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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 Authors' abstract: A multiparametric family of 2D Toda \(\tau\)-functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of \(S_n\). The coefficients \(F^{c_1,\dots,c_l}_{d_1,\dots,d_m}(\mu,\nu)\) in their series expansion over products \(P_\mu P'_\nu\) of power sum symmetric functions in the two sets of Toda flow parameters and powers of the \(l+m\) auxiliary parameters are shown to enumerate \(|\mu|=|\nu|=n\) fold branched covers of the Riemann sphere with specified ramification profiles \(\mu\) and \(\nu\) at a pair of points, and two sets of additional branch points, satisfying certain additional conditions on their ramification profile lengths. The first group consists of \(l\) branch points, with ramification profile lengths fixed to be the numbers \((n-c_1,\dots,n-c_l)\); the second consists of \(m\) further groups of ``coloured'' branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers \((d_1,\dots,d_m)\). The latter are counted with signs determined by the parity of the total number of such branch points. The coefficients \(F^{c_1,\dots,c_l}_{d_1,\dots,d_m}(\mu,\nu)\) are also shown to enumerate paths in the Cayley graph of the symmetric group \(S_n\) generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type \(\mu\) and ending in the class of type \(\nu\), with the first \(l\) consecutive subsequences of \((c_1,\dots,c_l)\) transpositions strictly monotonically increasing, and the subsequent subsequences of \((d_1,\dots,d_m)\) transpositions weakly increasing. Harnad, J.; Orlov, A. Yu., Hypergeometric \textit{ {\(\tau\)}}-functions, Hurwitz numbers and enumeration of paths, Commun. Math. Phys., 338, 267-284, (2015) Other hypergeometric functions and integrals in several variables, Coverings of curves, fundamental group Hypergeometric \(\tau\)-functions, Hurwitz numbers and enumeration of paths | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be a nonconstant semi-algebraic function which is continuous around a point \(\bar{x}\in \mathbb{R}^{n}.\) In non-differential case the role of the gradient is played by a set of vectors -- the subdifferential \(\partial f(\bar{x}).\) Using only subdifferential terms (by defining the tangence variety of \(f\) at \(\bar{x}\)) the author defines a finite sequence of real numbers \(a_{1},\ldots ,a_{p}\) such that
1. the point \(\bar{x}\) is a local minimizer of \(f\) if and only if \(a_{k}\geq 0\) for \(k=1,\ldots ,p,\)
2. the point \(\bar{x}\) is an isolated local minimizer of \(f\) if and only if \(a_{k}>0\) for \(k=1,\ldots ,p.\)
Moreover, if \(\bar{x}\) is an isolated local minimizer of \(f\) then he determines a ``tangent exponent'' \(\alpha _{\ast }>0\) such that for any \(\alpha >\alpha _{\ast }\) the following conditions are equivalent:
1. the point \(\bar{x}\) is an \(\alpha \)-th order sharp local minimizer of \(f\) (\(\iff \) there exist \(c>0\) such that \(f(x)\geq f(\bar{x})+c\left\vert \left\vert x-\bar{x}\right\vert \right\vert ^{\alpha }\) for \(x\) near \(\bar{x}),\)
2. the subdifferential \(\partial f\) is \((\alpha -1)\)-th order strongly metrically subregular at \(\bar{x}\) (\(\iff \) there exist \(c>0\) such that \(m_{f}(x)\geq c\left\vert \left\vert x-\bar{x}\right\vert \right\vert^{\alpha -1}\) for \(x\neq \bar{x}\) near \(\bar{x},\) where \(m_{f}(x):=\inf \{\left\vert \left\vert v\right\vert \right\vert :v\in \partial f(x)\}),\)
3. the function \(f\) satisfies the Łojasiewicz gradient inequality at \(\bar{x}\) with the exponent \(\frac{\alpha -1}{\alpha }\) (\(\iff \) there exist \(c>0\) such that \(m_{f}(x)\geq c\left\vert \left\vert f(x)-f(\bar{x})\right\vert\right\vert ^{\frac{\alpha -1}{\alpha }}\) for \(x\neq \bar{x}\) near \(\bar{x}).\) local minimizer; Łojasiewicz gradient inequality; optimality conditions; semi-algebraic; sharp minimality; strong metric subregularity; tangencies Semialgebraic sets and related spaces, Numerical optimization and variational techniques, Inequalities involving derivatives and differential and integral operators Local minimizers of semi-algebraic functions from the viewpoint of tangencies | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 scheme. Assume that we are given an action of the one dimensional split torus \(\mathbb {G}_{m,S}\) on a smooth affine \(S\)-scheme \(\mathfrak {X}\). We consider the limit (also called attractor) subfunctor \(\mathfrak {X}_{\lambda }\) consisting of points whose orbit under the given action `admits a limit at 0'. We show that \(\mathfrak {X}_{\lambda }\) is representable by a smooth closed subscheme of \(\mathfrak {X}\). This result generalizes a theorem of \textit{B. Conrad} et al. [Pseudo-reductive groups. Cambridge: Cambridge University Press (2010; Zbl 1216.20038)] where the case when \(\mathfrak {X}\) is an affine smooth group and \(\mathbb {G}_{m,S}\) acts as a group automorphisms of \(\mathfrak {X}\) is considered. It also occurs as a special case of a recent result by \textit{V. Drinfeld} on the action of \(\mathbb {G}_{m,S}\) on algebraic spaces [``On algebraic spaces with an action of \(\mathfrak G_m\)'', Preprint, \url{arXiv:1308.2604}] in case \(S\) is of finite type over a field. group schemes; torus action; limit subscheme Margaux, Benedictus, Smoothness of limit functors, Proc. Indian Acad. Sci. Math. Sci., 125, 2, 161-165, (2015) Group actions on varieties or schemes (quotients), Group schemes Smoothness of limit functors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article is devoted to the description of an algorithm for subdivision of a plane into nonintersecting domains by a finite set of simple Jordan arcs. Each resultant domain is defined via a set of its boundary arcs and its indicator (a bounded or an unbounded domain) which determines the characteristic domain function. In addition, an algorithm is obtained for implementation of a regularized set operations on domains without cutoffs. It is based on subdividing a plane by common boundaries on subdomains and constructing on this base the result of the operation. For computing the intersection points of the boundary arcs, the Newton method is applied whose square convergence is proven for the case of convex and monotone curves. geometric intersection problem; curve intersection; algorithm; subdivision; Jordan arcs; Newton method; convergence Numerical aspects of computer graphics, image analysis, and computational geometry, Polyhedra and polytopes; regular figures, division of spaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Subdivision of a plane and set operations on domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\phi\) be an endomorphism of an affine scheme \(X=\operatorname{Spec} A\) which is of finite type over a field \(k\), \(V\) be a Zariski-closed subset of \(X\), and let \(\alpha\) be a (not necessarily closed) point of \(X\). Then the author proves that ``the set of return time'' \(\{n: \phi^{\circ n}(\alpha)\in V\}\) is the union of finitely many arithmetic progressions and a set of Banach-density \(0\), affirmatively answering a question by \textit{L. Denis} [Bull. Soc. Math. Fr. 122, No. 1, 13--27 (1994; Zbl 0795.14008)]. This would solve the dynamical Mordell-Lang problem if the set of Banach-density \(0\) were a finite set.
As the author notes, the result is now superceded by [\textit{J. P. Bell}, \textit{D. Ghioca}, and \textit{T. J. Tucker}, ``The dynamical Mordell-Lang problem for Noetherian spaces'', Funct. Approx. Comment. Math. 53, No. 2, 313--328 (2015)] in which they prove the same result for continuous maps on Noetherian spaces (in particular, rational maps on quasi-projective varieties), and a similar result is mentioned in Gignac's thesis as a consequence of the ergodic theory on Zariski spaces developed in Favre's thesis (which also has a measure-theoretic proof [\textit{W. Gignac}, J. Geom. Anal. 24, No. 4, 1770--1793 (2014; Zbl 1320.37003)]). In the current paper under review, the author employs a different argument. Namely, \(\phi\) induces an endomorphism \(\Phi\) on the Berkovich space \(\mathcal M(A)\) of bounded seminorms on \(A\). When the Banach density of the return-time set is positive, the author creates, as a weak limit of probability measures on finitely many points, a \(\Phi\)-invariant measure which has a positive measure above \(V\). He then uses the Poincaré recurrence theorem on \(\mathcal M(A)\) to conclude that the return-time set must contain an arithmetic progression. The rest of the argument is a Noetherian induction. dynamical Mordell-Lang problem; Berkovich space; Poincaré recurrence Petsche, Clayton, On the distribution of orbits in affine varieties, Ergodic Theory Dynam. Systems, 35, 7, 2231-2241, (2015) Arithmetic dynamics on general algebraic varieties, Arithmetic properties of periodic points, Dynamical systems on Berkovich spaces, Dynamical aspects of measure-preserving transformations, Automorphisms of surfaces and higher-dimensional varieties, Rigid analytic geometry On the distribution of orbits in affine 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 The families of affine semi-algebraic sets over a real-closed field K and semi-linear sets over an ordered field enjoy many closure properties with algebraic and geometric significance. This paper studies the natural closure properties of Minkowski sums and scalar dilation. It gives an extension of the underlying vector space structure that enables the study of an arithmetic on the abstract points of their associated spectra. This arithmetic satisfies certain cancellation principles that motivates an investigation of an algebraic object weaker than a group and culminates with a version of the Jordan-Hölder theorem. With the subsequent definition of dimension we show that the collection of affine real ultrafilters in \(K^n\) is \(n\)-dimensional over the scalar ultrafilters. affine real ultrafilters; affine semi-algebraic sets; Minkowki sums; cancellation; Jordan-Hölder theorem Real algebraic sets, Semialgebraic sets and related spaces An algebraic study of affine real ultrafilters | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that the metric structure of morphisms \(f : Y \rightarrow X\) between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton \(\Gamma= (\Gamma_Y, \Gamma_X)\) of \(f\), the sets \(N_{f, \geq n}\) of points of \(Y\) of multiplicity at least \(n\) in the fiber are radial around \(\Gamma_Y\) with the radius changing piecewise monomially along \(\Gamma_Y\). In this case, for any interval \(l = [z, y] \subset Y\) connecting a point \(z\) of type 1 to the skeleton, the restriction \(f |_l\) gives rise to a \textit{profile} piecewise monomial function \(\varphi_y : [0, 1] \rightarrow [0, 1]\) that depends only on the type 2 point \(y \in \Gamma_Y\). In particular, the metric structure of \(f\) is determined by \(\Gamma\) and the family of the profile functions \(\{\varphi_y \}\) with \(y \in \Gamma_Y^{(2)}\). We prove that this family is piecewise monomial in \(y\) and naturally extends to the whole \(Y\). In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that \(\varphi_y\) coincides with the Herbrand function of \(\mathcal{H}(y) / \mathcal{H}(f(y))\). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions. Berkovich curves; wild ramification; Herbrand function Temkin, M.: Metric uniformization of morphisms of berkovich curves. (2014) Rigid analytic geometry Metric uniformization of morphisms of Berkovich curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper consists of notes about blowup-based tools for linear system.
Let \(p_1, \dots, p_r\) be general points in \(\mathbb{P}^2\) and let \(m_1, \dots, m_r\) be positive integers. By \(\mathcal{L}(d;m_1, \dots, m_r)\) we denote the system of plane curves of degree \(d\) with multiplicity at least \(m_j\) at \(p_j\), \(j=1, \dots, r\). \(\mathcal{L}(d;m_1^{\times s_1}, \dots, m_r^{\times s_r})\). is a system with repeated multiplicities. The expected dimension of \(\mathcal{L}(d;m_1, \dots, m_r)\) is
\[
\mathrm{edim}(\mathcal{L}(d;m_1, \dots, m_r)=\max \{\mathrm{vdim}(\mathcal{L}(d;m_1, \dots, m_r)), -1 \},
\]
where the virtual dimension, \(\mathrm{vdim}(\mathcal{L}(d;m_1, \dots, m_r))\), is \(\frac{d(d+3)}{2}-\sum_{j=1}^r {m_j+1 \choose 2}\). A system is special if its effective dimension is strictly greater than the expected one.
Recent years have seen significant advances in the understanding of linear systems with imposed multiple points. The case of points in general position is an important case, with several relevant contributions to the open conjectures of Nagata-Biran-Szemberg and Segre-Harbourne-Gimigliano-Hirschowitz. Most of these rely to some extent on semicontinuity and degeneration methods, which often allow setting up induction arguments on the multiplicity or the number of points.
The formalism of blowups has become an essential tool in te study of linear systems with multiple points, especially when using degeneration methods: the geometry of the variety blown up at the imposed points is important; induction arguments often lead to consider points that are not in general position, but ``infinitely near'', i.e. on blowups; useful degenerations are often built by blowing up the total space of some family.
The author aims to overview the set of blowup-based tools that are being used for specializing and degenerating linear systems, with emphasis on clusters of infinitely near points and Ciliberto-Miranda's \textit{blowup and twist}.
The author takes a rather elementary approach, which should serve as a friendly introduction and guide to original research articles. Sometimes full proof are not given or the exposition restricts to particular cases for the sake of simplicity; in such case the author includes references to the existing bibliography. In particular, the author deals only with linear systems of curves on smooth surfaces defined over the field of complex numbers. linear systems; degenerations methods, fat points Divisors, linear systems, invertible sheaves, Rational and birational maps, Fibrations, degenerations in algebraic geometry, Parametrization (Chow and Hilbert schemes) Blowup and specialization methods for the study of 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 We prove the central limit theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that if \(G\) is the Zariski closure of a group generated by the support of the distribution of our matrices, and if \(G\) is semisimple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition of \(G\). We present a detailed exposition of results announced by the authors [C. R. Acad. Sci., Paris, Sér. I 313, No. 5, 305-308 (1991; Zbl 0734.60012)]. For reasons explained in the introduction, this part is devoted to the case of \(SL(m, \mathbb{R})\) group. The general semisimple Lie group will be considered in the second part of the work.
The central limit theorem for products of independent random matrices is our main topic, and the results obtained complete to a large extent the general picture of the subject. The proofs rely on methods from two theories. One is the theory of asymptotic behaviour of products of random matrices itself. As usual, the existence of distinct Lyapunov exponents is the most important fact here. The other is the theory of algebraic groups. We want to point out that algebraic language and methods play a very important role in this paper. In fact, this mixture of methods has already been used for the study of Lyapunov exponents by the first author and \textit{G. A. Margulis} [Sov. Math., Dokl. 35, 309-313 (1987); translation from Dokl. Akad. Nauk SSSR 293, 297-301 (1987; Zbl 0638.15010) and Russ. Math. Surv. 44, No. 5, 11-71 (1989); translation from Usp. Mat. Nauk 44, No. 5(269), 13-60 (1989; Zbl 0687.60008)] and by the second author and \textit{A. Raugi} [Isr. J. Math. 65, No. 2, 165-196 (1989; Zbl 0677.60007)]. We believe that it is impossible to avoid the algebraic approach if one aims to obtain complete and effective answers to natural problems arising in the theory of products of random matrices.
In order also to present the general picture of the subject we describe several results which are well known. Some of these can be proven for stationary sequences of matrices, others are true also for infinite-dimensional operators [see e.g. \textit{P. Bougerol} and \textit{J. Lacroix}, ``Products of random matrices with applications to Schrödinger operators'' (1985; Zbl 0572.60001); \textit{V. I. Oseledets}, Trans. Mosc. Math. Soc. 19, 197-231 (1968), translation from Tr. Moskov. Mat. Obshch. 19, 179-210 (1968); \textit{F. Ledrappier}, in: École d'Eté de probabilités de Saint-Flour XII-1982. Lect Notes Math. 1097, 305-396 (1984; Zbl 0578.60029) and \textit{D. Ruelle}, Ann. Math., II. Ser. 115, 243-290 (1982; Zbl 0493.58015)]. But our main concern is with independent matrices, in which case very precise and constructive statements can be obtained. central limit theorem; products of i.i.d. random matrices; asymptotic behaviour of products of random matrices; Lyapunov exponents; algebraic language I. Y. Goldsheid and Y. Guivarc'h. Zariski closure and the dimension of the Gaussian law of the product of random matrices. I. Probab. Theory Related Fields 105 (1996) 109-142. Probability measures on groups or semigroups, Fourier transforms, factorization, Central limit and other weak theorems, Group actions on varieties or schemes (quotients) Zariski closure and the dimension of the Gaussian law of the product of random matrices. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In unpublished lecture notes, W. Kahan introduced a method of discretization applicable to any system of ordinary differential equations in \(\mathbb{R}^n\) that has a quadratic vector field. Such equations have the form \[\dot{x} = f(x) = Q(x) + Bx + c,\] where each component of \(Q : \mathbb{R}^n \rightarrow \mathbb{R}^n\) is a quadratic form, \(B\) is an \(n \times n\) real matrix and \(c \in \mathbb{R}^n\).
The discretization that Kahan uses has the following form: \[\frac{\tilde{x} - x}{2 \epsilon} = Q(x, \tilde{x}) + \frac{1}{2} B(x + \tilde{x}) + c,\] where \[Q(x, \tilde{x}) = \frac{1}{2} ((Q(x + \tilde{x}) - Q(x) - Q(\tilde{x}))\] is the symmetric bilinear form that corresponds to the quadratic form \(Q\).
The discretization is linear with respect to \(\tilde{x}\) and so defines a rational map \(\tilde{x} = \Phi_f(x, \epsilon)\). This can be written explicitly as \(\tilde{x} = \Phi_f (x, \epsilon) = x + 2 \epsilon (I - \epsilon f'(x))^{-1} f(x) \), where \(f'(x)\) represents the Jacobian of \(f\). This mapping approximates a time \(\epsilon\) shift along solutions of the original differential equation. Since the discretization is invariant with respect to interchange of \(x\) and \(\tilde{x}\), a reversibility property holds: \(\Phi^{-1}_f (x, \epsilon) = \Phi_f (x, -\epsilon)\).
In their prior work the authors studied properties of Kahan's method when applied to integrable systems. They determined that in a number of cases Kahan's approach preserves integrability in the sense that the mapping \(\Phi_f(x, \epsilon)\) has as many independent integrals of motion as the original system.
In [J. Phys. A, Math. Theor. 52, No. 4, Article ID 045204, 10 p. (2019; Zbl 1422.70012)] \textit{P. H. van der Kamp} et al. show that the Kahan map \(\Phi_f\) can be represented as a composition of two involutions on a pencil of conics in \(\mathbb{C}^2\). The current paper shows that the reverse is also true: a Kahan discretization can be reconstructed from a linear form and a pencil of conics.
For Part I, see [the authors et al., Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 475, No. 2223, Article ID 20180761, 13 p. (2019; Zbl 1501.37068)]. integrable map; Hamiltonian systems; birational maps; Kahan's discretization; pencil of conics Completely integrable discrete dynamical systems, Numerical methods for Hamiltonian systems including symplectic integrators, Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Rational and birational maps, Integrable difference and lattice equations; integrability tests Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. II: Systems with a linear Poisson tensor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathcal M = {\mathbb{A}}^{rs}\) denote the complex algebraic variety of \(r \times s\) complex matrices (\(r \leq s\)) and \(D^k\) the generic determinantal variety of rank \(k\), i.e., its points correspond to matrices in \(\mathcal M\) of rank \(\leq k\) (\(\leq r\)). In this paper the author studies the arc space and jet schemes of \(D^k\). An \(n\)-jet is a morphism \(\mathrm{Spec}({\mathbb{C}} [t]/(t^{n+1})) \to D^k\). These \(n\)-jets are represented by an algebraic \(\mathbb{C}\)-scheme \(D^k_n\). The inverse limit \(D^k_{\infty}\) of the schemes \(D^k_n\) (for \(n \to \infty\)) is a \({\mathbb{C}}\)-scheme (not necessarily of finite type) representing arcs on \(D^k\), i.e., morphisms \(\mathrm{Spec}({\mathbb{C}}[[t]]) \to D^k\).
The author proves that for \(k = 0\) or \(k=r-1\) the jet schemes \(D^k_n\) are irreducible, and gives a formula (in terms of \(n\) and \(k\)) for the number of irreducible components of \(D^k_n\) for \(1 \leq k \leq r-2\).
This analysis also yields a formula for the \textit{log canonical threshold} of the pair \((M,D^k)\), in terms of \(k\), \(r\) and \(s\). An expression for the topological zeta function of \((\mathcal M, D^k)\) (when \(r=s\)) is also obtained.
The main idea of this research is to exploit an action of the group \(G=\mathrm{GL}_r \times \mathrm{GL}_s\) on \(\mathcal M\), which induces similar group actions on \(D_k\), \({\mathcal M}_{\infty}\) and \({\mathcal M}_n\), for all \(n\).
The orbits under these actions are very important and, by a process of Gaussian elimination, each orbit in \({\mathcal M}_{\infty}\) is characterized by a sequence \(\lambda_1 \geq \dots \lambda _r \geq 0\) (where some \(\lambda _i\) could be infinite). This allows the author to use the Theory of Partitions (or a slight generalization thereof) to study, among other things, the question of incidence of closures of orbits. Other techniques that are used include the theory of \textit{contact loci} and \textit{motivic integration} (to compute motivic volumes of certain orbit closures). A number of auxiliary results, interesting in themselves, are obtained.
Although the material is technical the paper is well written and many necessary (known) results are briefly but clearly reviewed. Arc spaces; jet schemes; determinantal varieties; topological zeta functions; motivic integration; partitions Docampo, Roi Arcs on determinantal varieties \textit{Trans. Amer. Math. Soc.}365 (2013) 2241--2269 Math Reviews MR3020097 Arcs and motivic integration, Determinantal varieties, Group actions on varieties or schemes (quotients) Arcs on determinantal 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 Numerical Algebraic Geometry represents the irreducible components of algebraic varieties over \(\mathbb{C}\) by certain points on their components. Such witness points are efficiently approximated by Numerical Homotopy Continuation methods, as the intersection of random linear varieties with the components. We outline challenges and progress for extending such ideas to systems of differential polynomials, where prolongation (differentiation) of the equations is required to yield existence criteria for their formal (power series) solutions. For numerical stability we marry Numerical Geometric Methods with the Geometric Prolongation Methods of Cartan and Kuranishi from the classical (jet) geometry of differential equations. Several new ideas are described in this article, yielding witness point versions of fundamental operations in Jet geometry which depend on embedding Jet Space (the arena of traditional differential algebra) into a larger space (that includes as a subset its tangent bundle). The first new idea is to replace differentiation (prolongation) of equations by geometric lifting of witness Jet points. In this process, witness Jet points and the tangent spaces of a jet variety at these points, which characterize prolongations, are computed by the tools of Numerical Algebraic Geometry and Numerical Linear Algebra. Unlike other approaches our geometric lifting technique can characterize projections without constructing an explicit algebraic equational representation. We first embed a given system in a larger space. Then using a construction of \textit{D. Bates} et al. [to appear in RISC series in Symbolic computation], appropriate random linear slices cut out points, characterizing singular solutions of the differential system. Numerical computation of solutions to systems of equations, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Symbolic computation and algebraic computation, Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations Towards geometric completion of differential systems by 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 \(X\) be a \(C^\infty\)-manifold, and let \(\eta\) be a smooth \(1\)-form on \(X\). Its Radon transform is the function \(\tau(\eta)\) on the free loop space \(L(X)=C^\infty(S^1,X)\) whose value at \(\gamma:S^1\rightarrow X\) is \(\int_{S^1}\gamma^\ast\eta.\) The problem of describing the range of the transform \(\tau\) was studied by Brylinski who, in the case \(X=\mathbb{R}^N,\) characterized it by a system of differential equations. One part of these equations expresses that \(\tau(\eta)\) is invariant under reparametrizations of \(S^1.\) If \(X\) is a complex manifold and \(\eta\) is holomorphic, then \(\tau(\eta)\) satisfies two more properties, that \(\tau(\eta)=0\) on \(L^0(X)\), the space of loops extending holomorphically into the unit disc, and that \(\tau(\eta)\) is additive in holomorphic pairs of pants; the value on \(\tau(\eta)\) on the ``waist'' circle is equal to the sum of values on the ``leg'' circles.
In the present paper the authors assume that \(X\) is a smooth algebraic variety over \(\mathbb{C}\) and relate the Radon transform to the theory of vertex operator algebras and factorization algebras. \(L^0(X)\) and \(L(X)\) are replaced by the scheme \(\mathcal{L}^0(X)\) of formal arcs, and the ind-scheme \(\mathcal{L}(X)\) of formal loops. They are obtained by replacing the unit disk by \(\operatorname{Spec}\mathbb{C}[[t]]\) and the unit circle by \(\operatorname{Spec}\mathbb{C}((t)).\) For a regular \(1\)-form \(\eta\) there is a function \(\tau(\eta)\) on \(\mathcal{L}(X)\) vanishing on \(\mathcal{L}^0(X)\). For any closed \(2\)-form \(\omega\) the Radon transform \(\tau(d^{-1}(\omega))\) of any analytic primitive \(d^{-1}(\omega)\) of \(\omega\) is defined algebraically and depends only on \(\omega\). If \(\omega\) is nondegenerate, then \(\tau(d^{-1}(\omega))\) is a version of the symplectic action functional on the space of loops.
In an earlier paper in this series
[Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113--133 (2007; Zbl 1129.14022)], the authors showed how the space \(\mathcal{L}(X)\) provides a geometric construction of a particular sheaf of vertex algebras on \(X\), the chiral de Rham complex \(\Omega_X^{\operatorname{ch}}\). It was realized as a semiinfinite de Rham complex of a particular \(D\)-module on \(\mathcal{L}(X)\) of distributions supported on \(\mathcal{L}^0(X)\), and the sheaf \(\mathcal{O}_{\mathcal{L}(X)}\) of functions on \(\mathcal{L}(X)\) acts on \(\Omega_X^{\operatorname{ch}}\) by multiplication.
The main result of the article is the following: Let \(f\) be a function on \(\mathcal{L}(X)\) vanishing on \(\mathcal{L}^0(X)\). Then the following are equivalent:
(i) \(f\) has the form \(\tau(d^{-1}(\omega))\) for a closed \(2\)-form \(\omega\).
(ii) The operator of multiplication by \(\exp(f)\) in \(\Omega_X^{\operatorname{ch}}\) is an automorphism of vertex algebras.
The article is a part of a series devoted to interpreting, in geometric terms involving \(\mathcal{L}(X)\), the gerbe of chiral differential operators (CDO) on \(X\). Objects of this gerbe are sheaves of vertex algebras similar to \(\Omega_X^{\operatorname{ch}}\) but withouth the fermionic variables. For any such object \(\mathcal{A}\) the sheaf \(\underline{\operatorname{Aut}}(\mathcal{A})\) was found to be canonically isomorphic to \(\Omega^{2,\operatorname{cl}}_X\), the sheaf of closed \(2\)-forms. This is precisely the domain of definition of the Radon transform \(\tau\circ d^{-1}.\)
In a subsequent paper the authors will show that the anomaly inherent in the construction of CDO is related to the determinantal anomaly for the loop space by the Radon transform. Sheaves of CDO on \(X\) form a gerbe with lien \(\Omega^{2,\operatorname{cl}}_X\) while the determinantal gerbe of \(\mathcal{L}(X)\) has the lien \(\mathcal{O}^\times_{\mathcal{L}(X)}.\) On the level of liens the relation between the two gerbes associates to a \(2\)-form \(\omega\in\Omega^{2,\operatorname{cl}}_X\) the invertible function \(S(\omega)=\exp(\tau d^{-1}(\omega))\in\mathcal{O}_{\mathcal{L}(X)}^\times.\)
The authors give nice and thorough treatments of the essential, elementary parts. Most of the article is used to prove the main theorem, and the techniques are of general interest to a lot of applications of this theory, and also to other fields of geometry. vertex algebras; Radon transform; symplectic action; Gerbe Kapranov M., Vasserot É. , Formal loops III: Chiral differential operators, in preparation. Fibrations, degenerations in algebraic geometry, Analytic theory (Epstein zeta functions; relations with automorphic forms and functions), Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vertex operators; vertex operator algebras and related structures Formal loops. III. Additive functions and the Radon transform | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that if \(X_n\) is a variety of \(c\times n\)-matrices that is stable under the group \(\operatorname{Sym}([n])\) of column permutations and if forgetting the last column maps \(X_n\) into \(X_{n-1} \), then the number of \(\operatorname{Sym}([n])\)-orbits on irreducible components of \(X_n\) is a quasipolynomial in \(n\) for all sufficiently large \(n\). To this end, we introduce the category of affine \(\mathbf{FI}^{op}\)-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one \(\mathbf{FI}^{op} \)-scheme becomes of product form, where \(X_n=Y^n\) for some scheme \(Y\) in affine \(c\)-space. Furthermore, to any \(\mathbf{FI}^{op} \)-scheme of width one we associate a \textit{component functor} from the category \(\mathbf{FI}\) of finite sets with injections to the category \(\mathbf{PF}\) of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that \(\operatorname{Sym}([n])\)-orbits of components of \(X_n \), for all \(n\), correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem. We present applications of our methods to counting fixed-rank matrices with entries in a prescribed set and to counting linear codes over finite fields up to isomorphism. Components of symmetric wide-matrix 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 K be a finite extension of \({\mathbb{Q}}_ p\), X an affine space over K and G a reductive linear group acting on X in such a way that there is a hypersurface F in X such that \(Y:=X-F\) is a G-orbit. Then for any G(K)- orbit \(Y_ i\) of Y(K) the author defines a zeta-function \(Z_ i(\omega)(\Phi)\) on the set \(\Omega (K^*)\) of multiplicative \({\mathbb{C}}\)- valued characters of \(K^*\) by \(Z_ i(\omega)(\Phi):=\int_{Y_ i}\omega (f(x))\Phi (x)\mu (x),\) where f generates the ideal for F, \(\Phi\) is a \({\mathbb{C}}\)-valued locally constant function on X(K) with compact support, and \(\mu\) (x) denotes the Haar measure. He proves that \(Z_ i(\omega)(\Phi)\) is holomorphic for \(\sigma (\omega)>0\) (where \(| \omega | x\| =(x)^{\sigma (\omega)})\), has a meromorphic continuation to \(\Omega (K^*)\), satisfies a functional equation and is in fact a rational function of \(t=\omega(\pi)\), \(\pi\) generator for the valuation ideal of K. The sum of the zeta-functions of the orbits \(Y_ i\) gives the complex power \(\omega(f)\) of f studied earlier by the author.
The author calculates this zeta-function and its functional equation for numerous examples specifying X and f. prehomogeneous vector space; p-adic complex power; sum of the zeta- functions of the orbits; functional equation Igusa, Jun-ichi, Some results on \(p\)-adic complex powers, Amer. J. Math., 106, 5, 1013-1032, (1984) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Local ground fields in algebraic geometry, Homogeneous spaces and generalizations, Harmonic analysis on homogeneous spaces, Group actions on varieties or schemes (quotients), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) Some results on p-adic complex powers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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, as its name suggests, deals with the problem of approximating certain structures such as algebraic spaces, schemes and stacks. The fundamental example as to what approximation means here is the following: Every commutative ring \(R\) can be written as a direct limit of its subrings that are finitely generated \(\mathbb Z\)-algebras and every \(R\)-module \(M\) can be written as the direct limit of its finitely generated \(R\)-submodules. This means that every affine scheme \(X\) is an inverse limit of affine schemes of finite type over \(\mathrm{Spec}\mathbb Z\) and every quasi-coherent sheaf on \(X\) is direct limit of quasi-coherent sheaves of finite type. In the paper it is shown for example that every quasi- compact and quasi-separated Deligne-Mumford stack \(X\) is an inverse limit of Deligne-Mumford stacks of finite type over \(\mathrm{Spec}\mathbb Z\). Approximation here is in contrast with standard limit results which are relative: given an inverse limit \(X = \underset\longleftarrow\lim_{\lambda} X_{\lambda}\) describe finitely presented objects over \(X\) in terms of finitely presented objects over \(X_{\lambda}\) for sufficiently large \(\lambda\).
The following definitions are fundamental for understanding the paper: An algebraic stack \(X\) is said to have the completeness property if every quasicoherent sheaf on \(X\) is a filtered direct limit of finitely presented sheaves or, equivalently, if the abelian category \(QCoh(X)\) is compactly generated. Such an stack is called \textit{pseudo-noetherian} if it is quasi-compact, quasi-separated and \(X'\) has the completeness property for every finitely presented morphism \(X' \to X\) of algebraic stacks.
For \(S\) a pseudo-noetherian stack and \(X \to S\) a morphism of stacks, an approximation of \(X\) over \(S\) is a finitely presented \(S\)-stack \(X_0\) together with an affine \(S\)-morphism \(X \to X_0\). \(X/S\) can be approximated if there is such an approximation over \(S\).
A morphism \(f : X \to Y\) of algebraic stacks is of strict approximation type if f can be written as a composition of affine morphisms and finitely presented morphisms. The morphism \(f\) is of approximation type if there exists a surjective representable and finitely presented étale morphism \(p : X' \to X\) such that \(f \circ p\) is of strict approximation type. An algebraic stack \(X\) is said to be of approximation type if \(X \to\mathrm{Spec}\mathbb Z\) is so.
There are four main results in the paper dealing with above notions. The first one (Theorem A) asserts that every stack of approximation type is pseudo-notherian.
The second main result (Theorem B) shows that when \(X\) is a quasicompact stack with quasi-finite and separated diagonal, then there exists a scheme \(Z\) and a finite, finitely presented and surjective morphism \(Z \to X\) that is flat over a dense quasi-compact open substack \(U \subseteq X\).
Another main result of the paper (Theorem C) shows that certain sets of properties of morphisms of algebraic stacks (named PA, PC and PI in the paper) passes to a limit \(X \to S\) if these properties are satisfied by the inverse system \(\{X_{\lambda}\}\) for all \(\lambda\) sufficiently large, i.e. \(\lambda \geq \alpha\) for some \(\alpha\). See page 4 of the article for the statement and also page 30. Note that sometimes it is necessary that the morphisms \(X_\lambda \to S\) and the bonding maps \(X_\lambda \to X_\mu\) satisfy extra conditions such as closed immersion or of finite type.
The last main result (Theorem D) deals with the situation that \(S\) is a pseudo-noetherian algebraic stack and \(X \to S\) be a morphism of approximation type. Theorem D then says that there exists a finitely presented morphism \(X_0 \to S\) and an affine \(S\)-morphism \(X \to X_0\). The theorem shows moreover that the sets of properties mentioned earlier are satisfied for \(X_0 \to S\) provided that they are satisfied for \(X \to S\)
Many interesting tools and applications of the above results are proved in the paper. For example a version of étale d dévissage is shown in order to prove Theorem B. This is a statement of the form: Given \(X' \to X\) surjective and étale, then \(X\) can be approximated if \(X'\) can be approximated. This is a generalization of the older forms of dévissage because in those treatments it was always assumed that \(X \to X\) is an étale neighborhood and that \(X\) is quasi-affine. The present form of dévissage, however, is also reduced to the case that \(X' \to X\) is an étale neighborhood. As an application of the results, there is a generalization of a theorem of Chevalley. This generalization states that if \(X\) is affine and \(Y\) an algebraic space and \(X \to Y\) integral and surjective, then \(Y\) is also affine. Note that the notherianity assumption in Chevalley's theorem is dropped and also finite morphisms have been replaced by integral morphisms. noetherian approximation; algebraic spaces; algebraic stacks; Chevalley's theorem; Serre's theorem; global quotient stacks; global type; basic stacks Rydh, D., \textit{Noetherian approximation of algebraic spaces and stacks}, J. Algebra, 422, 105-147, (2015) Generalizations (algebraic spaces, stacks) Noetherian approximation of algebraic spaces and stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 work considers parametric systems of polynomial equalities and inequalities, the positive dimensional semialgebraic sets defined by them and how these depend on the values of the parameters. In particular, it focuses on the number of connected components that the semialgebraic set has for a given point in parameter space. It proposes algorithms to, on the one hand, decide the different possible numbers of connected components throughout the parameter space and provide selected points in that space for which each number of connected components is attained. On the other hand it deals with the construction of a connected path that brings any point in parameter space to one of the selected ones, such that the path stays in one of the connected regions of parameter space yielding constant number of connected components in the corresponding semialgebraic set. Such semialgebraic sets are of interest in real world applications, as illustrated here with the example of kinematic design of a particular class of robots.
The algorithms provided solve the former problems under certain regularity conditions on the ideals generated by the polynomials defining the system and the algebraic sets defined by these same polynomials, such as radicality of certain ideals, equidimensionality and codimension of the sets defined by them, and sometimes smoothness or codimensionality of the singular locus large enough. This work extends previous results in [\textit{J. Capco} et al., in: Proceedings of the 45th international symposium on symbolic and algebraic computation, ISSAC '20. New York, NY: Association for Computing Machinery (ACM). 62--69 (2020; Zbl 1486.14075)] by relaxing the regularity conditions on the polynomial systems and introducing a new complexity analysis and a new way of reducing connectivity queries in semialgebraic sets to closed bounded ones.
The algebraic and geometric tools used include, for example, Hardt's Theorem and semialgebraic trivialization, Sard's Theorem, the Jacobian criterion and Thom's isotopy Lemma. The computational tools used include elimination and the critical point method. As a main step of the strategy, the problem is reduced to computing roadmaps of closed and bounded semialgebraic sets.
The algorithms are implemented in Maple, and a full description and complexity analysis is provided.
The singularity-free space of the \textit{UR-series} robot is analyzed by means of the tools and algorithms proposed: the parametric system here is given by the determinant of the Jacobian matrix, which is a polynomial in four variables involving four parameters defining the kinematic singularities of the robot. parametric polynomial systems; positive dimensional semialgebraic sets; roadmaps; kinematic singularities Computational real algebraic geometry, Symbolic computation and algebraic computation, Kinematics of mechanisms and robots, Semialgebraic sets and related spaces Positive dimensional parametric polynomial systems, connectivity queries and applications in robotics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(f=(f_1,\ldots,f_k)\) be a finite sequence of polynomials \(f_j\in\mathbb{R}[x_1,\ldots,x_d]\), let \(K_f\) be the associated closed semi-algebraic subset defined by \(K_f=\bigcap_{i=1}^k\{x\in\mathbb{R}^d:f_i(x)\geq0\}\) and let \(T_f\) be the set of all finite sums of elements \(f_1^{\varepsilon_1}\cdots f_k^{\varepsilon_k}g^2\), where \(g\in\mathbb{R}[x_1,\ldots,x_d]\) and \(\varepsilon_1,\ldots,\varepsilon_k\in\{0,1\}\). For a sequence \(f\), consider the (MP) (resp.\ (SMP)) moment problem: For each linear functional \(L\) on \(\mathbb{C}[x_1,\ldots,x_d]\) such that \(L(T_f)\geq0\), is there a positive Borel measure \(\mu\) on \(\mathbb{R}^d\) (resp.\ a positive Borel measure \(\mu\) on \(\mathbb{R}^d\) with supp~\(\mu\subseteq K_f\)) such that \(L(p)=\int p(\lambda)d\mu(\lambda)\) for all \(p\in \mathbb{C}[x_1,\ldots,x_d]\)?
In the paper under review, in the case in which there exist polynomials \(h_1,\ldots,h_n\in \mathbb{R}[x_1,\ldots,x_d]\) which are bounded on the set \(K_f\), the author reduces the moment problem (MP) (resp.\ (SMP)) for the set \(K_f\) to the moment problem (MP) (resp.\ (SMP)) for the ``fiber sets'' \(K_f\cap C_\lambda\) for \(\lambda\in\Lambda\) with some \(\Lambda\subseteq\mathbb{R}^n\), where \(C_\lambda\) is real algebraic variety \(\bigcap_{i=1}^n\{x\in\mathbb{R}^d:h_i(x)=\lambda_i\}\).
There are known some other positive solutions of the moment problem for non-compact semi-algebraic sets, see, for example, \textit{J. Stochel} and \textit{F. H. Szafraniec} [J. Math. Soc. Japan 55, 405--437 (2003; Zbl 1037.47003)], \textit{J. Stochel} and \textit{F. H. Szafraniec} [J. Funct. Anal. 159, 432--491 (1998; Zbl 1048.47500)], \textit{T. B. Bisgaard} [Semigroup Forum 57, 397--429 (1998; Zbl 0923.47010)]. moment problem; (MP) properties; (SMP) properties; semi-algebraic sets Schmüdgen, K., On the moment problem of closed semi-algebraic sets, J. Reine Angew. Math., 588, 225-234, (2003) Linear operator methods in interpolation, moment and extension problems, Semialgebraic sets and related spaces, Moment problems On the moment problem of closed semi-algebraic 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 \textit{R. F. Brown} [Fixed Point Theory Appl. 2006, Spec. Iss., Article 29470, 10 p. (2006; Zbl 1093.55003)] defined the notion of \(\epsilon\)-Nielsen number for a self-map \(f\) on a Riemannian manifold. This number is difficult to compute since it is not homotopy-invariant. The author presents conditions under which an algebraic computation is possible.
Let \(M\) be a compact connected Riemannian manifold with or without boundary and let \(d\) be a Riemannian metric on \(M\). Choose an \(\epsilon>0\) so that any two points \(p,q\in M\) with \(d(p,q)<\epsilon\) are joined by a unique geodesic. Denote by \(M^\epsilon(f)\) the minimum number of fixed points of maps \(g\) with \(d(f,g)<\epsilon\). Two points \(x,y\) are called equivalent if there is a component of \(\Delta^\epsilon(f):=\{z\in X|\, d(z,f(z))<\epsilon\}\) which contains both \(x\) and \(y\). This leads to the notion of \(\epsilon\)-fixed point class. An \(\epsilon\)-fixed point class is called essential if \(\text{ind}(f;C)\not=0\) were \(C\) is the component of \(\Delta^\epsilon(f)\) that contains \(F\). The number of essential \(\epsilon\)-fixed point classes is called the \(\epsilon\)-Nielsen number \(N^\epsilon(f)\). The \(\epsilon\)-core of \(f\) is defined as the set of maps \(g\) such that \(d(f,g)<\epsilon\) and \(d(f(x),x)<\epsilon\) iff \(d(g(x),x)<\epsilon\). It is easy to see that \(N^\epsilon(f)=N(f)\) iff every fixed point class of \(f\) contains at most one essential \(\epsilon\)-fixed point class of \(f\). Now fix a covering \(p:\tilde{X}\to X\). Then \(p\) is called sheet-wise isometric if each \(x\in X\) possesses an evenly covered neighbourhood such that \(p\) maps each of \(p^{-1}(U)\) isometrically onto \(U\). Finally, let \(\tilde{f}\) be a lift of \(f\) and let \(F\) be an \(\epsilon\)-fixed point class of \(f\) and \(\tilde{F}\) an \(\epsilon\)-fixed point class of \(\tilde{f}\) with \(p(\tilde{F})=F\). Define \(J_\epsilon(F)\) to be the cardinality of \(p^{-1}(\{x\})\cap\tilde{F}\) for \(x\in F\). Similarly, define \(S_\epsilon(\tilde{f})\) to be the number of \(\epsilon\)-fixed point classes of \(\tilde{F}\) which are mapped onto \(F\). A Reidemeister class is a conjugacy class of lifts of \(f\) and a lift taken from a Reidemeister class is called a Reidemeister representative. In the following situation the author obtains a formula computing \(N^\epsilon(f)\): Assume that \(f\) has only finitely many fixed points and let \(\tilde{f}_1,\dotsc,\tilde{f}_r\) be the Reidemeister representatives of \(f\) corresponding to the nonempty Nielsen fixed point classes of \(f\). Assume that: (1) \(p\) is a sheet-wise isometric covering map, (2) the path components of \(\Delta^\epsilon(\tilde{f}_i)\) are simply connected for \(i=1,\dotsc,r\), (3) \(J_\epsilon(F)\) is the same for all \(\epsilon\)-fixed point classes in the same Nielsen fixed point class. Then \(N^\epsilon(f)=\sum_{i=1}^r\frac{N^\epsilon(\tilde{f}_i)}{S_\epsilon(\tilde{f}_i)}\).
Reviewer's remark: As a matter of fact, the author introduces a normal subgroup \(K\) of \(\pi_1(X)\) in the formulation of the theorem. There are, however, no conditions on \(K\), neither does the result depend on \(K\). A reader who wishes to find out what is really going on here should consult \textit{J. Jezierski}'s article [Fixed Point Theory Appl. 2006, Spec. Iss., Article 37807, 11 p. (2006; Zbl 1097.55002)]. \(\epsilon\)-Nielsen theory; Riemannian manifold; Reidemeister representatives; covering map Moh'd, F.: An algebraic method for computing N\(\varepsilon \)(f). Topol. appl. 161, 1-16 (2014) Fixed points and coincidences in algebraic topology, Coverings of curves, fundamental group An algebraic method for computing \(N^\epsilon(f)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 investigate components and singularities of Quot schemes and varieties of commuting matrices. The authors classify its components for any number of matrices of size at most \(7\). They prove that starting from quadruples of \(8\times 8\) matrices, this scheme has generically non reduced components, while up to degree \(7\) it is generically reduced.
Their approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Bialynicki-Birula decompositions to this setup. The authors include a thorough review of their methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. The results obtained in this paper give the corresponding statements for the Quot schemes of points, in particular the authors classify the components of Quot\(_d(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})\) for \(d\leq7\) and all \(r,n\).
This paper is organized as follows. The first Section is an introduction to the subject and statement of the results. Section 2 deals with notation and Section 3 with some preliminaries. Section 4 concerns structural results on the variety \(C_n(\mathbf{M}_d)\) of \(n\)-tuples of commuting \(d\times d\) matrices and Quot\(^d_r\). Section 5 is devoted to Bialynicki-Birula decompositions and components of Quot\(^d_r\) and Section 6 to some results specific for degree at most eight. The paper is supported by an appendix concerning a functorial approach to comparison between \(C_n(\mathbf{M}_d)\) and Quot\(^d_r\). Quot schemes; varieties of commuting matrices; components and singularities Parametrization (Chow and Hilbert schemes), Relationships between algebraic curves and integrable systems Components and singularities of Quot schemes and varieties of commuting matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider the algebraic subset \({\mathcal L}\subset\bigwedge^{2}\mathfrak{g}^\ast\otimes\mathfrak{g}= \mathbb{R}^{(n^3-n^2)/2}\) of all Lie brackets on a fixed real \(n\)-dimensional vector space \(\mathfrak{g}\). Each \(\mu\in{\mathcal L}\) defines a connected and simply connected Lie group \(G_\mu\) endowed with the left invariant Riemannian metric induced by a fixed inner product \(<.,.>\) on \(\mathfrak{g}\). The set of all left invariant Riemannian metrics on \(G_\mu\) is identified with the orbit \({\mathcal O}_\mu=\text{Gl}(n,\mathbb{R}).\mu\). By degeneration \(\mu\rightarrow\lambda\) of one Lie algebra \(\mu\in{\mathcal L}\) to another Lie algebra \(\lambda\in{\mathcal L}\) is meant that \(\lambda\) belongs to the closure of \({{\mathcal O}_\mu}\) in \(\mathbb{R}^{(n^3-n^2)/2}\).
On \({\mathcal L}\) a functional \(F\) is defined by \(\mu\rightarrow F(\mu)=\text{tr} \mathbf{R}_\mu^2\), where \(\mathbf{R}_\mu\) is a symmetric transformation defined in terms of the Ricci curvature operator of \(\mu\). Then, the limits of the flow lines of the gradient flow of \(F\) are degenerations of their starting points. This property of \(F\) is used to obtain interesting relations between the space of all left invariant metrics on \(n\)-dimensional connected and simply connected Lie groups and critical points of \(F:\)
(1) \(G_\mu\) has only one left invariant Riemannian metric up to isometry and scaling if and only if the only possible degeneration of \(\mu\) is \(0\).
(2) If \(\mu\) degenerates to \(\lambda\) and \(G_\lambda\) admits a left invariant Riemannian metric satisfying a pinched curvature condition then the same holds for \(G_\mu\).
(3) The orbit \(\text{Sl}(n,\mathbb{R}).\mu\) is closed if and only if \(G_\mu\) has left invariant Riemannian metric such that its curvature tensor is a multiple of the identity.
The closed \(\text{Sl}(n,\mathbb{R})\)-orbits of \({\mathcal L}\) are classified. Explicit 1-parameter families of mutually non-isometric Einstein solvmanifolds of dimensions 10 and 11 respectively are derived from curves of closed orbits of a representation of \(\bigwedge^2\text{Sl}(m,\mathbb{R})\otimes\text{Sl}(n,\mathbb{R})\). variety of Lie algebras; degeneration of Lie algebras; closed orbits of Lie algebras; left invariant Riemannian metrics J. Lauret, Degenerations of Lie algebras and geometry of Lie groups. \textit{Differential Geom}. \textit{Appl}. \textbf{18} (2003), 177-194. MR1958155 Zbl 1022.22019 Lie algebras of Lie groups, Differential geometry of homogeneous manifolds, Group actions on varieties or schemes (quotients), Manifolds of metrics (especially Riemannian) Degenerations of Lie algebras and geometry of Lie groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper concerns the asymptotic behavior of the number of charts in coverings of a given family of sets parameterized by \(\delta>0\). Four families of sets and two types of coverings are discussed. The main theorem is, for every family, the sets parameterized by \(\delta\) are covered by coverings having at most \(K_1\left(\log(1/\delta)\right)^{K_2}\) charts for some constants \(K_1\) and \(K_2\).
The first two families of sets are the \(\delta\)-complements \(G_{\delta} := B_1^n \setminus Y_0^{\delta}\) of a zero-level hypersurface \(Y_0\) and the regular level hypersurfaces \(Y_c\) of a polynomial \(P\) with complex coefficients. Here, \(B_r^n\) is a closed ball of radius \(r\) in \(\mathbb C^n\), \(Y_c=P^{-1}(c)\), and the \(\delta\)-neighborhood \(A^{\delta}\) is the set of all points in the ambient space that are at distance less than \(\delta\) from \(A\). Doubling coverings defined in [\textit{O. Friedland} et al., Pure Appl. Funct. Anal. 2, No. 2, 221--241 (2017; Zbl 1377.32008)] are treated in these settings. The main ideas behind the proofs are to make a reduction from the general case of a polynomial \(P\) to the monomial case applying the resolution of singularities and ``suspension'' construction for extending an \(n\)-dimensional chart into an \((n + 1)\)-dimensional one.
The remaining two families are a real semialgebraic set away from the \(\delta\)-neighborhood of the semialgebraic subset of lower dimension and level sets of a semi-algebraic function with parameter away from the \(\delta\)-neighborhood of a low dimensional set. The coverings considered in these settings are the covers by the images of a-charts defined in [\textit{Y. Yomdin}, J. Complexity 24, No. 1, 54--76 (2008; Zbl 1143.32007)]. The bounds are obtained also for real subanalytic and real power-subanalytic sets. Pre-parameterization result in [\textit{R. Cluckers} et al., Ann. Sci. Ecole Norm. Sup. 53, No. 1, 1--42 (2020; Zbl 1479.03016)] and its refinement are used for their proofs. smooth parametrization; Whitney covering; doubling covering; a-charts for semi-algebraic sets; pre-preparation of subanalytic sets Global theory and resolution of singularities (algebro-geometric aspects), Semialgebraic sets and related spaces, Model theory of ordered structures; o-minimality, Semi-analytic sets, subanalytic sets, and generalizations, Entropy and other invariants, Algebraic and analytic properties of mappings on manifolds, Rational points, Dynamical systems involving smooth mappings and diffeomorphisms Doubling coverings via resolution of singularities and preparation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 investigate the closure \(\overline{M}\) of a linear subvariety \(M\) of a stratum of meromorphic differentials in the multiscale compactification constructed by Bainbridge, Chen, Gendron, Grushevsky and Möller. Given the existence of a boundary point of \(M\) of a given combinatorial type, we deduce that certain periods of the differential are pairwise proportional on \(M\), and deduce further explicit linear defining relations. These restrictions on linear defining equations of \(M\) allow us to rewrite them as explicit analytic equations in plumbing coordinates near the boundary, which turn out to be binomial. This in particular shows that locally near the boundary \(\overline{M}\) is a toric variety, and allows us to prove existence of certain smoothings of boundary points and to construct a smooth compactification of the Hurwitz space of covers of \(\mathbb{P}^1\). As applications of our techniques, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of real linear subvarieties of meromorphic strata. Teichmüller dynamics; moduli of curves; flat surfaces; linear varieties Families, moduli of curves (analytic), Teichmüller theory; moduli spaces of holomorphic dynamical systems, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Equations of linear subvarieties of strata of differentials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 habilitationsschrift, the author introduces and develops the notion of real closed spaces. His basic idea is to generalize to arbitrary rings the notions of semi-algebraic sets and semi-algebraic continuous functions, in a similar way as schemes generalize algebraic varieties and regular functions. It is known that constructible subsets of the real spectrum of a ring provide a correct generalization of semi- algebraic sets, a semi-algebraic set M being naturally identified to a constructible subset \(\tilde M\) of the real spectrum of \({\mathbb{R}}[X_ 1,...,X_ n]\). In this geometric case the consideration of the germs of semi-algebraic continuous functions at a point of \(\tilde M\setminus M\) gives some nice applications.
The author defines for the real spectrum of any ring abstract semi- algebraic functions and so he gets the building blocks of his real closed spaces. This construction is analogous to that of semi-algebraic spaces starting with semi-algebraic sets. - In the last section the author uses the abstract approach to characterize semi-algebraic sets among semi- algebraic spaces. In particular he gets the following: M is a semi- algebraic set iff the space of closed points of \(\tilde M\) is Hausdorff. This means that many natural examples of semi-algebraic spaces are in fact semi-algebraic sets.
Up to now, abstract semi-algebraic continuous functions have not had significant applications except in the geometric context. Things could change with the study of the real spectrum of excellent rings, in the real analytic case for example. real closed spaces; semi-algebraic sets; semi-algebraic continuous functions; real spectrum of excellent rings N. Schwartz, Real closed spaces, Habilitationsschrift, München, 1984. Real algebraic and real-analytic geometry Real closed 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 \(\mathbb{C}^2 \to\mathbb{C}^2\) be a map given by a polynomial \(f \in C[X,Y]\) with finitely many critical points.
The paper under review gives estimates for the following invariants of \(f\) in terms of its Newton diagram at infinity: \(\mu (f)\) (the Milnor number of \(f\)), \(\lambda (f)\) (\(=\) jump of Milnor-numbers at infinity), \(r_{\infty}(f)\) (the number of branches at infinity) and the genus \(\gamma (f)\) of the generic fibre of \(f\). The Newton diagram \(\Delta _{\infty} (f)\) is the convex hull of \(\{ (0,0) \cup \text{supp}(f) \}\).
Main theorem: Let \(f:\mathbb{C}^2\to\mathbb{C}\) be a polynomial such that \(\mu (f)< \infty\). Suppose that \(\Delta _{\infty} (f)\) has a nonempty interior. Then the global Milnor number \(\mu (\Delta _{\infty} (f))\) and the number \(r(\Delta _{\infty} (f)) = \sum _{S\in\partial \Delta _{\infty} (f)} r(S)\) (where \(r(S)=\) \((\)number of integer points lying on the segment \(S)\) \(-1\)) satisfy the inequality
\[
\mu (\Delta _{\infty} (f))-(\mu (f) + \lambda (f)) \geq r(\Delta _{\infty} (f)) - r_{\infty} (f) \geq 0.
\]
Equality holds if \(f\) is nondegenerate on each segment of \(\partial \Delta _{\infty} (f)\) which is not included in a line passing through the origin. affine curves; singularities; Milnor number; invariants; Newton diagram at infinity Milnor fibration; relations with knot theory, Singularities of curves, local rings Invariants of singularities of polynomials in two complex variables and the Newton diagrams | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathcal{O}_{n}\) be the ring of holomorphic function germs at \(0\in \mathbb{C}^{n}\). The author characterizes ideals \(I\) of \(\mathcal{O}_{n}\) of finite colength whose integral closure \(\bar{I}\) is equal to the integral closure of an ideal generated by diagonal monomials i.e \(\bar{I}=\overline{(x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}})}.\) This characterization is given in terms of the log canonical threshold \(\text{lct}(I)\) (it is the supremum of those \(s\in \mathbb{R}_{>0}\) such that the function \((\left| g_{1}\right| ^{2}+\cdots +\left| g_{r}\right| ^{2})^{-s}\) is locally integrable around \(0\)) and in terms of mixed multiplicities \( e_{i}(I):=e(\underset{i}{\underbrace{I,\ldots ,I}},\underset{n-i}{ \underbrace{\mathfrak{m},\ldots ,\mathfrak{m}}}),\) \(i=1,\ldots,n\). Namely, \( I\) is diagonal if and only if
\[
\text{lct}(I^{0})=\frac{1}{e_{1}(I)}+\frac{e_{1}(I)}{e_{2}(I)}+\cdots + \frac{e_{n-1}(I)}{e_{n}(I)},
\]
where \(I^{0}\) is the ideal generated by monomials lying in \(\Gamma _{+}(I)\) -- the Newton polyhedron of \(I\). singularity; log canonical threshold; diagonal ideal; mixed multiplicity Singularities in algebraic geometry, Local complex singularities, Multiplicity theory and related topics Log canonical threshold and diagonal 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 G denote a group of transformations of \({\mathbb{R}}^ n\) generated by bijections of \({\mathbb{R}}^ n\) of the form \((f_ 1,...,f_ n),\) where each \(f_ i\) is a rational function of n variables with real coefficients. For a function \(\gamma:\quad G\to (0,\infty)\) a measure \(\mu\) on \({\mathbb{R}}^ n\) is \(\gamma\)-invariant if \(\mu (g[A])=\gamma (g)\mu (A)\) for every \(g\in G\) and \(\mu\)-measurable set A.
The main theorem of the paper is the following: Any \(\sigma\)-finite measure \(\mu\) has a proper \(\gamma\)-invariant extension whenever G contains an uncountable subset of rational functions H such that \(\mu (\{x:\quad h_ 1(x)=h_ 2(x)\}=0\) for all distinct \(h_ 1,h_ 2\in H.\)
For example, if G is any uncountable subgroup of affine transformations of \({\mathbb{R}}^ n\), \(\gamma\) (g) is the absolute value of the Jacobian of \(g\in G\), and \(\mu\) is a \(\gamma\)-invariant extension of n-dimensional Lebesgue measure, then \(\mu\) has a proper \(\gamma\)-invariant extension. invariant \(\sigma\)-finite measure; group of transformations of \({\mathbb{R}}^ n\); invariant extension Krzysztof Ciesielski, Algebraically invariant extensions of \?-finite measures on Euclidean space, Trans. Amer. Math. Soc. 318 (1990), no. 1, 261 -- 273. Set functions and measures on topological groups or semigroups, Haar measures, invariant measures, Classical measure theory, Classical groups (algebro-geometric aspects) Algebraically invariant extensions of \(\sigma\)-finite measures on Euclidean 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 \(f:\mathbb{K}^{n}\to \mathbb{K}\) be a polynomial function (\( \mathbb{K=R}\) or \(\mathbb{K=C}\))\(.\) The Łojasiewicz exponent at infinity of \(f\) near an unbounded fibre \(f^{-1}(t_{0})\) is defined by
\[
\mathcal{L}_{\infty }(f,t_{0})=\lim_{\delta \to 0}\mathcal{L} _{\infty }(\text{grad}f|f^{-1}(D_{\delta })),
\]
where \(D_{\delta }=\{t:| t-t_{0}| <\delta \}\) and
\[
\mathcal{L}_{\infty }(\text{grad}f|f^{-1}(D_{\delta }))=\sup \{\nu \in \mathbb{R}:\| \text{grad}f(x)\| \geq C\| x\| ^{\nu }\text{ for } | x| \to \infty ,\text{ }x\in f^{-1}(D_{\delta })\}.
\]
This exponent is always a rational number. The authors prove that in real case for every rational number \(\alpha \) there exists a \(f\in \mathbb{R[} x_{1},\dots ,x_{n}\mathbb{]}\) such that \(\mathcal{L}_{\infty }(f,t_{0})=\alpha \) (it is known that in complex case it is only possible for \( \alpha <-1\) or \(\alpha \geq 0).\) The second result is a formula (in real and complex case) for \(\mathcal{L}_{\infty }(f,t_{0})\) in terms of the Puiseux roots at infinity of \(f'_{x}(x,y)=0\) (under assumption that \(\mathcal{ L}_{\infty }(f,t_{0})<0\) and \(f(x,y)\) is monic in \(x;\) the complex case is known - Thm 3.4 in [\textit{J. Chadzyński} and the reviewer, Kodai Math. J. 26, No. 3, 317--339 (2003; Zbl 1051.32016)]). Łojasiewicz exponent; Puiseux expansion at infinity; Fedoryuk and Malgrange conditions; asymptotic critical value Affine fibrations, Complex singularities Łojasiewicz exponent of the gradient near the 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 In this paper, and in characteristic \(0\), the authors describe the smooth locus of the moduli space of linear series with prescribed vanishing sequences in at most two marked points. In the particular case of no marked points, this result specializes to the Gieseker-Petri theorem, proved previously in [\textit{D. Gieseker}, Invent. Math. 66, 251--275 (1982; Zbl 0522.14015)] and [\textit{ D. Eisenbud} and \textit{J. Harris}, Invent. Math. 74, 269--280 (1983; Zbl 0533.14012)], and recently in [\textit{S. Payne } and \textit{D. Jensen}, Algebra Number Theory 8, No. 9, 2043-2066 (2014; Zbl 1317.14139)].
The main idea of the proof is based on degeneration to a chain of elliptic curves and studying the corresponding the moduli space of limit linear series, introduced by Eisenbud and Harris. In parallel, the smoothness conditions of a point impose that the point must lie on the smooth locus of Schubert cycles and the smooth locus of Schubert varieties is characterized. degeneration; Grassmannian; Schubert varieties; almost-transverse flags; linear series; ramification Special divisors on curves (gonality, Brill-Noether theory), Grassmannians, Schubert varieties, flag manifolds The Gieseker-Petri theorem and imposed ramification | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is devoted to the approximation of continuous maps of nonsingular compact algebraic sets into a sphere by continuous rational maps.
Let \(X\subset \mathbb{R}^k\) and \(Y\subset \mathbb{R}^l\) be nonsingular algebraic sets and let \(X\) be compact.
A map \(f:X\to Y\) is said to be \textit{continuous rational} if it is continuous and there exist a Zariski open and dense subset \(U\) of \(X\) and a regular map \(\varphi:U\to Y\) such that \(f|_U=\varphi\). Denote by \(P(f)\) the indeterminacy locus of the rational map from \(X\) into \(Y\) represented by \(\varphi\). It is called the \textit{irregularity locus} of \(f\). Maps with \(f(P(f))\neq Y\) are said to be \textit{nice}.
The set of all continuous rational maps from \(X\) to \(Y\) is denoted by \(\mathcal{R}^0(X,Y)\). The space \(\mathcal{C}(X,Y)\) of all continuous maps from \(X\) into \(Y\) is endowed with the compact-open topology. By definition, a continuous map from \(X\) into \(Y\) can be \textit{approximated by continuous rational maps} if it belongs to the closure of \(\mathcal{R}^0(X,Y)\) in \(\mathcal{C}(X,Y)\).
A continuous map \(h:X\to Y\) is said to be \textit{transverse to a point} \(y\in Y\) if it is smooth in an open neighbourhood \(U\subset X\) of \(h^{-1}(y)\) and the restriction map \(h|_U:U\to Y\) is transverse to \(y\) in the usual sense. If \(h\) is transverse to some point \(y\in Y\) and \(h^{-1}(y)=\text{Reg}(V)\) for some algebraic subset \(V\) of \(X\), then \(h\) is said to be \textit{adapted}. The map \(h:X\to Y\) is said to be \textit{weakly adapted} if there exists a point \(y\in Y\) such that \(h\) is transverse to \(y\) and the submanifold \(h^{-1}(y)\) of \(X\) admits a weak algebraic approximation in \(X\), i.e., each neighbourhood of the inclusion map \(M \hookrightarrow X\) in the space \(\mathcal{C}^{\infty}(h^{-1}(y),X)\) contains a smooth embedding \(e:h^{-1}(y)\to X\) with \(e(h^{-1}(y))=\text{Reg}(V)\) for some algebraic subset \(V\) of \(X\).
The main result of this article is the following theorem.
Theorem 1.2. For a continuous map \(h:X\to \mathbb{S}^p\), where \(\mathbb{S}^p=\{(u_1,\ldots,\) \(u_{p+1})\in\mathbb{R}^{p+1}:u^2_1+\cdots+u^2_{p+1}=1\}\) is the unit sphere, the following conditions are equivalent:
(a) \(h\) can be approximated by nice continuous rational maps with irregularity locus of dimension at most \(\dim X-p\).
(b) \(h\) can be approximated by nice continuous rational maps.
(c) \(h\) can be approximated by adapted continuous maps.
(d) \(h\) can be approximated by adapted smooth maps.
(e) \(h\) can be approximated by weakly adapted continuous maps.
(f) \(h\) can be approximated by weakly adapted smooth maps.
As corollaries, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular the author proves Theorem 1.5 asserting that each continuous map between unit spheres can be approximated by continuous rational maps. real algebraic set; regular map; continuous rational map; semi-algebraic map; approximation W. Kucharz, Approximation by continuous rational maps into spheres, J. Eur. Math. Soc. (JEMS) 16 (2014), 1555-1569. Real algebraic sets, Topology of real algebraic varieties Approximation by continuous rational maps into spheres | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From the authors' abstract and introduction: This work considers semialgebraic subsets \(X\) of the unit ball in \(\mathbb{R}^{n}\) given by \(k\) quadratic equalities or inequalities. The main result provides a bound of type \(n^{O(k)}\) (i.e. polynomial in \(n\)) for the geodesic diameter. (For semialgebraic sets of general degree the bound is known to be exponential in \(n\).)
The proof borrows heavily from the idea of \textit{D. D'Acunto} and \textit{K. Kurdyka} [Bull. Lond. Math. Soc. 38, No. 6, 951--965 (2006; Zbl 1106.14046)] which consists in controlling the geodesic diameter by the length of trajectories of the gradient of a Morse function, which in turn are uniformly bounded by the length of the \textit{talweg} of this function. The main difficulty is to show that this talweg can be assumed of dimension \(1\), so that its length can be estimated via the Cauchy-Crofton formula, by counting intersection points with a hyperplane. This eventually leads to a system of equations containing a big (of size \(n\)) linear system in \(x\). This linearity comes from the fact that the gradients of quadratic functions are linear.
A method introduced by \textit{A. I. Barvinok} [Math. Z. 225, No. 2, 231--244 (1997; Zbl 0919.14034)] is hereby adapted to solve the linear system in \(x\) (as function of the parameters and a few free variables among \(x\)) and carry this solution in the other equations. An additional difficulty appears from the nonlinear dependence on the parameters of the matrix of the system. The co-rank of the linear system, that is, the number of free variables, is eventually shown to remain small, controlled by \(k\), when the parameters vary, which gives a bound polynomial in \(n\) of degree \(O(k)\) on the number of solutions of the complete system. semialgebraic set; geodesic diameter; quadratic equation; gradient orbit; talweg Coste, M., Moussa, S.: Geodesic diameter of sets defined by few quadratic equations and inequalities. arXiv: 1009.0452 Semialgebraic sets and related spaces, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Inequalities involving derivatives and differential and integral operators, Geodesics in global differential geometry Geodesic diameter of sets defined by few quadratic equations and 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 The paper under review presents a general method for expanding a truncated $G$-iterative Hasse-Schmidt (HS-) derivation, where $G$ is an algebraic group. The author introduces a concept of a canonical $G$-basis for a field equipped with an iterative HS-derivation (in the spirit of Definition 6.1 of \textit{K. Okugawa} [J. Math. Kyoto Univ. 2, 295--322 (1963; Zbl 0171.00302)]) and proves that the existence of such a basis implies strong integrability for an arbitrary iterativity condition. (The concept of strong integrability was introduced in [\textit{H. Matsumura}, Nagoya Math. J. 87, 227--245 (1982; Zbl 0458.13002)]; briefly, the strong integrability means that a truncated iterative HS-derivation can be expanded to a non-truncated one, satisfying the same iterativity conditions.) \par The main result of the paper under review (Theorem 4.17) states that, over an algebraically closed field, a connected and commutative linear algebraic group has a canonical basis if unipotent elements form a subgroup of dimension $\leq 2$. This fundamental result includes as particular cases similar statements about concrete algebraic groups (products of $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$) obtained in the mentioned paper by \textit{A. Tyc} [Nagoya Math. J. 115, 125--137 (1989; Zbl 0661.14034)], [\textit{M. Ziegler}, ``Canonical $p$-basis'', \url{http://home.mathematik.uni-freiburg.de/ziegler/preprints/canonical-p-bases.pdf}] and [\textit{M. Ziegler}, J. Symb. Log. 68, No. 1, 311--318 (2003; Zbl 1039.03031)]. Derivations and commutative rings, Group schemes, Linear algebraic groups over arbitrary fields On existence of canonical \(G\)-bases | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 describes the classification problem of continuous families of infinite dimensional linear systems. We are concerned with a family of controllable and observable linear systems (F(s), G(s), H(s)) in Banach spaces, which depend continuously on a parameter s in a Hausdorff topological space S.
Our problem is to classify such families of systems by isomorphism relation and parametrize them by ``moduli''. It is known that there exist ``moduli'' for finite dimensional controllable and observable systems. We show the result fails in the infinite dimensional case, and derive some conditions under which there exist ``moduli'', i.e., a fine-moduli theorem for infinite dimensional systems. The theory of Banach bundles plays an important role in our study. classification problem; moduli; fine-moduli theorem; Banach bundles; time-invariant; continuous-time Control/observation systems in abstract spaces, Controllability, Canonical structure, Observability, Algebraic moduli problems, moduli of vector bundles Fine moduli spaces of infinite dimensional 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 The paper under review studies the radius of convergence of a differential equation on a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. In particular, under the additional assumption that the curve has no boundary or that the differential equation is overconvergent, the authors gave a new proof of the continuity of the radius function (previously proved by \textit{F. Baldassarri} [Invent. Math. 182, No. 3, 513--584 (2010; Zbl 1221.14027)]) and that the radius function factorizes by the retraction through a locally finite graph (previously proved by the two authors [Acta Math. 214, No. 2, 307--355 (2015; Zbl 1332.12013); Acta Math. 214, No. 2, 357--393 (2015; Zbl 1332.12012)]). Roughly speaking, this new proof relies only on two technical inputs: (1) the radius functions on an annulus or a disc is continuous, concave, piecewise log-linear whose slopes have bounded denominators (this is proved by \textit{G. Christol} and \textit{B. Dwork} [Ann. Inst. Fourier 44, No. 3, 663--701 (1994; Zbl 0859.12004)] and \textit{K. S. Kedlaya} [\(p\)-adic differential equations. Cambridge: Cambridge University Press (2010; Zbl 1213.12009)]), and (2) the potential theory on Berkovich curves, as developed by A. Thuillier in his thesis. The key step lies in proving the radius of convergence is a super-harmonic function, at least locally in affinoid neighborhoods of type \(2\) points. The main theorems follow from general properties of super-harmonic functions. \(p\)-adic differential equations; Berkovich spaces; radius of convergence; potential theory Poineau, J., Pulita, A.: Continuity and finiteness of the radius of convergence of a \(p\)-adic differential equation via potential theory. J. Reine Angew. Math. \textbf{707}, 125-147 (English) (2015) \(p\)-adic differential equations, Rigid analytic geometry Continuity and finiteness of the radius of convergence of a \(p\)-adic differential equation via potential 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 \(\tilde{\mathcal A}_ m\) be the algebra of functions on \(\mathbb{R}^ n\) generated by polynomial functions and exponentials of linear forms. The subset \(S\) in \(\mathbb{R}^ n\) belongs to \({\mathcal P}_ n\) if and only if there exist \(m\) and \(F\) in \(\tilde{\mathcal A}_{n+m}\) for which \(S\) is the image of the zerosubset of \(F\) by the canonical projection of \(\mathbb{R}^{n+m}\) onto \(\mathbb{R}^ n\). Let \(\tilde{\mathcal P}_ n\) be the smallest subset of parts in \(\mathbb{R}^ n\) which contains \({\mathcal P}_ n\), their closures and the images by the canonical projection of the elements in \(\tilde{\mathcal P}_{n+m}\). --- This family of sets is defined by an induction in two steps. The main goal of this article is to prove that \(\tilde{\mathcal P}_{n+m}\) contains the complementary part of each element in \(\tilde{\mathcal P}_{n+m}\), the union and the intersection of every finite family in \(\tilde{\mathcal P}_{n+m}\). The key results for the proof are theorems by A. G. Khovanskij on the set of the solutions of a Pfaff system on a Pfaff manifold. polynomial functions; exponential; zeroset of a function; projection; A. G. Khovanskij's theorems; Pfaff system; Pfaff manifold Charbonnel, J. -Y.: Sur certains sous-ensembles de l'espace euclidien. Ann. inst. Fourier Grenoble 41, No. 3, 679-717 (1991) Nash functions and manifolds, Linear function spaces and their duals, Pfaffian systems, Real-analytic and Nash manifolds, Relevant commutative algebra About certain subsets of the Euclidean 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 For a smooth scheme X over a perfect field k of characteristic p, \(W\Omega_ X\) is the Hodge-Witt complex as defined by Etesse. The trace map \(Tr:W_ m\Omega^ N_ X\to f^ !_ mW_ m[-N]\) is defined \((N=\dim X)\) and shown to be an isomorphism. The pairing \(W_ m\Omega^ i\otimes_{W_ m{\mathcal O}}W_ m\Omega^{N-i}\to W_ m\Omega^ N\) results as the author proves in an isomorphism \(W_ m\Omega^ i\to R \underline{\Hom}_{W_ m{\mathcal O}}(W_ m\Omega^{N-i},W_ m\Omega^ N)\) which implies an isomorphism \(H^ i(X,W_ m\Omega^ j)\cong\Hom_{W_ m}(H^{N-i}(X,W_ m\Omega^{N-j}),W_ m)\). -- Here R is the Raynaud ring.
An auto-duality for \(R\Gamma(W\Omega^.)\) is obtained from this. The notion of de Rham-Witt systems on X is introduced in order to show that the auto-duality preserves R-module structures in a suitable sense. \(H^ i(D(M))\) for a coherent module M is computed and used via spectral sequence to analyze D(M) for a coherent complex M. Here \(D(\quad)=R\Hom_ R(-,\check R)\). For this one shows that if M is coherent so is D(M) and then M is isomorphic to D(D(M)). Using the first result above \(E_ 2^{1,3}\neq E_{\infty}^{1,3}\) for a supersingular abelian fourfold follows. A coherently commutative associative bifunctor \(_{*R}^ L\) on \(D^ b_{perf}(R)\) from which it follows that \(D^ b_{perf}(R)\) is a rigid additive tensor category is found. \(D^ b_{perf}(R)\) consists of the vertically bound perfect complex of \(R_ S\)-modules. Here \(R_ S\) denotes a certain graded W\({\mathcal O}_ S\)-ring related to the Raynaud ring and a complex of \({\mathcal O}\)-modules \(M^ 1\) (obtained by tensoring) is perfect if there is a finite interval d in \({\mathbb{Z}}\) such that the complex is locally isomorphic to a complex concentrated in d whose components are free \({\mathcal O}_ S\)-modules on finite sets. A Künneth formula for smooth maps \(f:X\to S, g:X\to S Rf_*W\Omega^._{X/S}\wedge_{*R}^ LRg_*W\Omega^._{X/S}=R(f\times_ Sg)_*W\Omega^._{X\times Y/S}\) and duality formula \(D(R_{f_*}W\Omega^._{X/S})[-N][- N]=R_{f_*}W\Omega^._{X/S}\) are obtained with suitable restriction. Various results on \(\text{Kün}^ R_ i(M,N)=H^{- i}(M_{*R}^ LN)\) are then obtained.
The thesis provides good motivation but could be better orchestrated. de Rham-Witt complex; crystalline cohomology; duality; Hodge-Witt complex; Künneth formula Ekedahl, T., On the multiplicative properties of the de Rham--Witt complex II (To appear inArk. Mat.). \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Duality in algebraic topology, Étale and other Grothendieck topologies and (co)homologies, Transcendental methods, Hodge theory (algebro-geometric aspects) On the multiplicative properties of the De Rham-Witt complex | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 theory of schemes, initially proposed and largely elaborated by A. Grothendieck as a rigorous foundation for algebraic geometry in the 1950s, then vastly amplified by many other algebraic geometers during the subsequent decades, has proved to be an extremely powerful and flexible tool in this area of mathematics. Many classical, long-standing concrete problems have been solved in this conceptually rigorous and unifying framework, and nowadays the theory of schemes is the well-established all-round instrument of algebraic and arithmetic geometry.
However, as for the average mathematician, and even for the algebraic geometer the theory of schemes is not even now a familiar language. This is not only due to the uncontested fact that the theory is extremely general and technically involved (that is, rather intimidating to impatient researchers), but also to the fact that there was actually no existing textbook, up to now, which provided an elementary, straightforward and technically uncomplicated introduction to scheme theory for non-specialists. Despite the existence of many recent standard textbooks on modern algebraic geometry, including both the theory of algebraic varieties and the framework of algebraic schemes, there still has been a need for a rather elementary text that would explain the use of schemes with a minimum of prerequisits and involved technicalities, however with lots of typical, instructive and motivating examples, just in order to make the essentials of scheme theory transparent and enjoyable to the beginner.
The present book is written in exactly this spirit. As the authors emphasize in the preface, this text is intended to fill the gap between texts on classical algebraic geometry and the ``full-blown'' accounts of general scheme theory. By focusing on the very basic definitions, which require a rather modest acquaintance with commutative algebra and classical algebraic varieties, and then providing many examples of affine schemes, projective schemes, morphisms, relative schemes, non-reduced schemes, flat families of schemes, and arithmetic schemes, the text leads the reader to a profound understanding of what general schemes are and what they are good for. The final chapter provides an introduction to the functorial point of view in the theory of schemes. This includes a brief discussion of such important and actual concepts like tangent spaces, group schemes, Hilbert schemes and their tangent spaces, and moduli spaces in algebraic geometry. Instead of detailed proofs, the authors concentrate on concrete examples, explaining the philosophy and suitable generalizations with the aid of them, and on hundreds of related exercises together with hints for their solution. This extremely inspiring text, written with exemplary, nearly affectionate didactic care, will certainly help the reader to overcome the barrier that separates classical algebraic geometry from contemporary algebraic geometry, and give him an idea of the falvour of the current research in both algebraic geometry and algebraic number theory. The reader who has worked through this book will be able to study the more advanced textbooks with much less dread of scheme theory. affine schemes; scheme theory; algebraic schemes; tangent space; group scheme; Hilbert scheme; moduli space D. Eisenbud and J. Harris. Schemes: The Language of Modern Algebraic Geometry. The Wadsworth \& Brooks/Cole Mathematics Series (1992). Schemes and morphisms, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Schemes: The language of modern algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The limit \(q\)-Bernstein operator \(B_q\) emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the \(q\)-boson theory to describe the energy distribution in a \(q\)-analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the \(q\)-operators. Over the past years, the limit \(q\)-Bernstein operator has been studied widely from different perspectives. It has been shown that \(B_q\) is a positive shape-preserving linear operator on \(C[0, 1]\) with \(||B_q|| = 1\). Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit \(q\)-Bernstein operator related to the approximation theory. A complete bibliography is supplied. Ostrovska, S.: A survey of results on the limit \(q\)-Bernstein operator. J. Appl. Math. \textbf{2013}, Article ID 159720, 7, (2013) Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Coherent states, Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials A survey of results on the limit \(q\)-Bernstein operator | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 classical uniformization theorem states that any Riemann surface has a universal covering conformally equivalent to either the Riemann sphere \(\mathbb P^1\), the complex plane \(\mathbb C\) or the hyperbolic upper half plane \(\mathbb H\). One of the consequences is that every smooth complex algebraic curve \(C\) of genus \(g>1\) is conformally equivalent to the quotient of \(\mathbb H\) by a Fuchsian group \(G\subset PSL_2(\mathbb R)\), and conversely.
The problem of how to realize this correspondence in an explicit fashion is, however, a difficult one, and it is still partially unsolved. In this paper, the authors consider one of the two possible directions of the correspondence, namely going from the Fuchsian group \(G\) to the algebraic curve in the case of real hyperelliptic curves.
A real curve \(C\) is an algebraic curve equipped with an anti-holomorphic involution \(\sigma\) whose fixed points comprise the set of real points \(C(\mathbb R)\) of \(C\). The set of real points has at most \(g+1\) components if \(g\) is the genus of \(C\), and the authors consider the case when this bound is attained. Thus \(C\) is hyperelliptic of genus \(g\) with \(C(\mathbb R)\) having \(g+1\) components. This amounts to say that the polynomial \(P\) in the equation \(y^2=P(x)\) is real with real roots \(x_1,\dots, x_{2g+1}\).
The authors' method is as follows. First, they show that for a curve \(C\) of the above type its period matrix can be reconstructed analytically from the data of hyperbolic capacities of a set of harmonic functions. The equation of the curve is then reconstructed from the period matrix via theta characteristics. The crucial point is to obtain for \(C\) a set of harmonic functions -- with their boundary conditions -- depending only on \(G\), without reference to the algebraic structure.
Let \(M\) be a domain in the hyperbolic plane, and let \(h\) be a harmonic function on \(M\). The capacity associated to it is
\[
\int_M {||\nabla h||}^2 dM = \int_{\partial M} h \nu [h] d\mu
\]
where \(\nu [h]\) is the normal derivative and \(d\mu\) is the measure induced on the boundary.
A hyperelliptic curve \(C\) with the above real structure can be obtained by gluing together two spheres with \(g+1\) disks removed. Each sphere can in turn be obtained by gluing together two \(2g+2\)-gons. The authors show that starting from the data of the hyperelliptic curve in the form \(y^2=P(x)\), a set of hyperbolic capacities \(\{h_i\}\) can be obtained satisfying very simple boundary conditions on the \(2g+2\)-gons.
At this point the algorithm works in the following way. Given the lengths of a \(2g+2\) and the boundary conditions, the capacities \(\{h_i\}\) can be reconstructed numerically via harmonic polynomials. In turn this determines the period matrix, and then the equation of the curve via theta characteristics.
The second half of the paper is devoted to a series of cases where the correspondence is actually exact. For certain specific families of real hyperelliptic genus \(2\) curves, the equation can be obtained explicitly in exact form. In particular, the authors consider the following subspace of the real moduli space of genus \(2\) real hyperelliptic curves. If \(g=2\), the \(2g+2\)-gon is a hexagon characterized by three lengths \(l_1,l_2,l_3\). If two of them are equal, say \(l_2=l_3\), an involution is obtained acting on the corresponding curve. Since there is also the hyperelliptic involution, the group \({\mathbb Z}/2 \times {\mathbb Z}/2\) acts on these curves by automorphisms preserving the real structure. Conversely, if \({\mathbb Z}/2 \times {\mathbb Z}/2\) is contained in the group of automorphisms of \(C\) preserving the real structure, the resulting hexagon will have two lengths equal. For families in this subspace, the authors construct an action of the dihedral group \(D_5\) and show in several explicit examples how for the various fixed points of the \(D_5\) action it is possible to obtain explicit equations starting from the lengths \(l_1\), \(l_2\).
A final section is devoted to a compilation of tables and examples. Other interesting points addressed in the body of the paper, concern the application of the procedure to one-punctured tori, and to curves with half twists. uniformization; hyperelliptic curves; real curves Buser, P., Silhol, R.: Geodesics, periods and equations of real hyperelliptic curves. Duke Math. J. 108, 211--250 (2001) Compact Riemann surfaces and uniformization, Real algebraic sets, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Period matrices, variation of Hodge structure; degenerations Geodesics, periods, and equations of real hyperelliptic 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 \(\Phi : F \to G^*\) be a generic map of locally free modules of rank \(m = \text{rk} F \geq n = \text{rk} G\) over a scheme \(X\) of the form \(X = \text{Spec(Sym} (F_ 0 \otimes_{{\mathcal O}_{X_ 0}} G_ 0))\), where \(X_ 0\) is a scheme defined over \(\mathbb{Q}\). For any partition \(\lambda\) there is an induced map \(\Phi_ \lambda : L_ \lambda F \to L_ \lambda G^*\) of Schur functors. The author proves that if \(\lambda\) is a rectangular partition, the homological dimension of the module \(M_ \lambda = \text{coker} (\Phi_ \lambda)\) is \(D(\lambda) (m - n) + 1\), where \(D(\lambda)\) is the size of the Durfee square of \(\lambda\), i.e. the largest square partition contained in \(\lambda\). This generalizes results of \textit{D. A. Buchsbaum} and \textit{D. Eisenbud} [Adv. Math. 18, 245-301 (1975; Zbl 0336.13007)] and of \textit{K. Akin}, \textit{D. A. Buchsbaum} and \textit{J. Weyman} [Adv. Math. 39, 1-30 (1981; Zbl 0474.14035)]. More precisely, the author determines the syzygy modules of \(M_ \lambda\) in the case \(\lambda = (r^ s)\) by pushing down a certain Schur complex on the Grassmannian \(Y = \text{Grass}_{n - r} (G)\) and by using the spectral sequences of hypercohomology and Bott's theorem to calculate the result. Schur functors; syzygy modules; Grassmannian; hypercohomology Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Cohen-Macaulay modules Syzygies of a certain family of generically imperfect 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 aim of this paper is to extend the theory of the discrete moving frame in two different ways.The first is to consider a discrete moving frame defined on a lattice variety, which can be thought of as the vertices, or 0-cells, together with their adjacency information, in a discrete approximation of a manifold. The authors describe their associated cross sections and define Maurer-Cartan invariants and local syzygies. They describe the equivalence classes of global syzygies that result from the first fundamental group of the variety. The authors consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, they construct multispace, a generalisation of the jet bundle that also generalises Olver's one-dimensional construction. Using interpolation to provide coordinates, they prove that it is a manifold containing the usual jet bundle as a submanifold and show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. The authors prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In their last chapter the authors discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, they discuss a more general result concerning the discretisation of smooth moving frames, and the continuum limit of equicontinuous families of discrete moving frames, with an example. More precisely, they use the Arzela-Ascoli theorem to give a general convergence result for an equicontinuous family of moving frames. This provides a rigorous foundation to a variety of examples involving the discretisation of a smooth frame. discrete moving frame; discrete invariants; local and global syzygies of invariants; multispace; discrete and smooth Maurer-Cartan invariants; finite difference calculus of variations; discrete integrable systems MaríBeffa, G; Mansfield, EL, Discrete moving frames on lattice varieties and lattice based multispaces, Found. Comput. Math., 18, 181-247, (2018) Relationships between algebraic curves and integrable systems, Applications of Lie algebras and superalgebras to integrable systems, Discrete approximations in optimal control, Differential invariants (local theory), geometric objects, Global differential geometry, Differential spaces, Exterior differential systems (Cartan theory) Discrete moving frames on lattice varieties and lattice-based multispaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Motivated by the problem of constructing fast matrix multiplication algorithms, \textit{V. Strassen} [J. Reine Angew. Math. 384, 102--152 (1988; Zbl 0631.68033); J. Reine Angew. Math. 375/376, 406--443 (1987; Zbl 0621.68026)] introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiring \(\mathcal{X}\) of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements of \(\mathcal{X}\) in terms of the asymptotic spectrum of \(\mathcal{X} \), which is defined as the collection of semiring homomorphisms from \(\mathcal{X}\) to the non-negative reals with a natural monotonicity property. The asymptotic properties characterized by this duality encompass fundamental problems in complexity theory, combinatorics and quantum information. Universal spectral points are elements in the asymptotic spectrum of the semiring of all tensors. Finding all universal spectral points suffices to find the asymptotic spectrum of any sub-semiring. The construction of non-trivial universal spectral points has been an open problem for more than thirty years. We construct, for the first time, a family of non-trivial universal spectral points over the complex numbers, called quantum functionals. We moreover prove that the quantum functionals precisely characterise the asymptotic slice rank of complex tensors. Our construction, which relies on techniques from quantum information theory and representation theory, connects the asymptotic spectrum of tensors to the quantum marginal problem and entanglement polytopes. matrix multiplication algorithms; semiring homomorphisms; complex tensors Multilinear algebra, tensor calculus, Geometric invariant theory, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Universal points in the asymptotic spectrum of tensors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 of interest to those studying number theory and algebraic geometry. The main objects of study are overconvergent isocrystals on a scheme over a field of characteristic \(p >0\). Overconvergent isocrystals can be roughly thought of as modules which are vector bundles with flat connection on a smooth dense open subset and have certain analytic behavior on the boundary of the open subset. The use of analytic methods on a scheme over a field of characteristic \(p >0\) is permitted by realizing the scheme as the special fiber of a scheme over characteristic \(0\). Let \((V,m)\) be a complete discrete valuation ring with \(\mathrm{char}(V/m)=p >0\) and \(\mathrm{char}(\mathrm{Quot}(V))=0\). If \(Z\) is a well-behaved scheme of finite type over \(V/m\) then (locally) there is a well-behaved scheme \(\mathfrak{Z}\) of finite type over \(V\) with special fiber \(Z = \mathfrak{Z} \times_{\mathrm{Spec}(V)} \mathrm{Spec}(V/m)\). Since \(V\) is complete, it is possible to ask analytic questions about functions on \(\mathfrak{Z}\) and its generic fiber. A theorem of Berthelot states that the definition of the category of (convergent) isocrystals for different choices of \(\mathfrak{Z}\) give equivalent categories and naturally respect passing to open subsets.
Let \(X \rightarrow Y\) be a dense open immersion of schemes of finite type over \(V/m\). Define \(\mathrm{Isoc}^{\dagger}(X,Y)\) to be the category of isocrystals on \(X\) overconvergent with respect to \(Y \setminus X\) and \(\mathrm{Isoc}(X)\) the category of (convergent) isocrystals on \(X\). An interesting question to ask is if the restriction on the analytic behavior on the boundary allows morphisms defined on \(X\) to extend uniquely to all of \(Y\). In categorical terms, is the pullback functor
\[
j^*: \mathrm{Isoc}^{\dagger}(X,Y) \rightarrow \mathrm{Isoc}(X)
\]
fully faithful? This conjecture was posed by \textit{N. Tsuzuki} as 6.2.1 in [Duke Math. J. 111, No. 3, 385--418 (2002; Zbl 1055.14022)] where a proof for the unit-root case was provided. Tsuzuki also posed a related version of this conjecture where \(\mathcal{O}_Y\) is replaced by the bounded Robba ring and \(\mathcal{O}_X\) by the Amice ring [2.3.1, loc. cit.]. Tsuzuki proved that Conjecture 2.3.1 [loc. cit.] implies Conjecture 6.2.1 [loc. cit.].
The reviewed paper contains two main results. The first result is a counterexample to Conjecture 2.3.1 [loc. cit.]. The paper proposes that Conjecture 2.3.1 [loc. cit.] may be true if the pullback functor is restricted to the subcategory of modules satisfying a certain condition called the DNL condition. The second result is that this new conjecture implies that the pullback functor is fully faithful when restricted to the category of overconvergent isocrystals satisfying the DNL condition, the analogous version of Conjecture 6.2.1 [loc. cit.]. It is worth noting that \textit{K. S. Kedlaya} [Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter. 819--835 (2004; Zbl 1087.14018)] has shown \(j^*\) is fully faithful if one requires that isocrystals also carry a Frobenius structure. Overconvergent isocrystals with a Frobenius structure satisfy the DNL condition. isocrystals; full faithfulness; counterexample; Tsuzuki's full faithfulness Rigid analytic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Varieties over finite and local fields, \(p\)-adic cohomology, crystalline cohomology Some notes on Tsuzuki's full faithfulness conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors consider an interpolation problem for linear systems \(L=L_{n,d}(m_1,\dots,m_s)\) of hypersurfaces of degree \(d\) in \(\mathbb P^n\), passing through \(s\) points \(P_1,\dots,P_s\) with multiplicities (at least) \(m_i\) at \(P_i\). Such systems have an expected dimension, and the interpolation problem mainly consists of determining the value of \(\dim(L)\), and finding conditions which imply that the value is as expected. Even in the case when the \(P_i\)'s are general points in \(\mathbb P^n\) a complete answer is unknown, for \(n\geq 2\), though there are conjectures and partial results, expecially for the case of points in the plane. When the \(P_i\)'s are located in special position the interpolation problem is widely open. The authors consider the case of points contained in a rational normal curve \(C\). The main result provides the value of the dimension of \(L\) for any \(n,d,m_1,\dots,m_s\) and for a general choice of \(s\) points in \(C\). The formula is expressed recursively in terms of the numerical data. In the case where all multiplicities \(m_i\) are equal to \(2\), the formula determines an Alexander-Hirschowitz type theorem for points on a rational normal curve, which can be applied to the study of some special decompositions of symmetric tensors. The authors also point out that the blow-up \(X_s^n\) of \(\mathbb P^n\) at \(s\) points in the curve \(C\) is a Mori dream space, and the formula for \(\dim(L)\) is connected with the description of the nef cone and the Mori chamber decomposition of the effective cone of \(X_s^n\). polynomial interpolation; linear systems; fat points; rational normal curves; special effect varieties Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry On linear systems with multiple points on a rational normal 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 Consider a system of differential equations defined by a quadratic vector field on \(\mathbb{R}^n\)
\[\dot{x} = f(x) := Q(x) + Bx + c, \]
where \(Q\) is an \(\mathbb{R}^n\)-valued quadratic form, \(B\in \mathbb{R}^{n\times n}\) and \(c\in \mathbb{R}^n\). The Kahan discretization method is defined by the numerical integration method \(x \mapsto \tilde{x}\) with step size \(\epsilon\) given by \[\frac{1}{2\epsilon} \left(\tilde{x} - x \right) = Q(x, \tilde{x}) + \frac{1}{2} B(x+\tilde{x}) + c. \]
Here \(Q(x, \tilde{x})\) is the bilinear form associated to \(Q(x)\). It was shown in [\textit{E. Celledoni} et al., J. Phys. A, Math. Theor. 45, No. 45, Article ID 025201, 12 p. (2012; Zbl 1278.65192)] that this method is equivalent to a Runge-Kutta method. However, the focus of the present article is on the so-called Kahan map defined by
\[ \tilde{x} = \Phi_\epsilon(x) = x + 2 \epsilon(\mathrm{Id} - \epsilon \nabla f)^{-1} f(x). \] Here \(\nabla f\) is the Jacobian of \(f\). The Kahan map is birational (when extended to \(\mathbb{C}^n\)) as can be seen from \(\Phi_\epsilon^{-1}(x) = \Phi_{-\epsilon}(x)\).
The main result of this article is a statement about the integrability of the Kahan map for certain quadratic vector fields, i.e., for a special form of \(f\). These are defined by
\[ \dot{x} = \ell_1^{1-\gamma_1}(x) \ell_2^{1-\gamma_2}(x) \ell_3^{1-\gamma_3}(x) J \nabla H(x), \quad x = (x_1, x_2) \in \mathbb{R}^2, \] where \(\ell_i(x_1, x_2) = a_i x_1 + b_i x_2\), \(J\) is the standard symplectic form on \(\mathbb{R}^2\) and \(\gamma_1, \gamma_2, \gamma_3 \in \mathbb{R} \setminus \{0\}\).
These quadratic vector fields (i.e., differential equations) are generalizations of the two-dimensional reduced Nahm systems, see [\textit{N. J. Hitchin} et al., Nonlinearity 8, No. 5, 662--692 (1995; Zbl 0846.53016)].
The integrability of the Kahan maps \(\Phi\colon \mathbb{C}^2 \to \mathbb{C}^2\) has only been considered for a few cases of \((\gamma_1,\gamma_2,\gamma_3)\) before [\textit{E. Celledoni} et al., Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 471, No. 2184, Article ID 20150390, 10 p. (2015; Zbl 1372.65201); \textit{M. Petrera} et al., Exp. Math. 18, No. 2, 223--247 (2009; Zbl 1178.14007); \textit{M. Petrera} and \textit{R. Zander}, J. Phys. A, Math. Theor. 50, No. 20, Article ID 205203, 13 p. (2017; Zbl 1377.37086)]. For example, for \((\gamma_1, \gamma_2, \gamma_3) = (1, 1, 1)\), one obtains the canonical Hamiltonian system on \(\mathbb{R}^2\) with homogeneous cubic Hamiltonian.
In this article, the author proves a strong statement about the non-integrabilty of the Kahan maps depending on \((\gamma_1, \gamma_2, \gamma_3)\). To do so, the Kahan maps are considered as birational maps \(\Phi\colon \mathbb{CP}^2 \to \mathbb{CP}^2\) and their singularity structures are studied in a precise way (based, e.g., on [\textit{J. Diller} and \textit{C. Favre}, Am. J. Math. 123, No. 6, 1135--1169 (2001; Zbl 1112.37308)]). In this way, the author proves whether the sequence \(d(m)\) of the degrees of \(\Phi^m\) grows exponentially, quadratically, linearly or is bounded (these are all possible cases). In the first case, \(\Phi\) is nonintegrable. Some of the integrable cases are considered in detail in the later sections of the article, thus exemplifying the general methods. birational map; integrable map; elliptic curve; elliptic pencil Completely integrable discrete dynamical systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria), Relationships between algebraic curves and integrable systems, Rational and birational maps, Integrable difference and lattice equations; integrability tests On the singularity structure of Kahan discretizations of a class of quadratic vector 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 Let \(\operatorname{SH}(X)\) denote the motivic stable homotopy category of a given scheme \(X\). We write \(\operatorname{SH}(X)^{\mathrm{cell}} \subset \operatorname{SH}(X)\) for the full subcategory of cellular motivic spectra; that is, the localizing subcategory generated by the bigraded spheres \(S^{p,q}\) for all integers \(p,q\in \mathbb{Z}\). In the present article, the authors establish a general method for exhibiting squares as below and provide a criterion for cartesianess in terms of étale and real étale cohomology. For instance, for every \(\mathcal{E}\in \operatorname{SH}(\mathbb{Z}[1/2])^{\mathrm{cell}}\), they prove that there is a pullback square relating \(\mathcal{E}(\mathbb{Z}[1/2])^{\wedge}_2\) with \(\mathcal{E}(\mathbb{R})^{\wedge}_2\) and \(\mathcal{E}(\mathbb{F}_3)^{\wedge}_2\) over \(\mathcal{E}(\mathbb{C})^{\wedge}_2\). Here, for \(X \in \operatorname{Sch}_{\mathbb{Z}[1/2]}\), we denote by \(\mathcal{E}(X)\) the spectrum of maps from \(\mathbf{1}_X \to p^*\mathcal{E}\) in \(\operatorname{SH}(X)\), where \(\mathbf{1}_X\) denotes the unit object and \(p:X \to \mathbb{Z}[1/2]\) the structure map. \par In particular, this is true when \(\mathcal{E}\) is the motivic spectra representing algebraic \(K\)-theory, \(\textbf{KGL}\), hermitian \(K\)-theory, \(\textbf{KO}\), Witt-theory, \(\textbf{KW}\), motivic cohomology or higher Chow groups, \(\textbf{HZ}\), and algebraic cobordism, \(\textbf{MGL}\).
An application of this result is that one can identify, up to odd-primary torsion, the endomorphism ring of \(\textbf{1}_{\mathbb{Z}[1/2]}\) with the Grothendieck-Witt ring of quadratic forms of the Dedekind domain \(\mathbb{Z}[1/2]\). More precisely, the unit map \(\textbf{1}_{\mathbb{Z}[1/2]}\to \textbf{KO}_{\mathbb{Z}[1/2]}\) induces an isomorphism \begin{center} \(\pi_{0,0}(\textbf{1}_{\mathbb{Z}[1/2]})\otimes \mathbb{Z}_{(2)} \simeq \operatorname{GW}(\mathbb{Z}[1/2])\otimes \mathbb{Z}_{(2)}\). \end{center}
This extends Morel's fundamental computation of \(\pi_{0,0}(\textbf{1})\) over fields to deeper base schemes.
In Section 2, the authors axiomatize some well-known facts about nilpotent completions in presentably symmetric monoidal stable \(\infty\)-categories with a \(t\)-structure. The proofs are straightforward generalizations of [\textit{L. Mantovani}, ``Localizations and completions in motivic homotopy theory'', Preprint, \url{arXiv:1810.04134}].
In Section 3, the authors prove `rigidity' results to the effect that if \(X\) is a suitable Henselian local scheme with closed point \(x\) and \(E\) is an appropriate motivic spectrum, then \(E(X) \simeq E(x)\). Many instances have been proved before when \(X\) is essentially smooth over a field; see, for example, the work of J. Hornbostel, S. Yagunov, A. Ananyevskiy and A. Druzhinin.
Section 4 contains the main results of the article. The authors exhibit pullback squares describing \(\operatorname{SH}(\mathcal{O}_F)[1/l]_l^{\wedge {\mathrm{cell}}}\) for suitable number fields \(F\) and prime numbers \(l\) in terms of \(\operatorname{SH}(k)_l^{\wedge {\mathrm{cell}}}\) for fields of the form \(k = \mathbb{C}, \mathbb{R}, \mathbb{F}_q\).
In Section 5, the authors apply the results in Section 4 to show slice completeness and compute the endomorphism ring of the motivic sphere over regular number rings. Their completeness result for Voevodsky's slice filtration is motivated by applications such as motivic generalizations of Thomason's étale descent theorem for algebraic \(K\)-theory, the solution of Milnor's conjecture on quadratic forms, or computations of universal motivic invariants and of hermitian \(K\)-groups. motivic homotopy theory; l-regular number fields; quadratic forms Homotopy theory and fundamental groups in algebraic geometry, Motivic cohomology; motivic homotopy theory, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Stable homotopy theory, spectra Topological models for stable motivic invariants of regular number 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 Ces notes contiennent les idées fondamentales des applications des systèmes différentiels (\({\mathcal D}\)-modules) à l'étude des singularités. Il s'agit d'un exposé de résultats sans démonstrations. On étudie les notions de fonction analytique multiforme et de fonction de la classe de Nilsson, tout d'abord dans le cas d'un diviseur à croissements normaux, puis dans le cas général par résolution. Après on étudie les équations différentielles ordinaires d'une variable complexe et les points singuliers réguliers, en donnant l'équivalence entre la version classique de Fuchs et la version cohomologique de Malgrange. En dimension supérieure on énonce le théorème de régularité de Nilsson et on donne quelques applications au cas des hypersurfaces à singularité isolée. Ceci se fait en rappelant la connexion de Gauss-Manin, le théorème de la monodromie et la définition des exposants de la singularité. Ceci laisse la porte ouverte à développements ultérieurs en rapport avec la théorie des \({\mathcal D}\)-modules (b-fonction) et la théorie de Hodge-Deligne.
Ces notes sont issues des exposés faits par l'A. en 1980, et donc elles manquent forcément des nombreux développements récents. Le lecteur intéressé peut consulter p.ex. le rapport ``Introduction to linear differential systems'', part l'A. et \textit{Z. Mebkhout} [Proc. Symp. Pure Math. 40, Part 2, 31-63 (1983; Zbl 0521.14006)] et l'article ``Le théorème de comparaison entre cohomologies de de Rham d'une variété algébrique complexe et le théorème d'existence de Riemann'', par \textit{Z. Mebkhout} [Pub. Math., Inst. Hautes Etud. Sci. 69, 47-89 (1989)]. multiform function; monodromy; regularity; Gauss-Manin Local complex singularities, Sheaves of differential operators and their modules, \(D\)-modules, Complex singularities, Singularities in algebraic geometry Monodromie et géométrie. (Monodromy and geometry) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors consider spectral curves \((C, B, x, y)\) that locally resemble the curve \(x y^2 = 1\) near some zeros of the differential form \(d x\). The local behavior of the invariants \(\omega_n^g\) are in general not determined by intersection numbers, as is the case for the Airy curve [\textit{B. Eynard} and \textit{N. Orantin}, J. Phys. A, Math. Theor. 42, No. 29, Article ID 293001, 117 p. (2009; Zbl 1177.82049)]. Instead, the authors relate this local behavior to the enumeration of dessins d'enfant.
Given a tuple of integers, \(\mu = (\mu_1, \ldots, \mu_n)\), one can count the number \(B_{g, n} (\mu)\) of dessins of genus \(g\) with ramification behavior \(\mu\) at infinity, weighted by their automorphism group. The authors show that when \(g\) and \(n\) are fixed, the generating function \(F_{g, n} = \sum_{\mu} B_{g, n} (\mu) \prod_i x_i^{\mu_i}\) is a rational function in \(z_i\) when writing \(x_i = z_i + z_i^{-1} + 2\). Together with \(x\), the derivative \(y\) of \(F_{0,1}\) with respect to \(x\) defines the spectral curve \(x y^2 - x y + 1\). The functions \(F_{g, n}\) then constitute the analytic expansion of the invariants \(\omega_n^g\) of this spectral curve at its point at infinity. This yields structure theorems and explicit formulae for the numbers \(B_{g, n}\).
By studying the asymptotic behavior of the functions \(F_{g, n}\) near the pole \((z_1, \ldots, z_n) = (-1, \ldots, -1)\), the authors then determine the one-point invariants \(\omega_1^g\) of the spectral curve \(x y^2 = 1\) in terms of a three-term recursion for the numbers \(B_{g, 1} (n)\) of dessins d'enfant with one face. This new recursion has analogues in [\textit{J. Harer} and \textit{D. Zagier}, Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)]. topological recursion; dessins d'enfant; spectral curves Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Dessins d'enfants theory Topological recursion for irregular spectral 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 This paper is one of a series of works by the authors [Proc. Natl. Acad. Sci. USA 108, No. 22, 8984--8989 (2011; Zbl 1256.37037), Invent. Math. 198, No. 3, 637--699 (2014; Zbl 1306.35109)]. For a given point \(A\) in the real Grassmannian, there is a soliton solution \(u_A(x, y, t)\) to the KP equation parameterized by \(A\). The contour plot of such a solution provides a tropical approximation to the solution by taking a large-scale limit of the variables \(x\), \(y\), and \(t\) for a fixed time \(t\). The tropical approximation of the contour plot is given as a graph which consists of piecewise linear segments and half lines.
In this paper, the authors show that the graphs are represented by several decompositions of the Grassmannian. {\parindent=0.5cm\begin{itemize}\item[1)] The positroid stratification of the real Grassmannian characterizes the unbounded line-solitons in the contour plots at \(y \gg 0\) and \(y \ll 0\). \item[2)] The Deodhar decomposition of the Grassmannian -- refinement of the positroid stratification -- characterizes the graphs at \(t\ll 0\). \item[3)] By indexing the components of the Deodhar decomposition of the Grassmannian in terms of Young tableaux which the authors call Go-diagrams, the Go-diagrams directly correspond to the graphs of \(t\ll0\) via generalized plabic graphs.
\end{itemize}} Using these results, the authors show that a soliton solution \(u_A(x, y, t)\) is regular for all times \(t\) if and only if \(A\) comes from the totally non-negative part of the Grassmannian. Deodhar decomposition; Grassmannian; KP equation; soliton solution Kodama, Y.; Williams, L., The deodhar decomposition of the Grassmannian and the regularity of KP solitons, Adv. Math., 244, 979-1032, (2013) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Relationships between algebraic curves and integrable systems, Grassmannians, Schubert varieties, flag manifolds The Deodhar decomposition of the Grassmannian and the regularity of KP solitons | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 presents a construction of the spectral curve and recursion kernel for the Eynard-Orantin recursion formula by taking Laplace transform of the disc and annulus amplitudes of the A-model. The construction is examined in four examples: dessins d'enfants, \(\psi\)-class intersection numbers of pointed stable curves, single Hurwitz numbers, and stationary Gromov-Witten invariants of \(\mathbb{P}^1\).
For instance, consider the weighted count \(D_{g,n}(\mu_1,\dots,\mu_n)\) of clean Belyi morphisms of smooth connected algebraic curves of genus \(g\) with \(n\) poles of order \((\mu_1,\dots,\mu_n)\). Then define
\[
W^D_{g,n} (t_1,\dots,t_n) = d_1 \dots d_n \sum_{\mu_1,\dots,\mu_n>0} D_{g,n}(\mu_1,\dots,\mu_n) e^{-(\mu_1 w_1 + \dots + \mu_n w_n)}
\]
where
\[
e^{w_j} = \frac{t_j+1}{t_j-1} + \frac{t_j-1}{t_j+1}.
\]
This is basically the Laplace transform. It is proved that \(W^D_{g,n} (t_1,\dots,t_n)\) satisfy the Eynard-Orantin recursion formula (Theorem 2). The spectral curve
\[
x = z + \frac{1}{z}
\]
\[
y = -z
\]
can be found by the Laplace transform of \(D_{0,1}\). See Section 3.
The construction in this paper is very interesting. There should be an intimate relation between this construction and SYZ with Fourier transform, which deserves further investigation. Eynard-Orantin recursion; spectral curve; dessins d'enfants; mirror symmetry; Hurwitz numbers; Gromov-Witten invariants O. Dumitrescu, M. Mulase, B. Safnuk, and A. Sorkin, ''The spectral curve of the Eynard-Orantin recursion via the Laplace transform,'' in: \textit{Algebraic and Geometric Aspects of Integrable Systems and Random Matrices}, Amer. Math. Soc., Providence, Rhode Island (2013), pp. 263-315. Mirror symmetry (algebro-geometric aspects), Dessins d'enfants theory, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The spectral curve of the Eynard-Orantin recursion via the Laplace transform | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 finiteness of non-symmetric and symmetric cohomologies associated with Jackson integrals of type \(BC_n\) is studied. The explicit bases of the cohomologies are given. These bases determine a parameter-dependent Jackson integral, and it is shown that they satisfy holonomic systems of linear \(q\)-difference equations with respect to the parameters. The main results of the paper are presented in four theorems. The first theorem is: Let \(Q=\{(\lambda_1,\dotsc,\lambda_n)\in\mathbb Z^n:-s-1-(n-1)l\leq \lambda_i\leq s+(n-1)l,\,i=1,\dotsc,n\}\) and the parameters \(a_1,a_2,\dotsc,a_m\) and \(t_1,t_1,\dotsc,t_l\) be generic. Then \(H^n(X,\Phi,\nabla_q)\) -- the non-symmetric rational de Rham cohomologies associated with the Jackson integrals of type \(BC_n\) (\(\nabla_q\) is the \(n\)-dimensional covariant \(q\)-difference operator) has dimension \(\tilde\kappa:=\{m+2(n-1)l\}^n\) and is spanned by the basis \(\{z^\lambda:\lambda\in Q\}\). The third theorem is: there exist invertible matrices \(Y_{a_k}\), \(Y_{t_j}\) whose entries \(\eta_{\lambda,\nu}^{(a_k)}\), \(\eta_{\lambda, \nu}^{(t_j)}\) are rational functions of \(a_1,a_2,\dotsc,a_m\) and \(t_1,t_1,\dotsc,t_l\) respectively, such that \(T_{a_k}\langle z^\lambda,\xi\rangle=\sum_{\nu\in Q}\eta_{\lambda,\nu}^{(a_k)} \langle z^\nu,\xi\rangle\), \(T_{t_j}\langle z^\lambda,\xi\rangle= \sum_{\nu\in Q}\eta_{\lambda,\nu}^{(t_j)} \langle z^\nu,\xi \rangle\), where \(T_u\) is the shift operator on a parameter and \(\lambda\) runs over the set \(Q\). cohomology; de Rham cohomology; symmetric cohomology; Jackson integrals; \(q\)-difference; basis Aomoto, K.; Ito, M., On the structure of Jackson integrals of \textit{BC}\_{}\{\(n\)\} type, Tokyo J. Math., 31, 449-477, (2008) \(q\)-gamma functions, \(q\)-beta functions and integrals, de Rham cohomology and algebraic geometry Structure of Jackson integrals of \(BC_n\) type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 of \(s\) points in \(\mathbb{P}^ d\) is called a Cayley-Bacharach scheme (CB-scheme), if every subset of \(s-1\) points has the same Hilbert function. We investigate the consequences of this ``weak uniformity.'' The main result characterizes CB-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a CB-scheme \(X\) has to satisfy growth conditions which are only slightly weaker than the ones gives by Harris and Eisenbud for points with the uniform position property [cf. \textit{J. Harris} (with the collaboration of \textit{D. Eisenbud}), ``Curves in projective space'', Sém. Math. Supér. 85 (1982; Zbl 0511.14014)].
We also characterize CB-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a CB-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains. CB-scheme; Cayley-Bacharach scheme; Hibert function; hyperplane sections; arithmetically Cohen-Macaulay curves A. Geramita, M. Kreuzer, and L. Robbiano, \textit{Cayley-Bacharach schemes and their canonical modules}, Trans. Amer. Math. Soc., 399 (1993), pp. 163--189, . Schemes and morphisms, Plane and space curves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Relevant commutative algebra, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special varieties, Computational aspects in algebraic geometry Cayley--Bacharach schemes and their canonical modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R=K[X_1,\dots,X_N]\), \(K\) being a field of characteristic zero. \(F\in R-\{0\}\) is a quasihomogeneous polynomial (with weights \(\lambda_i\)) if it satisfies the generalised Euler equation
\[
\sum \lambda_i X_i\frac{\partial F}{\partial X_i}=nF.
\]
Let \(A=\text{Spec }R/(F)\) be a reduced quasihomogeneous affine hypersurface with an isolated singularity at the origin. It is known that (for reduced hypersurfaces \(A\) with isolated singularities) the Kähler differentials \(\Omega^i_{A/K}\) are torsionfree for \(i\neq N-1\), \(N\) and \(\Omega^N_{A/K}\simeq A/I\), \(I=(\partial f/\partial x_1,\dots,\partial f/\partial x_N)\), \(\partial f/\partial x_i=\partial F/\partial X_i+(F)\). The main result of this paper is:
Theorem. The torsion submodule \(T(\Omega^{N-1}_{A/K})\) of \(\Omega^{N-1}_{A/K}\) is a cyclic \(A\)-module generated by
\[
\omega_0=\sum(-1)^{i+1}\bigl(\frac{\lambda_i} {n}\bigr)x_idx_1\wedge \dots\wedge \widehat{dx_i}\wedge\dots\wedge dx_N.
\]
Thus \(T(\Omega^{N-1}_{A/K})\approx \Omega^N_{A/K}\) as \(A\)-modules. -- This result is applied to deduce that for singularities corresponding to Dynkin diagrams with \(k\) vertices of type \(A_k\), \(D_k\), \(E_6\), \(E_7\) or \(E_8\) one has \(\dim_K T(\Omega^{N-1}_{A/K})=k\). Kähler differentials; torsion submodule; singularities DOI: 10.1216/rmjm/1181072113 Modules of differentials, Hypersurfaces and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Torsion of differentials of affine quasi-homogeneous 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 Let \(p_1,\dots,p_n\) be general points in the projective plane \(\mathbb P^2\), with corresponding homogeneous ideals \(P_1,\dots,P_n\). An ideal of the form \(I({\mathbf m},n) = P_1^{m_1} \cap \dots \cap P_n^{m_n}\) (where \(\mathbf m = (m_1,\dots,m_n)\) is an \(n\)-tuple of non-negative integers) defines a fat point subscheme \(Z \subset \mathbb P^2\). If \(m_1 = \dots = m_n = m\) (say), we say that \textbf{m} (or \(Z\)) is uniform and just write \(I(m,n)\). There are several conjectures about the Hilbert function and minimal free resolution of the ideal \(I(\mathbf m,n)\), and the answers are known in some cases (mostly uniform cases) and counterexamples are known in a few cases. However, all known failures involve \(n < 9\). The conjectures all involve some sort of maximal rank property.
In this paper the authors introduce the notion of quasiuniformity, saying that \(\mathbf m\) is quasiuniform if \({\mathbf m} = (m_1,m_2,\dots,m_n)\) with \(n \geq 9\) and \(m_1 = \dots = m_9 \geq m_{10} \geq \dots \geq m_n \geq 0\). They give conjectures for the Hilbert function and minimal free resolution of quasiuniform fat point schemes. Since quasiuniformity is an extension of the notion of uniformity, they note that their conjectures contain as special cases the existing conjectures (or in a few cases, theorems) for the uniform case. They also prove a number of solid results that give substantial evidence for their conjectures, especially for the uniform case. In particular, they prove the conjectures for infinitely many \(m\) for each of infinitely many \(n\), and for infinitely many \(n\) for every \(m>2\). They also show that in many cases the Hilbert function conjecture implies the resolution conjecture. As a by-product of their work, they get a strong bound on the regularity of \(I(m,n)\). ideal generation conjecture; symbolic powers; minimal free resolution; fat points; maximal rank; Hilbert function; regularity; quasiuniformity Harbourne, B.; Holay, S.; Fitchett, S., Resolutions of ideals of quasiuniform fat point subschemes of \(\mathbb P^2,\), Trans. Amer. Math. Soc., 355, 2, 593-608, (2003) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Multiplicity theory and related topics Resolutions of ideals of quasiuniform fat point subschemes of \(\mathbb{P}^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A continuous map \(\mathbb R^m \to \mathbb R^N\) or \(\mathbb C^m \to \mathbb C^N\) is called \(k\)-regular if the images of any \(k\) points are linearly independent. Given integers \(m\) and \(k\) a problem going back to Chebyshev and Borsuk is to determine the minimal value of \(N\) for which such maps exist. The methods of algebraic topology provide lower bounds for \(N\), but there are very few results on the existence of such maps for particular values \(m\) and \(k\). Using methods of algebraic geometry we construct \(k\)-regular maps. We relate the upper bounds on \(N\) with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for \(k \leq 9\), and we provide explicit examples for \(k \leq 5\). We also provide upper bounds for arbitrary \(m\) and \(k\). \(k\)-regular embeddings; secants; punctual Hilbert scheme; finite Gorenstein schemes Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces, Immersions in differential topology, Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Constructions of \(k\)-regular maps using finite local 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 The motivation of the paper comes from the Whitney problems for differentiable functions. It considers a generalization of differential calculus to more general settings than Euclidean spaces and manifolds. It focuses on (possibly fractal) subsets of \(\mathbb{R}^n\) (and a corresponding generalization of manifolds). More precisely, the studied sets are called \textit{weak \(k\)-Markov} for some \(k\in \mathbb{N}_0\). Before talking about weak Markov sets, we need to tell when a point is weak Markov; we let \(\mathcal{P}_{k,n}\) be the space of real polynomials on \(\mathbb{R}^n\) of degree \(k\) and \(Q_r(x)\subset \mathbb{R}^n\) be the closed cube centered at \(x\) of side length \(2r\).
Given a subset \(S\subset \mathbb{R}\), a point \(x\in S\) is said to be \textit{weak \(k\)-Markov} if
\[
\varliminf_{r\to 0} \left\{\sup_{p\in \mathcal{P}_{k,n}\setminus\{0\}}\left(\frac{\sup_{Q_r(x)}\left|p\right|}{\sup_{S\cap Q_r(x)}\left|p\right|}\right)\right\}<\infty.
\]
A closed set \(S\subset \mathbb{R}\) is said to be \textit{k-Markov} if it contains a dense subset of weak \(k\)-Markov points.
The term \textit{weak Markov set} is a reference to \textit{Markov set} (see [\textit{A. Jonsson} and \textit{H. Wallin}, Function spaces on subsets of \(\text{R}^ n\). Math. Rep. Ser. 2, No. 1 (1984; Zbl 0875.46003)]).
After introducing weak Markov sets, a section is dedicated to some of their properties. Then traces of differentiable functions to weak Markov sets are studied. Next, the focus shifts to extension problems. For example, a Hartogs-type problem is solved and also certain harmonic functions constructed.
The next section leaves the Euclidean setting by introducing \textit{weak Markov structures}. They are reminiscent of manifolds. Some properties are stated and a calculus of differential forms established. Generalizations of some classical theorems (the Poincaré \(d\)-lemma for differential forms on \(C^\infty\) manifolds, the de Rham theorem, and the Künneth formula) are obtained.
The remaining sections deal with the proofs of the statements in the first part of the paper. weak Markov set; Whitney problems; trace; extension; \(C^k\) function; de Rham cohomology Continuity and differentiation questions, Fractals, Abstract manifolds and fiber bundles (category-theoretic aspects), de Rham cohomology and algebraic geometry Differential calculus on topological spaces with weak Markov structure. 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 Let \(F\) be an analytic mapping defined on an open neighbourhood \(D\) of \(0 \in {\mathbb R}^n\),
\[
F:(D,0) \rightarrow ({\mathbb R}^m,0),\quad n>m,
\]
such that the origin is an isolated singular point of \(F^{-1}(0)\). For any regular value \(z \in {\mathbb R}^m\) which is close enough to the origin and a sufficiently small ball \(B_r=B(0,r)\), the fibre \(W_z=F^{-1}(z) \cap B_r\) is called the Milnor fibre. In complex case the topological type of the Milnor fiber does not depend on \(z\). In the real case this is no longer true. The author considers an analytic function \(g\) defined on \(B_r\) such that its restriction to the generic Milnor fibre is Morse. He studies the properties of the global index
\[
\lambda (g, W_z) = \sum (-1)^{\lambda(p)},
\]
where \(\lambda(p)\) is the Morse index of the critical point \(p\) of \(g_ {| W_z}\) and the sum extends over all such critical points. The main result of the paper is the proof that under some general assumptions there exist a matrix \(\Theta\) with analytic entries and analytic functions \(\theta_i\) such that
\[
\lambda (g, W_z)= \text{signature}(\Theta(z))= \sum \text{sgn} : \theta_i (z).
\]
Milnor fibration; singularities of real-analytic mappings; Morse functions Szafraniec, Z., Topological invariants of real Milnor fibres, Manuscripta Math, 110, 2, 159-169, (2003) Milnor fibration; relations with knot theory, Real-analytic and semi-analytic sets, Germs of analytic sets, local parametrization, Real-analytic and Nash manifolds, Real-analytic sets, complex Nash functions Topological invariants of real Milnor fibres | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 investigates fixed points for the action of \(U\) on the full flag manifold \(\text{ SL}(n,{\mathbb C})/B\), where \(U\) is the unipotent radical of a regular Jordan canonical matrix in \(\text{ SL}(n,{\mathbb C})\), and \(B\) consists of upper triangular matrices. Expressing the flag manifold as a union of Bruhat cells, the author produces conditions under which canonical matrices within a given Bruhat cell will be fixed by \(U\), and upon examining the conditions deduces that the fixed point set for \(U\) is actually the fixed point set for a single unipotent element \(\tilde u\in U\) (denoted \({\mathcal B}_{\tilde u}\)). Using existing knowledge of fixed point sets of unipotent elements (e.g., they are connected, and irreducible components all have the same dimension), the author shows that the intersection of \({\mathcal B}_{\tilde u}\) with a given Bruhat cell is connected and that each of its irreducible components is diffeomorphic to \({\mathbb C}^q\) for some \(q\). upper triangular matrices; fixed point sets of unipotent elements; components Shipman, B. A.: Fixed points of unipotent group actions in Bruhat cells of a flag manifold. JP J. Algebra number theory appl. 3, No. 2, 301-313 (2003) Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Fixed points of unipotent group actions in Bruhat cells of a flag manifold. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Take a (complex) analytic hypersurface germ \(f^{-1}(0)\subset(\mathbb{C}^{n+1},o)\), with an isolated singularity. The spectrum of the singularity is a strong numerical (topological) invariant. It is often encoded via the spectral polynomial, \(\chi_f(t):=\frac{1}{\mu}\sum T^{\alpha_i}\), with \(\alpha_1,\dots,\alpha_\mu\in \mathbb{Q}\cap (0,n+1)\). Thinking of this as a distribution of numbers on the interval \((0,n+1)\), K. Saito studied the asymptotic distribution of the spectrum. He observed that for many germs the asymptotic distribution is determined by the dimension \((n+1)\) only, and does not depend on \(f\). In this case the continuous (asymptotic) distribution is denoted by \(\mathcal{F}(N_{n+1})\).
The main result of the article is:
\textbf{Theorem 1.} Given a Newton diagram \(\Gamma\) take its scaled version, \(\omega\Gamma\), by the scaling factor \(\omega\). Take any Newton-non-degenerate function germ \(f\), whose Newton diagram is \(\Gamma\). Then \(lim_{\omega\to \infty}\chi_{f_\omega}=\mathcal{F}(N_{n+1})\).
Furthermore, K. Saito has defined the function \(\Phi_f:[0,1]\to\mathbb{R}\) as the difference of the continuous and the spectral distributions, \(r\to \int^r_0 N_{n+1}(s)-\frac{1}{\mu}\sum \delta(s-\alpha_i)ds\). A number \(0<r<\frac{n+1}{2}\) is called a dominating value if \(\Phi_f(r)>0\).
\textbf{Proposition 5.} Take a plane curve singularity, \(f^{-1}(0)\subset(\mathbb{C}^2,o)\), with one Puiseux pair \((p,q)\). Then:
(a) \(\Phi_f(\frac{1}{p}+\frac{1}{q})>0\) unless \(p=2\) and \(q\in \{3,5\}\). And \(lim_{p\to \infty}\Phi_f(\frac{1}{p}+\frac{1}{q})=0\).
(b) \(\Phi_f(1-\frac{1}{pq})<0\) with \(lim_{p\to \infty}\Phi_f(1-\frac{1}{pq})=0\).
\textbf{Theorem 6.} Take an irreducible plane curve singularity with value semigroup \(\{\beta_0,\beta_1,\dots\}\), such that \(\{\beta_0,\beta_1\}\) is not \(\{2,3\}\) or \(\{2,5\}\). Then \(\Phi_f(\frac{1}{\beta_0}+\frac{1}{\beta_1})>0\). In other words, \((\frac{1}{\beta_0}+\frac{1}{\beta_1})^2>\frac{2}{\mu}\). Moreover, \(lim_{n_g\to \infty}\Phi_f(\frac{1}{\beta_0}+\frac{1}{\beta_1})=0\). singularity spectrum; spectral distribution Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Mixed Hodge theory of singular varieties (complex-analytic aspects), Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type Limit spectral distribution for non-degenerate hypersurface 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 main result of this paper is the nonabelian generalization of proper base change theorem in étale cohomology, which works for étale sheaves valued in spaces that satisfy certain finiteness condition named truncated coherent étale sheaves. There are various application of this theorem, including a proof that profinite étale homotopy type commutes with finite product and symmetric power of proper algebraic spaces.
The contents in more detail:
In Section 1 the author states the main theorem and applications as well as the strategy of the proof.
In Section 2 the author gives a review of shapes and profinite completions in the infinity categorical context. Given an \(\infty\)-topos \(\mathcal{X}\), there is an essentially unique geometric morphism \(\pi_*:\mathcal{X}\longrightarrow\mathcal{S}\), the composition \(\pi_*\pi^*\) is a pro-space \(\mathrm{Sh}(\mathcal{X})\) named the shape of \(\mathcal{X}\). This is a generalization of the étale homotopy type to general topoi.
In Section 3 the author studies limits of \(\infty\)-topoi and proves that being a truncated pullback guarantees that the global section in the limit topos is equivalent to the colimit of global sections in each topos.
In Section 4 the author proves the proper base change theorem. The proof combines the standard technology in the abelian setting with the continuity properties of truncated objects.
In Section 5 and 6 the author proves that profinite shapes functor commute with finite products and symmetric powers for proper schemes by the main theorem. The basic idea is that the main theorem allows us to directly identify \(\mathrm{Sh}(X)\circ\mathrm{Sh}(Y)\) with \(\mathrm{Sh}(X\times Y)\) upon profinite completion, while \(\mathrm{Sh}(X)\circ\mathrm{Sh}(Y)\) is equivalent \( \mathrm{Sh}(X)\times \mathrm{Sh}(Y)\) on finite spaces by straightforward computation. proper base change; étale homotopy; infinity-categories; étale topologies; shape theory Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Homotopy theory and fundamental groups in algebraic geometry, \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories, Étale and other Grothendieck topologies and (co)homologies, Shape theory Proper base change for étale sheaves of 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 authors give global effective versions of the Briançon-Skoda-Huneke theorem:
Theorem. Let \(V\) be a germ of a reduced analytic set of pure dimension \(n\) at the origin in \(\mathbb{C}^N\). There is a number \(\mu_0\geq 0\) such that if \(a_1\), \dots, \(a_m\) are germs of holomorphic functions at \(0\), \(l\geq 1\), and \(|\phi| \leq C |a|^{\mu+\mu_0+l-1}\) in a neighborhood of \(0\) in \(V\), where \(C\) is a positive constant, \(\mu=\max\{m,n\}\) and \(|a|^2=|a_1|^2+\dots+|a_m|^2\), then \(\phi\) belongs to the ideal \((a_1, \dots, a_m)^l \subset \mathcal{O}_0\).
If \(V\) is smooth, then one can take \(\mu_0=0\) and this is the classical theorem of \textit{J. Briancon} and \textit{H. Skoda} [C. R. Acad. Sci., Paris, Sér. A 278, 949--951 (1974; Zbl 0307.32007)]. The general case is due to \textit{C. Huneke} [Invent. Math. 107, No. 1, 203--223 (1992; Zbl 0756.13001)], who proved it by purely algebraic methods. An analytic proof was given by \textit{M. Andersson} et al. [Ann. Inst. Fourier 60, No. 2, 417--432 (2010; Zbl 1200.32007)].
Effective versions of the Briançon-Skoda-Huneke theorem can be used to deduce effective versions of the Nullstellensatz. Let \(F_1, \dots, F_m\) and \(\phi\) be polynomials in \(\mathbb{C}[z_1, \dots, z_N]\) and assume that \(\phi\) vanishes on the common zero set of the \(F_j\). Then Hilbert's Nullstellensatz asserts that there are polynomials \(G_1, \dots, G_m\) such that
\[
F_1 G_1 + \dots + F_m G_m =\phi^\nu
\]
for some large enough integer \(\nu\). It is an interesting and intensively studied problem to bound the complexity of the solutions \((\nu, G_1, . . . , G_m)\), e.g., in the sense of finding bounds for \(\nu\) and the degrees of the \(G_j\) in terms of the degrees of the \(F_j\). We should mention some important contributions of \textit{J. Kollár} [J. Am. Math. Soc. 1, No. 4, 963--975 (1988; Zbl 0682.14001)], \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 137, No. 2, 427--448 (1999; Zbl 0944.14003)], and \textit{Z. Jelonek} [Invent. Math. 162, No. 1, 1--17 (2005; Zbl 1087.14003)]. An account of the state of the art can be found in the introduction of the paper under review, which contains new important contributions in the area.
The first main theorem of Andersson and Wulcan is as follows:
Theorem A. Assume that \(V\) is a reduced algebraic subvariety of \(\mathbb{C}^N\) of pure dimension \(n\) and let \(X\) be its closure in \(\mathbb{P}^N\). {\parindent=6mm \begin{itemize} \item[(i)] There exists a number \(\nu_0\) such that if \(F_1, \dots, F_m\) are polynomials of degree \(\leq d\) and \(\phi\) is a polynomial and
\[
|\phi| / |F|^{\mu+\mu_0} \text{ is locally bounded on } V,
\]
then there are polynomials \(Q_1, \dots, Q_m\) such that
\[
F_1 Q_1 + \dots + F_mQ_m=\phi
\]
holds on \(V\) and
\[
\deg(F_jQ_j) \leq \max\big\{\deg\phi + (\mu+\mu_0) d^{c_\infty} \deg X, (d-1) \min\{m,n+1\}+\text{reg}\;X\big\}.
\]
\item [(ii)] If \(V\) is smooth, then there is a number \(\mu'\) such that if \(F_1, \dots, F_m\) are polynomials of degree \(\leq d\) and \(\phi\) is a polynomial and
\[
|\phi|/|F|^\mu \text{ is locally bounded on } V,
\]
then there are polynomials \(Q_1, \dots, Q_m\) such that
\[
F_1Q_1 + \dots F_mQ_m=\phi
\]
on \(V\) and
\[
\deg(F_jQ_j) \leq \max\big\{\deg\phi + \mu d^{c_\infty} \deg X+\mu', (d-1) \min\{m,n+1\}+\text{reg}\;X\big\}.
\]
If \(X\) is smooth, then one can take \(\mu'=0\).
\end{itemize}}
Here, \(\mu=\max\{m,n\}\), \(d^{c_\infty}\) is, in the sense of Fulton-MacPherson, the maximal codimension of the so-called distinguished varieties of the sheaf \(\mathcal{J}_f\) generated by the sections in \(\mathcal{O}(d)|_X\) corresponding to \(F_1, \dots, F_m\), and \(\text{reg }X\) is the Castelnuovo-Mumford regularity of \(X\subset \mathbb{P}^N\). One has \(0\leq c_\infty \leq \mu\), and let \(c_\infty=-\infty\) if there are no distinguished varieties.
If one applies Theorem A to Nullstellensatz data, i.e., \(F_j\) with no common zeros on \(V\) and \(\phi=1\), then one almost gets back the optimal effective Nullstellensatz of Jelonek [Zbl 1087.14003]. One can use Theorem A also to deduce versions of the effective Nullstellensatz due to \textit{F. S. Macaulay} [The algebraic theory of modular systems. Cambridge: University press (1916; JFM 46.0167.01)] and \textit{M. Hickel} [Ann. Inst. Fourier 51, No. 3, 707--744 (2001; Zbl 0991.13009)].
The second main theorem in the paper under review is a generalization to non-smooth varieties of the geometric effective Nullstellensatz of \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 137, No. 2, 427--448 (1999; Zbl 0944.14003)]. To explain that, consider a projective variety \(X\) and an ample line bundle \(L \rightarrow X\). Then there is a smallest number \(\nu_L\) such that \(H^q(X,L^{\otimes s})=0\) for \(q\geq 1\) and \(s\geq \nu_L\). When \(X\) is smooth, then \(\nu_L\) is less than or equal to the least number \(\sigma\) such that \(L^{\otimes \sigma}\otimes K_X^{-1}\) is strictly positive by the Kodaira vanishing theorem. In particular, if \(X=\mathbb{P}^n\), then \(\nu_{\mathcal{O}(1)}=-n\).
Theorem B. Let \(X\) be a reduced projective variety of pure dimension \(n\). There is a number \(\mu_0\), only depending on \(X\), such that the following holds:
Let \(f_1, \dots, f_m\) be global holomorphic sections of an ample Hermitian line bundle \(L\rightarrow X\), and let \(\phi\) be a section of \(L^{\otimes s}\), where
\[
s\geq \nu_L + \min\{m,n+1\}.
\]
If
\[
|\phi| \leq C |f|^{\mu+\mu_0},
\]
then there are holomorphic sections \(q_1, \dots, q_m\) of \(L^{\otimes (s-1)}\) such that
\[
f_1 q_1 + \dots + f_m q_m =\phi.
\]
If \(X\) is smooth, then one can choose \(\mu_0=0\), and we get back the effective Nullstellensatz of Ein-Lazarsfeld.
Let us give some remarks on the proofs of Theorem A and Theorem B. They rely on known geometric estimates and some new results on multivariable residue calculus. In the situation of Theorem B, when \(X\) is smooth, the key idea is to construct from the Koszul complex of the \(f_j\) a residue current \(R^f\) with support on their common zero set such that its annihilator determines locally the ideal \((f_1, \dots, f_m)\): if \(\phi\) is a section of \(L^{\otimes s}\) such that \(R^f\phi=0\) and \(L^{\otimes s}\) is positive enough, so that a certain sequence of \(\overline\partial\)-equations can be solved on \(X\), one obtains a holomorphic solution \((q_1, \dots, q_m)\) to the division problem. The solvability of the involved \(\overline\partial\)-equations is achieved by taking \(s\) large enough. This approach was basically introduced in [\textit{M. Andersson}, Ann. Inst. Fourier 56, No. 1, 101--119 (2006; Zbl 1092.32002)], and further developed in [\textit{M. Anderesson} and \textit{E. Götmark}, Math. Ann. 349, No. 2, 345--365 (2011; Zbl 1216.32002)] and [\textit{E. Wulcan}, Math. Ann. 350, No. 3, 661--682 (2011; Zbl 1234.32001); J. Commut. Algebra 2, No. 4, 567--580 (2010; Zbl 1237.14061)].
Let us mention in this context also the following important result of \textit{M. Andersson} and \textit{E. Wulcan} [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 6, 985--1007 (2007; Zbl 1143.32003)], where they generalized the construction of the Coleff-Herrera product from complete intersection ideals to arbitrary ideals of holomorphic functions on complex manifolds. From a hermitian resolution of such an ideal \(\mathcal{J}\), they constructed explicitly a vector-valued current \(R^\mathcal{J}\) such that its annihilator in the holomorphic functions equals \(\mathcal{J}\): \(\text{ann}_{\mathcal{O}} R^\mathcal{J}=\mathcal{J}\).
The really new achievement in the paper under review is the extension of the framework from [Zbl 1092.32002; Zbl 1216.32002; Zbl 1234.32001; Zbl 1237.14061] to the singular setting. For this, one has to replace \(R^f\) by variations of a product \(R^f\wedge \omega_X\), where \(\omega_X\) is a structure form of \(X\). The important notion of structure forms was introduced in [\textit{M. Andersson} and \textit{H. Samuelsson}, Invent. Math. 190, No. 2, 261--297 (2012; Zbl 1271.32009)]. Structure forms are basically the generators of the Barlet-Henkin-Passare holomorphic \(n\)-forms on \(X\) (\(\overline\partial\)-closed meromorphic \((n,0)\)-currents with support on \(X\)). For the proofs of the main theorems, it is then essential to control the singularities of \(\omega_X\). effective Briançon-Skoda theorem; effective Nullstellensatz; residue currents; division problems Andersson M., Wulcan E., \textit{Global effective versions of the Brian\c{}}\textit{con--Skoda--} \textit{Huneke theorem}, Invent. Math. 200 (2015), 607--651. Residues for several complex variables, Effectivity, complexity and computational aspects of algebraic geometry, Polynomials in number theory, Vanishing theorems in algebraic geometry, Germs of analytic sets, local parametrization, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Integration on analytic sets and spaces, currents Global effective versions of the Briançon-Skoda-Huneke 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 \(C\) be a hyperbolic Riemann surface and \(H \rightarrow C\) be a local system over \(C\), this is the same as a linear representation of the fundamental group of \(C.\) Pick a random point \(x\) on \(C\) and a random direction \(\theta\) at \(x\), and let \(\gamma_{T}\) be the hyperbolic geodesic of length \(T\) starting at \(x\), in direction \(\theta\), and of length \(T\). Connect the endpoint of \(\gamma_{T}\) to \(x\) to get a closed loop on \(C\), and thus a monodromy matrix \(M_{\gamma_{T}}\). As \(T\) grows, what can one say about the eigenvalues of \(M_{\gamma_{T}}?\) The answer is given by the Oseledets theorem: there exists numbers \(\lambda_{1}\geq\dots\geq\lambda_{n}\) such that the eigenvalues of \(M_{\gamma_{T}}\) grow like \(e^{\lambda_{i}T}\) as long as the starting point \(x\) and direction \(\theta\), are chosen randomly. The above numbers are called Lyapunov exponents.
Consider a family of K3 surfaces over a hyperbolic Riemann surface. Their second cohomology groups form a local system, and the author shows that its top Lyapunov exponent is a rational number. One proof uses the Kuga-Satake construction, which reduces the question to Hodge structures of weight 1. A second proof uses integration by parts. The case of maximal Lyapunov exponent corresponds to modular families coming from the Kummer construction. family of K3 surfaces over a hyperbolic Riemann surface; Lyapunov exponent; monodromy matrix Compact Riemann surfaces and uniformization, Period matrices, variation of Hodge structure; degenerations, Variation of Hodge structures (algebro-geometric aspects) Families of K3 surfaces and Lyapunov exponents | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate [\textit{E. Cattani} et al., Ann. Math. (2) 123, 457--535 (1986; Zbl 0617.14005)] between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric introduced by \textit{R. Hain} [in: Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press. 527--578 (2015; Zbl 1322.14049)], analogs of the norm estimates of [\textit{K. Kato} et al., J. Algebr. Geom. 17, No. 3, 401--479 (2008; Zbl 1144.14005)] and the asymptotics of the naive limit Hodge filtration considered in [\textit{M. Kerr} and the second author, Ann. Inst. Fourier 64, No. 6, 2659--2714 (2014; Zbl 1327.14056)]. Hodge structure; nilpotent orbits Hayama, T.; Pearlstein, G., \textit{asymptotics of degenerations of mixed Hodge structures}, Adv. Math., 273, 380-420, (2015) Hodge theory in global analysis, Transcendental methods, Hodge theory (algebro-geometric aspects), Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) Asymptotics of degenerations of 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 \(\mathbf f=(f_{1}, \dots , f_{l}):U\rightarrow K^{l}\), with \(K=\mathbb R\) or \(\mathbb C\), be a \(K\)-analytic mapping defined on an open set \(U\subset K^{n}\), and let \(\Phi \) be a smooth function on \(U\) with compact support. In this paper, we give a description of the possible poles of the local zeta function attached to \((\mathbf f, \Phi )\) in terms of a log-principalization of the ideal \(\mathcal I_{f}=(f_{1}, \dots , f_{l})\). When \(\mathbf f\) is a non-degenerate mapping, we give an explicit list for the possible poles of \(Z_{\Phi}(s, \mathbf f)\) in terms of the normal vectors to the supporting hyperplanes of a Newton polyhedron attached to f, and some additional vectors (or rays) that appear in the construction of a simplicial conical subdivision of the first orthant. These results extend the corresponding results of \textit{A. N. Varchenko} [Funct. Anal. Appl. 10, 175-196 (1977); translation from Funkts. Anal. Prilozh. 10, No. 3, 13--38 (1976; Zbl 0351.32011)] to the case \(l\geqslant 1\), and \(K=\mathbb R\) or \(\mathbb C\). In the case \(l=1\) and \(K=\mathbb R\), \textit{J. Denef} and {P. Sargos} [J. Anal. Math. 53, 201--218 (1989; Zbl 0693.32003)] proved that the candidate poles induced by the extra rays required in the construction of a simplicial conical subdivision can be discarded from the list of candidate poles. We extend the Denef-Sargos result to arbitrary \(l\geqslant 1\). This yields, in general, a much shorter list of candidate poles, which can, moreover, be read off immediately from \(\Gamma (\mathbf f)\). local zeta functions; poles Edwin León-Cardenal, Willem Veys & Wilson A. Zúñiga-Galindo, Poles of Archimedean zeta functions for analytic mappings, J. Lond. Math. Soc.87 (2013), p. 1-21 Local complex singularities, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Poles of Archimedean zeta functions for analytic mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider multi-matrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over \(n\) fixed points \(z_i\), \(i = 1, \dots, n\), (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, \(z_1\) and \(z_n\). Ramifications at other \(n - 2\) points enter the sum with the length of the profile at \(z_2\) and with the total length of profiles at the remaining \(n - 3\) points. We find the spectral curve of the model for \(n = 5\) using the loop equation technique for the above generating function represented as a chain of Hermitian matrices with a nearest-neighbor interaction of the type tr \(M_i M_{i + 1}^{- 1}\). The obtained spectral curve is algebraic and provides all necessary ingredients for the topological recursion procedure producing all-genus terms of the asymptotic expansion of our model in \(1 / N^2\). We discuss braid-group symmetries of our model and perspectives of the proposed method. multi-matrix model; loop equations; braid-group action String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Coverings of curves, fundamental group, Relationships between algebraic curves and physics, Other special orthogonal polynomials and functions, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Braid groups; Artin groups Spectral curves for hypergeometric Hurwitz numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, the author extends and generalizes the approach to develop a completely geometric Voronoï theory which he began in [J. Reine Angew. Math. 482, 93--120 (1997; Zbl 1011.53035)] where he introduced the concept of generalized systoles. In the present article, the author introduces the notion of nondegenerate point and systematically studies points satisfying particular properties such as perfection and eutaxy whose definitions are motivated by those from the classical Voronoï theory of lattices. Particular emphasis is laid upon the study of onfigurations and of finiteness results. As applications, the author obtains new results on the invariants of Bergé-Martinet and of Hermite-Humbert that are attached to number fields, and on Riemann surfaces.
The theoretical framework of this theory is laid out in \S 1 of the present paper. Let \(E\) be a finite-dimensional \({\mathbb R}\)-vector space, \({\mathcal F}\) a finite set of vectors in \(E\) and \(K\) their convex hull. \({\mathcal F}\) is said to be perfect if it generates \(E\) affinely, and eutactic if the origin is in the affine interior of \(K\). Now let \(V\) be a smooth and connected variety with \({\mathcal C}^1\)-functions \(f_s:V\to{\mathbb R}\) indexed by some set \(C\). \((f_s)_{s\in C}\) is called a system of length functions if for every \(p\in V\), every neighborhood \(U\) of \(p\) and every \(L\in{\mathbb R}\) one has \(f_s >L\) on \(U\) for almost all \(s\in C\). The generalized systole \(\mu (p)\) in a point \(p\in V\) is now defined to be \(\mu (p)=\min_{s\in C}f_s(p)\). Then \(p\in V\) is called perfect resp. eutactic if the family of differentials \((df_s(p))_{s\in S_p}\) is perfect resp. eutactic in the cotangent space \(T^*_pV\), where \(S_p=\{ s\in C; f_s(p)=\mu (p)\}\), and (strictly) extreme if \(\mu\) has a (strict) local maximum in \(p\). \(V\) can be partitioned into so-called minimal classes, where two points \(p,q\in V\) belong to the same class iff \(S_p=S_q\).
Let now \(V\) be equipped with a connection (assumed to be geodesic). A \({\mathcal C}^1\)-function \(f:V\to{\mathbb R}\) is said to be (strictly) convexoïdal if any critical point of \(f\) restricted to a geodesic is a (strict) local minimum. Now suppose in addition that \((f_s)_{s\in C}\) is a family of convexoïdal length functions on \(V\). Then a point \(p\in V\) is said to be nondegenerate if for every geodesic emanating from \(p\) there is at least one \(f_s\) that is strictly convexoïdal on that geodesic in some neighborhood of \(p\). With these definitions, it follows readily that if \(V\) is equipped with a family of convexoïdal length functions, then perfect points are always nondegenerate (Proposition 1.4), and that Voronoï's theorem (i.e. a point is extreme iff it is perfect and eutactic) holds iff every extreme point is nondegenerate iff every extreme point is perfect. This holds in particular whenever the length functions are strictly convexoïdal (Proposition 1.5). Furthermore, the set of points that are eutactic and nondegenerate is discrete, so in particular also the set of points that are perfect and eutactic (Corollary 1.7).
The author then obtains certain somewhat technical results on isolation of points that are eutactic and nondegenerate by which he can prove that if \(V\) is a smooth subvariety of \(P_n\), the space of unimodular positive definite symmetric \(n\times n\) matrices, defined by polynomials with algebraic coefficients in \({\mathbb R}\), then there are only finitely many perfect points for \(\mu^D\) (relative to \(V\)) and they are all algebraic over \({\mathbb Q}\) as are all nondegenerate eutactic points under some additional assumption on \(V\) (Corollary 1.12). Here, \(\emptyset\neq D\subset{\mathbb Z}^n \setminus\{ 0\}\) and \(\mu^D(A)=\min_{s\in D}A[s]\) for \(A\in P_n\). The author points out that the interest in this result lies in the fact that it applies readily to all natural and interesting types of lattices, for example autodual lattices.
In \S 2, the author applies these foundational results in certain situations to obtain finiteness results, notably to \(P_n\) as defined above and connected complete totally geodesic subvarieties thereof, and in particular their \(\rho\)-invariant forms for some representation \(\rho :\Pi\to \text{GL}_n({\mathbb Z})\) for a finite group \(\Pi\) (such \(\rho\)-invariant forms correspond in a natural way to \(\Pi\)-lattices). The main results concern finiteness of minimal classes containing weakly eutactic points and finiteness of perfect or eutatic nondegenerate points (Theorem 1), criteria for nondegeneracy (Theorem 2), the rank of minimal vectors for perfect points (Theorem 3), finiteness of the number of minimal classes modulo a certain action in the hermitian, symmetric or antisymmetric bilinear case (Theorem 4).
Using the methods developed in this paper, the author shows in Theorem 5 that the invariant of Bergé-Martinet satisfies Voronoï's theorem (also generalized to the hermitian case). He then interprets the invariant of Hermite-Humbert for Humbert forms over the ring of integers \({\mathcal O}_L\) in some algebraic number field \(L\). In this new setting, he obtains Voronoï's theorem for a certain \(\mu_L\) which is defined in a way so that the notions of perfection and eutaxy coincide with the classical ones for Humbert forms.
A final application concerns systoles of Riemann surfaces (Theorem 6). Euclidean lattice; geometric Voronoï theory; perfect lattice; eutaxy; systole; length function; convexoïdal; configuration of minimal vectors; Humbert form; invariant of Hermite-Humbert; invariant of Bergé-Martinet; Riemann surface C. Bavard, ''Théorie de Voronoï géométrique. Propriétés de finitude pour les familles de réseaux et analogues,'' Bull. Soc. Math. France 133, 205--257 (2005). Quadratic forms (reduction theory, extreme forms, etc.), Quadratic forms over global rings and fields, Lattices and convex bodies (number-theoretic aspects), Minima of forms, Abelian varieties and schemes, Conformal metrics (hyperbolic, Poincaré, distance functions), Riemann surfaces, Connections (general theory), Global differential geometry Voronoï's geometric theory. Finiteness properties for families of lattices and similar objects | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 investigates the occurrence of free summands in maximal Cohen- Macaulay approximations over hypersurface rings \(R=S/(f)\) where \(S=k[[X_ 1,\dots,X_ n]]\) for some positive integer \(n\) and some field \(k\). One of the motivations for this study is the author's criterion: if \(R\) is an isolated singularity, then \(R\) is quasihomogeneous if and only if the maximal Cohen-Macaulay approximation of the moduli algebra \(R/\overline{j(f)}\) of \(R\) has no free summands.
In a study of this phenomenon over Gorenstein rings, \textit{S. Ding} proved the following result. Let \(R=S/(x)\), where \(S\) is a complete local Gorenstein ring with maximal ideal \({\mathfrak m}\). Let \(x\in{\mathfrak m}\) be an \(S\)-regular element and let \(N\) be a finitely generated \(R\)-module. Finally, let \(\delta(N)=\delta^ 0(N)\), resp. \(\delta^ i(N)\), denote the number of copies of \(R\) in the minimal Cohen-Macaulay approximation of \(N\), resp. of the \(i\)-th syzygy module \(\Omega^ iN\), of \(N\). Then if \(x\in{\mathfrak m} \text{Ann}_ SN\) then \(\delta^ i(N)=0\) for all \(i\geq 0\).
The main aim of this paper is to provide a necessary condition (in the noncyclic case) for the equality \(\delta(N)=0\) and a criterion for the equalities \(\delta^ i(N)=0\) for all \(i\geq 0\). If \(t\) is the Eisenbud operator of \(N\) \((t\) is a degree \(-2\) map on a minimal projective resolution \(F_ 0\) of \(N)\), then it appears that if \(\delta(N)=0\), then \(t_ 0:F_ 2\to F_ 0\) is epimorphic while \(\delta^ i(N)=0\) for all \(i\geq 0\) if and only if \(t\) is an epimorphism (and this is well known to be equivalent to \(t_ i:F_{i+2}\to F_ i\) being epimorphic for \(i\leq\dim(R)+1\). Note that if \(N\) is cyclic, then the above necessary condition on \(t_ 0\) is actually a criterion. free summands in maximal Cohen-Macaulay approximations; hypersurface rings; isolated singularity; Gorenstein rings; syzygy module Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Hypersurfaces and algebraic geometry, Cohen-Macaulay modules, Polynomial rings and ideals; rings of integer-valued polynomials Free summands in maximal Cohen-Macaulay approximations and Eisenbud operators over hypersurface 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 For a separated map \(\pi: S \to B\) of schemes, \textit{S. L. Kleiman} iteratively constructed schemes \(X_r\) which naturally parametrize ordered clusters of \(r\) points (possibly infinitely near) in the fibers of \(\pi\) [Acta Math. 147, 13--49 (1981; Zbl 0479.14004)]. One can think of an \(r\)-cluster as a sequence \((\sigma_r, \sigma_{r-1}, \dots, \sigma_0)\) of blow ups centered at points over a common image \(b \in B\), as seen in work of \textit{B. Harbourne} [Lect. Notes Math. 1311, 101--117 (1988; Zbl 0661.14030)]. To extend these ideas to the relative case, the author starts with a flat separated surjective family \(\pi: S \to B\) of finite type with \(B\) irreducible and generically irreducible fibers. For such a family, the relative notion of a 1-cluster is a continuous choice of points in each fiber of \(\pi\), in other words a section \(\sigma_0\) to \(\pi\); the notion of a 2-cluster (allowing infinitely near points) consists of a section \(\sigma_1\) of the composite \(\tilde S \to S \to B\), where \(\tilde S \to S\) is the blow up at \(\sigma_0 (B)\), and so on for higher ordered clusters. The author makes formal definitions to this effect, showing that under good conditions there exist schemes \(\text{Cl}_r\) which naturally parametrize \(r\)-relative clusters of \(\pi\), that is the corresponding moduli functor is represented by a scheme. In particular recovers Kleiman's schemes \(X_r\) in the absolute case when \(B = \operatorname{Spec}k\), where \(k\) is a field.
When \(B\) is smooth, the author gives a recursive construction, exhibiting \(\text{Cl}_{r+1}\) as a blow up of \(\text{Cl}_r^2 = \text{Cl}_r \times_{\text{Cl}_{r-1}} \text{Cl}_r\) along a closed subscheme which fails to be Cartier only along the diagonal. Some new phenomena arise in this relative construction, for example it is shown that there is an open subset \(V \subset \text{Cl}_{r+1}\) corresponding to relative clusters for which the last section is not infinitely near to the previous section. A flattening stratification of \(\text{Cl}_r^2\) is given in which the open set \(V\) is described as the union of certain admissible strata. The author closes with some examples of surfaces in which the \(\text{Cl}_r\) behave differently from Kleiman's iterated blow ups. Iterated blowups; infinitely near points; universal family of sections Fibrations, degenerations in algebraic geometry, Rational and birational maps On the universal scheme of \(r\)-relative clusters of a family | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main goal of this paper under review is to present a definition of relative trace formulas associated to certain double coset spaces \(\mathcal{X} = H_1 \backslash G / H_2\), where \(G\) is a reductive group over a number field \(k\). The viewpoint here is to regard \(\mathcal{X}\) as a smooth semi-algebraic stack, over the completion \(F = k_v\) at each place \(v\), and then study the Schwartz space of measures on \(\mathcal{X}(F)\). The construction is largely geometric without using any truncation; the resulting ``relative trace formula'' is a distribution \(\mathrm{RTF}_{\mathcal{X}}\) on the Schwartz space of \(\mathcal{X}(\mathbb{A}_k)\), which is automatically invariant. Note that despite its name, \(\mathrm{RTF}_{\mathcal{X}}\) is neither a trace nor a formula.
In \S 2, one introduces a notion of smooth semi-algebraic stacks over a local field \(F\) with \(\mathrm{char}(F) = 0\), which are defined as geometric stacks in the category of Nash manifolds. In particular, such a stack \(\mathcal{X}\) admits a presentation \(X \to \mathcal{X}\) which is smooth, i.e.\ submersive, and \(X\) is a Nash \(F\)-manifold.
In \S 3, the Schwartz space of a Nash stack is defined. The Schwartz measures form a cosheaf for the smooth topology. The construction is somewhat involved, but in the case of a quotient stack \(\mathcal{X} = X/G\) one recovers the spaces of co-invariants \(\mathcal{S}(X^T(F))\), where \(T\) ranges over the isomorphism classes of \(G\)-torsors over \(F\), and \(X^T = X \times^G T\).
Next, one moves to the global setting and consider algebraic stacks of the form \(X/G\) over \(k\). The adélic Schwartz space \(\mathcal{S}(\mathcal{X}(\mathbb{A}_k))\) is defined, which serves as the space of test functions appearing in the sought-after relative trace formula. In \S 4, one carefully develops a notion of stalks over \(k\)-points of the GIT-quotient \(\mathfrak{c} := X /\!/ G\). In \S 6 one defines the evaluation map \(\mathrm{ev}_x: \mathcal{S}(\mathcal{X}(\mathbb{A}_k))_\xi \to \mathbb{C}\) at each semisimple point \(x \in \mathcal{X}(k)\) lying over \(\xi \in \mathfrak{c}(k)\), assuming that \(\mathcal{X}\) has no critical exponents at \(x\). The precise definition of critical exponents involves Luna's étale slice \(V\) through \(x\), on which the stabilizer group \(H\) acts. After reducing the problem to \(V\), what remains is an in-depth analysis of asymptotically finite functions on toroidal compactifications of the automorphic quotient \([H]\), as done in \S 4. All in all, the evaluation map amounts to a regularized sum of \(G(\mathbb{A}_k)\)-orbital integrals over elements whose semi-simple part is isomorphic to \(x\).
Assuming that the exponents are non-critical at each semisimple \(k\)-point \(x\), the relative trace formula for \(\mathcal{X}\) is defined as
\[
\mathrm{RTF}_{\mathcal{X}}: f \longmapsto \sum_{x/\simeq} \mathrm{ev}_x(f),
\]
a linear functional on \(\mathcal{S}(\mathcal{X}(\mathbb{A}_k))\). The twists \(X^T\) by \(G\)-torsors in the stacky picture also ``explains'' the pure inner forms appearing in Langlands program.
Various examples of \(\mathrm{RTF}_{\mathcal{X}}\) are presented in \S 6. The highlights are Jacquet's \(S \backslash \mathrm{PGL}_2 / S\) where \(S\) is a non-split maximal torus in \(\mathrm{PGL}_2\), as well as the spaces \((X \times X) \big/ \mathrm{diag}(G)\) where \(X\) is the homogeneous space in the Gan-Gross-Prasad conjectures. It is expected that the resulting distribution is the same as that in [\textit{M.~Zydor}, Can. J. Math. 68, No. 6, 1382--1435 (2016; Zbl 1410.11041)]. Nonetheless, this formalism is not applicable to the Arthur--Selberg trace formula, corresponding to \(\mathcal{X}\) being the adjoint quotient stack of a reductive group \(H\). stacks; Schwartz spaces; orbital integrals Sakellaridis, Y., The Schwartz space of a smooth semi-algebraic stack, Sel. Math. (N.S.), 22, 2401-2490, (2016) Generalizations (algebraic spaces, stacks), Pseudogroups and differentiable groupoids, Topological groupoids (including differentiable and Lie groupoids), Representation-theoretic methods; automorphic representations over local and global fields, Geometric Langlands program (algebro-geometric aspects), Nash functions and manifolds The Schwartz space of a smooth semi-algebraic stack | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 initiates the author's Diophantine study of modulus spaces for local systems on surfaces and their mapping class group dynamics. The aim of the paper is to establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems. This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the author collects relevant background on surfaces and moduli of local systems. Section 3, deals with systoles of local systems. In this section, as a first step towards the main result, the author introduces the notion of systole for \(SL_2\)-local systems, and studies its boundedness properties. In Section 4, the author proves a compactness criterion for local systems, which extends Mumford's compactness criterion [\textit{D. Mumford}, Proc. Am. Math. Soc. 28, 289--294 (1971; Zbl 0215.23202)] on moduli of closed Riemann surfaces as well as a related work of [\textit{B. H. Bowditch} et al., Math. Ann. 302, No. 1, 31--60 (1995; Zbl 0830.57008)]. Section 5, deals with some remarks. Here the author collects further remarks on the main result, briefly visits the theory of arithmetic hyperbolic surfaces and derives elementary observations on the behavior of integral points on the moduli spaces \(X_k\). He gives an alternative proof of the finitude of mapping class group orbits for non degenerate faithful representations in \(X_k(\mathbb{Z})\) for surfaces with boundary. modulus spaces; local systems; systoles; Riemann surfaces Algebraic moduli problems, moduli of vector bundles, Group actions on varieties or schemes (quotients), 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.), Cubic and quartic Diophantine equations, Rational points Nonlinear descent on moduli of local systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a Riemann surface of genus \(g \geq 2\), and consider an ascending sequence \(\{S_n \rightarrow S\}\) of finite Galois covers converging to the universal cover. Then, \textit{D. A. Kazhdan} proved in [in: Lie Groups Represent., Proc. Summer Sch. Bolyai Janos math. Soc., Budapest 1971, 151--217 (1975; Zbl 0308.14007)] that the \((1,1)\)-forms on \(S\) inherited from canonical forms via \(\{S_n \rightarrow S\}\), converge uniformly to a multiple of the hyperbolic \((1,1)\)-form of the universal cover. In the present paper the authors generalize the result of Kazhdan, by replacing the universal cover with any infinite Galois cover. Their main result is Theorem A (Theorem 5.4), according to which if \(S'\) is any infinite Galois cover of \(S\), and \(\{S_n \rightarrow S\}\) is a sequence of finite Galois covers converging to \(S'\), then the sequence of \((1,1)\)-forms on \(S\) induced from the canonical forms on \(S_n\) converges uniformly to the \((1,1)\)-form induced from the canonical form on \(S'\). In order to prove it, Theorem 5.3 gives a weaker result, on the strong convergence of the associated measures attached to the \((1,1)\)-forms. Then, the uniform convergence of the forms follows from an analytic argument. For proving that Theorem 5.3, a Gauss-Bonnet type result (Theorem B) is relevant.
Results on metric graphs, very close to Theorems A and B, were obtained by the second and third listed authors in [Invent. Math. 215, No. 3, 819--862 (2019; Zbl 1440.14284)]. Riemann surfaces; infinity Galois cover; canonical forms; hyperbolic forms Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization, 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) Limits of canonical forms on towers of Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper builds upon earlier work of the author in [\textit{X. Faber}, ``Topology and geometry of the Berkovich ramification locus for rational functions'', Preprint, \url{arXiv:1102.1432v3}] on understanding the Berkovich ramification locus for a rational map \(\varphi: \mathbb{P}^1_{\mathrm{Berk}} \to \mathbb{P}^1_{\mathrm{Berk}}\) over a non-archimedean field \(k\). This locus is the subset of the source \(\mathbb{P}^1_{\mathrm{Berk}}\) where \(f\) is not locally injective. Give \(\mathbb{P}^1_{\mathrm{Berk}}\) a metric in the standard way. Let \(C \subseteq \mathbb{P}^1_{\mathrm{Berk}}\) be the hull of the critical locus of \(\varphi\). The paper has three main theorems. Loosely stated, they are:
(1) If \(k\) has characteristic zero, then a point in the ramification locus is not further than \(1/(p-1)\) away from \(C\) (and lies on \(C\) if the residue characteristic is zero or if \(\deg(\varphi) < p\)).
(2) If \(k\) has characteristic \(p\), then a theorem of type (1) holds if and only if \(\varphi\) is tamely ramified.
(3) Given a small enough Berkovich neighborhood \(U\) of a critical point, the intersection of \(U\) with the Berkovich ramification locus contains the intersection of \(U\) with \(C\) (the precise containment relation is given, and depends on the multiplicity of the point and the characteristic of \(k\)).
The proof proceeds by explicit methods involving Newton polygons of polynomials. In particular, an explicit function (called the visible ramification) is defined, which essentially measures, for a point in the Berkovich ramification locus and a direction away from critical points of \(\varphi\), how far one is from the border of the Berkovich ramification locus. This function is defined in two different ways, and careful study of it leads to the theorems above.
The theorems are applied to give, among other things, a non-Archimedean version of Rolle's theorem. Berkovich space; ramification locus; rational function; Newton polygon; non-archimedean curve Faber, X., Topology and geometry of the Berkovich ramification locus for rational functions, II. Math. Ann., 356, 819-844, (2013) Rigid analytic geometry, Ramification and extension theory, Algebraic functions and function fields in algebraic geometry, Non-Archimedean valued fields Topology and geometry of the Berkovich ramification locus for rational functions. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper deals with the generalization of the Castelnuovo-Mumford regularity for bigraded modules or sheaves on \(\mathbb{P}^m \times \mathbb{P}^n\). After defining suitable regions \(\text{St}_i\) and \(\text{Reg}_i(p,p')\) in \(\mathbb{Z}^2,\) the authors give a definition of \((p,p')\)-regularity for a coherent sheaf \({\mathcal F}\) on \(X=\mathbb{P}^m \times\mathbb{P}^n;\) precisely, \({\mathcal F}\) is \((p,p')\)-regular if for all \(i\geq 1\):
\[
H^i(X,{\mathcal F}(k,k'))=0 \text{ for all }(k,k') \in\text{St}_i(p,p')= (p,p')+\text{St}_i.
\]
This definition is related to the weakly \((p,p')\)-regularity of a bigraded module \(M\) on \(R=k[{\underline x},{\underline y}] \), i.e. modules for which
\[
H^i_{m}(M)_{k,k'}=0\text{ for all }(k,k')\in \text{Reg}_{i-1} (p,p')
\]
where \(m\) is the irrelevant ideal in \(R.\) In particular the authors prove that for a finitely generated bigraded \(R\)-module \(M,\) for all \((p,p')\):
\[
\text{If }H^i_{m}(M)=0,\;i\geq 1, \;(k,k')\in\text{St}_{i-1} (p,p')\text{ then }H^i_{m}(M)=0,\;i\geq 1, \;(k,k')\in \text{Reg}_{i-1}(p,p').
\]
Other definitions of regularity for bigraded modules are given and are related to the previous and the classical regularity. bigraded modules; cohomology Hoffman, J. W., Wang, H. H.: Castelnuovo-Mumford regularity in biprojective spaces, Adv. Geom., 4 (2004), no. 4, 513--536 Graded rings, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Ideals and multiplicative ideal theory in commutative rings, Projective techniques in algebraic geometry Castelnuovo-Mumford regularity in biprojective 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 [For the entire collection see Zbl 0607.00005.]
Let \(f(z_ 0,...,z_ n)\) be a germ of an analytic function with an isolated critical point at the origin; f is assumed to have a non- degenerate Newton boundary \(\Gamma\) (f). Let V be the germ of the hypersurface \(f^{-1}(0)\) at the origin. There is always a canonical resolution \(\pi: \tilde V\to V\) of V which is associated with a given simplicial subdivision \(\Sigma^*\) of the dual Newton diagram \(\Gamma^*(f)\). The main purpose of this paper (see in particular {\S} 5) is to study the topology of the exceptional divisors E(P) through a canonical simplicial subdivision \(\Sigma^*\), which is constructed in {\S} 3. If P is a strictly positive vertex of \(\Sigma^*\), then E(P) is always a compact divisor such that \(\pi (E(P))=\{0\}\). The topology of exceptional divisors E(P) of the two dimensional and the three dimensional singuarities is then studied in detail in {\S}{\S} 6 and 8 respectively. In section \(7\) the author shows that the fundamental group of E(P), where P is a strictly positive vertex of a fixed subdivision \(\Sigma^*\), is a finite cyclic group (with an order independent of the choice of \(\Sigma^*)\) if \(n>2\) and if the face \(\Delta\) (P) of \(\Gamma\) (f) is an n-simplex. In section \(9\) the canonical divisors of \(\tilde V\) and E(P) are considered. In particular, applying (9.1) and (9.2) one can calculate the signature of the Milnor fibre of f in the case \(n=2\) from the Newton boundary \(\Gamma\) (f). resolution of hypersurface singularities; simplicial subdivision; topology of the exceptional divisors; canonical; fundamental group; signature of the Milnor fibre; Newton boundary Oka (M.).- On the resolution of the hypersurface singularities. In Complex analytic singularities, volume 8 of Adv. Stud. Pure Math., pages 405-436. North-Holland, Amsterdam (1987). Zbl0622.14012 MR894303 Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Topological properties in algebraic geometry On the resolution of the hypersurface singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a complex analytically irreducible quasi-ordinary (q.o) singularity, defined by \(f\in \mathbb C\{x_1,x_2\}[z]\). It can be parametrized in the form \(x_1=x_1\), \(x_2=x_2\), \(z=\zeta(x_1,x_2)\) with \(\zeta\in \mathbb C\{x_1,x_2\}[z]\). [\textit{Y.-N. Gau}, Mem. Am. Math. Soc. 388, 109--129 (1988; Zbl 0658.14004)] as shown: A finite set of exponents in the support of the series \(\zeta\) -- they are called the characteristic exponents -- are complete invariants of the topological type of the singularity.
In the paper under review, the authors look for invariants for {\em all} types of singularities. They consider the set of jet schemes of \(X\). For \(m\in\mathbb N\), they define a functor \(F_m\colon \mathbb C\text{-Schemes}\to \text {Sets}\) which is representable by a \({\mathbb C}\)-scheme \(X_m\), the \(m\)th jet scheme. There is a canonical projection \(\pi_m \colon X_m\to X\). In section 4 q.o.\ surfaces with one characteristic exponent are considered. The irreducible components of the \(m\)-jet schemes through the singular locus of a such a surface are described in Th.\ 4.14. A graph \(\Gamma\) is constructed which represents the decomposition of \((\pi^{-1}_m(X_{\text{Sing}}))_{\text{red}}\) for every \(m\). The graph \(\Gamma\) is equivalent to the topological type of the singularity. In section 5 these results are generalized to the general case. quasi-ordinary; surface singularities; jet schemes Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Jet schemes of quasi-ordinary 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 Let \(\Delta\) be the partition of a region in affine space into a finite number of polyhedral cells. A \textit{multivariate spline} with smoothness \(\mu\) over \(\Delta\) is a \(C^\mu\)-function whose restriction to each cell of \(\Delta\) is a polynomial. A \(C^\mu\) \textit{piecewise algebraic variety} is the zero set of a collection of multivariate splines with smoothness \(\mu\). The article serves primarily as a review of the algebraic structure of multivariate spline spaces and of the concepts from algebraic geometry that translate to the more general setting of multivariate splines, namely the correspondence between ideals and varieties. In addition, although the Hilbert Nullstellensatz does not hold in general for splines, the authors provide some instances in which it does.
The article targets a general mathematical audience familiar with the basic concepts of algebraic geometry. piecewise algebraic varieties; algebraic varieties; multivariate splines; ideals Numerical computation using splines, Spline approximation, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects of algebraic curves The correspondence between multivariate spline ideals and piecewise algebraic 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 This paper is motivated by some of the problems that \textit{J. Nash} proposed in ``Arc structure of singularities'' in the midsixties. In this unpublished preprint, Nash considered parametrized formal arcs or curves on an algebraic variety \(V\) whose origin is in the singular set of \(V\). First he observed that, for each \(i\geq 0\), the space of their \(i\)-jets is a constructible set (in fact, this is a direct consequence of Artin's approximation theorem).
Here we get a constructive proof of this property when \(V\) is an hypersurface with an isolated singular point. Our method is algorithmic and provides a linear bound for Artin's \(\beta\)-function associated to the system of equations to solve, envolving the Milnor number and the multiplicity of the singular point. (Since then, this result has been generalized by M. Hickel.) Even if the singularity of the hypersurface \(V\) is not reduced to a point, the algorithm associates to each curve on \(V\) as above, a decreasing sequence of integers. If \(V\) is a branch of a curve in \(\mathbb{C}^ 2\), this sequence is equivalent to that of its Puiseux characteristic pairs. Artin's approximation theorem; hypersurface with an isolated singular point; \(\beta\)-function; Milnor number; multiplicity of the singular point Lejeune-Jalabert, M, Courbes tracées sur un germe D'hypersurface, Am. J. Math., 112, 525-568, (1990) Local deformation theory, Artin approximation, etc., Singularities of surfaces or higher-dimensional varieties Curves drawn on a hypersurface germ. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 author's previous paper [J. Math. Pures Appl., IX. Sér. 79, No. 1, 21--56 (2000; Zbl 0959.32015)], it was proved a characterization of the divisors of the Grassmannian \(\mathbb G_{q-1,n}\) which are Chow transforms of \((q,q)\)-currents on \(\mathbb P^n.\) This allowed the author to characterize the currents, associated to algebraic cycles, as being those for which the direct image by a Veronese embedding \(\mathbb P_n\to\mathbb P_{\binom{n+ d}{d}-1}\) of degree \(d\geqslant 2\) has a divisor for the Chow transform. The main results of the paper under review are best presented by the author's summary:
``We prove that a closed current on a projective manifold is cohomologous to an algebraic cycle with complex coefficients if and only if it is a weak limit of such cycles. This allows us to present two functional approaches of the problem of the algebraicity of cohomology classes. On the one hand, using the characterization of currents associated to algebraic cycles by the Chow transformation, we reduce the obstructions to an orthogonality condition with certain smooth functions on the Grassmannian, which are in general merely images of distributions by a suitable explicitly defined linear differential operator; this forces a convergence in the space of \(\mathcal C^k\) functions. On the other hand, by going onto the space of divisors of the Grassmannian, we introduce a scalar differential equation whose resolution gives the approximation''. Algebraic cycles; Currents, Chow transform Méo, M., Caractérisation fonctionnelle de la cohomologie algébrique d'une variété projective, C. R. Acad. Sci. Paris, Ser. I, 346, 1159-1162, (2008) Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Currents in global analysis Functional characterization of the algebraic cohomology of a projective manifold | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X\) and \(Y\) are closed subschemes of \(\mathbb{P}^n\) with respective ideal sheaves \(\mathcal{I}_X\) and \(\mathcal{I}_Y\), then one can define the residual scheme of \(X\) with respect to \(Y\). This subscheme is denoted by \(\mathrm{res}_Y X\) and is determined by the condition that its ideal sheaf is given by the colon ideal sheaf \((\mathcal{I}_X : \mathcal{I}_Y)\).
In the article under review, the authors fix closed subschemes \(X,Y \subseteq \mathbb{P}^n\) and are interested in the schemes \(\text{res}_Y X\) and \(\text{res}_{Y \cap V}(X \cap V)\) for \(V \subseteq \mathbb{P}^n\) a linear subspace. They prove two theorems.
To describe the first theorem, let \(H\subset \mathbb{P}^n\) denote a hyperplane and let \(I_X\), \(I_Y\), and \(I_H\) denote the saturated homogeneous ideals of \(X\), \(Y\), and \(H\) respectively. The authors prove that if \(H\) is general and if the ideals \(I_X+I_H\) and \((I_X : I_Y)+I_H\) are saturated, then the Castelnuovo-Mumford regularity of \(\text{res}_{Y \cap H}(X\cap H)\) and \(\text{res}_Y X\) are equal.
The second theorem concerns the concept of uniform position for a set of points in \(\mathbb{P}^n\). To state it, suppose that \(V \subseteq \mathbb{P}^n\) is a linear subspace of codimension \(r\) determined by hyperplanes in general position with respect to \(X\) and \(Y\). The authors prove that if \(\text{res}_Y X\) is irreducible of dimension \(r\), then the closed subscheme \(\text{res}_{Y\cap V}(X\cap V)\) is a set of points in uniform position. graded modules; schemes; regularity; uniform position principle Schemes and morphisms, Syzygies, resolutions, complexes and commutative rings, Graded rings, Divisors, linear systems, invertible sheaves On the regularity of the residual scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Zhang introduced semipositive metrics on a line bundle of a proper variety. In this paper, we generalize such metrics for a line bundle \(L\) of a paracompact strictly \(K\)-analytic space \(X\) over any non-archimedean field \(K\). We prove various properties in this setting such as density of piecewise \(\mathbb{Q}\)-linear metrics in the space of continuous metrics on \(L\). If \(X\) is proper scheme, then we show that algebraic, formal and piecewise linear metrics are the same. Our main result is that on a proper scheme \(X\) over an arbitrary non-archimedean field \(K\), the set of semipositive model metrics is closed with respect to pointwise convergence generalizing a result from Boucksom, Favre and Jonsson where \(K\) was assumed to be discretely valued with residue characteristic \(0\). Arakelov geometry; non-Archimedean geometry; model metrics; plurisubharmonic model functions; divisorial points Arithmetic varieties and schemes; Arakelov theory; heights, Rigid analytic geometry On Zhang's semipositive metrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Building on the abstract notion of prolongation developed by the authors [J. Inst. Math. Jussieu 9, No. 2, 391--430 (2010; Zbl 1196.14008)], the theory of iterative Hasse-Schmidt rings and schemes is introduced, simultaneously generalizing difference and (Hasse-Schmidt) differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse-Schmidt jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite-dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse-Schmidt variety is determined by its jet spaces at a point. theory of iterative Hasse-Schmidt rings and schemes; Hasse-Schmidt jet spaces Moosa, R.; Scanlon, T., Generalized Hasse-Schmidt varieties and their jet spaces, Proceedings of the London Mathematical Society, 103, 197-234, (2011), URL: http://dx.doi.org/10.1112/plms/pdq055 Schemes and morphisms, Differential algebra Generalized Hasse-Schmidt varieties and their jet 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 author gives an analogue of a theorem of \textit{W.-L. Chow} in [Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] for infinite dimensional vector spaces. He considers vector spaces \(X\) over division rings, the lattice \({\mathcal G}(X)\) of vector subspaces, and the graph \(\Gamma(X)\), whose set of vertices is \({\mathcal G}(X)\) with pairs \((U,V)\) of adjacent subspaces \(U\), \(V\) as edges; \(U\) is called adjacent to \(V\) (\(U\sim V\)) iff \(\dim U /(U\cap V) = \dim V / (U\cap V)\).
\({\mathcal C} \subset {\mathcal G}(X)\) is called a component iff \({\mathcal C}\) induces a connected component in the graph \(\Gamma(X)\).
For components \({\mathcal C} \subset {\mathcal G}(X)\) and \({\mathcal D} \subset {\mathcal G}(Y)\), where \(X\) and \(Y\) are vector spaces, a bijective map \(\Phi : {\mathcal C} \longrightarrow {\mathcal D}\) which preserves \(\sim\) in both directions is called isomorphism.
First, the author gives a careful description of components \({\mathcal C}\) and the sublattices \({\mathcal C}_\pm\) generated by them in \({\mathcal G}(X)\).
The main result of the paper describes isomorphisms in the case of infinite diameter of the connected components. First, it is shown that for diameters \(\geq 2\) any isomorphism \(\Phi : {\mathcal C} \longrightarrow {\mathcal D}\) of components may be uniquely extended to an isomorphism or antiisomorphism \(\rho : {\mathcal C}_\pm \longrightarrow {\mathcal D}_\pm\) of lattices. Furthermore, any such \(\rho\) which maps \({\mathcal C}\) bijectively onto \({\mathcal D}\) restricts to an isomorphism of components. Second, for components of infinite diameter such lattice isomorphisms or antiisomorphisms are characterized by two bijective semilinear maps between certain subspaces of \(X\) and \(Y\).
Let \({\mathcal G}_{\alpha,\beta}(X)\) denote the set of subspaces of dimension \(\alpha\) and of codimension \(\beta\) such that \(\alpha + \beta = \dim X\). Chow's theorem shows in the finite dimensional case that \(\alpha < \beta\) makes \({\mathcal G}_{\alpha,\beta}(X)\) a component, whose automorphisms may be described by bijective semilinear maps on \(X\). From his main result the author derives that an analogous description for infinite dimensional \(X\) and infinite \(\alpha\) and \(\beta\) is not possible.
Continuing the work of \textit{A. Blunck} and \textit{H. Havlicek} [Discrete Math. 301, No. 1, 46--56 (2005; Zbl 1083.51001)], the author applies his results to infinite dimensional vector spaces \(X\) and \(Y\), to characterize complementarity preserving bijections of \({\mathcal G}_{\alpha,\alpha}(X)\longrightarrow {\mathcal G}_{\gamma,\gamma}(Y)\) by certain bijective semilinear maps \(X\longrightarrow Y\). adjacent subspaces; Grassmann graph; complementary subspaces; distant graph; semilinear mapping Plevnik, L., Top stars and isomorphisms of Grassmann graphs, Beitr. algebra geom., 56, 703-728, (2015) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Distance in graphs, Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.), Grassmannians, Schubert varieties, flag manifolds, Linear transformations, semilinear transformations, Linear preserver problems, Homomorphism, automorphism and dualities in linear incidence geometry, Incidence structures embeddable into projective geometries Top stars and isomorphisms of Grassmann graphs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By generalizing the method of \textit{E. M. Friedlander} et al. [Math. Ann. 330, No. 4, 759--807 (2004; Zbl 1062.19004)] in the semi-topological Atiyah-Hirzebruch spectral sequence relating the morphic cohomology and the semi-topological K-theory of smooth complex variety, the authors constructed a version for smooth real varieties. Moreover, mimicing the complex analogue, they compare it with the known motivic Atiyah-Hirzebruch spectral sequence, the topological Atiyah-Hirzebruch spectral sequence and Dugger's spectral sequence for real varieties.
Generalizing Suslin's Conjecture on morphic cohomology of smooth complex varieties, the authors formulated its equivariant and real versions, and showed that they are all equivalent to each other. This enables them to prove equivariant Suslin's Conjecture in codimension one for all smooth real varieties and computes real morphic cohomology in codimension one.
As an application, they compute the semi-topological \(K\)-theory of certain real varieties. They also give another proof of the 2-adic Lichtenbaum-Quillen Conjecture over \(\mathbb{R}\), which was originally proved by Karoubi-Weibel and Rosenschon and Ostvaer.
In the end, they compare Poincaré Duality relating equivariant morphic cohomology and dos Santos' equivariant Lawson homology groups, and Poincaré Duality relating the Bredon cohomology and usual homology. DOI: 10.1090/S0002-9947-2012-05603-0 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Relations of \(K\)-theory with cohomology theories, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Equivariant semi-topological invariants, Atiyah's \(KR\)-theory, and real algebraic cycles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this article, we analyze the coercivity on \(\mathbb{R}^n\) of multivariate polynomials \(f\in \mathbb{R}[x]\) in terms of their so-called Newton polytopes at infinity. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions solely containing information about the geometry of the vertex set of the Newton polytope at infinity, as well as sign conditions on the corresponding polynomial coefficients. For all other polynomials, the so-called gem irregular polynomials, we introduce sufficient conditions for coercivity based on those from the regular case. For some special cases of gem irregular polynomials, we establish necessary conditions for coercivity, too. Using our techniques, the problem of deciding the coercivity of a polynomial can often be studied based on its Newton polytope at infinity. We relate our results to the context of polynomial optimization theory and the existing literature therein, and we illustrate our results with several examples. Newton polytope; Newton polytope at infinity; coercivity; polynomial optimization; noncompact semialgebraic sets Bajbar, T.; Stein, O., Coercive polynomials and their Newton polytopes, SIAM J Optim, 25, 1542-1570, (2015) Nonlinear programming, Real polynomials: analytic properties, etc., Polynomials in number theory, Special polynomials in general fields, Semialgebraic sets and related spaces, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Nonconvex programming, global optimization Coercive polynomials and their Newton polytopes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(f : X \rightarrow \Delta,\) \(\Delta = \{z\in {\mathbb C}: | z| < 1\},\) be a projective and semi-stable family of analytic spaces. Due to \textit{J. Steenbrink} [Invent. Math. 31, 229--257 (1976; Zbl 0303.14002)] a limit Hodge structure is defined for such a family as the limit of Hodge structures \(H^m(X_t, {\mathbb Z}),\) where \(m \in {\mathbb Z}\), \(t\in \Delta\setminus\{0\}.\) In his previous work [\textit{T. Matsubara}, Kodai Math. J. 21, 81--101 (1998; Zbl 1017.32017)], the second author gave an interpretation of the theory of Steenbrink in the framework of the log Hodge theory in the following manner: ``The higher direct images of \({\mathbb Z}_X\) on \(\Delta\) carry the natural variations of polarized log Hodge structure''.
The paper under review is devoted to a generalization of the theory of Steenbrink to log Hodge theory with coefficients. More precisely, general variations of polarized log Hodge structure \({\mathcal H}_{{\mathbb Z}}\) on \(X\) instead of \({\mathbb Z}_X\) are considered. In order to investigate this case the authors introduce and analyze log versions of basic notions of the theory including \(C^\infty\)-functions, degenerations of Hodge decompositions; \(\bar{\partial}\)-Poincaré lemma, Kähler metrics, harmonic forms, etc. [see also \textit{M. Harris} and \textit{D. H. Phong}, C. R. Acad. Sci., Paris, Sér. I 302, 307--310 (1986; Zbl 0597.32025)].
The authors underline that their main theorem gives new proofs of earlier results [\textit{T. Fujisawa}, Compos. Math. 115, No. 2, 129--183 (1999; Zbl 0940.14007); \textit{L. Illusie}, Duke Math. J. 60, No. 1, 139--185 (1990; Zbl 0708.14014); \textit{M. Cailotto}, C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 12, 1089--1094 (2001; Zbl 1074.14513)]. In addition, it is proved that the log Riemann-Hilbert correspondence [\textit{K. Kato} and \textit{C. Nakayama}, Kodai Math. J. 22, No. 2, 161--186 (1999; Zbl 0957.14015)] is, in fact, functorial relative to Hodge filtrations. variation of Hodge structures; limit of Hodge structures; nilpotent orbit; log geometry; log Riemann-Hilbert correspondence Kazuya Kato, Toshiharu Matsubara, and Chikara Nakayama, Log \?^{\infty }-functions and degenerations of Hodge structures, Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 269 -- 320. Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations Log \(C^\infty\)-functions and degenerations of 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 \(\mathcal S\) be the germ of a complex quasi-ordinary hypersurface of dimension \(d\). Let \(f\) be a quasi-ordinary polynomial defining \(\mathcal S\) and let \(\mathcal A\) be the local algebra of \(\mathcal S\).
The author constructs a morphism \(\theta:(\overline{\mathcal R}, P)\to (\mathcal S,0)\), where \((\overline{\mathcal R}, P)\) is a smooth space, and a special divisor with normal crossings \(\overline {\mathcal H}\) on \(\overline{\mathcal R}\) at \(P\). Thus, the author considers the functions \(h\in\mathcal A\) such that the components of the divisor \(\theta^*(h)\) either are components of \(\overline{\mathcal H}\) or do not contain the intersection of its components. For those functions the tuple formed by the orders of vanishing of \(\theta^*(h)\) along the components of \(\overline{\mathcal H}\) can be computed; this tuple is called the dominating exponent of \(\theta^*(h)\) with respect to \(\overline{\mathcal H}\) at \(P\). The set of these dominating exponents forms a semigroup that the author denotes by \(\Gamma'_P(\mathcal S)\).
Inspired by a previous construction of \textit{P. González-Pérez} [see J. Inst. Math. Jussieu 2, No. 3, 383--399 (2003; Zbl 1036.32020) or his Ph.D. thesis], the author defines in Section 12 a semigroup attached to \(f\), called {the reduced semigroup of \(\mathcal A\) with respect to \(f\)}, which is denoted by \(\Gamma'(f)\).
Then, the main result shown in this paper is the explicit construction of an isomorphism between the semigroups \(\Gamma'(f)\) and \(\Gamma'_P(\mathcal S)\) (see Theorem 13.2).
As pointed out by the author, in this paper it is shown a generalization to arbitrary dimension of previous results first obtained for surfaces in the author's thesis.
It is worth remarking that the exposition of the paper is quite clear and self-contained. toric geometry; quasi-ordinary singularities Popescu-Pampu, P.: On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity. Duke Math. J. 124(1), 67--104 (2004) Invariants of analytic local rings, Toric varieties, Newton polyhedra, Okounkov bodies On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(P_1,\dots,P_s\in{\mathbb P}^n\) be distinct points and let \(m_1,\dots,m_s\) be non-negative integers. The \(0\)-dimensional subscheme \(X\subset{\mathbb P}^n\) defined by the saturated ideal \(I_X=(I_{P_1})^{m_1}\cap\dots\cap (I_{P_n})^{m_s}\) is called scheme of fat points of multiplicities \(m_1,\dots,m_s.\)
The regularity index of \(X\) is
\[
\tau(X)=\min\{i\in\mathbb Z^+\mid H_X(i)=\deg X\}
\]
where \(H_X\) is the Hilbert function of \(X\).
The aim of the paper is to study upper bounds for the regularity index. They consider the Segre's Bound \(h_0\) for schemes of fat points of \({\mathbb P}^n\) whose support is in general position and the general Segre's bound for any scheme of fat points in \({\mathbb P}^n.\)
First the authors prove that the locus of all schemes of fat points with assigned multiplicities whose regularity index is bounded by \(h_0\) is open and strictly contains the non-empty open set given by the locus of the schemes of fat points whose support is in general position. Then they prove that the general Segre's bound holds for any scheme of \(n+2\) fat points in \({\mathbb P}^n.\) fat points; regularity index; Hilbert function; general position Benedetti, B; Fatabbi, G; Lorenzini, A, Segre's bound and the case of \(n+2\) points of \(\mathbb{P}^n\), Commun. Algebra, 40, 395-403, (2012) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Segre's bound and the case of \(n+ 2\) fat points of \(\mathbb P^{ n }\) | 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 This article is a continuation of the author's previous article [\textit{B.~Mesablishvili}, Appl. Categ. Struct. 12, No. 5--6, 485--512 (2004; Zbl 1084.14003)]. In that article it is shown that a quasi-compact morphism of schemes is a stable effective descent morphism for the scheme-indexed category of quasi-coherent modules if and only if it is pure, a notion also developed in that article. In this paper, the authors shows that this characterisation of (quasi-compact) effective descent morphisms by purity is also true for the scheme-indexed categories of (i) quasi-coherent modules of finite type, (ii) flat quasi-coherent modules, (iii) flat quasi-coherent modules of finite type and (iv) locally projective quasi-coherent modules of finite type [compare also \textit{B.~Mesablishvili}, Theory Appl. Categ. 10, 180--186, electronic only (2002; Zbl 1044.18006)].
Further results of this text are alternative descriptions of (quasi-compact) pure morphisms. The author proves the following two theorems:
A quasi-compact morphism of schemes is pure if and only if it is a stable regular epimorphism (i.e. a morphisms such that each of its pullbacks is an epimorphism that is a coequiliser).
A stable schematically dominant morphism of schemes [\textit{A.~Grothendieck} and \textit{J.~A.~Dieudonné}, Éléments de géométrie algébrique.~I (Die Grundlehren der mathematischen Wissenschaften. 166, Springer-Verlag, Berlin-Heidelberg-New York) (1971; Zbl 0203.23301)] is pure. If, in addition, the morphism is quasi-compact, the converse also holds.
In order to derive the characterisation for stable effective descent morphisms of the four examples (i)--(iii) of scheme-indexed category given above, the author also introduces the notion of a pure stack, which is a stack \(\mathcal F\) on the Zariski site of schemes (and in particular a scheme-indexed category) such that every pure morphism of affine schemes is an effective \(\mathcal F\)-descent morphism. The author then derives general results for pure stacks, in particular comparison results between two pure stacks. scheme; pure morphism; descent theory B. Mesablishvili, ''More on descent theory for schemes,'' Georgian Math. J., 11, No. 4, 783--800 (2004). Schemes and morphisms, Epimorphisms, monomorphisms, special classes of morphisms, null morphisms, Factorization systems, substructures, quotient structures, congruences, amalgams More on descent theory for schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is concerned with the drapeau theorem for differential systems. By a differential system \((R,D)\), we mean a distribution \(D\) on a manifold \(R\). The derived system \(\partial D\) is defined, in terms of sections, by \(\partial \mathcal{D}=\mathcal{D}+[\mathcal{D},\mathcal{D}]\). Moreover higher derived systems \(\partial^i D\) are defined by \(\partial^i D =\partial(\partial^{i-1} D)\). The differential system \((R,D)\) is called regular if \(\partial^i D\) are subbundles of \(TR\) for every \(i\geq 1\). We say that \((R,D)\) is an \(m\)-flag of length \(k\), if \((R,D)\) is regular and has a derived length \(k\), i.e., \(\partial^k D =TR\), such that \(\operatorname{rank}D=m+1\) and \(\operatorname{rank}\partial^{i} D=\partial^{i-1}D+m\) for \(i=1,\dots k\). Especially \((R,D)\) is called a Goursat flag (un drapeau Goursat) of length \(k\) when \(m=1\).
The main purpose of this paper is to clarify the procedure of ``rank 1 prolongation'' of an arbitrary differential system \((R,D)\) of rank \(m+1\), and to give good criteria for an \(m\)-flag of length \(k\) to be special. A generalisation of the drapeau theorem for an \(m\)-flag of length \(k\) for \(m\geq 2\) is proved. differential system; Goursat flag; \(m\)-flag of length \(k\); rank 1 prolongation K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems, Differential Geometry and its Applications 27 (2009), 793--808. Vector distributions (subbundles of the tangent bundles), General geometric structures on manifolds (almost complex, almost product structures, etc.), Exterior differential systems (Cartan theory), Jets in global analysis, Grassmannians, Schubert varieties, flag manifolds Drapeau theorem for differential systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review in devoted to the study of some basic properties of Berkovich's analytic spaces over non-archimedean fields [see \textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (AMS). (1990; Zbl 0715.14013)] and [Publ. Math., Inst. Hautes Étud. Sci. 78, 5--161 (1993; Zbl 0804.32019)]).
The first results concern the local rings of Berkovich spaces. The author shows that they are excellent when the space is affinoid (théorème 2.13). In the rigid analytic setting, this result is due to \textit{B. Conrad} [Ann. Inst. Fourier 49, No. 2, 473--541 (1999; Zbl 0928.32011)]. Next, the classical algebraic properties of the local rings (\(R_{m}\), \(S_{m}\), being regular, Cohen-Macaulay, Gorenstein, complete intersection) are investigated. The author studies their behaviours with regard to analytic extensions of the base field (théorème 3.1 and \S 3.4 in the non-good case). For this purpose, in the case of property \(R_{m}\) and regularity, he has to introduce the notion of analytically separable field extension (\S 1). Then he turns to the behaviour under analytification (théorème 3.4 and \S 3.4 in the non-good case). Let \({\mathcal X}\) be a scheme of finite type over an affinoid algebra and \(X\) be its analytification. For any of the aforementioned properties, the locus where it holds is a Zariski open subset of \({\mathcal X}\) and it holds at a point \(x\) of \(X\) if and only if it holds at its image in \({\mathcal X}\).
The rest of the paper is devoted to global properties of Berkovich spaces, especially normality, connectedness and irreductibility. The Zariski topology of an affinoid space being Noetherian, one may define its irreducible components in the classical way. This is no longer possible for more general analytic spaces. Let's mention that B. Conrad has defined the irreducible components of rigid spaces through normalization [loc. cit.]. Here the author proceeds in another way and gives a direct definition of the irreducible components of an analytic space \(X\) as the Zariski closures of the irreducible components of the affinoid domains of \(X\). He shows that they coincide with the maximal irreducible Zariski subsets (théorème 4.20). He also proves that analytic spaces admit normalizations (théorème 5.13) and that the connected components of the normalization of a space correspond bijectively to its irreducible components (théorème 5.17). A property is said to be geometric if it holds after any analytic extension of the base field. The author gives several characterizations of geometric connectedness (théorème 7.11) and geometric irreducibility (théorème 7.12).
Last, the author considers product of spaces and proves that if a geometric property holds for a couple of spaces, it still holds for their product (théorème 8.1 for local properties and 8.3 for global ones).
Let's also mention that there is a very nice and detailed introduction, which is all the more useful as the paper is quite long. At the end of it, the author explains how his results compare to what was known before, mainly the works of \textit{R. Kiehl} see [J. Reine Angew. Math. 234, 89--98 (1969; Zbl 0169.36501)], B. Conrad [loc. cit.] and V. Berkovich [loc. cit.].
The paper is thorough and self-contained (the author reproves most of what he needs). It will undoubtedly become a reference work for people interested in Berkovich theory. Berkovich spaces; non-archimedean analytic spaces; excellence, base field extension, irreducible components, normalization Antoine Ducros, ``Les espaces de Berkovich sont excellents'', Ann. Inst. Fourier59 (2009) no. 4, p. 1443-1552 Rigid analytic geometry, Foundations of algebraic geometry Berkovich spaces are excellent | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 new approach to étale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the big étale site, and in particular to all algebraic stacks. This approach also produces a more refined invariant than the original construction of \textit{M. Artin} and \textit{B. Mazur} [Etale homotopy. Cham: Springer (1969; Zbl 0182.26001)], namely we produce a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if \(\mathcal{X}\) is an arbitrary higher stack on the étale site of affine schemes of finite type over \(\mathbb{C}\), then the étale homotopy type of \(\mathcal{X}\) agrees with the homotopy type of the underlying stack \(\mathcal{X}_{\mathrm{top}}\) on the topological site, after profinite completion. In particular, if \(\mathcal{X}\) is an Artin stack locally of finite type over \(\mathbb{C}\), our definition of the étale homotopy type of \(\mathcal{X}\) agrees up to profinite completion with the homotopy type of the underlying topological stack \(\mathcal{X}_{\mathrm{top}}\) of \(\mathcal{X}\) in the sense of \textit{B. Noohi} [``Foundations of topological stacks. I'', Preprint, \url{arXiv:math/0503247}]. We also show this comparison is compatible in a suitable sense with the comparison theorem of \textit{E. M. Friedlander} for simplicial schemes [Etale homotopy of simplicial schemes. Princeton, NJ: Princeton University Press (1982; Zbl 0538.55001)]. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of \(\infty\)-categories which we believe to be of independent interest. étale homotopy; higher categories; stacks; topos theory Homotopy theory and fundamental groups in algebraic geometry, Generalizations (algebraic spaces, stacks), Homology with local coefficients, equivariant cohomology, Topoi On the étale homotopy type of higher stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a survey paper on Artin Approximation. Let us recall the Artin Approximation Theorem: Let \(f(x,y)\in\mathbb C\{x,y\}^p\) be a vector of convergent power series in two sets of variables \(x\) and \(y\). Let \(y(x)\in\mathbb C[[x]]^m\) be a given formal power series solution vanishing at~\(0\)
\[
f(x,y(x))=0.
\]
Then for any \(c\in\mathbb N\) there exists a convergent power series solution \(\tilde y(x)\in\mathbb C\{x\}^m\) vanishing at~\(0\)
\[
f(x,\tilde y(x))=0
\]
such that the terms of \(\tilde y(x)\) agree with the terms of \(y(x)\) up to degree \(c\):
\[
\tilde y(x)-y(x)\in (x)^c.
\]
The paper is organized as follows: in a first part the Artin Approximation Theorem and several variations of it are presented. In a second part the author ``presents a systematic argument which proves, with minor modifications, all theorems simultaneously'' (according to the author). After a presentation of the Weierstrass Division Theorem and some variations of it, the author reviews some results on algebraic power series. The end of the paper is devoted to review some results of the author concerning the Artin Approximation Theorem and the properties of algebraic power series.
Unfortunately, there are numerous mistakes and erroneous claims that may mislead the reader. The main problems concern the proofs of the several variations of the Artin Approximation Theorem (even if these results are correct since they have been proven by several authors before). These variations essentially assert the existence of a convergent solution \(\tilde y(x)\in\mathbb C\{x\}^m\) of \(f=0\) under some weaker hypothesis than in the Artin Approximation Theorem (namely one only requires that \(f=0\) has approximate solutions up to some well chosen orders). We explain here the most problematic mistakes:
\(\bullet\) The first proof is the proof of the classical Artin Approximation Theorem and is essentially the original proof of M.~Artin. But before giving the proof the authors presents the strategy of proof (at the end of page 603): ``We will construct from \(f\) [\dots] a new vector of power series \(f'=f'(x',w)\) in the first \(n-1\) components \(x'=(x_1,\ldots, x_{n-1})\) of the \(x\)-variables and in new variables \(w=(w_0,\ldots, w_{d-1})\) such that the existence of a formal or convergent solution \(y(x)\) to \(f(x,y)=0\) is \textit{equivalent} to the existence of a formal or convergent solution \(w(x')\) of \(f'(x',w)=0\).''
Then ``We will show that the formal solutions \(\widehat y(x)\) of the first system are in one-to-one correspondence with the formal solutions \(\widehat w(x')\) of the latter system, and that the same correspondence holds for the convergent solutions [\dots]''.
This claim is absolutely wrong as it can be checked already in Step 1 of the proof:
(a) Indeed on line 5 of Step 1 (p. 604) the author already replaces the ideal generated by \(f\) by the ideal of all convergent equations satisfied by \(\widehat y(x)\). This replacement depends on \(\widehat y(x)\) and will produce a set of equations whose solutions are not in correspondence with the solutions of \(f=0\). We provide an explicit counter-example: For instance let
\[
y(x)=(x_1,0,\ldots, 0)
\]
and \(f(x,y)=y_2\). We have that \(f(x,y(x))=0\). The ideal of all convergent equations satified by \(y(x)\) is the ideal generated by
\[
f':=(y_1-x_1, y_2,\ldots, y_m)
\]
which is really much bigger than the ideal generated by \(f\). Moreover \(y(x) =(2x_1,0,\ldots,0)\) is a solution of \(f=0\) but not of \(f'=0\).
(b) A bit further one more reduction destroys also this correspondence. From \(f\) (generating the ideal \(I\)) the author constructs a new system of equations \(f^*\) (generating an ideal \(I^*\)) for which there exist \(g\notin I\) and \(b\in\mathbb N\) and such that:
\[
I^*\subset I\text{ and }g^b\cdot I\subset I^*.
\]
The first inclusion shows that the solutions of \(f=0\) are solutions of \(f^*=0\). But the second inclusion only shows that the solutions of \(f^*=0\) \textit{which are not solutions of \(g=0\)} are also solutions of \(f=0\). So the solutions of \(f=0\) and \(f^*=0\) are not in one-to-one correspondence but \(f\) is replaced by \(f^*\) in the proof.
\(\bullet\) Then the author's attempt of providing a systematic argument to prove ``simultaneously'' the next results leads him to make fatal mistakes or improper shortcuts that make the proofs of the Parametrized Approximation Theorem, the Strong Approximation Theorem I, the Strong Approximation Theorem II and the Uniform Strong Approximation Theorem incorrect. Let us explain some of them (at least the most important ones from our point of view):
(a) Parametrized Approximation Theorem. Let \(\mathbb K\) be a valued field of characteristic zero and let \(f(x,y)\) be a vector of convergent power series in two sets of variables \(x\) and \(y\). Assume given a formal power series solution \(y(x)\) vanishing at 0,
\[
f(x,y(x)) = 0.
\]
Then there exists a convergent power series solution \(y(x, z)\) depending on \(x_1,\dots,x_n\) and new variables \(z = (z_1, \ldots, z_s)\),
\[
f(x,y(x,z)) = 0,
\]
and a vector of formal power series \(\widehat z(x) = (\widehat z (x), \ldots, \widehat z(x))\) vanishing at 0 such that
\[
\widehat y(x) = y(x, \widehat z(x)).
\]
The only proof is given by these two sentences (point (a) of Section 4 on page 607):``The proof of the parametrization theorem in both settings, the convergent and the algebraic one, does not need any substantial changes, provided that one refers in the induction step also to the stronger assertion as indicated in the theorem. The partial parametrization of the solution set of \(f (x, y) = 0\) nearby a formal solution \(\widehat y(x)\) is then constructed by working again backwards from the partial parametrization of the system \(f'(x', w) = 0\)''. But this is not enough. Indeed it does not explain why and how, at each step of the induction, one has to introduce new variables \(z\) which do not appear in the statement or in the proof of the ``Analytic Approximation Theorem''. For instance for the case \(n=0\) the solutions of \(f=0\) are power series in \(n=0\) variables, i.e. they are elements of the base field \(\mathbb K\). Thus formal and convergent solutions are the same and no variable \(z\) is required. So when one deduces the case \(n=1\) from the case \(n=0\) one has to explain why the variables \(z\) appear. This is in fact the whole difficulty of the proof of this theorem.
(b) In the proof of the \textit{Strong Approximation Theorem I} (point (b), page 607), the author introduces ``the ideal \(I\) of convergent power series \(h(x, y)\) for which the order of \(h(x, y^{(k)}(x))\) tends towards infinity for all sequences \(y^{(k)}(x)\) of power series vectors for which the order of \(f(x,y^{(k)}(x))\) tends towards infinity''. Then he claims that this ideal is prime, a fact that is essential for the rest of the proof. But this claim is wrong as the following example shows (even if the ideal \((f)\) of the equations is prime):
Let \(f=y_1^2y_2^2+xy_3^2y_4^2\) where \(x\) is one variable. If \(y^{(k)}(x)\) is sequence such that the order of \(f(x,y^{(k)}(x))\) tends towards infinity then the order of \((y_1^{(k)}y_2^{(k)})^2\) and the order of \(x(y_3^{(k)}y_4^{(k)})^2\) tend towards infinity since their initial terms cannot cancel (the order of the first one is even while the order of the second one is odd). So the orders of \(y_1^{(k)}y_2^{(k)}\) and \(y_3^{(k)}y_4^{(k)}\) tend also towards infinity. Thus \(y_1y_2\in I\) and \(y_3y_4\in I\).
But if we consider the sequence \(y^{(k)}(x)\) given by
\[
y^{(2l)}(x)=(x^{2l},1,x^{2l},1) \;\;\forall l\geq 0
\]
\[
y^{(2l+1)}(x)=(1,x^{2l+1},1, x^{2l+1})\;\;\forall l\geq 0
\]
we have that the order of \(f(x,y^{(k)}(x))\) tends towards infinity but none of the orders of the \(y_i^{(k)}(x)\) tends towards infinity. So neither \(y_1\) nor \(y_2\in I\) and \(I\) is not a prime ideal.
(c) A second problem concerns the proof of the Strong Approximation Theorem II. For the convenience of the reader we state this result:
\textit{Strong Approximation Theorem II:} Let \(f (x, y)\) be a vector of formal power series in two sets of variables \(x\) and \(y\). For any \(c\in\mathbb N\) there exists an integer \(e\in\mathbb{N}\) such that if \(f(x,y) = 0\) admits an approximate solution \(y(x)\) up to degree \(e\) and vanishing at 0,
\[
f(x,y(x)) \in (x)^e,
\]
then there exists an exact formal solution \(\widetilde y(x)\) to \(f (x, y) = 0\), as
\[
f(x,\widetilde y(x)) = 0,
\]
and such that \(\widetilde y(x) \equiv y(x)\) modulo \((x)^c\).
In the proof of this theorem the author writes (line -14 on page 608) ``So fix some \(e\), and consider an approximate solution \(y(x)\) of \(f(x,y) = 0\) up to degree \(e\).'' Then he writes ``If \(g(x,y(x)) \equiv 0\) modulo \((x)^e\), increase \(e\) by 1 and repeat.'' which does not seem possible since \(y(x)\) may not be an approximate solution of \(f(x,y)=0\) up to degree this new \(e\) (i.e. \(e+1\)) Nevertheless, a few lines further, he writes if \(g(x,y(x))\neq 0\) modulo \((x)^e\) for some \(e\), we may apply a generic linear triangular coordinate change to turn \(g(x, y(x))\) into an \(x_n\)-regular series of order \(e\).''
Thus the main problem here is that for this new integer \(e\), \(y(x)\) has no reason to be an approximate solution of \(f(x,y)=0\) up to degree \(e\) and this reduction cannot apply. For instance if we have
\[
f(x,y(x))\in (x)^{1000}\backslash (x)^{1001}
\]
\[
\text{ and }g(x,y(x))\in (x)^{2000}\backslash (x)^{2001}
\]
the argument consists in choosing \(e\) being simultaneously equal to an integer less than 1000 and to an integer greater than 2001.
Let us mention that the same problem appears also in the proof of the \textit{Uniform Strong Approximation Theorem}.
\(\bullet\) Finally let us mention another problem that may have consequences for the uninformed reader. On page 615 one reads: ``The set of algebraic series forms a subring \(\mathbb K\langle x\rangle\) of \(\mathbb K[[x]]\) which is closed under derivation but not under composition with other algebraic series.'' But it is well known that this ring is closed under composition with other algebraic series: if \(x\) is a set of variables and \(y\) is a single variable, and \(g(x,y)\) and \(f(x)\) are two algebraic power series, we denote by \(P(x,y,t)\) and \(Q(x,y)\) irreducible polynomials such that
\[
P(x,y,g(x,y))=0 \text{ and } Q(x,f(x))=0.
\]
Then we have that
\[
R(x,t):=\text{Res}_y(P(x,y,t),Q(x,y))
\]
is nonzero and
\[
R(x,g(x,f(x)))=0.
\]
So \(g(x,f(x))\) is algebraic. Then, in the general case, one deduces the algebraicity of \(g(x,f_1(x),\ldots, f_m(x))\), where \(g\) and the \(f_i\) are algebraic power series, by induction on \(m\).
Editorial remark: Meanwhile, an erratum has appeared [\url{doi:10.1090/bull/1606}] to clarify and complement various places in the article. Artin approximation; algebraic power series History of commutative algebra, Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces The classical Artin approximation theorems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author focuses on the generic Newton polygon of an \(n\)-dimensional parallelotope \(\Delta\). His main contribution in this paper is to prove the distribution of slopes of the generic Newton polygon GNP\((\Delta)\), when \(p\) is not necessary to be ordinary with respect to \(\Delta\). However, for technical reasons, it is difficult to prove in general that the slopes of the \(L\)-function associated to a non trivial finite character of \(\mathbb{Z}_ p\) and a generic polynomial \(f\) whose convex hull is an \(n\)-dimensional paralleltope \(\Delta\) form a union of arithmetic progression when \(f\) is a multi-variable polynomial. generic Newton polygon; \(L\)-functions Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta and \(L\)-functions in characteristic \(p\), Exponential sums Generic Newton polygon for exponential sums in \(n\) variables with parallelotope base | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author discusses the main ideas to a proof of the theorem:
Let \(p_1,\dots, p_n\in\mathbb{P}^2\) be general points. Suppose that there exists an integral plane curve of degree \(d\) which is singular only at \(p_1, \dots,p_n\) with ordinary multiple points of multiplicities \(m_i\geq 3\) at \(p_i\). Then
\[
\dim H^0\Bigl(\mathbb{P}^2, {\mathcal I}_{\sum^n_{i=1} m_ip_i} (d)\Bigr)= e(d,m_1, \dots,m_n),\quad i= \max\left\{0, {d+2\choose 2}-\sum{m_i+1 \choose 2}\right\}.
\]
He also sketches a proof of it under two further hypotheses:
(1) \(n=2\), \(m_1 \neq m_2\) and \(e(d,m_1,m_2)>0\);
(2) for the general member \(C\) of the system the pullback of the series to the normalization is complete; i.e., if \(n:C_0\to C\) is the normalization map, \(h^0(C_0,n^*{\mathcal O}_{\mathbb{P}^2}(1))=3\). expected dimension; multiple points Singularities of curves, local rings, Plane and space curves, Global theory and resolution of singularities (algebro-geometric aspects) On families of equisingular plane 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 \(f:\mathbb{R}^n\rightarrow \mathbb{R}\) be a \(C^{\infty}\)-smooth function. The smallest set \(B(f)\subset \mathbb{R}\) outside which the level sets of \(f\) define a locally trivial smooth fibration is its bifurcation set. The values of \(f\) outside \(B(f)\) are called typical. The author introduces the framework for finding out conditions implying that \(y\in \mathbb{R}\) is a typical value. This framework is adopted to obtain some currently known conditions for estimating the set of bifurcation values. Two new such conditions are added. It is also shown that the trivialization of \(f\) in a neighbourhood \(U\) of \(f^{-1}(y)\) can always be obtained by integrating its gradient with respect to some metric on \(U\). bifurcation values; fibrations; Malgrange condition; trivialization Fibrations, degenerations in algebraic geometry, Critical points of functions and mappings on manifolds Estimating the set of bifurcation values of a smooth function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce the notion of a projective hull for subsets of complex projective varieties parallel to the idea of a polynomial hull in affine varieties. With this concept, a generalization of J. Wermer's classical theorem on the hull of a curve in \(\mathbb C^n\) is established in the projective setting. The projective hull is shown to have interesting properties and is related to various extremal functions and capacities in pluripotential theory. A main analytic result asserts that for any point \(x\) in the projective hull \(\widehat{K}\) of a compact set \(K\subset\mathbb P^n\) there exists a positive current \(T\) of bidimension \((1,1)\) with support in the closure of \(\widehat {K}\) and a probability measure \(\mu\) on \(K\) with \(dd^cT= \mu-\delta_x\). This result generalizes any Kähler manifold and has strong consequences for the structure of \(\widehat{K}\).
We also introduce the notion of a projective spectrum for Banach graded algebras parallel to the Gelfand spectrum of a Banach algebra. This projective hull is shown to play a role (for graded algebras) completely analogous to that played by the polynomial hull in the study of finitely generated Banach algebras.
This paper gives foundations for generalizing many of the results on boundaries of varieties in \(\mathbb C^n\) to general algebraic manifolds. polynomial hull; Gelfand transformation; analytic varieties and their boundaries; Jensen measures; extremal functions; quasi-plurisubharmonic functions; pluripolar sets \beginbarticle \bauthor\binitsF. R. \bsnmHarvey and \bauthor\binitsH. B. \bsnmLawson, \bsuffixJr., \batitleProjective hulls and the projective Gelfand transform, \bjtitleAsian J. Math. \bvolume10 (\byear2006), page 607-\blpage646. \endbarticle \OrigBibText F. R. Harvey and H. B. Lawson, Jr., Projective hulls and the projective Gelfand transform , Asian J. Math., 10 (2006), pp. 607-646. \endOrigBibText \bptokstructpyb \endbibitem Holomorphic convexity, Commutative Banach algebras and commutative topological algebras, Cycles and subschemes, General pluripotential theory, Banach algebras of continuous functions, function algebras Projective hulls and the projective Gelfand transform | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article uses Wu's characteristic set algorithm [ \textit{W.-T. Wu},`` Mechanical theorem proving in geometries'' (1994; Zbl 0831.03003)] as a basic part of an algorithm to find a weak stratification of \({\mathcal F}\) semialgebraic sets. Let \(K\) be a computable ordered subfield of the field \({\mathbb{R}}\) of real numbers, \(n\) a natural number. Let \(\mathcal F=\{ f_1,...,f_s\}\) be a list of functions, \(f_i: {\mathbb{R}}^n\rightarrow {\mathbb{R}}\) such that in some neighbourhood of each point in \({\mathbb{R}}^n\), \(f_i\) is represented by an absolutely convergent power series. A polynomial in \(K[x_1,\ldots,x_n,f_1,\ldots,f_s]\) is called an \({\mathcal F}\)-polynomial. \({\mathcal F}\) is called Noetherian if all the partial derivatives of all the functions in \(\mathcal F\) are \(\mathcal F\)-polynomials.
The article deals with semialgebraic sets that is with sets defined by Boolean combinations of equalities and inequalities among \(\mathcal F\)-polynomials. For \(\alpha\in {\mathbb{R}}^n,\) \(\epsilon >0,\) denote by \(N_{\epsilon}(\alpha)\) the open ball of radius \(\epsilon\) around \(\alpha.\) A set \(S\subset {\mathbb{R}}^n\) is called a \(d-\)dimensional manifold if for every \(\alpha\in S\) there is a number \(\epsilon>0\) such that \( S\cap N_{\epsilon}(\alpha)\) is analytically diffeomorphic to \({\mathbb{R}}^d \times 0^{n-d}.\) A stratification of a set is a decomposition of it into finitely many disjoint manifolds.
Let the set \(S\) be given by a semialgebraic condition. According to the algorithm developed in the article a manifold \(M\) is removed from \(S,\) where \(M\) is defined by a so called basic condition. It is required that the problem of stratifying \(S\setminus M\) be in some sense less difficult than the problem of stratifying \(S,\) so this procedure will always terminate (provided that the notion of difficulty generates a well ordering).
See also the paper by \textit{A. Gabrielov} and \textit{N. Vorobjov}, Discrete Comput. Geom. 14, No.~1, 71-91 (1995; Zbl 0832.68056). weak stratification; semianalytic sets; semialgebraic sets Richardson, D.: Weak Wu stratification in rn. J. symbolic comput. 28, 213-223 (1999) Semialgebraic sets and related spaces, Symbolic computation and algebraic computation, Computational aspects in algebraic geometry Weak Wu stratification in \(\mathbb{R}^n\) | 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 This paper treats a generalization of the Briançon-Skoda theorem about integral closures of ideals for graded systems of ideals satisfying a certain geometric condition. This work divides in two sections, the first contains some definitions and basic properties of stable graded systems of ideals. The main result of section 2 is the following:
Let \(a_{\bullet}=\{a_n\}_{n\in \mathbb N}\) be a stable graded system of ideals. Then there exists an integer \(C\) so that for all \(n\gg 0\), \(\mathcal{J}(Cn.{a}_{\bullet})\subseteq {a}_{n},\) where \(\mathcal{J}(Cn.a_{\bullet})\) denotes the level \(n\) asymptotic multiplier ideal attached to graded system of ideal \(a_{\bullet}\).
As asymptotic multiplier ideals are integrally closed, the fact that \(\overline{a}_m \subseteq \mathcal{J}(m.a_{\bullet})\) then implies the following generalization of the aforementioned result of Briançon and Skoda.
Let \(a_{\bullet}=\{a_n\}_{n\in \mathbb N}\) be a stable graded system of ideals. Then there exists a positive integer \(C\), such that for all \(n\gg 0\), \({\overline{a}}_{Cn}\subseteq a_n\). asymptotic order of vanishing; ideal sheaf; level-\(n\) asympotic multiplier ideal; graded system of ideals; smooth complex variety; symbolic power of an ideal Ideals and multiplicative ideal theory in commutative rings, Integral closure of commutative rings and ideals, Structure, classification theorems for modules and ideals in commutative rings, Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry A Briançon-Skoda type theorem for graded systems of 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 The book is concerned with algorithms for proving inequalities involving multivariate polynomials with real coefficients over the reals. Although it briefly includes classical results, it is essentially devoted to describe the contributions of the authors to algorithmic polynomial optimization. As the authors assert, ``this book is far from a systematic or comprehensive introduction to all well-known theories and algorithms in the field. It is indeed a collection of practical algorithms for polynomial inequality proving and discovering developed by the authors and their collaborators in recent years''.
The contents are as follows. Chapter 1 is devoted to the basics of pseudo-division, resultants and subresultants of univariate polynomials with coefficients in a domain \(\mathcal{R}\). It is worth noting the proof of the subresultant chain theorem in the case where the degrees of both polynomials are equal.
Chapter 2 and 3 are concerned with decompositions of polynomial systems over an arbitrary field \(\mathcal{K}\) of characteristic zero and polynomial systems of equalities and inequalities over the reals. In Chapter 2, the decomposition of zero-dimensional ``parametric'' and ``constant'' systems into triangular systems are discussed, as well as related concepts such as successive pseudo-divisions and successive resultants (by a triangular system) and regular chains. A variant of \textit{W.-T. Wu}'s decomposition algorithm [Mechanical theorem proving in geometries. Berlin: Springer (1994; Zbl 0831.03003)] which keeps multiplicities is considered. Then, Chapter 3 considers decompositions of parametric and constant polynomial systems of equalities and inequalities (``semi-algebraic systems'', SAS for short) with rational coefficients towards the description of the set of real solutions. Here, a critical concept is that of ``border polynomial'' of a given system, which is obtained by a process of successive discriminants, and allows the authors to characterize the regions (of a zero-dimensional system) with a given number of real solutions. Both zero-dimensional and positive-dimensional systems are discussed.
Chapters 4, 5 and 6 discuss real root counting, isolation and classification. The main result of Chapter 4, the discrimination theorem, establishes the theoretical basis for the algorithmic determination of the number of real roots of a polynomial \(f\in\mathbb{R}[x]\) satisfying a sign condition \(g(x)>0\), where \(g\in\mathbb{R}[x]\). Real root isolation for SASs, considered in Chapter 5, is based on a combination of triangularizations of SASs and interval arithmetic. Finally, in Chapter 6 the real root classification problem is addressed. This problem asks for the determination of the sets of parametric instances of a parametric SAS corresponding to a system with real solutions, or positive-dimensional real solution sets, or zero-dimensional real solution sets with a prescribed number of elements. SASs with generic complex dimension zero is dealt with the tool of the previous chapters, while in the positive-dimensional case cylindrical algebraic decomposition (CAD) is applied. Then, the Maple program DISCOVERER for real root classification and isolation is presented and applied to problems concerning the automated discovering of geometric inequalities, the algebraic analysis of biological systems and program verification.
Chapter 7 is devoted to two variants of CAD called open CAD and open weak CAD, and their application to determine positive semi-definiteness and the global infimum of a polynomial over \(\mathbb{R}^n\). In both variants of CAD, a cylindrical algebraic sample of points is obtained, based on the computation of ``Brown-McCallum'' projections [\textit{C. W. Brown}, J. Symb. Comput. 32, No. 5, 447--465 (2001; Zbl 0981.68186)] and successive resultants.
Chapter 8 considers the SASs which have a triangularized form where all their main variables occur with degree at most 2. In such a case, ``dimension-decreasing'' algorithms are discussed, which deal with real inequalities involving radicals (square roots). Further, the MAPLE program BOTTEMA for inequality proving and polynomial optimization is presented.
In Chapter 9, the problem of deciding whether a given multivariate polynomial with real coefficients can be expressed as a sum of squares (SOS) is addressed. The approach relies on work of \textit{V. Powers} and \textit{T. Wörmann} [J. Pure Appl. Algebra 127(1), 99--104 (1998; Zbl 0936.11023)], which reduces the question to one of semi-definite programming, combined with strategies for reducing the size of the problem. Chapter 10 considers a related question, namely whether an homogeneous multivariate polynomial with real coefficients is positive semi-definite on the positive orthant \(\mathbb{R}_+^n\). In particular, a method, called ``successive difference substitution'', based on checking positivity of the coefficients of the polynomial under consideration under certain linear changes of variables, is discussed. Finally, Chapter 11 is devoted to a class of inequalities beyond the Tarsky model: checking positivity on \(\mathbb{R}_+^n\) of a symmetric homogeneous polynomial \(f(x_1,\ldots,x_n)\) for all positive integers \(n\).
Numerous examples are provided and a pseudo-code of the relevant algorithms is included. polynomial inequality; real polynomial; resultant; subresultant; cylindrical algebraic decomposition; sums of squares Research exposition (monographs, survey articles) pertaining to real functions, Research exposition (monographs, survey articles) pertaining to computer science, Inequalities for trigonometric functions and polynomials, Mechanization of proofs and logical operations, Semialgebraic sets and related spaces, , Symbolic computation and algebraic computation Automated inequality proving and discovering | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 semialgebraic set in \(\mathbb{R}^n\) is a finite union of sets of the form \(\{x\in \mathbb{R}^n : f(x)=0, g_1(x)>0,\dots ,g_k(x)>0\}\), where \(f, g_1,\dots ,g_k\) are polynomials on \(\mathbb{R}^n\). A continuous map of semialgebraic sets is said to be semialgebraic if its graph is semialgebraic. Let \(\Gamma\) denote the pseudogroup of all semialgebraic analytic diffeomorphisms between semialgebraic open subsets of \(\mathbb{R}^n\). An abstract Nash manifold is a manifold with \(\Gamma\)-structure and with a finite system of coordinate neighbourhoods. An abstract Nash manifold \(V\) is said to be affine if there exists an analytic semialgebraic imbedding \(V\subset \mathbb{R}^m\). One defines in the obvious way the \(C^r\) Nash manifolds and \(C^r\) Nash mappings between \(C^r\) Nash manifolds. If there exists a \(C^r\) Nash embedding of a \(C^r\) Nash manifold \(M\) into a Euclidean space, we say that \(M\) is affine. It is known that every affine Nash manifold is actually Nash diffeomorphic to a closed Nash submanifold of some finite dimensional real vector space. Chevalley's theorem is a fundamental result in the structure theory of algebraic groups. In the present paper, the authors formulate and prove an analogue of Chevalley's theorem in the setting of affine Nash groups. After presenting some preliminaries on algebraic groups and algebrizations, the authors state and prove Chevalley's theorem for affine Nash groups. Theorem 4.2. Let \(G\) be a connected affine Nash group. Then there exists a unique connected normal almost linear Nash subgroup \(L\) of \(G\) such that \(G/L\) is an abelian Nash manifold. Nash manifold; Nash group; algebraic group; Chevalley's theorem General properties and structure of real Lie groups, Group varieties, Nash functions and manifolds Chevalley's theorem for affine Nash groups | 0 |
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