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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 Motivic integration, a part of algebraic geometry, is a new approach to integration. Instead of working with measurable sets, this theory works directly with the underlying boolean formulas that define the sets. Motivic volumes are not positive real numbers, but elements of certain Grothendieck rings.
For a brief review of the development of motivic integration, we cite from the introduction: ``The idea of motivic integration was introduced by M. Kontsevich in 1995. It was quickly developed by \textit{J. Denef} and \textit{F. Loeser} in a series of papers [Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004); Compos. Math. 131, No. 3, 267--290 (2002; Zbl 1080.14001); J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)], and by others. This theory, whose applications are mostly in algebraic geometry over algebraically closed fields, now is often referred to as `geometric motivic integration', to distinguish it from the so-called arithmetic motivic integration that specifies to integration over \(p\)-adic fields. The theory of arithmetic motivic integration first appeared in the 1999 paper by \textit{J. Denef} and \textit{F. Loeser} [J. Am. Math. Soc. 14, No. 2, 429--469 (2001; Zbl 1040.14010)]. The articles [\textit{T. C. Hales}, Bull. Am. Math. Soc., New Ser. 42, No. 2, 119--135 (2005; Zbl 1081.14033)] and [\textit{J. Denef} and \textit{F. Loeser}, Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 13--23 (2002; Zbl 1101.14029)] together provide an excellent exposition of this work. In 2004, \textit{R. Cluckers} and \textit{F. Loeser} developed a different and very effective approach to motivic integration (both geometric and arithmetic) [Invent. Math. 173, No. 1, 23--121 (2008; Zbl 1179.14011)].''
The paper under review provides an introduction to some aspects of the theory of arithmetic motivic integration. The paper is written very nicely. The authors include in the exposition a number of examples, explaining motivations for the theory of motivic integration and also explaining applications. motivic integration; \(p\)-adic integration Julia Gordon and Yoav Yaffe, An overview of arithmetic motivic integration, Ottawa lectures on admissible representations of reductive \?-adic groups, Fields Inst. Monogr., vol. 26, Amer. Math. Soc., Providence, RI, 2009, pp. 113 -- 149. Arcs and motivic integration An overview of arithmetic motivic integration | 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 translation of the book ``Le problème des modules pour les branches planes'' of \textit{O. Zariski} [Hermann, Paris (1973; Zbl 0317.14004)] by Ben Lichtin. It is based on notes from a course of Zariski at Centre de Mathématiques de l'École Polytechnique 1973 and contains an appendix by B. Teissier considering the moduli problem from the point of view of deformation theory. Zariski's aim is to study the space of isormorphism classes of plane curve singularities (analytically irreducible curve germs) of given equsingularity type, i.e. the moduli space \(M(\Gamma)\) of plane curve singularities with fixed semigroup \(\Gamma\). His ideas and results were the basis for further research in this direction [cf. for example \textit{O. A. Laudal} and the reviewer, ``Local moduli and singularities'', Lect. Notes Math. 1310 (1988; Zbl 0657.14005)].
The first three chapters of the book introduce the basic notions, especially invariants as the semigroup, the conductor, the characteristic of a branch, short parametrizations, etc. Then the moduli space \(M(\Gamma)\) is studied. It is proved that \(M(\Gamma)\) is not seperated in general. The structure of \(M(\Gamma)\) is analyzed for special examples. The dimension of the generic component of the moduli space for \(\Gamma=\langle n, n+1\rangle\) is computed. This was generalized later on by C. Delorme.
The appendix of Teissier is based on the idea that each plane curve singularity with semigroup \(\Gamma\) appears as a deformation of the associated monomial curve, more precise in its negatively weighted part. It turns out that every plane branch has a miniversal equisingular deformation, over a smooth base, with an equisingular section. Among several results a ``natural'' compactification of \(M[\Gamma)\) is given, as well as an interpretation of the generic component. moduli problem; plane curve singularity; semi group; deformation Zariski, O., \textit{The Moduli Problem for Plane Branches (with an Appendix by Bernard Teissier)}, 39, (2006), AMS, Providence RI Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry The moduli problem for plane branches. With an appendix by Bernard Teissier. Transl. from the French by Ben Lichtin | 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 interesting article the authors discuss several versions of the Adams and Adams-Novikov spectral sequence in the framework of motivic stable homotopy theory. The spectral sequence constructions are considered in the \(2\)-complete motivic stable homotopy category over an algebraically closed field of characteristic \(0\). The authors establish especially here important convergence results for both spectral sequence types. Besides showing that these spectral sequences observe similar behavior as their classical topological counterparts, new and more algebro-geometric phenomena also show up, like the existence of permanent cycles related to the Tate twist.
Many interesting examples and applications are discussed, which nicely emphasize the interplay of topological and algebro-geometric methods. For example, they perform several interesting calculations on the cohomology of symmetric products of spheres in the motivic setting analogous to classical topological calculations obtained by Nakaoka in the 50s. As an interesting application they also discuss a \(2\)-complete version of the complex motvic \(J\)-homomorphism.
The appendix gives a thorough tour on the construction of motivic Adams spectral sequences and their main properties. The article also discusses nicely the different motivic cobordism spectra used in the construction of the motivic Adams-Novikov spectral sequences. motivic homotopy theory; Adams spectral sequences; Adams-Novikov spectral sequences; motivic cobordism theories, \(J\)-homomorphism Andrews, M., Miller, H.: Inverting the Hopf map in the Adams-Novikov spectral sequence (2014). \textbf{(preprint)} Motivic cohomology; motivic homotopy theory, \(J\)-homomorphism, Adams operations, Adams spectral sequences Remarks on motivic homotopy theory over algebraically closed 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 A set \(\mathcal L=\{l_1,\dots,l_r\}\), with \(r\ge n\), of hyperplanes is said to \textit{meet properly} if the intersection of any \(n\) of the hyperplanes in \(\mathcal L\) is a point and if any \(n+1\) hyperplanes are never concurrent. The point \(l_{i_1}\cap\dots\cap l_{i_r}\), for some \(i_1,\dots,i_r\) such that \(1\le i_1<\dots<i_n\le r\), is denoted by \(P_{i_1,\dots,i_n}\).
Given a set \(\mathcal L=\{l_1,\dots,l_r\}\), with \(r\ge n\), of hyperplanes meeting properly, a \textit{start configuration} of points in \(\mathbb P^n\) defined by \(\mathcal L\) is the set:
\[
\mathbb S(\mathcal L)=\bigcup_{1\le i_1<\dots<i_n\le r} P_{i_1,\dots,i_r}\subseteq \mathbb P^n.
\]
In this paper the authors introduce the notion of \textit{contact star configuration}. Given a rational normal curve \(\gamma\) in \(\mathbb P^n\), an osculating hyperplane to the curve \(\gamma\) at a point \(P\in \gamma\) is the hyperplane spanned by the length \(n\) scheme \(nP\cap \gamma\), where \(nP\) is a fat point of multiplicity \(n\).
Definition. Let \(P_1\),\dots,\(P_r\in \mathbb P^n\) be distinct point on a rational normal curve \(\gamma\) of \(\mathbb P^n\). Denote by \(\mathcal L=\{l_1,\dots,l_r\}\) the set of osculating hyperplanes to \(\gamma\) at \(P_1\),\dots,\(P_r\), respectively. The set \(\mathbb S(\mathcal L)\) is a \textit{contact star configuration} on \(\gamma\).
Since \(\gamma\) is a rational normal curve, the hyperplanes in \(\mathcal L\) always meet properly.
In a first part of the paper the authors show that the so called \textit{Hadamard star configurations} are contact star configurations. Then the authors focus on the union of two star configurations on the same conic in \(\mathbb P^2\), mainly focusing on the \(h\)-vector, and proving also that in some special cases this union is a complete intersetion. complete intersection; Hadamard product; star configuration; Gorenstein Linkage, complete intersections and determinantal ideals, Theory of modules and ideals in commutative rings described by combinatorial properties, Complete intersections, Special varieties, Configurations and arrangements of linear subspaces Rational normal curves and Hadamard products | 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 Considering the \(\Lambda_ p\)-behavior space defined by the author in J. Math. Kyoto Univ. 16, 101-112 (1976; Zbl 0333.30009), in this paper, we give a conformal mapping theorem of the interior of a compact bordered Riemann surface into a region. As a consequence of the Riemann-Roch theorem we know that there exists a univalent function on a compact Riemann surface S if and only if S is of genus zero. [\textit{J. Lehner,}, Discontinuous groups and automorphic functions (1964; Zbl 0178.429)]. Therefore our result can be regarded as a generalization of this fact. The result obtained is in contrast with those of \textit{M. Shiba} [(*) J. Math. Kyoto Univ. 11, 495-525 (1971; Zbl 0227.30022)] because the \(\Lambda_ p\)-behavior space is different than the \(\Lambda_ 0\)- behavior space. We use the same notation and the same classical techniques as in (*). conformal mapping theorem; Riemann-Roch theorem Differentials on Riemann surfaces, Compact Riemann surfaces and uniformization, Riemann-Roch theorems A conformal mapping 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 book is a classical reference on resolution of algebraic surfaces, embedded into a non-singular projective 3-fold over a perfect field \(k\). It gives a self-contained exposition on results of Zariski and leads up to the author's extensions to the case of characteristic \(p>0\).
The principal results are global resolution, global principalization, dominance, birational invariance of dimension of homology groups, as well as (for \(p\neq 2,3,5)\) uniformization and birational resolution. -- The first edition of this book appeared in 1966 (see Zbl 0147.20504). The second edition under review comes in a time of newly increasing interest in resolution of singularities [cf. the work of \textit{A. J. de Jong}, Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996) with an alternative approach, called ``alterations'']. For the recent generation of mathematicians, there may arise some difficulties with the terminology, like the direction of arrows in the introduction (the author is ``blowing down'' where others are ``blowing up'').
Here are some remarks on the new edition: It contains an appendix on analytic desingularization in characteristic zero, where a recent short proof is presented. -- From the author's abstract: ``It is hoped that this will remove the fear of desingularization from young minds and embolden them to study it further.'' The proof ``was inspired by discussion with the control theorist Hector Sussmann, the subanalytic geometer Adam Parusiński, and the algebraic geometer Wolfgang Seiler. Once again this illustrates the fundamental unity of all mathematics \dots By an inductive procedure incorporating the principalization lemma, the hypersurface \(f\) is approximated by a binomial hypersurface, i.e. a hypersurface of the form \(X_1^e+ X_2^{b_{e2}}\cdot\dots\cdot X_n^{b_{en}}\), where \(e\) is a positive integer and \(b_{e2},\dots, b_{en}\) are nonnegative integers. The reduction lemma enables us to further arrange matters so that \(\overline{b}_{e2}+\dots+ \overline{b}_{en}<e\), where \(\overline{b}_{e2},\dots, \overline{b}_{en}\) are the residues of \(b_{e2},\dots, b_{en}\) modulo \(e\). This then is a prototype of a good point of a surface.'' The final step is the reduction of multiplicity for such points using a ``good point lemma''. -- The book concludes with an extended bibliography, including also \textit{H. Hauser}'s recent article on ``Seventeen obstacles for resolution of singularities'' [in: Singularities. The Brieskorn anniversary volume, Prog. Math. 162, 289-313 (1998)] which the reader may wish to consult for further developments. resolution of algebraic surfaces; resolution of singularities; analytic desingularization S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, 2nd ed., Springer Monogr. Math., Springer, Berlin 1998. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Rational and birational maps, Modifications; resolution of singularities (complex-analytic aspects) Resolution of singularities of embedded algebraic surfaces. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book has its origin -- as explicitly stated by the authors in its introduction -- in the remarkable discovery by \textit{A. M. Odlyzko} [The \(10^{20}\)th zero of the Riemann zeta-function and 70 millions of its neighbors, ATT Bell Laboratories, 1989; Math. Comput. 48, 273-308 (1987; Zbl 0615.10049)], carried out by means of numerical experiments, that the distribution of the spacings between successive (nontrivial) zeros of the Riemann zeta-function is empirically the same as the so-called GUE measure, which is a probability measure on \(\mathbb{R}\) that arises in random matrix theory. Odlyzko's numerical experiments had been, on its turn, inspired by work of \textit{H. L. Montgomery} [Analytic Number Theory, Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 181-193 (1973; Zbl 0268.10023)], who had determined the pair correlation distribution between the zeros in a restricted range, and had already noted the compatibility of his results with the GUE prediction.
Recent new results have enforced the belief, explicitly conjectured in the book, that the distribution of the spacings between zeros of the Riemann zeta-function and also of quite general automorphic \(L\)-functions over \(\mathbb{Q}\) are, in fact, given by the GUE measure, satisfying the by now called Montgomery-Odlyzko law. The authors recognize that proving this in such generality, for arbitrary number fields, seems well beyond the range of existing techniques.
In the book, they restrain this scope to the case of finite fields, namely the authors establish the Montgomery-Odlyzko law for wide classes of zeta and \(L\)-functions over finite fields.
To fix ideas, the authors start by considering, already at the introduction, a special case, namely that of a finite field, \(\mathbb{F}_q\), and a proper, smooth, geometrically connected curve \(C/\mathbb{F}_q\), of genus \(g\) (the corresponding zeta function was introduced by E. Artin in his thesis). After reviewing in this example the defining of the normalized spacings between the zeros of the zeta function, they show that the spacing measure in this case is the probability measure which gives mass \(1/2g\) to each of the \(2g\) normalized spacings. Of course, the rest of the 420 page book is not that easy, but this example gives a clue to understand the essential phenomena, what is it all about, as the authors themselves like to remark. This pedagogical attitude is manifest throughout the book.
They go on by recalling the definition of the GUE measure on \(\mathbb{R}\) (that is, the Wigner measure, for physicists): the limit for big \(N\) of the probability measure \(A\in U(N)\). Using the Kolmogorov-Smirnov discrepancy function, one is able to have a numerical measure of how close two probability measures in \(\mathbb{R}\) are. The generalized Sato-Tate conjecture follows, as well as a number of conjectures, in particular, involving the low-lying zeros of \(L\)-functions of elliptic curves over \(\mathbb{Q}\).
In Chapter 1 we find statements of the main result in the book. It deals with the measures attached to spacings of eigenvalues and with the expected values of spacing measures. Three main theorems on the existence, universality and discrepancy for limits of expected values of spacing measures are given there, with some applications, corollaries, and an appendix on the continuity properties of the \(i\)th eigenvalue as a function on \(U(N)\).
Chapter 2 deals with a reformulation of the main results, under a different viewpoint, and provides several discussions on the combinatorics of spacings of finitely many points on a line. Chapter 3 deals with reduction steps in proving the main theorems, while the next chapters are devoted, respectively, to test functions (Ch. 4), the Haar measure (Ch. 5), tail estimates, a determinant-trace inequality, and multi-eigenvalue location measures (Ch. 6), large \(N\) limits and Fredholm determinants (Ch. 7), the case of several variables, with corresponding large \(N\) scaling limits (Ch. 8), equidistribution, with two versions of Deligne's equidistribution theorem (Ch. 9), monodromy of families of curves and monodromy of some other families (Chs. 10, 11), GUE discrepancies in various families of curves, abelian varieties, hypersurfaces, and Kloosterman sums (Ch. 12), and finally, on the distribution of low-lying Frobenius eigenvalues in various families, of curves, abelian varieties, hypersurfaces, and Kloosterman sums, according to the measures corresponding to different groups of the \(G(N)\), \(USp\), and \(SO\) types, with a passage to the large \(N\) limit (Ch. 13).
The book finishes with two appendices, on densities and large \(N\) limits, and on some graphs, how they were drawn and what they show, respectively. I've missed some more of these graphs, for different explicit examples, and also some specific applications (e.g. in physics, to honor Wigner's pioneering discoveries) of the results obtained in the book, but this is just a personal thought.
To summarize, a very complete and useful reference on the subject, by two well known specialists in the field. zeta functions; distribution of zeros; \(L\)-functions; finite fields; automorphic \(L\)-functions; GUE measure; Montgomery-Odlyzko law; normalized spacings; Wigner measure; Kolmogoroff-Smirnov discrepancy function; generalized Sato-Tate conjecture; low-lying zeros; \(L\)-functions of elliptic curves; spacings of eigenvalues; Haar measure; Fredholm determinants; Deligne's equidistribution theorem; monodromy; Kloosterman sums N.M. Katz and P. Sarnak. \textit{Random matrices, Frobenius eigenvalues, and monodromy, vol. 45 of American Mathematical Society Colloquium Publications}. American Mathematical Society, Providence, RI (1999). Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, \(\zeta (s)\) and \(L(s, \chi)\), Analytic computations, Structure of families (Picard-Lefschetz, monodromy, etc.), Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics, Limit theorems in probability theory Random matrices, Frobenius eigenvalues, and monodromy | 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 infinite field, whose characteristics is neither \(2\) nor \(3\). The paper deals with smoothable Gorenstein \(K\)-points in a punctual Hilbert scheme, getting the following main results
\begin{itemize}
\item every \(K\)-point defined by local Gorenstein \(K\)-algebras with Hilbert function \((1,7,7,1)\) is smoothable;
\item the Hilbert scheme \(\mathrm{Hilb}_{16}^7\) has at least five irreducible components.
\end{itemize}
A new elementary component in \(\mathrm{Hilb}_{15}^7\) is found, starting from the study of \(\mathrm{Hilb}_{16}^7\), The problem is studied via properties double-generic initial ideals and of marked schemes. We remark that the considered problem is deeply related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more in general, of the irreducibility of a Hilbert scheme, a relevant and open question. Gorenstein algebra; Hilbert scheme; strongly stable ideal; marked basis Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Smoothable Gorenstein points via marked schemes and double-generic initial 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 \(M(n,d)\) denote the moduli space of stable vector bundles of rank \(n\) and degree \(d\) on a smooth curve \(X\). The Poincaré and Hodge polynomials of \(M(n,d)\) for \(n\) and \(d\) coprime were determined by several authors. In case \(n\) and \(d\) are not coprime, a few partial results are known. In this paper, the author determines the virtual Poincaré polynomial of \(M(n,d)\) for arbitrary \(n, d\). The virtual Poincaré polynomial of a variety over a finite field \(k\) is defined in terms of the weight filtration on the compactly supported \(l\)-adic cohomology groups. To determine the virtual Poincaré polynomial of \(M(n,d)\), the author introduces a new \(\lambda\) ring called the ring of \(c\)-sequences. A Poincaré function is associated to each \(c\)-sequence. The virtual Poincaré polynomial of \(M(n,d)\) is the Poincaré function of a \(c\)-sequence \(a_{n,d}\), where \(a_{n,d}(k)\) denotes the number of stable sheaves over \(X\) which remain stable over all finite field extensions.
The author gives conjectures regarding virtual Hodge polynomials of \(M(n,d)\) and also virtual Poincaré and Hodge polynomials of the motive of \(M(n,d)\). virtual Poincaré polynomial; moduli of stable bundles; smooth curve Mozgovoy, S.: Invariants of moduli spaces of stable sheaves on ruled surfaces (2013). arXiv:1302.4134 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Enumerative problems (combinatorial problems) in algebraic geometry Poincaré polynomials of moduli spaces of stable bundles over curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The concept of cylindrical algebraic decomposition (CAD) of the Euclidean \(n\)-space \(E^n\) has been introduced by \textit{G. E. Collins} in [Lect. Notes Comput. Sci. 33, 134--183 (1975; Zbl 0318.02051)]. It is closely related to the classical simplicial and CW-complexes of algebraic topology and has several applications. Given a finite set \(A\) of \(n\)-variate integral polynomials, a CAD is \(A\)-invariant if the polynomials of \(A\) are sign-invariant in each of the regions of the decomposition. The definition of CAD is based on a sort of induction on the dimension of the Euclidean space and a key point for \(A\)-invariant CAD construction is the choice of a \textit{projection} of the polynomials of \(A\), i.e.~a certain set of \((n-1)\)-variate integral polynomials.
This paper is devoted to the investigation of \(A\)-invariant CAD construction focusing on the projection proposed by \textit{D. Lazard} [in: Algebraic geometry and its applications. Collections of papers from Shreeram S. Abhyankar's 60th birthday conference held at Purdue University, West Lafayette, IN, USA, June 1-4, 1990. New York: Springer-Verlag. 467--476 (1994; Zbl 0822.68118)], whose proof presented some flaws. Despite these flaws, the authors show that Lazard's projection \(P_L(A)\) is valid when the set \(A\) is well-oriented with respect to \(P_L(A)\) (see Definition 3.4).
In Section 2, the authors analyze the flaws of Lazard's projection and compare it with the projection introduced by \textit{S. McCallum} [J. Symb. Comput. 5, No. 1--2, 141--161 (1988; Zbl 0648.68054); in: Quantifier elimination and cylindrical algebraic decomposition. Proceedings of a symposium, Linz, Austria, October 6--8, 1993. Wien: Springer. 242--268 (1998; Zbl 0900.68279)] and with the Brown-McCallum projection [\textit{C. W. Brown}, J. Symb. Comput. 32, No. 5, 447--465 (2001; Zbl 0981.68186)], motivating the interest for a reinstatement of Lazard's method with several arguments (e.g., see Remark 3.2 and Section 4).
Section 3 contains the main result (Theorem 3.1) which leads to the development of a CAD algorithm using Lazard's projection (Section 3.2). The rest of Section 3 is devoted to the proof of the main result, especially to the study of a technical lemma (Lemma 3.12) obtained as a consequence of an important theorem of Abhyankar and Jung to which a final Appendix is also devoted. cylindrical algebraic decomposition; projection operation; theorem of Abhyankar and Jung McCallum, S.; Hong, H., On using Lazard's projection in CAD construction, J. Symb. Comput., 72, 65-81, (2016) Polynomials, factorization in commutative rings, Symbolic computation and algebraic computation, Cylindric and polyadic algebras; relation algebras, Computational aspects in algebraic geometry On using Lazard's projection in CAD construction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The results of the article under review lie in the arithmetic intersection theory of Arakelov, Faltings, and Gillet and Soulé. It contains three main statements, the first two concern effective upper bounds on the number of effective sections of an arbitrary Hermitian line bundle and for a nef arithmetic divisor over an arithmetic surface, they can be viewed as effective versions of the arithmetic Hilbert-Samuel formula. The third statement improves the upper bound substantially for nef line bundles of small degree on the generic fiber. As a consequence, the authors obtained effective lower bounds on the Faltings height and on the sef-intersection of the canonical bundle in terms of the number of singular points on the fibers of the arithmetic surface.
To be more precise, let \(\overline{\mathcal{L}}=(\mathcal{L},\parallel\cdot\parallel)\) be a Hermitian line bundle over an arithmetic surface \(X\) and let
\[
\widehat{H}^0(\overline{\mathcal{L}})=\{s\in H^0(X,\mathcal{L}) : \parallel s\parallel_{\text{sup}}\leq 1\}
\]
denote the set of effective sections. Define \(\widehat{h}^0(\overline{\mathcal{L}})=\log \# \widehat{H}^0(\overline{\mathcal{L}})\) and \(\widehat{\text{vol}}(\overline{\mathcal{L}})=\limsup_{n\to \infty}\frac{2}{n^2}\widehat{h}^0(\overline{\mathcal{L}})\), Huayi Chen has showed that the ``\(\limsup\)'' is actually a limit and hence
\[
\widehat{h}^0(n\overline{\mathcal{L}})=\frac{1}{2}\widehat{\text{vol}}(\overline{\mathcal{L}})n^2+\o(n^2)\quad\text{as}\quad n\to \infty.
\]
The authors of the article under review obtained an effective version of the above expansion in one direction:
Theorem 1. Let \(X\) be a regular and geometrically connected arithmetic surface of genus \(g\) over \(\mathcal{O}_K\), where \(\mathcal{O}_K\) stands for the ring of integers in a number field \(K\). Let \(\overline{\mathcal{L}}\) be a Hermitian line bundle on \(X\). Denote \(d^0=\text{deg}(\mathcal{L}_K)\), and denote by \(r'\) the \(\mathcal{O}_K\)-rank of the \(\mathcal{O}_K\)-submodule of \(H^0(\mathcal{L})\) generated by \(\widehat{H}^0(\overline{\mathcal{L}})\). Assume that \(r'\geq2\).
(i). If \(g>0\), then \(\widehat{h}^0(\overline{\mathcal{L}})\leq\frac{1}{2}\widehat{\text{vol}}(\overline{\mathcal{L}})+4d\log(3d)\). Here \(d=d^0[K:\mathbb{Q}]\);
(ii). If \(g=0\), then \(\widehat{h}^0(\overline{\mathcal{L}})\leq(\frac{1}{2}+\frac{1}{2(r'-1)})\widehat{\text{vol}}(\overline{\mathcal{L}})+4r\log(3r)\). Here \(r=(d^0+1)[K:\mathbb{Q}]\).
For nef Hermitian line bundle \(\overline{\mathcal{L}}\), \(d^0=\text{deg}(\mathcal{L}_K)\geq0\) and the self-intersection number \(\overline{\mathcal{L}}^2\geq0\), the authors also obtained similar results to Theorem 1 for such \(\overline{\mathcal{L}}\).
Theorem 2. Let \(\overline{\mathcal{L}}\) be a nef Hermitian line bundle on \(X\) with \(d^0=\text{deg}(\mathcal{L}_K)>0\).
(i). If \(g>0\) and \(d^0>1\), then \(\widehat{h}^0(\overline{\mathcal{L}})\leq\frac{1}{2}\overline{\mathcal{L}}^2+4d\log (3d)\). Here \(d=d^0[K:\mathbb{Q}]\);
(ii). If \(g=0\) and \(d^0>0\), then \(\widehat{h}^0(\overline{\mathcal{L}})\leq(\frac{1}{2}+\frac{1}{2d^0})\overline{\mathcal{L}}^2+4r\log(3r)\). Here \(r=(d^0+1)[K:\mathbb{Q}]\).
The authors also gave a proof of the fact that Theorem 2 implies Theorem 1, which is inspired by the arithmetic Zariski decomposition of Moriwaki.
Theorem 2 may be too weak if \(\text{deg}(\mathcal{L}_K)\) is very small. The authors presented in this article a substantial improvement of Theorem 2 for the line bundles of small degree on the generic fiber, that's the following:
Theorem 3. Let \(X\) be a regular and geometrically connected arithmetic surface of genus \(g>1\) over \(\mathcal{O}_K\). Let \(\overline{\mathcal{L}}\) be a nef Hermitian line bundle on \(X\). Assume that \(2\leq d^0\leq 2g-2\). Then
\[
\widehat{h}^0(\overline{\mathcal{L}})\leq(\frac{1}{4}+\frac{2+\varepsilon}{4d^0})\overline{\mathcal{L}}^2+4d\log(3d).
\]
Here \(d=d^0[K:\mathbb{Q}]\). The number \(\varepsilon=1\) if \(X_K\) is hyperelliptic and \(d^0\) is odd; otherwise, \(\varepsilon=0\).
Applying Theorem 3 to the case where \(\overline{\mathcal{L}}\) is the Arakelov canonical bundle \(\overline{\omega}_X=(\omega_X,\parallel\cdot\parallel_{\text{Ar}})\), it reads
Theorem 4. \(\widehat{h}^0(\overline{\overline{\omega}}_X)\leq\frac{g}{4(g-1)}\overline{\omega}_X^2+4d\log(3d)\). Here \(d=(2g-2)[K:\mathbb{Q}]\).
Theorem 4 can be combined with Faltings' arithmetic Noether formula to deduce effective lower bounds on the Faltings height and on the self-intersection of the Arakelov canonical bundle in terms of the number of singular points on fibers of \(X\).
Finally, the authors indicated that the three main results in this article can be viewed as arithmetic version of Noether-type inequalities. effective bound; linear series; arithmetic surface; arithmetic Hilbert-Samuel formula; Nother-type inequalities X. Yuan and T. Zhang, Effective bound of linear series on arithmetic surfaces, Duke. Math. J. 162 (2013), 1723-1770. Arithmetic varieties and schemes; Arakelov theory; heights, Heights Effective bound of linear series on arithmetic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The monograph represents a deep study of interplay between the Toda and Kac-van Moerbeke hierarchies of integrable nonlinear difference equations and a thorough construction of their algebro-geometric quasi-periodic solutions. The reader will find a substantial account which provides more essential details of this technique than one can usually find in the standard literature. The authors have obtained a number of new results, among which there is a complete derivation of all real-valued algebro-geometric quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy. Many of these results are obtained by original and non-standard methods which are often simpler and more streamlined than the existing ones, and this circumstance makes all exposition fresh and instructive.
In the introduction the authors summarize briefly main results of the work and explain its construction. In Chapter 2 the authors build the Toda hierarchy of integrable difference equations by means of a recursive approach first advocated by Al'ber. This approach, though equivalent to the conventional one, has preference to yield naturally the Burchnal -Chaundy polynomials and hence the underlying hyperelliptic curves, creating in such a way a basis for subsequent algebro-geometric construction.
The stationary Baker-Akhiezer function of the Toda hierarchy is constructed in Chapter 3 by the original method which goes back to a classic representation of positive divisors of an algebraic curve by Jacobi and which has been applied to algebraic integrable equations by Mumford and McKean. The authors express the Baker-Akhiezer function in terms of the Riemann theta-functions, write down accurately the expressions for the normalizing constants, and discuss numerous properties of these functions. Chapter 4 represents a digression into the spectral properties of self-adjoint Jacobi operators in \(l^2(\mathbb{Z})\) for the limit point case. The authors build the Green functions and the Weyl matrices of the Jacobi operators and describe their properties in details. Along with it a construction of appropriate spectra and trace formulas are derived also.
In Chapter 5 the authors construct the algebro-geometric finite-gap solutions for the stationary Toda hierarchy. The major result of this chapter is expression of the Toda variables \(a(n), b(n)\) in terms of the theta-function associated with the appropriate hyperelliptic curve. Some of these formulas are new. In conclusion of the chapter a criterion for the Toda variables to be periodic is presented. In Chapter 6 the authors build the algebro-geometric finite-gap solutions for the non-stationary Toda hierarchy with the stationary solution as initial condition. The authors obtain the equations for the time evolution of the Dirichlet eigenvalues and study properties of their solutions. Then the authors derive for the case of Toda flow the expressions for time-dependent Baker-Akhiezer function and Toda variables in terms of theta-functions.
A connection between the Toda hierarchy and the Kac-van Moerbeke hierarchy is considered in Chapter 7. It appears that these hierarchies are related in such a way as the Gel'fand-Dickey hierarchy and the Drinfeld-Sokolov hierarchy. In all these cases the hierarchies are counterparts of Miura-type transformation which is based on a factorization method. The discrete analog of the properly generalized Miura transformation in connection with the factorization method was employed systematically for the first time by Adler (1981) and developed further by Gesztesy, Holden, Simon and Zhao (1993).
In Chapter 8 the authors summarize basic facts on the finite-gap Dirac-type difference operators and study briefly their spectral properties. At last in chapter 9 the authors complete the main object of the work and construct all real-valued algebro-geometric quasi-periodic finite-gap solutions of the Kac-van Moerbeke hierarchy. Along with it they consider also appropriate isospectral manifolds. In conclusion of the chapter a number of possible applications of these results to different completely integrable lattice models is discussed shortly.
In order to make the exposition self-contained the authors include three Appendices on hyperelliptic curves and theta-functions, on periodic Jacobi operators and on the simplest explicit examples of algebro-geometric solutions for the genus 0 and 1. algebraically integrable non-linear evolution equations; Jacobi operators; Toda hierarchy W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, ''Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies,'' Mem. Amer. Math. Soc., vol. 135, iss. 641, p. x, 1998. Other completely integrable equations [See also 58F07], Difference operators, Research exposition (monographs, survey articles) pertaining to partial differential equations, Research exposition (monographs, survey articles) pertaining to difference and functional equations, Discrete version of topics in analysis, Soliton equations, Elliptic curves Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies | 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 deals with the problem of classifying complete \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\), the ring of formal power series in one variable.
A discrete invariant providing such a classification is a semigroup \(\Gamma\), in \(\mathbb{N}\), obtained by taking the orders of the elements of a given \(\mathbb{C}\)-subalgebra of \(\mathbb{C}[[t]]\). Hence, the problem is reduced to classify complete \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\) with a given semigroup \(\Gamma\). So, it is possible to define the space \(R_{\Gamma}\) of all \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\) with given \(\Gamma\).
As semigroups arising from unibranch curve singularities are exactly the so-called numerical semigroups, the Author focuses on \(R_{\Gamma}\) for this type of semigroup and the study of the space \(R_{\Gamma}\) is motivated by showing how it relates to the Zariski moduli space of curve singularities on the one hand and to a moduli space of global singular curves on the other.
In particular, \(R_{\Gamma}\) is proved to be an affine variety by providing an algorithm, which yields its defining equations in an ambient affine space in terms of the given semigroup; some examples show how to use these results to explicitly compute \(R_{\Gamma}\). Moreover, the question is addressed of whether or not \(R_{\Gamma}\) can always be identified with an affine space and whether is there a numerical criterion for this; although the problem remains open, certain types of semigroups are identified, for which \(R_{\Gamma}\) is always an affine space, and for general \({\Gamma}\) a finite stratification of \(R_{\Gamma}\) is described, by locally closed subsets corresponding to subalgebras with a fixed number of generators. The relationship between \(R_{\Gamma}\) and the Zariski moduli space \(\mathcal{M}_{\Gamma}\) is also explored, by explicitly computing the natural map from \(R_{\Gamma}\) to \(\mathcal{M}_{\Gamma}\) in some special cases.
The same algebraic problem addressed in the present paper was considered (yet only in an abstract and scheme-theoretic perspective) by \textit{S. Ishii} [J. Algebra 67, 504--516 (1980; Zbl 0469.14013)]. curve singularities; classification; numerical semigroups; Zariski moduli space Families, moduli of curves (algebraic), Singularities of curves, local rings Classifying complete \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\) | 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 Suppose \(k\) is an algebraically closed field of positive characteristic \(p>0\). An \(F\)-splitting of a \(k\)-scheme \(X\) is an \(\mathcal{O}_{X}\)-linear map splitting the map \(F^{*}: \mathcal{O}_{X}\longrightarrow F_{*}\mathcal{O}_{X}\) induced by the absolute Frobenius morphism; \(X\) is \(F\)-split if such a splitting exists.
This paper under review studies the \(F\)-splitting and \(F\)-regularity properties of normal varieties equipped with an effective action by a diagonalizable group. Let \(H\) be a diagonalizable group over \(k\) and \(X\) be a normal \(k\)-variety with an effective \(H\)-action. Let \(X^{\circ}\) be the open subvariety of \(X\) consisting of those points with finite stabilizers, and assume that \(X^{\circ}\) admits a geometric quotient \(\pi:X^{\circ}\longrightarrow Y, \) \(Y=X^{\circ}/H\), (e.g., \(H\) is a torus, or \(X\) is quasiprojective). An effective \(\mathbb{Q}\)-divisor \(\Delta\) on \(Y\) is defined by
\[
\sum_{P\subset Y}\frac{\mu(P)-1}{\mu(P)}P,
\]
where \(\mu(P)\) is the order of the stabilizer of the generic point of any irreducible component of \(\pi^{-1}(P)\subset X^{\circ}\). The main theorem (Theorem 4.1) in this paper states that \(X\) is \(F\)-split (\(F\)-regular) if and only if the pair \((Y,\Delta)\) is \(F\)-split (\(F\)-regular).
The machinery developed in the paper actually gives a bijection between \(H\)-invariant \(F\)-splittings of \(X\) and \(F\)-splittings of \((Y,\Delta)\), as well as giving a partial description of the set of all \(F\)-splittings of \(X\) in terms of the quotient pair \((Y,\Delta)\) (see Remark 4.11). The authors also consider some applications to analysis of the \(F\)-splitting and \(F\)-regularity of complexity-one \(T\)-varieties and toric vector bundles. Group actions on varieties or schemes (quotients), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure \(F\)-split and \(F\)-regular varieties with a diagonalizable group action | 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 studies the topology of singularities of regular functions on complex algebraic varieties with a help of their contact loci. Recall that an \(m\)-jet on a complex algebraic variety \(X\) is a morphism \(\gamma\colon \mathrm{Spec}\,\mathbb{C}[t]/(t^{m+1})\to X\) of schemes over \(\mathbb{C}\). The \(m\)-th jet scheme \(\mathcal{L}_m(X)\) is the variety parameterizing the \(m\)-jets on \(X\). For a regular function \(f\) on \(X\) and a Zariski closed subset \(\Sigma\) of \(X_0=f^{-1}(0)\) the \emph{\(m\)-th restricted contact locus} \(\mathcal{X}_m(f,\Sigma)\) of \(f\) is the subset of \(\mathcal{L}_m(X)\) consisting of jets \(\gamma\) such that \(\gamma(0)\in\Sigma\) and the composition \(f\circ\gamma\) is equivalent to \(t^m\) modulo \(t^{m+1}\).
Now let \(h\colon Y\to X\) be a log resolution of \((f,\Sigma)\) such that \(E=h^{-1}(X_0)\) and \(h^{-1}(\Sigma)\) are divisors with simple normal crossings and \(h\) is a composition of blowing ups of smooth centers. Denote by \(E_i\), \(i\in S\), the irreducible components of \(E\), \(m_i\) the multiplicity of \(f\) at \(E_i\), and \(E_{i}^{0}=E_i\setminus(\cup_{j\ne i} E_j)\). As a technical condition, the authors show that \(h\) can be chosen so that \(m_i+m_j>m\) if \(E_i\cap E_j\ne\emptyset\) for all \(i,j\in S\). Finally, the authors describe a certain unramified cyclic covering \(\tilde{E}_{i}^{0}\) of \(E_{i}^{0}\) of degree \(m_i\). As their main result, the authors construct under the condition that \(X\) is smooth a spectral sequence which converges to the integral cohomologies with compact support of \(\mathcal{X}_m(f,\Sigma)\) and where the first page consists of certain direct sums of integral homologies of the cyclic coverings \(\tilde{E}_{i}^{0}\).
In the case \(X=\mathbb{C}^d\) and \(f\) has isolated singularity at \(\Sigma\) the authors note that the first page of their spectral sequence coincides up to relabeling with the first page of the spectral sequence from \textit{M. McLean} [Geom. Topol. 23, No. 2, 957--1056 (2019; Zbl 1456.14042)], converging to Floer cohomology of the \(m\)-th iterate of the monodromy of \(f\). The authors conjecture that in fact the spectral sequences are isomorphic. When \(f\) has multiplicity \(m\) at the origin, the authors show that the cohomology groups \(H_{c}^{*}(\mathcal{X}_m(f,\Sigma),\mathbb{Z})\) are isomorphic to the homology groups \(H_{2(dm-1)-*}(F,\mathbb{Z})\) of the Milnor fiber \(F\simeq \{f_m=1\}\) of the initial form \(f_m\) of \(f\). Motivated by this observation, the authors propose a conjecture that if two germs of holomorphic functions on \(\mathbb{C}^d\) are embedded topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. contact locus; jet scheme; Milnor fiber; log resolution Arcs and motivic integration, Milnor fibration; relations with knot theory, Mixed Hodge theory of singular varieties (complex-analytic aspects) Cohomology of contact loci | 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 Denote with \(\mathcal{ M}(2;\mathbb{C})\) the set of \(2\times 2\) matrices with complex coefficients. In the paper under review, the authors consider several questions concerning the iteration of rational and polynomial self-maps \(\Phi\) of \(\mathcal{ M}(2;\mathbb{C})\). Such a self-map \(\Phi\) is called \textit{compatible with conjugation} if \(A\Phi(M)A^{-1}=\Phi(AMA^{-1})\) for a matrix \(A\), for all \(M\in\mathcal{ M}(2;\mathbb{C})\), whenever this makes sense. It is possible to associate a fibration by \(2\)-planes to the center \(\mathcal{ C}=\{\lambda\mathrm{ Id}: \lambda\in\mathbb{C}\}\) of the ring \(\mathcal{ M}(2;\mathbb{C})\). More precisely, for \(M\in\mathcal{ M}(2;\mathbb{C})\setminus\mathcal{ C}\), let \(\mathcal{ P}(M)\) denote the unique plane containing both \(M\) and \(\mathcal{ C}\): these are the matrices commuting with \(M\), so if \(\Phi\) is compatible with conjugation, then it preserves the fibration by the planes \(\mathcal{ P}(M)\). It is possible to consider the map \(\mathrm{ Inv}:\mathcal{ M}(2;\mathbb{C})\to \mathbb{C}^2\) given by \(\mathrm{ Inv}(M) =(\mathrm{tr}(M), \det(M))\), which takes a matrix \(M\) to its invariants. The authors prove in the present paper that \textsl{if \(\Phi\) is compatible with conjugation, then there is a map \(\mathrm{ Sq}~\Phi:\mathbb{C}^2\to \mathbb{C}^2\) such that \({\mathrm{ Sq}~\Phi\circ \mathrm{ Inv} }= \mathrm{ Inv}\circ \Phi\).} Thus \(\Phi\) is semi-conjugate to a lower-dimensional map, passing indeed to the quotient, modulo the invariant fibration \(\mathcal{ P}\). Moreover, \(\mathrm{ Sq}~\Phi\) contains a big part of the dynamics of \(\Phi\).
The authors study in particular the self-map \(\Phi_A\) of \(\mathcal{ M}(2;\mathbb{C})\) given by \(\Phi_A(M)=AM^2\). With such a notation, \(\Phi_{\mathrm{ Id}}\) is the squaring map \(M\mapsto M^2\). The paper contains several results about these maps for different choices of \(A\). Write \(\mathrm{ Aut}(\mathcal{ M}(2;\mathbb{C});{\Phi}_A)\) [resp. \(\mathrm{ Bir}(\mathcal{ M}(2;\mathbb{C});{\Phi}_A)\)] for the set of automorphisms [resp. birational maps] of \(\mathcal{ M}(2;\mathbb{C})\) commuting with \(\Phi_A\). It is clear that for any \(P\in\mathrm{GL}(2;\mathbb{C})\), the map \(\sigma_P(M) = PMP^{-1}\) commutes with \(\Phi_{\mathrm{ Id}}\), as well as the involution by transposition \(M\mapsto M^t\). The authors prove that \textsl{\(\mathrm{ Aut}(\mathcal{ M}(2;\mathbb{C});\Phi_{\mathrm{ Id}})\) is the semi-direct product of the group of maps \(\sigma_P\) with the transposition map}. Moreover, they show that, \textsl{by adding the birational involution \(M\mapsto M^{-1}\), we obtain \(\mathrm{ Bir}(\mathcal{M}(2;\mathbb{C});\Phi_{\mathrm{ Id}})\)}.
The authors investigate then veriuous other properties of \(\Phi_A\), particularly in the diagonal case \(A=\mathrm{ Diag}(\lambda,\lambda^{-1})\). They determine, for instance, the closure of the periodic points of \(\Phi_A\) for generic \(\lambda\). Moreovere, they describe the attracting basin of \(\Phi_A\) at the origin. They finally prove that \textsl{if \(\Phi_A\) is not conjugate to \(\Phi_{\mathrm{ Id}}\), then \(\mathrm{ Aut}(\mathcal{ M}(2;\mathbb{C});\Phi_A)\) is isomorphic to a \(\mathbb{C}^*\)-action on \(\mathcal{ M}(2;\mathbb{C})\).} iteration of rational maps; matrix spaces Cerveau, Dominique; Déserti, Julie, Itération d'applications rationnelles dans les espaces de matrices, Conform. Geom. Dyn., 1088-4173, 15, 72-112, (2011) Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Rational and birational maps, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Iteration for rational maps in matrix 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 The present article investigates the existence of separation theorems by polynomials. This is a variation of the complex Hahn-Banach separation theorem, which establishes that if \(K\subset\mathbb C^n\) is a closed convex set and \(z\) is a point in \(\mathbb C^n\setminus K\), then there exists a complex linear form \(f\) with sup\(_{w\in K}\{\mathrm{Re}(f(w))\}< \mathrm{Re}(f(z))\). If we assume that \(K\) is balanced we can replace the real part of \(f\) by the modulus of \(f\): sup\(_{w\in K}|f(w)|<|f(z)|\). In this work, the authors change the conditions on \(K\) by assuming that \(K\) is invariant under the action of a finite group of linear transformations. The function \(f\) will then be a polynomial. We say that a polynomial \(P\) separates a set \(K\) and a point \(z\) if sup\(_{w\in K}|P(w)|<|P(z)|\). In the main result, Theorem 2.3, is proven that if \(G\) is a finite subgroup of the general linear group \(\mathrm{GL}(n,\mathbb C)\), \(K\subset \mathbb C^n\) is invariant under the action of \(G\) and \(z\) is a point in \(\mathbb C^n\setminus K\) that can be separated from \(K\) by a polynomial \(Q\), then \(z\) can be separated from \(K\) by a \(G\)-invariant polynomial \(P\). Furthermore, if \(Q\) is homogeneous then \(P\) can be chosen to be homogeneous. As a corollary is obtained that if \(K\) is a symmetric polynomially convex compact set in \(\mathbb C^n\) and \(z\notin K\) then there exists a symmetric polynomial that separates \(z\) and \(K\). group invariant polynomials; separation theorem; symmetric polynomials; polynomial convexity Aron, R; Falcó, J; Maestre, M, Separation theorems for group invariant polynomials, J. Geom. Anal., 28, 393-404, (2018) Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Polynomial convexity, rational convexity, meromorphic convexity in several complex variables Separation theorems for group invariant polynomials | 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 introduction: If \(X\) is a \(C^{\infty}\) manifold with Poisson structure \(\alpha\) then Kontsevich has proved that there exists always a deformation quantization of the algebra of functions \(C^{\infty}(X)\) with first-order term \(\alpha\) [\textit{M. Kontsevich}, Lett. Math. Phys. 66, No. 3, 157--216 (2003; Zbl 1058.53065)]. Moreover, Kontsevich proved that such a deformation is unique in a suitable sense.
In the present paper, the author tries to see to what extent the methods applied in the case of \(C^{\infty}\) manifolds can be carried over to the algebro-geometric case. Let \(X\) be a smooth algebraic variety over a field of characteristic zero \(\mathbb K\). The author introduces the notion of Poisson structure on \(X\) and the notion of star product on \({\mathcal O}_X[[\hbar]]\) where \({\mathcal O}_X\) is the sheaf of functions and \(\hbar\) an indeterminate. A deformation quantization of \((X,\alpha)\) is then a star product on \({\mathcal O}_X[[\hbar]]\) whose associated star bracket is \(\alpha\).
The main result of the paper asserts that, under the assumptions that \(X\) is \(\mathcal D\)-affine and that \({\mathbb R}\subset {\mathbb K}\), there is a canonical map \(Q\) from the set of formal Poisson structures on \(X\) (up to equivalence) to the set of deformation quantizations on \({\mathcal O}_X\) (up to equivalence). The quantization map \(Q\) preserves first-order terms and commutes with étale morphisms \(X'\rightarrow X\). If \(X\) is affine then \(Q\) is bijective. The map \(Q\) is given by an explicit formula. This result is an algebraic analogue of \textit{M. Kontsevich} (loc. cit.), Theorem 3.1. deformation quantization; noncommutative algebraic geometry; DG Lie algebras; formal geometry A. Yekutieli, Deformation quantization in algebraic geometry, \textit{Adv. Math.}, 198 (2005), no. 1, 383--432.Zbl 1085.53081 MR 2183259 Deformation quantization, star products, Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory, Deformations of associative rings Deformation quantization in 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 Author's abstract: \textit{H. Rosengren} and \textit{M. Schlosser} [Compos. Math. 142, No. 4, 937--961 (2006; Zbl 1104.15009)] introduced notions of $R_{N}$ -theta functions for the seven types of irreducible reduced affine root systems, $R_{N}=A_{N_1}$, $B_{N}$, $B_{N}^{V}$, $C_{N}$, $C_{N}^{V}$, $BC_{N}$, $D_{N}$, $N\in N$, and gave the Macdonald denominator formulas. We prove that if the variables of the $R_{N}$ -theta functions are properly scaled with $N$, they construct seven sets of biorthogonal functions, each of which has a continuous parameter $t\in (0,t_{*})$ with given $0< t_{*} < \infty $. Following the standard method in random matrix theory, we introduce seven types of one-parameter $(t\in (0,t_{*}))$ families of determinantal point processes in one dimension, in which the correlation kennels are expressed by the biorthogonal theta functions. We demonstrate that they are elliptic extensions of the classical determinantal point processes whose correlation kernels are expressed by trigonometric and rational functions. In the scaling limits associated with $N\to \infty$, we obtain four types of elliptic determinantal point processes with an infinite number of points and parameter $t\in (0,t_{*})$. We give new expressions for the Macdonald denominators using the Karlin-McGregor-Lindstrom-Gessel-Viennot determinats for noncolliding Brownian paths and show the realization of the associated elliptic determinantal point processes as noncolliding Brownian brides with a time duration $t_{*}$, which are specified by the pinned configurations at time $t=0$ and $t=t_{*}$. Macdonald denominators; affine root systems, orthogonal theta functions; elliptic determinantal processes Point processes (e.g., Poisson, Cox, Hawkes processes), Theta functions and abelian varieties, Basic hypergeometric functions associated with root systems, Random matrices (probabilistic aspects), Brownian motion Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes | 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 \(C^\infty\)-manifold and let \(\phi: X \rightarrow X\) be a morphism. The author defines a \(\phi\)-graph trace kernel slightly generalizing the corresponding construction on manifolds introduced by \textit{M. Kashiwara} and \textit{P. Schapira} [J. Inst. Math. Jussieu 13, No. 3, 487--516 (2014; Zbl 1327.14083)]. He then discusses some basic properties of the microlocal Lefschetz class in this setting. Indeed, his main result is the functoriality of microlocal Lefschetz classes with respect to the composition of graph trace kernels. As an application, the microlocal Lefschetz fixed point formula for constructible sheaves on a real analytic manifolds is obtained [\textit{Y. Matsui} and \textit{K. Takeuchi}, Int. Math. Res. Not. 2010, No. 5, 882--913 (2010; Zbl 1198.32003)]. microlocal sheaf theory; trace kernels; Lefschetz classes; Lefschetz fixed point formulas; constructible sheaves; real analytic manifolds Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Microlocal Lefschetz classes of graph trace kernels | 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, in the reviewers opinion, a major contribution to singularity theory.
Based on the remarkable paper by \textit{V. V. Nikulin} [Math. USSR Izv. 14, 103-167 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111- 177 (1979; Zbl 0408.10011)] the authors attack the problem of characterizing the smoothing components and the intersection forms of the corresponding Milnor fiber, for isolated complex-analytic surface singularities.
Let (X,0) be any such singularity, i.e. a Stein space of dimension \(2\) with a smooth \((C^{\infty})\) boundary \(\partial X=L\), and a unique singular point 0. L is usually called the link of the singularity. Let \(\pi: \tilde X\to X\) be a good resolution, i.e. one with an exceptional curve \(E=\pi^{-1}(0)\) consisting of nonsingular irreducible components intersecting transversally. Consider a smoothing \(f: {\mathcal X}\to \Delta,\) where \(\Delta\) is an open disc in \({\mathbb{C}}\) containing 0. Here \({\mathcal X}\) is a Stein space with a partial \(C^{\infty}\)-boundary \(\partial {\mathcal X}\). There is an isomorphism of analytic spaces with boundary \(i: f^{- 1}(0)\overset \sim \rightarrow X\) and \(f| int({\mathcal X}-\{0\})\) and \(f| \partial {\mathcal X}\) are both submersions. The Milnor fiber M, i.e. the generic fiber of f, has a boundary \(\partial M\) which is diffeomorphic to L. Therefore the 3-dimensional manifold L bounds both of the two oriented 4- dimensional manifolds \(\tilde X\) and M.
The main result of this paper, (3.7), shows that the classical linking pairing of the link L, \(b: H_ 1(L)_ t\times H_ 1(L)_ t\to Q/Z\) where the subscript t denotes the torsion part, is in an obvious sense the link between the intersection pairings of \(H_ 2(\tilde X)\) and \(H_ 2(M)\). In fact, the complex structures of \(\tilde X\) and M induces an, up to homotopy, unique complex structure on the sum of the trivial bundle R and the tangent bundle \(\tau_ L\) of L.
Using the first Chern class of \(-\tau_ L\oplus R\) the authors construct a quadratic function \(q: H_ 1(L)_ t\to Q/Z\) such that the corresponding bilinear form \(b(x,y)=-(q(x+y)-q(x)-q(y))\) is the linking pairing. Now, let \(K_{\tilde X}:=-c_ 1(\tau_{\tilde X})\in H^ 2(\tilde X)=Hom(H_ 2(\tilde X),Z)\) then the bilinear form of the quadratic function \(Q_{\tilde X}: H_ 2(\tilde X)\to Z\) defined by \(Q_{\tilde X}(x)=(x\cdot x+K_{\tilde X}(x)),\) is the intersection form of \(H_ 2(\tilde X)\). Notice that since this form is negative definite, there is an element \(k\in H_ 2(\tilde X):=\{x\in H_ 2(\tilde X)\oplus Q| \forall y\in H_ 2(\tilde X), x\cdot y\in Z\}\) such that \(K_{\tilde X}(x)=k\cdot X\) for all \(x\in H_ 2(\tilde X).\)
To \(Q_{\tilde X}\) there is associated a discriminant quadratic function (DQF) \(q_{\tilde X}: H_ 2(\tilde X)/H_ 2(\tilde X)\to Q/Z\) and there is a natural isomorphism \(H_ 2(\tilde X)/H_ 2(\tilde X)\simeq H_ 1(L)_ t\) identifying \(q_{\tilde X}\) and q. Moreover, see (3.7) and (4.5), let \(K_ M\in Hom(H_ 2(M),Z)\) be the image of \(-c_ 1(\tau_ M)\), then the corresponding bilinear form of the quadratic function \(Q_ M: H_ 2(M)\to Z\) defined by \(Q_ M(x)=(x\cdot x+K_ M(x))\) is the intersection form of \(H_ 2(M)\), and the associated DQF is isomorphic to the quadratic function \(q_ I: I^{\perp}/I\to Q/Z\) induced by q, where \(I:=im(H_ 2(M,L)_ t\to H_ 1(L)_ t)\subseteq H_ 1(L)_ t\) is q-isotropic.
Specializing to Gorenstein singularities, the authors observe, see (4.8), that \(k\in H_ 2(\tilde X)\) and that q is a quadratic form. Moreover, \(K_ M=0\) implying that \(\bar H{}_ 2(M):=H_ 2(M)\) modulo torsion and the radical of the intersection form, is an even lattice with DQF canonically isomorphic to \(q_ I.\)
For Gorenstein singularities the formulas of \textit{J. H. M. Steenbrink} [in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 513-536 (1983; Zbl 0515.14003)] express the Sylvester invariants \((\mu_ 0,\mu_+,\mu_-)\) of the intersection form of \(H_ 2(M)\) as linear functions in the Betti number \(b_ 1(\tilde X)\), the genus p(X) and \(k\cdot k.\)
Consider now a smoothing component of the base space of a versal deformation. Any two smoothings contained in the same component will determine the same q-isotropic subgroup \(I\subseteq H_ 1(L)_ t\) and the same quadratic function \(q_ I\). Therefore \((\mu_ 0,\mu_+,\mu_-)\), I and \(q_ I\) are invariants of the smoothing component. The authors refine these invariants to what they call a smoothing datum.
The last part of the paper is concerned with the problem of characterizing the smoothing components, together with the intersection forms of the corresponding Milnor fibers, studying the subset of ''permissible'', see {\S} 6, smoothing data, and applying the classification of Nikulin, loc. cit. The discussion is confined to the minimally elliptic singularities, i.e. Gorenstein and \(p(X)=1\). For simple elliptic and triangle singularities this analyses gives a rather complete answer to the problems posed. smoothing components; intersection forms; Milnor fiber; isolated complex- analytic surface singularities; link of the singularity; discriminant quadratic function Looijenga (E.) & Wahl (J.).- Quadratic functions and smoothing surface singularities. Topology 25, p. 261-291 (1986). Zbl0615.32014 MR842425 Deformations of complex singularities; vanishing cycles, Local complex singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Complex singularities Quadratic functions and smoothing surface 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 book under review develops algebraic techniques which are especially adapted to the requirements of real algebraic geometry. A lot of function rings appear in real algebraic geometry as extensions (a sort of completions or closures) of the ring of polynomial functions on an algebraic set. The present work shows how these extensions can be described and treated in a general framework of monoreflectors of the category of reduced partially ordered rings.
A subcategory \({\mathbf D}\) of a category \({\mathbf C}\) is reflective if the inclusion functor has a left adjoint \(r:{\mathbf C} \to{\mathbf D}\). In this case \(r\) is called a reflector, and \(r_A:A\to r(A)\) (where \(A\) is an object of \({\mathbf C})\) is called the reflection morphism of \(A\). A reflector is a monoreflector if the reflection morphisms are monomorphisms. The monoreflective subcategories of the category POR/N of reduced partially ordered rings form a complete lattice. An important class of reflectors is composed by the \(H\)-closed monoreflectors. A monoreflector \(r:\text{POR/N} \to{\mathbf D}\) on the category of reduced partially ordered rings is called \(H\)-closed if for any partially ordered ring \((A,P)\) belonging to \({\mathbf D}\) and any surjective homomorphism \((A,P) \to(B,Q)\) in POR/N, the partially ordered ring \((B,Q)\) also belongs to \({\mathbf D}\). The \(H\)-closed monoreflective subcategories of POR/N also form a complete lattice.
The book contains 23 sections. Sections 1-5 are introductory. Sections 6-15 are devoted to rings of semi-algebraic functions and real closed rings. A systematic study of some intervals in the lattice of \(H\)-closed monoreflectors is performed in sections 16-22. Section 23 contains a summary of results on the lattice of monoreflectors of the category POR/N of reduced partially ordered rings and on the lattice of \(H\)-closed monoreflectors of POR/N. real algebraic geometry; complete lattice Schwartz, N., Madden, J.J.: Semi-algebraic Function Rings and Reflector of Partially Ordered Rings. Lecture Notes in Mathematics, vol. 1712. Springer, Berlin (1999) Semialgebraic sets and related spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to category theory, Research exposition (monographs, survey articles) pertaining to ordered structures, Ordered rings, algebras, modules, Ordered rings, Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Real-valued functions in general topology, Preorders, orders, domains and lattices (viewed as categories) Semi-algebraic function rings and reflectors of partially ordered 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 The paper under review is based on the author's Master`s Thesis, written under the supervision of S. Kobayashi (Tohoku University, Japan).
The author discusses interesting directions of the generalization of the theory of power series by \textit{R. F. Coleman} [Invent. Math. 53, 91--116 (1979; Zbl 0429.12010)] in the framework of \textit{B. Perrin-Riou} [Invent. Math. 115, No. 1, 81--149 (1994; Zbl 0838.11071)] and of \textit{S. Kobayashi} [Invent. Math. 191, No. 3, 527--629 (2013; Zbl 1300.11053)].
To generalize the theory of Kobayashi [loc. cit.] to general commutative formal groups over unramified rings, one needs the notion of \(Q\)-norm systems. Here, the author uses results by \textit{B. Perrin-Riou} [Invent. Math. 99, No. 2, 247--292 (1990; Zbl 0715.11030)].
The main result of the paper under review is Theorem 1.1. (cf. Theorem 4.6). This extends results by Coleman, Kobayashi and Perrin-Rion [loc. cit.] to \(d\)-dimensional commutative formal groups over \({\mathbb Z}_p\) of finite height \(h\).
The theorem is proved by a modification of a proof by Kobayashi [loc. cit.] with the help of mention result and the extension of Proposition 2.1 in [\textit{H. Knospe}, Manuscr. Math. 87, No. 2, 225--258 (1995; Zbl 0847.14026)] to the author's case.
The paper contains several lemmas and propositions which are of independent interest. Coleman power series; formal group; Dieudonne module Ota, Kazuto, A generalization of the theory of Coleman power series, Tohoku Math. J., 66, 309-320, (2014) Formal groups, \(p\)-divisible groups, Class field theory; \(p\)-adic formal groups, \(p\)-adic theory, Formal power series rings A generalization of the theory of Coleman power 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 The reviewed article is the first of a series of papers where the authors study the relationship between the asymptotic Hodge theory of a degeneration and the mixed Hodge theory of its singular fiber. The motivation to study this relationship comes from studying of compactifications of moduli spaces. The period map has been used to study moduli spaces of several examples, including abelian varieties, \(K3\) surfaces, hyper-Kähler manifolds, and cubic threefolds and fourfolds. In these examples, the period map embeds each moduli space as an open subset of a locally symmetric variety, which facilitates the comparison between Hodge-theoretic and geometric compactifications.
The authors are interested in extending the use of period maps in studying moduli of other examples, including surfaces of general type and Calabi-Yau threefolds, and establishing a similar strong connection between the compactifications. A key challenge is to compute the limiting mixed Hodge structures in the geometric boundary from the geometry of the fibers over this boundary.
With this goal in mind, the authors generalize the Clemens-Schmid sequence [\textit{C. H. Clemens}, Duke Math. J. 44, 215--290 (1977; Zbl 0353.14005)] in various ways. The first of these is in the case of a projective family of varieties over a disk.
{Theorem 1.} Let \(f:\mathcal{X}\to\Delta\) be a flat projective family of varieties over the disk, which is the restriction of an algebraic family over a curve, such that \(f\) is smooth over \(\Delta*\). If \(\mathcal{X}\) is smooth, then we have exact sequences of mixed Hodge structures \[ 0 \to H^{k-2}_{\lim}(X_t)_T(-1) \to H_{2n-k+2}(X_0)(-n-1) \to H^k(X_0) \to H^k_{\lim}(X_t)^T\to 0\] for every \(k\in\mathbb{Z}\), where the outer terms are the coinvariants and the invariants of the monododromy operator \(T\) on the limiting mixed Hodge structure.
The authors obtain more precise information when considering the Hodge numbers \(h^{p,q}\) with \(pq = 0\), namely, that these numbers are preserved under such degenerations. The first isomorphism of the following theorem extends a result from Steenbrink [\textit{J. H. M. Steenbrink}, Compos. Math. 42, 315--320 (1981; Zbl 0428.32017)], who proved it in the case when \(X_0\) has Du Bois singularities (using the result in [\textit{J. Kollár} and \textit{S. Kovács}, J. Am. Math. Soc. 23, No. 3, 791--813 (2010; Zbl 1202.14003 )].
{Theorem 2.} Let \(f:\mathcal{X}\to\Delta\) be a flat projective family of varieties over the disk, which is the restriction of an algebraic family over a curve, such that \(f\) is smooth over \(\Delta*\). Suppose that \(\mathcal{X}\) is normal and \(\mathbb{Q}\)-Gorenstein, and that the special fiber \(X_0\) is reduced.
\begin{itemize}
\item[1.] If \(X_0\) is semi-log-canonical, then \[ Gr^0_FH^k(X_0) \cong Gr^0_FH^k_{\lim}(X_t)\cong Gr^0_FH^k_{\lim}(X_t)^{T^{ss}}\] for all \(k\in\mathbb{Z}\), where \(T=T^{n}T^{ss}\) is the Jordan decomposition of the monodromy into the unipotent and (finite) semisimple parts.
\item[2.] If \(X_0\) is log-terminal, then additionally \[W_{k-1}Gr^0_FH^k_{\lim}(X_t) = \{0\}\] for all \(k\in\mathbb{Z}\).
\end{itemize}
The authors obtain a similar result one step higher in the Hodge filtration under stronger assumptions.
{Theorem 3.} Let \(f:\mathcal{X}\to\Delta\) be as in Theorem 2. Assume that the total space \(\mathcal{X}\) is smooth and the special fiber \(X_0\) is log-terminal (or more generally, has rational singularities). Then \[Gr^1_FH^k(X_0) \cong Gr^1_F(H^k(X_t))^{T^{ss}}.\]
The main technique used in the reviewed article is the theory of mixed Hodge modules from M. Saito [\textit{M. Saito}, Publ. Res. Inst. Math. Sci. 26, No. 2, 221--333 (1990; Zbl 0727.14004)], and especially the Decomposition Theory in this setting. The first part of the article reviews the result, with an emphasis on the Decomposition Theorem over a curve. In addition, several concrete geometric examples are discussed in Section 6.
The article ends with an Appendix by the third author, which discusses the Decomposition Theorem over a curve in the context of analytic spaces and includes the following general equivalence result.
{Theorem A.} Let \(f:X\to C\) be a proper surjective morphism of a connected complex manifold \(X\) to a connected non-compact curve \(C\). The Decomposition Theorem for \(\mathbf{R} f_*\mathbb{Q}_X\) is equivalent to the Clemens-Schmid exact sequence (or the local invariant cycle theorem) for every singular fiber of \(f\).
A condition for the Decomposition Theorem to hold in the context of Theorem 4 is described in terms of a resolution of singularities of \(X\) (Corollary A). limiting mixed Hodge structure; Clemens-Schmid sequence; mixed Hodge modules; moduli spaces Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects) Hodge theory of degenerations. I: Consequences of the decomposition 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 We are motivated by the question of when a convex semialgebraic set in \(\mathbb R^n\) is equal to the feasible set of a linear matrix inequality (LMI). Given a basic semialgebraic set, \(\mathcal{V}\), which is defined by quadratic polynomials, we restrict our attention to closure of its convex hull, namely \(\overline{\mathbf{co}{(\mathcal V)}}\). Our main result is that \(\overline{\mathbf{co}{(\mathcal V)}}\) is equal to the intersection of a finite number of LMI sets and the halfspaces supporting \(\mathcal V\) along a particular subset of the boundary of \(\mathcal V\). As a corollary, we show that in \(\mathbb R^2\), the halfspaces of concern are finite in number, so that an LMI representation for \(\overline{\mathbf{co}{(\mathcal V)}}\) always exists. Yıldıran, U; Kose, IE, LMI representations of the convex hulls of quadratic basic semialgebraic sets, J. Convex Anal., 17, 535-551, (2010) Semialgebraic sets and related spaces LMI representations of the convex hulls of quadratic basic semialgebraic 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 A finite unit norm tight frame (FUNTF) is a Hilbert space frame that is finite, unit norm, and tight. In other words, it is the collection of column vectors in a matrix whose row vectors are orthogonal with equal norm and whose column vectors each have unit norm. Let \({\mathcal F}^{R}_{N,d}\) and \({\mathcal F}^{C}_{N,d}\) denote the sets of real and complex FUNTFs of \(N\) vectors in \(d\) dimensions respectively. The frame homotopy problem asks for which \(N\) and \(d\) these spaces are path-connected. According to the authors this conjecture was first posed by D. R. Larson in 2002.
In the paper under review the authors solve the frame homotopy problem. Their main results are the following theorems.
Theorem 1.1. The space \({\mathcal F}^{C}_{N,d}\) is path-connected for all \(d\) and \(N\) satisfying \(d \geq 1\) and \(N \geq d\).
Theorem 1.2. The space \({\mathcal F}^{R}_{N,d}\) is path-connected for all \(d\) and \(N\) satisfying \(d \geq 2\) and \(N \geq d+2\).
These results affirm conjectures of \textit{K. Dykema} and \textit{N. Strawn} [Int. J. Pure Appl. Math. 28, No. 2, 217--256 (2006; Zbl 1134.42019)]. The authors also point out that the exceptional case \({\mathcal F}^{R}_{d+1,d}\) has \(2^{d+1}\) connected components by Theorem 3.1 of the above referenced work, that \({\mathcal F}^{R}_{d,d}\) is the set of orthogonal matrices, which has two components, and that \({\mathcal F}^{R}_{N,1} = \{-1,1\}^N\) for all \(N \geq 1\). They also show that the set of nonsingular points on these spaces is also connected, that spaces of FUNTFs are irreducible in the algebraic-geometric sense, and that generic FUNTFs are full spark, and hence the full spark FUNTFs are dense in the space of FUNTFs.
Their main technique involves explicit continuous lifts of paths from the polytope of eigensteps to spaces of FUNTFs (see [\textit{J. Cahill} et al., Appl. Comput. Harmon. Anal. 35, No. 1, 52--73 (2013; Zbl 1294.65117)]). frame theory; real algebraic geometry; tight frame; irreducible variety; full spark Cahill, J.; Mixon, D.; Strawn, N., Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames, SIAM J. Appl. Algebra Geom., 1, 38-72, (2017) General harmonic expansions, frames, Special classes of linear operators, Topology of real algebraic varieties Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We develop a method for computing all the generalized asymptotes of a real plane algebraic curve \(\mathcal C\) implicitly defined by an irreducible polynomial \(f(x,y)\in \mathbb R[x,y]\). The approach is based on the notion of perfect curve introduced from the concepts and results presented in \textit{A. Blasco} and \textit{S. Pérez-Díaz} [``Asymptotic behavior of an implicit algebraic plane curve'', \url{arXiv:1302.2522})]. In addition, we study some properties concerning perfect curves and in particular, we provide a necessary and sufficient condition for a plane curve to be perfect. Finally, we show that the equivalent class of generalized asymptotes for a branch of a plane curve can be described as an affine space \(\mathbb R^m\) for a certain \(m\). implicit algebraic plane curve; infinity branches; asymptotes; perfect curves Blasco, A.; Pérez-Díaz, S.: Asymptotes and perfect curves, Comput. aided geom. Des. 31, No. 2, 81-96 (2014) Computational aspects of algebraic curves, Computer-aided design (modeling of curves and surfaces) Asymptotes and perfect 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 In the present paper, the author studies the so-called generalized star configurations. Let \(\Lambda = \{\ell_{1}, \dots \ell_{n}\}\) be a collection of linear forms (possibly proportional) in \(R:= \mathbb{K}[x_{1},\dots, x_{k}]\) in any field and \( k\geq 2\). Suppose \(\langle \ell_{1},\dots, \ell_{n} \rangle = \langle x_{1},\dots, x_{k} \rangle := \mathfrak{m}\). Let \(a \in \{1,\dots, n\}\) and denote by \(I_{a}(\Lambda) \subset R\) the ideal generated by all \(a\)-fold products of the linear forms in \(\Lambda\). The projective scheme with the defining ideal \(I_{a}(\Lambda)\) will be called a generalized star configuration scheme of size \(a\) and support \(\Lambda\). The variety (subspace arrangement) of \(\mathbb{P}^{k-1}\) with the defining ideal \(\sqrt{I_{a}(\Lambda)}\) is called generalized star configuration variety and it will be denoted by \(\mathcal{V}_{a}(\Lambda)\). The main result of the paper is an interpolation result. Assuming that \(\mathbb{K}\) is infinite, denote by \(V = V_{1} \cup\dots\cup V_{m} \subset \mathbb{P}^{k-1}_{\mathbb{K}}\) a subspace arrangement of \(m\) irreducible components such that \(V_{1} \cap \dots \cap V_{m} = \emptyset\) and we denote by \(c_{i}\) the codimension of \(V_{i}\). In the coordinate ring \(R := \mathbb{K}[x_{1},\dots, x_{k}]\), the defining ideals \(I(V_{i})\) are prime ideals minimally generated by \(c_{i}\) linear forms, \(I(V) = I(V_{1}) \cap \dots \cap I(V_{m})\), and if \(\mathfrak{m}\) denotes the irrelevant ideal, then we have \(I(V_{1}) +\dots+ I(V_{m}) = \mathfrak{m}\). The last condition is a main motivation to say that \(V\) is \textit{essential}.
Theorem 1. Let \(V\) be an essential subspace arrangement as above. Then there exists a collection of linear forms \(\Lambda = (\ell_{1},\dots, \ell_{n})\) with \(\ell_{i} \in R\) generating \(\mathfrak{m}\) and an \(a \in \{1, \dots, n\}\) such that \(V = \mathcal{V}_{a}(\Lambda)\).
Another way of looking at this interpolation result is to use \([n,k]\)-linear codes, and this topic is investigated in Section 2.1. As a particular application of the main result, the author provides a natural idea to interpolate points in the projective plane with use of multi-arrangements of lines.
In Section 3, the author studies arithmetic ranks of generalized star configuration schemes. Let us recall that for any ideal \(I\) the arithmetic rank of \(I\), denoted by \(\mathrm{ara}(I)\), is the minimal number of elements in \(I\) that generate \(I\) up to its radical ideal.
Theorem 2. Let \(\Lambda = (\ell_{1},\dots \ell_{n})\) where \(\ell_{i}\) are linear forms generating the maximal ideal \(\mathfrak{m}\). Let \(a \in \{1,\dots,n\}\), then
\[
\mathrm{ara}(I_{a}(\Lambda)) \leq n - a + 1.
\]
subspace arrangement; star configuration, interpolation; arithmetic rank Configurations and arrangements of linear subspaces, Projective techniques in algebraic geometry, Numerical interpolation, Nil and nilpotent radicals, sets, ideals, associative rings Subspace arrangements as generalized star configurations | 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 subject of this note is the Vassiliev-Goodwillie-Weiss spectral sequence. This objects calculates an approximation of the homology of the space of knots. It has been proved by \textit{P. Lambrechts} et al. in [Geom. Topol. 14, No. 4, 2151--2187 (2010; Zbl 1222.57020)] that this spectral sequence degenerates at the \(E_2\)-page when working with rational coefficients. Besides, \textit{I. Volić} in [Compos. Math. 142, No. 1, 222--250 (2006; Zbl 1094.57017)] has shown that the knot invariant given by the \(E_2\)-page of this spectral sequence is the universal finite type invariant (or more precisely the associated graded one given by the filtration according to the grade). It has been conjectured that these two results remain true when working with integer coefficients.
In this note we are interested specifically in the first conjecture (to know whether the spectral sequence degenerates with integer coefficients). In joint work with Pedro Boavida de Brito we have introduced a new tool to study this problem. We construct a non-trivial action of the absolute Galois group of \(\mathbb Q\) on this spectral sequence. This action gives us information on the differentials. In this way we can show that a given differential is zero if we ignore the torsion for the first small integers (see Theorem 5.2 for the precise result).
\dots
Let us give some details on the contents of this note. The first section gives a quick introduction to the Goodwillie-Weiss embedding calculus. The second is a specialization of this theory to the case of embeddings from \(\mathbb R\) to \(\mathbb R^d\) following the work of \textit{D. P. Sinha} [J. Am. Math. Soc. 19, No. 2, 461--486 (2006; Zbl 1112.57004)]. The third section is an introduction to the Grothendieck-Teichmüller group and to the absolute Galois group of \(\mathbb Q\) and to their actions on the profinite completions of the pure braid groups. In the fourth section we recall the theory of completion of spaces at a prime number. This is an analog in homotopy theory of the completion of groups. Finally in the fifth section, we can give the main results together with a sketch of the proof. Note that only the results of this final part are original. They are due to Pedro Boavida de Brito and the author. A complete proof will be published soon. Vassiliev-Goodwillie-Weiss spectral sequence; space of knots; \(E_2\)-term; universal finite type invariant; Vassiliev invariant; integer coefficients; action; Galois group; differentials; completion Finite-type and quantum invariants, topological quantum field theories (TQFT), Spectral sequences in algebraic topology, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Galois group and knot space | 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 a field and \(A\) a \(k\)-domain. A set \(D=\{D_{n}\}_{n=0}^{\infty}\) of \(k\)-linear endomorphisms of \(A\) is called a \textit{higher \(k\)-derivation} on \(A\) if \(D_{0}\) is the identity map of \(A\) and \(D_{n}(ab) = \sum_{i+j=n}D_{i}(a)D_{j}(b)\) for any \(a, b\in A\), \(n\geq 0\). If, in addition, \(D_{i}\circ D_{j}={{i+j}\choose i}D_{i+j}\) (\(i, j\geq 0\)), then \(D\) is called \textit{iterative}. If \(D\) satisfies the condition that for any \(a\in A\), there exists \(n\geq 0\) such that \(D_{m}(a) = 0\) for all \(m\geq n\), then \(D\) is said to be \textit{locally finite}. In what follows a locally finite iterative higher \(k\)-derivation is abbreviated as lfihd; the set of all lfihds on \(A\) is denoted by \(\mathrm{LFIHD}(A)\). Furthermore, \(\mathrm{ML}(A)\) denotes the Makar-Limanov invariant of \(A\) defined as \(\mathrm{ML}(A) = \bigcap_{D\in\mathrm{LFIHD}}A^{D}\) where \(A^{D}=\{a\in A\mid D_{n}(a) = 0\) for all \(n\geq 1\}\). The main result of the paper is the following theorem where \(k^{[n]}\) denotes the polynomial ring in \(n\) variables over \(k\):
Let \(k\) be a field of characteristic \(p\geq 0\), \(R= k^{[3]}\) and \(\Delta = \{\Delta_{n}\}_{n=0}^{\infty}\) a nontrivial lfihd on \(R\). Assume that \(p = 0\) or \(\mathrm{ML}(R^{\Delta})\neq R^{\Delta}\). Then \(R^{\Delta}\cong k^{[2]}\).
As a consequence of this result, the author obtains another proof of the cancellation theorem, first proved in [\textit{S. M. Bhatwadekar} and \textit{N. Gupta}, J. Algebra Appl. 14, No. 9, Article ID 1540007, 5 p. (2015; Zbl 1326.14142)]: If \(k\) is a field and \(A\) is a \(k\) domain such that \(A^{[1]}\cong k^{[3]}\), then \(A\cong k^{[2]}\). cancellation problem; locally finite iterative higher derivation Actions of groups on commutative rings; invariant theory, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Derivations and commutative rings Notes on the kernels of locally finite higher derivations in polynomial rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((R, m)\) be a standard graded domain over a field \(K\) of positive characteristic \(p\), \(p\) prime. Let \(I\) be an ideal primary to the homogeneous maximal ideal \(m\) of \(R\). The function that associates \(\text{length}_R(R/I^{[q]})\) to \(q=p^e\), where \(I^{[q]} = (i^q : i \in I \}\), is called the Hilbert-Kunz function of \(I\) and is denoted here by \(\phi(q)\).
The main result of this paper provides a description of this function under the additional assumptions that \(R\) is normal, \(K\) is the algebraically closure of a finite field and \(I\) is a homogeneous ideal of \(R\). Under these conditions,
\[
\phi(q) = e_{HK}(I) q^2 + \gamma(q),
\]
where \(e_{HK}(I)\) is a rational number and \(\gamma(q)\) is an eventually periodic function.
The main contribution of the paper is the proof of the periodicity of the function \(\gamma(q)\). As the author remarks in the paper, the statement was previously proved in a number of particular cases by various authors.
The proof relies on a technique developed by the author based upon the theory of stable vector bundles over curves and used by him with great success in the study of tight closure theory in positive characteristic in dimension two. In particular, the paper under review can be seen as complementing Brenner's earlier work [Math. Ann. 334, No. 1, 91--110 (2006; Zbl 1098.13017)]. The main ingredient of the proof is the use of a stronger Harder-Narasimham filtration of a locally free sheaf on a smooth projective curve together with an argument first employed by \textit{H. Lange} and \textit{U. Stuhler} who studied the Frobenius pullback of a strongly semistable sheaf of degree \(0\) on a curve over a finite field [Math. Z. 156, 73--83 (1977; Zbl 0349.14018)]. Hilbert-Kunz function; Hilbert-Kunz multiplicity; stronger Harder-Narasimham filtration; stable vector bundles Brenner, Holger, The Hilbert-Kunz function in graded dimension two, Comm. Algebra, 35, 10, 3199-3213, (2007) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Finite ground fields in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The Hilbert-Kunz function in graded 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 A general ROTH system \(S=(\ell_1,\dots,\ell_n,c(1),\dots,c(n))\) is given by \(n\) linearly independent linear forms \((\ell_1,\dots,\ell_n)\) in \(n\) variables with real algebraic coefficients and \(n\) real numbers \(c(1),\dots,c(n)\) such that \(c(1)+\dots+c(n)=0\) and such that, for each \(\delta>0\), there are only finitely many integral solutions to the system of inequalities
\[
|\ell_q|<Q^{-c(q)-\delta}, \quad Q>1, \; 1\leq q\leq n.
\]
The special case \(n=2\), \(\ell_1(x_1,x_2)=x_1\), \(\ell_2(x_1,x_2)=\alpha x_1-x_2\) where \(\alpha\) is an irrational real algebraic number and \(c(1)=-1\), \(c(2)=1\) is called a classical ROTH system.
To a general ROTH system \(S\) is attached a filtration \(F_S^\cdot V\) over \(\overline{\mathbb Q}\) of the \(\mathbb Q\)-vector space \(V=\mathbb Q x_1+\dots + \mathbb Q x_n\), defined as
\[
F_S^{i}V=\sum_{c(q)\geq i}\overline{\mathbb Q} \ell_q \quad (i\in\mathbb R).
\]
In the case of a classical ROTH system attached to an irrational algebraic number \(\alpha\), this filtered space is denoted by \((\check{V}, F_\alpha^\cdot \check{V})\). The slope of a filtered vector space \((V, F^\cdot V)\) is
\[
\mu(V)=\frac{1}{\dim_\mathbb Q V} \sum_{w\in \mathbb R} w \dim_{\overline{\mathbb Q}} {\mathrm{gr}}^w(F^{\cdot} V)
\]
where \( {\mathrm{gr}}^w(F^{\cdot} V)=F^wV/F^{w+}V\), \(F^{w+}V=\bigcup _{j>w}F^jV\). The filtration is semi stable if \(\mu(W)=\mu(V)\) for any nonzero \(\mathbb Q\)-vector subspace \(W\) of \(V\). The category of finite dimensional vector spaces over \(\mathbb Q\) with semistable filtration of slope zero is denoted by \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\). See Chap.~VI, Theorem 2B of [\textit{W. M. Schmidt}, Diophantine approximation. Berlin etc.: Springer-Verlag (1980; Zbl 0421.10019)]. See also [\textit{G. Faltings}, Proc. ICM '94. Vol. I. Basel: Birkhäuser, 648--655 (1995; Zbl 0871.14010)] and [\textit{B. Totaro}, Duke Math. J. 83, No. 1, 79--104 (1996; Zbl 0873.14019)].
Here is the main result of the paper under review. Let \(\alpha\) be an irrational real algebraic number. If \(\alpha\) is not quadratic, then there exists a fully faithful tensor functor \(\iota\) of the category \({\mathrm{Rep}}_\mathbb Q {\mathrm{SL}}_2\) of finite dimensional representations over \(\mathbb Q\) of the special linear group \({\mathrm{SL}}_2\) of degree \(2\) into the tensor category \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\) such that the functor \(\iota\) commutes with the forgetful tensor functor to the tensor category \({\mathrm{Vec}}_\mathbb Q\) of finite dimensional vector spaces over \(\mathbb Q\) and such that the image of \(\iota\) contains the filtered vector space \((\check{V}, F_\alpha^\cdot \check{V})\) derived from a classical ROTH System.
On the other hand, if \(\alpha\) is quadratic, then there exists a fully faithful tensor functor \(\iota\) of the category \({\mathrm{Rep}}_\mathbb Q T_\alpha\) of finite dimensional representations over \(\mathbb Q\) of a one dimensional anisotropic torus \(T_\alpha\) over \(\mathbb Q\) into the tensor category \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\) such that the group \(T_\alpha(\mathbb Q)\) is isomorphic to the kernel of the norm map of the quadratic number field \(\mathbb Q(\alpha)\) over \(\mathbb Q\), such that the functor \(\iota\) is compatible with the forgetful tensor functor to \({\mathrm{Vec}}_\mathbb Q\) and such that the image of \(\iota\) contains the filtered vector space \((\check{V}, F_\alpha^\cdot \check{V})\). M. FUJIMORI, The algebraic groups leading to the Roth inequalities. J. TheÂor. Nombres Bordeaux, 24: pp. 257-292, 2012. Approximation to algebraic numbers, Simultaneous homogeneous approximation, linear forms, Rational points, Monoidal categories (= multiplicative categories) [See also 19D23] The algebraic groups leading to the Roth inequalities | 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 If \(f : S' \to S\) is a finite locally free morphism of schemes, we construct a symmetric monoidal ``norm'' functor \(f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)\), where \(\mathcal{H}_\bullet(S)\) is the pointed unstable motivic homotopy category over \(S\). If \(f\) is finite étale, we show that it stabilizes to a functor \(f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)\), where \(\mathcal{S}\mathcal{H}(S)\) is the \(\mathbb{P}^1\)-stable motivic homotopy category over \(S\). Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic \(E_\infty \)-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum \(H\mathbb{Z} \), the homotopy \(K\)-theory spectrum \(KGL\), and the algebraic cobordism spectrum \(MGL\). The normed spectrum structure on \(H\mathbb{Z}\) is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology. motivic homotopy theory, highly structured ring spectra, norms, multiplicative transfers Motivic cohomology; motivic homotopy theory, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Algebraic cycles, Research exposition (monographs, survey articles) pertaining to algebraic geometry Norms in motivic homotopy 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 \(B/\mathbb{Q}\) be a definite quaternion algebra, and \(\mathcal{H}\) universal the Hecke algebra acting on it. To each idempotent \(e\) of the \(\overline{\mathbb{Q}}\)-vector space of compactly supported \(\mathcal{C}^\infty\)-functions on the adelic points of the algebraic group of invertible elements of \(B\), one associates an eigenvariety \(\mathcal{D}(e)\). The \(\overline{\mathbb{Q}}_p\)-points of \(\mathcal{D}(e)\) embeds into the homomorphisms from \(\mathcal{H}\) to \(\overline{\mathbb{Q}}_p\) times the \(\overline{\mathbb{Q}}_p\)-valued points of the weight space \(\mathcal{W}=\mathrm{Hom}_\mathrm{cont}(\mathbb{Z}_p^\times,\overline{\mathbb{Q}}_p)\).
A point \(z\) in \(\mathcal{D}(e)(\overline{\mathbb{Q}})p)\) is called classical if there is a classical automorphic eigenform in the corresponding space of overconvergent forms, whose system of Hecke eigenvalues is that defined by \(z\); more generally, recall that points on the eigenvariety corresponds to Hecke packets of overconvergent modular forms.
The main result of this paper is the existence of Hecke packets coming from classical automorphic representations of an inner form of \(\mathrm{GL}(2)\) and two idempotents \(e_1\) and \(e_2\) such that the corresponding point in \(\mathcal{D}(e_1)\) is classical, while the corresponding point in \(\mathcal{D}(e_2)\) is not classical.
The proof of the theorem uses a \(p\)-adic version of a Labesse-Langlands transfer, developped by the author. \(p\)-adic Langlands programme; eigenvarieties; \(L\)-packets \(p\)-adic theory, local fields, Galois representations, Congruences for modular and \(p\)-adic modular forms, Automorphic forms, one variable, Rigid analytic geometry, Langlands-Weil conjectures, nonabelian class field theory \(L\)-indistinguishability on eigenvarieties | 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 put forward a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called ``nonarchimedean Milnor fiber'' that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. Then, applying \(T\)-convex integration after descent, matching the Euler characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries. Finally, we also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert-Loeser-Merle. Hrushovski-Kazhdan style motivic integration; equivariant Grothendieck ring; motivic zeta function; Denef-Loeser motivic Milnor fiber; Thom-Sebastiani formula; \(T\)-convex valued field Arcs and motivic integration, Non-Archimedean valued fields, Singularities in algebraic geometry Motivic integration and Milnor fiber | 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 an arbitrary scheme \(W\), the \(m\)-th jet scheme \(W_m\) parametrizes morphisms \(\text{Spec} \mathbb{C} [t]/(t^{m+1})\to W\). Main result:
If \(X\) is a smooth variety, \(Y\subset X\) a closed subscheme, and \(q>0\) a rational number, then:
(1) The pair \((X,q\cdot Y)\) is log canonical if and only if \(\dim Y_m\leq (m+1) (\dim X-q)\), for all \(m\).
(2) The pair \((X,q\cdot Y)\) is Kawamata log terminal if and only if \(\dim Y_m< (m+1)(\dim X-q)\), for all \(m\).
The main technique we use in the proof of this result is motivic integration, a technique due to Kontsevich, Batyrev, and Denef and Loeser. As a consequence of the above result, we obtain a formula for the log canonical threshold:
Corollary. If \(X\) is a smooth variety and \(Y\subset X\) is a closed subscheme, then the log canonical threshold of the pair \((X,Y)\) is given by \(c(X,Y)= \dim X-\sup_{m\geq 0} {\dim Y_m\over m+1}\).
We apply this corollary to give simpler proofs of some results on the log canonical threshold proved by \textit{J.-P. Demailly} and \textit{J. Kollár} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, No. 4, 525-556 (2001; Zbl 0994.32021)] using analytic techniques. jet schemes; log canonical threshold; motivic integration; Kawamata log terminal M. Mustaţǎ, Singularities of pairs via jet schemes, \textit{J. Amer. Math. Soc.} 15 no. 3 (2002) 599-615 (electronic). Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays) Singularities of pairs via jet 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 Definition. Let \(X,Y\) be analytic submanifolds of \(\mathbb{R}^n\) of dimensions \(k\) and \(l\) respectively. Let \(G_k (\mathbb{R}^n)\) denote the Grassmannian of \(k\)-dimensional vector subspaces of \(\mathbb{R}^n\), let \(T_{X,x}\) denote the tangent space of \(X\) at \(x\). Suppose that \(X \cap Y=\emptyset\) and \(Y\subset \overline X\). Let \(y\in Y\). We say that \((X,Y)\) satisfies Whitney's condition (a), respectively (b) at \(y\) if the following condition is satisfied:
(a) For any sequence \((x_\nu)_{\nu\in \mathbb{N}}\) of points of \(X\) with \(\lim x_\nu =y\), if \(\lim T_{X,x_\nu} =\tau\) in \(G_k (\mathbb{R}^n)\), then \(\tau \supset T_{Y,y}\).
(b) For any pair of sequences \((x_\nu)_{\nu \in\mathbb{N}}\), \(x_\nu \in X\), and \((y_\nu)_{\nu\in \mathbb{N}}\), \(y_\nu\in Y\), with \(\lim x_\nu= \lim y_\nu =y\), if \(\lim T_{X,x_\nu} =\tau\) and the sequence of lines \(\mathbb{R}(x_\nu-y_\nu)\) has a limit \(\lambda\) in \(G_1 (\mathbb{R}^n)\), then \(\tau\supset\lambda\).
The aim of the paper under review is to prove that every subset of \(\mathbb{R}^n\) definable from addition, multiplication and exponentiation admits a stratification satisfying Whitney's conditions (a) and (b). Whitney stratification; tangent space; real analytic set T L Loi, Whitney stratification of sets definable in the structure \(\mathbbR_{\exp}\), Banach Center Publ. 33, Polish Acad. Sci. (1996) 401 Real-analytic and semi-analytic sets, Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions Whitney stratification of sets definable in the structure \(\mathbb{R}_{\text{exp}}\) | 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 \(A(x)=A_0+x_1A_1+\cdots+x_nA_n\) be a linear matrix, or pencil, generated by given symmetric matrices \(A_0,A_1,\dots,A_n\) of size \(m\) with rational entries. The set of real vectors \(x\) such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality. We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree \(d\) of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bézout bound of \(d\). When \(m\) (resp., \(n\)) is fixed, such a bound, and hence the complexity, is polynomial in \(n\) (resp., \(m\)). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms. linear matrix inequalities; semidefinite programming; computer algebra algorithms; symbolic computation; polynomial optimization D. Henrion, S. Naldi, and M. Safey El Din, \textit{Exact algorithms for linear matrix inequalities}, SIAM J. Optim., 26 (2016), pp. 2512--2539. Semidefinite programming, Symbolic computation and algebraic computation, Effectivity, complexity and computational aspects of algebraic geometry Exact algorithms for linear matrix inequalities | 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 0745.00034.]
Let \(F\) be an algebraic curve in \(\mathbb{C}^ n\) defined by a system of polynomial equations \(f_ i(X)=0\), \(i=1,\dots,n-1\). Let \(X=(x_ 1,\dots,x_ n)=0\) be a singular point of \(F\), and let \(x_ i=\sum^ \infty_{k=1}b_{ik}t^{p_{ik}}\), \(i=1,\dots,n\) (where \(p_{ik}\) are integers, \(0>p_{ik}>p_{i,k+1}\), \(b_{ik}\) complex numbers, and the series converge for large \(| t|)\), be a local uniformization of a branch of \(F\) passing through the point \(X=0\). The authors give an algorithm for finding any initial parts of the above series, with the aid of Newton polyhedra and power transformations. local uniformization of a branch of an algebraic curve; Newton polyhedra Plane and space curves, Modifications; resolution of singularities (complex-analytic aspects), Computational aspects of algebraic curves The local uniformization of branches of an algebraic 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 Author's abstract: For a pair \((M, I)\), where \(M\) is finitely generated graded module over a standard graded ring \(R\) of dimension \(d \geq 2\), and \(I\) is a graded ideal with \(\ell(R / I) < \infty\) and generated by elements of the same degree, we prove that \(\lim_{q \rightarrow \infty} e_1(M, I^{[q]}) / q^d\) exists, where \(e_1(M, I^{[q]})\) denotes the first coefficient of the Hilbert-Samuel polynomial of \((M, I^{[q]})\).
We use this to get an expression for \(\lim_{k \rightarrow \infty} [e_{H K}(M, I^k) - e_0(M, I^k) / d!] / k^{d - 1}\), where \(e_{H K}\) denotes the Hilbert-Kunz multiplicity. In particular, if \(\dim M = d\) then we deduce that the difference \(e_{H K}(M, I^k) - e(M, I^k) / d!\) grows at least as a fixed positive multiple of \(k^{d - 1}\) as \(k \rightarrow \infty\).
This is proved using `renormalized' HK density functions. Hilbert-Kunz density; Hilbert-Kunz multiplicity; Hilbert-Samuel polynomial Trivedi, V., Asymptotic Hilbert-Kunz multiplicity, J. Algebra, 492, 498-523, (2017) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Asymptotic Hilbert-Kunz multiplicity | 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 \(p\) be an odd prime and \(R\) be a \(p\)-adically complete integral domain of characteristic zero satisfying the condition
\[
\bigcap_{s\in\mathbb{N}} p^{s}R=\{0\};
\]
write, for brevity, \(R[x, x^{-1}]:=R[x_{1},\dots, x_{n},x_{1}^{-1},\dots,x_{n}^{-1}]\) and
\[
x^{a}:=\prod_{i=1} ^{n} x_{i}^{a_{i}}\; \text{ for } a\in\mathbb{Z}^{n}.
\]
Let \(f(x)\in R[x, x^{-1}]\), let \(\Delta (f)\) be the Newton polyhedron of the Laurent polynomial \(f(x)\), let \(\mu\subseteq\Delta (f)\), and suppose that the set \(\Delta (f)\setminus\mu\) is a union of some faces of \(\Delta (f)\). The authors introduce an \(R\)-module
\[
\Omega_{f}(\mu):=\{(k-1)!g(x)f(x)^{-k}\mid k\in\mathbb{N}, g(x)\in R[x, x^{-1}], \operatorname{supp}(g)\subseteq k\mu (\mathbb{Z})\},
\]
where \(\mu (\mathbb{Z}):=\mu\cap\mathbb{Z}^{n}\) and
\[
\operatorname{supp}(g):=\{a\mid a\in\mathbb{Z}^{n}, g(x)=\sum_{a\in\mathbb{Z}^{n}}\gamma (a) x^{a}, \gamma (a)\neq 0\},
\]
let
\[
\widehat{\Omega}_{f}(\mu):=\varprojlim{\Omega}_{f}(\mu)/p^{s}{\Omega}_{f}(\mu),
\]
define, for every ``Frobenius lift'' on \(R\), i.e. a ring endomorphism \(\sigma: R\rightarrow R\) such that \(\sigma (r)=r^{p}\; (\operatorname{mod} p)\) for \(r\in R\), an R-linear ``Cartier operator''
\[
\mathfrak{C}_{p}: \widehat{\Omega}_{f}(\mu)\rightarrow\widehat{\Omega}_{\sigma f}(\mu),
\]
let
\[
U_{f}(\mu):=\{\omega\mid\omega\in\widehat{\Omega}_{f}(\mu), \mathfrak{C}_{p}(\omega)=0\; (\operatorname{mod} p^{s}\widehat{\Omega}_{(\sigma^{s} f)}(\mu)), s\in\mathbb{N}\},
\]
define a ``Hasse-Witt matrix'' \(\beta_{f} (m, \mu):=(b_{uv})\), where \(b_{uv}\) is the coefficient of \(x^{mu-v}\) in \(f(x)^{m-1},\{u,v\}\subseteq\mu (\mathbb{Z})\), \(m\in\mathbb{N}\), and prove that if the Hasse-Witt matrix \(\beta_{f} (p, \mu)\) is invertible, then the ``unit root crystal''; \(\widehat{\Omega}_{f}(\mu)/U_{f}(\mu)\) is a free \(R\)-module with a basis \(\{x^{u}f(x)^{-1}\mid u\in\mu (\mathbb{Z})\}\) and that the Hasse-Witt matrices \(\beta_{f} (p^{s}, \mu), s\in\mathbb{N}\), are invertible. Making use of those results, the authors obtain a version of the main theorem in the work of \textit{N. M. Katz} [Ann. Sci. Éc. Norm. Supér. (4) 18, 245--285 (1985; Zbl 0592.14021)]. According to the authors, some of the ideas of their paper are already present in the cited work of N. M. Katz, ``but in a very different language''. The paper under review uses the language of the classical \(p\)-adic analysis in the style of B. Dwork, and the authors reprove some of Dwork's results by their methods. For example, let \(R:=\mathbb{Z}_{p}[[z]], \sigma (g(z))=g(z^{p})\) for \(g\in R\),
\[
f(x, y)=y^{2}-x(x-1)(x-z),
\]
let \(\mu\) be the interior of \(\Delta (f)\), so that \(\mu (\mathbb{Z})=\{(1, 1)\}\), and consider a hypergeometric function
\[
F(z):=\sum_{k=0}^{\infty}\Gamma (k+1/2)^{2}(\Gamma (1/2)\Gamma (k+1))^{-2}z^{k},
\]
the authors prove then that
\[
(-1)^{p-1}F(z)/F(z^{p})=\lim_{s\to\infty}\beta_{f} (p^{s}, \mu)/\sigma\beta_{f} (p^{s-1}, \mu).
\] \(p\)-adic analysis; toric hypersurfaces; Newton polyhedron; formal groups; Cartier operator; Hasse-Witt matrix Local ground fields in algebraic geometry, Fourier coefficients of automorphic forms, \(p\)-adic theory, local fields, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Varieties over finite and local fields, Toric varieties, Newton polyhedra, Okounkov bodies, Formal groups, \(p\)-divisible groups, Non-Archimedean analysis, Generalized hypergeometric series, \({}_pF_q\) Dwork crystals. 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 A collection of functions \((f_{F,\psi})_{F,\psi}\), indexed by non-Archimedean local fields \(F\) and additive characters \(\psi\) on \(F\), is called a function of motivic exponential class, or a \(\mathcal C^{\exp}\)-function, if the functions \(f_{F,\psi}\) can be given descriptions that are uniform in \(F\) and \(\psi\) in a particular sense. The class of \(\mathcal C^{\exp}\)-functions is closed under certain ``integration along fibers'' operations. Also, certain statements in terms of \(\mathcal C^{\exp}\)-functions satisfy a transfer principle like the one of Ax-Koshen/Ershov. More precisely, whether or not all points of a given definable set satisfy a given \(\mathcal C^{\exp}\)-condition depends only on the residue field of \(F\), as long as that residue field has sufficiently large characteristic.
The authors prove that various statements involving bounds, approximations, and limits of \(\mathcal C^{\exp}\)-functions can be described in terms of \(\mathcal C^{\exp}\)-conditions and \(\mathcal C^{\exp}\)-functions. As a consequence, transfer principles apply to many statements involving continuity and existence of limits. As another application, the authors prove that if a \(\mathcal C^{\exp}\)-function is \(L^2\), then its Fourier transform is a \(\mathcal C^{\exp}\)-function.
The authors also use these ideas to show that if a \(\mathcal C^{\exp}\)-function is bounded for each local field \(F\), then that function has a uniform bound in terms of the cardinality of the residue field of \(F\). As an application, the authors obtain uniform bounds for Fourier transforms of orbital integrals. transfer principles for motivic integrals; uniform bounds; motivic integration; motivic constructible exponential functions; loci of motivic exponential class; orbital integrals; admissible representations of reductive groups; Harish-Chandra characters 5 R. Cluckers, J. Gordon and I. Halupczok, 'Uniform analysis on local fields and applications to orbital integrals', \textit{Trans. Amer. Math. Soc.}, Preprint, 2018, arXiv:1703.03381 (version 2). Arcs and motivic integration, Representations of Lie and linear algebraic groups over local fields, Summability in abstract structures Uniform analysis on local fields and applications to orbital integrals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This survey is devoted to the classical and modern problems related to the entire function \({\sigma({\mathbf{u}};\lambda)} \), defined by a family of nonsingular algebraic curves of genus 2, where \({\mathbf{u}} = (u_1,u_3)\) and \(\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})\). It is an analogue of the Weierstrass sigma function \(\sigma({{u}};g_2,g_3)\) of a family of elliptic curves. Logarithmic derivatives of order \(2\) and higher of the function \({\sigma({\mathbf{u}};\lambda)}\) generate fields of hyperelliptic functions of \({\mathbf{u}} = (u_1,u_3)\) on the Jacobians of curves with a fixed parameter vector \(\lambda \). We consider three Hurwitz series \(\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}, \sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}\) and \(\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!} \). The survey is devoted to the number-theoretic properties of the functions \(a_{m,n}(\lambda), \xi_k(u_1;\lambda)\) and \(\mu_k(u_3;\lambda)\). It includes the latest results, which proofs use the fundamental fact that the function \({\sigma ({\mathbf{u}};\lambda)}\) is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space. abelian functions; two-dimensional sigma functions; Hurwitz integrality; generalized Bernoulli-Hurwitz number; heat equation in nonholonomic frame Bernoulli and Euler numbers and polynomials, Coverings of curves, fundamental group, Jacobians, Prym varieties, Other functions coming from differential, difference and integral equations, Research exposition (monographs, survey articles) pertaining to special functions Analytical and number-theoretical properties of the two-dimensional sigma function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The well known Lojasiewicz inequality \((\|\text{grad} f\|>f^\alpha\), \(\alpha<1\), in short \(L\)-inequality) concerns real analytic functions \(f\) in a neighborhood of a point \(a\in R^n\), \(f(a)=0\). A trajectory of the vector field \(-\text{grad} f\) is defined as a maximal differentiable curve \(\gamma\) verifying \(\gamma'(t)=-\text{grad} f(\gamma(s))\).
Lojasiewicz proved that all trajectories of \(-\text{grad} f\) are of finite length, when \(f\) is real analytic in a neighborhood of a compact \(U\).
The notion of ``\(o\)-minimal structure on the real field'' describes abstractly [see \textit{L. Van den Dries} and \textit{C. Miller}, Duke Math. J. 84, No. 2, 497-540 (1996; Zbl 0889.03025)] different kinds of geometric categories of sets which appear in semialgebraic and subanalytic geometries. For instance \((R\), exp)-definable sets define a ``\(o\)-minimal structure'' (theorem of Wilkie).
Actually the generalizations of the \(L\)-inequality for ``\(o\)-minimal structures'' is of great interest. In this paper a new \(o\)-minimal version of the \(L\)-inequality is obtained. A study of all trajectories of \(-\text{grad} f\) is given too, proving the theorem that the length of mentioned trajectories is bounded by a constant independent of the trajectory. Another interesting author's result says that the flow of \(-\text{grad} g\) with a non negative definable \(g\), determines a deformation retract onto \(g^{-1}(0)\). flows of gradient; \(o\)-minimal structure; subanalytic sets; Łojasiewicz inequalities; trajectories of gradient Krzysztof Kurdyka, ``On gradients of functions definable in o-minimal structures'', Ann. Inst. Fourier48 (1998) no. 3, p. 769-783 Semi-analytic sets, subanalytic sets, and generalizations, Analytic algebras and generalizations, preparation theorems, Real-analytic and semi-analytic sets, Real-analytic functions, \(C^\infty\)-functions, quasi-analytic functions, Model theory On gradients of functions definable in o-minimal structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(m = (m_ 1, \dots, m_ n)\) be a fixed list of positive numbers. Let \(\widehat {\mathcal C}_ n(m)\) denote the space of (possible degenerate) \(n\)-gons in \(\mathbb{R}^ 3\) with sides of length \(m_ i\), up to rigid motion of \(\mathbb{R}^ 3\). At the same time, consider a configuration of points \(p_ 1, \dots, p_ n\) in the complex projective line \(P^ 1\), with the point \(p_ i\) assigned the weight \(m_ i\). The configuration is said to be semistable if the sum of weights at any multiple point \(p_{i_ 1} = \cdots = p_{i_ k}\) does not exceed half the sum of all weights. Let \(\overline {\mathcal C}_ n (m)\) denote the space of all semistable configurations, modulo the action of \(PSL_ 2 (C)\).
In the first part of this paper the author carries out a differential geometric analysis of the space \(\widehat {\mathcal C}_ n (m)\) of polygons. The singularities of \(\widehat {\mathcal C}_ n (m)\) are located (they occur at degenerate polygons of zero area) and identified. It is shown that this space has the structure of a complex variety, and carries a Kähler metric. The author proceeds to study the cohomology of \(\widehat {\mathcal C}_ n (m)\). There is a Hodge-like decomposition of \(H^* (\widehat {\mathcal C}_ n (m))\) which is shown to be ``pure'', and a recursion formula for the Betti numbers is established. The structure of \(H^* ({\mathcal C}_ n (m))\) as a representation of the symmetric group \(S_ n\) is also given. Results on the structure of the space of semistable configurations, with applications to invariants of binary forms, are deduced from a natural identification of \(\widehat {\mathcal C}_ n (m)\) with \(\overline {\mathcal C}_ n (m)\). space of \(n\)-forms; configuration; projective line; semistable; singularities; Kähler metric; cohomology; Hodge-like decomposition; Betti numbers Klyachko, A. A., Spatial polygons and stable configurations of points in the projective line, \textit{Algebraic Geometry and Its Applications (Yaroslaevl' 1992), Vieweg}, 67-84, (1994) Polyhedra and polytopes; regular figures, division of spaces, Configuration theorems in linear incidence geometry, Classical real and complex (co)homology in algebraic geometry, Local differential geometry of Hermitian and Kählerian structures, Plane and space curves Spatial polygons and stable configurations of points in the projective line | 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 closed manifold of positive dimension \(n\), we estimate the expected volume and Euler characteristic for random submanifolds of codimension \(r \in \{1, \ldots, n \}\) in two different settings. On one hand, we consider a closed Riemannian manifold and some positive \(\lambda\). Then we take \(r\) independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than \(\lambda\) and consider the random submanifold defined as the common zero set of these \(r\) functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as \(\lambda\) goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle \(\mathcal{L}\) and a rank \(r\) holomorphic vector bundle \(\mathcal{E}\) that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of \(\mathcal{E} \otimes \mathcal{L}^d\) as \(d\) goes to infinity. The same techniques apply to both settings. Euler characteristic; Riemannian random wave; random polynomial; real projective manifold Letendre, Thomas, Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal., 270, 8, 3047\textendash3110 pp., (2016) Integration on manifolds; measures on manifolds, Topology of real algebraic varieties, Integral representations; canonical kernels (Szegő, Bergman, etc.), Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators, Geometric probability and stochastic geometry, Random fields Expected volume and Euler characteristic of random submanifolds | 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,0) be a two-dimensional rational singularity, and \(\pi: ({\mathfrak X},0)\to (S,0)\) be the universal deformation of (X,0). If (X,0) is not a double point, the parameter space S has many components \(S_ 1,S_ 2,..\). in general. Among them a component \(S_ i\) is called a smoothing component if for some point \(s\in S_ i\) sufficiently near to 0 the fiber \({\mathfrak X}_ s=\pi^{-1}(s)\) over s is smooth. Let \(\{S_{\lambda}\}_{\lambda \in \Lambda}\) be the set of smoothing components. On the other hand, we consider partial resolutions of X. Let f: \(M\to X\) be the minimal resolution of X. Let \(E=f^{-1}(0)\) be the exceptional curve. Let \(\{D_{\gamma}\}_{\gamma \in \Gamma}\) be the set of connected curves contained in E such that the contraction of \(D_{\gamma}\) on M defines a rational quadruple point and, for the corresponding canonical cycle \(K_{\gamma}\) of \(D_{\gamma}\), \(2K_{\gamma}\) is a Cartier divisor. Then, for every \(S_{\lambda}\) there exists \(D_{\gamma}\) with the following relation (this is the main result of this article and solves a conjecture of \textit{Kollár}):
Let \(D'_{\gamma}\) be the union of components of E disjoint from \(D_{\gamma}\). Let \(\sigma_{\gamma}: M\to Y_{\gamma}\) be the contraction of the union \(D_{\gamma}\cup D'_{\gamma}\) to a normal space. \(Y_{\gamma}\to X\) is a partial resolution. There is a one- parameter smoothing \(\tau: {\mathfrak Y}\to T\) with \(\tau^{-1}(0)\cong Y_{\gamma}\) for \(0\in T\) such that some multiple of the canonical class of \({\mathfrak Y}\) is Cartier. We can contract \(\sigma_{\gamma}(E)\) in \({\mathfrak Y}\) and obtain a smoothing \({\bar\tau}: \bar {\mathfrak Y}\to T\) of X. The image of the induced morphism \(T\to S\) is contained in \(S_{\gamma}.\)
As the sub-main result the following characterization is given: A two- dimensional quadruple rational singularity (Y,0) has a one-parameter smoothing \(\tau: {\mathfrak Y}\to T\) with \(\tau^{-1}(0)\cong Y\) for \(0\in T\) such that some multiple of the canonical class of \({\mathfrak Y}\) is Cartier if and only if, for the canonical cycle K on the minimal resolution \(M\to Y\), 2K is Cartier. minimal resolution; two-dimensional quadruple rational singularity Stevens, Jan. Partial resolutions of rational quadruple points. Internat. J. Math. 2 (1991), 205--221.DOI: 10.1142/S0129167X91000144 Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Partial resolutions of rational quadruple 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 In the paper under review, the author studies particular loci of the Hilbert scheme \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) of \(r\) points in the affine space \(\mathbb{A}^n\). In a previous paper [J. Commut. Algebra 3, No. 3, 349--404 (2011; Zbl 1237.14012)], the author introduced the functor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}: (k\mathrm{-Alg}) \rightarrow (\mathrm{Sets})\) that associates to any algebra \(B\) over a ring \(k\) the set of reduced Gröbner bases in the ring \(B[x_1,\ldots,x_n]\) with respect to the lexicographic order with a given standard set \(\Delta\) of \(r\) monomials. He proved that this functor is representable and represented by a locally closed subscheme of \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) called Gröbner stratum. In this paper, the author studies the subfunctor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) of \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}\) that considers only the reduced Gröbner bases of ideals defining reduced points. The main result of the paper is that \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) is representable and, in the case of a ring \(k\) such that \(\mathrm{Spec}\, k\) is irreducible, all the connected components of the representing scheme have the same dimension. Moreover, the number of connected components and their dimension are nicely described in terms of combinatorial properties of the standard set \(\Delta\). Hilbert scheme of points; Gröbner stratum; lexicographic order; reduced points Mathias Lederer (2014). Components of Gröbner strata in the Hilbert scheme of points. \textit{Proc. Lond. Math. Soc}. (3) \textbf{108}(1), 187-224. ISSN 0024-6115. URL http://dx.doi.org/10.1112/plms/pdt018. Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomials over commutative rings, Enumerative problems (combinatorial problems) in algebraic geometry Components of Gröbner strata in the Hilbert scheme of 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 In the present paper we introduce a recursive family of functions \(\{F_n\}){n=1,2,\dots}\) generated by the relations
\[
F_{n+1} F_{n-1}= \lambda(F_n\partial_{xy} F_n- \partial_x F_n- \partial_x F_n\partial_y F_n)+\mu f(x, y)F^2_n\tag{1}
\]
with initial conditions \(F_0= 1\) and \(F_1= f(x,y)\). It is shown that, for any \(n\), the function \(F_n(x, y)\) is a polynomial with integer coefficients in a function \(f(x, y)\), its partial derivatives, and the parameters \(\lambda\) and \(\mu\). The differential polynomials \(F_n(x, y)= D_n(f(x, y))= D_n(f(x, y);\lambda,\mu)\) are representable as leading principal minors of a matrix, and the family \(\{D_n(f)\}\) itself is generated by the Sylvester identity for compound determinants.
We apply this recursion to the classical problem of generalizing the relation
\[
{\sigma(u+ v)\sigma(u- v)\over \sigma(u)^2\sigma(v)^2}= \wp(v)- \wp(u)
\]
for the elliptic Weierstrass functions which plays a key role in the theory and applications of elliptic functions (here the genus is \(g= 1\)), to the case of hyperelliptic Kleinian \(\sigma\)-functions (with genus \(g> 1\)).
A hyperelliptic \(\sigma\)-function of genus \(g\) is defined as an element of the ring of Riemann \(\theta\)-functions that is automorphic with respect to the action of the modular group \(\text{Sp}(2g,\mathbb{Z})\). The logarithmic derivatives of the \(\sigma\)-function,
\[
\wp_{ij}=- {\partial^2\over\partial u_i\partial u_j}\ln \sigma(u),\;\wp_{i,j,k}=-{\partial^2\over\partial u_i\partial u_j\partial u_k}\ln\sigma(u),\;i,j,k= 1,\dots, g,
\]
and so on, are hyperelliptic Abelian functions, i.e., \(2g\)-periodic meromorphic functions.
If the genus \(g\) is one, then the field of elliptic functions is generated by Weierstrass elliptic functions \(\wp\) and \(\wp'\), which uniformize an elliptic curve. In the case \(g> 1\), the hyperelliptic functions \(\wp_{ij}\) and \(\wp_{ijk}\) uniformize the Jacobian variety \(\text{Jac}(V)\) of a hyperelliptic curve \(V\), while the even functions \(\wp_{ij}\) themselves uniformize the Kummer variety \(\text{Kum}(V)= \text{Jac}(V)/(u\to -u)\), defined as the quotient of the Jacobian with respect to the hyperelliptic involution.
The following assertion is one of the main results of the paper.
For a hyperelliptic Kleinian \(\sigma\)-function of genus \(g\geq 1\), the following relation holds:
\[
{\sigma(u+ v)\sigma(u- v)\over \sigma(u)^2\sigma(v)^2}= D_g\Biggl(\wp_{gg}(v)- \wp_{gg}(u);\,{1\over 4},{1\over 2}\Biggr),
\]
where \(\wp_{gg}(v)- \wp_{gg}(u)\) is regarded as a function of \(x= u_g+ v_g\) and \(y= u_g- v_g\), and \(D_g(\;;\;)\) is the differential polynomial defined by the recursion (1). Buchstaber V.M., Enolskii V.Z., Leykin D.V.: A recursive family of differential polynomials generated by the Sylvester identity and addition theorems for hyperelliptic Kleinian functions. Funct. Anal. Appl. 31, 240--251 (1997) Elliptic functions and integrals, Analytic theory of abelian varieties; abelian integrals and differentials, Theta functions and curves; Schottky problem, KdV equations (Korteweg-de Vries equations), Jacobians, Prym varieties, Theta functions and abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) A recursive family of differential polynomials generated by the Sylvester identity, and addition theorems for hyperelliptic Kleinian 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 Assume that \(f,g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) are continuous functions, and \(\emptyset \neq K\subset\mathbb{R}^{n}\) is closed and convex. Consider the generalized variational inequality,
denoted by \(GVI{(g,f,K)}\); find an \(\mathbf{x}\in\mathbb{R}^{n}\) such that
\[
g\left( \mathbf{x}\right) \in K,\ \ \ \left \langle f\left( \mathbf{x}
\right) ,\mathbf{y}-g\left( \mathbf{x}\right) \right \rangle \geq
0\ \ \ \forall \mathbf{y}\in K\text{.}\tag{GVI}
\]
If in (GVI), both \(f\) and \(g\) are polynomials, then \(GVI(g,f,K)\) is
called the generalized polynomial variational inequality and is denoted by GPVI.
In the paper under review, the authors organize an existence and uniqueness
theorem for the GPVI by using the properties of the involved polynomials. By
using the exceptional family of elements and degree theory, they bring up the
existence of solutions for GPVI. Also, they present an example that affirms
their theoretical detections. generalized variational inequality; polynomial function; strongly monotone function Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming), Sensitivity, stability, parametric optimization, Semialgebraic sets and related spaces Existence and uniqueness of solutions of the generalized polynomial variational inequality | 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 Poincaré-Hopf theorem can be expressed as: (*) The Euler characteristic of a compact manifold is the self-intersection number of the zero section of the tangent vector bundle.
Starting this point of view, the author extends the formula for singular varieties through the micro-local languages. Let X be a complex manifold and let \({\mathcal F}\) be a constructive sheaf on X. \(\chi\) (X,\({\mathcal F})\) is defined in the usual way. Let \(\{X_ a\}\) be a stratification such that \({\mathcal F}\) is locally constant on each stratum. The characteristic cycle S(\({\mathcal F})\) is defined as a cycle of the conormal bundle \(T^*X\), and it takes the form \(S({\mathcal F})=\sum m_ a({\mathcal F})T^*_{X_ a}X\) where \(T^*_{X_ a}X\) is the closure of the conormal bundle of the stratum \(X_ a\) and \(m_ a({\mathcal F})\) is a suitable integer. Let \(\Lambda_ 1\) and \(\Lambda_ 2\) be Lagrangean cycles in \(T^*X\). Then the intersection index \(I(\Lambda_ 1,\Lambda_ 2)\) is defined in a suitable way. Then the main theorem is: Let \({\mathcal F}_ 1\) and \({\mathcal F}_ 2\) be constructive sheaves on X. Then \(\chi (X,{\mathcal F}_ 1\otimes {\mathcal F}_ 2)=I(S({\mathcal F}_ 1),S({\mathcal F}_ 2)).\) Putting \({\mathcal F}_ 2=C_ X\), one has \(\chi({\mathcal F})=I(S({\mathcal F})\), \(T_ X^*X)\). The characteristic cycle can be interpreted as the singular support cycle of the corresponding holonomic D-module. \(D_ x\)-module; complex manifold with singularities; Euler characteristic; constructive sheaf; stratification; characteristic cycle; conormal bundle; singular support Ginzburg, V. A.: A theorem on the index of differential systems and the geometry of manifolds with singularities. Soviet math. Dokl. 31, No. 2, 309-313 (1985) Sheaves and cohomology of sections of holomorphic vector bundles, general results, Complex singularities, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Riemann-Roch theorems A theorem on the index of differential systems, and the geometry of manifolds with 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 problem of resolving the singularities of an algebraic variety by a sequence of birational transformations has a long history. A ground- breaking step forward was made in 1940, when O. Zariski developed an ingenious method for uniformizing hypersurface singularities (in characteristic zero) by a process of successive substitutions of variables in polynomial or power series [cf. \textit{O. Zariski}, Ann. Math., II. Ser. 41, 852-896 (1940; Zbl 0025.21601)]. Then, in 1964, H. Hironaka gave an affirmative answer to the whole problem of resolving singularities in characteristic zero, essentially by generalizing Zariski's approach to a general process of successive ``permissible'' blow-up transformations, expressible in the full scheme-theoretic framework [cf. \textit{H. Hironaka}, Ann. Math., II. Ser. 79, 109-326 (1964; Zbl 0122.386)]. After that celebrated paper of Hironaka's, many attempts have been undertaken to analyse the constructiveness of his process of desingularization in various concrete situations. Among them are the papers of \textit{S. S. Abhyankar} [cf. ``Weighted expansions for canonical desingularization'', Lect. Notes Math. 910 (1982; Zbl 0479.14009)], \textit{O. E. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], \textit{E. Bierstone} and \textit{P. Milman} [J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007)] and others.
In the present paper, the author provides another approach to the presentation and uniformization of hypersurface singularities. Generalizing Zariski's method and systematizing Hironaka's ``quasi- canonical resolution procedure'' for hypersurface singularities with a normal crossing factor, he constructs a numerical sequence for any hypersurface singularity, which classifies the singularity completely and, moreover, describes a permissible resolution procedure in a very concrete and effective way. As the author points out, his systematized approach has the advantage of being applicable to the study of hypersurface singularities in positive characteristic, too [cf. the author, Publ. Res. Inst. Math. Sci. 23, No. 6, 965-973 (1987; Zbl 0657.14002)]. resolving the singularities; uniformization of hypersurface singularities [M]Moh, T. T., Canonical uniformization of hypersurface singularities of characteristic zero.Camm. Algebra 20 (1992), 3207--3251. Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Quasi-canonical uniformization of hypersurface singularities of characteristic zero | 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 provide the very first example of a finite set of points in \(\mathbb{P}^{3}_{\mathbb{C}}\) that has the so-called geproci property, but it is neither a grid nor a half-grid. Recall that a finite set of points in \(\mathbb{P}^{3}_{\mathbb{C}}\) having the property that its general projection to a plane is a complete intersection is called a geproci set (which comes from a general projection complete intersection set). For example, grids have the geproci property; recall that by a grid in \(\mathbb{P}^{3}_{\mathbb{C}}\) we mean a set \(Z\) consisting of \(a\cdot b\) points such that there exist two sets of lines \(L_{1}, \dots, L_{a}\) and \(M_{1}, \dots, M_{b}\) with the properties that lines in each of the sets are pairwise skew and such that \(Z = \{L_{i}\cap M_{j} : i \in \{1, \dots,a\}, j \in \{1, \dots,b\}\}.\) Moreover, half-grids have also the geproci property; here by a half-grid we mean a set \(Z\) consisting of \(a\cdot b\) points in \(\mathbb{P}^{3}_{\mathbb{C}}\) such that there exists a set of mutually skew lines \(L_{1}, \dots, L_{a}\) covering the set \(Z\) that a general projection of \(Z\) to a hyperplane is a complete intersection of the images of the lines with a possibly reducible curve of degree \(b\).
The main result of the paper tells us that the set of \(60\) points in \(\mathbb{P}^{3}_{\mathbb{C}}\) that comes from the root system \(H_{4}\) has the geproci property, but it is not a grid, neither a half grid. This result is proved by direct computations. geproci sets; grids; half-grids; root systems Divisors, linear systems, invertible sheaves, Configurations and arrangements of linear subspaces, Ideals and multiplicative ideal theory in commutative rings Generic projections of the \(\mathrm{H}_4\) configuration of 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 In [Lond. Math. Soc. Lect. Note Ser. 242, 85--95 (1997; Zbl 0902.14019)], \textit{T. Oda} showed that the the profinite completions of the étale homotopy type of the moduli stack of hyperbolic curves of genus \(g\) with \(n\) marked points over \(\overline{\mathbb Q}\) and of the Eilenberg-Maclane space \(K(\Gamma_{g,n},1)\) of the corresponding mapping class group \(\Gamma_{g,n}\) are weakly equivalent. Here the authors prove an analogous result for the moduli stack of principally polarized abelian varieties of dimension \(g\geq 1\) over \(\overline{\mathbb Q}\): the profinite completion of its étale homotopy type is weakly equivalent to the profinite completion of the Eilenberg-Maclane space \(K(\text{Sp}(2g,{\mathbb Z}),1)\). In fact, the result is stated for more general moduli stacks of polarized abelian varieties but the argument is the same.
The strategy of the proof is the same as that of Oda: combining the Artin-Mazur comparison theorem between étale and complex analytic homotopy types together with a homotopy descent theorem of Cox, the authors reduce the computation of the étale homotopy type of the moduli stack to that of the homotopy type of its complex analytification. They then compute the latter by means of the analytic hypercovering obtained as the Čech nerve of the uniformization map by the Siegel upper half space.
The paper is very clearly written and recalls a lot of background material for the benefit of the reader. étale homotopy theory; algebraic stacks; moduli of abelian schemes; principally polarised abelian varieties Homotopy theory and fundamental groups in algebraic geometry, Families, moduli of curves (algebraic), Families, moduli of curves (analytic) Étale homotopy types of moduli stacks of polarised abelian schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0653.00005.]
The diagonal of a formal Laurent series \(F=\sum a_{n_ 1,...,n_ s}X_ 1^{n_ 1}...X_ s^{n_ s} \) over a field k is defined as \(\Delta (F)=\sum a_{n,...,n}\lambda^ n \in k((\lambda))\). A formal Laurent series \(f\in k((\lambda))\) is called diagonal of a rational function if \(f=\Delta (P/Q)\) for some polynomials P and Q. The author first reviews some known results on the algebra \({\mathcal D}(k)\subset k((\lambda))\) of diagonals of rational functions. He shows in particular that in characteristic 0 every \(f\in {\mathcal D}(k)\) is a solution of a linear differential equation with polynomial coefficients.
In the second part of the paper it is conjectured that a formal power series with integer coefficients is in \({\mathcal D}\) if and only if it has nonzero radius of convergence over \({\mathbb{C}}\) and solves a nontrivial differential equation. This conjecture is shown to be equivalent to a more geometric formulation involving differential modules over the function field of a smooth projective curve. Assuming conjectures of Y. André, Bombieri and Dwork on G-functions and related topics, the conjecture is reduced to the case of Picard-Fuchs differential equations. The main result of the paper shows that the dimension of a Picard-Fuchs type differential module M in the conjecture is at least the dimension of the r-th homology of the intersection complex of the inverse image of P; here M is defined by the Gauß-Manin connection for a \((r+1)\)- dimensional variety, projective over a smooth curve C, and a point P on C. formal Laurent series; diagonals of rational functions; G-functions; Picard-Fuchs differential equations; Picard-Fuchs type differential module; Gauß-Manin connection G. Christol , Diagonales de fractions rationnelles , Séminaire de Théorie des Nombres , ( 1986 /87), 65 - 90 . Progr. Math. , 75 , Birkhäuser Boston , Boston, MA , 1988 . MR 990506 | Zbl 0694.13013 Formal power series rings, Global ground fields in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Modules of differentials Diagonales de fractions rationelles. (Diagonals of rational fractions) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and residue field \(k\) of characteristic \(p>0\).
{ Theorem (Raynaud).} There is an equivalence of categories between {\parindent=6mm \begin{itemize} \item[{\(\cdot\)}] the category of quasi-paracompact admissible formal \(R\)-schemes, localized by the class of formal blow ups, and \item [{\(\cdot\)}] the category of quasi-separated quasi-paracompact \(K\)-rigid spaces.
\end{itemize}} The main purpose of the present paper is to prove an analogue of Raynaud's theorem: with formal \(R\)-schemes replaced by weak formal \(R\)-schemes, and with \(K\)-rigid spaces replaced by \(K\)-dagger spaces.
Weak formal \(R\)-schemes have been introduced by \textit{D. B. Meredith} [Nagoya Math. J. 45, 1--38 (1972; Zbl 0207.51502)]. They allow a sheafification and globalization of the constructions of Monsky and Washnitzer who used overconvergent function algebras to define a \(p\)-adic cohomology theory for smooth affine \(k\)-schemes. On the other hand, \(K\)-dagger spaces have been introduced by \textit{E. Grosse-Klönne} [J. Reine Angew. Math. 519, 73--95 (2000; Zbl 0945.14013)]. They can be thought of as \(K\)-rigid spaces, but with the usual Tate algebras (and their quotients) replaced by certain dense subalgebras (and their quotients) consisting of 'overconvergent' (rather than just 'convergent') power series.
Thus, the main theorem here reads:
{ Theorem.} There is an equivalence of categories between {\parindent=6mm \begin{itemize} \item[{\(\cdot\)}] the category of quasi-paracompact admissible weak formal \(R\)-schemes, localized by the class of weak formal blow ups, and \item [{\(\cdot\)}] the category of quasi-separated quasi-paracompact \(K\)-dagger spaces.
\end{itemize}} weak formal scheme; dagger space; generic fibre Rigid analytic geometry, \(p\)-adic cohomology, crystalline cohomology An analogue of Raynaud's theorem: weak formal schemes and dagger 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 \(\{X_t\}_{t \in\mathbb{R}}\) be the family of fibers of a polynomial function \(f\) on a non-compact non-singular real algebraic surface. The authors characterize the regular fibers of \(f\) which are atypical due to their asymptotic behaviour at infinity. More precisely, the authors prove the following statement.
Let \(t_0\) be a regular value of \(f\). Then the value \(t_0\) is typical (i.e., the map \(f\) is a \(C^\infty\)-fibration at \(t_0)\) if and only if the Euler characteristic \(\chi(X_t)\) is constant when \(t\) varies in some neighbourhood of \(t_0\) and there is no component of \(X_t\) which vanishes at infinity as \(t\) tends to \(t_0\).
This criterion is compared in the paper with the situation in the complex case. For a family \(\{X_t\}_{t\in\mathbb{C}}\) of fibers of a polynomial function \(f:\mathbb{C}^2 \to\mathbb{C}\), \textit{Hà Huy Vui} and \textit{Lê Dũng Tráng} [Acta Math. Vietnam. 9, 21-32 (1984; Zbl 0597.32005)] proved that a reduced curve \(X_{t_0}\) is typical (i.e., \(t_0\) is typical) if and only if its Euler characteristic \(\chi(X_{t_0})\) is equal to the Euler characteristic of a general fiber of \(f\). On the other hand, a regular fiber \(X_{t_0}\) of a complex polynomial function is typical if and only if there are no vanishing cycles at infinity corresponding to this fiber [see \textit{D. Siersma} and \textit{M. Tibăr}, Duke Math. J. 80, 771-783 (1995; Zbl 0871.32024)]. The authors show that in the real case the two conditions should be considered together: Neither the constancy of Euler characteristic no non-vanishing condition implies that \(X_{t_0}\) is typical. real algebraic surface; Euler characteristic Tibăr, M; Zaharia, A, Asymptotic behaviour of families of real curves, Manuscripta Math., 99, 383-393, (1999) Families, moduli of curves (algebraic), Topology of real algebraic varieties, Controllability of vector fields on \(C^\infty\) and real-analytic manifolds, Fibrations, degenerations in algebraic geometry Asymptotic behaviour of families of real 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 \(G(k,n)\) denote the Grassmannian parametrizing the \(k\)-dimensional vector subspaces of the \(\mathbb C\)-vector space \({\mathbb C}^n\). An \textit{ind-Grassmannian} \(\mathbf X\) is the direct limit of an infinite sequence of embeddings \({\varphi}_i : G(k_i,n_i) \to G(k_{i+1},n_{i+1})\), \(i \geq 1\). One has \({\varphi}_i^\ast{\mathcal O}(1) \simeq {\mathcal O}(d_i)\) for some \(d_i \geq 1\). \(\mathbf X\) is called \textit{twisted} if \(d_i > 1\) for infinitely many \(i\) and \textit{linear}, otherwise.
In the first part of the paper under review, the authors describe the embeddings \(\varphi : G(k,n) \to G(k^\prime,n^\prime)\) with \(\varphi^\ast{\mathcal O}(1) \simeq {\mathcal O}(1)\). As a consequence of this description, it follows that a linear ind-Grassmannian is either isomorphic to \(G(l,\infty)\) for some \(l \geq 1\), or \(\lim_{i\to \infty} k_i = \infty = \lim_{i\to \infty}(n_i - k_i)\). In the first case, the vector bundles of finite rank on \(G(l,\infty)\) were described by \textit{E. Sato} [J. Math. Kyoto Univ. 19, 171--189 (1979; Zbl 0423.14034)] (using previous results of Barth, Van de Ven, and Tyurin), while in the later case \textit{J. Donin} and \textit{I. Penkov} [Int. Math. Res. Not. 2003, No. 34, 1871--1887 (2003; Zbl 1074.14530)] showed that every finite rank vector bundle on \(\mathbf X\) is a direct sum of line bundles.
If \(\mathbf X\) is a twisted ind-Grassmannian then it is conjectured that every finite rank vector bundle on \(\mathbf X\) is trivial. This has been proven by Donin and Penkov for twisted ind-projective spaces and for a special class of twisted ind-Grassmannians. In the second part of the paper under review, the authors prove the conjecture for rank-2 vector bundles. The proof is based on the following proposition: there exists no rank-2 vector bundle \(E\) on \(G(2,4)\) with \(c_2(E)\) a positive multiple of the class of an \(\alpha\)-plane and whose restriction to a general \(\beta\)-plane is trivial. Recall that an \(\alpha\)-plane consists of the 2-subspaces of \({\mathbb C}^4\) containing a fixed 1-subspace and a \(\beta\)-plane consists of the 2-subspaces of \({\mathbb C}^4\) contained in a fixed 3-subspace. The proof of the proposition looks quite involved. ind-Grassmannian; vector bundle Penkov, I.; Tikhomirov, A. S., No article title, Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, Progr. Math., 270, 555-572, (2009) Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Rank-2 vector bundles on ind-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 The notion of Castelnuovo-Mumford regularity is a very useful tool in the study of vector bundles on projective spaces. In particular it allows to characterize direct sums of line bundles (Evans-Griffith splitting criterion). Regularity has been first introduced by Mumford in the 60s and more recently it has been generalized by many authors to other ambient spaces, such as Grassmannians, products of projective spaces, quadrics.
The authors of the paper under review propose a new notion of regularity in the case of Grassmannians of lines. Their definition is based on the analogue of the Koszul exact sequence, obtained taking the exterior powers of the universal exact sequence on the Grassmannian. This notion is less restrictive than the other previous definitions of regularity, and allows to obtain a characterization of the direct sums of line bundles on the Grassmannian. Moreover the authors find a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian. universal bundles on Grassmannians; Castelnuovo-Mumford regularity Costa, L., Miró-Roig, R.M.: Homogeneous Ulrich bundles on Grassmannians of lines (2012) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cohomological characterization of vector bundles on Grassmannians of lines | 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 There has been recently a renewal of the theory of operads motivated by the study of moduli spaces of curves [\textit{V. Ginzburg} and \textit{M. Kapranov}, ``Koszul duality for operads'', Duke Math. J. 76, No. 1, 203-272 (1994; Zbl 0855.18006)]. In fact, the operadic formalism is closely related to curves of genus 0. The authors introduce in this paper a higher genus analogue of the theory of operads. The resulting objects are called modular operads; their definition relies on the combinatorics of graphs in the same way as the definition of usual operads relies on the combinatorics of trees. Modular operads are then studied systematically and various examples coming from the theory of moduli spaces of curves are given.
The bar construction is one of the key tools in the theory of operads. An analogous functor, the Feynman transform, is constructed on the category of dg-modular operads, which generalizes Kontsevich's graph complexes [\textit{M. Kontsevich}, ``Formal (non)-commutative symplectic geometry'', in: The Gelfand mathematics seminars, 1990-1992 , 173-187 (1993; Zbl 0821.58018)].
Using the theory of symmetric functions, the (Euler) characteristics of cyclic and modular operads are defined and studied [for what concerns cyclic operads, see \textit{E. Getzler} and \textit{M. Kapranov}, ``Cyclic operads and cyclic homology'', in: Geometry, topology and physics for Raoul Bott, Conf. Proc. Lect. Notes Geom. Topol. 4, 167-201 (1995; Zbl 0883.18013)]. For such purposes, powerful analogues of the Legendre and Fourier transforms are introduced in the setting of symmetric functions. Euler characteristics of Feynman transforms are then calculated. This calculation is closely related to Wick's theorem and to the computation by Harer-Zagier and Kontsevich of the Euler characteristics of moduli spaces [\textit{M. Kontsevich}, ``Intersection theory on moduli spaces of curves and the matrix Airy function'', Commun. Math. Phys. 147, No. 1, 1-23 (1992; Zbl 0756.35081)]. operad; bar-construction; graph; symmetric function; Feynman diagram; modular operads; moduli spaces of curves; Euler characteristics; Feynman transforms E. Getzler and M. M. Kapranov, Modular operads, \textit{Compositio Math.}, 110 (1998), no. 1, 65--126.Zbl 0894.18005 MR 1601666 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads, Families, moduli of curves (algebraic), Relational systems, laws of composition Modular operads | 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 continuum is a compact connected metric space. A continuum \(X\) is homogeneous if for any two elements \(x,y \in X\) there exists an autohomeomorphism \(h\) of \(X\) such that \(h(x)=y\); in the case that \(X\) admits a compatible metric \(d\) such that for all \(x,y \in X\) the homeomorphism \(h\) can be chosen to be an isometry, \(X\) is called isometrically homogeneous.
Continua which are topological groups and continua which are isometrically homogeneous are two particular but very important classes of continua. This paper is devoted to studying these two classes of continua and the structure of their subcontinua having nonempty interior.
A continuum \(X\) is: (a) indecomposable if each of its proper subcontinua has empty interior; (b) aposyndetic if for every pair of distinct points \(x,y \in X\), there exists a subcontinuum of \(X\) containing one of the points in its interior and non containing the other; (c) semi-indecomposable if for any two disjoint subcontinua of \(X\), at least one has an empty interior; and (d) mutually aposyndetic if for every pair of distinct points \(x\) and \(y\) in \(X\), there exist disjoint subcontinua \(M\) and \(N\) of \(X\) such that \(x\) is in the interior of \(M\) and \(y\) is in the interior of \(N\).
The main result of this paper is: \newline THEOREM. Let \(G\) be a non-degenerate continuum which is either isometrically homogeneous or a topological group. Then, exactly one of the three following statements holds: \newline (a) \(G\) is indecomposable. \newline (b) \(G\) is semi-indecomposable and aposyndetic. \newline (c) \(G\) is mutually aposyndetic.
Each of the three classes (a), (b) and (c) is composed of continuum-many mutually non-homeomorphic members. Clearly, if \(M\) and \(N\) are in two different classes, then the subcontinua of \(M\) with nonempty interior play an extremely different role than the subcontinua of \(N\) with nonempty interior.
All the solenoids belong to class (a), a big family of products of two solenoids belong to class (b), and class (c) is very large, in particular, contains all locally connected continua which are topological groups and another big family of products of two solenoids.
The paper also contains some characterizations of solenoids and the author proves that path connected isometrically homogeneous continua are locally connected. aposyndetic; ample; continuum; homogeneous; indecomposable; isometrically homogeneous; semi-indecomposable; topological group Prajs, J., Isometrically homogeneous and topologically homogeneous continua, Indiana Univ. Math. J., 65, 1289-1306, (2016) Continua and generalizations, Topological groups (topological aspects), Homogeneous spaces and generalizations, Structure of general topological groups, Topological spaces of dimension \(\leq 1\); curves, dendrites Isometrically homogeneous and topologically homogeneous continua | 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 an \(l\)-dimensional vector space over a field of characteristic 0, and let \(A\) be an arrangement in \(V\), i.e. a finite set of \((l-1)\)-dimensional subspaces. Let \(L(A)\) be the intersection lattice, i.e. the set of all intersections of elements in \(A\). An arrangement \(A\) is called essential if the intersection of all elements in \(A\) is 0. A hyperplane \(H\) is \(k\)-generic to \(A\) if \(X\nsubseteq H\) for every \(X\in L(A)\) with \(\text{codim} X<k\). \(A\) is \(k\)-generic if \(H\) is \(k\)-generic to \(A\setminus\{H\}\) for every \(H\in A\).
Let \(S=k[x_1,\dots,x_l]\), let \(R\) be the coordinate ring \(S/Q\), where \(Q= \prod_{H\in A}H\), and let \(D(A)\) be the \(S\)-module of \(k\)-derivations of \(R\), i.e. \(\{\eta\in \text{Der}_k (S);\eta (Q)\subseteq QS\}\). It is shown that, if \(A\) is an essential \((l-1)\)-generic arrangement, then the Hilbert function of \(D(A)\) is a polynomial in degrees \(\geq| A|-l\). As a consequence it follows that \(D(A)\) is generated as an \(S\)-module by derivations of degree \(\leq d-l+1\), i.e. by derivations \(\eta\) such that \(\deg(\eta (x_i))\leq d-l+1\) if \(A\) contains \(d\) hyperplanes. essential arrangement; intersection lattice; Hilbert function Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Configurations and arrangements of linear subspaces On Hilbert polynomials of modules of logarithmic vector fields of arrangements | 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 gives a survey on geometric applications of the residue theorem in the \(n\)-dimensional affine or projective space. Let \(f:=(f_ 1,\dots,f_ n)\) be a sequence of polynomials in the polynomial ring \(K[X_ 1,\dots,X_ n]\) over an algebraically closed field \(K\) such that the homogeneous components form a regular sequence \(Gf=(Gf_ 1,\dots,Gf_ n)\). Then \((f_ 1,\dots,f_ n)\) is also a regular sequence, and the algebraic set \({\mathcal V} (f_ 1, \dots, f_ n) \subset \mathbb{A}^ n\) consists of finitely many closed points \(P_ i\). For any differential form \(\omega=hd X_ 1 \dots dX_ n\) and each point \(P \in {\mathcal V} (f_ 1,\dots,f_ n)\) there is the Grothendieck residue \(\text{Res}_ P [{\omega \over f}]\), for which the author cites several sources in the literature, especially the elementary construction of Scheja and Storch. (In the classical case \(K=\mathbb{C}\) they coincide with the analytically defined residues.) One defines \(G \omega:=Gh \cdot dX_ 1 \dots dX_ n\) and \(\det \omega:=\deg h+ \sum^ n_{i= 1} \deg X_ i\). Then the residue theorem states that for \(\deg \omega \leq \sum^ n_{i=1} \deg X_ i\) one has
\[
\sum_{P \in {\mathcal V} (f)} \text{Res}_ P \left[ {\omega \over f}\right]=\text{Res}_ O \left[ {G \omega \over Gf} \right].
\]
The right hand side of this equation vanishes for \(\deg \omega< \sum^ n_{i=1} \deg X_ i\). If one takes the standard graduation \(\deg X_ i=1\) for all \(i\) then the left hand side of the residue theorem depends only on the points at infinity of the hypersurfaces \(h=0\) und \(f_ i=0\), \(i=1,\dots,n\).
By choosing suitable differential forms \(\omega\) which allow a geometric interpretation one can get many classical results in intersection theory of hypersurfaces in \(\mathbb{A}^ n\) and generalizations of them as corollaries of the residue theorem. The author gives several impressing examples. Among others he proves a theorem of Newton, a formula of Reiss (1837), a formula of Jacobi (1835), a theorem of Humbert (1885) and even the well known theorems of Pappus and Pascal. He also gives generalizations of some of these theorems. differential form; Grothendieck residue E. Kunz, Über den \?-dimensionalen Residuensatz, Jahresber. Deutsch. Math.-Verein. 94 (1992), no. 4, 170 -- 188 (German). Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Relevant commutative algebra, Modules of differentials Über den \(n\)-dimensionalen Residuensatz. (On the \(n\)-dimensional residuum 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 Let \(H=(f,g):\mathbb{C}^ 2\to\mathbb{C}^ 2\) be a polynomial map and
\[
N(H)=\bigl\{v\in\mathbb{R}:\exists A>0,\exists B>0,\forall| z|>B,A| z|^ v\leq| H(z)|\bigr\}.
\]
The Łojasiewicz exponent \({\mathcal L}_ \infty(H)\) at infinity of \(H\) is defined as the supremum of the set \(N(H)\) if \(N(H)\neq\emptyset\) and \(- \infty\) otherwise. The authors give a description of \({\mathcal L}_ \infty(H)\) for general \(H\) as well as for \(H=(h_ x',h_ y')\) where \(h(x,y)\) is a polynomial. They also use this definition to characterize polynomials \(h\) such that \((h,g)\) is an automorphism of \(\mathbb{C}^ 2\). A necessary and sufficient condition for this is that \((h_ x',h_ y')\) does not vanish anywhere in \(\mathbb{C}^ 2\) and \({\mathcal L}_ \infty(h_ x',h_ y')>-1\). Lojasiewicz exponent Jacek Chądzyński and Tadeusz Krasiński, Sur l'exposant de Łojasiewicz à l'infini pour les applications polynomiales de \?² dans \?² et les composantes des automorphismes polynomiaux de \?², C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 13, 1399 -- 1402 (French, with English and French summaries). Automorphisms of curves, Power series, series of functions of several complex variables On the Lojasiewicz exponent at infinity for polynomial mappings from \(\mathbb{C}^2\) to \(\mathbb{C}^2\) and components of polynomial automorphisms of \(\mathbb{C}^2\) | 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 Let \(X\) be a modular curve and consider a sequence of Galois orbits of CM points in \(X\), whose \(p\)-conductors tend to infinity. Its equidistribution properties in \(X(\mathbf{C})\) and in the reductions of \(X\) modulo primes different from \(p\) are well understood. We study the equidistribution problem in the Berkovich analytification \(X^{\mathrm{an}}_p\) of \(X_{\mathbf{Q}_p}\).
We partition the set of CM points of sufficiently high conductor in \(X_{\mathbf{Q}_p}\) into finitely many explicit \textit{basins} \(B_V\), indexed by the irreducible components \(V\) of the \(\bmod\)-\(p\) reduction of the canonical model of \(X\). We prove that a sequence \(z_n\) of local Galois orbits of CM points with \(p\)-conductor going to infinity has a limit in \(X^{\mathrm{an}}_p\) if and only if it is eventually supported in a single basin \(B_V\). If so, the limit is the unique point of \(X^{\mathrm{an}}_p\) whose \(\bmod\)-\(p\) reduction is the generic point of \(V\).
The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin-Tate space. CM points; equidistribution; Berkovich spaces; arithmetic dynamics Complex multiplication and moduli of abelian varieties, Complex multiplication and abelian varieties \(p\)-adic equidistribution of CM 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 authors propose a rather general definition of an embedding \(\varepsilon\) of a point-line geometry \(\Gamma=(\mathcal{P,L})\) to the set \(\mathcal P_V\) of points of a projective space \(\mathrm{PG}(V)\); they define that \(\varepsilon\) is an injective mapping of \(\mathcal P\) to \(\mathcal P_V\) that satisfies three axioms, the most remarkable one is:
E(1) \; For every line \(\ell\in\mathcal L\) the image \(\varepsilon(\ell):=\{\varepsilon(p)\}_{p\in\ell}\) of \(\ell\) spans a \(d\)-dimensional subspace of \(\mathrm{PG}(V)\).
The authors speak of a locally \(d\)-dimensional embedding of \(\Gamma\) in \(\mathrm{PG}(V)\). This definition includes many different situations considered in literature.
The authors ``focus on a class of \(d\)-embeddings, which they call Grassmann embeddings, where the points of \(\Gamma\) are firstly associated to the lines of a projective geometry \(\mathrm{PG}(V)\), next they are mapped onto the points of \(\text{PG}(V\wedge\,V)\) via the usual projective embedding of the line-Grassmannian of \(\mathrm{PG}(V)\) in \(\mathrm{PG}(V\wedge\,V)\).'' The authors present a tool which allows them to describe polar line-Grassmannians by matrix equations, using the isomorphism of \(V\wedge V\) to the space \(S_n(\mathbb F)\) of skew-symmetric \((n\times n)\)-matrices over \(\mathbb F\). The authors consider ``Grassmann embeddings of a number of generalized quadrangles. They also compare those embeddings with other embeddings, which they call quadratic Veronesean embeddings, obtained by composing a projective embedding \(\varepsilon\) with the usual quadratic Veronesean mapping of the projective space hosting \(\varepsilon\).'' embeddings; polar space; generalized quadrangle; exterior products; tensor products; Grassmann variety; Veronese variety Cardinali, I.; Pasini, A., Embeddings of line-Grassmannians of polar spaces in Grassmann varieties. groups of exceptional type, Coxeter groups and related geometries, Springer Proc. Math. Stat., vol. 82, 75-109, (2014), Springer New Delhi Incidence structures embeddable into projective geometries, Polar geometry, symplectic spaces, orthogonal spaces, Generalized quadrangles and generalized polygons in finite geometry, Exterior algebra, Grassmann algebras, Multilinear algebra, tensor calculus, Varieties and morphisms Embeddings of line-Grassmannians of polar spaces in Grassmann 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 Let \(X\subset \mathbb {CP}^{n+1}\) be an \(n\)-dimensional complex manifold (e.g. the smooth part of a hypersurface). For each \(x\in X\) let \(C_{q,x} \subset \mathbb {P}T_xX\) be the set of all tangent directions to osculating lines of order \(q\) with \(X\) at \(x\) (if \(X\) is the smooth part of a degree \(q\) hypersurface, then \(C_{q,x}\) is the cone of tangent directions of lines in \(X\) through \(x\)).
Under mild genericity assumptions on \(X\) and \(x\), the authors show that \(C_{q,x}\) is the transversal intersection of \(q-1\) smooth hypersurfaces of degree \(2,\dots ,q\) in \(\mathbb {CP}^{n-1}\). Call \(M_q\) the space of all such complete intersections. They fix \(X\) and move \(x\). They get an open subset \(U\) of \(X\) and a morphism \(f_q: U \to M_q\). They prove that the differential of this map has rank \(n = \dim (X)\) (the maximal possible one). They prove that the map \(f_q\) is not arbitrary or general, because it satisfies a system of first-order partial differential equations. This system is obtained expressing the condition that the Gauss map of \(f_q\) must lie in a smaller variety (explicitly described). As a corollary they get conditions which assures that \(X\) is uniruled. If \(q\geq 4\) they prove the existence of a nonempty Zariski open subset \(A_q\) of \(M_q\) such that \(f_q\) has rank \(n\) for any \(X\) for which \(f_q\) is defined and sends a general \(x\in X\) into \(A_q\). The proofs use moving frames. complex hypersurface; osculating line; Gauss map; moving frame Landsberg, JM; Robles, C, Lines and osculating lines of hypersurfaces, J. Lond. Math. Soc., 82, 733-746, (2010) Projective and enumerative algebraic geometry, Exterior differential systems (Cartan theory) Lines and osculating lines of 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 The author gives:
(1) a biduality theorem for the perfect complexes of left (or right) modules over the ring of differential operators \({\mathcal D}^\dag_X\), introduced by \textit{P. Berthelot} [in: \(p\)-adic Analysis, Proc. Int. Conf., Trento 1989, Lect. Notes Math. 1454, 80-124 (1990; Zbl 0722.14008)], for \(X\) a smooth scheme over a field of characteristic \(p>0\), or a smooth formal scheme over a complete principal valuation ring of characteristic \((0,p)\) [see also \textit{Z. Mebkhout} and the reviewer, ibid. 267-308 (1990; Zbl 0727.14011) for the case of weakly formal schemes];
(2) the commutation of the duality functor with the inverse image functor by the Frobenius morphism; and
(3) a homological characterization of holonomicity for the coherent \({\mathcal D}_{{\mathcal X}, Q}^\dag\)-modules equipped with a Frobenius structure [\textit{P. Berthelot}, ``\({\mathcal D}_Q^\dag\)-modules cohérents, II, III'' (in preparation)], where \({\mathcal X}\) is a smooth formal scheme over the Witt ring of a perfect field of characteristic \(p\).
Points (1) and (3) are conceptually similar to the corresponding results in the classic cases of complex smooth analytic varieties or smooth algebraic varieties over a field of characteristic 0. Key points in the proof are the structure of \({\mathcal D}^\dag_X\) as a direct limit of the rings of differential operators of finite level, and the fact that the dualizing sheaf \(\omega_X\) has a canonical structure of right \({\mathcal D}_X^\dag\)-module [\textit{P. Berthelot} (loc. cit.); see also \textit{B. Haastert}, Manuscr. Math. 62, No. 3, 341-354 (1988; Zbl 0673.14012) and \textit{Z. Mebkhout} and the reviewer (loc. cit., 1.1.4)]. Frobenius morphism; ring of differential operators VIRRION (A.) . - Théorème de bidualité et caractérisation des F-D\dagger X,Q- modules holonomes , C. R. Acad. Sciences Paris, t. 319, Série I, 1994 , p. 1283-1286. MR 96e:14019 | Zbl 0829.14010 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, \(p\)-adic cohomology, crystalline cohomology, Rings of differential operators (associative algebraic aspects) Biduality theorem and characterisation of holonomic \(F\)-\({\mathcal D}_{{\mathcal X},\mathbb{Q}}^ \dagger\)-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 author computes the irregularity of cyclic multiple planes associated to the branch curve with arbitrary singularities. The paper is very well written and has numerous examples and applications.
In more detail, let \(B=\{f(x,y)=0\}\) be a (complex, reduced) plane curve, transversal to the infinite line. The corresponding surface \(S_0=\{z^n=f(x,y)\}\) is in general singular, let \(S\to S_0\) be its resolution. The irregularity \(q(S):=h^1(\mathcal{O}_S)=h^0(\Omega^1_S)\) was first computed by Zariski, in the case when \(B\) has nodes and ordinary cusps only. The irregularity is expressed via \(h^1(\mathbb{P}^2,I_Z)\), where \(Z\) is the support of cusps.
Generalizations and applications of Zariski's formula were studied by many people (Esnault, Libgober, Artal-Bartolo). The author generalizes the formula to the case of \(B\) with arbitrary singularities. The formula is in terms of multiplier ideals of \(B\) and the corresponding jumping numbers..
The proof follows Zariski's ideas. The resolution \(S\to S_0\to\mathbb{P}^2\) is expressed as a standard cyclic covering. Then the theory of cyclic coverings [\textit{R. Pardini}, J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)] is applied to express the irregularity. Finally the Kawamata-Viehweg-Nadel vanishing theorem is used.
In \S2 (preliminaries) the cyclic covers and multiplier ideals are nicely introduced. In \S3 the main theorem is proved. \S4 contains examples and applications. In particular, the author uses the main theorem to reconstruct the examples of Zariski pairs (by Artal-Bartolo). In Appendix the author computes in details the case when \(B\) has singularities of the form \(x^m+y^n=0\). cyclic coverings; irregularity; Zariski pairs Naie D.: Irregularity of cyclic multiple planes after Zariski. L'enseignement mathématique 53, 265--305 (2008) Special surfaces, Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Coverings in algebraic geometry, Ramification problems in algebraic geometry, Vanishing theorems in algebraic geometry The irregularity of cyclic multiple planes after Zariski | 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 authors are motivated by Zannier's proof of Pisot's \(d\)th root conjecture, for linear recurrence sequences defined over number fields \textit{U.~Zannier} [Ann. Math. (2) 151, No.~1, 375--383 (2000; Zbl 0995.11007)]. Their main result establishes a function field analogue of Zannier's result.
Specifically, in the present article, the authors consider the case of exponential polynomials \[b(n) = \sum_{i=1}^\ell B_i(n) \beta_i^n\text{,}\] where \(B_i \in K[T]\) and \(\beta_i \in K^*\text{,}\) for \(K = k(C)\) the function field of a smooth projective algebraic curve \(C\), defined over an algebraically closed characteristic zero field \(k\).
To formulate the main result, let \(L\) be the smallest finite extension field of \(K\) that has the property that the roots \(\beta_1,\dots,\beta_\ell\) and the leading coefficient of the polynomial \(\sum_{i=1}^\ell B_i(T)\) all have \(d\)th roots in \(L\). Further, assume that the multiplicative subgroup of \(K^*\) that is generated by the roots \(\beta_1,\dots,\beta_\ell\) has trivial intersection with \(k^*\). Finally, assume that \(b(n)\) is a \(d\)th power in \(K\) for infinitely many natural numbers \(n\).
Within this context, the authors establish existence of an exponential polynomial \[ a(m) = \sum_{i=1}^r A_i(m) \alpha_i^m \text{,} \] where \(A_i \in L[T]\) and \(\alpha_i \in \overline{L}^* \text{,}\) together with a polynomial \(R[T] \in k[T]\) for which \[b(m) = R(m) a(m)^d\text{,}\] for all natural numbers \(m\).
The proof of this result is of an independent interest. The idea is to establish function field analogues for the generalized greatest common divisor results of \textit{A.~Levin} [Invent. Math. 215, No. 2, 493--533 (2019; Zbl 1437.11094)] and \textit{A.~Levin} and \textit{J.~T.~-Y. Wang} [J. Reine Angew. Math. 767, 77--107 (2020; Zbl 1454.14070)]. Another point involves an application of the main theorem for one variable function fields from \textit{H.~Pasten} and \textit{J.~T.~-Y. Wang} [Int. Math. Res. Not. 2015, No. 11, 3263-3297 (2015; Zbl 1375.11078)]. Pisot's \(d\)-th root conjecture; function fields; linear recurrences; GCD estimates Algebraic functions and function fields in algebraic geometry, Exponential Diophantine equations, Recurrences On Pisot's \(d\)-th root conjecture for function fields and related GCD estimates | 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 simplest version of the main result of this paper is the following. Let \(f\) be a polynomial on a finite dimensional vector space \(W\). Then the Fourier transform of the absolute value of \(f\) is smooth on a dense open subset of \(W^*\). This result holds uniformly over all local fields \(F\) of characteristic \(0\): If the inputs are defined over some field \(K\) (i.e., \(W = W_K \otimes_K F\) for some \(K\)-vector space \(W_K\) and \(f\) is obtained from a polynomial on \(W_K\) with coefficients in \(K\)), then the authors obtain a dense open sub-variety \(L \subset W_K^*\) such that the Fourier transform is smooth on \(L(F)\), for every \(F\) as above and for every embedding of \(K\) into \(F\).
The full main result strengthens the above in several ways. In particular, the absolute value of a polynomial can be replaced by a more general distribution on \(W\), e.g., by the direct image \(\phi_*(|\omega|)\) of the measure induced by a regular top differential form \(\omega\) on a smooth algebraic variety \(X\), under a proper map \(\phi: X \to W\). Moreover, the result about the Fourier transform is more precise, stating that its wave front set is contained in an isotropic algebraic sub-variety of \(T^*(W^*) = W \times W^*\) (which again can be given uniformly in \(F\)).
Several of the weaker versions of the main result have been known before. What is new is a uniform (in \(F\)) description of the wave front set of Fourier transforms. Moreover, this description is specified explicitly in terms of a desingularization. (Desingularization is a key ingredient to the proof, in contrast to older proofs which used \(D\)-modules in the archimedean case and model theory in the non-archimedean case.) wave front set; distribution; Fourier transform; local field; desingularization 2 A. Aizenbud and V. Drinfeld, 'The wave front set of the fourier transform of algebraic measures', \textit{Israel J. Math.}207 (2015) 527-580 (English). Local ground fields in algebraic geometry, Measures on groups and semigroups, etc., Global theory and resolution of singularities (algebro-geometric aspects), Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups The wave front set of the Fourier transform of algebraic measures | 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 projective scheme over \(\mathbb{C}\) with a very ample line bundle \(\mathcal{L}\), and let \(\mathcal{C}\) be the cone of \(X\) associated with \(\mathcal{L}\). Let \(x_{1},\dots,x_{n}\) be the projective coordinates of \(\mathbb{P}^{n-1}(\mathbb{C})\) such that \(X\subseteq \mathbb{P}^{n-1}(\mathbb{C})\) and \(\mathcal{O}_{\mathbb{P}^{n-1}(\mathbb{C})}(1)\vert_{X}=\mathcal{L}\), and let \(\mathfrak{a} \subseteq R=\mathbb{C}[x_{1},\dots,x_{n}]\) be the defining ideal of \(\mathcal{C}\subseteq \mathbb{A}^{n}_{\mathbb{C}}\). Then the Lyubeznik numbers of \(\mathcal{C}\) are defined as
\[\lambda_{i,j}(\mathcal{C}):= \dim_{\mathbb{C}} \mathrm{Ext}_{R}^{i}\left(\mathbb{C},H_{\mathfrak{a}}^{n-j}(R)\right)\]
for every \(i,j \geq 0\). The authors specify a technical condition that implies the dependence of the Lyubeznik numbers \(\lambda_{i,j}(\mathcal{C})\) on the choice of \(\mathcal{L}\). Using this result, they prove that if \(k\) is a field of characteristic \(0\), then there exist projective schemes over \(k\) such that the Lyubeznik numbers \(\lambda_{i,j}(\mathcal{C})\) of their cone \(\mathcal{C}\) depend on their projective embeddings. This answers a question of [\textit{G. Lyubeznik}, Lect. Notes Pure Appl. Math. 226, 121--154 (2002; Zbl 1061.14005), p. 133] negatively. The striking feature of this result is that it contrasts entirely the positive characteristic case where the machinery of Frobenius endomorphism is available; see [\textit{L. Núñez-Betancourt} et al., Contemp. Math. 657, 137--163 (2016; Zbl 1346.13035)] and [\textit{W. Zhang}, Adv. Math. 228, No. 1, 575--616 (2011; Zbl 1235.13012)]. local cohomology; Lyubeznik numbers; projective schemes Local cohomology and commutative rings, Local cohomology and algebraic geometry, Deformations of complex singularities; vanishing cycles, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings | 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 \(B\) be a Noetherian separated scheme of finite Krull dimension. For a functor \(E: {\mathcal S}m/B^{op} \rightarrow {\mathcal S}pt\) from smooth \(B\)-schemes of finite type to spectra the author constructs the homotopy coniveau tower
\[
\cdots \rightarrow E^{(p+1)}(X, -) \rightarrow E^{(p)}(X,-) \rightarrow \cdots \rightarrow E^{(0)}(X, -).
\]
The \(E^{(p)}(X,-)\) are simplical spectra, whose \(n\)-simplices are limits of spectra with support \(E^W(X\times\Delta^n)\), where \(W\) runs through closed subsets of \(X\times\Delta^n\) with the property that \(\text{codim}_{X\times F} (W\cap(X\times F)) \geq p\) for all faces of \(\Delta^n\). Here \(\Delta^n = \text{Spec}(\mathbb Z[t_0,\cdots,t_n]/(\sum_jt_j-1))\). Let \(E^{(p/p+1)}(X,-)\) denote the homotopy cofiber of the map \(E^{(p+1)}(X, -) \rightarrow E^{(p)}(X,-)\). An immediate consequence of the homotopy coniveau tower is the spectral sequence
\[
E_1^{p.q} :=\pi_{-p-q}(E^{(p/p+1)}(X, -)) \Longrightarrow \pi_{-p-q}(E^{(0)}(X,-)),\tag{\(*\)}
\]
which in case of the \(K\)-theory spectrum yields the Bloch-Lichtenbaum spectral sequence from motivic cohomology to \(K\)-theory [cf. \textit{S. Bloch} and \textit{S. Lichtenbaum}, ``A spectral sequence for motivic cohomology'', preprint (1995); \textit{E. M. Friedlander} and \textit{A. Suslin}, ``The spectral sequence relating algebraic \(K\)-theory to motivic cohomology'', Ann. Sci. Éc. Norm. Supér. (4) 35, 773--875 (2002; Zbl 1047.14011)].
The main result of the paper yields functoriality of the spectral sequence \((*)\) on \({\mathcal S}m/B\). In order to achieve functoriality the author first constructs a functorial model of the presheaf of spectra \(U \mapsto E^{(p)}(U,-)\) on the Nisnevich site \(X_{\text{Nis}}\) of \(X\), which is denoted by \(E^{(p)}(X_{\text{Nis}},-)\). The construction uses a generalization of the classical method used to prove Chow's moving lemma for cycles modulo rational equivalence. This yields functoriality for a similar spectral sequence with \(E^{(p)}(X,-)\) replaced by a fibrant model of \(E^{(p)}(X_{\text{Nis}},-)\). Functoriality of \((*)\) then follows from the localization properties of \(E^{(p)}(X,-)\), which were developed by \textit{M. Levine} [``Techniques of localization in the theory of algebraic cycles'', J. Algebr. Geom. 10, 299--363 (2001; Zbl 1077.14509)]. Bloch-Lichtenbaum spectral sequence; algebraic cycles; coniveau tower 10.1007/s10977-006-0004-5 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Motivic cohomology; motivic homotopy theory, Algebraic cycles Chow's moving Lemma and the homotopy coniveau tower | 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 Consider a homogeneous ideal \(I\) in a polynomial ring \(R = K[x_1,\dots,x_m]\) with the standard grading and some fixed term order. The generic initial ideal of \(I\), \(\mathrm{gin}(I)\), is a coordinate-independent version of the initial ideal; as a monomial ideal, there is a Newton polytope \(P_{\mathrm{gin}(I)}\) of \({R}^m\) associated to it. In this article, the author introduces the generic initial system \(\{\mathrm{gin}(I^n)\}_n\) of \(I\), and defines the limiting shape of this system to be the limit of the polytopes \(\frac{1}{n}P_{\mathrm{gin}(I^n)}\) and studies the case where \(I\) is a complete intersection with minimal generators of degrees \(d_1,d_2,\dots,d_r\) with \(d_1\leq d_2\leq \dots\leq d_r\). The main result of the paper is Theorem 1.1 that describes the limiting shape of the reverse lexicographic generic initial system of such an ideal \(I\). This result easily extends to the case where \(I\) is an \(r\)-complete intersection in \(R\).
The structure of \(\mathrm{gin}(I^n)\) is not so simple; for example, it is not clear that the ideals \(\mathrm{gin}(I^n)\) depend only on the degrees of the generators of \(I\), even when \(n = 1\). The focus of this article is on understanding the behavior of the generic initial ideals of the powers of a fixed ideal.
In Section 2, the author introduces some notation, definition and preliminary results. In particular, in Subsection 2.3 they discuss the geometric interpretation of the volume and of the asymptotic multiplier ideal associated to a graded system of monomial ideal.
Section 3 is devoted to describe the reverse lexicographic generic initial ideals \(\mathrm{gin}(I^n)\) for a complete intersection of type \((d_1,\dots,d_r)\) with \(d_1\leq d_2\leq \dots\leq d_r\). In particular, in Lemma 3.1., it is shown that, no matter how many variables the ambient ring \(R\) has, the minimal generators of these generic initial ideals only involve the first \(r\) variables.
Given this fact, it is natural to wonder whether we can restrict our attention to the case where \(r = m\) and Proposition 3.3 gives us a positive answer, that is, if \(I\) is a type \(d_1,d_2,\dots,d_r\) complete intersection in any polynomial ring \(K[x_1,\dots,x_m]\), then there exists a complete intersection \(L\subset K[x_1,\dots,x_r]\) of the same type such that the minimal generators of \(\mathrm{gin}(I^n)\) are the same as the minimal generators of \(\mathrm{gin}(I^n)\).
Now, fix an \(r\)-complete intersection \(I\) of type \(d_1,d_2,\dots,d_r\). Let \(p_i(n)\) denote the smallest power of \(x_i\) contained inside of \(\mathrm{gin}(I^n)\) so that \((x_1^{p_1(n)}, x^{p_2(n)}_2,\dots,x^{p_r (n)}_r )\) is the largest ideal generated by variable powers that is contained inside of \(\mathrm{gin}(I^n)\). Lemma 3.4, 3.5, and 3.6 give some results on \(p_i(n)\).
In Section 4, the author proves Theorem 1.1 describing the asymptotic behavior of the reverse lexicographic generic initial system of a complete intersection \(I\) of type \(d_1,d_2,\dots,d_r\), and \(p_i(n)\) is the minimal power of \(x_i\) contained in \(\mathrm{gin}(I^n)\).
Thus, by Proposition 3.3, Theorem 1.1 gives the limiting shape \(P\) of any complete intersection.
Moreover, from Subsection 2.3, knowing the limiting shape of a graded system is enough to compute its asymptotic multiplier ideal (see Corollary 4.2). asymptotic; complete intersection; generic initial ideal; graded system; multiplier ideal S. Mayes, The limiting shape of the generic initial system of a complete intersection, Comm. Algebra, 42, 2299, (2014) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Linkage, complete intersections and determinantal ideals, Graded rings, Multiplier ideals The limiting shape of the generic initial system of a complete intersection | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Dans cet article, nous avons essayé de décrire de manière très explicite un idéal à gauche I de l'anneau \({\mathcal D}\) des germes d'opérateurs différentiels analytiques d'une variable complexe; pour cela nous utilisons essentiellement un théorème de division adapté à \({\mathcal D}\). Nous commençons par donner une présentation de I de la forme \(0\to {\mathcal D}^{q-p}\to {\mathcal D}^{q-p+1}\to I\to 0,\) puis deux générateurs canoniques \(F_ p\) et \(F_ q\) de \({\mathcal D}\). Ensuite nous associons à I le couple \(E(I)\rightleftharpoons^{u}_{v}F(I)\) formé par les solutions analytiques de \({\mathcal D}/I\) dans un disque coupé, les solutions microfonctions, le morphisme canonique et le morphisme de variation; et nous démontrons que ce couple détermine l'idéal I. Nous démontrons alors que la catégorie des \({\mathcal D}\)-modules holonomes à singularité régulière est isomorphe à la catégorie des couples \(E\rightleftharpoons^{u}_{v}F\) ce qui nous permet de retrouver le théorème de structure de L. Boutet de Monvel; cela grâce à une idée de B. Malgrange. Enfin, nous précisons ce théorème dans le cas où \(I={\mathcal D}P\) est principal, P opérateur à singularité régulière. germ of analytic differential module; regular singularity; holonome D- module; microfunctions J. Briançon et P. Maisonobe , Idéaux de germes d'opérateurs différentiels à une variable , Enseign. Math. 30 (1984). Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Power series rings, General theory of ordinary differential operators, Ordinary differential equations in the complex domain Idéaux de germes d'opérateurs différentiels à une variable | 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 subset \(S\) of \(\mathbb{R}^n\), denote by \({\mathcal B}_d(S)\) the set of all polynomials in \(\mathbb{R}[x_1,\ldots,x_n]\) whose square is bounded on \(S\) by a polynomial of degree at most \(2d\). Each set \({\mathcal B}_d(S)\) is a module over \({\mathcal B}_0(S)\) and their direct sum \(\mathcal{B}(S)\) is a graded algebra. In this article, the authors systematically construct examples and counterexamples to the following statements:
(1) \({\mathcal B}(S)\) is finitely generated.
(2) \({\mathcal B}_0(S)\) is finitely generated.
(3) Every \({\mathcal B}_d(S)\) is a finitely generated \({\mathcal B}_0(S)\) module.
(4) \({\mathcal B}_0(S) = \mathbb{R}\) implies that every \({\mathcal B}_d(S)\) is a finite dimensional vector space.
The ``interesting'' examples and counterexamples pertain to semialgebraic sets without low-dimensional components and with isolated points at infinity.
Above statements are logically related to the ``moment problem'': Given a functional \(\varphi: \mathbb{R}[x_1,\ldots,x_n] \to \mathbb{R}\), is there a measure \(\mu\) such that \(\varphi(p) = \int p\; d\mu\) for all polynomials \(p \in \mathbb{R}[x_1,\ldots,x_n]\)? In some cases, this problem can be decided by criteria of \textit{E. K. Haviland} [Am. J. Math. 58, 164--168 (1936; Zbl 0015.10901)] and \textit{K. Schmüdgen} [Math. Ann. 289, No. 2, 203--206 (1991; Zbl 0744.44008), J. Reine Angew. Math. 558, 225--234 (2003; Zbl 1047.47012)]. On the other hand, it follows from [\textit{C. Scheiderer,} J. Complexity 21, No. 6, 823--844 (2005; Zbl 1093.13024)] that in case of \({\mathcal B}_0(S) = \mathbb{R}\) and validity of (4) the moment problem cannot be decided by these criteria.
The main insight of this article is that there are sets that are neither amenable to Schmüdgen's nor to Scheiderer's criteria. In other words: Existing methods are insufficient to decide the moment problem for sets constructed in this paper. semialgebraic set; compactification; graded algebra; moment problem Mondal, M.; Netzer, T., How fast do polynomials grow on semialgebraic sets?, J Algebra, 413, 320-344, (2014) Semialgebraic sets and related spaces, Graded rings and modules (associative rings and algebras), Moment problems, Toric varieties, Newton polyhedra, Okounkov bodies, Compactifications; symmetric and spherical varieties How fast do polynomials grow on semialgebraic 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 author relates the geometry of the maximum dimension locus of a real irreducible analytic germ \(X_ 0\subset {\mathbb{R}}^ n_ 0\) and the space of orders of the field of germs of meromorphic functions over \(X_ 0\), \({\mathcal O}(X_ 0)\). Results of the same nature are already known in the real algebraic case.
A formal half branch is defined to be a germ of non-constant \(C^{\infty}\)-mapping from \(]0,\epsilon[\) to \(X_ 0,c\), such that, if \(\hat c\) is the jet of c, an analytic function f is such that \(f(\hat c)=0\) iff for \(\epsilon\) sufficiently small for all \(t\in]0,\epsilon [\): \(f(c(t))=0\) \((t\to(t,e^{-1/t}))\) is not a formal half branch of \({\mathbb{R}}^ 2_ 0\), for example). The dimension of a formal half-branch is the dimension of the smallest analytic germ containing the image of c. - An order \(\alpha\) of \({\mathcal O}(X_ 0)\) is centered at a formal half- branch of \(X_ 0\) if every germ of analytic function over \(X_ 0\) positive on the image of c is positive for \(\alpha\). This is a geometric illustration of the theory of specialization in the real spectrum of the ring of germs of analytic functions on \(X_ 0\). Let \(\Omega\) be the set of orders on \({\mathcal O}(X_ 0)\), \(\Omega^*\) (resp. \(\Omega_ e\), \(e=1,...,d=\dim X_ 0)\) be the set of orders centered at a formal half- branch (resp. of dimension e). The author shows that \(\Omega_ 1,...,\Omega_ d\), \(\Omega-\Omega^*\) are disjoint and dense in \(\Omega\). Then he establishes a bijection between the clopens of \(\Omega\) and the regularly closed semi-analytic germs of the maximum dimension locus \(X^*_ 0\) of \(X_ 0\), which gives, as in the real algebraic case, a solution to Hilbert 17th problem. analytic germ; semi-analytic real spectrum; space of orders of the field of germs of meromorphic functions; formal half branch; maximum dimension locus; Hilbert 17th problem Ruiz, J.: Central orderings in fields of real meromorphic function germs. Preprint (1984) Real algebraic and real-analytic geometry, Real-analytic manifolds, real-analytic spaces, Germs of analytic sets, local parametrization, Real-analytic functions, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) Central orderings in fields of real meromorphic function germs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal{X}\) be a proper scheme over the field \(F\) of functions meromorphic in an open neighborhood of zero in the complex plane. The scheme \(\mathcal {X}\) gives rise to a proper morphism of complex analytic spaces \(\mathcal{X}^h \to D^\ast\) and, if the radius of the open disc \(D\) is sufficiently small, the cohomology groups of the fibers \(\mathcal{X}^h\_t\) at points \(t \in D^\ast\) form a variation of mixed Hodge structures on \(D^\ast\), which admits a limit mixed Hodge structure. The purpose of the paper is to construct a canonical isomorphism between the weight zero subspace of this limit mixed Hodge structure and the rational cohomology group of the non-Archimedean analytic space \(\mathcal{X}^{\text{an}}\) associated to the scheme \(\mathcal{X}\) over the completion of the field \(F\). complex and non-archimedean analytic spaces; limit mixed Hodge structures V. G. Berkovich. A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures. In Algebra, Arithmetic and Geometry - Manin Festschrift (to appear). Boston: Birkhäuser. Variation of Hodge structures (algebro-geometric aspects), Rigid analytic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Non-Archimedean analysis, Period matrices, variation of Hodge structure; degenerations, Mixed Hodge theory of singular varieties (complex-analytic aspects) A non-archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(U\in\text{SO}(N)\), such that \(n\) of its eigenvalues are equal to 1. Let \(\Lambda_U\) be its characteristic polynomial and let \(\Lambda^{(n)}_U\) be its \(n\)-th derivative. The author shows that, for small \(\epsilon>0\), the probability that \(| \Lambda_U^{(n)}(1)| <\varepsilon\) is approximately \(\frac2{2n+1}\,\varepsilon^{\frac{2n+1}2}\,f(n,M)\), where \(M=(N-n)/2\) and
\[
f(n,M)=\frac{\displaystyle\prod_{j=1}^M \frac{\Gamma(j)\Gamma(M+n+j-1)}{\Gamma(n-1/2+j)\Gamma(M+j-3/2)}}{(n!)^{\frac{2n+1}2}\,2^{(2n+1)M}\,\Gamma(M)}\,.
\]
The computation is related to an ongoing program that examines how random matrix theory can be used to predict the distribution of values of the Riemann zeta function and other \(L\)-functions. The paper contains a well-written and thorough discussion of these relations. elliptic functions; random matrices; L-functions; eigenvalues; Riemann zeta function; elliptic curves Snaith, N. C. (2005). Derivatives of random matrix characteristic polynomials with applications to elliptic curves. J. Phys. A 38 10345-10360. Random matrices (algebraic aspects), \(\zeta (s)\) and \(L(s, \chi)\), Elliptic functions and integrals, Elliptic curves, Elliptic curves over global fields Derivatives of random matrix characteristic polynomials with applications to 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 \(F:\mathbb{K}^{n}\mathbb{\rightarrow K}^{n}\) be a polynomial mapping such that \(F^{-1}(0)\) is compact, where \(\mathbb{K=R}\) or \(\mathbb{C}\). The authors give an estimation of the Łojasiewicz exponent at infinity of \(F\) (defined as the supremum of those real numbers \(\alpha \) for which there exist constants \(C,M>0\) such that \(\left| \left| x\right| \right| ^{\alpha }\leq C\left| \left| F(x)\right| \right| \) for \(\left| \left| x\right| \right| \geq M),\) under some non-degeneracy conditions for some Newton polyhedra. This estimation allows a uniform estimation of the Łojasiewicz exponent at infinity for the mappings in a homotopy \(F+tG,\) \(t\in [ 0,1]\), where \(G \) is another polynomial mapping satisfying appropriate conditions. This, in turn, implies the invariance of the global index of the mappings in the homotopy \(F+tG.\) In particular, \(F\) and \(F+G\) have the same global index. polynomial mapping; topological index; Newton polyhedron; Łojasiewicz exponent at infinity Bivià-Ausina, C; Huarcaya, JAC, Growth at infinity and index of polynomial maps, J. Math. Anal. Appl., 422, 1414-1433, (2015) Singularities of vector fields, topological aspects, Singularities of holomorphic vector fields and foliations, Affine geometry, Real algebraic and real-analytic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Growth at infinity and index 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 If \(k\) is a field of characteristic zero and \(A:=k[x]\) the polynomial ring in the variable \(x\), we may define the \(l\)th module of principal parts \(P(A,l):=A\otimes_k A/I^{l+1}\) where \(I\) is the ideal of the diagonal in \(A\otimes_k A\). The module \(P(A,l)\) is canonically a left and right \(A\)-module. We may define \(dx:=1\otimes x-x\otimes 1 \in A\otimes_k A\), and it follows
\[P(A,l)\cong A\{1,dx,dx^2,\dots,dx^l\} \tag{1}\]
is free as left \(A\)-module on the basis \(dx^i\) for \(i=0,\dots,l\). There is a canonical map \(T^l: A \rightarrow P(A,l)\) defined by \(T^l(f(x)):= 1\otimes f(x)\) and one proves that \(T^l(f(x))=\sum_{i\geq 0}\frac{1}{i!}f^{(i)}(x)dx^i\) where \(f^{(i)}(x)\) is the \(i\)th formal derivative of the polynomial \(f(x)\) with respect to the \(x\)-variable. Hence the map \(T^l\) Taylor expands the polynomial \(f(x)\) in the basis \(dx^i\). The Taylor map is a differential operator of order \(l\) -- the universal differential operator for \(E\). The module of principal parts \(P(A,l)\) may be defined in greater generality, and in the case of a smooth scheme \(X\) of finite type over a field \(k\) of characteristic zero and \(E\) a locally trivial finite rank \(\mathcal O\)-module, one may define \(P(E,l)\) -- the \(l\)th module of principal parts of \(E\). It follows \(P(E,l)\) is a locally trivial \(\mathcal O\)-module from the left and right, and there is a Taylor morphism \(T^l:E \rightarrow P(E,l)\) which locally gives a Taylor expansion of a local section \(s\) of \(E\) over an open set \(U\) in \(X\) similar to the situation for the polynomial ring in one variable. General properties of the modules of principal parts may be found in the paper [\textit{A. Grothendieck}, Publ. Math., Inst. Hautes Étud. Sci. 32, 1--361 (1967; Zbl 0153.22301)].
In the paper under review, the authors claim they prove some general properties (including the isomorphism in (1)) of this construction that are not presented in the literature. The reviewer believes that much of this may be found in [loc. cit.], too.
Furthermore, the authors study the computational properties of the principal parts with the aim of formalizing the notion ``fractional derivative'' in an algebro-geometric setting. Given a real analytic function \(f(x)\) on an interval \(I:=(a,b)\) and any positive real number \(\alpha \in (0,\infty)\) they ``define'' in 1.2.1 the non-integer Riemann-Liouville fractional derivative as
\(D{\alpha}_{a+}f(x):= \sum_{n\geq 0} \binom{\alpha}{n} \frac{(x-a)^{(n-\alpha)}}{\Gamma(n-\alpha+1)}f{(n)}(x).\)
In section 3 of the paper, the authors do something similar for algebraic varieties for sections of line bundles on the projective line. If \(X\) is the projective line over the field of real numbers with homogeneous coordinates \(x,y\) and \(J(\infty)\) is the module of principal parts of infinite order, with \(t:=y/x\) a coordinate on the open set \(D(x)\) they introduce the infinite Taylor expansion for a section \(f \in \Gamma(D(x), O_X)\):
\[J^{\infty}(f(t))(t_0):=(f(t_0), D(f)(t_0), \frac{1}{2}D^2(f)(t_0),\dots, \frac{1}{n!}D^n(f)(t_0),\dots),\]
called the algebraic infinitesimal neighborhood of the real number \(t_0\). It appears \(f(t)\) is a polynomial or a rational function, hence for a polynomial \(f\) it follows if \(n\geq deg(f)\) we have \(D^n(f)=0\). Hence what appears to be an infinite sequence of numbers in \(J{\infty}(f(t))(t_0)\) is a finite set of numbers when \(f\) is a polynomial. The authors do something similar to define the ``Riemann-Liouville fractional derivative'' in the algebraic geometric setting. The authors remark that in the algebraic geometric setting the Zariski topology has large open sets, that sections of structure sheaves of algebraic varieties are polynomials and rational functions and that it may be better to work with sheaves of real valued smooth functions and sheaves of sections of real smooth vector bundles on smooth manifolds. Since the definition of the sheaves of principal parts is done using the sheaf of real valued smooth functions on the manifold and sheaves of sections of vector bundles, this should be possible. In fact there is already such a theory of sheaves in differential geometry called ``\(C\)-infinity algebraic geometry'' and the authors may want to study their construction of ``fractional derivative'' in this theory -- papers may be found on the arXiv preprint server or in published journals. The authors end the paper with some comments on possible generalizations and some conjectures. principal parts; left and right module; Taylor morphism; differential operator; real manifold; real vector bundle; \(C\)-infinity algebraic geometry; fractional derivative Schemes and morphisms, Projective and free modules and ideals in commutative rings, Relevant commutative algebra, Fractional derivatives and integrals, Global differential geometry of Finsler spaces and generalizations (areal metrics) Principal parts of a vector bundle on projective line and the fractional derivative | 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 \(f\colon X \rightarrow S\) be a flat family of (reduced) varieties of dimension \(d\) with normal crossing singularities. Assume that \(X\) and \(S\) are smooth, \(\Delta \subset S\) and \(Y = f^{-1}(\Delta) \subset X\) are divisors with normal crossings such that \(f\colon X\setminus Y \rightarrow S\setminus\Delta\) and \(f\colon (X, \log Y) \rightarrow (S, \log\Delta)\) are smooth and log smooth morphisms, respectively. Let \({\mathcal L}\) be a line bundle on \(X.\) The author defines projective heat operators which are logarithmic analogues of those from [\textit{B. van Geemen, A. J. de Jong}, J. Am. Math. Soc. 11, No. 1, 189--228 (1998; Zbl 0920.32017)] and describes sufficient conditions for the existence of a projective logarithmic heat operator on \({\mathcal L}\) over \(S\) similarly [loc. cit.] which gives a logarithmic projective connection on \(f_\ast{\mathcal L}.\) This definition is based essentially on the notion of the sheaf of logarithmic differential operators on log schemes introduced by the author. It should be remarked that in a more general context (without assumption of normal crossings) sheaves of logarithmic differential operators were considered and studied in detail in earlier works [\textit{F. J. Calderon-Moreno}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, No. 5, 701--714 (1999; Zbl 0955.14013)]. In conclusion the author proves that a family of generalized Jacobians (moduli spaces of torsion-free sheaves of rank one over nodal curves) satisfies the above mentioned sufficient conditions; in particular, this implies the existence of a projective logarithmic heat operator in this case. flat connections; heat operators; logarithmic differential operators; logarithmic schemes; logarithmic heat equation; moduli spaces; theta divisors; generalized Jacobians Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Heat equation, Algebraic moduli problems, moduli of vector bundles Logarithmic heat projective operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V\) be a vector space of dimension \(\aleph\) and let \(\mathcal S\) be the set of proper subspaces of \(V\). The orbits of GL\((V)\) acting on \( \mathcal S\) are called Grassmannians. For \(\aleph < \infty\) the Grassmannians are given by \({\mathcal G}_k = \{ S \in {\mathcal S}\, |\, \text{ dim} S = k \}\), where \(1 \leq k < \aleph\). Let \(\mathcal G\) be a Grassmannian. Each base of \(V\) defines a base subset of \(\mathcal G\) which consists of all elements of \(\mathcal G\) spanned by vectors of this base. A block space \(\mathcal B(G)\) of \(\mathcal G \) can be defined by taking the elements of \(\mathcal G\) as the set of points and the set of base subsets of \(\mathcal G\) as the set of blocks.
In a previous paper [J. Geom. 75, 132--150 (2002; Zbl 1035.51013)], the author proved for \(3\leq \aleph < \infty\) and \(1 < k < \aleph - 1\) that each automorphism of \({\mathcal B}({\mathcal G}_{k})\) is induced by a semilinear isomorphism of \(V\) to itself or to its dual. For \(k=1, \aleph-1\) this is essentially the fundamental theorem of projective geometry. For finite dimensions this proof depends on a result of \textit{W. L. Chow} about automorphisms of Grassmannian graphs [Ann. Math. 50, 32--67 (1949; Zbl 0040.22901)].
In the paper under review the infinite dimensional case is investigated. For any cardinal number \(\alpha \leq \aleph\) set \({\mathcal G}_{\alpha} = \{ S \in {\mathcal S}\,|\, \text{ dim} S = \alpha, \text{ codim} S = \aleph\}\) and let \({\mathcal B}_{\alpha}\) be the family of all base subsets of \({\mathcal G}_{\alpha}\). The following theorem is proved. If \(\alpha < \aleph\), then any automorphism of \( {\mathcal B}({\mathcal G}_{\alpha})\) is induced by a semilinear autmorphism of \(V\).
The elegant proof of this result does not depend on Chow's result, since the latter does not generalize to Grassmannian graphs of infinite dimensional vector spaces. base subset; Grassmannian; infinite dimensional Pankov, M, Base subsets of Grassmannians: infinite-dimensional case, Eur. J. Combin., 28, 26-32, (2007) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds Base subsets of Grassmannians: infinite dimensional case | 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 firstly gives an overview of results concerning strongly canonical expansions and canonical expansions of \(\omega\)-categorical structures. There are certain such structures which have no nontrivial strongly canonical expansions or no nontrivial canonical expansions, respectively.
Throughout the paper \(\infty = \aleph_ 0\) is assumed as the dimension of the considered vector spaces. A symplectic space \(\text{SPG} (\infty, q)\) is a projective space \(\text{PG} (\infty, q)\) corresponding to an infinite dimensional vector space \(V(\infty, q)\) over the field \(\text{GF}(q)\) which is equipped with the ternary relation of collinearity and the binary relation of orthogonality. By Witt's Lemma a symplectic space \(\text{SPG} (\infty, q)\) is \(\omega\)-categorical.
A subgroup \(G\) of the group \(\text{P} \Gamma \text{Sp}(\infty, q)\) of all projective transformations induced on \(\text{PG} (\infty, q)\) by the group of semilinear transformations which preserve the symplectic form \(\sigma\) is said to be Witt homogeneous if \(G\) and \(\text{P} \Gamma \text{Sp}(\infty, q)\) have the same orbits on the set of finite-dimensional subspaces of \(\text{SPG} (\infty, q)\). So we are able to formulate the main result of the paper:
Let \(G \leq \text{P} \Gamma \text{Sp} (\infty, q)\) be Witt homogeneous and let \(\Gamma\) be a non-degenerate \(2t\)-dimensional subspace of \(\text{SPG} (\infty, q)\). Then the group \(\text{PSp} (\infty, q)\) of projective transformations induced by the linear transformations on \(V(\infty,q)\) preserving \(\sigma\) is a subgroup of \(G_{\{T\}}/G_{(T)}\). Moreover, if \(\mathcal M\) is a canonical expansion of \(\text{SPG} (\infty, q)\) and \(G = \text{Aut} ({\mathcal M})\) then \(\text{PSp} (\infty, q) \subseteq \text{P} \Gamma \text{Sp} (\infty, q)\). Hence, \(\text{SPG} (\infty, q)\) has finitely many canonical expansions. group actions; algebraic closures; totally isotropic subspaces; radical actions; canonical expansions of \(\omega\)-categorical structures; symplectic spaces; projective spaces; projective transformations; group of semilinear transformations; symplectic forms; Witt homogeneous Linear algebraic groups over finite fields, Categoricity and completeness of theories, Linear transformations, semilinear transformations, Polar geometry, symplectic spaces, orthogonal spaces, Applications of logic to group theory, Geometry of classical groups, Classical groups (algebro-geometric aspects) Canonical expansions of countably categorical structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For an arbitrary finite family of semialgebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies the Mangasarian-Fromovitz constraint qualification. Using the Milnor-Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of ``regular'' problems approximating an arbitrary semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided. constraint qualification; Mangasarian-Fromovitz constraint quailification; Arrow-Hurwicz-Uzawa; Lagrange multipliers; optimality conditions; tame programming J. Bolte, A. Hochart, and E. Pauwels, \textit{Qualification Conditions in Semi-Algebraic Programming,}, 2017. Variational inequalities, Time-scale analysis and singular perturbations in control/observation systems, Quadratic programming, Optimality conditions for problems involving relations other than differential equations, Nonsmooth analysis, Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems, Real-analytic and semi-analytic sets Qualification conditions in semialgebraic programming | 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 prove a well-known conjecture on the regularity index of \(n+3\) almost equimultiple fat points in \(\mathbb{P}^n\).
Let \( R = k[x_0 ,\dots, x_n]\) be the standard graded polynomial ring over an algebraically closed field \(k\).
Let \( X = \{P_1, \dots , P_s\} \subset \mathbb{P}^n\) be a set of \(s\) distinct points, and let \(m_1,\ldots,m_s\) be positive integers. Denote \(Z:=m_1P_1+\ldots+m_sP_s\) the zero-scheme corresponding to the ideal of all forms of \(R\) vanishing at \(P_i\) with multiplicity at least \(m_i\), for \(i=1,\ldots,s.\) This zero-scheme \(Z\) is called a set of fat points and its defining ideal is \(I_Z =\wp_1^{m_1} \cap \dots \cap \wp_s^{m_s}\).
It is well known that the Hilbert function \(H_{R/I}(t) := \dim_{k}(R/I)_t=\dim_{k}R_t-\dim_{k}I_t\) is strictly increasing until it reaches the multiplicity \({e(R/I)}: =\sum_{i=1}^{s}{{m_i+n-1}\choose{n}}\) at which it stabilizes. The least integer \(t\) for which \(H_{R/I}(t)= {e(R/I)}\) is called the regularity index of \(Z\) and it is denoted by \(\mathrm{reg}(Z)\). It is well-known that \(\mathrm{reg}(Z)\) is equal to the Castelnuovo regularity index \(\mathrm{reg}(R/I)\).
In 1996, N. V. Trung formulated the following conjecture (independently given also by G. Fatabbi and A. Lorenzini):
Conjecture. Let \(Z=m_1P_1+\ldots+m_sP_s\) be a set of fat points in \(\mathbb{P}^n\). For \(j=1,\ldots,n\), let
\[
T_j:=\max\{[{\frac {1}{n}}(m_{i_1}+m_{i_2}+\ldots+m_{i_q}+j-2)]| P_{i_1},P_{i_2},\dots,P_{i_q} \text{ lie~ on~ a \(j\)-plane}\}
\]
then \(\mathrm{reg}(Z)\leq T_{Z}:=\max\{T_j|j=1,\dots,n\}\).
A set of fat points \(Z\) is called almost equimultiple if the multiplicities of the points are equal to \(m\) or \(m-1\) for a given integer \(m\geq 2\). In this paper the authors prove the conjecture for any set of \(n+3\) almost equimultiple, non degenerate fat points in \(\mathbb{P}^n\). The authors also show several results which will be used to prove the conjecture, and explain why the case of \(n+3\) fat points is more complicated than the case of \(n+2\) fat points. regularity index; zero-scheme; Hilbert function; fat points Tu, N. C.; Hung, T. M.: On the regularity index of n+3 almost equimultiple fat points in pn. Kyushu J. Math. 67, 203-213 (2013) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Polynomials, factorization in commutative rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Computational aspects in algebraic geometry, Multiplicity theory and related topics On the regularity index of \(n+3\) almost equimultiple fat points in \(\mathbb P^n\) | 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 There is a number of open problems concerning Nash functions, and a central one is the ``separation problem'', which asks whether Nash functions are sufficient to separate the global analytic components of real algebraic sets. The authors introduce a new problem (``equal complexities''): if a semialgebraic set is described by \(s\) simultaneous global analytic inequalities, can it be described by \(s\) Nash inequalities? The main result of the paper is that ``separation'' and ``equal complexities'' are equivalent for compact Nash manifolds. There are other interesting results, such as the fact that ``separation'' implies ``extension'' (a Nash function on a Nash subset may be extended to a Nash function on the ambient manifold). The introduction of the paper gives informations about related results in the literature about Nash functions. Since the writing of the paper under review, the authors and the reviewer (to appear in Am. J. Math.) have proven that the separation problem has a positive answer on a compact Nash manifold; hence ``equal complexities'' also hold.
The problem of ``equal complexities'' is in the line of the works initiated by \textit{L. Bröcker} about the number of inequalities needed to describe semialgebraic sets; the main tool in this activity is the notion of fan, which comes from the algebraic theory of quadratic forms. The main result of the paper is translated in terms of fans, and then the proof uses algebraic tools. separation problem; problem of equal complexities; Nash functions; number of inequalities; fans Ruiz, J.M. and Shiota, M. : On global Nash functions , Ann. scient. Éc. Norm. Sup. 27 (1994) 103-124. Nash functions and manifolds, Real-analytic sets, complex Nash functions On global Nash 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 Let \(X \subset \mathbb{P}^N\) be a smooth nondegenerate projective variety of dimension \(n\) (\(\geq 2\)), codimension \(e\) and degree \(d\) (over an algebraically closed field of characteristic zero). It is known that the linear system \(|{\mathcal O}_X(d-n-2)\otimes \omega_X^\vee|\) is base point free, as their elements can be seen as the double point divisors of linear projections onto hypersurfaces of \(\mathbb{P}^{n+1}\). Moreover, it has been proved (see the Introduction of the paper under review and references therein) that they separate points unless \(X \subset \mathbb{P}^N\) is of a particular type (a projection of a so called \textit{Roth variety}). The main purpose of this paper is to prove further positivity results on these linear systems. To be precise: It is shown that the base locus of \(|{\mathcal O}_X(d-n-e-1)\otimes \omega_X^\vee|\) is a finite set except if \(X \subset \mathbb{P}^N\) belongs to a finite list of projective varieties (completely stated, see Thm. 4). For the study of these exceptions, varieties whose generic inner projections have exceptional divisors are classified.
Some applications of these results are also provided: inequalities for the delta and sectional genera, property \((N_{k-d+e})\) for \({\mathcal O}_X(k)\) and new evidences of the regularity conjecture, in fact that \({\mathcal O}_X\) is \((d-e)\)-regular. double point divisors; base locus; linear projections; regularity Noma, A., Generic inner projections of projective varieties and an application to the positivity of double point divisors, Trans. Amer. Math. Soc., 336, 4603-4623, (2014) Classical problems, Schubert calculus, Projective techniques in algebraic geometry Generic inner projections of projective varieties and an application to the positivity of double point divisors | 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 Quasi-periodic solutions of the Belov-Chaltikian (BC) lattice hierarchy associated with a \(3\times 3\) discrete matrix spectral problem are constructed in terms of Riemann theta functions.
The BC lattice is the coupled lattice system
\[
\begin{aligned} u_t=&u(u^+-u^-)+v-v^+, \\ v_t=&u(u^+-u^{--}), \end{aligned}
\]
where \(u^\pm(n,t)=(n\pm 1,t), v^\pm(n,t)=v(n\pm 1,t)\).
It was found in the study of lattice analogues of \(W\)-algebras [\textit{A. A. Belov} and \textit{K. D. Chaltikian}, Phys. Lett., B 309, No. 3--4, 268--274 (1993; Zbl 0905.17032); \textit{K. Hikami} and \textit{R. Inoue}, J. Phys. A, Math. Gen. 30, No. 19, 6911--6924 (1997; Zbl 1042.17510)].
In Section 2, a \(3\times 3\) discrete matrix spectral problem involving \(u\) and \(v\) as potentials is considered. Then, introducing Lenard recursion relations, a Lenard equation is derived. Thanks to this equation and a zero-curvature equation, the BC lattice hierarchy associated with a \(3\times 3\) discrete matrix spectral problem is constructed. The characteristic polynomial of the Lax matrix for the BC hierarchy defines the trigonometric curve \(\mathcal{K}_{m-1}\) and the Baker-Akhiezer function is given as an algebraic function carrying the data of the divisors of \(\mathcal{K}_{m-1}\). In Section 4 meromorphic functions on \(\mathcal{K}_{m-1}\) and asymptotic behaviors of the Baker-Akhiezer functions are studied. \(\mathcal{K}_{m-1}\) is an algebraic curve of genus \(m-1\). In Section 5, after studying the homology basis of \(\mathcal{K}_{m-1}\), the Abel map \(\underline{\mathcal{A}}:\mathcal{K}_{m-1}\to \mathcal{J}_{m-1}\) (\(\mathcal{J}_{m-1}\) is the Jacobian variety of \(\mathcal{K}_{m-1}\)) is introduced. It straightens out the continuous flow and the discrete flow in the Jacobian variety, from which the quasi-periodic solutions of the entire BC lattice hierarchy are obtained in terms of Riemann theta functions.
The author concludes giving some trivial examples in the lowest-genus case. classical and quantum field theory; quadratic nonlinearity; vertex approximation; Belov-Chaltikian lattice Frederiksen, J. S., Quasi-diagonal inhomogeneous closure for classical and quantum statistical dynamics, J. Math. Phys., 58, (2017) 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, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems Quasi-diagonal inhomogeneous closure for classical and quantum statistical dynamics | 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 the integrable Neumann system as a particular case of the Hitchin system with marked points. The family of the genus \(g\) spectral curves of this system has the form
\[
y^2 - \prod_{k=1}^g (x-b_k) \prod_{\alpha=1}^{q+1} (x-q_\alpha)
\]
where \(q_\alpha\) are fixed and \(b_k\) are parameters on the base \(B\). Therewith we have a holomorphic family of curves over \(B\).
The author derives the residue formula for the derivative \(dp(X,Y,Z)\) of the period map of this family of curves. For a certain choice of a vector field \(e\) on \(B\) the formula \(dp(X,Y,e)\) gives a flat nondegenerate metric near a generic point of \(B\) and this metric gives rise to an associative multiplication \(\ast\) on the holomorphic tangent bundle to \(B\): \(c(X,Y,Z) = (X \ast Y, Z) = dp(X,Y,Z)\). By the construction, \(c\) is symmetric in \(X,Y,Z\) and it is proved that \(\nabla c(X,Y,Z,W)\) is symmetric in \(X,Y,Z,W\) where \(\nabla\) is the Levi-Civita connection corresponding to the metric. Hence it is proved that the base \(B\) endowed with these metric, multiplication and the unit field \(e\) meets all axioms of a Frobenuis manifold except the existence of the Euler vector field. Frobenius manifold; period matrices; Neumann system Hoevenaars, L. K.: Associativity for the Neumann system. Proc. sympos. Pure math. 78, 215-238 (2008) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests Associativity for the Neumann 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 [For the entire collection see Zbl 0681.00016.]
The aim of this paper is the study of singly-exponential techniques for stratification of real semi-algebraic sets, that means to find a decomposition of \({\mathfrak R}^ d\) into simple-shaped cells of dimensions ranging from 0 to d for n d-variate polynomials \(f_ 1,...,f_ n\) (d fixed) with rational coefficients, such that each \(f_ i\) has constant sign over each cell. Collins has given in 1975 a sign-invariant decomposition with \(O(n^{2^ d-1})\) cells. The authors are given a sign-invariant decomposition with \(O(n^{2d-1}*\beta (n))\) cells where \(\beta\) (n) is a slow-growing function. As application of this study is the generalized point location problem as by Chazelle and Sharir in 1989. The both problems stratification and generalized point location are restricted problems related to the theory of reals and they can be solved in singly-exponential time as by Canny 1987 or Caniglia et co., Grigor'ev, et co., Renager 1988. From the logical point of view are these problems quantifier elimination which can be solved into doubly- exponential time as by Davenport and Heintz 1988. stratification of real semi-algebraic sets; decomposition; generalized point location problem B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir, A Singly-Exponential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications,Proc. 16th ICALP, Lecture Notes in Computer Science, Vol. 372, Springer-Verlag, Berlin, 1989, 179--193. Analysis of algorithms and problem complexity, Semialgebraic sets and related spaces, Symbolic computation and algebraic computation, Computational aspects of higher-dimensional varieties A singly-exponential stratification scheme for real semi-algebraic varieties and its 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 The aim of this paper is to study the generalization of local homology and derived completion to comodules over a Hopf algebroid \((A, \Psi)\) with respect to an invariant ideal \(I \lhd A\). The discrete case \(\Psi =A\) corresponds to the usual setting of commutative algebra; working with comodules introduces new phenomena, for instance local homology can be non-zero in both negative and positive degrees.
The authors commence by \(I\)-adic completion of comodules in the non-derived setting. The subtlety is that the inverse limit of a diagram of \(\Psi\)-comodules is not in general created in \(A\)-modules. Under their hypothesis that \((A, \Psi)\) is true-level, they give an explicit treatment of \(I\)-adic completion; this extends previous work of other authors.
They then turn to derived completion and local homology. For this, the derived category of \(\Psi\)-comodules is not adequate; the solution (under the appropriate hypotheses), based upon their earlier work [\textit{T. Barthel} et al., Adv. Math. 335, 563--663 (2018; Zbl 1403.55008)], is to work with the monoidal stable \(\infty\)-category \(\mathrm{Stable}_\Psi\); this can be interpreted as passing from quasi-coherent to Ind-coherent sheaves. The \(I\)-torsion category \(\mathrm{Stable}^{I-\mathrm{tors}}_\Psi\) is then defined as the localizing tensor ideal of \(\mathrm{Stable}_\Psi\) generated by \(A/I\).
They construct a local homology functor \(\Lambda^I\) for comodules and show that, in general, local homology cannot be calculated as the derived functors of completion. They give a criterion for a comodule to be \(\Lambda^I\)-local, which can be interpreted as a generalization of Bousfield and Kan's Ext \(p\)-completeness criterion.
Finally they consider \(I\)-torsion objects in the derived category of comodules as well as complete objects.
There are at least three candidate stable \(\infty\)-categories of torsion modules; the main result relates these under the appropriate hypotheses. In the discrete case they characterize \(I\)-completion at the derived level, leading to a tilting-theoretic version of local duality.
This work is motivated by problems from stable homotopy theory, notably ongoing work on the algebraic chromatic splitting conjecture. Hopf algebroid; comodule; local homology; derived completion Localization and completion in homotopy theory, Local cohomology and commutative rings, Local cohomology and algebraic geometry, Abstract and axiomatic homotopy theory in algebraic topology Derived completion for comodules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a reduced Gorenstein projective curve. A generalized linear system on \(C\) is a datum \((\mathcal {F},e)\), where \(\mathcal {F}\) is a rank 1 torsion free sheaf on \(C\) and \(e: V \to H^0(C,\mathcal {F})\) a linear map of vector spaces; it is called strongly non-degenerate if for each irreducible component \(C_i\) of \(C\) the composition of \(e\) with the map \(H^0(C, \mathcal {F}) \to H^0(C_i,\mathcal {F}| C_i)\) is injective. This is a strong restriction for reducible curves, essential for almost everything here, but satisfied in many very interesting cases (not by \(H^0(C,\omega _C)\) for most reducible curves). The authors associate to \((\mathcal {F},e)\) two intrinsic objects: a zero-dimensional subscheme \(Z(\mathcal {F},e)\subset C\) and a \(0\)-cycle \(R(\mathcal {F},e)\). Now assume that \((C,(\mathcal {F},e))\) is a one-dimensional limit of a family \((C_t,L_t)\) of true linear systems. They prove that the Weierstrass divisors of \((C_t,L_t)\) parametrizing weighted Weierstrass points of \(L_t\) converge to a subscheme of \(C\) containing \(Z(\mathcal {F},e)\) and with \(R(\mathcal {F},e)\) as its associated \(0\)-cicle (the limit may depends on the family). The proofs use Wronskians for families of curves ([\textit{E. Esteves}, Ann. Sci. Éc. Norm. Supér. (4) 29, No. 1, 107--134 (1996; Zbl 0872.14025)]) and limits of Cartier divisors ([\textit{E. Esteves}, J. Pure Appl. Algebra 214, No. 10, 1718--1728 (2010; Zbl 1184.14011)]). A key part of the paper is a very useful comparison of all concepts with respect to a partial normalization \(C' \to C\). The results are proved in characteristic zero and it is observed that it works if the characteristic is \(> \dim (V)\), where \(V\) is the vector space of the given generalized linear system. curves; degenerations; linear systems; Weierstrass points; ramification point; Gorenstein curve Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry Generalized linear systems on curves and their 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 \(S\) be a noetherian scheme defined over an algebraically closed field \(k\) of arbitrary characteristic and \(\pi:X\to S\) a smooth projective morphism of relative dimension \(2\). The object of this paper is to define a functor compactifying the stack \(\mathrm{Bun}(r,d)\) of vector bundles of rank \(r\) with \(c_2=d\) along the fibres of \(\pi\). The author calls this functor the Uhlenbeck moduli functor or functor of quasibundles; it should be noted that the term quasibundle has previously been used with two different meanings, but the author clarifies this and gives a precise definition so there is no risk of confusion.
When \(k=\mathbb{C}\) and \(S=\mathrm{Spec}(\mathbb{C})\), the Uhlenbeck compactness theorem applies giving a compact moduli space with a set theoretic decomposition \(M^U(r,d)=\coprod_{k\geq0}M(r,d-k)\times\mathrm{Sym}^kX\), where \(M(r,d)\) is the usual moduli space of semistable bundles. In this paper, however, the author works with an arbitrary algebraically closed field and defines a functor \(\mathrm{QBun}(r,d)\) which contains \(\mathrm{Bun}(r,d)\) as an open subfunctor and has a compactness property.
The functors \(\mathrm{Pic}(X)\to\mathrm{Pic}(S)\) corresponding to \(c_2\) and families of zero cycles (actually \(\mathrm{PIC}(X/S)\to\mathrm{PIC}(S)\), where the \(\mathrm{PIC}\) are larger categories taking base change into account) have a certain multiplicative property (they are compatible with tensor products of line bundles and also with base change). One then considers quadruples \((Z,E,N,D)\), where \(Z\) is a closed subset of \(X\) which is finite over \(S\), \(E\) is a vector bundle on \(X\setminus Z\), \(N\) is a multiplicative functor and \(D\) is a line bundle on \(X\) extending \(\det E\). There is a concept of \(E\)-localisation of \(N\) at \(Z\), which says essentially that \(N\) is identified with \(c_2^E\) on \(X\setminus Z\). This \(E\)-localisation is effective if a certain rational section of a line bundle is regular. A quasibundle is now defined to be a quadruple as above with an effective \(E\)-localisation and the corresponding functor is the Uhlenbeck functor.
There is an extensive introduction containg much background and motivational material as well as describing the main results of the paper. Multiplicative functors are introduced in Section 2. Section 3 is concerned with localisation and Section 4 with effectiveness of localisations. The Uhlenbeck functor \(\mathrm{QBun}(r,d)\) is constructed in Section 5 and its properties established; in particular, it is shown that the existence part of the valuative criterion for properness holds (of course, it is too much to expect uniqueness). A conjectural local covering by schemes is described as well as a morphism of functors from the Gieseker compactification to \(\mathrm{QBun}(r,d)\). Finally, in Section 6, a similar functor is constructed for torsors over split semisimple simply-connected groups. An indication is also given of how to extend the definition of quasibundle to higher dimensions. vector bundles on surfaces; moduli space; Uhlenbeck compactification Baranovsky, V, Uhlenbeck compactification as a functor, Int. Math. Res. Not., 2015, 12678-12712, (2015) Vector bundles on surfaces and higher-dimensional varieties, and their moduli Uhlenbeck compactification as a functor | 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 constructs a moduli space \({\mathcal M}_0\) for torsion-free sheaves \(\mathcal A\) of rank \(r\) and degree 0 on a complex irreducible curve \(X\) with one node. Each sheaf \(\mathcal A\) is endowed with a homomorphism \({\mathfrak d}: \bigwedge^r {\mathcal A}\to {\mathcal O}_X\) which is an isomorphism away from the node. The curve \(X\) is considered as a degeneration \({\mathcal X}_0\) in a family \({\mathcal X}\to S\) of irreducible projective curves whose generic fibre is smooth curve \({\mathcal X}_{\eta}\) and whose base \(S\) is a spectrum of a discrete valuation ring. The corresponding moduli space \({\mathcal M}_0\) to be constructed is also a degeneration in the family \({\mathcal M}_S \to S\) of moduli spaces (of semistable torsion-free sheaves of rank \(r\) and degree 0 on \(\mathcal X\)) with same base and its generic fibre is moduli \({\mathcal M}_{S,\eta}\) of semistable vector bundles of rank \(r\) and degree 0 on the smooth curve \({\mathcal X}_{\eta}.\)
Let \({\mathcal N}_0\) be the set of all polystable torsion-free sheaves \({\mathcal E}\) of rank \(r\) and degree 0 for which there is an homomorphism \(\bigwedge^r {\mathcal E}/\mathrm{Torsion} \to {\mathcal O}_{{\mathcal X}_0}\) which is an isomorphism away from the node. The problem is to give a modular interpretation to this subset. In particular, this will give it a scheme structure.
The author interprets the theory of vector bundles of rank \(r\) as a theory of principal \(\mathrm{GL}_r(\mathbb C)\)-bundles. There is a closed subvariety \({\mathcal N} \subset {\mathcal M}_{S,\eta}\) which is isomorphic to moduli space of semistable principal \(\mathrm{SL}_r(\mathbb C)\)-bundles. Let \({\mathcal N}_S\) be the closure of \({\mathcal N}\) in \({\mathcal M}_S\), so that \({\mathcal N}_{S,\eta}={\mathcal N}\). He constructs a (relative) moduli space \(\widehat {\mathcal N}_S \to S\) together with a surjective \(S\)-morphism \(\widehat N_S \to \mathcal N_S\) isomorphic on the generic fibre and such that \(\widehat {\mathcal N}_{S,0}\) parametrizes trosion-free sheaves \(\mathcal A\) with a homomorphism \(\mathfrak d\) as required.
A part of the theory is developed for any semisimple linear algebraic group \(G\) instead of \(\mathrm{SL}_r(\mathbb C).\) torsion-free sheaves; nodal curve; moduli space; Nagaraj-Seshadri locus Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles On the modular interpretation of the Nagaraj-Seshadri locus | 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 investigates the following question: given n points in \({\mathbb{P}}^ 2\), subject to some condition of ''genericity'', what is the least integer \(\delta\) such that there is a nonsingular curve of degree less or equal to \(\delta\) passing through the points? An evaluation for \(\delta\) is given when no three of the points are on a line. The value of \(\delta\) is also obtained if different notions of general position for n points are considered [see for instance \textit{A. V. Geramita} and \textit{F. Orecchia}, J. Algebra 70, 116-140 (1981; Zbl 0464.14007)]. Special attention is paid to points in uniform position: it is shown that if \(n=\left( \begin{matrix} d+2\\ 2\end{matrix} \right)+h\), \(0\leq h\leq 2\) and n points are in uniform position, then \(\delta =d+1\). Analogous results are proved for other ranges of h. Many examples are also discussed. [For part II see the following review.] general position of points Renato Maggioni and Alfio Ragusa, Nonsingular curves passing through points of \?² in generic position. I, II, J. Algebra 92 (1985), no. 1, 176 -- 193, 194 -- 207. Curves in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry Nonsingular curves passing through points of \({\mathbb{P}}^ 2\) in generic position. 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 [For part I and II see Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 373-413 (1989; Zbl 0717.14006) and Sci. Pap. Coml. Arts Sci., Univ. Tokyo 40, No.1, 1-25 (1990; Zbl 0717.14007).]
Consider a group theoretic abelian scheme (GTAS) \(\pi: A\to V=\Gamma \setminus {\mathcal D}\). The space \(H^{(p,p)}(A_{\lambda})\cap H^ r(A_{\lambda},{\mathbb{Q}})\) \((r=2p)\) of Hodge cycles in a generic fibre \(A_{\lambda}\) is controlled by an invariant theory of G, the Q- semisimple algebraic group attached to V. When A is rigid, the space \(H^{(p,p)}(A_{\lambda})\cap H^ r(A_{\lambda},{\mathbb{Q}})\) coincides with the space \(H^ r(A,{\mathbb{Q}})^ G=\bigwedge^ r(F)^ G\) of G- invariant elements in \(H^ r(A_{\lambda},{\mathbb{Q}})\), where \(F=H^ 1(A_{\lambda},{\mathbb{Q}})\). In the paper under review the asymptotic behavior of \(\dim (\bigwedge^ r(\mu F)^ G)\) (as \(\mu\to \infty)\) is studied. group theoretic abelian scheme; Hodge cycles; invariant theory M. Kuga, W. Parry, and C.-H. Sah: Invariants and Hodge cycles. III. Proc. Japan Acad., 66A, 22-25 (1990). Transcendental methods, Hodge theory (algebro-geometric aspects), Geometric invariant theory, Group actions on varieties or schemes (quotients) Invariants and Hodge cycles. III | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For this reviewer, the proof by Tognoli of the Nash conjecture is one of the most beautiful pieces in the history of real algebraic geometry. This theorem says that each compact \(C^\infty\)-submanifold \(M\subset \mathbb{R}^n\) is diffeomorphic to some non-singular algebraic subset of \(\mathbb{R}^p\), for some \(p\geq n\). Moreover, the diffeomorphism can be chosen as close as needed to the inclusion \(M\hookrightarrow \mathbb{R}^p\). This result uses ``approximation techniques'' and the paper under review provides new results in this direction. More precisely, it is introduced the Whitney topology on the set of sections of a coherent sheaf \({\mathcal F}\) of \({\mathcal O}_X\)-modules over the coherent real analytic space \((X,{\mathcal O}_X)\), and it is proved that for any open subset \(U\) of \(X\), the set of sections \(\Gamma(U,{\mathcal F})\) is dense, for the Whitney topology, in the set \(\Gamma (U, {\mathcal F}^\infty)\) of sections of the sheaf \({\mathcal F}^\infty ={\mathcal F} \otimes_{{\mathcal O}_X} {\mathcal E}_X\). As an application, the authors prove the result which motivates the title of the paper:
Given analytic functions \(g_i\), \(a_{ij}\) on an open subset \(U\) of \(\mathbb{R}^n\) and \(C^\infty\) functions \(f_j\) on \(U\), \(i=1, \dots, m\); \(j=1, \dots, p\), such that \(\sum^p_{j=1} a_{ij} (x)f_j(x) =g_i(x)\), \(i=1,2, \dots, m\), then, in any neighbourhood of \(f= (f_1, \dots, f_p)\) in the Whitney topology of \(C^\infty (U)^p\), there exists \(h= (h_1, \dots, h_p)\) such that each \(h_j\) is analytic on \(U\) and \(\sum^p_{j=1} a_{ij} (x)h_j(x) =g_i(x)\), \(i=1,2, \dots, m\). analytic linear systems; approximation; Nash conjecture; coherent sheaf Real-analytic and semi-analytic sets, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) Smooth and analytic solutions for analytic linear systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on \(n\) darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index \(n\) in \(\Delta^+ = \mathbb{Z}_2\ast\mathbb{Z}_2\ast\mathbb{Z}_2\) via a simple bijection between pavings and finite index subgroups which can be deduced from the action of \(\Delta^+\) on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in \(\Delta^+\). Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on \(n\le 16\) darts. cellular complexes; pavings; Tutte's problem in dimension three Combinatorial aspects of simplicial complexes, Enumeration in graph theory, Enumerative problems (combinatorial problems) in algebraic geometry, Subgroup theorems; subgroup growth, Fuchsian groups and their generalizations (group-theoretic aspects), Generalized hypergeometric series, \({}_pF_q\) Three-dimensional maps and subgroup growth | 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 \(f : \mathbb{C}^d \to \mathbb{C}^d\) be a birational map and \(\delta(f) := \lim_{n \to \infty} (\deg f^n)^{1/n}\) be its dynamical degree. We note again \(f\) the meromorphic extension to \(\mathbb{C}\mathbb{P}^d\). The authors compute \(\delta(f)\) for a class of maps of the form \(f = L \circ J\), where \(J(x_1,\ldots,x_d) = (x_1^{-1},\ldots,x_d^{-1})\) and \(L\) is an invertible linear map of \(\mathbb{C}^d\). The idea is to look at the action of \(f^*\) on the cohomology group \(H^{1,1}(\mathbb{C}\mathbb{P}^d)\). Indeed, when the identity \((f^*)^n = (f^n)^*\) holds on \(H^{1,1}(\mathbb{C}\mathbb{P}^d)\) (we say that \(f\) is \(1\)-regular), then \(\delta(f)\) is the spectral radius of the linear map \(f^*\). When \(f\) is not \(1\)-regular, one may regularize \(f\), i.e., find a birational equivalence \(h : \mathbb{C}\mathbb{P}^d \to X\) such that \(\tilde f := h \circ f \circ h^{-1}\) is \(1\)-regular on \(X\). \textit{J. Diller} and \textit{C. Favre} proved that this operation is always possible in dimension \(2\) [Am. J. Math. 123, 1135--1169 (2001; Zbl 1112.37308)]. The authors consider the problem in higher dimension. They regularize some maps of the form \(L \circ J\), called ``elementary''. The interplay between the dynamics of the indeterminacy set and the exceptional set is crucial. They focus on the exceptional varieties that are mapped by some iterate of \(f\) on a point of inderminacy : it gives rise to a so-called ``singular'' orbit. A map \(f\) is said elementary if it is a local biholomorphism at each point of the singular orbit. The regularization of an elementary map is obtained by blowing up the points of the singular orbits. The spectral radius of the regularized map depends only on two lists \(\mathcal{L}^o , \mathcal{L}^c\) of positive integers determined by the set of singular orbits. Note that most of the maps considered in this article appear naturally in mathematical physics literature. birational mappings; dynamical degree E. Bedford and K. Kim, ''On the degree growth of birational mappings in higher dimension,'' J. Geom. Anal., vol. 14, iss. 4, pp. 567-596, 2004. Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Rational and birational maps On the degree growth of birational mappings in higher dimension | 0 |
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