisor \(S\) and rational parabolic weights. Then there is a natural equivalence between the following two classes: {\parindent=6mm \begin{itemize}\item[(1)] Triples \(( Y, \varphi, V_* )\), where \(\varphi : Y \to X\) is a ramified covering map [\(Y\) not necessarily being connected], and \(V_*\) is a parabolic vector bundle on \(Y\), such that \(\varphi_* V_* = E_*\). \item[(2)] Ramified torus bundles for \(E_*\).
\end{itemize}}
[A \textit{ramified torus subbundle for \(E_*\)} is a torus subbundle of \(\mathrm{Ad} (E)\) satisfying certain conditions at the points of the parabolic divisor of \(E_*\).]
{Theorem 6.6.} The equivalence in Theorem 6.3 takes a connection on the parabolic vector bundle \(V_*\) on \(Y\) to a connection on \(E_*\) that preserves the ramified torus sub-bundle for \(E_*\) corresponding to \((Y, \varphi, V_*)\). Conversely, a connection on \(E_*\) preserving a ramified torus sub-bundle \({\mathcal T}\) for \(E_*\) is taken to a connection on the parabolic vector bundle \(V_*\) on \(Y\), where \((Y, \varphi, V_* )\) corresponds to \({\mathcal T}\). parabolic bundle; parabolic connection; ramified torus bundle; parabolic direct image Coverings in algebraic geometry, Vector bundles on curves and their moduli, Other connections On the direct images of parabolic vector bundles and parabolic connections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using the ``\(k\)-ampleness'' (in the sense of Sommese) of the vector bundle associated with a branched covering, \(f : X \to {\mathcal Q}_n\), of the smooth quadric \({\mathcal Q}_n\), we prove that \(f\) induces \(\mathbb{C}\)-cohomology isomorphisms between \(X\) and \({\mathcal Q}_n\) in a certain range of cohomology and homotopy dimensions, provided the degree of the covering is small enough. Lefschetz type theorems; \(k\)-ampleness; branched covering; smooth quadric Kim, M.,On Branched Coverings of Quadrics, à paraître Coverings in algebraic geometry, Surfaces and higher-dimensional varieties On branched coverings of quadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``Given a finite Galois extension \(E\) of the rational function field \(\mathbb Q(T)\), and a point \(t_0 \in \mathbb P^1(\mathbb Q)\), there is a well-known notion of specialization \(E_{t_0} /\mathbb Q\).''
``The specialization process has been much studied towards the inverse Galois problem, which asks whether every finite group \(G\) occurs as the Galois group of a finite Galois extension \(\mathbb F/\mathbb Q\). In that case, we shall say that such an extension \(F/\mathbb Q\) is a \(G\)-extension.''
``Recent progress has been made on the set \(\text{Sp}(E)\) of all specializations of a given regular \(G\)-extension \(E/\mathbb Q(T)\). For example, for many groups \(G\), no regular \(G\)-extension \(E/\mathbb Q(T)\) is parametric, i.e., \(\text{Sp}(E)\) does not contain all \(G\)-extensions of \(\mathbb Q\).''
``We provide evidence for this conclusion: given a finite Galois cover \(f: X \to \mathbb P^{1}_{\mathbb Q}\) of group \(G\), almost all (in a density sense) realizations of \(G\) over \(\mathbb Q\) do not occur as specializations of \(f\). We show that this holds if the number of branch points of \(f\) is sufficiently large, under the \(abc\)-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of \(\mathbb Q\) of given group and bounded discriminant. This widely extends a result of Granville on the lack of \(\mathbb Q\)-rational points on quadratic twists of hyperelliptic curves over \(\mathbb Q\) with large genus, under the \(abc\)-conjecture (a diophantine reformulation of the case \(G = \mathbb Z/2\mathbb Z\) of our result). As a further evidence, we exhibit a few finite groups \(G\) for which the above conclusion holds unconditionally for almost all covers of \(\mathbb P^{1}_{\mathbb Q}\) of group \(G\). We also introduce a local-global principle for specializations of Galois covers \(f: X\to\mathbb P^{1}_{\mathbb Q}\) and show that it often fails if \(f\) has abelian Galois group and sufficiently many branch points, under the \(abc\)-conjecture. On the one hand, such a local-global conclusion underscores the ``smallness'' of the specialization set of a Galois cover of \(\mathbb P^{1}_{\mathbb Q}\). On the other hand, it allows to generate conditionally ``many'' curves over \(\mathbb Q\) failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.'' Galois theory; specializations; \(abc\)-conjecture; Malle conjecture; uniformity conjecture; hyperelliptic and superelliptic curves; rational points; twisted covers; Hasse principle Galois theory, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Coverings in algebraic geometry, Rational points Density results for specialization sets of Galois covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Positionsänderungen eines, von aktiven Polygonen \(P_i\) (Manipulatoren) eingeklemmten passiven Polygons \(W\) (Werkstück) durch kontrollierte Veränderungen der Lage und Orientierung der Manipulatoren effektiv zu planen, erfordert volles Verständnis der kinematischen Kontakt- (Zwangs-)bedingungen. Anhand eines ebenen Systems mit einem Werkstück und drei Manipulatoren werden, durch Anwendung einfacher Ergebnisse der algebraischen Geometrie, vier fundamentale Typen der Kontakte beschrieben und jeweils die Berechnung von Lage und Orientierung in geschlossener Form durchgeführt. Die Ergebnisse werden mit früheren Ansätzen in Verbindung gebracht [\textit{H. Asada} and \textit{A. B. By}, Kinematic analysis of workpart fixturing for flexible assembly with automatically reconfigurable fixtures, IEEE J. Robotics Autom. 1, No. 2, 86-94 (1985)]. Ein von \textit{R. C. Brost} in seiner Dissertation (Analysis and planning of planar manipulation tasks, Carnegie Mellon University School of Computing Sciences, January 1991) nicht behandelter Spezialfall zeigt, daß für ein spezielles Werkstück alle drei Manipulatoren mit drei gegenseitig nicht parallelen Seiten Kontakt aufrecht erhalten können, das Werkstück aber trotzdem beweglich bleiben kann. contact constrains; planning of motion Kinematics of mechanisms and robots, Coverings in algebraic geometry On the algebraic geometry of a class of contact formation cells | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a field \(k\) which is complete with respect to a discrete valuation and for a finite Galois extension \(K/k\) with Galois group \(G\), the Swan character \(\text{sw}_G\) was introduced in chapter 19 of the book of \textit{J. P. Serre} [Représentations linéaires des groupes finis (Hermann, Paris) (1971; Zbl 0223.20003)]. It is a character of a representation which is realizable over \(\mathbb{Q}_l\) for any prime number \(\ell\) which is different from the residue characteristic \(p\) of \(k\) and it is \(0\) if and only if \(K/k\) is tamely ramified.
Moreover there is a projective \(\mathbb{Z}_l[G]\)-module \(\text{Sw}_G\), uniquely determined up to isomorphism, such that \(\mathbb{Q}_l\otimes \text{Sw}_G\) has character \(\text{sw}_G\). For any \(\mathbb{F}_l[G]\)-module \(M\) the Swan invariant \(l(M):= \dim\Hom^G(\text{Sw}_G/l\cdot \text{Sw}_G,M)\) is \(0\) if and only if the first ramification subgroup \(G_1\) of \(G\) acts trivially on \(M\). These concepts and results have been useful in various contexts of arithmetic algebraic geometry and some of them have been generalized considerably. Motivated by earlier work of P. Deligne, L. Illusie and others, the author of the paper under review obtains the following result: Let \(X\) be a normal scheme of finite type over a local field of residue characteristic \(p\), let \(l\) be a prime number \(\neq p\) and let \(F\), \(F'\) be two locally constant constructible sheaves of \(\mathbb{F}_l\)-vector spaces over \(X\) of the same rank and with the same wild ramification at infinity (in a sense which is made precise in Section 2); then the alternating sum of the Swan invariants of certain \(l\)-adic cohomology groups with values in \(F\) and \(F'\) are the same. Some parts of this paper make use of modular representation theory. schemes; sheaves; ramification; modular; representation theory Vidal, Isabelle, Théorie de {B}rauer et conducteur de {S}wan, Journal of Algebraic Geometry, 13, 349-391, (2004) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Étale and other Grothendieck topologies and (co)homologies, Ordinary representations and characters, Modular representations and characters Brauer theory and Swan conductor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is an obituary remembering the work and life of Shreeram Abhyankar. He was a leading algebraist and geometer who for almost sixty years made important contributions to Mathematics. He passed away on November 2, 2013. He was active in research until his death.
In this note there is a non technical but accurate description of Abhyankar's work in different fields, such as resolution of singularities, valuation theory, ramified coverings, some difficult problems of affine geometry, etc. Some of his most noteworthy work was in resolution of singularities over fields of positive characteristic, where some of the best results available at present are still essentially due to him. The author makes interesting remarks on the preferred methods of Abhyankar (explicit techniques, such as fine calculations with polynomials and series) and on aspects of his life and personality. A brief but useful bibliography including some of his more important works and papers by other authors related to them is included. resolution of singularities; valuation; ramification; affine geometry History of algebraic geometry, Biographies, obituaries, personalia, bibliographies, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Jacobian problem The mathematical life of Shreeram Abhyankar | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the article is the following: Let \(F=(P,G)\) be a polynomial map of the complex plane with finite fibers. Suppose that all the members of the pencil spanned by \(P\) and \(Q\) are irreducible and rational. Then the following facts are equivalent: (a) \((0,0)\) is a regular value of \(F\); (b) det\(DF\) is a nonzero constant; (c) \(F\) is invertible. As a consequence, the author also proves the following statement. Let \(F\) be a polynomial map of the plane with finite fibers. If for generic points \(q\) of the plane the inverse images \(F^{-1}(l)\) of all complex lines passing through \(q\) are irreducible and rational curves, then \(F\) is invertible. Jacobian conjecture; pencils of plane curves; rational curves Jacobian problem, Affine fibrations, Coverings in algebraic geometry Pencils of irreducible rational curves and plane Jacobian conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is well known that if \(f:X\rightarrow Y\) is a covering projection from a path-connected space \(X\) on a path-connected and locally path-connected topological group \((Y,e)\), then for a point \(x_0\in f^{-1}(e)\) there exists a group operation on \(X\) which makes \((X,x_0)\) a topological group and \(f\) a homomorphism of groups. But in the last time, some authors have investigated other properties, weaker for \(X\) and \(Y\), but reinforcing the properties of the map \(f\), so that \((X, x_0)\) is still a group and \(f\) a homomorphism. In this direction the subject was approached by \textit{S. A. Grigorian} and \textit{R. N. Gumerov} in a series of three articles, [Lobachevskii J. Math. 10, 9--16 (2002; Zbl 1010.22007); ibid. 6, 39--46 (2000; Zbl 0968.22002); Topology Appl. 153, No. 18, 3598--3614 (2006; Zbl 1110.57001)], in which the authors proved that the answer is positive if \((Y,e)\) is an arbitrary compact connected group and \(f\) is a finite-sheeted covering projection. The finite-sheeted case for arbitary compact connected groups was als proved by the authors of the present paper [Topology Appl. 153, No. 7, 1033--1045 (2006; Zbl 1104.22005)], which considered the map \(f\) presented as the inverse limit of a pull-back expansion of finite-sheeted covering maps \(f_\lambda:X_\lambda\rightarrow Y_\lambda\) over compact connected Lie groups \(Y_\lambda\), and the group operation on \(X\) is induced by those on \(X_\lambda\). Subsequently the same authors [Fundam. Math. 221, No. 1, 69--82 (2013; Zbl 1276.22001)] proved that any covering homomorphism between topological groups is a special covering map, namely an overlay map.
The aim of the present paper is to investigate the existence and uniqueness of a group structure on the total space of a covering projection \(f:X\rightarrow Y\) over a group \((Y,e)\) which makes \(f\) a covering homomorphism, \(f:(X,x_0)\rightarrow (Y,e)\), generalizing the results obtained for overlay maps over compact groups and covering maps over locally path-connected groups. For this purpose the authors give the following definition. ``For a covering map \(f:X\rightarrow Y\) we say that the space \(X\) is \(f\)-compactly connected (\(f\)-compactly openly connected) if for any points \(a,b\in X\) there exist a (an open) connected subset \(C\) of \(X\) such that \(a,b\in C\) and the closure \(\overline{f(C)}\) is compact. A space \(X\) is said to be compactly connected if \(X\) is \(\mathrm{id}_X\)-compactly connected''.
The main results can be summarized as follows.
Supposing that \(f:X\rightarrow Y\) is an overlay map (respectively a covering map) from a connected space \(X\) over a topological group \((Y,e)\), \(x_0\in f^{-1}(e)\), and \(X\) being \(f\)-compactly openly connected (respectively locally \(f\)-compactly connected), then there exists a group operation on \(X\) such that \((X,x_0)\) is a topological group, and \(f\) is a homomorphism of groups. For another point \(x_0^{\#}\in f^{-1}(e)\) the groups \((X,x_0)\) and \((X,x_0^{\#})\) are isomorphic. (Theorems 1.1 and 1.2).
In addition, the authors prove that in the above conditions, for a fixed point \(x_0\in f^{-1}(e)\) , the group structure on \(X\) with \(x_0\) the identity and \(f\) a homomorphism is unique. (Theorem 1.3).
At the end of the article, the authors formulate the following two questions:
Question 1. Let \(f:X\rightarrow Y\) be an overlay map from a connected space \(X\) to a topological group \(Y\). Is there a group operation on \(X\) such that \(X\) is a topological group and \(f\) is a homomorphism?
Question 2. Does there exist an overlay map \(f:X\rightarrow Y\) from a connected space \(X\) to a topological group \(Y\) which admits two different group operations on \(X\) such that \(X\) is a topological group and \(f\) is a homomorphism? topological group; compact group; locally compact group; abelian group; compactly connected space; locally compactly connected space; covering map; overlay map; covering homomorphism General properties and structure of locally compact groups, Covering spaces and low-dimensional topology, Coverings in algebraic geometry Existence and uniqueness of group structures on covering spaces over groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Artin-Schreier coverings \(\sigma:W\to V\) where the function field extension of \(k(W)\) over \(k(V)\) is of degree \(p>0\) are studied first. Let \(V=\mathbb{A}^ 2\), the affine 2-space, \(\sigma\) be étale and then \(W\) be defined by \(\tau^ p-\tau-f(x,t)=0\). There are three cases:
(i) If \(\deg(f)\equiv 0\pmod p\), certain conditions on \(f\)'s non- singularity imply that \(W\) is non-singular.
(ii) Let \(\deg(f)\equiv p-1\pmod p\). Here \(W\) has exactly \(d\) rational double points of type \(A_{p-1}\), given that the homogenization of \(f\) meets the line at infinity transversally. The same properties that hold for sheaves and their invariants in case (ii) hold here.
(iii) Let \(\deg(f)\not\equiv 0, p-1\pmod p\). For a Gorenstein scheme \(W'\) there is a map \(\Psi:W'\to\mathbb{P}^ 2\). There is also a normalization map \(\pi:W''\to\mathbb{P}^ 2\) corresponding to the extension of \(k(W)\) by \(k(\mathbb{P}^ 2)\). The author fills in the exact sequence \(0\to\Psi_ *{\mathcal O}\to\pi_ *{\mathcal O}\to{\mathcal H}\to 0\) of sheaves over \(\mathbb{P}^ 2\) under the conditions of case (ii). The canonical sheaf of the minimal resolution \(N\) of singularities of \(W''\) is also determined. Independent of the cases above, if \(1+e=mp (0\leq e<p)\) and \(e=p-1\), then, with the hypothesis of case (ii) above, one has \(H^ 1(N,{\mathcal O}_ N)=0\) for \(m\geq p-1\). etale Galois coverings; normalization; cohomology; Artin-Schreier coverings; Gorenstein scheme; minimal resolution of singularities Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Étale and other Grothendieck topologies and (co)homologies Etale Galois coverings of degree \(p\) of the affine plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{M. Namba} [``Branched coverings and algebraic functions'', Pitman Res. Notes Math. Ser. 161 (1987; Zbl 0706.14017); chapter 1], constructed various examples of Galois coverings over affine and projective planes. Among them, the Galois coverings over \(\mathbb{C}^2\) with branch locus \(B_3: =\{(v,w)\in \mathbb{C}^2 \mid v^3= w^2\}\) are studied in detail [loc. cit., pp. 43-52], and as an application, the existence or non-existence of some maximal Galois coverings over \(\mathbb{P}^2\) with branch locus \(\overline{B_3} \cup l_\infty\) is shown, where \(\overline {B_3}\) is the projective closure of \(B_3\) and \(l_\infty\) is the infinite line.
In this note, we extend Namba's results to Galois coverings over \(\mathbb{C}^2\) with branch locus \(B_q:= \{(v,w)\in \mathbb{C}^2 \mid v^q= w^2\}\), where \(q\) is a positive odd integer, under the condition that the maximal Galois group \(G(\mathbb{C}^2, eB_q)\) of \((\mathbb{C}^2, eB_q)\) is finite. It turns out that we have five cases in all, three cases of which appear on p. 43 of the book cited above. As an application we determine when there exists the maximal Galois coverings over \(\mathbb{P}^2\) with branch locus \(\overline {B_q} \cup l_\infty\), and also describe the explicit structure of \(G(\mathbb{P}^2,e \overline {B_q}+ ml_\infty)\) in these cases. Galois coverings; branch locus Nakano, T., Tamai, K.: On some maximal Galois coverings over affine and projective planes. Osaka J. Math. 33, 347--364 (1996) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Complex singularities, Low-dimensional topology of special (e.g., branched) coverings On some maximal Galois coverings over affine and projective planes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review deals with the following question:
Let \(\{f_n\colon X_n\to X_{n+1}\}_{n=1,2,\dots}\) be an infinite tower of nonisomorphic finite étale coverings between smooth projective \(k\)-folds \(X_n\)'s with \(0\leq\kappa(X_n)=a<k\). It is true that for every \(n\), a suitable étale covering \(\widetilde X_n\to X_n\) has the structure of a smooth abelian scheme over a nonsingular projective variety \(W_n\) with \(0\leq\dim(W_n)<k\)?
The author shows that the answer to this question is affirmative if \((k,a)=(3,2)\) (in general), or if \((k,a)=(3,0)\) (under the assumption that all \(X_n\)'s are birationally isomorphic). In the first case, \(\widetilde X_n\cong W_n\times E_n\), with \(E_n\) an elliptic curve and \(W_n\) a surface of general type. In the second case \(\widetilde X_n\) can be: an abelian threefold, or a product \(S_n\times E_n\), with \(E_n\) an elliptic curve and \(S_n\) a surface which is birationally isomorphic to an abelian, or a \(K3\) surface. threefolds; Kodaira dimension; relative abelian schemes Y. Fujimoto: Tower problem on finite étale coverings of smooth projective 3-folds, International Journal of Math. 15 (2004), 135--150. Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), \(3\)-folds Tower problem on finite étale coverings of smooth projective 3-folds. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{V. G. Drinfel'd} [Funct. Anal. Appl. 10, 107-115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29-40 (1976; Zbl 0346.14010)] constructed a formal scheme \(\Omega\) over \(\mathbb{F}_ q[[t]]\) and gave a universal group over the base \(\Omega\otimes\mathbb{F}_{q^ 2}[[t]]\). The torsion points of this formal group give rise to a sequence of étale Galois coverings of rigid analytic spaces:
\[
\Omega\leftarrow\Omega\otimes\mathbb{F}_{q^ 2}[[t]]= \Sigma_ 0\leftarrow\Sigma_ 1\leftarrow\dots .
\]
In the present note a sketch of the proof of the following statement is given: ``\(\Sigma_ n\) are irreducible analytic spaces''. Drinfel'd space; formal group; rigid analytic space A. Genestier, Irréductibilité du revêtement de Drinfeld, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 91-94. Coverings in algebraic geometry, Local ground fields in algebraic geometry, Formal power series rings Irréductibilité du revêtement de Drinfel'd. (Irreducibility of the Drinfel'd covering) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0626.00011.]
The paper under review is an exposition (with many proofs included) of the theory of mixed Hodge structures (MHS) on the (truncations of the group ring of the) fundamental group of a pointed algebraic variety (V,x). It starts with a self-contained account of Chen's theory of iterated integrals [cf. \textit{K.-T. Chen}, Bull. Am. Math. Soc. 83, 831- 879 (1977; Zbl 0389.58001)]; an elementary proof of Chen's \(\pi_ 1\quad de Rham\) theorem is provided. Then the construction of the MHS on \(\pi_ 1(V,x)\) is explained; the Hodge and the weight filtrations are defined in terms of length and weights of iterated integrals representing elements of the truncated group ring \(\pi_ 1.\)
Various applications are discussed such as: Torelli for pointed varieties (especially pointed curves, cf. work of the author and \textit{M. Pulte}), link with B. Harris' harmonic volume [cf. \textit{B. Harris}, Acta Math. 150, 91-123 (1983; Zbl 0527.30032)] and link with the Riemann-Hilbert problem [cf. the author, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 609-27 (1986; Zbl 0616.14004)]. fundamental group; mixed Hodge structures; iterated integrals; filtrations; Torelli; harmonic volume; Riemann-Hilbert problem R. M. Hain, ''The geometry of the mixed Hodge structure on the fundamental group,'' in Algebraic Geometry, Bowdoin, 1985, Providence, RI: Amer. Math. Soc., 1987, vol. 46, pp. 247-282. Transcendental methods, Hodge theory (algebro-geometric aspects), Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry The geometry of the mixed Hodge structure on the fundamental group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For part I and II see Int. Math. Res. Not. 1996, 753-768 (1996; Zbl 0888.14012) and ibid. 1999, 685-716 (1999; Zbl 0939.14014).
A projective structure on a Riemann surface \(X\) is a covering of \(X\) by holomorphic coordinate charts whose transition functions are Möbius transformations. Let \(\Delta\) denote the diagonal divisor of \(X\times X\) and let \(p_i: X\times X\to X\), \(i=1,2\), be the projection onto the \(i\)-th factor. Let \(K_X\) be the holomorphic cotangnet bundle of \(X\), \({\mathcal M}=p_1^*K_X\otimes p^*_2 K_X\otimes {\mathcal O}_{X\times X} (2\Delta)\), and denote by \(\Delta_n\) the \(n\)-th order infinitesimal neighborhood of \(\Delta\) in \(X\times X\). The authors introduce the notion of generalized projective structure on \(X\) as the choice of an extension of the trivialization of the line bundle \({\mathcal M}\) on \(\Delta_1\) to a trivialization on \(\Delta_n\) for \(n\geq 2\), and they provide several canonically equivalent descriptions of the space \({\mathcal S}_n(X)\) of generalized projective structures on \(X\). Moreover, they construct a canonical involution on \({\mathcal S}_n(X)\), and interpret it in the isomorphic spaces introduced above. The paper under review represents a rather interesting progress of the previous work (part I and II, loc. cit.) by both authors. covering of Riemann surface; generalized projective structures Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Coverings in algebraic geometry, Coverings of curves, fundamental group Projective structures on a Riemann surface. III | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f\in \mathbb{Q}[y]\) be a polynomial of degree \(n\) over the rationals. Assume \(f\) is indecomposable and consider the splitting field \(\Omega_f\) of \(f(y)-x\) over \(\mathbb{Q}(x)\). Denote the constants of \(\Omega_f\) by \(\widehat\mathbb{Q}_f\). Then, \(\widehat \mathbb{Q}_f \subset \mathbb{Q} (\zeta_n)\) where \(\zeta_n\) is a primitive \(n\)th root of 1. When \(n=p\), a prime, and \(f= x^p\) (cyclic polynomial), \(\widehat \mathbb{Q}_f= \mathbb{Q} (\zeta_p)\). When \(f=T_p\), the \(p\)th Chebyshev polynomial, \(\widehat \mathbb{Q}_f= \mathbb{Q} (\zeta_p+ \zeta_p^{-1})\). Cohen raised the following question. If \(\widehat\mathbb{Q}_f\) is nontrivial \((f\) has nontrivial extension of constants), it is then true that \(f\) is linearly equivalent over \(\overline\mathbb{Q}\) to a cyclic or Chebyshev polynomial? We show this is false for each non-square odd integer \(n\). This uses elementary group theory and the branch cycle argument. Such \(f\) also give counterexamples to a conjecture of Chowla and Zassenhaus: For all sufficiently large \(p\) (dependent on the degree of \(f)\), \(f(x)-b\) is irreducible for some \(b\in \mathbb{F}_p\). That is, we show for these particular \(f\)'s, for infinitely many \(p\), there is no \(b\in \mathbb{F}_p\) so that \(f(x)-b\) is irreducible over \(\mathbb{F}_p\). Also, for these \(p\), there is no \(b\in \mathbb{F}_p\) so that \(f(x)-b\) splits completely over \(\mathbb{Z}/p\). Further, using Müller's classification of geometric monodromy groups of polynomials we show \(n\) must be odd for such counterexamples. These are \((A_n,S_n)\) realizations by polynomials over \(\mathbb{Q}\). More delicate examples require rigidity applied to non-Galois covers. These contrast the arithmetic of covers with and without using braid operations on branch cycle descriptions. Braid operations describe four families of covers that include the renowned Davenport polynomials of degree 7. Chowla-Zassenhaus conjecture; cyclic polynomial; \(p\)th Chebyshev polynomial; extension of constants; branch cycle; Davenport polynomials Fried M D. Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture. Finite Fields Appl, 1995, 1: 326--359 Galois theory, Separable extensions, Galois theory, Coverings in algebraic geometry, Arithmetic theory of algebraic function fields Extension of constants, rigidity, and the Chowla-Zassenhaus conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article is a report about a joint paper of the author and his brother [Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 6, 65--106 (2000; Zbl 1012.14004)]. Generic coverings \(f:X\to \mathbb P^2\) of the projective plane by surfaces \(X\) with A-D-E-singularities are studied. Results in the case of smooth surfaces are generalized and numerical inva\-riants of a generic covering in terms of intersection numbers on a minimal resolution of the surface are computed. The results are applied to prove the Chisini conjecture for generic \(m\)-canonical coverings of surfaces of general type: Let \(S_1\) and \(S_2\) be minimal models of surfaces of general type with the same \((K^2_S)\) and \(\chi(S)\), and let \(f_1: X_1\to \mathbb P^2, f_2: X_2\to \mathbb P^2\) be generic \(m\)-canonical coverings with the same discriminant curve. Then for \(m\geq 5\) the coverings are equivalent. generic projections; generic coverings Valentine S. Kulikov, On a conjecture of Chisini for coverings of the plane with A-D-E-singularities, Real and complex singularities 232 (2003), 175-188, Dekker, New York Zbl1081.14050 MR2075064 Singularities of surfaces or higher-dimensional varieties, Coverings in algebraic geometry On a conjecture of Chisini for coverings of the plane with A-D-E-singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From the author's abstract: For the Galois closure \(X_{\text{gal}}\) of a generic projection from a surface \(X\), it is believed that \(\pi_1(X_{\text{gal}})\) gives rise to new invariants of \(X\). However, in all examples this group is surprisingly simple. In this article, we offer an explanation for this phenomenon: We compute a quotient of \(\pi_1(X_{\text{gal}})\) that depends on \(\pi_1(X)\) and data from the generic projection only. In all known examples, this quotient is in fact isomorphic to \(\pi_1(X_{\text{gal}})\). As a byproduct, we simplify the computations of Moishezon, Teicher and others. fundamental group; algebraic surface; Galois closure; generic projection Liedtke, C.: Fundamental groups of Galois closures of generic projections, Trans. amer. Math. soc. 362, 2167-2188 (2010) Coverings in algebraic geometry, Surfaces of general type Fundamental groups of Galois closures of generic projections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Dans Ann. Math., II. Ser. 32, 485-511 (1931; Zbl 0001.40301), \textit{O. Zariski} étudie les revêtements de \(\mathbb{P}^ 2\) ramifiés le long d'une courbe \(C\) et d'une droite à l'infini \(L\) transverse à \(C\). Par définition, l'irrégularité \(q\) d'un tel revêtement \(\widehat{X}\) est la dimension du groupe de cohomologie \(H^ 1(Z,{\mathcal O}_ Z)\), où \(Z\) est une résolution des singularités de \(\widehat{X}\). Grâce à la théorie de Hodge, l'irrégularité est liée au premier nombre de Betti de la surface \(Z\) par l'égalité \(2q = b_ 1(Z) = \dim H^ 1(Z,\mathbb{C})\). Dans le cas où la courbe \(C\) a pour seules singularités des points doubles ordinaires et des points cuspidaux, Zariski calcule l'irrégularité du revêtement en fonction de la surabondance des systèmes linéaires des courbes de certains degrés fixés passant par les points cuspidaux de \(C\), c'est à dire en fonction des groupes de cohomologie \(H^ 1(\mathbb{P}^ 2,{\mathcal A}(n))\), où \(\mathcal A\) est l'idéal des fonctions holomorphes sur \(\mathbb{P}^ 2\) s'annulant aux points cuspidaux de \(C\).
Dans cet article nous nous proposons de généraliser ces résultats au cas d'une surface non singulière projective \(X\) et d'une courbe réduite \(C\) sur \(X\) ayant des singularités quelconques. Si nous notons \(\omega_ X\) le faisceau dualisant sur \(X\) et \({\mathcal A}_ \alpha\) le faisceau sur \(X\) constitué des fonctions holomorphes dont l'exposant de Hodge en chaque point singulier de \(C\) est strictement supérieur à \(\alpha\), nous avons les résultats:
Théorème 3.1. Soit \(H\) un diviseur très ample sur une surface non singulière projective \(X\), soit \(C\) une courbe réduite sur \(X\) appartenant au système linéaire \(| mH|\) et soit \(L\) une courbe non singulière appartenant à \(| H|\) transverse à \(C\); alors l'irrégularité \(q\) du revêtement cyclique de \(X\) de degré \(n\), ramifié le long de \(C \cup L\) est égale à:
\[
q = \sum \dim H^ 1(X,{\mathcal A}_ \alpha (\lceil pm/n\rceil H) \otimes \omega_ X)
\]
où \(\alpha = p/n - 1\), \(p = 0,1,\dots,n - 1\), et où pour tout nombre réel \(x\) on note \(\lceil x\rceil\) le plus petit entier supérieur ou égal à \(x\).
Théorème 3.6. Si \(\beta\) est le plus grand exposant strictement négatif appartenant au spectre d'une singularité de la courbe \(C\), le groupe \(H^ 1(X,{\mathcal A}_ \beta(sH) \bigotimes \omega_ X)\) est nul pour tout entier \(s\) tel que \(m \geq s > m(\beta + 1)\). irregularity; divisor; cyclic coverings; resolution of singularity; vanishing of first cohomology Vaquié, Michel, Irrégularité des revêtements cycliques des surfaces projectives non singulières, Amer. J. math., 114, 6, 1187-1199, (1992), MR 1198299 (94d:14015) Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry Irregularity of cyclic coverings of nonsingular projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper deals with normal generic covers branched over \(\{x^n=y^m\}\), i.e., finite holomorphic maps \(\pi : S \rightarrow \mathbb{C}^2\) from a connected normal surface \(S\) to complex plane \(\mathbb{C}^2\), which is an analytic covering branched over a curve \(B=\{(x,y) \in \mathbb{C}^2 : x^n=y^m\}\), such that the fiber \(\pi^{-1}(p)\) for \(p \in B\setminus \{(0,0)\}\) is supported on \(\deg \pi -1\) points. The authors show that the germ of such a map \(\pi\) is equivalent to a germ of the branched covering defined on a surface \(X\) obtained by contracting a section \(C_0\) of a ruled surface \(\widetilde{X}\) on a smooth curve \(C\) and taking a quotient by an action of a finite cyclic group \(G\). Basing on the above they give the numerical characterization of all the smooth normal generic covers branched over \(\{x^n=y^m\}\) and state also the rationality criteria. branched coverings; normal surface singularities Complex surface and hypersurface singularities, Singularities in algebraic geometry, Coverings in algebraic geometry, Local complex singularities, Rational and ruled surfaces Ruled surfaces and generic coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This talk is concerned with the proof of Abhyankar's conjecture after \textit{M. Raynaud} [Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013)]. Abhyankar's conjecture; covering of affine line; Sylow \(p\)-group; Galois group of covering Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry Introduction to the Abhyankar's conjecture for \(\mathbb{P}^1_{\{\infty\}}\) for the case \(G\not={G(S)}\) after M. Raynaud | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S_ 0\) be a finite subset of \({\mathbb{P}}^ 1({\mathbb{C}})\) containing \(\{0,1,\infty\}\), \(r:=card(S_ 0)-1\), \(k_ 0:={\mathbb{Q}}(S_ 0\setminus \{\infty \})\), \(\bar k_ 0\) the algebraic closure of \(k_ 0\) in \({\mathbb{C}}\), and \(G_{k_ 0}:=Gal(\bar k_ 0/k_ 0)\). In the first part of this paper [cf. \textit{G. Anderson} and \textit{Y. Ihara}, Ann. Math., II. Ser. 128, 271-293 (1988; Zbl 0692.14018)], the authors established a relationship between the actions of \(G_{k_ 0}\) on the pro-\(\ell\) fundamental group \(\pi_ 1\) of \({\mathbb{P}}^ 1({\mathbb{C}})\setminus S_ 0\) and on the group \(E^{\cdot}\) of higher circular \(\ell\)-units of \(S_ 0\), respectively, where \(\ell\) is a prime number.
In the present second part of their investigations, they provide a strengthening of this relationship, in that they assign certain \({\mathbb{Z}}_{\ell}\)-valued invariants to the actions of \(G_{k_ 0}\) on \(\pi_ 1\) and the group \(E^{\cdot}\), respectively, and compare them modulo powers of the prime number \(\ell.\) Basically, the invariants associated with the Galois action on \(\pi_ 1\) are matrices with entries in the completed group algebra \({\mathbb{Z}}_{\ell}[[\pi_ 1]]\), whereas the invariants associated with the \(G_{k_ 0}\)-action on \(E^{\cdot}\) are certain finite trees parametrized by sequences of elements of \(E^{\cdot}\cup \{0\}\) which, on their own, arise from branched coverings \(f: {\mathbb{P}}^ 1\to {\mathbb{P}}^ 1\) that are totally ramified over \(\infty\). The main result then consists in establishing congruence relations (modulo powers of \(\ell)\) between these invariants. pro-\(\ell \) fundamental group; ramification; Galois action; branched coverings Anderson, GW; Ihara, Y., Pro-\(l\) branched coverings of \({ P}^1\) and higher circular \(l\)-units. II, Int. J. Math., 1, 119-148, (1990) Coverings of curves, fundamental group, Arithmetic theory of algebraic function fields, Coverings in algebraic geometry Pro-\(\ell\) branched coverings of \({\mathbb{P}}^ 1\) and higher circular \(\ell\)-units. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author introduces new birational invariants for a nonsingular complete variety \(X\) defined over a finite field \(\mathbb F_q\), \(q=p^r\), which are defined in terms of the étale and crystalline cohomologies of \(X\). Then he applies these invariants to show that the fundamental group of a unirational, nonsingular, complete variety defined over an algebraically closed field of characteristic \(p>0\) has a trivial \(p\)-Sylow group. If the variety is projective instead of being complete, the result is due to N. Suwa. unirational variety; fundamental group; prime characteristic; trivial p-Sylow group EKEDAHL (T.) . - Sur le groupe fondamental d'une variété unirationnelle , C. R. Acad. Sci. Paris, t. 297, 1983 , p. 627-629. MR 85f:14010 | Zbl 0591.14010 Coverings in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Rational and unirational varieties, Finite ground fields in algebraic geometry On the fundamental group of a unirational variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Over the past 25 years, the study of cohomological support varieties and representation-theoretic rank varieties has led to numerous results in the modular representation theory of finite groups, restricted Lie algebras, and other related structures. This work follows previous work of the authors [Am. J. Math. 127, No. 2, 379-420 (2005; Zbl 1072.20009), Erratum ibid. 128, No. 4, 1067-1068 (2006; Zbl 1098.20500)] in attempting to formulate a unifying theory for arbitrary finite group schemes (equivalently finite-dimensional cocommutative Hopf algebras) over arbitrary fields of prime characteristic. The paper contains several foundational results as well as a number of illuminating examples.
Let \(G\) be a finite group scheme over a field \(k\) of prime characteristic \(p\). Extending their previous notion of a \(p\)-point (defined over algebraically closed fields), the authors introduce the notion of a \(\pi\)-point of \(G\) which is a flat map of \(K\)-algebras \(K[t]/t^p\to KG\) which factors through the group algebra of a unipotent Abelian subgroup scheme for a field extension \(K/k\). The \(\Pi\)-points of \(G\), denoted \(\Pi(G)\), is the set of equivalence classes of such \(\pi\)-points under a certain specialization relation.
The first of several important results is that \(\Pi(G)\) is homeomorphic to the projectivized prime ideal spectrum of the (even-dimensional) cohomology ring of \(G\) over \(k\). For a finite-dimensional \(G\)-module \(M\), the \(\Pi\)-support of \(M\) is defined as a certain subset of \(\Pi(G)\) and can be identified cohomologically. Moreover, the authors extend this definition to arbitrary (i.e., even infinite-dimensional) \(G\)-modules. For an arbitrary module, the \(\Pi\)-support does not have a direct cohomological interpretation. The authors show that it does satisfy a number of nice properties and that every subset of \(\Pi(G)\) can be identified with the \(\Pi\)-support of some module. Another fundamental result is that the projectivity of a module can be detected by restriction to \(\pi\)-points which extends several known results in special cases. Further, the \(\Pi\)-support is used to determine the tensor-ideal thick subcategories of the stable module category of finite-dimensional \(G\)-modules, thus verifying a conjecture of \textit{M. Hovey, J. H. Palmieri}, and \textit{N. P. Strickland} [Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. Using this stable module category information, the authors give a scheme structure to \(\Pi(G)\) and show that the aforementioned homeomorphism of varieties can be extended to an isomorphism of schemes. group schemes; support varieties; rank varieties; \(p\)-points; thick subcategories; stable module categories; cohomology rings; finite-dimensional cocommutative Hopf algebras Friedlander, E. M.; Pevtsova, J., \({\Pi}\)-supports for modules for finite group schemes, Duke Math. J., 139, 2, 317-368, (2007) Representation theory for linear algebraic groups, Group schemes, Modular representations and characters, Cohomology theory for linear algebraic groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act \(\Pi\)-supports for modules for finite group schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0626.00011.]
The paper is devoted to the construction of some remarkable examples of minimal surfaces of general type with finite commutative (or even trivial) fundamental group and with positive (or vanishing) index [cf. also the authors, Invent. Math. 89, 601-643 (1987; Zbl 0627.14019)]. They are obtained as Galois coverings corresponding to generic projections \(X\to P^ 2\), where X is either \(P^ 1\times P^ 1\) or \(P^ 2\) embedded via complete linear systems. The examples above disprove two conjectures which have been circulating for some time, namely that for any minimal surface of general type (1) if the index is positive then the fundamental group is infinite; and (2) if the index is zero then the universal covering is the polydisk. minimal surfaces of general type; fundamental group; index B. Moishezon and M. Teicher, Galois coverings in the theory of algebraic surfaces, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 47 -- 65. Surfaces of general type, Coverings in algebraic geometry Galois coverings in the theory of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems R. Brauer has shown that every complex character \(\chi\) of every finite group \(G\) can be written as an integral linear combination of characters induced from one-dimensional characters of subgroups of \(G\). Canonical versions of such ``induction formulae'' were given by \textit{V. Snaith} [Invent. Math. 94, No. 3, 455-478 (1988; Zbl 0704.20009)], \textit{R. Boltje} [Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)] and the author [Comment. Math. Helv. 66, No. 2, 169-184 (1991; Zbl 0797.20008)]. The paper under review gives two analogs of such induction formulae in a \(p\)-modular setting; the approach is geometric. The first of these formulae deals with Brauer characters. In this case the formula is not integral, in general, but rational with \(p\)-power denominators. It differs from the formula obtained by \textit{R. Boltje} [in J. Algebra 206, No. 1, 293-343 (1998; Zbl 0913.20001)] which has integral coefficients and was derived algebraically. The second formula deals with the Green ring of trivial source modules. In this case the formula is integral and coincides with the one obtained by \textit{R. Boltje} [in Proc. Symp. Pure Math. 63, 7-30 (1998; Zbl 0892.20006)]. induction formulae; trivial source modules; Brauer characters; Brauer maps; Euler characteristic; Green rings; modular representations Symonds, P.: A splitting principle for modular group representations. Bull. London math. Soc. 34, 551-560 (2002) Modular representations and characters, Frobenius induction, Burnside and representation rings, Étale and other Grothendieck topologies and (co)homologies, Finite groups of transformations in algebraic topology (including Smith theory) A splitting principle for modular group representations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A short introduction is given (without proofs) to coverings, their Galois theory and Riemann surfaces up to the Riemann-Hurwitz formula and certain of its applications. Galois coverings; Galois theory of Riemann surfaces Reyssat, E.: In: Waldschmidt, M., Moussa, P., Louck, J.M., Itzykson, C. (eds.) From Number Theory to Physics. Springer, Berlin Heidelberg New York (1992) Coverings of curves, fundamental group, Separable extensions, Galois theory, Compact Riemann surfaces and uniformization, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Riemann surfaces; Weierstrass points; gap sequences, Coverings in algebraic geometry, Inverse Galois theory Galois theory for coverings and Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here the author proves (in characteristic 0 and using algebraic tools) the following results of Lefschetz-Barth-Larsen type.
Let \(f : X \to \mathbb{P}^n \times \mathbb{P}^n\) be a finite morphism with \(X\) smooth and irreducible, \(\dim (X) \geq n + 2\). Let \(\Delta \subset \mathbb{P}^n \times \mathbb{P}^n\) be the diagonal and \(Y : = \dim (f^{-1} (\Delta))\). Assume \(\dim (Y) = \dim (X) - n\) and \(Y\) normal. Then the natural maps \(\text{Pic}^0 (X) \to \text{Pic}^0 (Y)\) and \(\text{Alb} (Y) \to \text{Alb} (X)\) are isomorphisms and if \(\dim (X) \geq n + 3\) there is a canonical exact sequence
\[
0 \to \mathbb{Z} \to \text{Pic}^0 (X) \to \text{Pic}^0 (Y) \to 0.
\]
If \(Y \subset X_{\text{reg}}\), then there is a canonical exact sequence of profinite groups
\[
\mathbb{Z}^\wedge \to \pi_1^{\text{alg}} (Y) \to \pi_1^{\text{alg}} (X_{\text{reg}}) \to 0.
\]
Corollary: Let \(g : A \to \mathbb{P}^n\) be a finite morphism with \(A\) irreducible and \(B\) a closed subvariety of \(\mathbb{P}^n\). Set \(a : = \dim (A)\) and \(b : = \dim (B)\). If \(a + b \geq n + 2\), \(2b \geq n + 2\) and \(C : = g^{-1} (B)\) is normal of dimension \(a + b - n\), then the natural maps \(\text{Pic}^0 (A) \to \text{Pic}^0 (C)\) and \(\text{Alb} (C) \to \text{Alb} (A)\) are isomorphisms. If also \(a + b \geq n + 3\), then the natural restriction map \(\text{Pic} (A) \to \text{Pic} (C)\) is an isomorphism. ramified covering; Lefschetz theorem; Picard group; Albanese variety; algebraic fundamental group; finite morphism Topological properties in algebraic geometry, Picard schemes, higher Jacobians, Coverings in algebraic geometry, Picard groups Lefschetz type results for proper intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We compute fundamental groups of the complements of a class of real curves in the complex projective plane. As a result, we obtain a new Zariski pair for arrangements of conics. As an application, we give a method for the computations of the fundamental groups of resolutions of Galois covering spaces of the projective plane ramifying along a special type of curves. fundamental groups; complements of real curves; Zariski pair; arrangements; Galois covering Namba, M; Tsuchihashi, H, On the fundamental groups of Galois covering spaces of the projective plane, Geom. Dedicata, 104, 97-117, (2004) Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Plane and space curves, Topology of real algebraic varieties On the fundamental groups of Galois covering spaces of the projective plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The reviewer finds the abstract of the paper under review concise and clear, so it is basically reiterated here, simply adding the references:
``\textit{E. Horikawa} [Invent. Math. 31, 43--85 (1975; Zbl 0317.14018)] introduced a method of resolving singularities of double covers of a smooth surface, called the canonical resolution. \textit{Ashikaga} [Tôhoku Math. J. 44, 177--200 (1992; Zbl 0801.14011)] gave a similar method for certain triple covers and \textit{S.-L. Tan} [AMS/IP Stud. Adv. Math. 29, 143--164 (2002; Zbl 1018.14003)] constructed the canonical resolution for any triple covers. These methods are useful for the global or local study of branched covers of surfaces.
In this paper, we consider similar resolution for 4-fold covers over a smooth surface, which is based on Lagrange's method to solve quartic equations. By using this method, we compute the Chern numbers \(c_1^2\) and \(c_2\) of certain 4-fold covers over a smooth projective surface.'' 4-fold cover; canonical resolution; Chern numbers Shirane, T, On 4-fold covers of algebraic surfaces, Kyushu J Math, 64, 297-322, (2010) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On 4-fold covers of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a short announcement (containing hardly any proof) of results concerning the birational structure of double covers of the projective plane. It also contains an application to the problem of the existence of plane curves with assigned singularities. birational structure of double covers of the projective plane Yoshihara, H.: Double coverings of P2. Proc. Japan Acad.66, 233--236 (1990) Coverings in algebraic geometry, Surfaces and higher-dimensional varieties, Projective techniques in algebraic geometry Double coverings of \(\mathbb{P}^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The original main theorem in the the author's paper [ibid. 154, No. 3--4, 279--284 (2017; Zbl 1388.14130)] is revised based on a referee's
request. Especially, a corrected proof of statement (1) of the theorem is given by proving a slightly stronger statement. Algebraic theory of abelian varieties, Coverings in algebraic geometry Erratum to: ``A note on Galois embeddings of abelian varieties'' | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0504.14019. Galois wildly ramified covers of the projective line; characteristic p; monodromy; branch cycles; supersingular p-covers; fundamental group Coverings of curves, fundamental group, Local ground fields in algebraic geometry, Coverings in algebraic geometry Ordinary and supersingular covers in characteristic p | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present an algorithm for finding the homotopy type of the complement to an algebraic curve in \({\mathbb{C}}^ 2\) with arbitrary singularities. More precisely we describe a presentation of the fundamental group of the complement [in terms of the braid monodromy introduced by \textit{B. G. Moishezon} in Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 107-192 (1981; Zbl 0476.14005)] such that the associated 2-dimensional complex has the same homotopy type as the complement to the curve. As a corollary we describe how the homotopy type of the complement to a plane curve changes when this curve degenerates and acquires new singularities. In particular in a regeneration of a curve in which one cusp changes into a node from the homotopy point of view amounts to taking wedge with a 2-dimensional sphere. As an example we show that the complement in an affine plane to the sextic with 6 cusps on conic has the homotopy type of the wedge of the complement to the trefoil knot in 3- sphere and 13 copies of 2-sphere. Zariski problem; homotopy type of the complement to an algebraic curve; fundamental group A.Libgober, ``On the homotopy type of the complement to plane algebraic curves.'' J. Reine Angew. Math. 367 (1986): 103--114. Homotopy theory and fundamental groups in algebraic geometry, Singularities of curves, local rings, Classification of homotopy type, Formal methods and deformations in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Coverings in algebraic geometry On the homotopy type of the complement to plane algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe the application of the lattice covering problem to the placement of templates in a search for continuous gravitational waves from the low-mass x-ray binary Scorpius X-1. Efficient placement of templates to cover the parameter space at a given maximum mismatch is an application of the sphere covering problem, for which an implementation is available in the \texttt{LatticeTiling} software library. In the case of Sco X-1, potential correlations, in both the prior uncertainty and the mismatch metric, between the orbital period and orbital phase, lead to complications in the efficient construction of the lattice. We define a shearing coordinate transformation which simultaneously minimizes both of these sources of correlation, and allows us to take advantage of the small prior orbital period uncertainty. The resulting lattices have a factor of about three fewer templates than the corresponding parameter space grids constructed by the prior straightforward method, allowing a more sensitive search at the same computing cost and maximum mismatch. gravitational waves; lattices; Scorpius X-1 Radiative transfer in astronomy and astrophysics, Gravitational waves, Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory, Quantum optics, Coverings in algebraic geometry, Measures of association (correlation, canonical correlation, etc.), Orbital mechanics, Uncertainty relations, also entropic Template lattices for a cross-correlation search for gravitational waves from Scorpius X-1 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This survey article contains the lecture notes for the author's mini-course taught at the Graduate Summer School of the IAS/Park City Mathematics Institute in July 2011. As the author points out, the main goal of his five lectures is to explain why the absolute Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) and its action on the profinite completion of the fundamental group of a \(n\)-pointed Riemann surface of genus \(g\) appear quite naturally in the study of the corresponding mapping class group, and therefore also in the geometric description of the moduli space \(M_{g,n}\) of such surfaces.
This approach is based on Grothendieck's theory of the algebraic fundamental group of an algebraic variety, and consequently this abstract framework is briefly introduced in Lecture 1. In this context, the algebraic fundamental group of an algebraic variety \(K\) defined over a subfield \(K\) of \(\mathbb{C}\) appears as the profinite completion of the classic topological fundamental group, and its connection with the Galois group \(\mathrm{Gal}(\overline K/K)\) is explained via Grothendieck's seminal work developed in the famous seminar notes ``SGA 1'' [Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag. XXII, 447 p. (1971; Zbl 0234.14002)]. In the special case of the Teichmüller description of the moduli space \(M_{g,n}(\mathbb{C})\) it is shown that the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) naturally appears in a short exact sequence containing the algebraic fundamental groups of \(M_{g,n}(\mathbb{C})\) and \(M_{g,n}(\mathbb{Q})\), on the one hand, and in an exact sequence containing the so-called arithmetic mapping class group \(\mathrm{AMCG}_{g,n}\) and the profinite completion of the ordinary mapping class group \(\mathrm{MCG}_{g,n}\) alternatively.
Lecture 2 briefly recalls the monodromy representation on topological fundamental groups, thereby pointing out that the above Galois representation of \(\mathrm{Gal}(\mathbb{Q}/\mathbb{Q})\) may be viewed as an analogue of the latter in the arithmetic/algebraic case.
Lecture 3 provides a sketchy description of the arithmetic mapping class groups \(\mathrm{AWCG}_{g,n}\), with the main reference being the related work of \textit{T. Oda} [Étale homotopy type of the moduli spaces of algebraic curves. Geometric Galois actions 1. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 85--95 (1997; Zbl 0902.14019)].
Lecture 4 points to some more recent results concerning the properties of various Galois representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) with regard to outer fundamental groups of \((g,n)\)-surfaces, or to the profinite completion of the mapping class group \(\mathrm{MCG}_{g,n}\), respectively. One of the main references, in this context, is the author's own work [\textit{M. Matsumoto} and \textit{A. Tamagawa}, Am. J. Math. 122, No. 5, 1017--1026 (2000; Zbl 0993.12002)].
Finally, Lecture 5 discusses the question of how these Galois actions of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) can be used to obtain a bound for the size of the mapping class group \(\mathrm{MCG}_{g,n}\) in the group of automorphisms of the Lie algebraization of the topological fundamental group of a \((g,n)\)-surface by giving Galois obstructions to the surjectivity of the so-called higher Johnson homomorpbisms. In this context, the geometry of the mixed Hodge structure on the topological fundamental group of a \((g,n)\)-surface plays a crucial role, just as the recent progress concerning related conjectures of Deligne-Ihara and of T. Oda does. The paper ends with an appendix, the purpose of which is to describe algebraic fundamental groups in terms of étale coverings and fiber functors à la A. Grothendieck. algebraic fundamental group; arithmetic mapping class group; Galois representations; Johnson homomorphisms; graded Lie algebras M. Matsumoto, Introduction to arithmetic mapping class groups, Moduli spaces of Riemann surfaces, IAS/Park City Math. Ser. 20, American Mathematical Society, Providence (2013), 319-356. Homotopy theory and fundamental groups in algebraic geometry, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Coverings in algebraic geometry, Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Introduction to arithmetic mapping class groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We adapt \textit{M. F. Atiyah}'s \(L^2\)-index theory [Astérisque 32-33, 43-72 (1976; Zbl 0323.58015)] to treat coherent sheaves on algebraic manifolds and use it as a tool to investigate certain questions posed by \textit{J. Kollár} [``Shafarevich maps and automorphic forms'' (1995; Zbl 0871.14015); chapter 18]. Let \(X\) be a connected projective algebraic compact complex manifold. We prove that, if \(L\) is a big and nef divisor on \(X\), such that the restriction of \(K_X+L\) to the general fiber of a Shafarevich map is effective, \(K_X+L\) is effective. Let \(X\) be a connected Kähler manifold such that some big cohomology class of type \((1,1)\) is in the image of \(H^2(\pi_1(X),\mathbb{R})\). We prove that \(\chi(X, K_X) \geq 0\). Furthermore, if \(\chi(X, K_X)\) is not \(0\), the universal covering space of \(X\) carries a non trivial \(L^2\) holomorphic form of maximal degree. If \(\chi(X, K_X)\) is zero, we prove that zero belongs to the spectrum of the Laplace-Beltrami operator on the middle degree forms, provided the fundamental group has subexponential growth. \(L^2\) index theorem; Nadel's vanishing theorem; adjoint bundle; adjoint linear systems; coherent sheaves; Shafarevich map; fundamental group DOI: 10.5802/aif.1670 Divisors, linear systems, invertible sheaves, Homotopy theory and fundamental groups in algebraic geometry, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vanishing theorems, Coverings in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects) \(L^2\) adjoint linear systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a plane projective curve with singularities; the fundamental group \(\pi_ 1(\mathbb{P}^ 2-C)\) was studied by many authors [e.g. \textit{O. Zariski}, Am. J. Math. 51, 305-328 (1929), \textit{P. Deligne} in Sém. Bourbaki, 32e Année, Vol. 1979/80, Exposé 543, Lect. Notes Math. 842, 1-10 (1981; Zbl 0478.14008) and others, including the author of the present paper]. Not much is known about cuspidal curves in general.
In the paper under review the author constructs systematically plane curves with nodes and cusps, defined by symmetric polynomials and gives a lot of examples of cuspidal curves of small degree; he computes their fundamental group and the fundamental group of their complement and discusses their degenerations, obtaining in particular a new proof of some result of Zariski. plane curves; nodes; cusps; fundamental group Oka, Symmetric plane curves with nodes and cusps, J. Math. Soc. Japan 44 ((3)) pp 375-- (1992) Singularities of curves, local rings, Coverings of curves, fundamental group, Coverings in algebraic geometry Symmetric plane curves with nodes and cusps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A Hurwitz space \(H\) is the space of (ramified) coverings of compact Riemann surfaces with their genera, covering degree and ramification fixed. The author constructs a combinatorial model of a compactification \(\overline {H}\) of \(H\) and uses it to give a cellular decomposition of \(\overline{H}\). This generalizes the cellular decomposition of the moduli space \(M_{g,n}\) of punctured Riemann surfaces (Harer-Kontsevich theorem) [\textit{M. Kontsevich}, Commun. Math. Phys. 147, 1--23 (1992; Zbl 0756.35081)]. The author also gives an analogy between Hurwitz spaces of cyclic Galois coverings and moduli spaces of spin curves. ramified coverings; Galois coverings; compactification; moduli spaces of spin curves Coverings of curves, fundamental group, Coverings in algebraic geometry, Planar graphs; geometric and topological aspects of graph theory, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (algebraic) Cellular decomposition of compactified Hurwitz spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A closed Riemann surface \(S\) of genus \(g \geq 2\) is called cyclic \(n\)-gonal if it admits an order \(n \geq 2\) conformal automorphism \(\omega\) so that \(S/\langle \omega \rangle\) has genus zero. It is well known that these surfaces can be described by algebraic curves of the form
\[
y^{n}=f(x)=\prod_{j=1}^{s}(x-a_{j})^{t_{j}},
\]
where (i) \(a_{1},\ldots, a_{s} \in {\mathbb C}\) are distinct, (ii) \(t_{j} \in \{1,\ldots,n-1\}\) and (iii) \(\gcd(n,t_{1},\ldots,t_{s})=1\). In such a model, the automorphism \(\omega\) is given by \(\omega(x,y)=(x,e^{2 \pi i/n}y)\). In many cases, the cyclic group \(C=\langle \omega \rangle\) is a normal subgroup of \(A=\mathrm{Aut}(S)\); for instance:
(1) If \(n=p\) is a prime integer and \(g>(p-1)^{2}\), then \(C\) is a normal subgroup. If moreover, \(f(x)\) is square free, then \(C\) is central [\textit{R. D. M. Accola}, Proc. Am. Math. Soc. 26, 315--322 (1970; Zbl 0212.42501)].
(2) If \(t_{1}+\cdots+t_{s} \equiv 0 \mod(n)\), \(\gcd(n,t_{j})=1\), for all \(j\), and \(g>(n-1)^{2}\), then \(C\) is a normal subgroup [\textit{A. Kontogeorgis}, J. Algebra 216, No. 2, 665--706, Art. No. jabr.1998.7804 (1999; Zbl 0938.11056)].
(3) If \(C\) is a weakly malnormal subgroup and \(g>(n-1)^{2}\), then \(C\) is a normal subgroup [\textit{S. A. Broughton} and \textit{A. Wootton}, Full automorphism groups of generalized superelliptic Riemann surfaces'', to appear].
If \(C\) is not a normal subgroup and \(N\) denotes the normalizer of \(C\) in \(A\), then every element of \(A-N\) is called an \textit{exceptional automorphism} of \(S\). In general, to determine those exceptional automorphisms is a difficult problem. The authors restrict to those cyclic \(n\)-gonal curves, called \textit{generalized superelliptic surfaces}, satisfying the conditions \(t_{1}+\cdots+t_{s} \equiv 0 \mod(n)\) and \(\mathrm{gcd}(n,t_{j})=1\), for all \(j\) (these are examples for which \(C\) is malnormal). The cases needed to consider, because of (2) above, are when \(g \leq (n-1)^{2}\). The weakly malnormal condition produces restrictions on the normalizer \(N\) and this, together Singerman's lists of maximal Fuchsian groups, permits to describe the possible signatures of \(S/A\) and \(S/N\), together the index of \(N\) in \(A\) and the group \(A\). Riemann surface; automorphisms of Riemann surfaces; p-gonal curve; super-elliptic curve Broughton, S. A.; Wootton, A.: Exceptional automorphisms of (generalized) super elliptic surfaces. Contemp. math. 629, 29-42 (2014) Automorphisms of curves, Compact Riemann surfaces and uniformization, Classification theory of Riemann surfaces, Coverings in algebraic geometry, Plane and space curves Exceptional automorphisms of (generalized) super elliptic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper considers the fundamental group \(\pi_Y\) of the complement of a branch curve of a generic projection of an algebraic surface \(Y\) to \(\mathbb{C}\text{P}^2\). There is no precise description of this group but there is a precise description of the kernels of homomorphisms of \(\pi_Y\) into \(\widetilde{\text{Sp}}(n,\mathbb{Z}/3)\). finite quotients of braid groups; symplectic groups; fundamental groups; complements of branch curves; algebraic surfaces B. Moishezon,Finite quotients of braid groups related to symplectic groups over \(\mathbb{Z}\)/3, preprint. Braid groups; Artin groups, Coverings of curves, fundamental group, Coverings in algebraic geometry Finite quotients of braid groups related to symplectic groups over \(\mathbb{Z}/3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We develop a quadratic technique for proving the birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove the birational rigidity of generic fibrations into Fano double spaces of dimension \(M\geq 4\) and index \(1\) over a rationally connected base of dimension at most \((M-2)(M-1)/2\). We obtain a near-optimal estimate for the codimension of the set of hypersurfaces of a given degree in projective space that have positive-dimensional singular sets. Fano-Mori fibre space; Fano variety; maximal singularity; birational map; linear system Fano varieties, Rational and birational maps, Coverings in algebraic geometry, Rationally connected varieties Birational geometry of algebraic varieties fibred into Fano double spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is about the following question: given a branched covering \(X_ K\to P^ 1_ K\), defined over a number field K, what can one say about the specializations of it to points of \(P^ 1_ K\) defined over K?
The main result describes the precise ramification degrees for most primes of K in extensions arising from specializations of the branched covering to points of \(P^ 1_ K\) defined over K. These ramification degrees are described in terms of the branch cycle description of the covering and the intersection of its branch locus with the point to which the covering is specialized, when these points are ``spread out'' to divisors on the arithmetic surface \(P^ 1_{{\mathcal O}}\), where \({\mathcal O}\) is the ring of integers of K. One corollary is an explicit version of Hilbert's irreducibility theorem that applies to certain cases. In particular it applies to the case of a Galois branched covering with group G whose branch cycles \((g_ 1,...,g_ r)\) satisfy the condition that if \(h_ i\) is conjugate to \(g_ i\) in G for \(i=1,...,r\), then \(h_ 1,...,h_ r\) generate G. branched covering; ramification degrees; Hilbert's irreducibility theorem Beckmann, S., On extensions of number fields obtained by specializing branched coverings, Journal für die reine und angewandte Mathematik, 419, 27-53, (1991) Algebraic number theory: global fields, Ramification and extension theory, Coverings in algebraic geometry, Ramification problems in algebraic geometry On extensions of number fields obtained by specializing branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a commutative ring, \(G\) a finite abelian group. The group \(\mathrm{Gal}(R,G)\) of isomorphism classes of Galois extensions of \(R\) with group \(G\), as developed by Auslander, Goldman, Chase, Harrison and Rosenberg in the 1960s, has important connections with algebraic number theory: if \(R\) is the ring of integers of a number field \(K\), then by class field theory \(\mathrm{Gal}(R,G)\) is closely related to the class group of \(K\); more recently, work of Kersten, Michalichek and the author has found important and subtle connections with Leopoldt's and Vandiver's conjectures [c.f. \textit{I. Kersten} and \textit{J. Michalichek} [J. Number Theory 32, 371--386 (1989; Zbl 0709.11058)].
The main purpose of these notes is to describe \(\mathrm{Gal}(R,G)\) in a noncohomological way. Since \(\mathrm{Gal}(R,G)\) respects products in the second variable, the problem immediately reduces to \(G\) cyclic of prime power order \(p^n\), which we assume henceforth.
Chapter 0 gives a useful description of the basic theory and a description of \(\mathrm{Gal}(R,G)\) and the subgroup \(\mathrm{NB}(R,G)\) of Galois extensions with normal basis, when \(R\) is connected and contains \(p^{-1}\) and a primitive \(p^n\)-th root of unity \(\zeta\). This generalized Kummer theory goes back to the reviewer [Ill. J. Math. 15, 273--280 (1971; Zbl 0211.37102)] and \textit{A. Z. Borevich} [J. Sov. Math. 11, 514--534 (1979); transl. from Zap. Nauchn. Semin. LOMI Steklova 57, 8--30 (1976; Zbl 0379.13003)].
Chapter I then describes \(\mathrm{Gal}(R,G)\) when \(R\) contains \(p^{-1}\) but not \(\zeta\), by letting \(S=R[\zeta]\) and ``descending'' the Kummer theory over \(S\) to \(R\). This chapter is based on the author's paper [Trans. Am. Math. Soc. 326, 307--343 (1991; Zbl 0743.11060)], and has useful applications to number theory, as is shown in chapter IV.
Chapter II describes an alternate approach to obtaining \(\mathrm{Gal}(R,G)\) from \(\mathrm{Gal}(S,G)\) using corestriction, which leads to a proof that the map from \(\mathrm{NB}(R,G)\) to \(\mathrm{NB}(R/N,G)\) is surjective if \(N\subseteq\mathrm{Rad}(R)\).
In Chapter III the author specializes to number fields. He obtains the order of \(\mathrm{NB}(R,G)\) (as always, \(G\) is cyclic of order \(p^n)\) if \(p\) is odd and either \(R\) is a finite extension of \(\mathbb Q_p\) or \(R={\mathfrak O}_K(p^{-1})\), where \(K\) is a finite extension of \(\mathbb Q\) with ring of integers \({\mathfrak O}_K\). In the latter case, \(\mathrm{NB}(R,G)=O(1)p^{n(s+1)}\) where \(s\) is the number of pairs of nonreal complex embeddings of \(K\).
Chapter IV then presents the author's description of \(\mathrm{Gal}(R,\mathbb Z_ p)\) when \(R={\mathfrak O}_K(p^{-1})\), and its connection with Leopoldt's conjecture in cyclotomic fields. The results are adapted from the author's paper cited above, except for some additional information relating the theory with extensions obtained by adjoining torsion points of abelian varieties of CM type.
Chapter V studies \(\mathbb Z^p\)-extensions of the function field \(K\) of a variety defined over a number field. Among the results obtained is that all such extensions arise from \(\mathbb Z_p\)-extensions of the largest number field contained in \(K\). The results in this chapter are previously unpublished. The author views the approach in this chapter as an alternative approach to the geometric class field theory of \textit{N. Katz} and \textit{S. Lang} [Enseign. Math., II. Sér. 27, 285--314 (1981; Zbl 0495.14011)].
The final chapter gives an exposition of the author's paper in [Manuscr. Math. 64, 261--290 (1989; Zbl 0705.13004)] which reformulates and generalizes \textit{H. Hasse}'s description [J. Reine Angew. Math. 176, 174--183 (1936; Zbl 0016.05204)] using the ``Artin-Hasse exponential'', of those \(b\) in \(K^*/(K^*)^q\), \(q=p^n\), \(K\) a local field containing \(\mathbb Q_p[\zeta]\), so that the Kummer extension \(K[\root q \of {b}]\) is unramified over \(K\): this is equivalent to describe \(\mathrm{Gal}(R,G)\) where \(R\) is the valuation ring of \(K\). The chapter concludes with the (previously unpublished) construction of a generic \(G\)-Galois extension.
The author's work on \(\mathrm{Gal}(R,G)\) which is presented in this monograph is fundamental to the subject. These notes will become a standard reference in the area. extensions of function field; generic Galois extension; Kummer theory; Leopoldt's conjecture; cyclotomic fields; geometric class field theory C. Greither, Cyclic Galois extensions of commutative rings. Lecture Notes in Mathematics, vol. 1534. Springer, Berlin-Heidelberg-New York, 1992. Zbl0788.13003 MR1222646 Galois theory and commutative ring extensions, Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to number theory, Extension theory of commutative rings, Cyclotomic extensions, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Ramification and extension theory, Ramification problems in algebraic geometry, Coverings in algebraic geometry Cyclic Galois extensions of commutative rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper investigates the variety of minimal rational tangents (WMRT) of cyclic coverings of porjective spaces. The main theorem states that if \(X\to \mathbb{P}^n\) is the cyclic covering of degree \(d\) branched along a hypersurface \(Y\) of degree \(d\), when \(3\leq d\leq n\) and \(Y\) is general, then the VMRT of \(X\) has maximal variation. Since such \(X\) can be embedded into \(\mathbb{P}^{n+1}\) as a hypersurface of degree \(d\), a corollary of the main theorem tells that if \(X\) is a general hypersurface of \(\mathbb{P}^{n+1}\) of degree \(d\), where \(3\leq d\leq n\), then the VMRT of \(X\) has maximal variation. The method of proof is to identify the minimal rational curves on \(X\) with \(d\)-tangent lines on \(Y\) and find the explicit defining equations of the VMRT at a general point of \(X\). cyclic covering; Fano manifolds; varieties of minimal rational tangents Fano varieties, Coverings in algebraic geometry, Rational and unirational varieties Variation of the variety of minimal rational tangents of cyclic coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author sets the scene for his paper in the context of a paper of \textit{M. Raynaud} [ibid. 32, 87--126 (1999; Zbl 0999.14004)], who gave a criterion for good reduction of Galois covers of the projective line that are ramified at three points. He introduced the notion of an auxiliary cover, which enables one to analyze Galois covers with bad reduction in characteristic \(p\), in terms of subgroups \(H\) with the same type of bad reduction, in which \(H\) is a solvable quotient of a subgroup of \(G\). For example if \(p\) is a proper divisor of the order of \(G\), then \(H\) is a metacyclic group, isomorphic to \(\mathbb{Z}/p\times \mathbb{Z}/m\). Thus the study of bad reduction can be reduced to the study of covers with certain solvable (in the case under discussion, metacyclic) Galois groups.
The foregoing is a motivation for the study of metacyclic covers of \(\mathbb{P}_1\) with Galois group isomorphic to \(\mathbb{Z}/p\times \mathbb{Z}/m\), which arise as auxiliary covers of \(\mathbb{P}^1\), having three branch points and prime-to-\(p\) ramification.
The paper begins with a characterisation of metacyclic covers, \(f: Y\to \mathbb{P}^1\) which are the composition of a ramified \(m\)-cyclic cover \(Z\to \mathbb{P}^1\) and an étale \(p\)-cyclic cover \(Y\to Z\), with Galois group \(\mathbb{Z}/p\times\mathbb{Z}/m\). The theory of such special metacyclic covers is developed very clearly and geometrical examples are given, concluding with a connection between the existence of étale Galois covers of the affine line in characteristic \(p\), related to Abhyankar's conjecture.
The author then investigates the problem of the reduction of a special cover to characteristic \(p\) which is determined by what he calls special degeneration data, which are given essentially by an \(m\)-cyclic cover \(\overline Z_0\to \mathbb{P}^1_k\) of the projective line in characteristic \(p\), together with a logarithmic differential form \(\omega_0\) on \(Z_0\), where \(Z_0\) is defined in the proof of the author's Theorem A and, together with \(\omega_0\), determines a pair \((Z_0,\omega_0)\) where \(\overline Z_0\to \mathbb{P}^1_k\) is an \(m\)-cyclic cover branched at certain points \(\tau_1,\dots, \tau_n\).
Finally, the author shows that any special degeneration datum can be lifted to a special cover.
The geometrical motivation for the study of Galois covers is clearly presented, and the author promises further developments in subsequent papers. coverings; fundamental groups; Galois theory Stefan Wewers, Reduction and lifting of special metacyclic covers, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 1, 113 -- 138 (English, with English and French summaries). Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Coverings in algebraic geometry Reduction and lifting of special metacyclic covers. | 0 |
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