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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 The paper is a survey on multivariable operator model theory and Fourier analysis, and continues the study of indefinite Hardy spaces on a finite bordered Riemann surface. The classical case of the closed unit disk is analysed in the first section of the paper. In the second section a model for a commuting pair of contractions in a general context is obtained. A more concrete model for the Riemann surface is described in the third section, arriving at a model theory for a commuting pair of nonunitary operators. In the last section,the Z-transform along a curve and the corresponding spaces are analysed. Applying the Z-transform to a certain system, a precise sense of the transfer function of the system is given. finite bordered Riemann surface; Hardy spaces; operator model theory; Fourier analysis; multivariable systems Ball, J.A., Vinnikov, V.: Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory, and Fourier analysis along a unimodular curve. In: Borichev, A.A., Nikolskii, N.K. (eds.) Systems, Approximation, Singular Integral Operators, and Related Topics. Oper. Theory Adv. Appl., vol. 129, pp. 37-56. Birkhäuser, Basel (2001) Compact Riemann surfaces and uniformization, Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc., Dilations, extensions, compressions of linear operators, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Multivariable systems, multidimensional control systems, Research exposition (monographs, survey articles) pertaining to operator theory, Several-variable operator theory (spectral, Fredholm, etc.), Spaces of bounded analytic functions of one complex variable Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory and Fourier analysis along a unimodular curve. | 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 paper is to give the following characterization of a smooth compact connected curve \(K\) in \(\mathbb{R}^2\): (1) \(K\) is algebraic; (2) \(K\) satisfies a tangential Markov inequality with exponent one, i.e. there exists \(M= M(K) >0\) such that \(|D_T p|_K\leq M(\deg p) |p|_K\) for all polynomials \(p\) where \(D_T\) denotes the unit tangential derivative along \(K\); (3) For some \(0< \alpha<1\), \(K\) satisfies a Bernstein theorem: there exists \(B>0\) such that for \(f\in {\mathcal C} (K)\), if \(\text{dist}_K (f, {\mathcal P}_n )\leq n^{- \alpha}\), then \(f\in \text{Lip} (\alpha)\) and \(|f|_\alpha \leq B\) where \({\mathcal P}_n\) is the space of all polynomials of degree at most \(n\) in two variables and \(|f|_\alpha\) denotes the \(\text{Lip} (\alpha)\) norm of \(f\).
Reviewer's remark. Here smooth should be read \(({\mathcal C}^\infty)\) and not \(({\mathcal C}^1)\) as is misprinted in the paper. An extension of the above result to a smooth compact \(m\)-dimensional submanifold of \(\mathbb{R}^n\) without boundary \((1\leq m\leq N-1)\) has been given in \textit{L. Bos}, \textit{N. Levenberg}, \textit{P. Milman} and \textit{B. A. Taylor} [Indiana Univ. Math. J. 44, No. 1, 115-138 (1995; Zbl 0824.41015)]. algebraic curves; Bernstein theorem for Lipschitz functions; tangential Markov inequality L. P. Bos, N. Levenberg, and B. A. Taylor, ``Characterization of smooth, compact algebraic curves in \Bbb R2,'' in Topics in Complex Analysis (Warsaw, 1992), P. Jakóbczak and W. Pleśniak, Eds., vol. 31 of Banach Center Publ., pp. 125-134, Polish Academy of Sciences, Warsaw, 1995. Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities), Real algebraic sets Characterization of smooth, compact algebraic curves in \(\mathbb{R}^ 2\) | 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 aim of this paper is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a \(\hbar\)-difference equation :
\[
\psi(x+\hbar)=\left(e^\hbar\frac{d}{dx}\right)\psi(x)=L(x;\hbar)\psi(x),\quad L(x;\hbar)\in \mathrm{GL}_2((\mathbb{C}(x))[\hbar]).
\]
In particular, the author extends the notion of determinantal formulas and topological type property proposed for formal WKB solutions of \(\hbar\)-differential systems to this setting. He applies his results to a specific \(\hbar\)-difference system associated to the quantum curve of the Gromov-Witten invariants of \(\mathbb{P}^1\) for which he proves that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve \(y=\coth^{-1}\frac{x}{2}\). Finally, identifying the large \(x\) expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of \(\mathbb{P}^1\). In other words, the purpose of this paper is to prove on the simple example of the quantum curve arising in the enumeration of Gromov-Witten invariants of \(\mathbb{P}^1\) that the determinantal formulas and the topological type property may be used in the context of \(\hbar\)-difference systems rather than \(\hbar\)-differential systems. However, as presented in this paper, it seems that the construction might be adapted for more general situations. WKB expansion; topological recursion; Gromov-Witten invariants; difference systems; determinantal formulas; topological type property Marchal, O.: WKB solutions of difference equations and reconstruction by the topological recursion (2017). arXiv:1703.06152 [math-ph] Relationships between algebraic curves and integrable systems, Difference equations, scaling (\(q\)-differences), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds WKB solutions of difference equations and reconstruction by the topological recursion | 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 notion of shtukas was introduced by \textit{V. G. Drinfel'd} [Funct. Anal. Appl. 21, 107--122 (1987; Zbl 0665.12013); J. Sov. Math. 46, No. 2, 1789--1821 (1989; Zbl 0672.14008)] in his proof of the Langlands correspondence for \(\mathrm{GL}_2\) over function fields. It is also used in \textit{L. Lafforgue}'s proof of the Langlands correspondence for \(\mathrm{GL}_r\) over function fields [Invent. Math., 147, No. 1, 1--241 (2002; Zbl 1038.11075)]. In the paper under review, the author gives an introduction to the theory of the stacks of shtukas and a short review on the author's recent results on the problem of the compactification of the stacks of shtukas.
In this survey article, the definition of the stacks \(\mathrm{Sht}^r\) of Drinfeld shtukas of rank \(r\) is given. One of Drinfeld's theorem says that the stack \(\mathrm{Sht}^r\) is an algebraic Deligne-Mumford stack. A proof of this theorem is presented. The stack \(\mathrm{Sht}^r\) has an infinite number of connected components \(\mathrm{Sht}^{r, d}\) for \(d\) running through all integers. The notion of Harder-Narasimhan polygons \(p : [0,r] \to {\mathbb R}\) is reviewed. Then, a result of Lafforgue says that for each integer \(d\) and for any given Harder-Narasimhan polygon \(p\) satisfying certain property, there exists a unique open substack \(\mathrm{Sht}^{r,d,p}\) of \(\mathrm{Sht}^{r,d}\) which is of finite type. Moreover, \(\mathrm{Sht}^{r,d}\) is the union of these open substacks \(\mathrm{Sht}^{r,d,p}.\) The compactification of the open substack \(\mathrm{Sht}^{r,d,p}\) is obtained by Drinfeld for the rank 2 case. The remaining part which is also the main purpose of this paper is to review Drinfeld's construction on the compactification of \(\mathrm{Sht}^{2,d,p}\) Another approach using geometric invariant theory obtained by the author [Astérisque 313. Paris: Société Mathématique de France (2007; Zbl 1144.11047)] is also sketched in the paper. shtuka; stack; vector bundle; function field; finite field; Frobenius map; level structure; compactification T. Ngô Dac, \textit{Introduction to the stacks of shtukas}, in: \textit{Algebraic Cycles, Sheaves, Shtukas, and Moduli}, Trends in Mathematics, Birkhäuser, Basel, 2008, pp. 217-236. Generalizations (algebraic spaces, stacks), Arithmetic theory of algebraic function fields, Lie groups, Discontinuous groups and automorphic forms Introduction to the stacks of shtukas | 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 the 1-dimensional boundary in a fiber of a subanalytic mapping bounds, in the fiber, a surface of area estimated by a linear function of the length of the boundary. The result is a generalization of the Teissier/Denkowska/Kurdyka theorem about a uniform bound on the length of curves in the fibers of a subanalytic set, which in turn generalizes, in a sense, the Gabrielov theorem about the bound on the number of connected components.
The reviewer likes very much the proof, very geometrical, not using the algebraic topology, giving a good insight into the structure of subanalytic sets as well as the \(L\)-regular set of Parusiński. The theorems \(A\), \(B\), \(C\) are very interesting and the proof really worth reading, along with the papers in Lect. Notes Math. 1524, 316-322 (1992; Zbl 0779.32006) by the author and in Ann. Inst. Fourier 38, No. 4, 189- 213 (1988; Zbl 0661.32015) by \textit{A. Parusiński}. metric properties; subanalytic sets Semi-analytic sets, subanalytic sets, and generalizations, Singularities in algebraic geometry Linear bounds for resolutions of 1-boundaries in subanalytic sets | 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 notion of differential graded scheme (DG scheme) was introduced by M. Kontsevich about twenty years ago, in a first approach to what is now called derived algebraic geometry. By definition, a DG scheme is a usual scheme \((X,\mathcal{O}_X)\) together with a sheaf \(\mathcal{O}_X^\bullet\) of differential graded \(\mathcal{O}_X\)-algebras such that the natural map \(\mathcal{O}_X\to H^0(X,\mathcal{O}_X^\bullet)\) is surjective. The theory of DG schemes was further developed by M. Kapranov, I. Ciocan-Fontanine, K. Behrend, and others, mainly in order to study derived moduli spaces and derived moduli stacks in algebraic geometry.
On the other hand, there is the notion of indschem due to A. Beilinson and V. Drinfeld, which is defined as follows: An indscheme is a presheaf on the category of affine schemes that is representable as an inductive limit of closed embeddings of schemes. A prominent example of an indscheme is the infinite Grassmannian Gr\(_G\) corresponding to an algebraic group \(G\), which plays a crucial role in the geometric Langlands program.
The main motivation for the paper under review is to develop a suitable framework for the study of the category QCoh(Gr\(_G\)) of quasi-coherent sheaves on Gr\(_G\), and that in the context of \textit{D. Gaitsgory}'s multi-volume series [``Notes on geometric Langlands'', \url{http://www.math.harvard.edu/~gaitsgde/GL/}]. To this end, the authors introduce the conceptual framework of so-called DG indschemes, in which the afore-mentioned theories of DG schemes and indschemes are combined to create a new and powerful abstract machinery in derived algebraic geometry. As the entire approach is heavily based on the language, the methods, and the results of D. Gaitsgory's ``Notes on geometric Langlands'' ([loc. cit.] as well as of \textit{J. Lurie}'s large preprint series [``Derived algebraic geometry'', \url{http://www.math.harvard.edu/~lurie}], we here confine ourselves to briefly indicate the topics covered in the ten sections, instead of undertaking the hopeless attempt to explain any of the countless technical details. After a comprehensive introduction to the motivation, the structure, and the main results of the present paper, Section 1 is devoted to the definition of DG indschemes and their basic properties. Section 2 provides a detailed study of quasi-coherent sheaves and ind-coherent sheaves on DG indschemes, respectively, together with their fundamental functorial properties. Closed embeddings into a DG indscheme and push-outs are described in Section 3, whereas Sections 4 and 5 elaborate a characterization of DG indschemes via deformation theory. Thereafter, formal completions of DG indschemes are analyzed in Section 6, and the study of quasi-coherent and ind-coherent sheaves on formal completions of DG indschemes follows in Section 7. The notion of formal smoothness of DG indschemes is introduced in Section 8, and a characterization of formally smooth DG indschemes via deformation theory is provided there, too. Section 9 is devoted to a comparison between classical and derived formal smoothness of DG indschemes, with an outlook to applications with respect to loop groups and infinite Grassmannians, thereby coming back to the motivation for entire paper as indicated in the introduction above. Finally, Section 10 establishes a functorial equivalence between the categories of quasi-coherent sheaves and ind-coherent sheaves on a DG indscheme, respectively. As the authors point out, this main result of theirs was also proved by J. Lurie using a different method. DG schemes; indschemes; DG indschemes; quasi-coherent sheaves; deformation theory; formal smoothness; prestacks; stacks; infinite Grassmannians Gaitsgory, D.; Rozenblyum, N.: DG indschemes Generalizations (algebraic spaces, stacks), Geometric Langlands program (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Double categories, \(2\)-categories, bicategories, hypercategories, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) DG indschemes | 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 book is devoted to the study of \(C^ r\) Nash maps and \(C^ r\) Nash manifolds, \(r=0,1,...,\omega\). Let U, V be open semialgebraic subsets of \({\mathbb{R}}^ n\) and \({\mathbb{R}}^ m\) respectively. A \(C^ r\) map \(U\to V\) is a \(C^ r\) Nash map if its graph is semialgebraic in \({\mathbb{R}}^ n\times {\mathbb{R}}^ m\). A \(C^ r\) Nash manifold is a \(C^ r\) manifold with a finite system of coordinate neighborhoods \(\{\psi_ i: U_ i\to {\mathbb{R}}^ m\}\) such that the corresponding transition maps \(\psi_ j\circ \psi_ i^{-1}: \psi_ i(U_ i\cap U_ j)\to \psi_ j(U_ i\cap U_ j)\) are \(C^ r\) Nash diffeomorphisms. Starting from these definitions, one can develop a natural analogue of many basic facts of differential manifold theory. An important new concept is that of affine manifold. A \(C^ r\) Nash manifold is affine if there exists a \(C^ r\) Nash imbedding of it into some \({\mathbb{R}}^ n\). The main results of the book are grouped around this notion and the regularity properties of noncompact Nash manifolds. They are the following.
(1) Let \(0<r<\infty\). Then a \(C^ r\) Nash manifold is affine and admits a unique affine \(C^{\omega}\) Nash manifold structure (Chapter III).
(2) Every compact \(C^{\omega}\) differential manifold admits many pairwise nonisomorphic non-affine \(C^{\omega}\) Nash manifold structures. Moreover there exists a continuum number of such structures. Similar results hold for interiors of compact \(C^{\omega}\) manifolds with boundary (Chapter IV).
(3) Every \(C^ 0\) Nash manifold has a natural compactification to a \(C^ 0\) Nash manifold with boundary. A compact \(C^ 0\) Nash manifold possibly with boundary admits a unique PL manifold structure. Moreover, there is a 1-1 correspondence between isomorphism classes of compact PL manifolds possibly with boundary and isomorphism classes of \(C^ 0\) Nash manifolds without boundary (given by \(M\mapsto int M)\) (Chapter V).
(4) Similarly, one can uniquely compactify a noncompact affine \(C^{\omega}\) Nash manifold by attaching boundary. This \(C^{\omega}\) Nash compactification can be realized by nonsingular algebraic varieties in a natural way (Chapter VI).
There are similar results on \(C^ r\) Nash vector and fibre bundles. The results on affine Nash manifolds are based on an approximation theorem of \(C^ r\) Nash maps by \(C^{\omega}\) Nash maps in an appropriate \(C^ r\) topology. Such a theorem is the main result of Chapter II. Chapter I is devoted to some preliminaries. affine Nash manifold; Nash maps; Nash manifolds; PL manifold; algebraic varieties; fibre bundles; approximation theorem Shiota, M., Nash Manifolds, Lect. Notes Math., \textbf{1269}, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987. Real-analytic and Nash manifolds, Research exposition (monographs, survey articles) pertaining to global analysis, Real algebraic and real-analytic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry Nash 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 consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map. integrable maps; Poisson algebra; Laurent property; cluster algebra; algebraic entropy; tropical Fordy, A.P., Hone, A.N.W.: Symplectic maps from cluster algebras. SIGMA \textbf{7}, 091 (2011) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Poisson algebras, Poisson manifolds; Poisson groupoids and algebroids, Symplectic maps from cluster 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 paper under reviewed concerns the Gromov-Hausdorff limit of Kähler manifolds. The main results are the verification of a weakened version of \textit{G. Tian}'s partial \(\mathcal{C}^0\)-estimates conjecture [Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 587--598 (1991; Zbl 0747.53038)] (Theorem 1) and an extremely beautiful theorem connecting Gromov-Hausdorff limits and limits in algebraic geometry (Theorem 2). Precisely, the authors consider for \(n \in \mathbb{N}\) and \(c,V > 0\), the class \(\mathcal{K}(n,c,V)\) of compact Kähler manifolds \((X,\omega)\) of complex dimension \(n\) with volume \(V\), together with a Hermitian line bundle \(L\) endowed with a connection \(A\) with curvature \(-j\omega\), such that the Ricci tensor is uniformly bounded and the ratio of the volume of any metric ball to that of an Euclidean ball with the same radius is \(\geq c\). After proving a weakened version of Tian's partial \(\mathcal{C}^0\)-estimates conjecture for \(\mathcal{K}(n,c,V)\), they show that there exists some integer \(k\), depending only on \(n,c\) and \(V\), such that every \(X\) in \(\mathcal{K}(n,c,V)\) can be embedded into \(\mathbb{C}\mathbb{P}^N\) by the linear system \(|L^{\otimes k}|\). Moreover, they show that if \(X_j\) is a convergente sequence in \(\mathcal{K}(n,c,V)\), then its Gromov-Hausdorff limit \(X_\infty\) is homeomorphic to a normal projective variety \(W\) such that, up to taking a subsequence and projective transformations, \(X_j\) converges to \(W\) as points in the Chow variety parameterizing projective varieties in \(\mathbb{C}\mathbb{P}^N\).
The case where the \(X_j\)'s are Kähler-Einstein manifolds is also discussed. In this case the authors show that the limit space \(W\) has only log-terminal singularities. The paper ends with a detailed study of the structure of the tangent cones in the complex three-dimensional case, and a conjecture on the uniform bounds of the Bergman function of Kähler manifolds in \(\mathcal{K}(n,c,V)\). Gromov-Hausdorff convergence; Hörmander's \(L^2\) estimations; Kähler manifolds Donaldson, S.K., Sun, S.: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. arXiv:1206.2609 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces, Kähler manifolds, Global differential geometry of Hermitian and Kählerian manifolds, Fano varieties, Holomorphic bundles and generalizations Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry | 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 paper is to give a survey of certain types of theorems and their applications to a broad range of topics in topology and geometry. More precisely, the author discusses various families \((M_d)_{d\in\mathbb{Z}}\) of specific classifying spaces or moduli spaces \(M_d\) indexed by some geometrically defined integer \(d\in\mathbb{Z}\), where this characteristic number is referred to as the ``degree'' of the corresponding moduli space \(M_d\), and he describes some crucial ``stability theorems'' with respect to the homology and/or homotopy of these varying spaces. Generally, two basic stability questions about the topology of these spaces \(M_d\) naturally occur:
1. Is there a ``stability range'' for their homology or homotopy groups, i.e., is there an unbounded and non-decreasing arithmetic function \(r(d)\) such that the \(k\)th homology and/or homotopy groups of \(M_d\) and \(M_{d+1}\) are isomorphic for all \(k< r(d)\)?
2. Is there a naturally defined, more easily accessible limiting homotopy type, as the degree \(d\) gets sufficiently large?
In fact, the central theme of the present survey article is to explain how these two basic stability problems are addressed in a variety of concrete, both classical and more recent examples.
In Section 1, some classical stability theorems of this kind are depicted, including the Freudenthal Suspension Theorem (1938), the Whitney Embedding Theorem (1944), and the Bott Periodicity Theorem (1959). Section 2 turns to stability theorems concerning configuration spaces of points in manifolds [\textit{D. McDuff}, Topology 14, 91--107 (1975; Zbl 0296.57001)], whereas Section 3 discusses some more recent stability theorems that lie in the intersection of topology, algebraic geometry, and differential geometry. These are stability results regarding moduli spaces of holomorphic maps, holomorphic curves, Yang-Mills connections on Riemann surfaces, holomorphic vector bundles, and instantons, respectively, as they were established by G. Segal, Atiyah-Bott, Cohen-Galatius-Kitchloo, Atiyah-Jones, C. Taubes, and many others. Section 4 deals with stability theorems in the context of stable topology of general linear groups and algebraic \(K\)-theory (à la R. Charney, W. G. Dwyer, and W. van der Kallen), on the one hand, and pseudo-isotopies of differentiable manifolds (à la J. Cerf, K. Igusa, and F. Waldhausen) on the other. Section 5 is devoted to important stability theorems concerning moduli spaces \(M_{g,n}\) of compact Riemann surfaces of genus \(g\) with \(n\) boundary components. After a review of the allied Teichmüller theory, the stability theorems of \textit{J. Harer} [Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)] and \textit{N. V. Ivanov} [Leningr. Math. J. 1, No. 5, 1177--1205 (1990; Zbl 0766.32023); translation from Algebra Anal. 1, No. 5, 115--143 (1989)] on the homology of the mapping class groups \(\Gamma_{g,n}\) are briefly explained, together with their very recent improvements and generalizations by S. K. Boldsen, Cohen-Madsen, and Madsen-Weiss, respectively. In this context, the spectacular stability theorem by \textit{I. Madsen} and \textit{M. Weiss} [cf. Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)], which implies a generalization of a long-standing conjecture of \textit{D. Mumford} about the stable cohomology of the moduli spaces \(M_{g,1}\) [Towards an enumerative geometry of the moduli space of curves. Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)], is analyzed in greater detail, especially with regard to its basic topological ingredients. This section ends with the discussion of some similar stability theorems, this time with respect to the stable homology of classifying spaces of automorphism groups of free groups (after \textit{A. Hatcher} and \textit{K. Vogtmann} [Algebr. Geom. Topol. 4, 1253--1272 (2004; Zbl 1093.20020)] and \textit{S. Galatius} [Stable homology of automorphism groups of free groups, Preprint: \url{arXiv:math/061021} (2008)]). Finally, in Section 6, the author makes some final comments regarding possible strategies for finding general criteria under which stability theorems for certain types of classifying spaces or moduli spaces hold.
Altogether, this survey article provides a very lucid, enlightening, versatile, and utmost topical introduction to the subject of stability phenomena in the homology and homotopy theory of classifying spaces in topology and geometry, with numerous hints and references to both the current research and the vast literature in the related fields. moduli spaces; Riemann surfaces; Teichmüller spaces; mapping class groups; holomorphic curves; Thom spaces; \(K\)-theory; stable homology; stable homotopy; stable topology; gauge theory Cohen, R.: Stability phenomena in the topology of moduli spaces, Surveys in differential geometry XIV, 23-56 (2010) Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Compact Riemann surfaces and uniformization, Stable homotopy theory, spectra, Topological \(K\)-theory, Moduli problems for differential geometric structures Stability phenomena in the topology of moduli 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 In this groundbreaking paper, the authors lay the foundation for motivic infinite loop space theory. Developing ideas of V.~Voevodsky (outlined in unpublished notes), they define framed correspondences and prove that they provide computationally accessible models for infinite loop spaces of suspension spectra of smooth schemes.
What does this mean? Motivic homotopy theory is a homotopy theory of smooth schemes introduced by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)]. It has a stable variant in which smash product with the projective line \(\mathbb{P}^1\) becomes invertible. This forms a category of motivic spectra which is related to motivic spaces in the same way that topological spectra (the objects of classical stable homotopy theory) are related to topological spaces. In particular, we have an adjunction \((\Sigma^\infty_{\mathbb{P}^1},\Omega^{\infty}_{\mathbb{P}^1})\) between pointed motivic spaces and motivic spectra. As such, understanding the (nonnegative) homotopy groups (or sheaves) of \(\Omega^\infty_{\mathbb P^1}E\) is equivalent to understanding the (nonnegative, stable) homotopy groups (or sheaves) of \(E\).
Motivic spectra represent cohomology theories of algebro-geometric interest -- motivic cohomology, (homotopy) algebraic \(K\)-theory, Hermitian \(K\)-theory, and algebraic cobordism, to name a few -- and thus it is very desirable to have good models for motivic infinite loop spaces. The authors verify that framed correspondences provide such models for \(\mathbb P^1\)-suspension spectra of simplicial smooth schemes under mild hypotheses. Indeed, a mild reinterpretation of Theorem 1.3 (see also Theorem 10.7) says that for a smooth scheme \(X\) over an infinite perfect field, the Nisnevich localization of the \(\mathbb{A}^1\)-localization of the group completion of the presheaf taking \(U\in Sm/k\) to the space of framed correspondences from \(U\) to \(X\) is an equivalence of simplicial presheaves on \(Sm/k\). This theorem is the computational basis for the \(\infty\)-categorical elaboration of motivic infinite loop space machines (in the spirit of Segal's \(\Gamma\)-spaces) due to \textit{E. Elmanto} et al. [Camb. J. Math. 9, No. 2, 431--549 (2021; Zbl 07422194)].
The definition of framed correspondences is too technical for this review, and I also will not elaborate on the theory of `big framed motives' introduced by the authors. An important warning, though, is in order: to the contrary of the authors' claims, framed correspondences do \emph{not} form the morphism sets of a category. One usually composes correspondences via pullback, but the data of a framed correspondence (speicifically the closed subset \(Z\) of \(\mathbb A^n_X\) from Definition 2.1) is not preserved by this construction. See [loc. cit., Paragraph 1.3.7] for an elaboration of this point, noting that those authors refer to Garkusha-Panin's framed correspondences as \emph{equationally} framed correspondences. This categorical frustration does not impact the arguments in the reviewed paper. motivic homotopy theory; framed correspondences; motivic infinite loop spaces Motivic cohomology; motivic homotopy theory, Topological properties in algebraic geometry, Stable homotopy groups, Infinite loop spaces Framed motives of algebraic varieties (after V. Voevodsky) | 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 Inspired by Besser's work on Coleman integration, we use \(\nabla\)-modules to define iterated line integrals over Laurent series fields of characteristic \(p\) taking values in double cosets of unipotent \(n\times n\) matrices with coefficients in the Robba ring divided out by unipotent \(n\times n\) matrices with coefficients in the bounded Robba ring on the left and by unipotent \(n\times n\) matrices with coefficients in the constant field on the right.
We reach our definition by looking at the analogous theory for Laurent series fields of characteristic 0 first, and re-interpreting the classical formal logarithm in terms of \(\nabla\)-modules on formal schemes. To illustrate that the new \(p\)-adic theory is nontrivial, we show that it includes the \(p\)-adic formal logarithm as a special case. \(p\)-adic integration; Laurent series fields Arithmetic ground fields for abelian varieties, \(p\)-adic cohomology, crystalline cohomology, Homotopy theory and fundamental groups in algebraic geometry Iterated line integrals over Laurent series fields of 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 \(k\) be an algebraically closed field and let GradAlg\((H)\) denote the scheme parametrizing graded quotients of \(R = k[x_0,\dots,x_n]\) with Hilbert function \(H\). Given integers \(a_0 \leq \dots \leq a_{t+c-2}\) and \(b_1 \leq \dots \leq b_t\), denote by \(W_s(\underline{b};\underline{a})\) the Hilbert scheme parametrizing the standard determinantal rings defined by the \(t \times t\) minors of a homogeneous \(t \times (t+c-1)\) matrix whose entries have degree \(a_j - b_i\). The author extends previous results on the dimension and codimension of \(W_s(\underline{b};\underline{a})\) in GradAlg\((H)\), including to the case of artinian determinantal rings. Among these, under the condition \(n-c > 1\) and \(a_0 > b_t\), the author determines when \(\overline{W_s(\underline{b};\underline{a})}\) is an irreducible component of GradAlg\((H)\), finds the codimension of \(W_s(\underline{b};\underline{a})\) in GradAlg\((H)\) if its closure is not a component, and determines when GradAlg\((H)\) is generically smooth along \(W_s(\underline{b};\underline{a})\). In addition, he proves a result that implies that the general element of an irreducible component \(W\) of the Hilbert scheme of \(\mathbb P^n\) is \textit{glicci} provided \(W\) contains a standard determinantal scheme satisfying some conditions. (Recall that a result of the author, the reviewer, Miró-Roig, Nagel and Peterson [\textit{J. O. Kleppe} et al., Mem. Am. Math. Soc. 732, 116 p. (2001; Zbl 1006.14018)] gives that standard determinantal schemes are glicci.) Finally, the author examines the behavior of ghost terms in the minimal free resolution under deformation in GradAlg\((H)\). standard determinantal ring; artinian determinantal ring; Hilbert scheme; glicci; ghost terms; minimal free resolution; deformation Kleppe, J.O.: Families of artinian and low dimensional determinantal rings. arXiv:1506.08087 Linkage, complete intersections and determinantal ideals, Parametrization (Chow and Hilbert schemes), Deformations and infinitesimal methods in commutative ring theory, Determinantal varieties, Commutative Artinian rings and modules, finite-dimensional algebras, Syzygies, resolutions, complexes and commutative rings, Homological functors on modules of commutative rings (Tor, Ext, etc.), Graded rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Families of Artinian and low dimensional determinantal 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 From the authors' abstract: ``Algebraic topological methods are especially well suited for determining the non-existence of continuous mappings satisfying certain properties. In combinatorial problems it is sometimes possible to define a mapping from a space \(X\) of configurations to a Euclidean space \(\mathbb R^m\) in which a subspace, a discriminant, often an arrangement of linear subspaces \(A\), expresses a target condition on the configurations. Add symmetries of all these data under a group \(G\) for which the mapping is equivariant. If we remove the discriminant from \(\mathbb R^m\), we can pose the problem of the existence of an equivariant mapping from \(X\) to the complement of the discriminant in \(\mathbb R^m\). Algebraic topology may sometimes be applied to show that no such mapping exists, and hence the image of the original equivariant mapping must meet the discriminant.''
In this paper, the authors introduce a general framework, based on a comparison of Leray-Serre spectral sequences. This comparison can be related to the theory of the Fadell-Husseini index. They apply the framework to: {\parindent=6mm\begin{itemize}\item[-] solve a mass partition problem (antipodal cheeses) in \(\mathbb R^d\), \item[-] determine the existence of a class of inscribed 5-element sets on a deformed 2-sphere, \item[-] obtain two different generalizations of the theorem of Dold for the non-existence of equivariant maps which generalizes the Borsuk-Ulam theorem.
\end{itemize}} mass partition problems; subspace arrangements; equivariant cohomology; Serre spectral sequence; Borel construction; Borsuk-Ulam type theorems Blagojević, Spectral sequences in combinatorial geometry: cheeses, inscribed sets, and Borsuk-Ulam type theorems, Topol. Appl. 158 pp 1920-- (2011) Equivariant homology and cohomology in algebraic topology, Serre spectral sequences, Computational aspects related to convexity, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Equivariant operations and obstructions in algebraic topology, Configurations and arrangements of linear subspaces Spectral sequences in combinatorial geometry: cheeses, inscribed sets, and Borsuk-Ulam type theorems | 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 \(X\) be a smooth projective curve of genus \(\ge 1\) over \(\mathbb{C}\), and \(e,\infty \in X(\mathbb{C})\) be distinct points. Let \(L_n\) be the mixed Hodge structure of functions on \(\pi_1(X-\{\infty \},e)\) given by iterated integrals of length \(\le n\) (as defined by Hain). Building on a work of Darmon et al. [Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2012/2, 19--46 (2012; Zbl 1332.11054)], we express the mixed Hodge extension \(\mathbb{E}^\infty_{n,e}\) given by the weight filtration on \(\frac{L_n}{L_{n-2}}\) as the Abel-Jacobi image of a null-homologous algebraic cycle on \(X^{2n-1}\). This algebraic cycle is constructed using the different embeddings of \(X^{n-1}\) into \(X^n\). As a corollary, we show that the extension \(\mathbb{E}^\infty_{n,e}\) determines the point \(\infty \in X-\{e\}\). When \(n=2\), our main result is a strengthening of a theorem of Darmon et al. (loc. cit.). In the final section we assume that \(X,e,\infty\) are defined over a subfield \(K\) of \(\mathbb{C}\). Generalizing a construction in (Darmon et al., (loc. cit.), we use the extension \(\mathbb{E}^\infty_{n,e}\) to define a family of \(K\)-rational points on the Jacobian of \(X\) parametrized by \((n-1)\)-dimensional algebraic cycles on \(X^{2n-2}\) defined over \(K\). Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic cycles, Jacobians, Prym varieties, Mixed Hodge theory of singular varieties (complex-analytic aspects) Algebraic cycles and the mixed Hodge structure on the fundamental group of a punctured curve | 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 give a construction of the moduli space of stable maps to the classifying stack \(B\mu_r\) of a cyclic group by a sequence of \(r\)-th root constructions on \({\overline{\mathcal{M}}}_{0, n}\) and prove a closed formula for the total Chern class of \(\mu_r\)-eigenspaces of the Hodge bundle. From this, linear recursions for all genus-zero Gromov-Witten invariants of the stacks \([{\mathbb{C}}^N/\mu_r]\) are deduced.
More precisely, let \({\overline{\mathcal{M}}}_{0, n}(e_1, \dots, e_n; B\mu_r)\) be the moduli space of twisted stable maps to \(B\mu_r \cong [0/\mu_r] \subset [{\mathbb{C}}^N/\mu_r]\), where the branching behavior at the \(i\)-th section is prescribed by \(e_i \in \mu_r\). For every proper subset \(T \subset \{1, \dots, n-1\}\) having at least 2 elements, let \(D^T\) be the boundary divisor of \({\overline{\mathcal{M}}}_{0,n}\) consisting of curves having a node which separates the marking labels \(1,\ldots, n\) into \(T\) and \(\{1,\dots,n\}\setminus T\), and let \(r_T=\prod_{i \in T} e_i\). The authors show that \({\overline{\mathcal{M}}}_{0, n}(e_1, \dots, e_n; B\mu_r)\) is a \(\mu_r\)-gerbe over the stack constructed from \({\overline{\mathcal{M}}}_{0, n}\) by successively doing the \(r_T\)-th root construction at the boundary divisor \(D^T\) for all proper subsets \(T \subset \{1,\dots,n-1\}\) having at least 2 elements.
To determine a formula for the Chern class of the obstruction bundle, the authors introduce an ad-hoc definition of a ``moduli space of weighted stable maps to \(B\mu_r\)'', inspired by by the notion of weighted stable curves [\textit{B.~Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] and weighted stable maps [\textit{V.~Alexeev, G.~M.~Guy}, J. Inst. Math. Jussieu 7, No. 3, 425--456 (2008; Zbl 1166.14034); \textit{A.~M.~Mustaţă, A.~Mustaţă}, J. Reine Angew. Math. 615, 93--119 (2008; Zbl 1139.14043); \textit{A.~Bayer, Yu I.~Manin}, Mosc. Math. J. 9, No. 1, 3--32 (2009; Zbl 1216.14051)]. When the weights of the linear action of \(\mu_r\) on \({\mathbb{C}}^N\) are chosen such that all fibers of the universal curve are irreducible, the obstruction bundle can easily be computed from general facts about the \(r\)-th root construction [\textit{C.~Cadman}, Am. J. Math. 129, No. 2, 405--427 (2007; Zbl 1127.14002)]. In particular, this is done for the weight data that give a moduli space isomorphic to \({\mathbb{P}}^{n-3}\). Next, by a careful analysis of the wall--crossing for changing weights, the authors are able to lift this to a closed formula for the equivariant top Chern class of the obstruction bundle for the standard (non-weighted) stable maps.
Finally, by a generalized inclusion-exclusion principle, the Chern class formula leads to linear recursions for all Gromov-Witten invariants of \([{\mathbb{C}}^N/\mu_r]\) by a sum over partitions, where every partition corresponds to a moduli space of comb curves, and an explicit formula for the non-equivariant invariants of \([{\mathbb{C}}^3/\mu_3]\) is deduced. stable maps; root construction; Gromov-Witten invariants; stacks; quantum orbifold cohomology A. Bayer and C. Cadman, Quantum cohomology of \([\mathbb{C}^N/\mu_r]\), Compos. Math. 146 (2010), no. 5, 1291-1322. MR2684301 (2012d:14095) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stacks and moduli problems, Families, moduli of curves (algebraic) Quantum cohomology of \([\mathbb C^N/ \mu _r]\) | 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 \(f:({\mathbb C}^n,0)\to({\mathbb C},0)\) be an isolated quasihomogeneous singularity of degree \(d\) and weights \(\alpha_1,\dots,\alpha_n\). It is possible to construct a quasihomogeneous miniversal deformation of \(f\), by taking \(F(x,\lambda)=f(x)+\sum_{j=1}^\mu \lambda_j e_j\), where \(e_1,\dots,e_\mu\) are monomials which provide a basis of a supplementary of the jacobian ideal in \({\mathcal O}_n\). We can assign weight \(\beta_i=d-\deg e_i\) to each \(\lambda_i\) so that \(F\) becomes quasihomogeneous in both \(x\) and \(\lambda\).
Let us denote by \(\overline\Psi\) the \({\mathbb C}^*\) action on \({\mathbb C}^n\times {\mathbb C}^\mu\) given by
\[
\overline\Psi(t,x,\lambda)= (\Psi(t,x),\psi(t,\lambda))= (t^{\alpha_1}x_1,\dots,t^{\alpha_n}x_n,t^{\beta_1}\lambda_1,\dots,t^{\beta_\mu}\lambda_\mu).
\]
Then, \(F(\overline\Psi(t,x,\lambda))=t^dF(x,\lambda)\).
Let \(G(y,\gamma)\) be another miniversal deformation of \(f\). Locally we have \(G(y,\gamma)=F(H_\gamma(y),h(\gamma))\) for some diffeomorphisms \(H_\gamma\) and \(h\). The pull back through \(H_\gamma\) and \(h\) of the \({\mathbb C}^*\) action \(\overline\Psi\), gives another \({\mathbb C}^*\) action \(\overline \Phi=(\Phi,\phi)\), whose base part is defined by \(\phi(t,\gamma)=h^{-1}(\psi(t,h(\gamma)))\) and is called the induced \({\mathbb C}^*\) action on \({\mathbb C}^\mu\).
The main result of this paper is that locally any two induced \({\mathbb C}^*\) actions \(\phi_1\) and \(\phi_2\) on the base of the same miniversal deformation \(G\) coincide on some neighbourhood of the origin. The proof is based on the use of the Euler vector field. miniversal deformation; quasihomogeneous singularity; Euler vector field Deformations of complex singularities; vanishing cycles, Deformations of singularities On the uniqueness of the quasihomogeneity | 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 Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on \(\mathbb Z\), a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of \(\mathbb T\) and \(\mathbb Z[X]\), we define analytic spectra and sheaves of analytic functions on them.
We propose in this text a new kind of analytic spaces that gives a natural answer to this simple open problem. The construction is made in several steps. We start in Section 1 by studying the category of halos, which is the simplest category that contains the category of rings, and such that Krull valuations and seminorms are morphisms in it. In Section 2, we define a new notion of tempered generalized valuation which entails a new notion of place of a ring. In Section 3, we use this new notion of place to define a topological space called the harmonious spectrum of a ring. In Section 4, we give a definition of the analytic spectrum and define analytic spaces using local model similar to Berkovich's [\textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-archimedean fields. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)]. We finish by computing in detail the points of the global
analytic a ne line over \(\mathbb Z\) and proposing another approach to the definition of analytic functions. valuation; seminorm; algebraic numbers; analytic functions; number fields Paugam, F., Global analytic geometry, J. Number Theory, 129, 10, 2295-2327, (2009) Valuations and their generalizations for commutative rings, Model theory (number-theoretic aspects), Rigid analytic geometry, Foundations of algebraic geometry Global analytic geometry | 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 finitely generated graded module \(E\) over a polynomial ring \(R\) has a filtration by its finitely generated graded submodules over polynomial subrings satisfying the conditions related with a certain kind of direct sum decomposition, if and only if the variables of \(R\) form a filter-regular sequence with respect to \(E\) when arranged reversely. We obtain this fact as a variation of the existence theorem for a system of Weierstrass polynomials or a standard basis with respect to generic coordinates, without using the ordering on the initial terms. We also show that the properties of the filtration mentioned above are reflected naturally upon the free complex constructed from a free resolution of \(E\) by a method described in the author's previous work [cf. \textit{M. Amasaki}, J. Math. Kyoto Univ. 33, No. 1, 143-170 (1993; Zbl 0794.13009)]. Gröbner basis; graded module; polynomial ring; filtration; Weierstrass polynomials Amasaki, M.: Generators of graded modules associated with linear filter-regular sequences. J. pure appl. Algebra 114, 1-23 (1996) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects of algebraic curves Generators of graded modules associated with linear filter-regular sequences | 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 On note \({\mathcal M}[X]=\oplus _{n\in {\mathbb{N}}}M_ nX\) n\ un anneau gradué alors \(M_ 0=A_ 0\) est un anneau unitaire et pour tout n, M est un \(A_ 0\)-module. On suppose que \(\cup _{n\in {\mathbb{N}}}M_ n\quad est\) un anneau commutatif unitaire A. On donne des conditions necessaires et suffisantes pour que le spectre de \({\mathcal M}[X]\) ne depende que de \(A_ 0\) et de A, de façon précise qu'il soit la réunion disjointe d'un fermé homéomorphe au spectre de \(A_ 0\) et d'un ouvert homéomorphe à un ouvert de A[X]. Dans ce cas tout premier de \({\mathcal M}[X]\) se relève de façon unique dans l'anneau des polynômes de A[X] dont le terme constant appartient à \(A_ 0\). Ces deux anneaux ont bien sûr même dimension mais leur spectres ne sont pas nécessairement homéomorphes. On donne de nombreux exemples, notamment celui de l'anneau des polynômes dont les k premières dérivées appartiennent à \({\mathcal M}[X]\). spectrum of polynomial ring over a sequence of modules Y. Haouat, Spectre d'anneaux de polynômes sur une suite de modules. Arch. Math.51, 303-307 (1988). Ideals and multiplicative ideal theory in commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Structure, classification theorems for modules and ideals in commutative rings, Relevant commutative algebra Spectre d'anneaux de polynômes sur une suite de modules. (Spectrum of polynomial rings over a sequence of modules) | 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 a previous article [\textit{E. M. Feichtner} and \textit{D. N. Kozlov}, Int. Math. Res. Not. 2003, No. 32, 1755--1784 (2003; Zbl 1046.14025)], the authors described the abelianization, given by the maximal De Concini-Procesi model of the braid arrangement, of the permutation action of \(S_{n}\) (the symmetric group) on real \(n\)-space. An interesting property of this abelianization is that the stabilizers of points on the arrangement model are elementary abelian \(2\)-groups. An abelianization with this property is called a digitalization. The purpose of this paper is to extend this construction, first to finite groups of orthogonal matrices acting on \(\mathbb{R}^{n},\) then to finite groups acting on smooth manifolds.
Let \(G\) be a finite subgroup of \(\text{O}(n)\) acting on \(\mathbb{R} ^{n}.\) For any \(H\leq G\) define \(L\left( H\right) \) to be the span of lines in \(\mathbb{R}^{n}\) invariant under \(H\). Let \(\mathcal{A}\) be the arrangement given by proper subspaces \(L\left( H\right) \) of \(\mathbb{R}^{n}\) for various \(H\leq G,\) and let \(Y_{\mathcal{A}}\) denote the maximal De Concini-Procesi model for \(\mathcal{A}\) \ The abelianization of the linear action is defined to be this \(Y_{\mathcal{A}}.\) It is proved that for effective actions of \(G\) on \(\mathbb{R}^{n}\) we have that \(Y_{\mathcal{A}}\) is a digitalization of \(G,\) which justifies applying the term ``digitization'' in this case.
Next, let \(G\) be a finite group acting differentiably on a smooth manifold \(X \). For any \(x\in X\) and \(H\) a subgroup of the stabilizer of \(x\), define \( L\left( x,H\right) \) to be the space generated by all \(\ell \in T_{x}X\) which are invariant under \(H\). Here a stratification of \(X\) is constructed, called the \(\mathcal{L}\)-stratification of \(X\), and a model \(Y_{X,\mathcal{L} }\) is constructed from it as well. In the case where \(G\) is also acting effectively this model is a digitalization of the action.
The final section of this paper deals with examples of these constructions. Specifically, the digitalization of the action of \(S_{n}\) on \(\mathbb{R}^{n}\) is given, and is shown to be equivalent to the one in the authors' previous work cited above. Also, the digitalization of \(S_{n}\) acting on \(\left( n-1\right) \)-dimensional projective space is described, with a very explicit description in the case \(n=3\). finite group actions; digitizations E. Feichtner - D. Kozlov, A desingularization of real di\?erentiable actions of finite groups, Int. Math. Res. Not. (2005), 881-898. Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) A desingularization of real differentiable actions of finite 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 Let \(V\) be a Fano hypersurface of degree \(M\) in \({\mathbb P}^{M}\) with \(M \geq 5\). In the paper under review the author considers a class of singular Fano hypersurfaces which are regular, where by regular the author means the following. Let \(V\) be defined by an equation \(f=0\) and let \(x\in V\). One can assume that in an affine open set of \({\mathbb P}^{M}\) the point \(x\) has coordinates \(x=(0, \dots, 0)\) and in such an affine set \(f= q_0+\dots +q_M\), where \(q_i\) are homogeneous polynomials of degree \(i\). If \(x\in V\) is a smooth point then \(x\) is said to be regular if \(q_1, \dots , q_M\), form a regular sequence in \(\mathcal {O}_{x,{\mathbb P}^{M}}\). If \(x\in V\) is a singular point then \(x\) is said to be regular if the quadric \(\{q_2=0\}\) is smooth (that is \(x\) is a non-degenerate double point) and if \(q_2, \dots , q_M\), form a regular sequence in \(\mathcal {O}_{x,{\mathbb P}^{M}}\). The hypersurface \(V\) is said to be regular if all of its points are regular.
The main result in the paper under review is that a general Fano hypersurface \(V\) of degree \(M\) in \({\mathbb P}^{M}\), \(M \geq 5\) with at least one singular point which is a non-degenerate double point, is birationally superrigid [see \textit{A. V. Pukhlikov}, Invent. Math. 134, No.2, 401-426 (1998; Zbl 0964.14011), for definition of birationally superrigid]. In particular \(V\) cannot be fibered into uniruled varieties by a non-trivial rational map. Pukhlikov, AV, Birationally rigid singular Fano hypersurfaces, J. Math. Sci. (N. Y.), 115, 2428-2436, (2003) Fano varieties, Hypersurfaces and algebraic geometry, Birational automorphisms, Cremona group and generalizations Birationally rigid singular Fano hypersurfaces | 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 Numerical analysis computational problems such as Cholesky decomposition of a positive definite matrix, or unitary transformation of a complex matrix into upper triangular form (for instance by the Householder algorithm), require algorithms that use also ``non-arithmetical'' operations such as square roots. The aim of this paper is twofold:
1. Generalizing the notions of arithmetical semi-algebraic decision trees and computation trees (that is, with outputs) we suggest a definition of Nash trees and Nash straight line programs (SLPs), necessary to formalize and analyze numerical analysis algorithms and their complexity as mentioned above. These trees and SLPs have a Nash operational signature \(N^R\) over a real closed field \(R\). Based on the sheaf of abstract Nash functions over the real spectrum of a ring as introduced by \textit{M.-F. Roy} [Géométrie algébrique réelle et formes quadratiques, Journ. S.M.F., Univ. Rennes 1981, Lect. Notes Math. 959, 406-432 (1982; Zbl 0497.14009)], we propose a category \(\text{nash}_R\) of partial (homogeneous) \(\text{N}^R\)-algebras (in the sense of universal algebra) in which these Nash operations \(\nu\in \text{N}^R\) make sense in a natural way.
2. Using this framework, in particular the execution of \(\text{N}^R\)-SLPs in appropriate \(\text{N}^R\)-algebras, we extend the degree-gradient lower bound from \textit{T. Lickteig} [(*) J. Pure Appl. Algebra 110, No. 2, 131-184 (1996)] to Nash decision complexity of the membership problem of co-one-dimensional semi-algebraic subsets \(E\subset U\) of open semi-algebraic subsets \(U\subseteq R^n\). Up to a constant factor the lower bounds given in (*) for several concrete applications remain valid in the Nash framework as well. We assume the reader to be familiar with the real spectrum, abstract Nash functions, and semi-algebraic complexity as treated in (*). Nash trees; Nash complexity; Cholesky decomposition of a positive definite matrix , Analysis of algorithms and problem complexity, Symbolic computation and algebraic computation, Semialgebraic sets and related spaces, Nash functions and manifolds Nash trees and Nash complexity | 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 \(K \subset {\mathbb {R}}^n\) be a compact definable set in an o-minimal structure over \(\mathbb {R}\), for example, a semi-algebraic or a subanalytic set. A definable family \(\{ S_\delta | 0< \delta \in {\mathbb {R}} \}\) of compact subsets of \(K\) is called a \textit{monotone family} if \(S_\delta \subset S_\eta\) for all sufficiently small \(\delta > \eta >0\). The main result of the paper is that when \(\dim K \leqslant 2\), there exists a definable triangulation of \(K\) such that, for each (open) simplex \(\Lambda\) of the triangulation and each small enough \(\delta >0\), the intersection \(S_\delta \cap \Lambda\) is \textit{equivalent} to one of the five \textit{standard} families in the standard simplex (the equivalence relation and a standard family will be formally defined). The set of standard families is in a natural bijective correspondence with the set of all five lex-monotone Boolean functions in two variables. As a consequence, we prove the two-dimensional case of the topological conjecture in [the second and the third author, J. Lond. Math. Soc., II. Ser. 80, No. 1, 35--54 (2009; Zbl 1177.14097)] on approximation of definable sets by compact families. We introduce most technical tools and prove statements for compact sets \(K\) of \textit{arbitrary} dimensions, with the view towards extending the main result and proving the topological conjecture in the general case. Semialgebraic sets and related spaces, Real-analytic and semi-analytic sets, Topology of real algebraic varieties Triangulations of monotone families. I: two-dimensional families | 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 system of polynomial equations \(F_0= \cdots =F_l =0,\) where \(F_0, \ldots , F_l\) are Laurent polynomials on the complex torus \((\mathbb{C} \backslash 0)^k.\) The coefficients of \(F_0, \ldots , F_l\), in their turn are assumed to be Laurent polynomials on \((\mathbb{C} \backslash 0)^n\) which is regarded as the parameter space. Let \(\Sigma\) be the subset in \((\mathbb{C} \backslash 0)^n\) made up by the values of the parameters for which the system \(F_0= \cdots =F_l =0\) defines a singular set in \((\mathbb{C} \backslash 0)^k.\) Usually the closure of \(\Sigma\) is a hypersurface; its defining equation is called the discriminant of the system. The author explicitly computes the Newton polyhedron of the discriminant in terms of the Newton polyhedra of the polynomial coefficients of \(F_0, \ldots , F_l\). The computation was not made before in such generality; it puts in a coherent context scattered results previously known: [\textit{I. M. Gelfand, M. M. Kapranov} and \textit{A. N. Zelevinsky}, Discriminants, resultants and multidimensional determinants. Mathematics: Theory \& Applications. Boston, MA: Birkhäuser. (1994; Zbl 0827.14036)]; \textit{B. Sturmfels} [J. Algebr. Comb. 3, No. 2, 207--236 (1994; Zbl 0798.05074)]; \textit{J. McDonald} [Discrete Comput. Geom. 27, No. 4, 501--529 (2002; Zbl 1067.52013)]; \textit{P. D. González Pérez} [Can. J. Math. 52, No. 2, 348--368 (2000; Zbl 0970.14027)]; \textit{A. Esterov} and \textit{A. Khovanskii} [Funct. Anal. Other Math. 2, No. 1, 45--71 (2008; Zbl 1192.14038)]. discriminant; Newton polyhedron; mixed fiber polyhedron; mixed volume; elimination theory; Euler obstruction; dual defectiveness; Cayley trick Esterov, A., \textit{Newton polyhedra of discriminants of projections}, Discrete Comput. Geom. 44 (2010), no. 1, 96--148. Toric varieties, Newton polyhedra, Okounkov bodies, Mixed volumes and related topics in convex geometry, Solving polynomial systems; resultants, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Newton polyhedra of discriminants of 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 This paper is based on a previous investigation of the author [J. Approximation Theory 76, No. 3, 326-350 (1994; Zbl 0802.41022)] of the classical problem of best approximation of \(l_\infty(n)\) (the Euclidean space \(\mathbb{R}^n\), \(n\in \mathbb{N}\), endowed with the maximum norm \(\|\cdot\|_\infty\)) by a linear subspace \(U\) using Plücker-Grassmann coordinates and a classification corresponding to the set of vertices of the polyhedron \(Q= U^\perp\cap \overline{b^l_1(0)}\). The author shows that:
(a) the extremal points of \(Q\) are precisely the \(l_1(n)\)-normed elementary vectors in \(U^\perp\), which in turn correspond to all circuit vectors of a matrix the columns of which form a basis of the subspace,
(b) the classification is discrete and divides the Grassmann manifold into finitely many strata using ideas from approximation theory, and
(c) the stratification evidenced by (b) is identical with the decompositions of the Grassmannian given by \textit{I. M. Gel'fand}, \textit{R. M. Goresky}, \textit{R. D. MacPherson} and \textit{V. V. Serganova} [Adv. Math. 63, 301-316 (1987; Zbl 0622.57014)]. metric projection; stratification of Grassmannian; best approximation; extremal points; Grassmann manifold Norms of matrices, numerical range, applications of functional analysis to matrix theory, Best approximation, Chebyshev systems, Grassmannians, Schubert varieties, flag manifolds, Convex sets in \(n\) dimensions (including convex hypersurfaces), Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) Metric projection and stratification of the Grassmannian | 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 moduli space \(\mathcal{M}_{g,1}^N\) of pointed algebraic curves of genus \(g\) with a prescribed semigroup \(N\) whose complement w.r.t. \(\mathbb{N}_0:=\{0,1,2, \cdots\}\) is the gap sequence at the marked point, when the underlying base field is of positive characteristic. In zero characteristic case, there is a nice description of such a moduli space in terms of the negative part of miniversal deformation space of the monomial curve of the semigroup corresponding to the given Weierstrass gap sequence, due to the well-known work of \textit{H. C. Pinkham} [Astérisque 20, 1--131 (1974; Zbl 0304.14006)].
However in positive characteristic case, the isomorphism established by a theorem of Pinkham may not hold in general. In this paper the author establishes criteria for validity of the theorem of Pinkham up to genus 4 in positive characteristic case. Specifically it is proved that Pinkham's statement hold for any numerical semigroup \(N\) of genus \(g\leq 4\) if the characteristic of the base field does not divide any exponent of the generating monomials which defines the negative miniversal deformation of the monomial curve \(X\) w.r.t \(N\), unless \(N=\{5,6,7, 8, \cdots\}\) and \(g=4\). In the case \(N=\{5,6,7, 8, \cdots\}\) (\& \(g=4\)) - which is the semigroup of non-gap sequence at non Weierstrass points - together with additional assumption that the characteristic \(p\) is not five, it is proved that the same statement also holds. This paper has a nice review on the work of Pinkham which is easily accessible to the reader.
For part I, II, see [\textit{T. Nakano} and \textit{T. Mori}, ibid. 27, No. 1, 239--253 (2004; Zbl 1077.14037); [\textit{T. Nakano}, ibid. 31, No. 1, 147--160 (2008; Zbl 1145.14025)]. pointed algebraic curves; moduli space; monomial curves; Weierstrass point Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences On the moduli space of pointed algebraic curves of low genus. III: Positive characteristic. | 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 map \(X:[0,1]\rightarrow {\mathbb R}^d\), and its projection \(X_i\) on the \(i\)-th coordinate, for a fixed \(k\in{\mathbb N}\), \textit{K. T. Chen} defined in [Trans. Am. Math. Soc. 89, 395--407 (1958; Zbl 0097.25803)] the \(k\)-th signature of \(X\) as the \(k\)-tensor \(\sigma^{(k)}(X)\in ({\mathbb R}^d)^{\otimes k}\) whose \((i_1,\ldots,i_k)\)-th entry is the iterated integral \[\int_0^1\int_0^{t_k}\cdots \int_0^{t_3}\int_0^{t_2} \dot{X}_{i_1}(t_1)\cdots \dot{X}_{i_k}(t_k)\, dt_1\cdots dt_k\] and \(\sigma^{(0)}(X)=1\). The sequence \(\sigma(X)=(\sigma^{(k)}(X): k\geq 0)\) is the \textit{signature} of the path \(X\) and there is a truncated version of it \(\sigma^{\leq m}(X)=(\sigma^{(k)}(X): 0\leq k\leq m)\). For smooth \(X\), the iterated integrals do not give any extra information not derived from \(X\), however when \(X\) is not smooth, there can exist sequences of smooth paths \(X^n, Y^n\) both point-wise converging to \(X\) and with corresponding sequences of iterated integrals also converging but to different limits; these limits are no longer iterated integrals, but they satisfy the so-called Chen identities [loc. cit.]
Fixing the class of paths and an integer \(k\), the \(k\)-th signature \(\sigma^{(k)}\) is an algebraic map into \(({\mathbb R}^d)^{\otimes k}\) and the (Zariski) closure of its image is called the \textit{\(k\)-th signature variety}. In the paper under review, the authors consider signature varieties for two classes of paths. First, they consider the signature variety of rough paths, which are first defined as the Zariski closure of the image of rough paths of order \(m\) in the tensor space \(({\mathbb R}^d)^{\otimes k}\), but since this is only a semi-algebraic subset, they complexify first and then take the projectivization in \(({\mathbb C}^d)^{\otimes k}\). The corresponding signature variety has some similarities with the classical Veronese variety and is known as the \textit{rough Veronese variety} \({\mathcal R}_{d,k,m}\). One such similarity considers the fact that the classical Veronese variety is the image of the map given by degree \(k\) monomials in the usual grading, and it is known that the rough Veronese variety \({\mathcal R}_{d,k,m}\) is the closure of the image of a weighted projective space by a map given by all monomials of weighted degree \(k\). Looking for more similarities, and since the classical Veronese variety is defined by quadrics, the corresponding property for the rough Veronese variety is not true by a counterexample in Proposition 28 of [\textit{F. Galuppi}, Linear Algebra Appl. 583, 282--299 (2019; Zbl 1432.14041)]. The first main result of the paper under review is that, in general, does not even exist a bound to the degree of the generators of the ideal defining the rough Veronese variety (Proposition 2.9). On the other hand, the second main result (Proposition 2.11) shows that \({\mathcal R}_{d,k,m}\) is defined by quadrics outside of a coordinate linear subspace of large codimension. Using toric geometry, the authors characterize the cases in which \({\mathcal R}_{d,k,m}\) is an embedding of the weighted projective space and conditions that make it (projectively) normal and examples when it is not. Lastly, they obtain formulas for the dimension and degree of \({\mathcal R}_{d,k,m}\) in Proposition 2.21.
The second family of signature varieties that the authors study correspond to the class of axis-parallel paths, for which they obtain a combinatorial parametrization in Lemma 3.3 and using this description they show that the variety is toric in some cases. signature varieties; rough paths; axis-parallel paths; toric varieties Computational aspects of higher-dimensional varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Stochastic analysis Toric geometry of path signature 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 We propose a Hodge theory for the spaces \(E_{p,q}^{2}\) featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold \(X\) or in the spectral sequence associated with a pair \((N,F)\) of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on \(X\) whose kernel in every bidegree \((p,q)\) is isomorphic to \(E_{p,q}^{2}\) in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for \(E_{p,q}^{2}\) to give sufficient conditions for the degeneration at \(E_{2}\) of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on \(X\). For example, in the Frölicher case, we prove degeneration at \(E_{2}\) if there exists an SKT metric \(\omega\) (i.e., such that \(\partial \bar{\partial}\omega =0\)) whose torsion is small compared to the spectral gap of the elliptic operator \(\Delta ^{\prime }+\Delta ^{\prime \prime }\) defined by \(\omega\). In the foliated case, we obtain degeneration at \(E_{2}\) under a hypothesis involving the Laplacians \(\Delta_{N}^{\prime }\) and \(\Delta_{F}^{\prime } \) associated with the splitting \(\partial =\partial _{N}+\partial _{F}\) induced by the foliated structure. SKT metrics; Frölicher spectral sequence; pseudo-differential operators; Hodge theory; commutation relations for Hermitian metrics Popovici, D, Degeneration at \(E_2\) of certain spectral sequences, Int. J. Math., 27, 1650111, (2016) Global differential geometry of Hermitian and Kählerian manifolds, Transcendental methods, Hodge theory (algebro-geometric aspects), de Rham cohomology and algebraic geometry Degeneration at \(E_2\) of certain spectral sequences | 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 introduces a general method for relating characteristic classes to singularities of a bundle map. The method is based on the notion of geometric atomicity. This is a property of bundle maps \(\alpha:E\to F\) which universally guarantees the existence of certain limits arising in the theory of singular connections. Under this hypothesis, each characteristic form \(\Phi\) of \(E\) or \(F\) satisfies an equation of the form
\[
\Phi=L+dT,
\]
where \(L\) is an explicit localization of \(\Phi\) along the singularities of \(\alpha\) and \(T\) is a canonical form with locally integrable coefficients. The method is constructive and leads to explicit calculations. For normal maps (those transversal to the universal singularity sets) it retrieves classical formulas of R. MacPherson at the level of forms and currents [see Part I, the authors, Asian J. Math. 4, 71--95 (2000; Zbl 0981.58003)]. It also produces such formulas for direct sum and tensor product mappings. These are new even at the topological level The condition of geometric atomicity is quite broad and holds in essentially every case of interest, including all real analytic bundle maps. An important aspect of the theory is that it applies even in cases of ``excess dimension,'' that is, where the singularity sets of \(\alpha\) have dimensions greater than those of the generic map. The method yields explicit calculations in this general context. A number of examples are worked out in detail. Global theory of complex singularities; cohomological properties, Singularities of differentiable mappings in differential topology, Characteristic classes and numbers in differential topology, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other connections Singularities and Chern-Weil theory. II: Geometric atomicity | 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 Grassmannians of an infinite-dimensional vector space \(V\) are defined as the orbits of the action of the general linear group \(\mathrm{GL}(V)\) on the set of all subspaces. Let \(\mathcal{G}\) be one of these Grassmannians. An apartment in \(\mathcal{G}\) is the set of all elements of \(\mathcal{G}\) spanned by subsets of a certain basis of \(V\) . The author shows that every bijective transformation \(f\) of \(\mathcal{G}\) such that \(f\) and \(f^{-1}\) send apartments to apartments is induced by a semilinear automorphism of \(V\) .
Elements \(X,Y\in \mathcal{G}\) are called adjacent if
\[
\dim X/ X\cap Y=\dim Y/ X\cap Y=1,
\]
that is, \(X\cap Y\) is a hyperplane in both \(X\) and \(Y.\) The Grassmann graph \(\Gamma(\mathcal{G})\) is the graph whose vertex set is \(\mathcal{G}\) and the edges are pairs of adjacent subspaces.
In the case when \(\mathcal G\) consists of subspaces whose dimension and codimension both are infinite, the author shows the following. Let \(\mathcal{C},\mathcal{C}'\) be two connected components of \(\Gamma(\mathcal{G})\). If \(f\) is a bijection of \(\mathcal{C}\) to \(\mathcal{C}'\) such that \(f\) and \(f^{-1}\) send apartments to apartments, then \(f\) is induced by a semilinear automorphism of \(V.\)
The paper is clear, well written and easy to follow. The proofs and arguments are mostly set-theoretic. Grassmannian; infinite-dimensional vector space; semilinear isomorphism Grassmannians, Schubert varieties, flag manifolds, Linear transformations, semilinear transformations Apartments preserving transformations of Grassmannians of infinite-dimensional vector 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 Let \(H\) be a complex Hilbert space. Denote by \(\mathcal{G}_k(H)\) the Grassmannian consisting of \(k\)-dimensional subspaces of \(H\). Every orthogonal apartment of \(\mathcal{G}_k(H)\) is defined by a certain orthogonal base of \(H\) and consists of all \(k\)-dimensional subspaces spanned by subsets of this base. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of \(\mathcal{G}_k(H)\). In the case when \(H\) is infinite-dimensional, we prove the following: if \(f\) is a bijective transformation of \(\mathcal{G}_k(H)\) such that \(f\) and \(f^{- 1}\) send orthogonal apartments to orthogonal apartments (in other words, \(f\) preserves the compatibility relation in both directions), then \(f\) is induced by an unitary or antiunitary operator on \(H\). Suppose that \(\dim H = n\) is finite and not less than 3. For \(n \neq 2 k\) (except the case when \(n = 6\) and \(k\) is equal to 2 or 4), we show that every bijective transformation of \(\mathcal{G}_k(H)\) sending orthogonal apartments to orthogonal apartments is induced by an unitary or antiunitary operator on \(H\). Our third result is the following: if \(n = 2 k \geq 8\) and \(f\) is a bijective transformation of \(\mathcal{G}_k(H)\) such that \(f\) and \(f^{- 1}\) send orthogonal apartments to orthogonal apartments, then there is a unitary or antiunitary operator \(U\) such that, for every \(X \in \mathcal{G}_k(H)\), we have \(f(X) = U(X)\) or \(f(X)\) coincides with the orthogonal complement of \(U(X)\). Hilbert Grassmannian; compatibility relation; orthogonal apartment Pankov, M., Orthogonal apartments in Hilbert grassmanians, Linear algebra appl., 506, 168-182, (2016) Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product), Grassmannians, Schubert varieties, flag manifolds Orthogonal apartments in Hilbert Grassmannians | 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 \(U\) be a connected noetherian scheme of finite étale cohomological dimension \(d\) with the property that every finite set of points of \(U\) is contained in an affine open subscheme. For \(\alpha \in H^{2}(U_{\text{ét}}, {\mathbb G}_{m})\) define the étale index \(\mathrm{eti}({\alpha})\) to be the positive generator of the rank map \({\mathbf{K}}_{0}^{{\alpha},{\text{ét}}}(U) \rightarrow {\mathbb Z}\), where \({\mathbf{K}}^{{\alpha},{\text{ét}}}\) denotes \(\alpha\)-twisted étale \(K\)-theory.
The author prove the following (except the divisibility property, which is proved in the other work of his)
{[Theorem 1.3]} Let \(\alpha \in H^{2}(U_{\text{ét}}, {\mathbb G}_{m})_{\mathrm{tors}}.\) Then, \(\mathrm{eti}(\alpha)\) kas the following properties:
1. computability: in the descent spectral sequence
\[
E_{2}^{s,t}=H^{s}(U_{{\text{é}}t}, {\mathcal K}_{t}^{\alpha}) \Rightarrow {\mathbf{K}}_{t-s}^{{\alpha},{\text{ét}}}(U)
\]
for \({\alpha}\)-twisted étale \(K\)-theory, the integer \(\mathrm{eti}(\alpha)\in {\mathbb Z}\cong H^{0}(U_{\text{ét}}, {\mathcal K}_{0}^{\alpha})\) is the smallest positive integer such that \(d_{k}^{\alpha}(\mathrm{eti}(\alpha))=0\) for all \(k\geq 2,\) where \(d_{k}^{\alpha}\) is the k-th differential in the spectral sequence;
2. divisibility: \(\mathrm{per}({\alpha})| \mathrm{eti}(\alpha)\), where \(\mathrm{per}({\alpha})\) is the order of \(\alpha\) in \(H^{2}(U_{{\text{é}}t}, {\mathbb G}_{m})_{\mathrm{tors}} ;\)
3. obstruction; if \(\mathcal A\) is an Azumaya algebra in the class of \({\alpha}\), then \(\mathrm{eti}(\alpha)| deg({\mathcal A}),\) where \(deg({\mathcal A})\), the degree of \({\mathcal A}\), is the positive square-root of the rank of \(\mathcal A\);
4. bound; if \(\mathrm{per}({\alpha})\) is prime to the characteristics of the residue fields of \(U\), then
\[
\mathrm{eti}(\alpha) | {{{\Pi}}_{j\in \{1,\dots, d-1\}}} l_{j}^{\alpha}
\]
where \(l_{j}^{\alpha}\) is the least common multiple of the exponents of \({\pi}_{j}^{s}\) and \({\pi}_{j}^{s}(B{\mathbb Z}/(\mathrm{per}({\alpha})))\)
In particular, \(\mathrm{eti}(\alpha)\) is finite even if \(\alpha\) is not representable by an Azumaya algebra. Brauer groups; twisted sheaves; higher algebraic \(K\)-theory; stable homotopy theory DOI: 10.1017/is010011030jkt136 Brauer groups of schemes, Brauer groups (algebraic aspects), Symmetric monoidal categories, Stable homotopy groups, Stable homotopy of spheres Cohomological obstruction theory for Brauer classes and the period-index problem | 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 Based upon studies of the ``formal geometry'' [J. Differ. Geom. 51, No. 3, 431--469 (1999; Zbl 1065.14512)] over \(\mathbb C\), the authors investigated the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. They extended the Krichever map defined over a finite covering of pointed Riemann surfaces and trivializations at the punctures \textit{M. R. Adams} and \textit{M. J. Bergvelt}, [Commun. Math. Phys. 154, No. 2, 265--305 (1993; Zbl 0787.35085)], \textit{Y. Li} and \textit{M. Mulase}, [Commun. Anal. Geom. 5, No. 2, 279--332 (1997; Zbl 0920.14010)] to the case of the ``formal spectral cover'' over the formal geometry.
As its application, they characterised the existence of certain linear series on a smooth curve in terms of soliton equations, e.g., multicomponent KP hierarchy, Hamiltonian flows of the Hitchin system and so on. Infinite Grassmannians, Hurwitz schemes, KP hierarchies, Krichever map José M. Muñoz Porras and Francisco J. Plaza Martín, Equations of Hurwitz schemes in the infinite Grassmannian, Math. Nachr. 281 (2008), no. 7, 989 -- 1012. Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Formal power series rings, Grassmannians, Schubert varieties, flag manifolds, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems Equations of Hurwitz schemes in the infinite Grassmannian | 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 \(V\) be a vector space of dimension n over a field \(K\). Let \(N\) be the set of all nilpotent endomorphisms of \(V\) and let \(Y\) be the set of all flags \(F_*=(F_ i)_{0\leq i\leq n}\) in \(V\). Let \(N(F_*)=\{x\in N| xF_ i\subset F_{i-1}\text{ for all } i>0\}.\) Note that \(x\in N(F_*)\) if and only if x is strictly upper triangular with respect to a basis of V corresponding to \(F_*\). Let \(X=\{(x,F_*)\in N\times Y|\) \(x\in N(F_*)\}\) and let \(\pi: X\to N\) be the morphism given by the first projection. Let \(Y(x)\) denote the fiber \(\pi^{-1}(x)\). This paper considers a classification of the elements of \(X\), which can be thought of as strictly upper triangular matrices, and of the fiber \(Y(x)\), which is the conjugacy class of \(x\).
A nilpotent endomorphism \(x\) is characterized by a partition \(\lambda (x)=(\lambda_ 1,...,\lambda_ r)\), where \(\lambda_ 1\geq...\geq \lambda_ r\) are the sizes of the Jordan blocks of \(x\). Given \((x,F_*)\), \(x\) induces nilpotent endomorphisms on the subquotients \(F_ q/F_ p\), hence a system \(t=(t[p,q])_{p<q}\) of partitions. The information contained in such a system of partitions is reorganized in the form of a strict upper triangular matrix with entries zeros or ones. Such a matrix is called a typrix.
A typrix is said to occur if it arises from some \((x,F_*)\). The author obtains separate sets of conditions which are either necessary or sufficient for a typrix to occur. The sufficient conditions are used to describe certain dense subsets of the irreducible components of the fibers \(Y(x)\), in case K is infinite. The necessary conditions are of a combinatorial nature and are amenable to verification by a computer, if \(n\) is not too large. In this way the author obtains a list of possibly occurring typrices for \(n\leq 7\). For larger \(n\), a number of occurring typrices are described in several special cases. nilpotent endomorphisms; flags; classification; strictly upper triangular matrices; fiber; Jordan blocks; typrix Hesselink, W. H.: A classification of the nilpotent triangular matrices. Compositio math. 55, 89-133 (1985) Canonical forms, reductions, classification, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields A classification of the nilpotent triangular matrices | 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 Man bezeichne die ursprünglichen Moduln der hyperelliptischen Functionen erster Ordnung mit \(\kappa^2,\lambda^2,\mu^2\), ihre Argumente mit \(u_1,u_2\), ihre Perioden mit \(2K_{11},2K_{21},2K_{12},2K_{22}\), ferner die entsprechenden mit Hülfe einer rationalen Transformation \(n^{\text{ten}}\) Grades transformirten Grössen mit \(c^2,l^2,m^2,u_1',u_2',2C_{11},2C_{21},2C_{12}, 2C_{22}\) beziehlich. Die transformirten Argumente \(u_1',u_2'\) sind dann homogene lineare Functionen:
\[
\begin{aligned} & u_1'=M_0u_1+M_1u_1,\\& u_2'=M_2u_1+M_3u_2\end{aligned}
\]
der ursprünglichen, und die dabei auftretenden Coefficienten \(M_0,M_1,M_2,M_3\) sind die sogenannten Multiplicatoren. Diese Multiplicatoren genügen, als Functionen von \(\kappa,\lambda,\mu\) oder von \(c,l,m\) betrachtet, einer Reihe von Differentialgleichungen erster Ordnung und einer Reihe von Differentialgleichungen zweiter Ordnung. Die ersteren werden in \S 1, die letzteren in \S 3 der vorliegenden Abhandlung von dem Herrn Verfasser abgeleitet. In \S 2 werden ferner Differentialgleichungen aufgestellt, denen die Perioden \(2C_{11},2C_{21}\), als Functionen von \(c,l,m\) aufgefasst, Genüge leisten, und in \S 4 endlich werden Differentialgleichungen zwischen den ursprünglichen und den transformirten Moduln abgeleitet. Die Resultate der \S\S 2 und 4 sind zum grössten Teile auch in die \S\S 31, 45 und 43 des Werkes des Herrn Verfassers: ``Die Transformation der hyperelliptischen Functionen erster Ordnung'' (Leipzig 1886) aufgenommen. theta functions Theta functions and abelian varieties On some differential equations in the realm of the theta functions of two variables. | 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 Generalizing a construction of [\textit{A. Girand}, Bull. Soc. Math. Fr. 144, No. 2, 339--368 (2016; Zbl 1344.14011)], the author introduces an \(n\)-parameter family \(\nabla_{\lambda}\) of flat, regular singular \(\mathfrak{sl}_{2}\)-connections on the trivial bundle of complex projective space \(\mathbb{P}^{n}\). Let \(D\) be the polar divisor of \(\nabla_{\lambda}\).\\
The monodromy representation of \(\nabla_{\lambda}\) is computed for generic values of \(\lambda\). The author describes the local monodromy explicitly for a specific choice of generators for the fundamental group of \(\mathbb{P}^{n}\setminus D\), see Proposition 3.3. This shows that the monodromy group lies inside the infinite dihedral group \(\mathbf{D}_{\infty}\). In addition the monodromy representation is proven to be virtually abelian, that is it is abelian after passing to a finite cover of \(\mathbb{P}^{n}\setminus D\), see Theorem 1.3. \\
Using \(\nabla_{\lambda}\) the author constructs via pullback an isomonodromic family of connections on \(\mathbb{P}^1 \times T\) where \(T\subset \mathbb{A}^{2(n-1)}\) is a certain Zariski open subset parametrizing generic lines in \(\mathbb{P}^{n}\). From this isomonodromic family the author obtains an algebraic solution to the Garnier system in \(2(n-1)\) variables, see Theorem 1.4. algebraic function; Garnier system; isomonodromic deformation Algebraic functions and function fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies A family of flat connections on the projective space having dihedral monodromy and algebraic Garnier solutions | 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 study the asymptotic behavior of the point-counting function of configuration spaces over finite fields by means of homological and representation stability. The main result is the following (Theorem A): let \(X\) be a scheme smooth over \(\mathbb{Z}[1/N]\) for some integer \(N\), with geometrically connected fibers of finite type. Let \(p\) be a prime not dividing \(N\) and let \(q\) be a power of \(p\). Let \(P\) be a character polynomial, i.e., a polynomial with rational coefficients in \(\mathbb{Q}[X_1,X_2,\cdots]\), where \(X_i\) is the class function on all symmetric groups \(S_n\) that sends a permutation to the number of \(i\)-cycles in its cycle decomposition. Then the following equality holds:
\[
\underset{n\to\infty}{\text{lim}}q^{-\text{dim}X} \sum_{y\in UConf_n(X)(\mathbb{F}_q)}P(y) = \sum_{i=0}^\infty (-1)^i\text{Tr}(\text{Frob}_q|H^i_{\mathrm{et}}(PConf(X))_P)
\]
where \(\text{Frob}_q\) is the \(q\)-th power Frobenius, \(UConf_n(X)\) (resp. \(PConf_n(X)\)) is the configuration space of unordered (resp. ordered) \(n\)-tuples of points over \(X\), and \(H^i_{\mathrm{et}}(PConf(X))_P\) is the stable \(P\)-isotypic part of the étale cohomology of the co-FI-scheme \(PConf_\bullet(X)_{\overline{\mathbb{F}}_q}\), whose existence is asserted by Theorem B. The proof of Theorem A consists of two main steps: firstly the authors show the étale homological and representation stability, which states that the isomorphism class as representation of the Galois group of the étale cohomology groups of the sequence of schemes \((PConf_n(X))\) becomes stabilized, and as such one can decompose the whole cohomology into stable and unstable parts (Theorem B); in the second step the authors give subexponential bounds on the unstable part in the case of configuration spaces (Theorem C) and deduce the main result. asymptotic point counting; arithmetic statistics; homological stability Farb, B.; Wolfson, J., Étale homological stability and arithmetic statistics, preprint Étale and other Grothendieck topologies and (co)homologies, Manifolds and cell complexes Étale homological stability and arithmetic statistics | 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 investigate the topology of the discriminant of stable perturbations \(f_t\) of multi-germs \(f:({\mathbb K}^n,S) \to ({\mathbb K}^p,0)\) with \(n \geq p-1\) (when \(n=p-1\) discriminant means image), where \(S \subset{\mathbb K}^n\) is a finite set, and where \({\mathbb K}={\mathbb R}\) or \(\mathbb C\).
When \({\mathbb K}={\mathbb C}\), the discriminant \(D(f_t)\) has the homotopy type of a wedge of \((p-1)\)-spheres [\textit{J. Damon} and \textit{D. Mond}, Invent. Math. 106, No. 2, 217-242 (1991; Zbl 0772.32023); \textit{D. Mond}, Lect. Notes Math. 1462, 221-234 (1991; Zbl 0745.32020)]. The number of these spheres is called the discriminant Milnor number when \(n\geq p\) and the image Milnor number when \(n=p-1\), and denoted \(\mu_\Delta\) and \(\mu_I\) respectively. When \(n \geq p\) and \((n,p)\) are in Mather's range of nice dimensions [\textit{J. N. Mather}, Proc. Liverpool Singularities-Symp. 1969-1970, 207-253 (1971; Zbl 0211.56105)], it is known (Damon and Mond, loc. cit.) that \(\mu_\Delta(f)\) and the \({\mathcal A}_e\)- codimension of \(f\) satisfy the Milnor-Tjurina relation:
\[
\mu_\Delta(f) \geq {\mathcal {A}}_e\text{-codimension} (f)
\]
with equality if \(f\) is weighted homogeneous in some coordinate system. In case \(n=p-1\), the same relation, with \(\mu_I\) in place of \(\mu_\Delta\), is only known to hold when \(n=1\) [\textit{D. Mond}, Math. Proc. Camb. Philos. Soc. 117, No. 2, 213-222 (1995; Zbl 0869.57027)] and \(n=2\) [\textit{T. de Jong} and \textit{D. van Straten}, Lect. Notes Math. 1462, 199-211 (1991; Zbl 0734.32020) and \textit{D. Mond}, loc. cit.], and there is evidence that it holds in higher dimensions as well [\textit{K. Houston} and \textit{N. Kirk}, Lond. Math. Soc. Lect. Notes Ser. 263, 325-351 (1999; Zbl 0945.58030)].
Let \(g:(\mathbb R^n,S)\to (\mathbb R^p,0)\) be a real analytic map germ of finite \(\mathcal {A}\)-codimension, with a stable perturbation \(g_t\). Suppose also that the complexification \(g_{\mathbb C,t}\) of \(g_t\) is a stable perturbation of the complexification \(g_\mathbb C\) of \(g\). By definition \(g_t\) is a good real perturbation of \(g\) if \(\text{rank} H_{p-1} (D(g_t);\mathbb{Z})=\text{rank} H_{p-1}(D(g_{\mathbb{\mathbb C},t});\mathbb{Z})\) (in which case the inclusion of real in complex induces an isomorphism on the vanishing homology of the discriminant).
The authors state the following conjectures:
Conjecture I. The Milnor-Tjurina relation holds in all nice dimensions \((n,n+1)\).
Conjecture II. For every \({\mathcal A}_e\)-codimension 1 equivalence class of map-germs in the nice dimensions, there exists a real form with a good real perturbation. That is, the vanishing topology of all codimension 1 complex singularities is visible over \(\mathbb{\mathbb R}\).
Conjecture II holds for mono-germs of maps \(\mathbb C^n\to\mathbb C^p\) (with \(n\geq p\) and \((n,p)\) nice dimensions) of corank 1 [\textit{D. Mond}, Prog. Math. 134, 259-276 (1996; Zbl 0851.32033)], and for germs of maps \(\mathbb C^2\to\mathbb C^3\) [\textit{V. V. Goryunov}, Funct. Anal. Appl. 25, 174-180 (1991; Zbl 0744.32019)].
Note that every real germ \(\mathbb C\to\mathbb C^2\) has a good real perturbation [\textit{N. A'Campo}, Math. Ann. 213, 1-32 (1975; Zbl 0316.14011); \textit{S. M. Gusein-Zade}, Funct. Anal. Appl. 8, 295-300 (1974; Zbl 0309.14006)], but once \(n>1\), map-germs \(\mathbb C^n\to\mathbb C^{n+1}\) with good real perturbations become the exception [\textit{W. L. Marar} and \textit{D. Mond}, Topology 35, No. 1, 157-165 (1996; Zbl 0870.32013)]. The main results in this paper provide evidence for both conjectures.
Theorem 7.2. Every multi-germ \(f:(\mathbb C^n,S)\to (\mathbb C^{n+1},0)\) of corank 1 and \({\mathcal A}_e\)-codimension 1 has \(\mu_I(f)=1\).
Theorem 7.3. Every \({\mathcal A}\)-equivalence class of multi-germ \(f:(\mathbb C^n,S)\to(\mathbb C^p,0)\) (\(n\geq p-1, (n,p)\) nice dimensions) of corank 1 and \({\mathcal A}_e\)-codimension 1 has a real form with a good real perturbation.
Moreover the proofs give as well an inductive classification of multi-germs of codimension 1. singularity; stable perturbation; vanishing topology; discriminant Cooper, T; Mond, D; Wik-Atique, RG, Vanishing topology of codimension 1 multi-germs over \({\mathbb{R}}\) and \({\mathbb{C}}\), Compos. Math., 131, 121-160, (2002) Local complex singularities, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry, Topology of real algebraic varieties, Other operations on complex singularities, Topological properties of mappings on manifolds Vanishing topology of codimension 1 multi-germs over \(\mathbb R\) and \(\mathbb C\) | 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 According to the publication standards of Cambridge Tracts in Mathematics, the present book provides a thorough yet reasonably concise treatment of a specific topic by taking up a single thread in a wide subject, following various ramifications of it, and illuminating thus its different aspects from a central point of view. The topic of the treatise under review is the local theory of singularities of algebraic varieties, analytic spaces, or morphisms, and the thread taken up is the investigation of singularities via the cohomology theory for sheaves of differential forms, i.e., by means of methods of Hodge theory and its diverse generalizations.
Thus, in a body, this book provides both an introduction to, and a survey of, some central aspects of singularity theory, such as they have been intensively studied over the past thirty years. The text consists of three chapters, each of which is subdivided into several sections. To chapter I, which is preceded by a very thorough, detailed, motivating and masterly written introduction to the contents of the book, has been given the title ``The Gauss-Manin connection''. Here the author explains, always in a very concise but comprehensive and lucid manner, many of the fundamental ideas, methods, techniques, and results centred around this crucial concept. This includes: Milnor fibration, Picard-Lefschetz monodromy transformations, locally constant sheaves and systems of linear differential equations, integrable connections, relative De Rham cohomology sheaves, meromorphic connections, Brieskorn lattices, quasi-homogeneous singularities, singular points of systems of linear differential equations, regularity of the Gauss-Manin connection, period matrices, the geometry of the Picard-Fuchs equation, the monodromy theorem, the Gauss-Manin connection in the case of non-isolated hypersurface singularities, and many other related topics.
Chapter II deals with the various kinds of Hodge structures and their variation behavior under deformations. This chapter is entitled ``Limit mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singularity''. Here some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and their associated period maps, are developed in the local situation of isolated singularities of holomorphic functions. The main topics of this chapter are, among others, the following ones: mixed Hodge structures, polarized Hodge structures, Deligne's theorem, limit Hodge structures in the sense of W. Schmid, limit mixed Hodge structures in the sense of J. Steenbrink, the Hodge theory of smooth hypersurfaces (after Griffiths-Deligne), the Gauss-Manin system of an isolated singularity, decompositions of meromorphic connections, the limit Hodge filtration due to Varchenko and Scherk-Steenbrink, and spectra of various types of singularities.
The concluding chapter III, entitled ``The period map of a Milnor-constant deformation of an isolated hypersurface singularity associated with Brieskorn lattices and mixed Hodge structures'', discusses the glueing of Milnor fibrations, meromorphic connections of Milnor-constant deformations of singularities, root components of geometric sections, the period map for embeddings of Brieskorn lattices and various types of singularities, the period map associated with the mixed Hodge structure on the vanishing cohomology, the infinitesimal Torelli theorem, the period map for quasi-homogeneous singularities, the Picard-Fuchs singularity, and the recent results of C. Hertling for hypersurface singularities.
Without any doubt, the author has covered a wealth of material on a highly advanced topic in complex geometry, and in this regard he has provided a great service to the mathematical community, first and foremost with a view to the systematic, comprehensive understanding of the vast realm of singularity theory. Although he had to relinquish almost all proofs of the numerous deep theorems, he has succeeded in providing a brilliant introduction to, and a comprehensive overview of, this contemporary central subject of complex geometry. This book is designed for researchers whose interests are closely bound up with singularity theory, algebraic geometry, and complex analysis, and in that it is an excellent source and guide for them. Also, the entire text represents an irresistible invitation to the subject, and may be seen as a dependable pathfinder with regard to the vast existing original literature in the field. local theory of singularities; cohomology theory; sheaves of differential forms; Gauss-Manin connection; Milnor fibration; Picard-Lefschetz monodromy; De Rham cohomology; Brieskorn lattices; period matrices; Picard-Fuchs equation; monodromy; mixed Hodge structure; hypersurface singularity Kulikov, V. S.: Mixed Hodge structures and singularities, (1998) Singularities in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local complex singularities, Complex surface and hypersurface singularities Mixed Hodge structures and 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 Let \(X\) be a surface fibered over \(\mathbb{P}^1\) (\(t\)-line) with its general fiber defined by \(y^p=x^a(1-x)^b(t^{\ell}-x)^{p-b}\), where \(\ell\) and \(p\) are distinct prime numbers. Let \(H\) denote its second cohomology modulo the image of classes supported at singular fibers. It admits an action of \(\mu_{p\ell}\) and is shown to be one-dimensional over \(K=\mathbb{Q}(\mu_{p\ell})\). For each embedding \(\chi :K\rightarrow\mathbb{C}\), let \(H^{\chi}\) be the eigencomponent. The first main result of the present paper concerns with the period \(\text{Per}(H^{\chi})\) of \(H^{\chi}\) and shows that \(\text{Per}(H^{\chi})\sim_{K^{'\times}}B(\beta,\mu)B(1-\beta,\beta-\alpha+\mu)\), where \(K'=\mathbb{Q}(\mu_{2p\ell}), \chi(\zeta_p)=\zeta_p^n, \chi(\zeta_{\ell})=\zeta_{\ell}^m, \alpha=\{na/p\}, \beta=\{nb/p\}, \mu=\{m/{\ell}\}\). The second main result concerns with the Beilinson regulator \(r_{\mathcal{D}}:H_{\mathcal{M}}^3(X,\mathbb{Q}(2))\rightarrow H_{\mathcal{D}}^3(X_{\mathbb{C}},\mathbb{Q}(2))\) from the motivic cohomology to the Deligne cohomology, and shows that if \(\chi\) is an embedding such that \(H_{\text{dR}}^{\chi}\subset F^1H_{\text{dR}}\), then for any \(\omega\in H_{\text{dR}}^{\chi}\) and \(z\in H_{\mathcal{M},Z_1}(X,\mathbb{Q}(2))\), where \(Z_1\) denotes the union of the fibers over \(\mu_{\ell}\), the image \(r_{\mathcal{D}}(z)(\omega)\) is expressed through Gauss' hypergeometric function \(_3F_2\). Moreover the authors show the non-vanishing of the regulator image under a mild assumption. period; regulator; complex multiplication; hypergeometric function Asakura, M., Otsubo, N.: CM periods, CM regulators, and hypergeometric functions, I. Can. J. Math. \textbf{(to appear)} Variation of Hodge structures (algebro-geometric aspects), Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Generalized hypergeometric series, \({}_pF_q\), Complex multiplication and moduli of abelian varieties, Complex multiplication and abelian varieties CM periods, CM regulators and hypergeometric functions, I | 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 the article under review remarkable local monomialization theorems for morphisms of complex and real analytic spaces are proved. They are analogous to similar ones that had been previously obtained by the author in the context of algebraic varieties over a field of characteristic zero. Recall that a morphism \(\phi : Y \to X\) of analytic manifolds is \textit{monomial} at \(p \in Y\) if there are local coordinates \(x_1, \ldots, x_m, x_{m+1}, \ldots, x_t\) at \(\phi(p)\), \(y_1, \ldots, y_n\) at \(p\), an \(m\) by \(n\) matrix \((c_{ij})\) with \(\geq 0\) integral coefficients and rank \(m\) such that \(\phi ^{*}(x_i)= \prod_{j=1}^{n}y_j ^{c_{ij}} \) for \(1 \leq i \leq m\) while \(\phi ^{*}(x_i)=0\) for \(m < i \leq t\).
The main result of the paper also involves the notion of \textit{étoile} over an analytic space \(X\). This notion, introduced by Hironaka around 1970, is an analogue of the concept of valuation on the function field of an algebraic variety.
An étoile \(e\) over a complex analytic space \(X\) is a category of morphisms \(f:X' \to X\) which are finite sequences of local blow ups, satisfying certain properties. If \(f \in e\) then there is a well defined point \(e_{X'} \in X'\), called the centre of \(f\) at \(X'\). Then the author proves the following result (Theorem 1.2).
If \(\phi:Y \to X\) is a morphism of reduced complex analytic spaces and \(e\) is an étoile over \(Y\), then there are morphisms of complex analytic spaces \(\alpha:X_e \to X\), \(\beta : Y_e \to Y\), \(\phi_e:Y_e \to X_e\), such that \(\alpha \phi_e = \phi \beta\), \(\beta \in e\), \(\alpha\) and \(\beta\) are finite products of local blow ups with non singular centres, \(Y_e\) and \(X_e\) are manifolds and \(\phi_{e}\) is monomial at the centre \(e_{Y_e}\). Moreover, off a closed analytic subspace \(F_e\) of \(X_e\) the map induced by \(\alpha\) is an open embedding and \(\phi ^{-1}(F_e)\) is nowhere dense in \(Y_e\).
A seemingly stronger version of this result (Theorem 8.13 of the paper) can be derived using properties of the \textit{voûte étoilée}, i.e., the collection of all the étoiles of an analytic space, which can be endowed with a useful topology (another contribution of Hironaka).
Using methods of complexification of real analytic spaces (also due to Hironaka) the author obtains an analogue of Theorem 8.13 for real analytic spaces (Theorem 9.7), although the precise statement is somewhat more complicated.
The proofs of the previous results in the case of algebraic varieties, obtained elsewhere by the author using (among others) techniques of Valuation Theory, do not directly generalize to the present analytic context. New methods are introduced, e.g., the notion of ``independence of variables'' for an étoile, which replaces the concept of rational rank of a valuation of the algebraic situation. The proofs of the theorems of the present paper are quite involved. local monomialization; analytic map; local blow up; étoile; voûte étoilée; local monomial transformation; complexification Modifications; resolution of singularities (complex-analytic aspects), Other operations on complex singularities, Local analytic geometry, Complex singularities, Global theory and resolution of singularities (algebro-geometric aspects) Local monomialization of analytic maps | 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 the paper under review, the authors study the Fano scheme \(F_{k}(Y)\) parametrizing \(k\)-dimensional linear subspaces contained in a complete intersection \(Y \subset \mathbb{P}^{m}\) of multi-degree \(\mathbf{d} = (d_{1},\dots,d_{s})\) with \(1 \leq s \leq m-2\) and such that \(\prod_{i=1}^{s} d_{i} > 2\). Let \(S_{\mathbf{d}} = \bigoplus_{i=1}^{s}H^{0}(\mathbb{P}^{m}, \mathcal{O}_{\mathbb{P}^{m}}(d_{i}))\) and consider its Zariski open subset \(S^{*}_{\mathbf{d}}:= \bigoplus_{i=1}^{s}H^{0}(\mathbb{P}^{m}, \mathcal{O}_{\mathbb{P}^{m}}(d_{i})\setminus\{0\})\). For any \(u = (g_{1},\dots, g_{s}) \in S^{*}_{\mathbf{d}}\), let \(Y_{u} = V(g_{1},\dots, g_{s}) \subset \mathbb{P}^{m}\) be the closed subscheme defined by the vanishing of polynomials \(g_{1},\dots,g_{s}\). For any integer \(k\geq 1\), we define the locus
\[W_{\mathbf{d},k}:=\bigg\{u \in S^{*}_{\mathbf{d}} \, | \, F_{k}(Y_{u}) \neq \emptyset \bigg\} \subseteq S^{*}_{\mathbf{d}}\]
and set
\[t = t_{m,k,\mathbf{d}} :=\sum_{i=1}^{s} {d_{i} +\frac{k}{k}} - (k+1)(m-k).\]
The main result of the paper can be formulated as follows.
Main Result. Let \(m,k,s\) and \(\mathbf{d} = (d_{1},\dots, d_{s})\) be such that \(\prod_{i=1}^{s} d_{i} > 2\) and \(t>0\). Then \(W_{\mathbf{d},k} \subseteq S^{*}_{\mathbf{d}}\) is non-empty, irreducible and rational with \(\mathrm{codim}_{S^{*}_{\mathbf{d}}} \,(W_{\mathbf{d},k}) =t\). Furthermore, for a general point \(u \in W_{\mathbf{d},k}\), the variety \(Y_{u} \subset \mathbb{P}^{m}\) is a complete intersection of dimension \(m-s\) whose Fano scheme \(F_{k}(Y_{u})\) is a zero-dimensional scheme of length one. Moreover, \(Y_{u}\) has singular locus of dimension \(\max\{-1,2k+s-m-1\}\) along its unique \(k\)-dimensional linear subspace (in particular, \(Y_{u}\) is smooth if and only if \(m-s \geq 2k\)). Fano schemes; complete intersections; enumerative results; rationality results; linear subspaces Complete intersections, Grassmannians, Schubert varieties, flag manifolds, Configurations and arrangements of linear subspaces, Parametrization (Chow and Hilbert schemes), Rational and unirational varieties On complete intersections containing a linear subspace | 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 \(\text{Hilb}^{p(t)} (\mathbb P^n)\) be the Grothendieck Hilbert scheme that parametrizes closed subschemes \(X \subset \mathbb P^n\) with Hilbert polynomial \(p(t)\). \textit{A. Reeves} and \textit{M. Stillman} showed that the point in \(\text{Hilb}^{p(t)} (\mathbb P^n)\) corresponding to the lexicographic ideal is smooth [J. Algebr. Geom. 6, 235--246 (1997; Zbl 0924.14004)] and determines a canonical \textit{lexicographic component} of \(\text{Hilb}^{p(t)} (\mathbb P^n)\). \textit{I. Peeva} and \textit{M. Stillman} gave a version of this theorem for toric Hilbert schemes [Duke Math. J. 111, 419--449 (2002; Zbl 1067.14005)]. With \textit{M. Haiman} and \textit{B. Sturmfels} extending the Grothendieck Hilbert schemes to standard graded Hilbert schemes \(\mathcal H^{\mathfrak h} (R)\) parametrizing homogeneous ideals \(I\) with fixed Hilbert function \(\mathfrak h\) in a graded ring \(R\) [J. Algebr. Geom. 13, 725--769 (2004; Zbl 1072.14007)] it becomes natural to ask what the lexicographic points look like in that setting. \textit{D. Maclagan} and \textit{G. G. Smith} gave an analog to the Reeves-Stillman for standard multigraded Hilbert schemes in two variables [Adv. Math. 223, 1608--1631 (2010; Zbl 1191.14007)].
The authors give examples showing that the lexicographic point does not behave so nicely in \(\mathcal H^{\mathfrak h} (R)\) for more variables. They show that for \(S = k[x,y,z]\) and \(\mathfrak h = (1,3,4,4,3,3,3, \dots)\), the Hilbert scheme \(\mathcal H^{\mathfrak h} (S)\) is the union of two irreducible components of dimension \(8\) containing the lexicographic point in their intersection, so the lexicographic point is singular and does not correspond to a canonical component. They also give an example where the lexicographic point is not even Cohen-Macaulay. They also show for the exterior algebra \(E = \bigwedge k^5\) and \(\mathfrak h = (1,5,7,2)\) that the standard graded Hilbert scheme \(\mathcal H^{\mathfrak h} (E)\) is the union of two irreducible components of dimensions \(14\) and \(15\) which contain the lexicographic point in their intersection. standard graded Hilbert scheme; lexicographic component; lexicographic ideal; reducible scheme; exterior algebra Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Exterior algebra, Grassmann algebras On the smoothness of lexicographic points on Hilbert 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 Let \(X\), \(Y\) be spaces and \(f\) a map whose domain \(\mathrm{dom}(f)\) is contained in \(X\) and whose range is \(Y\). Such \(f\) will be called a partial function on \(X\). The closure \(\overline\Gamma(f)\) of the graph of \(f\) in \(X\times Y\) is called the graphoid of \(f\). The graphoid determines a function \(\overline f:X\to\mathcal{P}(Y)\) (the power set of \(Y\)), assigning to a point \(x\in X\), \(\overline f(x)= \{y\in Y\,|\,(x,y)\in\overline\Gamma(f)\}\). This \(\overline f\) is called the graphoid extension of \(f\). The domain, \(\mathrm{dom}(f)\), equals \(\{x\in X\,|\,\overline f(x)\neq\emptyset\}\).
The authors are interested in the graphoids of rational functions of \(k\)-variables, \(f(x_1,\dots,x_k)=\frac{p(x_1,\dots,x_k)}{q(x_1,\dots,x_k)}\), where \(p\) and \(q\) are relatively prime polynomials. The domain of such a map is the open dense subset \(\mathrm{dom}(f)=\mathbb{R}^k\setminus (p^{-1}(0)\cap q^{-1}(0))\) of \(\mathbb{R}^k\). Using \(\overline{\mathbb{R}} =\mathbb{R}\cup\{\infty\}\) to denote the projective real line, one may treat \(\overline{\mathbb{R}}^k\) as a \(k\)-dimensional torus; so \(\mathrm{dom}(f)\) is an open dense subset of \(\overline{\mathbb{R}}^k\). If \(\mathbb{R}(x_1,\dots,x_k)\) denotes the field of such rational functions, then the elements of \(\mathbb{R}(x_1,\dots,x_k)\) will take values in \(\overline{\mathbb{R}}\). A subset \(\mathcal{F}\subset\mathbb{R}(x_1,\dots,x_n)\) is called a rational vector-function.
In this setting, in case \(\mathcal{F}\) is countable, then one can define \(\mathrm{dom}(\mathcal{F})=\bigcap\{\mathrm{dom}(f)\, |\,f\in\mathcal{F}\}\); the latter is a dense \(\mathrm{G}_\delta\)-set in \(\overline{\mathbb{R}}^k\). One may think of \(\mathcal{F}:\mathrm{dom}(\mathcal{F})\to\overline{\mathbb{R}}^\mathcal{F}\) via \(x\mapsto(f(x))_{f\in\mathcal{F}}\). Its graphoid is a closed subset of the compact Hausdorff space \(\overline{\mathbb{R}}^k\times \overline{\mathbb{R}}^\mathcal{F}\), and it has a graphoid extension \(\overline{\mathcal{F}}\) with \(\mathrm{dom}(\overline{\mathcal{F}})= \overline{\mathbb{R}}^k\), i.e., \(\overline{\mathcal{F}} :\overline{\mathbb{R}}^k\to\mathcal{P}(\overline{\mathbb{R}}^\mathcal{F})\).
For uncountable families \(\mathcal{F}\), a different approach is needed, and such is described by the authors. One then similarly obtains \(\overline{\mathcal{F}}:\overline {\mathbb{R}}^k\to\mathcal{P}(\overline{\mathbb{R}}^\mathcal{F})\) as just indicated. The good properties that such a ``set-valued'' function has are listed as (1)--(4) on page 25.
The paper is devoted to:
{ Problem 1.1.} Given a family of rational functions \(\mathcal{F}\subset\mathbb{R}(x_1,\dots,x_k)\), study topological (and dimension) properties of the graphoid \(\overline{\Gamma}(\mathcal{F}) \subset\overline{\mathbb{R}}^k\times\mathbb{R}^\mathcal{F}\) of \(\mathcal{F}\). A precise question: Has \(\overline{\Gamma}(\mathcal{F})\) the topological dimension \(k\)?
The main result of the paper is,
{ Theorem 1.2.} For any family of rational functions \(\mathcal{F}\subset\mathbb{R}(x,y)\), its graphoid \(\overline{\Gamma}(\mathcal{F})\subset\overline{\mathbb{R}}^2\times \overline{\mathbb{R}}^\mathcal{F}\) has covering dimension \(2\). graphoid; graph; rational vector-function; topological dimension; extension dimension; cohomological dimension; Pontryagin space Banakh T., Potyatynyk O., Dimension of graphoids of rational vector-functions, Topology Appl., 2013, 160(1), 24--44 Unicoherence, multicoherence, Dimension theory in algebraic topology, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Real algebraic sets, Real rational functions, Degree, winding number, Topology of special sets defined by functions Dimension of graphoids of rational vector-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 This paper is concerned with the construction of a Bloch-Kato exponential map for \(\mathbb{Q}_p\)-valued representations of the absolute Galois group of a discrete valuation field \(L\) of mixed characteristic \((0,p)\) whose residue field \(k_L\) is not necessarily perfect but verifies \([k_L:k_L^p]<\infty\): an example is the field of fractions of the completed local ring of a finite-type \(\mathbb{Z}_p\)-scheme at a point. The author follows Brinon's construction of the relevant period rings in this setting and generalizes a previous work of Kato for the construction of the exponential maps: she finishes with a discussion of an explicit reciprocity law.
More precisely, fix a prime number \(p\) and a discrete valuation field \(L\) of mixed characteristic \((0,p)\) with residue field \(k_L\) and denote by \(\mathcal{G}_L\) the absolute Galois group of \(L\). In Section 2 the author recalls the construction of \textit{O. Brinon} [``Cristalline representations in the case of an imperfect residue field'', Ann. Inst. Fourier 56, No. 6, 919--999 (2006; Zbl 1168.11051)] of a suitable ``big period ring'' \(\mathbb{B}_{dR}\) which is endowed with a connection \(\nabla\) and with a filtration \(\mathrm{Fil}^i\), as well of some of its subrings \(\mathbb{B}_{dR}^\nabla,\mathbb{B}_{dR}^+,\mathbb{B}_{dR}^{\nabla +}\): with this at hand, the definition of a functor \(\mathbb{D}_{dR}(-)\) and of ``de Rham representations'' is straightforward. Suffices here to emphasize that the main difference from the classical situation (i.e. when \(k_L\) is perfect) is the existence of the connection \(\nabla\) which induces a complex of filtered vector spaces
\[
0\to\mathbb{D}_{dR}(V)\rightarrow{\nabla}\mathbb{D}_{dR}(V)\otimes_K\Omega_K^1\rightarrow{\nabla}\cdots\rightarrow{\nabla}\mathbb{D}_{dR}(V)\otimes_K\Omega^{d-1}\to 0\;. \tag{complex}
\]
Analogously, a theory of fields of norms is developed and the notion of (étale) \((\phi,G_K)\)-modules is presented in subsection 2.3, where \(G_K\) is the Galois group of a certain extension \(K_\infty/K\) of a subfield \(K\subseteq L\) which plays the role of the maximal unramified subfield. If \(K\) had perfect residue field, the theory of the fields of norms as developed, for instance, in [\textit{J.-M. Fontaine} and \textit{J.-P. Wintenberger}, ``Le ''corps des normes'' de certaines extensions algébriques de corps locaux'', C. R. Acad. Sci., Paris, Sér. A 288, 367--370 (1979; Zbl 0475.12020)] would construct a field of characteristic \(p\) and an extension of it whose Galois group is canonically isomorphic to the subgroup \(H_K\) of \(\mathcal{G}_K\) fixing the cyclotomic extension of \(K\): in the present setting the theory needs to be modified by replacing this cyclotomic extension by the extension \(K_\infty=K(\mu_{p^\infty},\sqrt[p^\infty]{X_1},\hdots,\sqrt[p^\infty]{X_{d-1}})\) where \([k_L:k_L^p]=p^{d-1}\) and the \(X_i\) are indeterminates: since the Galois group \(G_K=\mathrm{Gal}(K_\infty/K)\) is not abelian, this adds a layer of technicalities to the whole theory but similar results as in the perfect residue field case can be obtained.
Section 3 is the core of the paper where the exponential map (and its dual) is constructed: as a matter of fact, the author constructs \(d\) higher exponential maps (see Definition 3.11) as morphisms
\[
\text{exp}_{(i),L,V}:H^i_{dR}(V)\oplus H^{i-1}_{dR}(V)\longrightarrow H^{i+1}(L,V)
\]
for a de Rham representation \(V\) and for \(1\leq i\leq d\), where the de Rham cohomology \(H^i_{dR}\) is the cohomology of ({complex}).
Sections 4 and 5 provide the counterpart in the setting of imperfect residue field of the work [\textit{F. Cherbonnier} and \textit{P. Colmez}, ``Théorie d'Iwasawa des représentations \(p\)-adiques d'un corps local'', J. Am. Math. Soc. 12, No. 1, 241--268 (1999; Zbl 0933.11056)] by Cherbonnier and Colmez: the notion of overconvergent representation (and the definition of a related functor \(\mathbb{D}_{ind}^\dagger\)) is introduced in subsection 4.2 and two crucial operators \(\phi^{-n}\) and \(\psi\) are defined and studied (see Corollary 4.18 and Proposition 4.25, respectively). These operators (in particular the kernel of \(\psi\)) turn out to be the key for studying the Iwasawa theory of Galois representations in the spirit of the paper by Cherbonnier and Colmez (see also [\textit{L. Berger}, ``Bloch and Kato's exponential map: three explicit formulas'', Doc. Math., J. DMV Extra Vol., 99--129 (2003; Zbl 1064.11077)] for a beautiful account in the perfect residue field setting) and Theorems 5.11 and 5.12 focus on the Galois cohomology of \({ker}(\psi)\).
Finally, Section 6 contains a discussion of an explicit reciprocity law as conjectured (for crystalline representations in the case of perfect residue field) by \textit{B. Perrin-Riou} in [``Iwasawa theory of \(p\)-adic representations over a local field'', Invent. Math. 115, No. 1, 81--149 (1994; Zbl 0838.11071)]: the main result is the following
\textbf{Theorem 6.8} Let \(V\) be a de Rham representation of \(\mathcal{G}_L\) and let \(n\in\mathbb{N}\) such that \(p^{-(n+1)}<R\), where \(R\) is a constant (see Remark 6.6) related to the overconvergence of \(V\). Then, for all \(i\leq i\leq d\), we have a commutative diagram
\[
\begin{tikzcd} H^i \left(G_L,\mathbb{D}_{\mathrm{ind}}^\dagger(V)^{\psi=1}\right) \rar["\phi^{-n}"] \dar["\delta^{(i)}" '] & H^i (G_L,\left(V\otimes\mathbb{B}_{dR}^\nabla)^{\mathcal H}\right) \dar["\text{inf}"] \\ H^i(L,V) \rar["V\to V\otimes\mathbb{B}_{dR}^\nabla" '] & H^i\left(L,V\otimes \mathbb{B}_{dR}^\nabla \right) \rlap{\,,}\end{tikzcd}
\]
where \(\mathcal{H}_L={Gal}(\overline{K}/LK_\infty)\) and \(\delta^{(i)}\) is a canonical homomorphism defined in Proposition 6.5.
The paper closes with two appendices discussing generalities about overconvergence (Appendix A) and about analytic \(p\)-groups and their cohomology (Appendix B). Bloch-Kato exponential; p-adic Hodge theory; local fields Kedlaya, K., Liu, R.: Relative \(p\)-adic Hodge theory, II: imperfect period rings. arXiv:1602.06899 Galois cohomology, \(p\)-adic cohomology, crystalline cohomology Bloch-Kato exponential maps for local fields with imperfect residue fields | 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 study analytic families of germs of isolated complete-intersection singularieties (ICIS), or ICIS germs. Their main goal is to develop some new algebraic tools and a geometric point of view that enables them to describe some standard equisingularity conditions (Whitney's condition A and Thom's Condition \(A_f)\) in terms of suitable numerical invariants of isolated singularities.
The basic numerical invariants used in this context are certain homological Buchsbaum-Rim multiplicities for modules, the general theory of which is recalled and considerably advanced in Sections 1 and 2 of this paper, culminating in a generalization (from ideals to modules) of B. Teissier's so-called ``principle of specialization of integral dependence'' [cf. \textit{B. Teissier}, Astérisque 7-8, 285-362 (1974; Zbl 0295.14003)]. Originally, B. Teissier had established this principle as an equivalent condition for Whitney equisingularity, and formulated it in terms of the Jacobian ideal of an analytic family of hypersurface germs with isolated singularities.
In the sequel, the authors apply their generalization of this principle of specialization of integral dependence to the study of ICIS germs, i.e, to higher codimension. For this purpose, they introduce the more general concept of the Jacobian module of an ICIS germ, the integral closure of which is then used to describe equisingularity conditions based on the behavior of the limit tangent hyperplanes to the general member of the family.
After a thorough treatment of strict dependence (à la M. Lejeune-Jalabert and B. Teissier) in the more general context, which is carried out in Section 3 of the paper, the authors study Whitney's Equisingularity Condition A and establish a generalization of it to families of ICIS germs in Section 4.
The concluding Section 5 concerns Thom's Condition \(A_f\) for function germs \(f\) and culminates in a generalization of the Lê-Saito theorem to families of ICIS germs. This generalization is based upon a more recent construction by \textit{A. J. Parameswaran} [Compos. Math. 80, No. 3, 323-336 (1991; Zbl 0751.14005)] and yields, among other important results, a refinement of an earlier theorem of \textit{J. Briançon}, \textit{P. Maisonobe} and \textit{M. Merle} [Invent. Math. 117, 531-550 (1994; Zbl 0920.32010)].
Altogether, the methods and results developed in the paper under review must be seen as a major step toward the general study of equisingularity. hypersurface singularities; invariants of singularities; analytic families; Buchsbaum-Rim multiplicities; equisingularity; Jacobian module Terence Gaffney & Steven L. Kleiman, ``Specialization of integral dependence for modules'', Invent. Math.137 (1999) no. 3, p. 541-574 Equisingularity (topological and analytic), Singularities in algebraic geometry, Multiplicity theory and related topics Specialization of integral dependence for modules | 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 consists of four sections, and it is based on a series of lectures given at KAWA 2015, the sixth annual school and workshop in complex analysis, held in Pisa, Italy. It reviews some earlier results of the author, introduces a general proof for some known results in the literature, and introduces interesting questions on the dynamics of a family of rational functions on the Riemann sphere.
Let \(\mathbb{P}^1\) denote \(\mathbb{P}^1(\mathbb{C})\), let \(X\) be a complex manifold, let \(k\) be the function field \(\mathbb{C}(X)\), and let \(k(z)\) be the field of rational functions in \(z\) with coefficients in \(k\). If \(f(z)\in k(z)\) and \(t\in X\), we shall denote by \(f_t\) the function \(: \mathbb{P}^1 \to \mathbb{P}^1\) obtained by evaluating the coefficients of \(f(z)\) at \(t\). Given a rational function \(f(z)\in k(z)\), we also consider one of its critical points \(c\), and it can be viewed as a holomorphic function \(: X \to \mathbb{P}^1\). Given a pair \((f,c)\) and \(t\in X\), the behavior of \(f^n_t(c(t))\) as \(n\) varies is a main subject in complex dynamics. The classic example is the setting where \(X=\mathbb{P}^1\), \(f(z) = z^2+t\), and \(c(t) = 0\), and in this context, the Mandelbrot set is considered as a subset of \(X\). The boundary of the Mandelbrot set is an interesting subject in complex dynamics, and its generalization for a complex manifold \(X\) and a rational function \(f\) in \(k(z)\) is called the \textit{bifurcation locus} \(B(f,c)\). The above notions can be also generalized for an arbitrary point \(P\in \mathbb{P}^1(k)\) as follows. A rational point \(P\in \mathbb{P}^1(k)\) can be viewed as a function \(: X \to \mathbb{P}^1\), and given \(t\in X\), we may study the behavior of \(f^n_t( P(t) )\) as \(n\) varies.
Extending
\textit{C. T. McMullen}'s result [Complex dynamics and renormalization. Princeton, NJ: Univ. Press (1995; Zbl 0822.30002)]
\textit{R. Dujardin} and \textit{C. Favre} [Am. J. Math. 130, No. 4, 979--1032 (2008; Zbl 1246.37071)]
recently proved that the bifurcation locus \(B(f,c)\) is empty if and only if \((f,c)\) is preperiodic, as long as \(f\) is non-isotrivial and \(\mathrm{deg}(f)\geq 2\). In Section 3 the author introduces one of her earlier works that generalizes the Dujardin-Favre theorem for \((f,P)\) where \(P\) is an arbitrary point in \(\mathbb{P}^1(k)\), and the idea of the proof of this theorem is discussed at the end of the section. For the case when a bifurcation locus is nonempty, the author raises the following question in Section 2: \textit{For two critical points \(c_1\) and \(c_2\), if \(B(f,c_1)=B(f,c_2)\) is not empty, what can we conclude about the orbits of \(f^n_t(c_1(t))\) and \(f^n_t(c_2(t))\) as \(n\) varies?} She observes that there are no known examples where the bifurcation loci will coincide without some orbit relations, and also formulates the question in terms of \textit{bifurcation currents}.
In the final section, the author introduces an arithmetic dynamical question, which is similar to Manin-Mumford conjecture and the André-Oort conjecture. Let \(M_d\) be the moduli space of rational maps \(f : \mathbb{P}^1 \to \mathbb{P}^1\) of degree \(d\geq 2\). A map \(f : \mathbb{P}^1 \to \mathbb{P}^1\) of degree \(d\) is said to be \textit{postcritically finite} if each of its \(2d-2\) critical points has a finite forward orbit, and let PCF denote the subset of \(M_d\) consisting of the postcritically finite maps. In the Appendix, the author proves that the subset of PCF consisting of \textit{hyperbolic postcritically finite maps} is Zariski dense in \(M_d\), and hence, PCF is Zarski dense in \(M_d\). In Section 4, she asks, \textit{for which algebraic curves \(C\) in \(M_d\), the intersection of \(C\) and PCF is infinite.}
The author believes that the answer must be formulated in terms of some ``critical orbit relations'', and introduces some supportive results in the literature. In Subsection 4.3, she offers a general strategy that will prove the supportive results.
Inspired by the Masser-Zannier's theorem on the Legendre family of elliptic curves, the author introduces the following conjecture in Subsection 4.2:
Let \(f\in k(z)\) be non-isotrivial, and let \(P,Q\) be points in \(\mathbb{P}^1(k)\) such that \(f^n_{t_1}(P(t_1))\) and \(f^n_{t_2}(P(t_2))\) have infinite orbits for some \(t_1,t_2\in X\) as \(n\) varies. Let \(S_P\) be the subset of \(X\) consisting of \(t\) such that \(P(t)\) has finite forward orbit under \(f_t\), and let \(S_Q\) be similarly defined for \(Q\). Then, the following are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \(S_P\cap S_Q\) is infinite; \item[(2)] \(S_P=S_Q\); \item[(3)] there exist \(A,B\in \overline k(z)\) and an integer \(\ell \geq 1\) such that
\[
f^\ell \circ A = A\circ f^\ell,\;f^\ell \circ B = B\circ f^\ell, \quad\text{and } A(P) = B(Q).
\]
\end{itemize}} complex dynamics; bifurcation locus; postcritically finite; dynamical moduli spaces Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Families and moduli spaces in arithmetic and non-Archimedean dynamical systems, Elliptic curves over global fields, Elliptic curves Dynamical moduli spaces and elliptic 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 Let \(M\) be a compact cusped hyperbolic three-manifold with finite volume whose interior \(N\) is complete. If \(n\) denotes the number of cusps of \(M\), for every integer \(i\) with \(1 \leq i \leq n\), we can choose a meridian \(\mu_i\) and a longitude \(\lambda_i\) on each torus of \(\partial M\). By Thurston's hyperbolic Dehn surgery theorem, if \(\mathbf{k}\) consists of \(n\) couples \((p_i,q_i)\) of coprime integers, the manifold \(M_{\mathbf{k}}\) obtained from \(M\) by a hyperbolic Dehn filling of shape \(p_i / q_i\) at each cusp remains an hyperbolic manifold except for a finite number of \(\mathbf{k}\).
The hyperbolic structures on \(N\) associated to these Dehn fillings form a zero-dimensional subset of the deformation space \(\mathcal{D}(N)\), the latter consisting of deformed incomplete hyperbolic structures on \(N\). By results of Neumann-Zagier and Thurston, the space \(\mathcal{D}(N)\) is a complex variety. Furthermore, if \(l_i : \mathcal{D}(N) \rightarrow \mathbb{C}^{\times}\) and \(m_i : \mathcal{D}(N) \rightarrow \mathbb{C}^{\times}\) are the derivatives of the holonomy along \(\lambda_i\) and \(\mu_i\), and \(u_i, v_i\) are logarithms of \(m_i\) and \(l_i\) in the neighborhood of the initial hyperbolic structure, then \((u_1, \dots, u_n)\) are local holomorphic coordinates on \(\mathcal{D}(N)\). The functions \(u_i\) and \(v_i\) are related by the equations \(\frac{\partial \phi}{\partial u_i}=2v_i\), where \(\phi\) is the Neumann-Zagier potential function.
In this article, \(M\) is supposed to have only \textit{one} cusp; the irreducible component \(Y\) of \(\mathcal{D}(N)\) containing the initial hyperbolic structure of \(N\) is called the \textit{deformation curve} of \(N\). The authors give a geometric realization of \(\phi\) as follows: let \(H\) be the Heisenberg group and \(P=H(\mathbb{Z}) \setminus H(\mathbb{C})\). The space \(P\) is naturally a holomorphic principal \(\mathbb{C}^{\times}\)-bundle over \(\mathbb{C}^{\times} \times \mathbb{C}^{\times}\) an can be endowed with a holomorphic connection \(\theta\). The first result of the article is that the Neumann-Zagier potential allows to produce explicitly a flat section of the holomorphic line bundle with connection \((l,m^2)^{*}\{P, \theta \}\) near the origin of \(Y\).
In the second part of the article, the authors extend this flat section to the smooth locus \(X\) of the whole deformation curve \(Y\), replacing the Neumann-Zagier potential by the \(\mathrm{SL}(2, \mathbb{C})\) Chern-Simons invariant (the link between the two objects is due to Yoshida and Kirk-Klassen). As a corollary, they construct a unipotent variation of mixed Hodge structures on \(X\).
The bundle \((l,m^2)^{*}\{P, \theta \}\) on \(X\) can be interpreted in terms of Deligne cohomology: it is exactly the cup-product \(l \cup m^2\) in the Deligne cohomology group \(H^2_D (X,\mathbb{Z}(2))\), where \(l\) and \(m^2\) are considered as classes in \(H^1_D (X,\mathbb{Z}(1))\). hyperbolic three-manifolds; deformation of hyperbolic structures; Neumann-Zagier potential, Chern-Simons invariant; Deligne cohomology; variation of mixed Hodge structures. Eta-invariants, Chern-Simons invariants, Invariants of knots and \(3\)-manifolds, Knots and links in the 3-sphere, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Chern-Simons variations and Deligne cohomology | 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 \(R = \mathbb{Z}[x,y]\) denote the ring of polynomials in \(x\) and \(y\) with integer coefficients and fix a grading by some abelian group \(A\). The main result of this paper establishes the smoothness and irreducibility of the multigraded Hilbert scheme parametrizing those ideals of \(R\) with a given Hilbert function.
The paper's generality is remarkable on two fronts: one is that the polynomial ring only assumes integer coefficients, which implies that the theorem extends to schemes over any fixed scheme via change of base; the other is that the the group \(A\) is permitted to be any abelian group, even one with non-zero torsion. As is pointed out in the paper, the major restriction is the use of two variables, which is essential. Indeed it is known that in general multigraded Hilbert schemes may fail to even be connected and can be highly non-singular.
The proof is broken down into several steps. The fundamental idea is to connect any point on the Hilbert scheme to a distinguished point and show that the tangent space to any point along the connecting path has constant dimension. The paper is well written with thorough proofs. The authors do an excellent job of bringing the reader up to the current state of knowledge on relevant issues concerning the multigraded Hilbert scheme. Hilbert schemes; multigraded rings; combinatorial commutative algebra DOI: 10.1016/j.aim.2009.10.003 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Smooth and irreducible multigraded Hilbert 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 This is a carefully written paper on the geometry of spectral curves and all order dispersive integrable system. After a summary of algebraic geometry, the authors review in the reconstruction of an isospectral Lax system from its semiclassical spectral curve which is time-independent. The techniques for this reconstruction are closely related to those developed by Krichever to produce the algebro-geometric solutions of the Zakharov-Shabat hierarchies. The authors propose a definition for a Tau function and a spinor kernel closely related to Baker-Akhiezer functions, where times parametrize slow (of order \(1/N\)) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of \(1/N\), where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when \(N\) is large. The large \(N\) limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. The authors check that their conjectural Tau function satisfies the Hirota equations to the first two orders, and they conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. They analyze its consequences, namely the possibility of reconstructing order by order in \(1/N\) an isomonodromic problem given by a Lax pair, and the relation between correlators, the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry. topological recursion; tau function; Sato formula; Hirota equations; Whitham equations G. Borot and B. Eynard, Geometry of spectral curves and all order dispersive integrable system. SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), Article Id. 100. Relationships between algebraic curves and integrable systems, Theta functions and curves; Schottky problem, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Riemann surfaces, KdV equations (Korteweg-de Vries equations) Geometry of spectral curves and all order dispersive integrable system | 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 show first a vanishing theorem for families of linear series with base ideal being a fat points ideal. They then apply this result in order to give a partial proof of a conjecture raised by the reviewer, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.
One of the central problems in the theory of linear series is the study of linear systems of hypersurfaces in projective spaces with assigned base schemes. This problem is related to various other topics, for example to the polynomial interpolation, the Waring problem, the classification of defective higher secant varieties and to the problem of containment relations between ordinary and symbolic powers of ideals.
Given a finite number \(s\) of points \(P_1,\dots, P_s \in \mathbb{P}^n\) and positive integers \(m_1, \dots ,m_s\), let \({\mathcal{L}}={\mathcal{L}}_{n}(t; m_1,\dots, m_s)\) be the linear system of hypersurfaces of \(\mathbb{P}^n\) of degree \(t\) vanishing in the given set of points with prescribed multiplicities. One is interested in determining the dimension of \({\mathcal{L}}\). The virtual dimension of this space
\[
\mathrm{vdim}({\mathcal{L}}):=\binom{n+t}{n}-1-\sum{i=1}^s \binom{n+m_i-1}{n}
\]
arises by assuming that the conditions imposed by the underlying set of points are independent. The expected dimension is defined as
\[
\mathrm{edim}({\mathcal{L}}):=\text{max}\{\mathrm{vdim}({\mathcal{L}}),-1\}.
\]
If the conditions imposed by the assigned points are not linearly independent, the actual dimension of \({\mathcal{L}}\) is greater than the expected one: in that case we say that \({\mathcal{L}}\) is special.
A subscheme \(Z\) of \(\mathbb{P}^n\), defined by an ideal of the form
\[
I_Z={\mathbf m}_{P_1}^{m_1}\cap \cdots \cap {\mathbf m}_{P_s}^{m_s}
\]
where \({\mathbf m}_P\) denotes the maximal ideal of a point \(P \in \mathbb{P}^n\) is called a fat points scheme and the ideal \(I_Z\) is called a fat points ideal. It follows from the long cohomology sequence attached to the twisted structure sequence of \(Z\)
\[
0 \to \mathcal{I}_Z(t) \to \mathcal{O}_{\mathbb{P}^n}(t)\to \mathcal{O}_Z(t) \to 0
\]
that the system \(\mathcal{L}\) is non-special exactly when the cohomology group \(H^1(\mathbb{P}^n,\mathcal{I}_(t))\) vanishes.
The first main result of the paper the following vanishing theorem.
{Theorem A}. Let \(P_1,\dots, P_s\) be \(s\geq 4\) general planar points. Let \(m_1\geq m_2\geq \cdots \geq m_s\geq 1\) be fixed integers. If \(t \geq m_1+m_2\) and \(\mathrm{vdim}({\mathcal{L}}_{2}(t; m_1,\dots, m_s))\geq \frac{1}{2}(3m^2_4-7m_4+2)\) then
\[
h^1(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}\otimes {\mathbf m}_{P_1}^{m_1}\otimes \cdots \otimes {\mathbf m}_{P_s}^{m_s})=0
\]
that is the system \({\mathcal{L}}_{2}(t; m_1,\dots, m_s)\) is non-special.
Moving to the algebraic side, let \(\mathcal{I}\subset \mathbb{C}[\mathbb{P}^n]=\mathbb{C}[x_0, \dots, x_n]\) be a homogeneous ideal. The \(m\)th symbolic power \(\mathcal{I}^{(m)}\) of \(\mathcal{I}\) is defined as
\[
\mathcal{I}^{(m)}=\mathbb{C}[\mathbb{P}^n]\cap \left(\bigcap_{p\in \mathrm{Ass}(\mathcal{I})} \mathcal{I}^m \mathbb{C}[\mathbb{P}^n]_p\right)
\]
where the intersection is taken in the field of fractions of \(\mathbb{C}[\mathbb{P}^n]\). If \(\mathcal{I}\) is a fat points ideal then the symbolic power is simply
\[
\mathcal{I}^{(m)}=\bigcap_{i=1}^s {\mathbf m}_{P_i}^{m\cdot m_i}.
\]
There has been considerable interest in containment relations between usual and symbolic powers of homogeneous ideals over the last two decades. The most general results in this direction have been obtained with multiplier ideal techniques in characteristic zero by Ein, Lazarsfeld and Smith [\textit{L. Ein} et al., Invent. Math. 144, No. 2, 241--252 (2001; Zbl 1076.13501)] and using tight closures in positive characteristic by \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)]. Applying these results to a homogeneous ideal \(\mathcal{I}\) in the coordinate ring \(\mathbb{C}[\mathbb{P}^n]\) of the projective space we obtain the following containment statement: \(\mathcal{I}^{(nr)}\subset \mathcal{I}^r\) for all \(r\geq 0\).
There are examples showing that one cannot improve the power of the ideal \(\mathcal{I}\) on the right-hand side of the previous inequality. Nevertheless, it is natural to wonder to what extent this result can be improved; for example, under additional geometrical assumptions on the zero-locus of \(\mathcal{I}\). In particular, if \(\mathcal{I}\) is a fat points ideal, it is natural to wonder for which non-negative integers \(m\), \(r\) and \(j\) there is the containment
\[
\mathcal{I}^{(m)}\subset \mathcal{M}^j\mathcal{I}^r,
\]
where \(\mathcal{M}\) denotes the irrelevant ideal. Harbourne and Huneke suggested the following {
Conjecture} Let \(\mathcal{I}\) ne a fat points ideal in \(\mathbb{P}^n\). Then
\[
\mathcal{I}^{(nr)}\subset \mathcal{M}^{r(n-1)}\mathcal{I}^r,
\]
for all \(r\geq 1\).
This conjecture has been proved recently by \textit{B. Harbourne} and \textit{C. Huneke} [J. Ramanujan Math. Soc. 28A, 247--266 (2013; Zbl 1296.13018)] for general points in \(\mathbb{P}^2\) and by the first author for general points in \(\mathbb{P}^3\).
In this paper, the authors extend these results to a large family of fat points ideals in \(\mathbb{P}^2\). More specifically they show the following theorem.
{Theorem B}. Let \(\mathcal{I}={\mathbf m}_{P_1}^{m_1}\cap \cdots \cap {\mathbf m}_{P_s}^{m_s}\) be a fat points ideal supported on \(s\geq 9\) points in \(\mathbb{P}^2\). If one of the following conditions holds: {\parindent=6mm \begin{itemize} \item[(a)] at least \(s-1\) among the \(m_i\) are equal (almost homogeneous case); \item [(b)] \(m_1\geq \cdots \geq m_s\geq m_1/2\) (uniformly fat case);
\end{itemize}} then the conjecture above holds, that is, there is the containment
\[
\mathcal{I}^{(2r)}\subset \mathcal{M}^r\mathcal{I}^r,
\]
for all \(r\geq 1\). fat points; linear systems; polynomial interpolations; vanishing theorems Dumnicki, M.; Szemberg, T.; Tutaj-Gasińska, H., A vanishing theorem and symbolic powers of planar point ideals, LMS J. Comput. Math., 16, 373-387, (2013) Divisors, linear systems, invertible sheaves, Plane and space curves, Rational and ruled surfaces, Ideals and multiplicative ideal theory in commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials A vanishing theorem and symbolic powers of planar point ideals | 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 \({\phi}\) be a rational function of degree at least two defined over a number field \(k\). Let \({a \in \mathbb{P}^1(k)}\) and let \(K\) be a number field containing \(k\). We study the cardinality of the set of rational iterated preimages Preim\({(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a \text{ for some } N \geq 1\}}\). We prove two new results (Theorems 2 and 4) bounding \({|\mathrm {Preim}(\phi, a, K)|}\) as \({\phi}\) varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim\({(\phi, a, K)}\) and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics. arithmetic dynamics; iterated preimages; Runge's method; unit equations Dynamical systems over global ground fields, Global ground fields in algebraic geometry, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps Rational preimages in families of dynamical 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 The article provides a survey on the heat kernel from the arithmetic geometry point of view, explaining what the heat kernel is and how it is used in the Arakelov geometry. The paper starts with recalling the concepts of a differential form and a current, after that the Hodge \(*\)-operator and the Laplace operator \(\Delta\) on a Riemannian manifold \(X\) are introduced. The heat kernel \(P_{t}\) in the case of compact manifolds \(X\) is given as an infinite sum over expressions involving the eigenvalues and eigenfunctions of the Laplacian, and its standard properties are then explicitly formulated. As an application, it is shown that if \(Y\) is a closed submanifold with the harmonic projection \(\mu_{Y}:=\lim_{t\to\infty} P_{t}(\delta_{Y})\) of the current \(\delta_{Y},\) then \(G(Y):= G(\delta_{Y})\) may be obtained by \(G(Y) = \int_0^\infty (P_{t}(\delta_{Y})-\mu_{Y})\,dt\) where \(G\) is the Green operator. A short introduction is further given into Arakelov geometry in the form developed in [\textit{H.~Gillet} and \textit{C.~Soulé}, Publ. Math., Inst. Hautes Étud. Sci. 72, 93--174 (1990; Zbl 0741.14012)]. The author's own contribution to the theory is mentioned, namely, the study of the canonical Green's current given by \(g_{Y}:=4\pi\Lambda G(Y)\) in case of a Kähler manifold \(X.\) Some remarks are also given on how this current can be effectively computed for Grassmannians. Kähler manifold; Green current; Arakelov geometry Hein, G.: Green currents on Kähler manifolds. The ubiquitous heat kernel. Contemp. Math., 398, pp. 245--256. Amer. Math. Soc., Providence (2006) Heat and other parabolic equation methods for PDEs on manifolds, Kähler manifolds, Heat kernels in several complex variables, Arithmetic varieties and schemes; Arakelov theory; heights Green currents on Kähler 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 The Hilbert scheme of \(n\) points in \({\mathbb P}^2\) has a natural stratification by strata \(H(\phi)\) obtained from its Hilbert functions (or series) \(\phi\). The precise inclusion between the closures of the strata is in general unknown. The present paper gives necessary and sufficient conditions for the inclusion of \(H(\phi)\) in \(\overline {H(\psi)}\) where the Hilbert functions \(\phi\) and \( \psi\) are as close as possible and \(\phi \leq \psi\). This generalizes a result of \textit{F. Guerimand} [Thesis, University of Nice (2002)] which has the same result under an extra condition. In the case \(\phi\) is generic in some Brill-Noether locus, one may get the inclusion by using results of e.g. \textit{J. Brun} and \textit{A. Hirschowitz} [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 2, 171--200 (1987; Zbl 0637.14002)].
Parts of the paper are a little technical, but the main ideas of the authors are clear and nice. They show that the necessary and sufficient conditions above are equivalent to a strict inequality of the dimension of certain \(\text{Ext}^1\)-groups where the smallest one is the dimension of the tangent space of deformations at some ideal of a non-commutative algebra. It turns, however, out that one may avoid ``non-commutative deformations'' and instead look to the Hilbert-flag scheme (of ``commutative'' deformations), or the incidence correspondence, of pairs of graded quotients of a polynomial ring and one of its projections. In the present case one needs to handle quotients for which the depth is zero, thus treat a slightly more general case than considered in several papers the reviewer [e.g., J. Algebra 311, No. 2, 665--701 (2007; Zbl 1129.14009)]. postulation Hilbert scheme; incidence; stratification; deformation Denaeghel, K.; Den Bergh, M. Van: On incidence between strata of the Hilbert scheme of points on P2, Math. Z. 255, 897-922 (2007) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Deformations and infinitesimal methods in commutative ring theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On incidence between strata of the Hilbert scheme of points on \(\mathbb{P}^{2}\) | 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 author is concerned with an algebraic curve \(X \subset \mathbb{P}^N\) \((N\geq 3)\) over a finite field of characteristic \(p>0\) lying in an \(N\)-dimensional projective space \(\mathbb{P}^N\) which possesses an order-sequence in the sense of \textit{A. Garcia} and \textit{M. Homma} [in: Proc. Conf. Inst. Experimental Math., Essen 1992, 27-41 (1994; Zbl 0824.14019)] or of \textit{K.-O. Stöhr} and \textit{J. F. Voloch} [Proc. Lond. Math. Soc., III. Ser. 52, 1-19 (1986; Zbl 0593.14020)]. Considering such a curve \(X \subset \mathbb{P}^N\) over a finite field \(\mathbb{F}_{q'}\) with \(q'\) elements (\(q'\) is a power of \(p\)), it possesses the \(q'\)-Frobenius order-sequence in the sense of the papers cited above, too. Let \(x_0: x_1: x_2:\cdots: x_N\) be the coordinate functions, \(k(X)\) be the function-field of \(X \subset \mathbb{P}^N\), and \(\{D^{(r)}\); \(0\leq r\in \mathbb{Z}\}\) be the system of Hasse-Schmidt derivatives on \(k(X)\). Put \({\mathfrak x}= (x_0, x_1, x_2,\dots, x_N)\). Then each of both sequences which are denoted by \(0= \varepsilon_0< \varepsilon_1< \varepsilon_2<\dots< \varepsilon_N\) (order-sequence) and \(0= \nu_0< \nu_1<\dots< \nu_N\) (\(q'\)-Frobenius order-sequence), means the minimal sequence consisting of integers, in the lexicographic order, such that the \(N+1\) row-vectors \(D^{(\varepsilon_i)} \cdot {\mathfrak x}\) \((0\leq i\leq N)\) are linearly independent over \(k(X)\), the \(N+1\) row-vectors \({\mathfrak x}^{q'}\), \(D^{(\nu_i)} \cdot{\mathfrak x}\) \((0\leq i\leq N-1)\) are so, respectively. By proposition 2.1 in the paper by \textit{K.-O. Stöhr} and \textit{J. P. Voloch} cited above, for the relationship between both sequences, it is known that there exists an integer \(I\) depending on \(q'\) \((1\leq I\leq N)\) such that \(\nu_i= \varepsilon_i\) (whenever \(i<I\)), \(\varepsilon_{i+1}\) (whenever \(i\geq I\)). This value \(I\) is named Frobenius index. The author denotes by \(\iota(q';X)\) this value for the curve \(X \subset \mathbb{P}^N\) over \(\mathbb{F}_{q'}\).
Now, let an integer \(N\geq 3\) and an odd prime number \(p\) be arbitrarily given. Take arbitrarily an integer \(I\) \((1\leq I\leq N)\). Then the author gives an example of \(X \subset \mathbb{P}^N\) over \(\mathbb{F}_{q'}\) satisfying \(\iota (q';x)= I\), where \(q'\) is some power of \(p\). This example is obtained by a complete intersection in \(\mathbb{P}^N\) of \(N-I\) Fermat equations and \(I-1\) Artin-Schreier equations. Concerning this, the author notes that
(a): in case \(N=3\), an example of \(X \subset \mathbb{P}^3\) with ``\(\iota (q'X)= 1\) for some \(q'\)'' is given in the Garcia-Homma paper (see above) or by \textit{A. Garcia} and \textit{J. F. Voloch} [Bol. Soc. Bras. Mat., Nova Sér. 21, No. 2, 159-175 (1991; Zbl 0766.14021)] which is a complete intersection of Fermat equations. The author's example for \(\iota (q';X)= 1\) in \(N=3\) in the same one as (a).
(b): For any \(N\), an example of \(X \subset \mathbb{P}^N\) with ``\(\iota (q';X)= N\) for any \(q'\)'' which is an image of \(\mathbb{P}^1\) is given in the Garcia-Homma paper cited above.
(c): For any \(N\), an example of \(X \subset \mathbb{P}^N\) with ``\(\iota (q';X)= N-1\) for some \(q'\)'' which is an image of \(\mathbb{P}^1\) is given in the same paper by Garica and Hamma.
In the case of the author's example for \(\iota (q';X)= 1\), since it is seen that \(X\) is smooth, the author calculates the number of \(\mathbb{F}_{q'}\)-rational points on the curve \(X\), by applying the formula of theorem 1 in a paper by \textit{A. Hefez} and \textit{J. F. Voloch} [Arch. Math. 54, No. 3, 263-273 (1990; Zbl 0662.14016)].
Let \(q= p^e\), \(q'= q^2\), where \(e\) is a positive integer satisfying \(N\leq p^e\). Consider the curve \(X \subset \mathbb{P}^N\) over \(\mathbb{F}_{q'}\) of the author's example for \(\iota (q';X)= 1\). Then the genus of \(X\) equals \(1+ \frac 12 (q+1)^{N-1} [(N-1) q-2]\) and moreover \(\# X(\mathbb{F}_{q'})\) equals \((q+1)^{N-1} [q^2+ 1- (N-1)q]\). curves over finite fields; order-sequence; Frobenius index Finite ground fields in algebraic geometry, Arithmetic ground fields for curves, Enumerative problems (combinatorial problems) in algebraic geometry Frobenius indices of certain curves over finite fields | 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 aim of this paper is to present explicit closed algebraic formulas for Orlov-Scherbin \(n\)-point functions. More precisely, the authors give an explicit closed formula for \[H_{g,n}=\sum_{m_1\cdots m_n=1}^\infty h_{g,(m_1,\dots,m_n)}X_1^{m_1}\cdots X_n^{m_n},\] where \(h_{g,(m_1,\dots,m_n)}\) are weighted double Hurwitz numbers for each pair \((g, n)\). They derive a new explicit formula in terms of sums over graphs for the \(n\)-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). They use a change of variables deduced from the structure of the associated spectral curve, and obtain a formula that turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.
The paper is organized as follows. The first section is an introduction to the subject. In Section 2, the authors give some notations and recall the basic formalism of operators in the bosonic Fock space that they use throughout the paper. In Section 3 they compute \(H_{g,n}\) as a series in \(X_1,\dots,X_n\), which, in particular, leads to a formula giving each particular formal weighted double Hurwitz number \(h_{g,(m_1,\dots,m_n)}\) in a closed form. Strictly speaking, this section is not necessary for the rest of the paper, but it sets up the notation and clarifies the logic of the calculations. In Section 4 the authors derive an explicit closed formula for \(D_1\cdots D_nH_{g,n}\), where \(D_i=X_i\partial_{X_i}\). In Section 5 they prove the main result of the paper, which explicitly represents \(H_{g,n}\) for given \(g\) and \(n\) in a closed form. Section 6 deals with the slightly exceptional cases of \(n=1\) for any \(g\) and \((g, n)=(0, 2)\). Section 7 is devoted to some applications of main general formula obtained in this paper, thus deriving explicit expressions for \(H_{g,n}\) for small \(g\) and \(n\) in terms of small number of basic functions. Hurwitz numbers; KP tau functions; Fock space Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Coverings of curves, fundamental group, Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions Explicit closed algebraic formulas for Orlov-Scherbin \(n\)-point 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 authors study constructive embedded resolutions of irreducible quasi-ordinary singularities in \(\mathbb{C}^3\). A surface germ \((V,p)\subset (\mathbb{C}^3,0)\) is a quasi-ordinary singularity if it admits a finite projection \(\pi:(V,p) \to(\mathbb{C}^2,0)\) such that the discriminant locus (i.e., the plane curve over which \(\pi\) ramifies) has only normal crossings. If \(f\) is such a singularity and is irreducible, it admits a parametrization (analogous to the Puiseux series of an irreducible algebroid plane curve) from which certain pairs of numbers, called the characteristic pairs, can be extracted. They are important in the study of the germ. For instance, explicit resolutions of such a singularity have been studied by \textit{J. Lipman} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 2, 161-172 (1983; Zbl 0521.14014)]. In this process, which does not lead to embedded resolutions, an important role is played by the characteristic pairs. Recall that, informally, ``embedded resolution'' means a process where along which the singular variety \(V\) one transforms the ambient space where it is defined, so that eventually the strict transform of \(V\) is non-singular and its union with the exceptional divisor is locally defined by simple, ``nice'' equations. Recently, several (closely related) methods to constructively (or canonically) obtain embedded resolutions of general singularities have been proposed. That is, procedures which involve a finite sequence of blowing-ups and which tell us, at each stage of the resolution process, how to choose the center of the transformation.
More precisely, in this note the authors explicitly study what results from the application of the general canonical process devised by \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207-302 (1997; Zbl 0896.14006)] to an irreducible quasi-ordinary singularity. It turns out that the process depends on the (suitably normalized) characteristic pairs of the singularity only. The description of the algorithm given in the paper is very explicit. The authors apply it to a non-trivial example, and they affirm that the method can be actually implemented by a computer.
The authors plan to apply similar techniques to the problem of simultaneous desingularization of a family of quasi-ordinary singularities in a future work. canonical resolution; quasi-ordinary singularities; characteristic pairs; embedded resolutions C. Ban and L. J. McEwan, Canonical resolution of a quasi-ordinary surface singularity , Canad. J. Math. 52 (2000), 1149--1163. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex surface and hypersurface singularities Canonical resolution of a quasi-ordinary surface singularity | 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 considers a complete discrete valuation ring \(R\) with a maximal ideal \(\mathfrak m\) generated by a prime number \(p\). Suppose that \(k=R/\mathfrak m\) is an algebraically closed field. A map \(\delta: R\rightarrow R\) defined by \(\delta x= (\phi (x) - x^p ) / p\) where \(\phi : R\rightarrow R\) is a map obtained by lifting of the Frobenius automorphism on a field \(k\), is called a \(\delta\)-function. In the paper it is shown that if one replaces regular functions by \(\delta\)-functions than one can obtain a new nontrivial semi-invariant for a remarkable class of regular self maps of the projective plane. Here a complicated algebraic technique of a discrete dynamical systems and spaces of \(p\)-jet is used. flat dynamical system; complete discrete valuation ring; scheme; \(p\)-jet space Local ground fields in algebraic geometry, Algebraic number theory: local fields, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Differential orbit spaces of discrete dynamical 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 numerical criteria for the smoothness of the degeneracy locus of a general map between two vector bundles. Precisely we prove the following theorem:
Let \(E^ m\supsetneqq E_{\ell}\supset \cdot \cdot \cdot \supset E_ 1\) and \(F^ n\supsetneqq F_{\ell}\supset \cdot \cdot \cdot \supset F_ 1\) be filtrations of vector bundles over a smooth variety \(X\) with \(\text{rk}(E_ i)=m_ i\) and \(\text{rk}(F_ i)=n_ i\), and let \(\alpha_ i=n_ i- m_ i\), \(\alpha =n-m\). Assume \(\alpha_ i\geq \alpha \geq 0\) for all \(i\) and the subbundle \({\mathcal B}\subset E^{\vee}\otimes F\) defined by \({\mathcal B}(x)=\{\phi_ x\mid \phi_ x(E_ i(x))\subset F_ i(x)\) for all \(i\}\) is generated by global sections. Let \(Y\) be the degeneracy locus of a general \(\phi\in \Hom(E,F)\). Then we have
(i) \(\text{cod}(Y)\geq \alpha +1;\)
(ii) \(\text{cod}_ Y(\text{Sing}(Y))\geq \min_{j}\{2\alpha_ j-2\alpha +1,\alpha_ j+2,\alpha +3\}.\)
As an application, we are able to give an algorithm to analyze the case in which \(E^ r=\oplus {\mathcal O}(-a_ i) \) and \(F^{r+1}=\oplus \Omega^{pj}(-k_ j) \) in terms of two sequences of integers. This algorithm leads to a classification of nonsingular Buchsbaum subvarieties of codimension 2 in \(\mathbb P^ n.\)
[See also the author's paper in J. Differ. Geom. 31, No. 2, 323--341 (1990; Zbl 0663.14034)]. smoothness of the degeneracy locus of a general map between two vector bundles; nonsingular Buchsbaum subvarieties of codimension 2 Chang M.-C.: A filtered Bertini-type theorem. J. Reine Angew. Math. 397, 214--219 (1989) Rational and birational maps, Low codimension problems in algebraic geometry A filtered Bertini-type theorem | 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 proves equidistribution of periodic points under regular automorphisms on \(\mathbb A^n\). This generalizes earlier results for Hénon maps on \(\mathbb A^n\), as well as providing a non-archimedean counterpart to [\textit{T.-C. Dinh} and \textit{N. Sibony}, ``Density of Positive Currents and Dynamics of Hénon type automorphism of \(\mathbb C^k\)'', Preprint, \url{arXiv:1203.5810}]. More precisely, let \(K\) be a number field, \(v\) a place, \(f:\mathbb A^n \to \mathbb A^n\) an affine regular automorphism, and \(x_m \in \mathbb A^n(\overline K)\) a sequence of \(f\)-periodic points that are generic (i.e. any infinite subsequence is Zariski-dense). Letting \(\mu_m\) be the discrete probability measure supported on the Galois orbit of \(x_m\), Theorem A shows that \(\mu_m\) converges weakly to an \(f\)-invariant probability measure on the \(v\)-adic Berkovich projective space.
The author actually introduces a generalization of affine regular automorphisms: a commuting pair \(S = \{f_1,f_2\}\) of polynomial self-maps on \(\mathbb A^n\) is \textit{strongly regular} if (1) \(f_i\)'s do not have a common indeterminacy point in \(\mathbb P^n\), (2) iterates \(f_i^m\) and \(f_1\circ f_2\) satisfy various degree conditions, and (3) \(f_i\)'s satisfy the height growth condition \(h(f_i(P))\gg h(P)\). The technical heart of the paper is Theorem 6.5: if \(S = \{f_1,f_2\}\) is strongly regular and in addition \(\deg f_1 = \deg f_2\), then a sequence of adelic metrics converges uniformly, say to \(\|\cdot \|_S\). To construct this metric, the author shows that \(\{f_1^m, f_2^m\}\) is also strongly regular (Corollary 6.4), so \((f_1^m, f_2^m)\) defines a morphism \(\phi_m: \mathbb P^n \rightarrow \mathbb P^{2n}\). Then the pullback via \(\phi_m\) of the \(v\)-adic sup norm on the degree-one line bundle is adelic for each \(m\). By extending the theory of Green functions and good reduction developed for affine regular automorphisms [\textit{S. Kawaguchi}, Algebra Number Theory 7, No. 5, 1225--1252 (2013; Zbl 1302.37067)] to the case of strongly regular pairs, the uniform convergence is proved.
As a result of Theorem 6.5, \textit{X. Yuan}'s theory [Invent. Math. 173, No. 3, 603--649 (2008; Zbl 1146.14016)] immediately implies that given a strongly regular pair \(S = \{f_1,f_2\}\) such that \(\deg f_1 = \deg f_2\) and a generic sequence \(x_m\) which is small with respect to \(\|\cdot \|_S\), the Galois orbits of \(x_m\)'s are equidistributed (Theorem B). As for an affine regular automorphism \(f\), Proposition 7.1 shows that there exist \(l_1\) and \(l_2\) such that \(\deg f^{l_1} = \deg f^{-l_2}\). The author also proves that the \(f\)-periodic points over \(\overline K\) are Zariski-dense (Theorem C), by adapting the proof for polarizable morphisms [\textit{N. Fakhruddin}, J. Ramanujan Math. Soc. 18, No. 2, 109--122 (2003; Zbl 1053.14025)]. From these results, Theorem A follows from Theorem B. equidistribution; affine regular automorphism; small point; periodic point; canonical height; Green function; adelic metric Fornæss, J.-E.: Dynamics in several complex variables. CBMS Regional Conference Series in Mathematics, vol. 87. American Mathematical Society, Providence, RI (1996) Heights, Applications to coding theory and cryptography of arithmetic geometry, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems The equidistribution of small points for strongly regular pairs of polynomial maps | 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 and the preceding one are separate, but it is convenient to consider them together because they more or less follow the same method in the proofs, so repetitions can be avoided, leaving the reviewer more space for other comments. The method in question uses the slope properties of a locally free sheaf (or vector bundle), semistability and the so-called Harder-Narasimhan filtration of the sheaf.
So, let us consider the paper above (Zbl 1070.13008). The introduction states that: ``Let \((R,{\mathfrak m})\) denote a local Noetherian ring or an \(\mathbb N\)-graded algebra of dimension \(d\) of positive characteristic \(p\). Let \(I\) denote an \({\mathfrak m}\)-primary ideal, and set \(I^{[q]}=(f^q :f \in I)\) for a prime power \(q = p^e\). Then the Hilbert-Kunz function of \(I\) is given by \(e\mapsto \lambda(R/I^{[p^e]}),\) where \(\lambda\) denotes the length. The Hilbert-Kunz multiplicity of \(I\) is defined as the limit \(e_{HK}(I)= \lim_{e\to\infty}\lambda(R/I^{[p^e]})/p^{ed}.\) This limit exists as a positive real number, as shown by Monsky. It is an open question whether this number is always rational.
The Hilbert-Kunz multiplicity is related to the theory of tight closure. Recall that the tight closure of an ideal \(I\) in a Noetherian ring \(R\) of characteristic \(p\) is by definition the ideal
\[
I^*=\{f\in R :\exists c \text{ not in any minimal prime}: cf^q\in I^{[q]}\text{ for almost all }q=p^e\}
\]
[\textit{C. Huneke}, ``Tight closure and its applications'', Reg. Conf. Ser. Math. 88 (1996; Zbl 0930.13004)].''
For an analytically unramified and formally equidimensional local ring \(R\) the equation \(e_{HK}(I) = e_{HK}(J)\) holds if and only if \(I^* = J^*\) holds true for ideals \(I\subseteq J\). Hence \(f \in I^*\) if and only if \(e_{HK}(I) = e_{HK}((I,f))\). This is the Hilbert-Kunz criterion for tight closure in positive characteristic.
The aim of this paper is to give a characteristic zero version of this relationship between Hilbert-Kunz multiplicity and tight closure for \(R_+\)-primary homogeneous ideals in a normal two-dimensional graded domain \(R\). There are several notions for tight closure in characteristic zero, defined either by reduction to positive characteristic or directly. We will work with the notion of solid closure [\textit{M. Hochster}, in: Commutative Algebra; Szyzygies, multiplicities, and birational algebra: AMS-IMS-SIAM Summer Res. Conf., Contemp. Math. 159, 103--172 (1994; Zbl 0812.13006)].''
The solid closure of an ideal coincides with tight closure in positive characteristic and gives a satisfactory notion for characteristic zero in dimension two.
Let \(I^*\) denote the solid closure of an ideal \(I\). The main result of the paper is the following:
Theorem: Let \(K\) denote an algebraically closed field. Let \(R\) denote a standard-graded two-dimensional normal \(K\)-domain. Let \(I\) be a homogeneous \(R_+\)-primary ideal and let \(f\) denote a homogeneous element. Then \(f\in I^*\) if and only if \(e_{HK}(I)=e_{HK}((I,f))\).
As mentioned above, the proof uses the slope properties of a locally free sheaf.
The paper under review (Zbl 1070.13009) deals with the computational aspect of the tight closure of an ideal, using the new approach of the above-said slope properties. There are different approaches to the problem of computing the tight closure due to K. E. Smith, M. Katzman and S. Sullivant.
To be more precise, drawing again from the introduction: ``We want to attack the problem of computing the tight closure of an ideal from another point of view: using the slope criteria for vector bundles. This rests upon the geometric interpretation of tight closure via vector bundles and projective bundles which we have developed in previous papers. This paper will emphasize the computational usefulness of this approach.
The main object to consider in this approach is the sheaf of relations of total degree \(m\) for homogeneous ideal generators \(f_1,\dots,f_n\). This is a locally free sheaf \({\mathcal R}(m)\) on the smooth projective curve \(Y= \text{Proj\,}R\). The slope properties of this sheaf are crucial for the underlying tight closure problem. So we may forget the definition of tight closure and struggle instead with the notions of slope, minimal and maximal slope, semistability and the Harder-Narasimhan filtration of this sheaf of relations. This is still a difficult task, however we can use many more tools from algebraic geometry to attack the tight closure problem.'' tight closure of an ideal; solid closure of an ideal; Hilbert-Kunz multiplicity Brenner, H, Computing the tight closure in dimension two, Math. Comput., 74, 1495-1518, (2005) Integral closure of commutative rings and ideals, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Vector bundles on curves and their moduli Computing the tight closure in dimension two | 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 the article under review, the author studies algebraic dynamical system on a smooth projective surface defined over a number field \(K\), generalizing the results of \textit{J. H. Silverman} [Invent. Math. 105, No. 2, 347--373 (1991; Zbl 0754.14023)] for \(K3\) surfaces. Let \(X\) be such a surface and let \(f:X\rightarrow X\) be an automorphism. Any algebraic point \(x\in X(\overline K)\) is said to be \(f\)-periodic if \(f^n(x)=x\) for some positive integer \(n\). Similarly, an integral curve \(C\) in \(X\) is said to be \(f\)-periodic if \(f^n(C)=C\) for some positive integer \(n\).
In the first part of the article, the author establishes the finiteness of \(f\)-periodic curve in \(X\), under the condition that the topological entropy of \(f\), defined as the spectral radius of \(f_{\mathbb C }^*\) acting on \(H^{1,1}(X_{\mathbb C},\mathbb R)\), is positive. The positivity of topological entropy allows to construct a nef and big divisor \(D\) such that any integral curve \(C\) in \(X\) is \(f\)-periodic if and only if \(([C],D)=0\), which implies the finiteness result.
In the second part of the article, always under the positive entropy hypothesis, the author constructs a canonical height function \(\widehat h_D\) which adapts to the Weil's height machine with respect to the divisor \(D\) above. Furthermore, for any algebraic point \(x\in X(\overline K)\), \(\widehat h(x)=0\) if and only if \(x\) is \(f\)-periodic or is contained in an \(f\)-periodic curve. Moreover, if we denote by \(E\) the union of all \(f\)-periodic curves, then \(\widehat{h}\) satisfies the Northcott finiteness property on \(X\setminus E\). The author then uses this canonical height to study the arithmetic properties of dynamical system \(f\). He proves that, for any Weil height associated to an ample divisor, the set \(\{f\text{-periodic points in }(X\setminus E)(\overline K)\}\) is of bounded height. He also estimates the density of non-periodic points. dynamical system; projective surface; canonical height; periodic points S. Kawaguchi, \textit{Projective surface automorphisms of positive entropy from an arithmetic viewpoint}, Am. J. Math. \textbf{130}(2008), no. 1, 159-186. Arithmetic varieties and schemes; Arakelov theory; heights, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Varieties over global fields, Automorphisms of surfaces and higher-dimensional varieties Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint | 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 \(X_1,\dots ,X_n\) be indeterminates over \(\mathbb{Q}\) and let \(X:=(X_1,\ldots , X_n\)). Let \( F_1,\ldots ,F_p\) be a regular sequence of polynomials in \( \mathbb{Q}[X]\) of degree at most \( d\) such that for each \( 1\leq k \leq p\) the ideal \( (F_1,\ldots , F_k\)) is radical. Suppose that the variables \( X_1,\ldots ,X_n\) are in generic position with respect to \( F_1,\ldots ,F_p\). Further, suppose that the polynomials are given by an essentially division-free circuit \( \beta \) in \( \mathbb{Q}[X]\) of size \( L\) and non-scalar depth \(\ell\).
We present a family of algorithms \(\Pi_i\) and invariants \(\delta_i\) of \(F_i,\dots,F_p, 1 \leq i \leq n-p\), such that \(\Pi_i\) produces on input \(\beta\) a smooth algebraic sample point for each connected component of \(\{x \in \mathbb{R}^n \mid F_1(x) = \ldots = F_p(x) = 0\}\) where the Jacobian of \(F_1 = 0,\dots,F_p = 0\) has generally rank \(p\).
The sequential complexity of \(\Pi_i\) is of order \(L(nd)^{O(n)}(\min\{(nd)^{cn},\delta_i\})^2\) and its non-scalar parallel complexity is the order of \(O(n(\ell+\log nd)\log \delta_i)\). Here \(c>0\) is a suitable universal constant. Thus, the complexity of \(\Pi_i\) meets the already known worst case bounds. The particular feature of \(\Pi_i\) is its pseudo-polynomial and intrinsic complexity character and this entails the best runtime behavior one can hope for. The algorithm \(\Pi_i\) works in the non-uniform deterministic as well as in the uniform probabilistic complexity model. We also ehibit a worst case estimate of order \((n^n d)^{O(n)}\) for the invariant \(\delta_i\). The reader may notice that this bound overestimates teh extrinsic complexity of the \(\Pi_i\), which is bounded by \((nd)^{O(n)}\). real polynomial equation solving; intrinsic complexity; singularities; polar; copolar and bipolar varieties; degree of variety Bank, Bernd; Giusti, Marc; Heintz, Joos, Point searching in real singular complete intersection varieties - algorithms of intrinsic complexity, Math. comp., 83, 286, 873-897, (2014) Symbolic computation and algebraic computation, Singularities in algebraic geometry, Real algebraic sets, Deformations of singularities, Parallel algorithms in computer science Point searching in real singularcomplete intersection varieties: algorithms of intrinsic complexity | 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 formal scheme \(\mathfrak{X}\) of finite type over a complete rank-one valuation ring, we construct a specialization morphism
\[
\pi^{\text{dJ}}_1(\mathfrak{X}_\eta) \to \pi^{\text{proét}}_1(\mathfrak{X}_k)
\] from the de Jong fundamental group of the rigid generic fiber to the Bhatt-Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of \(\mathfrak{X}\) and normalizations of the irreducible components of \(\mathfrak{X}_k\), and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent. rigid spaces; de Jong fundamental group; pro-étale fundamental group; specialization; fundamental group; geometric covering Rigid analytic geometry, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry Specialization for the pro-étale 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 In this work, \(\tau\)-functions of the Kadomtsev-Petviashvili hierarchy are studied in terms of abelian group actions on finite dimensional Grassmannians. These functions are viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal, and Wilson. A determinantal formula of Gekhtman and Kasman involving exponentials of finite-dimensional matrices is shown to follow naturally from such reductions. All reduced flows of exponential type generated by matrices with arbitrary nondegenerate Jordan forms are derived, both in the Grassmannian setting and within the fermionic operator formalism. A slightly more general determinantal formula involving resolvents of the matrices generating the flow, valid on the big cell of the Grassmannian, is also derived. An explicit expression is deduced for the Plücker coordinates appearing as coefficients in the Schur function expansion of the \(\tau\)-function. finite Grassmannians; fermionic operators; Shur function Balogh, F.; Fonseca, T.; Harnad, J., Finite dimensional KP tau functions. I. finite grassmanians, J. Math. Phys., 55, 8, 083517, (2014) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Grassmannians, Schubert varieties, flag manifolds Finite dimensional Kadomtsev-Petviashvili {\(\tau\)}-functions. I: Finite Grassmannians | 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,x) be a germ of an isolated singularity of an n-dimensional analytic space. To investigate a normal isolated singularity (X,x) K. Watanabe introduced pluri-genera \(\{\delta_ m(X,x)\}_{m\in {\mathbb{N}}}\). For a normal isolated Gorenstein singularity (X,x), it is known that either \(\delta_ m(X,x)=0\) for any m, \(\delta_ m(X,x)=1\) for any m or \(\delta_ m(X,x)\) grows in order n as a function in m.
In this article, it is shown that for a normal isolated Gorenstein singularity (X,x), \(\delta_ m(X,x)\leq 1\) holds for every \(m\in {\mathbb{N}}\) if and only if \(H^ i(\tilde X,{\mathcal O}_{\tilde X})\cong H^ i(E,{\mathcal O}_ E)\) for any \(i>0\), where \(f:\tilde X\to X\) is a resolution of the singularity (X,x) with \(E=f^{-1}(x)_{red}\) simple normal crossings. The singularity with the second property is called a Du Bois singularity (Steenbrink). - Let E be as above and decompose it into irreducible components \(E_ i (i=1,2,...,r)\). Then it is also shown that a normal isolated Gorenstein singularity (X,x) is Du Bois if and only if the canonical divisor \(K_{\tilde X}=\sum^{r}_{i=1}m_ iE_ i\) satisfies \(m_ i\geq -1\) for every i. More precisely, \(''\delta_ m(X,x)=0\) for every m'' iff \(m_ i\geq 0\) for every i, and \(''\delta_ m(X,x)=1\) for every m'' iff \(m_ i\geq -1\) for every i and \(m_ i=-1\) for some i. In the former case, (X,x) is rational. We call (X,x) purely elliptic in the later case. - We classify the set of n-dimensional purely elliptic singularities into n-types by the Hodge structure of the exceptional divisors, and consider the configuration of the exceptional divisor of a certain resolution of a singularity of each type. Next, we construct purely elliptic singularities of each type of any dimension \(n\geq 2\), by means of blowing down. germ of an isolated singularity; Du Bois singularity; n-dimensional purely elliptic singularities; Hodge structure of the exceptional divisors S. Ishii, On Isolated Gorenstein Singularities. Math. Ann.270, 541--554 (1985). Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects) On isolated Gorenstein 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 The paper by establishes utilities of numerical semigroups in studying stratifications of \(\mathcal{M}_{g, 1}\). For a genus \(g\) numerical semigroup \(S\), by relating the effective weight of \(S\) and the codimension of some strata of \(\mathcal{M}_{g, 1}\), the paper reports strong results in concern with components of \(\mathcal{M}^S_{g, 1}\).
In section \(2\), Pflueger studies the cases where \(\mathrm{codim} \mathcal{M}^S_{g,1}\) is known, with insistence on clear proofs in the case of numerical semigroup with two generators. After reviewing the works of Eisenbud-Harris, he relates, through an inequality, the effective weight of a numerical semigroup of genus \(g\) to the Deligne bound. He, moreover, determines the possibilities for the numerical semigroups \(S\) where this inequality turns to be an equality. If \(S\) is a numerical semigroup with two number of generators or if \(S=N_{d, \delta}\), where \(S=N_{d, \delta}\) is an specific type of numerical semigroups, then it is proved in Propositions \((2.14)\) and \((2.18)\) that \(\mathcal{M}^S_{g, 1}\) is irreducible of codimension equal to the effective weight of \(S\). He finishes section two by a nice example, and indeed the smallest genus, of a numerical semigroup for which \(\mathrm{codim} \mathcal{M}^S_{g, 1}\) differs from the effective weight of \(S\).
The main result of section \(3\) is a characteristic free proof of the Eisenbud-Harris' ``regeneration theorem''. In order to do this, he distinguishes an open subset of the scheme of limit linear series consisting of the so called ``refined limit linear series''.
In section \(4\), the author refines his study once more by introducing the effective subsequences of a numerical semigroup of genus \(g\). If \(T\) is an effective subsequence of a numerical semigroup \(S\) of genus \(g\), then he obtains an equivalent statement for existence of \textit{effectively proper} points for \(\mathcal{M}^S_{g, 1}\) in terms of the existence of \textit{dimensionally proper} points for \(\tilde{\mathcal{G}}^r_{g, d}(d-T)\).
Using the machinery produced in section \((4)\), the author proves that, for \(S\) a numerical semigroup of genus \(g\), if \(\mathcal{M}^S_{g,1}\) is nonempty and if \(X\) is an irreducible component of it, then
\[
\dim X \geq \dim \mathcal{M}_{g, 1}-\mathrm{ewt}(S),
\]
where \(\mathrm{ewt}(S)\) denotes the effective weight of \(S\). This is the first main result of this paper.
The second main result of the paper states that if \(S\) is a genus \(g\) numerical semigroup with \(\mathrm{ewt}(S)\geq g-2\), then \(\mathcal{M}^S_{g, 1}\) has an effectively proper component; furthermore, if \(\mathrm{char} k=0\), then the assertion holds for all numerical semigroups with \(\mathrm{ewt}(S)\geq g-1\). This is proved in section \(6\) by using the machinery produced in previous sections alongside an analogue of Eisenbud-Harris' Proposition \(5.2\) in secant loci; tangent cone; very ample line bundle Curves in algebraic geometry, Special divisors on curves (gonality, Brill-Noether theory) On nonprimitive Weierstrass points | 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 \(f : S \rightarrow T\) be a relatively minimal projective morphism between a smooth projective surface \(S\) and a smooth curve \(T\) such that the genus \(g\) of the general fiber of \(f\) is positive. The slope of \(f\), \(\lambda_f\), is defined to be the ratio \(K_{S/T}^2/\deg (f_{\ast}K_{S/T})\), where \(K_{S/T}\) is the relative canonical sheaf of \(f\). Part of its significance is that if \(T=\mathbb{P}^1\), \(g \geq 2\), and \(f\) is semistable, then \(\lambda_f\) is related to the the number \(\sigma\) of the singular fibers of \(f\) by the relation \(K_{S/T}^2<(\sigma-2)(2g-2)\).
In this paper the authors study the slope of a morphism \(f : S \rightarrow T\) when \(S\) is a rational surface and \(T=\mathbb{P}^1\). In particular they obtain lower bounds for it under restrictions on the genus \(g\) and the gonality of the general fiber of \(f\). Their method is the following. It is known that in the case of the paper, \(\deg (f_{\ast}K_{S/T})=g\) and hence in order to bound the slope it suffices to bound \(K_{S/T}^2\). In order to do that the authors show that under certain restrictions on the genus \(g\) and the gonality of the general fiber \(C\) of \(f\), \(C+nK_S\) is effective for \(n=2,3\). Then there is a Zariski decomposition \(C+nK_S=P+N\) where \(P\) is a nef divisor and \(N\) an effective divisor. They calculate explicitely the divisors \(P\) and \(N\) and then the inequalities for \(K_{X/T}^2\) are obtained from the fact that \(P^2\geq 0\). fibration; minimal; slope Fibrations, degenerations in algebraic geometry, Rational and ruled surfaces On the slope of relatively minimal fibrations on rational complex 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 Let \(X\) be a smooth projective surface and \(\mathcal{F}\) a rank one foliation with canonical singularities. This paper studies various results for the adjoint divisor \(K_{\mathcal{F}}+\epsilon K_X\) with \(0<\epsilon \ll 1\).
The main results are the following:
\begin{itemize}
\item (MMP) If \(0<\epsilon \leq 1/5\), then there exists a minimal model or a Mori fiber space for \(K_{\mathcal{F}}+\epsilon K_X\);
\item (Existence of canonical model) If \(0<\epsilon \leq 1/5\) and \(K_{\mathcal{F}}\) is big, then there exists a canonical model for \(K_{\mathcal{F}}+\epsilon K_X\);
\item (Boundedness) There exists a constant \(\tau>0\) such that if \(0<\epsilon <\tau\), then the set of \(\epsilon\)-adjoint canonical model \((X, \mathcal{F})\) with \((K_{\mathcal{F}}+\epsilon K_X)^2\leq C\) form a bounded family for any \(C>0\).
\item (Effective birationality) There exists a constant \(\tau>0\) such that if \(0<\epsilon <\tau\), there exists a positive integer \(M=M(\epsilon)\) such that \(|M(K_{\mathcal{F}}+\epsilon K_X)|\) defines a birational map as long as \(K_{\mathcal{F}}\) is big. As corollaries, there is also a uniform positive lower bound for the volume \((K_{\mathcal{F}}+\epsilon K_X)^2\) and an linear upper bound for the birational automorphism group in terms of the volume.
\end{itemize} surface; foliation; minimal model program Divisors, linear systems, invertible sheaves, Minimal model program (Mori theory, extremal rays) Effective generation for foliated surfaces: results and 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 Let \(X\) be a smooth complex projective surface, \(H\) an ample class on \(X\) and \(c\in K(X)_{\mathrm{num}}\). Denote by \(M^\mu(c)\) the corresponding moduli space of slope stable bundles on \(X\). This moduli space can be compactified by taking its closure in the Gieseker-Maruyama moduli space \(M^{\mathrm{ss}}(c)\). The space \(M^\mu(c)\) can also be regarded as a moduli space of bundles carrying a Hermitian-Yang-Mills metric and, as such, has a differential-geometric compactification \(N(c)\), called the Uhlenbeck-Donaldson compactification. The space \(N(c)\) can be given the structure of a scheme and there is a morphism \(M^{\mathrm{ss}}(c)\to N(c)\) which restricts to an isomorphism on \(M^\mu(c)\).
In this paper, the authors consider the corresponding problem for framed sheaves, that is for pairs \((\mathcal{E},\phi:\mathcal{E}\to\mathcal{F})\), where \(\mathcal{F}\) is a fixed coherent sheaf on \(X\). There is a notion of slope stability for framed sheaves depending on a parameter \(\delta\), which is a polynomial with rational coefficients. This gives rise to a GIT moduli space \(M=M^{\mathrm{ss}}(c,\mathcal{F})\). The main result of the paper is the construction of an analogue \(M^{\mu ss}\) of the Uhlenbeck-Donaldson compactification under the condition that \(\mathcal{F}\) is an \(\mathcal{O}(D)\)-module for some big and nef curve \(D\) on \(X\). The authors prove also that there is a projective morphism \(M\to M^{\mu \mathrm{ss}}\) which is birational on the components of \(M\) which contain locally free framed sheaves.
The first step in the proof is, as expected, a boundedness result for framed sheaves; this is not restricted to surfaces. The moduli space \(M^{\mu \mathrm{ss}}\) is then defined as a sort of quotient of a locally closed subset of a suitable Quot scheme. The space \(M^{\mu \mathrm{ss}}\) has a natural set-theoretic stratification which allows a comparison with the moduli spaces of framed ideal instantons. framed sheaves; moduli spaces; Uhlenbeck-Donaldson compactification; stable pairs; instantons Bruzzo, U.; Markushevich, D.; Tikhomirov, A., Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces, Math. Z., 275, 1073-1093, (2013) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Uhlenbeck-Donaldson compactification for framed sheaves on 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 Let\(\mathcal{\;H}\) be a complex Hilbert space and \(\mathcal{B(H)}\) the algebra of all bounded linear operators on \(\mathcal{H}\). For \(A\in \mathcal{B(H)}\), the numerical range of \(A\) is defined by \(W(A)=\left\{ \left\langle Ax,x\right\rangle :x\in \mathcal{H}\text{ and }\left\| x\right\| =1\right\} \). The diameter of \(W(A)\) is defined as
\[
d(W(A))=\sup \left\{ \left| \lambda -\mu \right| :\lambda ,\mu \in W(A)\right\} .
\]
We say that a map \(\Psi :\mathcal{B(H)\rightarrow B(H)}\) preserves the diameter if
\[
d(\Psi (W(A)))=d(W(A))
\]
for all \(A\in \mathcal{B(H)}\). In the paper under review, authors study linear maps on \(\mathcal{B(H)}\) that preserve the diameter. They prove the following result.
Theorem. Let \(\Psi :\mathcal{B(H)}\rightarrow\mathcal{B(H)}\) be a surjective linear map that preserves the diameter. Then there exist \(\lambda \in \mathbb{C}\) with \(\left| \lambda \right| =1\), a unitary operator \(U\in \mathcal{B(H)}\), and a linear functional \(\Lambda :\mathcal{B(H)}\rightarrow \mathbb{C}\) satisfying \(\Lambda (I)+\lambda \neq 0\) such that \(\Psi \) is either of the form
\[
\Psi (A)=\lambda UAU^{\ast }+\Lambda (A)I\quad \text{for every }A\in \mathcal{B(H)}
\]
or of the form
\[
\Psi (A)=\lambda UA^{t}U^{\ast }+\Lambda (A)I\quad \text{for every }A\in \mathcal{B(H)},
\]
where \(I\) is the identity operator on \(\mathcal{H}\) and \(A^{t}\) denotes the transpose of \(A\) taken with respect to an orthonormal basis fixed in advance.
The proof of this theorem is divided into two main parts. Let \(\left\langle I\right\rangle \) denote the linear span of \(I\). Consider the dual space \(\left( \mathcal{B(H)}/\left\langle I\right\rangle \right) ^{\ast }\) of the quotient space \(\mathcal{B(H)}/\left\langle I\right\rangle \). In the first part of the proof, authors give a characterization of the extreme functionals, i.e., the extreme points of the closed unit ball of \(\left( \mathcal{ B(H)}/\left\langle I\right\rangle \right) ^{\ast }\). They use this characterization in the second part of the proof where another main tool is used -- namely, a result of \textit{L. Molnár} and \textit{M. Barczy} [J. Funct. Anal. 205, No. 2, 380--400 (2003; Zbl 1047.47029)] which describes the form of all bijective linear maps that preserve the diameter on the set of all self-adjoint operators in \(\mathcal{B(H)}\). numerical range; preserver; diameter Transformers, preservers (linear operators on spaces of linear operators), Numerical range, numerical radius, Linear preserver problems, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Linear mappings preserving the diameter of the numerical range | 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 in positive (or finite) characteristic additional problems arise in the comparison of the infinitesimal and local points of view. For example trying to ``exponentiate'' the derivation \({\mathcal D}\) of a Lie algebra \(L\) to a formal automorphism \(\varphi=1+t{\mathcal D}+t^2\varphi_2+\cdots\), you meet a problem because the usual formula \(\varphi_n= \frac{1}{n!} {\mathcal D}^n\) stops working for \(n=p\), where \(p\) is the characteristic of the ground field. One can make a cocycle for \(H^2(L,L)\):
\[
\psi(x,y)= \sum_{i=1,\dots,p-1} \frac{1}{i!(p-i)!} [{\mathcal D}^ix,{\mathcal D}^{p-i}y],
\]
which is denoted \(\text{Sq }{\mathcal D}\) and called obstruction. The author makes careful considerations of these problems and of the whole situation with deformation in this context. Following Gerstenhaber, he distinguishes geometric rigidity and formal analytic rigidity, provides several criteria for a finite-dimensional Lie or associative algebra \(L\) to be rigid in that sense, and discusses properties of the obstruction subspace in \(H^2(L,L)\). His Theorem 2 shows the special role of the automorphism scheme \(\Aut(L)\) in this context. As an application the scheme theoretic description of deformations of the Jacobson-Witt algebras \(W_n\) is given. deformations; obstructions; Lie algebra; group scheme; geometric rigidity; formal analytic rigidity; automorphism scheme; deformations of the Jacobson-Witt algebras Skryabin, S, Group schemes and rigidity of algebras in characteristic zero, J. Pure Appl. Algebra, 105, 195-224, (1995) Cohomology of Lie (super)algebras, Formal methods and deformations in algebraic geometry, Group schemes, Deformations of associative rings Group schemes and rigidity of algebras in positive characteristic | 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 the distribution of the polygonal paths joining partial sums of classical Kloosterman sums \(\mathrm{Kl}_{p}(a)\), as \(a\) varies over \(\mathbb{F}_{p}^{\times}\) and as \(p\) tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications. Kloosterman sums; Kloosterman sheaves; Riemann hypothesis over finite fields; random Fourier series; short exponential sums; probability in Banach spaces Exponential sums, Gauss and Kloosterman sums; generalizations, Étale and other Grothendieck topologies and (co)homologies, Functional limit theorems; invariance principles, Sample path properties, Sums of independent random variables; random walks Kloosterman paths and the shape of exponential sums | 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 Itoh-Narita-Bogoyavlensky lattice hierarchy associated with a discrete \(3\times 3\) matrix spectral problem is derived by using Lenard recursion equations. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a three-sheeted Riemann surface \(\mathcal{K}_{m-1}\) of arithmetic genus \(m-1\) and construct the corresponding Baker-Akhiezer function and meromorphic function on it. On the basis of the theory of Riemann surface, the continuous flow and discrete flow related to the lattice hierarchy are straightened with the help of the Abel map. Quasi-periodic solutions of the lattice hierarchy in terms of the Riemann theta function are constructed by using the asymptotic properties and the algebro-geometric characters of the meromorphic function and Riemann surface. Itoh-Narita-Bogoyavlensky lattice hierarchy; three-sheeted Riemann surface; quasi-periodic solutions Lattice dynamics; integrable lattice equations, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Theta functions and curves; Schottky problem, Relationships between algebraic curves and integrable systems, Integrable difference and lattice equations; integrability tests Three-sheeted Riemann surface and solutions of the Itoh-Narita-Bogoyavlensky lattice hierarchy | 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 examine the Poisson geometry of moduli spaces of local systems.
Let \(X\) be a smooth complex variety of dimension \(d\) and \(G\) a reductive group. If \(X\) is curve than it is well-known that the moduli space of \(G\)-local systems on \(X\) carries a canonical Poisson structure, whose symplectic leaves are moduli of \(G\)-local systems whose monodromy at infinity is fixed (up to conjugacy).
The authors give a natural extension of this result to the case of higher dimensions, however even the correct formulation now invariably involves derived geometry. If \(d > 1\) then the moduli space of \(G\)-local systems is naturally replaced by a derived moduli stack \(\mathrm{Loc}_G(X)\) of \(G\)-local systems (note that this takes into account the whole homotopy type of \(X\) as opposed to just the fundamental group). It follows from earlier results of the authors and their collaborators that for any compact oriented (real) manifold \(M\) there is a \((2 - \dim_{ \mathbb R} M)\)-shifted symplectic structure on \(\mathrm{Loc}_G(M)\) and thus a non-degenerate \((2 - \dim_{ \mathbb R} M)\)-shifted Poisson structure.
In this new paper the authors show that if \(X\) is a smooth complex variety, not necessarily proper, \(\mathrm{Loc}_G(X)\) carries a canonical \((2-2d)\)-shifted Poission structure.
They moreover describe some generalized symplectic leaves of the foliation if \(X\) admits a smooth compactification whose divisor at infinity is simple normal crossing with at most double intersections. Analogously to the case of curves these leaves are given by derived moduli of \(G\)-local system with fixed local monodromy at infinity as long as a technical condition the authors call `strictness' is satisfied.
As the authors say this is a first step towards understanding moduli of local systems on higher dimensional open varieties, with nonproper generalizations of Simpson's nonabelian Hodge theory as a key long term motivation.
The key ingredient in the proof is the restriction map to the boundary at infinity. Any smooth complex algebraic variety has well-defined boundary at infinity \(\partial X\) that is a compact manifold of real dimension \(2d-1\). Now \(\mathrm{Loc}_G(\partial G)\) has a shifted symplectic structure and by results of Calaque the restriction map is Lagrangian [\textit{D. Calaque}, Contemp. Math. 643, 1--23 (2015; Zbl 1349.14005)]. Then this induces the Poisson structure on \(\mathrm{Loc}_G(X)\) by \textit{V. Melani} and \textit{P. Safronov} [Sel. Math., New Ser. 24, No. 4, 3061--3118 (2018; Zbl 1461.14006); Sel. Math., New Ser. 24, No. 4, 3119--3173 (2018; Zbl 1440.14004)]. The characterization of symplectic leaves is more involved, already the notion of `fixing the monodromy at infinity' is more complicated than in the 1-dimensional case.
The paper includes useful discussions of algebraic descriptions of the moduli of \(G\)-local systems and of the boundary at infinity of a smooth complex variety. The special case that \(X\) is a curve is discussed in detail. The authors expect (but do not show) that in this case the 0-shifted symplectic structure they construct agrees with the one that is known from the literature. local systems; shifted symplectic structures; derived moduli stacks Families, moduli, classification: algebraic theory, Stacks and moduli problems, Symplectic structures of moduli spaces Poisson geometry of the moduli of local systems on smooth 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 A finite set of morphisms
\[
\phi_1,\ldots,\phi_n:\mathbb{A}^N\to\mathbb{A}^M
\]
defined over \(\overline{\mathbb{Q}}\) is jointly regular if the canonical extensions \(\overline{\phi_i}:\mathbb{P}^N\to\mathbb{P}^M\) satisfy \(\bigcap_{i=1}^n\,Z(\overline{\phi_i})=\emptyset\), where \(Z(\overline{\phi_i})\) denotes the locus of indeterminacy of \(\overline{\phi_i}\). The main result, Theorem 1, is the following estimate on the canonical height \(h\): In the above situation, write \(d_i:= \deg(\phi_i)\). Then there is a constant \(C\) such that
\[
\sum_{i=1}^n\frac{1}{d_i}h(\phi_i(P))\geq h(P)-C \text{ for all } P\in\mathbb{A}^N(\overline{\mathbb{Q}}).
\]
J. H. Silverman, Height bounds and preperiodic points for families of jointly regular affine maps, Pure Appl. Math. Q. 2 (2006), no. 1, Special Issue: In honor of John H. Coates. Part 1, 135--145. Heights, Arithmetic varieties and schemes; Arakelov theory; heights, Birational automorphisms, Cremona group and generalizations, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Height bounds and preperiodic points for families of jointly regular affine maps | 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 multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity \(C\) and for the so called divisorial filtration for a modification of \(({\mathbb C}^2,0)\) by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ \(C\) or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious ``motivic version'' of an A'Campo type formula. Campillo, A., Delgado, F., Gusein-Zade, S.: Multiindex filtrations and motivic Poincaré series. Monatshefte. Math. 150, 193-209 (2007) Singularities of curves, local rings, Singularities in algebraic geometry, Complex singularities Multi-index filtrations and generalized Poincaré 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 the entire collection see Zbl 0721.00009.]
These notes represent a survey of the main points of the authors' explicit proof of canonical desingularization of an algebraic subvariety (resp. analytic subspace) \(X\) of an algebraic (resp. analytic) manifold \(M\), in characteristic zero. The full details of this result are planned for a forthcoming work `Canonical desingularization in characteristic zero: a simple constructive proof'. The authors' result is essentially a new proof of Hironaka's theorem, although they also give an explicit resolution algorithm. The centres of the blowings-up used in the desingularization are determined by a local invariant of the singularity of \(X\), defined over a sequence of blowings-up.
The first section of this paper describes the general strategy of the proof and ends with a precise statement of the main theorem. --- The second section recalls the definitions and basic notions involved in resolution of singularities (analytic space, blowing-up, strict transform, normal crossings, etc.). --- The third section introduces the local invariant of a singularity of \(X\) and begins an analysis of this invariant. --- The fourth section gives a key part of the proof that it is in fact invariant. Much of these notes deal with the special case where \(X\) is a hypersurface, and the general case involves a reduction to this case. canonical desingularization of an algebraic subvariety; resolution algorithm Zhou, X. Y., Zhu, L. F.: An optimal \(L\)\^{}\{2\} extension theorem on weakly pseudoconvex Kähler manifolds. To appear in J. Differential Geom. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Effectivity, complexity and computational aspects of algebraic geometry A simple constructive proof of canonical resolution of 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 The author proves that if \(\sum_{d}I_{d}Q^{d}\) where \(I_{d}(z,z^{-1})\) are cohomology valued Laurent \(z\)-series (representing a point on the graph of d\(\mathcal{F}\), the differential of genus-0 descendent potential, in symplectic loop space \(\mathcal{H}\)) and if \(\Phi_{\alpha}\) are polynomials in \(p_{1},\ldots , p_{r}\) then the family
\[
I(\tau)=\sum_{d}T_{d}Q^{d}\text{exp}\Big\{ \frac{1}{z}\sum_{\alpha}\tau_{\alpha} \Phi_{\alpha}(p_{1}-zd_{1},\ldots , p_{r}-zd_{r}) \Big \}
\]
lies on the graph of d\(\mathcal{F}\). Here \(Q^{d}\) stands for the element corresponding to \(d\) in the semigroup ring of Mori cone \(\mathcal{M}\) of the compact Kähler manifold \(X\). Moreover, for arbitrary scalar power series \(C_{\alpha}(z)=\sum_{k\geq 0}\tau_{\alpha,k}z^{k}\), the linear combination \(\sum_{\alpha}c_{\alpha}(z)z\partial_{\tau_{\alpha}}I\) of the derivatives also lies on the graph. Furthermore, in case when \(p_{1}\ldots ,p_{r}\) generate \(H^{*}(X,\mathbb{Q})\), and \(\Phi_{\alpha}\) represents a linear basis, such linear combinations comprise the whole graph. genus-0 descendent potential; symplectic loop space; dilation equation; Novikov ring; \(K\)-theory Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Riemann-Roch theorems, Chern characters, Kähler manifolds Explicit reconstruction in quantum cohomology and \(K\)-theory | 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 the paper under review, the authors investigate the minimal number of generators \(\mu\) and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result, based on combinatorial arguments, is that for each \(C\in \mathbb{Z}_+\) there exist, up to isomorphism, only finitely many divisorial ideals \(I\) such that \(\mu(I)\leq C\) (see theorem 5.1). It then follows by Serre's numerical Cohen-Macaulay criterion that there exist only finitely many divisor classes representing Cohen-Macaulay modules (see corollary 5.2). Moreover, the authors determine the minimal depth of all divisorial ideals and the behaviour of \(\mu\) and depth in `arithmetic progressions' in the divisor class group. The results of the paper are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations. In section 7 of the paper under review, a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals is given; it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions (see theorem 7.2). We refer the reader to the paper, since the results have much more details than can be reproduced here. normal semigroup ring; Cohen-Macaulay module; minimal number of generators; depth of divisorial ideals; multigraded Hilbert functions Bruns, W., Gubeladze, J.: Divisorial linear algebra of normal semigroup rings. Alg. Represent. Theory 6, 139--168 (2003) Commutative rings and modules of finite generation or presentation; number of generators, Cohen-Macaulay modules, Diophantine inequalities, Linear algebraic groups over arbitrary fields, Class groups, Toric varieties, Newton polyhedra, Okounkov bodies, Semigroup rings, multiplicative semigroups of rings, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Divisorial linear algebra of normal semigroup 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 A reflexive sheaf \(F\) of rank \(r\) on a complex manifold \(X\) of dimension \(n\) is called smooth if \(\text{Ext}^ q(F,{\mathcal O})=0\) for \(q\geq 2\), i.e. \(\text{pd} F\geq 1\) and for any point \(x\) of the singular locus of \(X\), \(\text{Ext}^ 1(F,{\mathcal O})_ x\cong{\mathcal O}_ x/(t_ 1,\dots,t_{r+1})\) for some choice of a regular system \((t_ 1,\dots,t_ n)\) of \({\mathcal O}_ x\). In the case \(r=2\) these sheaves were introduced by \textit{A. Hirschowitz} and \textit{R. Marlin} in Math. Ann. 267, 83-89 (1984; Zbl 0545.14026) under the name of convenient reflexive sheaves.
It is shown (theorem 1) that if \(\dim X\leq 2r+1\) and \(F\) is a smooth reflexive sheaf of rank \(r\) generated by global sections, then the zero scheme of a general section \(s\) of \(F\) is smooth of codimension \(r\) or empty and contains \(\text{Sing} F\). For \(r=2\) and 3 one obtains formulae for the Chern classes. This generalizes formulae of \textit{C. Okonek} in Math. Ann. 260, 211-237 (1982; Zbl 0523.14018). -- The second result shows that smooth sheaves are somehow generic reflexive sheaves of \(\text{pd}=1\): If \(E\) and \(F\) are vector bundles of rank \(e\) and \(f\geq e+2\) on the projective manifold \(X\) such that \(\text{Hom}(E,F)\) is globally generated and \(\dim X<2(f-e+2)\), then the cokernel of a general morphisms \(E\to F\) is a smooth reflexive sheaf. -- Various examples of smooth projective subvarieties are constructed as zero sets of sections of reflexive sheaves. smooth reflexive sheaf Bănică, Constantin, Smooth reflexive sheaves, Proceedings of the Colloquium on Complex Analysis and the Sixth Romanian-Finnish Seminar, 0035-3965, 36, 9-10, 571-593, (1991) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Smooth reflexive sheaves | 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 proves two theorems:
Theorem 1. Each affine Nash manifold \(M\) with \(\dim M \geq 1\) has an infinite family of nonsingular algebraic subsets \(\{X_n : n \in \mathbb{N}\}\) of some Euclidean space such that each \(X_n\) is Nash diffeomorphic to \(M\) and that \(X_n\) is not birationally equivalent to \(X_n\) for \(n \neq m\).
Theorem 2. Every compactifiable \(C^\infty\) manifold \(M\) with \(\dim M \geq 1\) has an infinite family of nonsingular algebraic subsets \(\{X_n : n \in \mathbb{N}\}\) of some Euclidean space such that each \(X_n\) is \(C^\infty\) diffeomorphic to \(M\) and that \(X_n\) is not birationally equivalent to \(X_m\) for \(n \neq m\).
Theorem 1 is a very interesting generalization of the well-known fact that any affine Nash manifold admits an algebraic model [\textit{M. Shiota}, ``Nash manifolds'', Lect. Notes Math. 1269 (1987; Zbl 0629.58002)]. Its proof is reduced to the noncompact case. The author makes a \(C^\infty\)-manifold with border imbedded in \(\mathbb{R} \mathbb{P}^\ell\), and he applies the techniques of \textit{J. Bochnak} and \textit{W. Kucharz} [Invent. Math. 97, No. 3, 585-611 (1989; Zbl 0687.14023)], and \textit{S. Akbulut} and \textit{H. C. King} [Ann. Math., II. Ser. 113, 425-446 (1981; Zbl 0494.57004)] in order to achieve the proof. The author obtains the theorem 2 by means of the relative Nash theorem and the work of \textit{M. Shiota} on Nash manifolds [Proc. Am. Math. Soc. 96, 155-162 (1986; Zbl 0594.58006)]. affine Nash manifold Nash functions and manifolds, Real-analytic and Nash manifolds Nonisomorphic algebraic models of Nash manifolds and compactifiable \(C^ \infty\) 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 Given a set \(\mathbb{X}\) of \(s\) points in \(\mathbb{R}^n\) whose coordinates are known only with limited precision \(\epsilon\), each `admissible perturbation' \({\tilde{\mathbb{X}}}\) of \(s\) points whose coordinates differ from those of the original set by less than the specified precision are possible candidates for the `true' set of points. Thus there are multiple possibilities for specifying a set of polynomials to describe \(\mathbb{X}\), depending upon one's goals.
In this paper, the author presents an algorithm which produces a set of polynomials \(\mathcal{J}\), with support \(\mathcal{O}\), that describes \(\mathbb{X}\) and all of its admissible perturbations in the sense that the polynomials are `almost vanishing', w.r.t. the norm of their coefficient vectors, at the original set of points and each of its admissible perturbations. Additionally, if there is a `nice' geometrical structure upon which all of the points of some admissible perturbation lie, it is \textit{possible} (though not guaranteed) that the algorithm will find the polynomial(s) describing the structure.
The algorithm, appropriately named the ``numerical Buchberger-Möller (NBM) algorithm'', is based on Buchberger and Möller's algorithm [\textit{H.M. Möller} and \textit{B. Buchberger}, in: Computer algebra, EUROCAM '82, Conf. Marseille/France 1982, Lect. Notes Comput. Sci. 144, 24--31 (1982; Zbl 0549.68026)] which, given a term order \(\sigma\), computes the \(\sigma\)-Gröbner basis GB of the vanishing ideal of \(\mathbb{X}\). The NBM algorithm is appropriately modified to handle approximate data: A key requirement of the Buchberger/Möller algorithm is the ability to assert the linear independence of power products evaluated at a set of points. The NBM algorithm accomplishes this in the presence of limited precision by using ``the sensitivity of the least squares problem''.
The modified algorithm does not, in general, produce a Gröbner basis of the vanishing ideal of \(\mathbb{X}\). However, the author gives several examples where it produces `approximately vanishing' polynomials describing the geometric structure(s) from which the test points were perturbed. The algorithm has been implemented in the \texttt{CoCoALib} (\texttt{CoCoA} release 4.7, available at \url{http://apcocoa.org}) under the name \texttt{StableBBasisNBM5}. vanishing ideal; border bases; Gröbner bases; limited precision data; stable order ideal; \texttt{CoCoA}; approximate vanishing ideal Buchberger, B., Möller, H.M.: The construction of multivariate polynomials with preassigned zeros. In: Proceedings of EUROCAM '82, LNCS, vol. 144, pp 24-31. Springer, Berlin Heidelberg New York (1982) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Numerical computation of solutions to systems of equations, Computational aspects in algebraic geometry, Real polynomials: location of zeros Almost vanishing polynomials for sets of limited precision points | 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 book under review is geared toward upper-level undergraduate students who are familiar with the basics of real analysis and linear algebra. Its main goal is to provide both a panoramic overview and a profound introduction concerning a number of modern, more advanced mathematical concepts permeating contemporay mathematics as a whole. In this regard, the presentation of the interrelation between various mathematical disciplines is particularly emphasized, in the course of which the discussion of several fundamental mathematical structures serves as the guiding methodological principle.
As for the precise contents, the book consists of three parts, each of which is divided into several chapters and subsections.
Part I is titled ``Algebraic structures'' and contains four chapters. Chapter 1 is devoted to basic ring theory, whereas chapter 2 discusses modules over a ring,some of their fundamental structural properties, and concrete applications of the latter to linear mappings, including the Jordan normal form of matrices. Chapter 3 develops the principles of multilinear algebra for modules over a ring, with the focus on tensor products and their universal properties, tensor algebras, symmetric algebras, and exterior algebras. Chapter 4 explains how the concrete algebraic structures in chapters 2 and 3 are reflected in the more general conceptual framework of universal algebra and category theory, with a special view toward limits, colimits, and adjoint functors.
Part II is superscribed ``Local structures'' and contains the subsequent three chapters. Chapter 5 provides an introduction to the ubiquitous toolkit of sheaf theory, both from the categorical and from the topological point of view. This includes étale spaces, ringed spaces, and sheaves of modules, thereby offering a glimpse of modern algebraic geometry along the way. Chapter 6 turns to special topological structures, more precisely to differentiable manifolds, tangent and tensor bundles, differential forms, integration theory on real manifolds, and the rudiments of complex analytic functions of one variable. Finally, chapter 7 returns to algebraic geometry by discussing algebraic sets, algebraic varieties, and algebraic schemes in greater detail.
Part III offers an outlook to some concepts obtained by combining different types of structures. Its only chapter (Chapter 8) is titled ``Additional structures'' and touches upon Riemannian manifolds, symplectic manifolds, Kähler manifolds, and Poisson structures. Furthermore, affine connections on manifolds, differential fiber bundles, and group objects in categories serve as supplementing, illustrating examples in this context.
All together, this is a very useful book for students seeking orientation for their further specialization in mathematics. The presentation of the material is utmost lucid, sufficiently detailed, versatile, and didactically refined. As such, this excellent primer is a perfect source for further, more detailed reading, and a highly useful companion for students in general. algebraic structures; rings; modules; multilinear algebra; sheaves; manifolds; algebraic varieties; algebraic schemes Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Mathematics in general, Mathematics for nonmathematicians (engineering, social sciences, etc.), Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis Mathematical structures. From linear algebra over rings to geometry with sheaves | 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 geometry of the expansion \({\mathcal R}_{Q}\) of the real field~\(\mathbb R\) by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński--Lion--Rolin). To this end, we study non-standard models~\(\mathcal R\) of the universal diagram~\(T\) of~\({\mathcal R}_{Q}\) in the language \(\mathcal L\) augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our earlier article [Ann. Polon. Math. 96, No. 1, 65--74 (2009; Zbl 1165.14039)], combines modifications by blowing up with a suitable partitioning. Via an analysis of \(\mathcal L\)-terms and infinitesimals, we prove the valuation property for functions given by \(\mathcal L\)-terms, and next the exchange property for substructures of a given model~\(\mathcal R\). Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for~\(T\) is established through model-theoretic compactness. The universal theory~\(T\) is thus complete and o-minimal, and \({\mathcal R}_{Q}\)~is its prime model. Under the circumstances, every definable function is piecewise given by \(\mathcal L\)-terms, and therefore the previous results concerning \(\mathcal L\)-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux's theorem with parameter is demonstrated. quasianalytic functions; special cubes; special modifications; analytically independent infinitesimals; active and non-active infinitesimals; valuation property; quantifier elimination; preparation theorem [5]K. J. Nowak, Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings, Ann. Polon. Math. 96 (2009), 247--282. Real-analytic and semi-analytic sets, Modifications; resolution of singularities (complex-analytic aspects), Semi-analytic sets, subanalytic sets, and generalizations, Quantifier elimination, model completeness, and related topics, \(C^\infty\)-functions, quasi-analytic functions, Model theory of ordered structures; o-minimality Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings | 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 \(G(\alpha; n,d,k)\) be the moduli space of stable coherent systems \((E,V)\) where \(E\) is a vector bundle of rank \(n\), degree \(d\); \(V\) is a \(k\)-dimensional subspace of sections of \(E\) and \(\alpha\) is a positive real parameter. For large \(\alpha\), the moduli spaces are independent of \(\alpha\) and are denoted by \(G_L(n,d,k)\). This paper introduces the new useful technique of 'flips' which enables one to deduce information about \(G(\alpha; n,d,k)\) from that on \(G_L(n,d,k)\). The main results are:
(1) For \(k=n\geq 2, \, G_L\) is nonempty if and only if \(d>n\). When nonempty, it is irreducible, smooth and of expected dimension equal to the Brill-Noether number \(\beta= \beta(n,d,k)\).
Henceforth assume that \(X\) is a generic curve of genus \(g\geq 2\).
(2) For \(k=n+1, \, G_L\) is nonempty if and only if \(\beta \geq 0\). If nonempty, it has dimension \(\beta\) and is irreducible if \(\beta >0\).
(3) For \(n\geq 2, k= 2, 3\), if \(G(\alpha; n,d,k)\) is nonempty, then it is irreducible and of dimension \(\beta\). \(G(\alpha; n,d,k)\) is nonempty (i) for \(n=k=2\) if and only if \(d>2\), (ii) for \(n=2, k=3\) if and only if \(d\geq 2g/3 +2\), (iii) for \(n=3, k=3\) if and only if \(d>3\), (iv) for \(n>2, k=2\) or \(n>3, k=3\) if and only if \(d>0\). The relation between the Brill-Noether loci \(B(n,d,k)\) and \(G(\alpha; n,d,k)\) with \(\alpha\) small is used to deduce the irreducibility of \(B(n,d,k)\) for \(k=1,2,3\) and to compute the Picard group of the smooth part of \(B(n,d,1)\) for \(n\geq 3, (n,d)=1=(n-1,d), \, 0\leq d \leq n(g-1)\). The case \(k<n\) was studied by \textit{S. B. Bradlow} and \textit{O. Garcia-Prada} [J. Reine Angew. Math. 551, 123--143 (2002; Zbl 1014.14012)]. general curve; Brill-Noether loci. Bradlow, B.; García-Prada, O.; Mercat, V.; Muñoz, V.; Newstead, P., Coherent systems and brill-Noether theory, Int. J. math., 14, 683-733, (2003) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Special divisors on curves (gonality, Brill-Noether theory) Coherent systems and Brill-Noether theory. | 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 \(V\) be a locally Cohen-Macaulay, equidimensional closed subscheme of \(\mathbb P^N\) of dimension \(v \geq 1\). The index of speciality of \(V\) is \(e(V) = \max \{ t \in {\mathbb Z} \;| \;h^{v+1}({\mathcal I}_V(t)) \neq 0 \}\). The scheme \(V\) is arithmetically Cohen-Macaulay if \(h^i ({\mathcal I}_V(t)) = 0\) for all \(t\) and all \(1 \leq i \leq v\), and it is subcanonical if \(\omega_V \cong {\mathcal O}_V(\gamma)\) for some integer \(\gamma\). Furthermore, \(V\) is arithmetically Gorenstein if it is arithmetically Cohen-Macaulay and subcanonical. An effective divisor \(D\) on a smooth connected, equidimensional closed subvariety \(V \subset \mathbb P^N\) is said to be lone if \(h^0({\mathcal O}_V(D)) = 1\). It is said to be minimal if \(D\) is linearly equivalent to \(H\), or if \(h^0({\mathcal O}_V(D-H)) = 0\), where \(H\) is a general hyperplane section of \(V\).
Let \(D \neq 0\) be an arithmetically Gorenstein effective divisor on a smooth, connected, closed arithmetically Gorenstein subscheme \(V \subset \mathbb P^N\). Assume that \(D\) is not linearly equivalent to any \(tH\). The purpose of this paper is to show that such \(D\) are very special. First it is shown that \(-v \leq e(D) \leq 2e(V) + v\). A connection is also given between divisors whose index of speciality is in the lower set \(\{ -v, \dots, e(V)-1\}\) and divisors whose index of speciality is in the upper set \(\{ e(V) +1, \dots, 2e(V)+v \}\): the former are minimal and lone, and the latter are obtained, up to linear equivalence, from the former by adding a suitable number of hyperplane sections. In the central value \(e(V)\) the author finds minimal, not lone, divisors. The author also shows that the ``extremal'' cases \(2e(V) +v-1 \leq e(D) \leq 2e(V)+v\) correspond to divisors whose corresponding minimal divisors are reduced quadric subvarieties or linear subvarieties. As a consequence he finds all possible arithmetically Gorenstein integral curves on an arithmetically Gorenstein smooth surface \(S\) in the case \(e(S) = 1\). (The author had previously shown the case \(e(S) \leq 0\).) subcanonical; arithmetically Cohen-Macaulay; lone divisor; minimal divisor Dolcetti A.: Index of speciality and Arithmetically Gorenstein subschemes. Adv. Geom. 5, 347--352 (2005) Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Index of speciality and arithmetically Gorenstein subschemes | 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 \(D\) be a \(g\)-holed planar domain with analytic boundary. For \(g_ 0\) in \(D\) let \(M_ 0\) denote the set of nonnegative representing measures for evaluation at \(g_ 0\). Thus if \(m\in M_ 0\), then \(f(g_ 0)=\int_{\partial D}f dm\) for every \(f\) in \(R(\overline D)\), the uniform closure in \(C(\partial D)\) of the algebra of rational functions with poles off \(\overline D\). The author describes a smooth parametrization for the set \(M_ 0\). He uses this parametrization to study the geometry of the convex set \(M_ 0\) of representing measures and their critical values. Applying the above result the author answers some of the questions posed by \textit{D. Sarason} [J. Funct. Anal. 7, 359- 385 (1971; Zbl 0223.30057)] and \textit{D. Nash} [Trans. Am. Math. Soc. 192, 129-138 (1974; Zbl 0283.30024)]. For example, he shows that when \(D\) is a domain with \(g>2\) holes, then the set \(M_ 0\) is not strictly convex. This paper also illustrates the use of theta functions and Abel-Jacobi theory in the investigation of rational functions on multiply connected domains. measures; critical values; theta functions; Abel-Jacobi theory; multiply connected domains K. Clancey, The geometry of representing measures and their critical values , Oper. Theory: Adv. Appl. 41 , 1989. Spaces of bounded analytic functions of one complex variable, Banach algebras of continuous functions, function algebras, Theta functions and curves; Schottky problem The geometry of representing measures and their critical values | 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 \(I\) be an ideal \(I\subset S = k[x_1,\dots,x_r]\) such that \(A= S/I\) is Artinian; then \(A\) is said to have the Weak Lefschetz Property (WLP) if \(\exists l\in S_1\) such that \(\forall m\) the multiplication map \(\mu_l:A_m \rightarrow A_{m+1}\) has full rank. Notice that if such an \(l\) exists, then for a generic form \(l\in S_1\), \(\mu_l\) has full rank.
Previous results show that WLP always holds for \(r\leq 2\), while for \(r=3\) it holds in many cases, e.g. when \(I\) is generated by powers of linear forms. This last result fails for \(r\geq4\).
In this paper WLP of \(A\) is studied in general when \(I\) is generated by powers of linear forms; by the well known correspondence (via inverse systems) between such ideals and ideals of fat points, the problem can be studied from a geometric point of view, and several results are proved, namely:
- If \(I=(l_1^t,\dots,l_n^t)\), where the \(l_i\)'s are general in \(S_1\), the map \(\mu_l\) has full rank if and only if \((r,t,n) \notin \{(4,3,5),(5,3,9),(6,3,14),(6,2,7)\}\) (this depends on Alexander-Hirschowitz classification of irregular linear system defined by 2-fat points in \({\mathbb P}^n\)).
- Let \(I \in k[x_1,\dots,x_4]\), be generated by general linear forms. For \(n\in \{5,6,7,8\}\) the WLP fails for, respectively, \(t\geq 3,27,140,704\).
- For \(n=r+1\) and \(r=2k\geq 4\), WLP fails in degree \({r \over 2}(t-1)-1\), for \(t\gg 0\).
By using Gelfand-Tsetlin patterns, partial results are obtained to extend the last result for \(r\) odd, thus suggesting the following conjecture:
\[
\text{For \(n \geq r+1\geq 5\), WLP fails for \(t\gg 0\).}
\]
inverse systems; powers of linear forms; fat points; Lefschetz property R. M. Miró-Roig, Ordinary curves, webs and the ubiquity of the Weak Lefschetz Property, Algebras and Representation Theory (2014) to appear. 10.1007/s10468-013-9460-9. Syzygies, resolutions, complexes and commutative rings, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Other special types of modules and ideals in commutative rings, Linkage, complete intersections and determinantal ideals, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property | 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 \(N\) be an integer greater than 1 and let \(\Gamma(N)\) denote the subgroup of the modular group \(\Gamma(1) = \mathrm{SL}_2 (\mathbb Z)\) consisting of the matrices \(\gamma\equiv 1\pmod N\). \(\Gamma(1)\) acts on the upper half plane \(\mathfrak H\) and the modular function \(j\) defines a complex analytic isomorphism \(\Gamma(1)\backslash\mathfrak H \to \mathbb P^1(\mathbb C) - \infty\) whose one-point compactification is the Riemann surface \(\mathbb P^1(\mathbb C)\). The subgroup \(\Gamma(N)\) acts on \(\mathfrak H\) and there is a complex analytic isomorphism \(\Gamma(N)\backslash\mathfrak H \to Y(N)\) lifting that defined by \(j\), where \(Y(N)\) is an affine curve, whose compactification \(X(N)\) is obtained by adding the inverse image of \(\infty\) on the \(j\)-line: thus \(X(N) =Y(N)\cup X^\infty(N)\), where the points in \(X^\infty(N)\) are called cusps, after the shape of the fundamental domain for \(\Gamma(N)\). If \(\mathfrak H^* = \mathfrak H \cup \mathbb Q\cup \{\infty\}\), then \(X^\infty(N)\) is the set of equivalence classes of \(\mathbb Q\cup \{\infty\}\) with respect to the action of \(\Gamma(N)\).
The affine ring of regular functions on \(Y(N)\) over \(\mathbb C\) is the integral closure of \(\mathbb C[j]\) in the function field of \(X(N)\) over \(\mathbb C\) and one can also consider, for example, curves defined over \(\mathbb Q(\mu_N)\) (the cyclotomic field of \(N\)-th roots of unity), in which case one obtains the integral closure of \(\mathbb Q[j]\). In what follows we are concerned primarily with functions defined over \(\mathbb Q(\mu_N)\).
The cuspidal divisor class group \(\mathcal C(N)\) consists of the group of divisors of degree 0 whose support is the set \(X^\infty(N)\) of cusps, modulo the group of divisors of functions on \(X(N)\) having neither poles nor zeros outside the cusps; that is, the subgroup \(\mathrm{Pic}^\infty X(N)\) of \(\mathrm{Pic } X(N)\). The units consist of those functions having no zeros nor poles in the upper half plane.
The primary object of the theory presented in this book is the study of \(\mathcal C(N)\) as a module over a certain Cartan group \(C(N)\); namely the reduction \(\bmod N\) of a subgroup of \(\mathrm{GL}_2 (\mathbb Z_N)\), where \(\mathbb Z_N = \prod_{p\mid N} \mathbb Z_p\). The theory is analogous to the study of the ideal class group of \(\mathbb Q(\mu_N)\) as a module over the group ring \(\mathbb Z[G]\), \(G\approx (\mathbb Z/N\mathbb Z)^*\), in the cyclotomic case. Unlike our present knowledge of the cyclotomic case, \(\mathcal C(N)\) admits a complete description. In chapter 5, the authors show that the divisor class group generated by the cusps can be represented as a quotient of the group ring of the Cartan group \(C(N)\) by an analogue of the Stickelberger ideal. Full use is made of the characterization of the units given in chapters 2 and 3 and there are beautiful connections with algebraic geometry. The order of the cuspidal divisor class group is computed in a manner analogous to that used by Iwasawa in the cyclotomic case, the role of the Bernoulli numbers \(B_{1,\chi}\) in the latter case being played by the second Bernoulli numbers \(B_{2,\chi}\) in the case of \(\mathcal C(N)\). Chapter 5 closes with an analysis of the eigenspace decompositions on \(X(p)\), \(p\) a prime, and it turns out that they involve ordinary Bernoulli numbers and Gauß sums.
Chapter 6 deals with the cuspidal divisor class group for the modular curve \(X_1(N)\) obtained as before from the subgroup \(\Gamma_1(N)\) of \(\Gamma(1)\) of matrices \(\gamma\equiv \begin{pmatrix} 1 & b \\ 0 & 1\end{pmatrix}\pmod N\), where \(b\) is arbitrary.
Chapters 7 to 13 are more specialised. In Chapter 7 the authors study the modular units on Tate curves; that is on the elliptic curves \(Y^2 -XY = X^3 - h_2X - h_3\) where \(h_2 =5\,\sum_{n=1}^\infty q^n/(1-q^n)\), \(h_3 = \sum_{n=1}^\infty (5n^3+7n^5)\cdot q^n/12 \cdot (1-q^n)\).
Chapter 8 is concerned with applications to Diophantine equations and in particular to
\[ \frac{X_3-X_1}{X_2-X_1} + \frac{X_2-X_3}{X_2-X_1} = 1 \]
which is satisfied by the \(\lambda\)-function.
The remaining chapters contain an exposition of the theory of Robert's elliptic units in arbitrary class-fields with a unit index computation due to Kersey. The modular units are an example of a universal even distribution. That is, of a mapping \(\varphi\colon \mathbb Q/\mathbb Z\to A\) to an Abelian group such that for every \(N\) and some positive integer \(k\), \(N^k \sum_{j=1}^{N-1} \varphi(x+\tfrac{j}{N}) = \varphi(Nx)\). Distributions of that kind occur in a number of contexts in number theory and the book begins with an account of the basic theory. Not only is the geometric theory of the cuspidal divisor class group analogous to the theory of cyclotomic fields, but also the most exciting developments in recent years have their origins in work of \textit{A. Wiles} [Invent. Math. 58, 1--35 (1980; Zbl 0436.12004)], who first showed how the connection between the two theories can be made.
The present book is to be welcomed both as an exposition of a fascinating subject, much of which appeared first in a series of papers in the Math. Ann. and also as an introduction to the dramatic developments due to Mazur and Wiles [see \textit{S. Lang}, Bull. Am. Math. Soc., New Ser. 6, 253--316 (1982; Zbl 0482.12002)]. modular units; class groups; modular forms; cyclotomic fields; divisor class group; cusp; Cartan group; Stickelberger ideal; Bernoulli numbers; Tate curve Kubert D and Lang S 1981 Modular Units \textit{Grundlehren Wissenschaften} vol 244 (Berlin: Springer) Elliptic and modular units, Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Cyclotomic extensions, Global ground fields in algebraic geometry, Holomorphic modular forms of integral weight Modular units | 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 goal of the article is to explore the structure of singularities that occur in generic fibres in positive characteristic. The author determines which rational double points do and which do not occur on generic fibres.
Let \(k\) be an algebraically closed ground field of characteristic \(p>0\), and suppose \(f:S \rightarrow B\) is a morphism between smooth integral schemes. Then the generic fiber \(S_\eta\) is a regular scheme of finite type over the function field \(E=\kappa(\eta).\) In characteristic \(0,\) this implies that \(S_\eta\) is smooth over \(E\). The absolute Galois group \(G=\operatorname{Gal}(\bar E/E)\) acts on the geometric generic fiber \(S_{\bar\eta}\) with quotient isomorphic to \(S_\eta\), so to understand the generic fiber it suffices to understand the geometric generic fiber which is again smooth over an algebraically closed field, together with its Galois action.
The situation is more complicated in characteristic \(p>0\). The reason is that over nonperfect fields the notion of regularity is weaker than the notion of geometric regularity, which coincides with formal smoothness. Here it easily happens that the geometric generic fiber \(S_{\bar\eta}\) acquires singularities. As is proved by Bombieri and Mumford, there are quasi-elliptic fibrations of \(p=2\) and \(p=3\), which are analogous to elliptic fibrations but have a cusp on the geometric generic fiber. A proper morphism \(f:S\rightarrow B\) of smooth algebraic schemes is called a \textit{quasifibration} if \(\mathcal O_B=f_\ast(\mathcal O_S)\) and if the generic fiber \(S_\eta\) is not smooth. Quasi-fibrations involve some fascinating geometry and offer new freedom to achieve geometrical constructions that are impossible in characteristic \(0\).
Nonsmoothness of the generic fiber \(S_\eta\) leads to unusual complications. However, singularities appearing on the geometric generic fiber \(S_{\bar\eta}\) are not arbitrary. First, they are locally of complete intersection; hence many powerful methods from commutative algebra apply. They also satisfy far more restrictive conditions, and the goal of the paper is to analyze these. Hirokado started an analysis, characterizing those rational double points in odd characteristic that appear on geometric fibres. His approach was to study the closed fibres \(S_b\) \((b\in B)\) and their deformation theory. In this article, the author looks at the generic fibre \(S_\eta\) and work over the function field \(\kappa(\eta).\) The author works in the following abstract setting: Given a field \(F\) in characteristic \(p>0\) and a subfield \(E\) such that the field extension \(E\subset F\) is purely inseparable. Then the author considers \(F\)-schemes \(X\) of finite type that descend to regular \(E\)-schemes \(Y\), that is, \(X\backsimeq Y\otimes_E F.\) The first results of such schemes are: In codimension 2, the local fundamental groups are trivial and the torsion of the local class groups are \(p\)-groups. Moreover, the Tjurina numbers are divisible by \(p\), the stalks of the Jacobian ideal have finite projective dimension, and the tangent sheaf \(\Theta_X\) is locally free in codimension \(2\). These conditions give strong conditions on the singularities. The first main result of the article is a surprising restriction on the cotangent sheaf: If an \(F\)-scheme \(X\) descends to a regular scheme then, for each point \(x\in X\) of codimension \(2\), the stalk \(\Omega^1_{X/F,x},\) contains an invertible direct summand.
As an application of these results the author determines which rational double points appear on surfaces descending to regular schemes and which do not. It turns out that the situation is most challenging in characteristic 2: Besides the \(A_n\)-singularities, which behave as in characteristic \(0\), there are the following isomorphism classes: \(D_n^r\) with \(0\leq r\leq \lfloor n/2\rfloor-1\) and \(E_6^0, E_6^1,E_7^0,\dots,E_7^3,E_8^0,\dots,E_8^4.\) Notice that all the members of the list have a tangent sheaf that are locally free, although there are other rational double points with locally free tangent sheaf.
The author sets up notation and gives some elementary examples and results. He analyzes \(F\)-schemes that \(X\) descend to regular \(E\)-schemes \(Y\), and treats the local fundamental groups. He proves that integer-valued invariants are multiples of \(p\), and he treats the finite projective dimension of sheaves obtained from the cotangent sheaf \(\Omega^1_{X/F}.\)
This article is written in an easy to understand language, it is more or less self contained with good references when needed. Finally, it treats a computable field of singularity theory which is of importance to the study in positive characteristic. singularities on generic fibers; absolute Galois group; positive characteristic; descent; descends to regular scheme; quasi-fibration Stefan Schröer, Singularities appearing on generic fibers of morphisms between smooth schemes, Michigan Math. J. 56 (2008), no. 1, 55 -- 76. Arithmetic ground fields (finite, local, global) and families or fibrations, Fibrations, degenerations in algebraic geometry Singularities appearing on generic fibers of morphisms between smooth 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 Let \(X\) be a scheme. The author defines groups \(KV^ j_ m(X)\) which can be thought of as the Karoubi-Villamayor theory of modules having codimenson \(\geq j\), and also groups \(KV_ m^{j/j+1}(X)\). This is done by lifting Bloch's higher Chow construction [\textit{S. Bloch}, Adv. Math. 61, 267-304 (1986; Zbl 0608.14004)] from abelian groups to topological spectra. The \(KV\) groups are the homotopy groups of simplicial spectra \(K^ j(X,\bullet)\) (respectively \(K^{j/j+1}(X,\bullet))\). There is a spectral sequence \(E_ 1^{pq}=KV^{p/p+1}_{-p-q}(X,n)\Rightarrow K_{-p-q}(X)\) \((K_ n\) being Quillen's higher \(K\)-functor) which leads to a filtration on higher \(K\)-theory. Additional invariants \(KH_ m^{j/j+1}(X,n)\) are defined which are the \(E^ 2\) terms of spectral sequences \(E^ 2_{p,q}=KH_ q^{j/j+1}(X,p)\Rightarrow KV^{j/j+1}_{p+q}(X)\). Under additional hypotheses there are spectral sequences \('E_ 1^{p,q}=KH_{r-p}^{p/p+1}(X,-q)\Rightarrow H^{p+q}(X,{\mathcal K}_ r)\) (where \({\mathcal K}_ r\) is the sheafification in the Zariski topology of the functor \(K_ r)\).
These various spectral sequences are applied to the case \(X=\text{Spec}(k)\), \(k\) a field, in order to compute some of the \(KV\) and \(KH\) groups of a field. For example \(KH_ 3^{0/1}(X,0)\cong K_ 3(k)^{\text{ind}}\) and the \(KV^ j_ 3(X)\) \((0\leq j\leq 3)\) are described in terms of Bloch's higher Chow groups and the image of \(K^ M_ 3(k)\) (Milnor's \(K\)-group) in \(K_ 3(k)\). The paper concludes with a speculative discussion of how these invariants might relate to motivic cohomology. \(KV\) groups; higher \(K\)-theory; Karoubi-Villamayor theory; higher Chow construction; spectral sequences; motivic cohomology Landsburg S. , Some filtrations on higher K-theory and related invariants , K-theory 6 ( 1992 ) 431 - 457 . MR 1194843 | Zbl 0774.14006 Applications of methods of algebraic \(K\)-theory in algebraic geometry, Parametrization (Chow and Hilbert schemes), \(K\)-theory of schemes Some filtrations on higher \(K\)-theory and related invariants | 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 discusses some results on kernels of locally nilpotent derivations of a polynomial ring over an algebraically closed field of characteristic zero \(k\). Locally nilpotent derivations of \(k[x_1,\ldots,x_n]\) correspond to algebraic actions of the additive group \((k,+)\) on affine \(n\)-space. The kernel of such a locally nilpotent derivation corresponds to the invariant set of polynomials with respect to the corresponding \((k,+)\) action. An interpretation of the Cancellation problem in terms of the kernel of a locally nilpotent derivations is given. The Cancellation problem asks : if \(X\) is an affine variety and \(X\times{\mathbb{A}}^1\) is isomorphic to affine \(n\)-space, then is \(X\) isomorphic to affine \(n-1\)-space. A slice of a locally nilpotent derivation \(D\) is an element \(f\) such that \(D(f)=1\).
The Cancellation problem can be interpreted as the following question: Let \(D\) be a locally nilpotent \(k\)-derivation on \(k[x_1,\ldots,x_n]\) and suppose that \(D\) has a slice. Then is the kernel of \(D\) isomorphic to a polynomial ring in \(n-1\) variables?
The theorems 3.2 and 3.4 of the present article claim the following.
(1) Let D be a locally nilpotent derivation with a slice on \(k[x_1,...,x_n]\) with \(n\geq 4\) and such that \(x_1,...,x_{n-3}\) are in the kernel of \(D\). Then the kernel of \(D\) is a polynomial ring of dimension \(n-1\).
(2) Let D be a triangulable locally nilpotent derivation with a slice on \(k[x_1,...,x_n] \). Then the kernel of \(D\) is a polynomial ring of dimension \(n-1\). However, the proofs of these two results are not complete. cancellation problem; \(k^*\)-action; fix-pointed action; triangular derivation Masuda, K.: Torus actions and kernels of locally nilpotent derivations with slices, Affine algebraic geometry in honor of Professor masayoshi miyanishi (2007) Group actions on affine varieties, Group actions on varieties or schemes (quotients), Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Torus actions and kernels of locally nilpotent derivations with slices | 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 Throughout this paper, we will work in \(\mathbb{P}^r= \mathbb{P}_k^r\), where \(k\) is an algebraically closed field with characteristic zero and \(r\geq 3\). Let \(R:=k [x_0,\dots, x_r]\). A scheme will be taken to mean an equidimensional, locally Cohen-Macaulay closed subscheme of \(\mathbb{P}^r\). Given a scheme, \(V\), \({\mathcal I}_V\) will denote its associated ideal sheaf. Recall that the Hartshorne-Rao modules (or deficiency modules) of \(V\) in \(\mathbb{P}^r\) are defined for \(1\leq i\leq\dim(V)\) as \(M_i(V)=\bigoplus_{n\in \mathbb{Z}}H^i (\mathbb{P}^r,{\mathcal I}_V(n))\). Each \(M_i(V)\) is a graded \(R\)-module of finite length and various properties of \(V\) can be deduced from the structure of the Hartshorne-Rao modules.
Let \(I\subset R\) be a homogeneous ideal and let \(V\) be the scheme defined by \(I\) with associated ideal sheaf \({\mathcal T}_V\). The saturation, \(\overline I\), of \(I\) is defined as \(\overline I=\bigoplus_{t \in\mathbb{Z}} H^0(\mathbb{P}^r, {\mathcal I}_V(t))\). We say \(I\) is saturated if \(I=\overline I\). If \(J\) is any other ideal defining the same scheme as \(I\), then \(J\subseteq\overline I\) and \(\overline J=\overline I\). We hence call \(\overline I\) the homogeneous ideal of \(V\).
Let \(V\) be an arithmetically Cohen-Macaulay scheme of codimension two in \(\mathbb{P}^r\). The homogeneous ideal of \(V\) will be denoted by \(I\). Assume \(V\) is a generic complete intersection but not a complete intersection. A theorem by \textit{Huneke} and \textit{B. Ulrich} [Math. Sci. Res. Inst. 15, 339-346 (1989; Zbl 0731.13008)] then states that for all \(n\geq 2\) either \(I^n\) is not saturated or the scheme defined by \(I^n\) is not arithmetically Cohen-Macaulay. Let \(V_n\) denote the scheme defined by \(I^n\). If \(V\) is a local complete intersection and \(I^n\) is saturated then this theorem implies that for some \(i\), \(\dim_k(M_i (V_n))>0\). In a similar vein, we are concerned in this paper with the question: If \(V\) is a local complete intersection codimension two subscheme of \(\mathbb{P}^r\) what can we say about \(M_i(V_n)\) in terms of \(M_i(V)\)?
The ideal \(I\) is said to have analytic spread three if \(\dim_k(I^s/ I^s{\mathfrak m})\) is given by a quadratic polynomial in \(s\) for \(s\gg 0\) (here \({\mathfrak m}\) denotes the maximal ideal in \(R)\). Using a result of Migliore and a result of Huckaba and Huneke we show
Proposition. Let \(C\) be a reduced, equidimensional, local complete intersection curve in \(\mathbb{P}^3\) with homogeneous ideal \(I_C\). If the analytic spread of \(I_C\) is three then \(\dim_k(M_1 (C_n))\geq s\cdot n^2\) for some \(s>0\) and all \(n\gg 0\).
If \(V\) is a codimension two scheme then we say \(V\) is a quasi-complete intersection if \(V\) can be cut out as a scheme by three equations. Using approximation complexes we show:
Theorem. Let \(C\) be a quasi-complete intersection, local complete intersection, codimension two subscheme of \(\mathbb{P}^r\). Then \(\dim_k(M_i(C_n)) ={n+1\choose 2}\cdot \dim_k(M_i(C))\) for \(1\leq i\leq r-3\) and \(\dim_k (M_{r-2}(C_n)) \geq{n+2 \choose 2}\cdot \dim_k(M_{r-2}(C)) \).
Finally we show how information about the first infinitesimal neighborhood of a scheme provides useful information for the entire even liaison class of the scheme. Hartshorne-Rao modules; arithmetically Cohen-Macaulay scheme; local complete intersection; codimension two subscheme; analytic spread three; quasi-complete intersection; first infinitesimal neighborhood; liaison class Complete intersections, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, Low codimension problems in algebraic geometry Quasi complete intersections, powers of ideals and deficiency modules | 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 compact set in \(\mathbb R^d\) and \(\rho\) be a measure defined on \(E\). Under some assumptions on the structure of \(E\) and \(\rho\), the author provides a methodology to construct wavelets to the space \(L^2(E, \rho)\). The methodology includes four steps: defining the measure \(\rho\) on \(E\); finding a map from \(E\) onto the unit in \(\mathbb R^d\); constructing the maps along with a refinable vector field; and constructing the initial wavelet space. Examples of the construction for a rectangular surface, a triangular surface, and a smooth simplex in \(\mathbb R^{3}\) are given. An extension of the theory to triangulated manifolds of finite distortion in \(n\) dimensions is also explained. wavelets; fractals; manifolds; compact domain; Jacobian problem; rectangular surface; triangular surface; smooth simplex; triangulated manifold Numerical methods for wavelets, Jacobian problem, Compact (locally compact) metric spaces, Dynamics induced by flows and semiflows, Symplectic manifolds (general theory), Nontrigonometric harmonic analysis involving wavelets and other special systems The construction of wavelets adapted to compact domains | 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 the paper under review, the author establishes a version for parametrization of power-subanalytic sets. Here is the precise formulation:
If \(X\subset [-1,1]^n\) is power-subanalytic of dimension \(m\leq n\), then for any positive integer \(r\) there exists a \(C^r\)-parametrization of \(X\) consisting of \(cr^{m^3}\) maps whose \(C^r\)-norm is bounded by \(1\). Morover, if \(X\) belongs to a power-subanalytic family of such sets, the constant \(c\) holds for all members in the family.
This generalizes a result by \textit{R. Cluckers} et al. [Ann. Sci. Éc. Norm. Supér. (4) 53, No. 1, 1--42 (2020; Zbl 1479.03016)] by giving an effective description of the exponent. Note that it is not as good as in the case of subanalytic sets (see [\textit{G. Binyamini} and \textit{D. Novikov}, Ann. Math. (2) 190, No. 1, 145--248 (2019; Zbl 1448.14056)]) where the exponent is just given by the dimension.
The proof of the result relies on careful analysis of the derivatives of a composite function using ideas from the theory of Gevrey functions. Gevrey functions; mild functions; subanalytic sets; \(C^r\)-parameterization; preparation of power-subanalytic functions; Weierstrass systems Real-analytic and semi-analytic sets, Applications of model theory, Semi-analytic sets, subanalytic sets, and generalizations, \(C^\infty\)-functions, quasi-analytic functions, Analytic algebras and generalizations, preparation theorems Smooth parameterizations of power-subanalytic sets and compositions of Gevrey 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 joint spectrum \(\sigma(A_1, \dots, A_n)\) of a tuple \((A_1, \dots, A_n)\) of operators acting on a Hilbert space \(H\) consists of all \((x_1, \dots, x_n)\in \mathbb{C}^n\) such that \(x_1A_1 + \dots + x_nA_n\) is not invertible on \(H\). This paper is aimed to investigate the relationship between the geometry of the spectrum and the mutual behaviour of the operators. If the spectrum contains an algebraic hypersurface passing through an isolated spectral point of one of the operators, then the authors give some necessary and sufficient geometric conditions for the operators in the tuple to have a common reducing subspace. In addition, they study the issue of spectral continuity, that is, if two hypersurfaces are close in a neighbourhood of a point, and both have self-adjoint spectral representation of which one is decomposable, how far from being decomposable is the other? Further, they obtain a norm estimate for the commutant of a pair of self-adjoint matrices in terms of the Hausdorff distance of their joint spectrum to a family of lines. joint spectrum; algebraic curve; hypersurface; commutant; decomposable; spectral continuity Several-variable operator theory (spectral, Fredholm, etc.), Real-analytic and semi-analytic sets, Hypersurfaces and algebraic geometry Geometry of joint spectra and decomposable operator tuples | 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 stratification of a variety \(V\) is an expression of \(V\) as the disjoint union of a locally finite sets of connected analytic manifolds, called strata, such that the boundary of each stratum is the union of a set of lower-dimensional strata. The most important notion in stratification theory is the regularity condition between strata.
In this paper, the author investigates regularity conditions relative to a fixed Newton filtration in the context of stratification theory. First, the author shows a characterization of the (\(c\))-regularity condition defined by \textit{K. Bekka} [Lect. Notes Math. 1462, 42--62 (1991; Zbl 0733.58003)]. Next, the author presents a criterion for regularity conditions in terms of the defining equations of the strata and, after introducing a pseudo-metric adapted to the Newton polyhedron, obtains versions relative to the Newton filtration of the Fukui-Paunescu theorem [\textit{T. Fukui} and \textit{L. Paunescu}, Can. J. Math. 53, No. 1, 73--97 (2001; Zbl 0983.32006)]. In this approach it is possible to consider the version relative to a Newton filtration of the \((w)\)-regularity condition. It is shown that this condition implies \((c)\)-regularity condition. Finally, the author shows that the Demon-Gaffney condition [\textit{J. Damon} and \textit{T. Gaffney}, Invent. Math. 72, 335--358 (1983; Zbl 0519.58021)] implying the topological triviality of an analytic deformation of an analytic function can be expressed also in terms of Newton filtrations and proves that it implies \((c)\)-regularity condition. singularities; stratified sets; Newton polyhedron; regularity condition Abderrahmane, OM, Stratification theory from the Newton polyhedron point of view, Ann. Inst. Fourier Grenoble, 54, 235-252, (2004) Stratified sets, Singularities in algebraic geometry The theory of stratification relative to a Newton polyhedron. | 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 shown that in the class of compact sets \(K\) in \(\mathbb R^n\) with an analytic parameterization of order \(m\) (\(1\leq m\leq n\)), the sets with Zariski dimension \(m\) are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on \(K\). In particular, by Hironaka's rectilinearization theorem, such a parameterization is admitted by compact subanalytic subsets of \(m\)-dimensional real-analytic submanifolds of \(\mathbb R^n\). Hence the above result covers a similar characterization of compact real-analytic manifolds given by \textit{L. Bos, N. Levenberg, P. Milman} and \textit{B. A. Taylor} [Indiana Univ. Math. J. 44, No. 1, 115-138 (1995; Zbl 0824.41015)]. Bernstein and van der Corput-Schaake type inequalities; traces of polynomials on semialgebraic sets; Zariski dimension Baran, M.; Pleśniak, W., Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities, Studia math., 141, 3, 221-234, (2000) Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities), Rate of convergence, degree of approximation, Semialgebraic sets and related spaces, Real algebraic sets Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities | 0 |
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