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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main object of the paper under review is a linear subspace \(A\subset S^2V^*\), where \(V\) denotes a complex vector space of dimension \(m\). One can naturally identify \(A\) with a vector space of symmetric \(m\times m\) matrices. If every non-zero matrix from \(A\) has rank \(r\), the space \(A\) is said to be of constant rank \(r\). One can ask how large such an \(A\) can be. The answer is given by theorem 2.14:
The biggest possible dimension of \(A\) equals \(m-r+1\). [This result is classical for \(r\) odd and is due to \textit{R.~Meshulam}, Linear Algebra Appl. 216, 93-96 (1995; Zbl 0827.15026) for \(r\) even.]
The authors give a new proof based on algebraic-geometric methods. Their main observation [going back to the paper of \textit{L.~Ein}, Invent. Math. 86, 63-74 (1986; Zbl 0603.14025)] is the following one: Linear spaces of constant rank can arise from a smooth variety \(X\) whose dual variety \(X^*\) is degenerate. To be more precise, the authors define the defect of an \(n\)-dimensional variety \(X\subset \mathbb P^N\) as \(\delta = N-1-\dim (X^*)\) and show (theorem 3.4) that the linear system of quadrics generated by the second fundamental form of \(X^*\) at a smooth point is of projective dimension \(\delta\) and constant rank \(n-\delta\). This yields a \(\delta +1\) dimensional linear subspace \(A\subset S^2\mathbb C^{N-1-\delta}\) of constant rank \(n-\delta\). As a by-product, the authors get a new proof of a theorem of \textit{F.~L. Zak} [Funct. Anal. Appl. 21, 32-41 (1987); translation from Funkts. Anal. Prilozh. 21, No. 1, 39-50 (1987; Zbl 0623.14026)] stating that \(\dim (X^*)\geq\dim (X)\) provided \(X\) is smooth. symmetric matrix; dual variety; degeneracy loci B. Ilic and J. M. Landsberg, \textit{On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties}, Math. Ann., 314 (1999), pp. 159--174. Determinantal varieties, Matrix pencils On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual 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 R be a Noetherian ring. An R-sequence \(x_ 1,...,x_ n\) is built by always choosing \(x_ i\) to be not in the union of the primes in Ass R/(x\({}_ 1,...,x_{i-1})R\). In the terminology of this paper, \(A(I)=Ass R/I\) is a grade scheme and its associated grade function is \(f(I)=the\) (classical) grade of I. If instead, we take \(A(I)=\{P| P\quad is\quad an\) essential prime (respectively, asymptotic prime) of \(1\}\), and construct a sequence in the same way, we get the essential (respectively asymptotic) grade function. This paper studies the abstract nature of such grade schemes and their associated grade functions, in particular, finding which properties of the above examples are abstract in nature, and which are indigeneous to the given example.
The main theorem says that if f is a function from the set of all ideals in all localizations of R, to the set of nonnegative integers, then f is the grade function of some grade scheme A(I) if and only if f satisfies (i) f(I\({}_ S)=\min \{f(P_ P)| P_ S\) is
\[
a prime
\]
containing \(I/_ S\}\), (ii) f(P\({}_ P)\leq height P\) for all \(P\in Spec R\), and \((iii)\quad if\quad (Q,U)\) is a conforming pair in R, and if \(f(P_ P)\leq n\) for all \(P\in U\), then \(f(Q_ Q)\leq n-1\). - Here, (Q,U) is a conforming pair if \(Q\in Spec R\) and U is an infinite set of primes, each properly containing Q, such that if W is any infinite subset of U, then \(\cap \{P\in W\}=Q\). asymptotic grade; essential grade; Noetherian ring; grade function; grade scheme Commutative ring extensions and related topics, Commutative Noetherian rings and modules, Relevant commutative algebra Grade schemes and grade functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(J=(f_{1},\ldots ,f_{p})\) and \(I=(g_{1},\ldots ,g_{q})\) be proper ideals in the ring \(\mathcal{O}_{n}\) of analytic function germs \((\mathbb{C}^{n},0)\rightarrow \mathbb{C}\). The Łojasiewicz exponent \(\mathcal{L}_{J}(I)\) of \(I\) with respect to \(J\) is the infimum of \(\alpha \in \mathbb{R}_{\geq 0}\) such that \(\left\vert \left\vert f\right\vert \right\vert^{\alpha }\leq C\left\vert \left\vert g\right\vert \right\vert \) in a neighbourhood of \(0\in \mathbb{C}^{n}.\) It is known \(\mathcal{L}_{J}(I)\) is characterized algebraically as
\[
\mathcal{L}_{J}(I)=\inf \{r/s:r,s\in \mathbb{N},\ J^{r}\subset \overline{I^{s}}\}=\inf \{r/s:r,s\in \mathbb{N},\ e(I^{s})=e(I^{s}+J^{r})\},
\]
where \(\overline{I}\) means the algebraic closure of the ideal \(I\) and \(e(I)\) the Samuel multiplicity of \(I\). The author generalizes the Łojasiewicz exponent for systems of ideals \((I_{1},\ldots ,I_{n})\) instead of one \(I\) (and even in general Noetherian local rings \((R,\mathfrak{m)}\) instead of \(\mathcal{O}_{n})\). He defines
\[
\mathcal{L}_{J}(I_{1},\ldots ,I_{n})=\inf \{r/s:r,s\in \mathbb{N},\ \sigma(I_{1}^{s},\ldots ,I_{n}^{s})=\sigma (I_{1}^{s}+J^{r},\ldots
,I_{n}^{s}+J^{r})\},
\]
provided \(\sigma (I_{1},\ldots ,I_{n})<\infty ,\) where \(\sigma (I_{1},\ldots,I_{n})\) is the Ree's mixed multiplicity of ideals \(I_{1},\ldots ,I_{n}\). He studies the sequence of the Łojasiewicz exponents \(\mathcal{L}_{J}^{\ast}(I):=(\mathcal{L}_{J}^{(n)}(I),\ldots,\mathcal{L}_{J}^{(1)}(I)),\) where \(\mathcal{L}_{J}^{(i)}(I):=\mathcal{L}_{J}(I,\ldots ,I,J,\ldots ,J)\) with \(I\) repeated \(i\) times and \(J\) repeated \((n-i)\) times in the case \(I,J\) are monomial ideals of finite colength in \(\mathcal{O}_{n}\). The author relates this notion with the combinatorial objects associated to \(I\) and \(J\) -- the Newton polyhedra. ring of analytic function germ; Łojasiewicz exponent; integral closure of ideal; mixed multiplicities of an ideal; monomial ideal; Newton polyhedron Multiplicity theory and related topics, Local complex singularities, Integral closure of commutative rings and ideals, Toric varieties, Newton polyhedra, Okounkov bodies The sequence of mixed Łojasiewicz exponents associated to pairs of monomial 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 \(X\) be an arithmetic surface defined over \(\mathbb{Z}\), \(X_\infty \) the compact Riemann surface associated to the generic fiber \(X_\mathbb{Q}\) and \(F_\infty\) the complex conjugation of \(X_\infty\). A metrized line bundle on \(X\) is a pair \((L,h)\) of a line bundle \(L\) on \(X\) and a smooth \(F_\infty\)-invariant Hermitian form \(h\) on \(L\otimes\mathbb{C}\). Then the intersection pairing of two metrized line bundles on \(X\) is established by S. Arakelov and many deep results about this pairing have been obtained.
In the paper under review the author presents a generalization of the notion of metrized line bundle on \(X\). He calls it a logarithmically singular metrized line bundle. It is a pair of a line bundle with a metric which is smooth, \(F_\infty\)-invariant and Hermitian on the complement of a finite set of points on \(X_\infty\) and has certain growth conditions around each point. The intersection pairing can also be extended to logarithmically singular metrized line bundles.
As an application, the author treats the case of modular curves. The metric on the line bundle of modular forms on it which gives rise to the Petersson inner product becomes logarithmically singular at cusps and elliptic points. He shows that its self-intersection number is expressed by the value of the Riemann zeta function. intersection pairing; logarithmically singular metrized line bundle; modular curves; self-intersection number; Riemann zeta function U. Kühn, Generalized arithmetic intersection numbers , C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 283--288. Arithmetic varieties and schemes; Arakelov theory; heights, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Generalized arithmetic intersection 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 Levin's method produces a parameterization of the intersection curve of two quadrics in the form
\[
p(u)=a(u)\pm d(u)\sqrt{s/(u)}
\]
where \(a(u)\) and \(d(u)\) are vector valued polynomials, and \(s(u)\) is a quartic polynomial. This method, however, is incapable of classifying the morphology of the intersection curve, in terms of reducibility, singularity, and the number of connected components, which is a critical structural information required by solid modeling applications. We study the theoretical foundation of Levin's method, as well as the parameterization \(p(u)\) it produces. The following contributions are presented in this paper: (1) It is shown how the roots of \(s(u)\) can be used to classify the morphology of an irreducible intersection curve of two quadric surfaces. (2) An enhanced version of Levin's method is proposed that, besides classifying the morphology of the intersection curve of two quadrics, produces a rational parameterization of the curve if the curve is singular. (3) A simple geometric proof is given for the existence of a real ruled quadric in any quadric pencil, which is the key result on which Levin's method is based. These results enhance the capability of Levin's method in processing the intersection curve of two general quadrics within its own self-contained framework. quadric surface; intersection; stereographic projection Wang, W., Goldman, R., Tu, C.: Enhacing Levin's method for computing quadric-surface intersections. Comput. Aided Geom. Des. 20(7), 401--422 (2003) Numerical aspects of computer graphics, image analysis, and computational geometry, Computational aspects of algebraic curves, Computer-aided design (modeling of curves and surfaces), Computer graphics; computational geometry (digital and algorithmic aspects), Computer science aspects of computer-aided design Enhancing Levin's method for computing quadric-surface intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present new results on standard basis computations of a 0-dimensional ideal \(I\) in a power series ring or in the localization of a polynomial ring over a computable field \(K\). We prove the semicontinuity of the ``highest corner'' in a family of ideals, parametrized by the spectrum of a Noetherian domain \(A\). This semicontinuity is used to design a new modular algorithm for computing a standard basis of \(I\) if \(K\) is the quotient field of \(A\). It uses the computation over the residue field of a ``good'' prime ideal of \(A\) to truncate high order terms in the subsequent computation over \(K\). We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps \(A={\mathbb{Z}}\) and \(A=k[t], k\) any field and \(t\) a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for \(A={\mathbb{Z}} \), since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular and we present several examples illustrating its power. standard bases; algorithm for zero-dimensional ideals; semicontinuity; highest corner Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Effectivity, complexity and computational aspects of algebraic geometry Using semicontinuity for standard bases computations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series \( f\) in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call \( \mathcal {U}\) and \( \mathcal {D}\) operators, whose action on the Schur symmetric function \( s_{\lambda }\) can be simply expressed in terms of powers of the contents of the cells in \( \lambda \). Among our results, we construct the \( \mathcal {U}\) and \( \mathcal {D}\) operators explicitly as partial differential operators in the underlying power sum symmetric functions. We also give a combinatorial interpretation for the content series in terms of the Jucys-Murphy elements in the group algebra of the symmetric group. This leads to an interpretation for the content series as a generating series for branched covers of the sphere by a Riemann surface of arbitrary genus \( g\). As particular cases, by suitable choice of the underlying series \( f\), the content series specializes to the generating series for three known classes of branched covers: Hurwitz numbers, monotone Hurwitz numbers, and \( m\)-hypermap numbers of Bousquet-Mélou and Schaeffer. We apply our pde to give new and uniform proofs of the explicit formulas for these three classes of numbers in genus 0. generating functions; transitive permutation factorizations; symmetric functions; Jucys-Murphy elements; contents of partitions Exact enumeration problems, generating functions, Permutations, words, matrices, Symmetric functions and generalizations, Combinatorial aspects of partitions of integers, Coverings in algebraic geometry Contents of partitions and the combinatorics of permutation factorizations in genus 0 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 submanifold M of \({\mathbb{R}}^ n\) of class \(C^ r\) \((r=0,...,\infty,\omega)\) is called a \(C^ r\)-Nash manifold, and a \(C^ r\)-Nash map is a \(C^ r\)-map of a semialgebraic graph. Using partitions of unity with Nash functions and stratifications by Nash manifolds, several results of fundamental importance are derived: Every \(C^ r\)- Nash map between \(C^ r\)-Nash manifolds can be approximated by a \(C^{\omega}\)-Nash map in the \(C^ r\)-topology. (A limit \(f_ k\to 0\) in this topology means the uniform convergence \(v_ 1...v_ sf_ k\to 0\) for any \(C^ r\)-Nash vector fields \(v_ 1,...,v_ s\) in the number \(s\leq r.)\) Every \(C^ r\)-Nash manifold M in \({\mathbb{R}}^ n\) \((1\leq r<\infty)\) can be approximated by \(C^{\omega}\)-Nash manifolds in \(C^ r\)-topology, and for any compact \(C^{\omega}\)-Nash submanifold \(N\subset M\), the approximations can be chosen identical on N. As a result, \(C^ r\)-Nash diffeomorphism classes are identical with \(C^{\omega}\)-Nash diffeomorphism classes of \(C^ r\)-Nash manifolds. (For the whole class of abstract Nash manifolds, an analogous result is not true.) Moreover, a \(C^ 0\)-vector bundle over a \(C^ r\)-Nash manifold (0\(\leq r\leq \omega)\) possesses a unique \(C^ r\)-Nash bundle structure. Nash functions; Nash manifolds; Nash diffeomorphism \beginbarticle \bauthor\binitsM. \bsnmShiota, \batitleApproximation theorems for Nash mappings and Nash manifolds, \bjtitleTrans. Amer. Math. Soc. \bvolume293 (\byear1986), no. \bissue1, page 319-\blpage337. \endbarticle \endbibitem Real-analytic and Nash manifolds, Real algebraic and real-analytic geometry Approximation theorems for Nash mappings and Nash manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal M}_{n}\) denote the vector space of all \((n,n)\)-matrices over an algebraically closed field \(F\) of characteristic zero. The author is interested in the structure of algebraic subsets of \({\mathcal M}_{n}\) which are invariant under the action of the general linear group \(GL_{n}(F)\) on \({\mathcal M}_{n}\) by conjugation, in particular he is interested in a characterization of all invariant algebraic sub-cones of \({\mathcal M}_{n}\). Standard examples are the algebraic subvarieties
\[
{\mathcal S}^{k}_{n}=\{A\in {\mathcal M}_{n}\mid \text{rank }A^{n}\leq k\}
\]
and more generally for every rank function \(r:\mathbb N\to\mathbb N\) (a weakly decreasing function \(r\) with the property \(r(j)+r(j+2)\geq 2r(j+1)\) for all \(j\in \mathbb N\)) the rank varieties
\[
\chi_{r}=\{A\in{\mathcal M}_{r(0)}\mid \text{rank }A^{j}\leq r(j) \text{ for all } j\in \mathbb N\}.
\]
The author uses information about conjugacy classes of nilpotent matrices to prove basic properties of rank varieties. He characterizes invariant irreducible algebraic sub-cones of \({\mathcal S}^{2}_{n}\) and presents some results about the linear capacity (the maximal dimension of an \(F\)-linear subspace) of these sub-cones. rank varieties; invariant cones Determinantal varieties, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Algebraic systems of matrices On \(GL_n\)-invariant cones of matrices with small stable ranks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a bounded open subset \(G\) of \(\mathbb{R}^{n}\) with algebraic boundary \(\Gamma =\partial G\) of degree \(d,\) the authors describe an algorithmic procedure for obtaining a real polynomial \(p\in I(\Gamma )\) (i.e. vanishing on \(\Gamma \)) by means of finitely many moments of the Lebesgue measure on \(G\) (up to the order \(3d\) for the general case, and \(2d\) for the convex case). The main result is Theorem 2.2, which uses a Whitney's generalization of Stokes's theorem applied over a semi-algebraic triangulation of \(\Gamma\). The authors prove this result under the technical assumption that \(x=0\) does not lie on the Zariski closure of \(\Gamma \) (i.e. the polynomials of the ideal \(I(\Gamma )\) do not vanish at \(0\)), and treat the general case by a change of variable (Section~2.3). In addition to the Lebesgue measure, the authors' technique is easily adapted to include measures of the form \(d\mu :=\exp (p(x))dx\) where \(p\in \mathbb{R}[x]\) (in this case the required number of moments takes into account the degree of \(p\)). The authors also deal with general (non-algebraic) boundaries, using approximations and illustrate their results with several examples. moment problem; semi-algebraic set; finite determination; generalized Stoke's formula; Hankel matrices Lasserre, J. B.; Putinar, M., Algebraic-exponential data recovery from moments, Discrete Comput. Geom., 54, 993-1012, (2015) Semialgebraic sets and related spaces, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Moment problems Algebraic-exponential data recovery from moments | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to study congruence properties of Taylor coefficients of algebraic functions. Let \({\mathbb{C}}_ p\) be the completion of some algebraic closure of \({\mathbb{Q}}_ p\). Let E be the completion of \({\mathbb{C}}_ p(x)\) for the Gauss norm. An analytic function in the unit disk of \({\mathbb{C}}_ p\) will be called algebraic function if it is algebraic over the field E (then it is bounded), and it will be called algebraic element if it is a uniform limit (on the unit disk) of algebraic functions.
The first step is to find a ''fine'' primitive element for each finite extension of the field E. Then we can prove finiteness properties for the ring generated by the Taylor coefficients of an algebraic function (or element). Afterwards, although \({\mathbb{C}}_ p\) is a non discrete valuation field, we can use the discrete valuation technique.
We define a Frobenius operator \(\phi\) over the ring \({\mathcal B}\) of bounded analytic functions in the unit disk of \({\mathbb{C}}_ p\) (\(\phi\) (f) looks like \(f(x^ p))\). For a function f of \({\mathcal B}\), we consider the condition (C): the vector space generated over E by the \(\phi^ n(f)\) is of finite dimension. It is shown that the algebraic functions fulfill (C) (the vector space being E(f)) and that any function satisfying (C) is an algebraic element. Translated for a linear differential equation with coefficients in E, this result gives a deep link between the existence of a strong Frobenius structure (something like to be an F-crystal) and the algebraicity of the solutions.
Finally, it is proved that a function of \({\mathcal B}\) is an algebraic element if and only if, for every n, the sequence of its Taylor coefficients can be obtained modulo \(p^ n\) by a p-automata.
Many examples of algebraic elements are given (exponential, hypergeometric, Bessel functions, diagonals of rational fractions of several variables,...), showing that this notion appears in several branches of mathematics: algebraic geometry, combinatorics,... congruence properties of Taylor coefficients; algebraic functions; algebraic element; Frobenius operator; bounded analytic functions; linear differential equation; F-crystal; p-automata; Gauss norm Christol, Gilles, Fonctions hypergéométriques bornées, Groupe d'étude d'analyse ultramétrique, 8, 1-16, (1986/1987) Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), \(p\)-adic differential equations, Formal languages and automata, Linear differential equations in abstract spaces, \(p\)-adic cohomology, crystalline cohomology Fonctions et éléments algébriques. (Algebraic functions and elements) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 under review deals with an important problem in semialgebraic geometry. Let us start with the fundamental definitions in this setting. A subset \(M \subset \mathbb{R}^m\) is basic semialgebraic if it can be expressed as \(\{x \in \mathbb{R}^m, f(x)=0, g_1(x)>0, \dots, g_{\ell}(x)>0\}\), where \(f, g_1, \dots, g_{\ell}\) are polynomials of \(\mathbb{R}[x_1, \dots, x_m]\). Then, a semialgebraic set is a finite union of basic semialgebraic sets. Given two semialgebraic sets \(M \subset \mathbb{R}^m\) and \(N \subset \mathbb{R}^n\), a continuous map \(f : M \rightarrow N\) is semialgebraic if its graph is a semialgebraic subset of \(\mathbb{R}^{m+n}\). Now, let \(\mathcal{S}(M)\) be the set of semialgebraic functions on \(M\). By means of the sum and product of functions, \(\mathcal{S}(M)\) is endowed with a structure of commutative \(\mathbb{R}\)-algebra with unit. Calling \(\mathcal{S}^*(M)\) the subset of \(\mathcal{S}(M)\) consisting of bounded functions, \(\mathcal{S}^*(M)\) is an \(\mathbb{R}\)-subalgebra of \(\mathcal{S}(M)\). Throughout the paper, the authors study jointly \(\mathcal{S}(M)\) and \(\mathcal{S}^*(M)\), and they call either of them \(\mathcal{S}^{\diamond} (M)\). Then, \(\text{Spec}^{\diamond} (M)\) is the Zariski spectrum of \(\mathcal{S}^{\diamond} (M)\) with the Zariski topology, and \(\beta^{\diamond}(M)\) its set of closed points. Call \(\texttt{r}_M : \text{Spec}^{\diamond} (M) \rightarrow \beta^{\diamond} (M)\) the natural retraction. Then, take a semialgebraic map \(\pi : M \rightarrow N\). It has associated a homomorphism of \(\mathbb{R}\)-algebras \(\varphi_{\pi}^{\diamond} : \mathcal{S}^{\diamond} (N) \rightarrow \mathcal{S}^{\diamond} (M)\) defined by \(g \mapsto g \circ \pi\). This determines two continuous morphisms, \(\text{Spec}^{\diamond}(\pi) : \text{Spec}^{\diamond} (M) \rightarrow \text{Spec}^{\diamond} (N)\), \(\mathfrak{p} \mapsto (\varphi_{\pi}^{\diamond})^{-1}(\mathfrak{p})\), and \(\beta^{\diamond} (\pi) = \texttt{r}_N \circ \text{Spec}^{\diamond}(\pi) |_{\beta^{\diamond}(M)} : \beta^{\diamond}(M) \rightarrow \text{Spec}^{\diamond} (N) \rightarrow \beta^{\diamond}(N)\).
A long list of papers by the authors of the present article during the last ten years studies the relationship between the above maps \(\pi\) and \(\text{Spec}^{\diamond}(\pi)\). Here, they consider the case in which \(\pi : M \rightarrow N\) is a semialgebraic branched covering. Consider a map \(\pi : X \rightarrow Y\). It is a finite quasi-covering if it is separated, open, closed, surjective, and its fibers are finite. Then, the branching locus of \(\pi\) is the set \(\mathcal{B}_{\pi}\) of points in \(X\) at which \(\pi\) is not a local homeomorphism. The ramification set of \(\pi\) is \(\pi (\mathcal{B}_{\pi}) = \mathcal{R}_{\pi}\), and the regular locus of \(\pi\), \(X_{\text{reg}}\), is \(X \setminus \pi^{-1}(R_{\pi})\).
The authors state Definition 2.11, according to which \(\pi\) is a branched covering if \(X_{\text{reg}}\) is dense in \(X\) and each \(y \in Y\) admits a so-called special neighborhood, which they also define in Section 2 of the paper. This specific definition of branched covering is made in the article in order to avoid anomalous cases as shown in Example 2.26. In these conditions, the main Theorem 1.1 of the paper is the following:
Let \(\pi : M \rightarrow N\) be a semialgebraic map. Then, \(\pi\) is a branched covering if and only if \(\text{Spec}^{\diamond}(\pi)\) is a branched covering, if and only if \(\beta^{\diamond} (\pi)\) is a branched covering. In that case, \(\mathcal{B}_{\text{Spec}^{\diamond}(\pi)} = \text{Cl}_{\text{Spec}^{\diamond}(M)}(\mathcal{B}_{\pi})\), \(\mathcal{B}_{\beta^{\diamond}(\pi)} = \text{Cl}_{\beta^{\diamond}(M)}(\mathcal{B}_{\pi})\), \(\mathcal{R}_{\text{Spec}^{\diamond}(\pi)} = \text{Cl}_{\text{Spec}^{\diamond}(N)}(\mathcal{R}_{\pi})\), \(\mathcal{R}_{\beta^{\diamond}(\pi)} = \text{Cl}_{\beta^{\diamond}(N)}(\mathcal{R}_{\pi})\), where \(\text{Cl}_X(A)\) stands for the closure of \(A\) in \(X\).
The other main goal of the article is to study the collapsing set of the spectral map \(\text{Spec}^{\diamond}(\pi)\) when \(\pi\) is a \(d\)-branched covering, that is to say, a branched covering such that the fibers of the points outside \(\mathcal{R}_{\pi}\) have constant cardinality \(d\). The collapsing set \(\mathcal{C}_{\pi}\) of \(\pi\) is defined as the set of points such that the fiber \(\pi^{-1}(\pi(x))\) is a singleton. Then, the authors study \(\mathcal{C}_{\text{Spec}^{\diamond}(\pi)}\) and \(\mathcal{C}_{\beta^{\diamond}(\pi)}\). In order to do that, a map \(\mu ^{\diamond} : \mathcal{S} ^{\diamond} (M) \rightarrow \mathcal{S} ^{\diamond} (N)\) is defined in Section 4 (Definition 4.1). The result is Theorem 1.2, according to which, given a semialgebraic \(d\)-branched covering \(\pi : M \rightarrow N\), then \(\mathcal{C}_{\text{Spec}^{\diamond}(\pi)}\) is the set of prime ideals of \(\mathcal{S}^{\diamond} (M)\) containing \(\text{ker}(\mu ^{\diamond})\), and it equals \(\text{Cl}_{\text{Spec}^{\diamond}(M)}(\mathcal{C}_{\pi})\), and \(\mathcal{C}_{\beta^{\diamond}(\pi)}\) is the set of maximal ideals of \(\mathcal{S}^{\diamond} (M)\) containing \(\text{ker}(\mu ^{\diamond})\), and it equals \(\text{Cl}_{\beta^{\diamond}(M)}(\mathcal{C}_{\pi}).\)
Section 2 of the paper is devoted to define and study branched coverings, and Section 3 to analyze the properties of spectral maps associated to quasi-finite coverings and branched coverings. Then, the proof of Theorem 1.2 is developed in Section 4, and that of Theorem 1.1 in Section 5. Finally, an example is detailed in an Appendix at the end of the paper. It is worth to note that the article is very carefully written, and its introduction is very enlightening both on the state-of-the-art, and on the purpose and procedures of the work. semialgebraic set; semialgebraic function; branched covering; branching locus; ramification set; ramification index; Zariski spectra; spectral map; collapsing set Semialgebraic sets and related spaces, Real-valued functions in general topology, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Chain conditions, finiteness conditions in commutative ring theory Spectral maps associated to semialgebraic branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper considers the problem of determining the asymptotic family of a plane algebraic curve \(\mathcal C\) which is assumed to have only regular points at infinity. The asymptotic family of \(\mathcal C\) is the set of algebraic plane curves that have the same asymptotic behaviour as \(\mathcal C\). The main result states that these curves form the vector space of all algebraic plane curves of the degree \(d = \deg \mathcal C\) whose defining equations share the homogeneous forms of degree \(d\) and of degree \(d-1\). This generalizes results of [\textit{A. Blasco} and \textit{S. Pérez-Díaz}, Comput. Aided Geom. Des. 31, No. 2, 81--96 (2014; Zbl 1301.14033)] for curves with only one regular infinity branch.
The characterization of asymptotic families simplifies the computation of generalized asymptotes of \(\mathcal C\) (plane algebraic curves of minimal degree, one for each infinity branch, whose distance to \(\mathcal C\) converges to zero as the curves tend to infinity) as it is possible to ignore the irrelevant coefficients of the low-degree homogeneous forms. It also allows to compute in a straightforward way an asymptotic family from its set of regular \(g\)-asymptotes by simply multiplying the \(g\)-asymptotes' equations and extracting the two homogeneous forms of highest degree.
This paper is self-contained with precise definitions of central concepts, collections of previous results, and examples illustrating results and theories. Proofs are collected in a concluding section. An obvious question for future research is the generalization of the results to the case of curves with non-regular points at infinity. implicit algebraic plane curve; parametric plane curve; infinity branches; asymptotes; perfect curves; approaching curves Computational aspects of algebraic curves, Plane and space curves, Computer-aided design (modeling of curves and surfaces), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) Determining the asymptotic family of an implicit 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 We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of ``hyperbolic measures'' on universal covers of metric graphs. Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss-Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a ``trace formula''. In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry. Arithmetic aspects of tropical varieties, Distance in graphs, Infinite graphs, Coverings of curves, fundamental group, Geometric group theory, Group actions on manifolds and cell complexes in low dimensions Canonical measures on metric graphs and a Kazhdan's 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 For an \(n\)-by-\(n\) complex matrix \(A\) and a positive integer \(k\), \(1\leq k\leq n\), the rank-\(k\) numerical range \(\Lambda_k(A)\) of \(A\) is defined as the subset \(\{\lambda\in\mathbb{C}:PAP= \lambda P\) for some rank-\(k\) orthogonal projection \(P\}\) of the complex plane. It is known that \(\Lambda_k(A)\) is a compact convex set and may be empty. When \(k= 1\), \(\Lambda_1(A)\) is exactly the classical numerical range \(W(A)\). The present paper is devoted to the study of the boundary curve of \(\lambda_k(A)\) via the homogeneous form
\[
F_A(t,x,y)= \text{det}(tI_n+ x\cdot\text{Re\,}A+ y\cdot\text{Im\,}A)
\]
associated with \(A\), where \(\text{Re\,}A= (A+ A^*)/2\) and \(\text{Im\,}A= (A- A^*)/(2i)\) are the real and imaginary parts of \(A\), respectively.
In Section 2, the authors propose a method to generate \(\partial\Lambda_k(A)\) via symbolic calculations and analysis of the singularities of the associated curve. They illustrate this by a 4-by-4 real tridiagonal matrix and a 6-by-6 complex symmetric matrix.
Then, in Section 3, they consider an explicit \(2m\)-by-\(2m\) \((m\geq 2)\) matrix \(A\) with the algebraic curve \(F_A(1,x,y)= 0\) given by the roulette curve
\[
z(s)= \exp(-i(m- 1)s)+ a\exp(ims)\quad\text{for }0\leq s\leq 2\pi,
\]
where \(a> 1\). It is shown that the total number of line segments on \(\partial\Lambda_k(A)\) for \(1\leq k\leq m\) is \((2m- 1)(m- 1)\). This shows that the upper bound \((2m- 1)(m- 1)\) for such numbers of \(2m\)-by-\(2m\) matrices is attained by such an \(A\).
Finally, in Section 4, the authors give a 4-by-4 matrix \(A\) with \(\Lambda_2(A)\) not equal to the classical numerical range \(W(B)\) of any matrix \(B\). This is in sharp contrast to the case for \(c\)-numerical ranges. It concludes with a sufficient condition that the \(n\)-by-\(n\) matrix \(A\) is such that \(\partial\Lambda_k(A)\) \((k\geq 2)\) has sharp points and \(F_A(t,x,y)\) is irreducible for the nonattainability of \(\Lambda_k(A)\) by any \(W(B)\). rank-\(k\)-numerical range; flat portion; singular point; roulette curve DOI: 10.1016/j.laa.2011.05.020 Norms of matrices, numerical range, applications of functional analysis to matrix theory, Computational aspects of algebraic curves The boundary of higher rank numerical ranges | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 real and complex \((\mu,S)\)-frames, finite frames having a given frame operator \(S\) and consisting of vectors \(\{f_1, f_2, \dots, f_N\}\) having specified lengths \(\{\mu_1, \mu_2, \dots, \mu_N\}\), respectively. In terms of the synthesis operator \(F=[f_1 f_2 \cdots f_N]\), a \((\mu,S)\)-frame satisfies the quadratic equations \(FF^*=S\) and \(\mathrm{diag}(F^* F)=\mu\). From a geometric viewpoint, the condition involving \(S\) means that \(F\) is in an ellipsoidal, deformed Stiefel manifold, while the norm condition means that it is in a product of spheres. The intersection of these two manifolds forms a variety with possible singular points. The intersection of these two matrix manifolds forms a real algebraic variety with possible singular points.
The paper characterizes the singular points in this variety as orthodecomposable frames. This result is the generalization of earlier work with Dykema on spherical tight frames [\textit{K. Dykema} and \textit{N. Strawn}, ``Manifold structure of spaces of spherical tight frames'', Int. J. Pure Appl. Math. 28, No. 2, 217--256 (2006; Zbl 1134.42019)]. A finite frame is orthodecomposable if it can be partitioned into subsequences which are spanning for mutually orthogonal subspaces. If a \((\mu,S)\)-frame is not orthodecomposable, then it is at a non-singular point in the variety. An explicit, real-analytic parameterization for the neighborhood of any non-singular point is constructed. The method of construction begins with a suitable permutation of the frame vectors, assigning the order inductively so that the last \(d\) vectors form a basis \(B\) for the Hilbert space. The parameterization is realized by perturbing the first \(N-d\) vectors in a norm-preserving way and by compensating the change in the frame operator with the remaining \(d\) vectors in \(B\). There is some residual freedom in the choice of the remaining vectors which amounts to choosing the entries below the subdiagonal of the synthesis operator of \(B\). The construction is explained in detail for \((\mu, S)\)-frames in real Hilbert spaces. The appendix of the paper explains a similar strategy for complex Hilbert spaces. finite frames; real and complex Hilbert space; Stiefel manifold; tangent space; nonsingular points; local coordinates Strawn, N.: Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. J. Fourier Anal. Appl. 17, 821--853 (2011) General harmonic expansions, frames, Computational aspects of higher-dimensional varieties, Special varieties, Special matrices, Differentiable manifolds, foundations Finite frame varieties: Nonsingular points, tangent spaces, and explicit local parameterizations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 devoted to a finer analysis of a certain class of algebraic schemes generalizing quasi-projective schemes. More precisely, the authors reconsider the class of divisorial schemes, which was introduced by \textit{M. Borelli} exactly forty years ago [Pac. J. Math. 13, 375--388 (1963; Zbl 0123.38102)]. Recall that a quasi-compact and quasi-separated scheme \(X\) is called divisorial if there is a collection \(L_1, \dots, L_r\) of invertible sheaves on \(X\) satisfying the following condition: The open sets \(X_F\) for any \(f\in \Gamma(X, L_1^{d_1}\otimes\cdots\otimes L^{d_r}_r)\) and any multiindex \(d:=(d_1, \dots, d_r)\in\mathbb Z^r\) form a base of the Zariski topology of the scheme \(X\). Such a collection \(L_1, \dots, L_r\) is then called an ample family on \(X\).
For instance, all quasi-projective schemes, all smooth varieties, and all locally \(\mathbb Q\)-factorial varieties are known to be divisorial in this sense.
In the paper under review, the authors first generalize Grothendieck's construction of homogeneous spectra for \(\mathbb N\)-graded rings [\textit{A. Grothendieck}, EGA III, premiére partie, Publ. Math., Inst. Hautes Étud. Sci. 11, 349--511 (1962; Zbl 0118.36206)] to so-called multihomogeneous spectra of multigraded rings, that is to rings graded by an arbitrary, finitely generated abelian group \(D\). For such a ring \(S=\bigoplus_{d\in D}S_d\), the multihomogeneous spectrum \(\text{Proj}(S)\) is obtained by patching certain affine open pieces \(D_+(f)=\text{Spec}(S_{(f)})\), where \(f\in S\) are special homogeneous elements in \(S\). In the particular case of \(\mathbb N^r\)-graded rings, a similar construction has been carried out by \textit{P. Roberts} [``Multiplicities and Chern classes in local algebra'', Camb. Tracts Math. 133 (1998; Zbl 0917.13007)]
As the authors point out, their multihomogeneous spectra share many properties of the classical homogeneous spectra, but are possibly non-separated. In the sequel, the authors relate multihomogeneous spectra to divisorial schemes and simplicial toric varieties. In this context, one of their main results states that a scheme is divisorial if and only if it admits an embedding into a suitable multihomogeneous spectrum of a multigraded ring. This follows from a characterization of ample families on a scheme \(X\) in terms of the multihomogeneous spectrum \(\text{Proj}(S)\) for the multigraded ring \(S:=\bigoplus_{d\in\mathbb N}^r\Gamma(X,L^{d_1}_1\dots L^{d_r}_r)\). Moreover, generalizing \textit{H. Grauert}'s classical criterion of ampleness [Math. Ann. 146, 331--368 (1962; Zbl 0173.33004)] to families of invertible sheaves, the authors also characterize ample familiea, and therefore divisorial schemes, in terms of affine hulls and contraction maps. In addition, they provide a cohomological characterization of divisoriality, which may be regarded as an analogue of Serre's criterion of ampleness. Finally, as an application of the foregoing results, Grothendieck's algebraization theorem for formal schemes [\textit{A. Grothendieck}, ``Techniques de construction et théorèmes d'existence en géométrie algébrique. III: Préschémas quotients.'' Sem. Bourbaki 13 (1960/61), No.212, 20 p. (1961; Zbl 0235.14007)] is generalized as follows:
A proper formal scheme \({\mathcal X}\to\text{Spf}(R)\) is algebraizable if there is a finite collection of invertible formal sheaves restricting to an ample family on the closed fibre and satisfying an additional technical condition.
All these very substantial results are comprehensively derived in the six sections of the present paper, and that in an utmost lucid, detailed and rigorous manner, with numerous clarifying remarks and bibliographical references. schemes and morphisms; divisorial schemes; ample sheaves; graded rings; formal schemes; group actions; geometric invariant theory; ampleness criteria; algebraization Brenner, H.; Schröer, S.: Ample families, multihomogeneous spectra, and algebraization of formal schemes. Pacific J. Math. (2001) Schemes and morphisms, Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Geometric invariant theory, Formal methods and deformations in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Ample families, multihomogeneous spectra, and algebraization of formal schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a scheme locally of finite type over an algebraically closed field \(k\) of characteristic 0 and \(\mathcal M\) a coherent sheaf on \(X\). Recall that \(\text{Ass}(\mathcal M)\) consists of the irreducible closed subsets \(Y\) of \(X\) such that, for every affine open subset \(U\) of \(X\), \({\mathcal I}_Y(U) \in \text{Ass}_{{\mathcal O}_X(U)}({\mathcal M}(U))\). The first main result of the paper under review asserts that if \(\sigma : G \times X \rightarrow X\) is an action of a connected linear algebraic group over \(k\) on \(X\), if \(\mathcal M\) is \(G\)-equivariant and if \(\mathcal N \subset \mathcal M\) is a \(G\)-equivariant coherent subsheaf then: (1) every \(Y \in \text{Ass} (\mathcal M/\mathcal N)\) is \(G\)-invariant, and (2) there exists, for every \(Y \in \text{Ass}(\mathcal M/\mathcal N)\), a \(G\)-equivariant coherent subsheaf \({\mathcal Q}_Y\) of \(\mathcal M\) such that \(\text{Ass}(\mathcal M/{\mathcal Q}_Y) = \{ Y\}\) and \(\mathcal N = \bigcap_{Y\in \text{Ass}(\mathcal M)}{\mathcal Q}_Y\). This is called a \(G\)-equivariant primary decomposition of \(\mathcal N\) in \(\mathcal M\).
Let, now, \(\sigma : G\times Z \rightarrow Z\) be an action of \(G\) on an integral \textit{affine} scheme \(Z\), \(W \subset Z\) a union of \(G\)-invariant affine open subsets of \(Z\) and \(H \subset G\) a diagonalizable closed normal subgroup of \(G\) such that there exists a good quotient \(\pi : W \rightarrow W//H =: X\). If \(\mathcal F\) is a \(H\)-equivariant coherent sheaf on \(Z\) then one has a natural decomposition \({\pi}_{\ast}(\mathcal F\, | \, W) = \bigoplus_{\chi \in X(H)}{\pi}_{\ast}(\mathcal F\, | \, W)_{\chi}\), where \(X(H)\) is the group character of \(H\).
The second main result of the paper asserts that if \(H\) \textit{acts freely} on \(W\), if \(\mathcal F \subset \mathcal E\) are \(G\)-equivariant coherent sheaves on \(Z\) and if \(\mathcal F = \bigcap_{Y\in \text{Ass}(\mathcal E/\mathcal F)} {\mathcal Q}_Y\) is a \(G\)-equivariant primary decomposition of \(\mathcal F\) in \(\mathcal E\) then:
\[
{\pi}_{\ast}(\mathcal F\, | \, W)_0 = \bigcap _{\substack{ Y\in \text{Ass}(\mathcal E/\mathcal F)\\ Y\cap W\neq \emptyset }} {\pi}_{\ast}({\mathcal Q}_Y\, | \, W)_0 \subset {\pi}_{\ast}(\mathcal E\, | \, W)_0
\]
is a \(G/H\)-equivariant primary decomposition on \(X\).
The authors illustrate this result with some examples of equivariant primary decompositions on toric varieties. toric variety; toric sheaf; equivariant primary decomposition; sheaves of local cohomology Perling, M., Trautmann, G.: Equivariant primary decomposition and toric sheaves. Manuscripta Math. \textbf{132}(1-2), 103-143 (2010). arXiv:0802.0257. MR 2609290 (2011c:14051) Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Local cohomology and algebraic geometry, Graded rings Equivariant primary decomposition and toric sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth irreducible projective curve of genus g defined over the field of complex numbers. The ramification sequence of a linear system \(g^r_d\) at \(P \in X\) is the sequence \(0 \leq \alpha_0 \leq \cdots \leq \alpha_r \leq d - r\) such that the coefficients of \(P\) at the divisors of \(g^r_d\) are exactly \(\{\alpha_i + i : 0 \leq i \leq r\}\); \(P\) is said to be an \textit{inflection point} (respectively, a \textit{normal inflection point}) of \(g^r_d\) if \(\sum_{i = 0}^r \alpha_i > 0\) (respectively, \(\sum_{i = 0}^r \alpha_i = 1\)). A point \(P\) is said to be a \textit{generalized inflection point} of an invertible sheaf \(L\) if for some integer \(n > 0\) it is an inflection point of the linear system determined by \(L^{\otimes n}\). A generalized inflection point is said to be \textit{normal} if for any \(n > 0\) the ramification sequence for \(L^{\otimes n}\) satisfies \(\sum_{i = 0}^r \alpha_i \leq 1\), if furthermore \(\sum_{i = 0}^r \alpha_i = 1\) for exactly one value of \(n\) then \(P\) is said to be \textit{strongly normal}. In the present paper the author proves that if \(L\) is a very general invertible sheaf of degree \(d > 0\) on \(X\) then all generalized inflection points are strongly normal. In particular, the statement holds for the generic invertible sheaf of degree \(d\) on \(X\). Inflection point; Linear system; Divisor; Curve; Invertible sheaf Coppens M.: Generalized inflection points of very general line bundles on smooth curves. Ann. Mat. Pura Appl. 187(4), 605--609 (2008) Riemann surfaces; Weierstrass points; gap sequences Generalized inflection points of very general line bundles on smooth curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors extend their work from [Trans. Am. Math. Soc. 364, No. 9, 4663--4682 (2012; Zbl 1279.14072)]. Given a normal affine real varity \(V\) and a semi-algebraic subset \(S\) of \(V\), the idea of that paper was to analyze the ring \(B_V(S)\) of regular functions on \(V\) which are bounded on \(S\) by a completion of \(V\) that have been called \(S\)-compatible. Here are the definitions: A \textit{completion} of \(V\) is an open dense embedding \(V\hookrightarrow X\) into a normal complete real variety. The completion \(X\) is said to be \textit{\(S\)-compatible} if, for every irreducible component \(Z\) of \(X\setminus V\), the set \(Z(\mathbb{R})\cap\overline{S}\) is either empty or Zariski-dense in \(Z\). In the present paper the authors use toric completions: Let \(V\) be an affine toric varity. A \textit{toric completion} of \(V\) is an open embedding of \(V\) into a complete toric varity \(X\) which is compatible with the torus actions. Given a semi-algebraic subset \(S\) of \(V\), the existence of a toric \(S\)-compatible completion allows to describe the asymptotic growth of polynomial functions on \(S\) in terms of combinatorial data. In particular, the ring \(B_V(S)\) can be made explicitely in this case. real algebraic varieties, semialgebraic sets; bounded functions; toric completions Real algebraic and real-analytic geometry, Divisors, linear systems, invertible sheaves, Toric varieties, Newton polyhedra, Okounkov bodies, Semialgebraic sets and related spaces Toric completions and bounded functions on real 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 The origin of this paper lies in the question on étale cohomology for rigid analytic spaces posed in a paper by \textit{P. Schneider} and \textit{K. Stuhler} [Invent. Math. 105, No. 1, 47-122 (1991; Zbl 0751.14016)]. In that paper an étale site and a corresponding cohomology theory for analytic varieties are defined. We prove here that the axioms for an `abstract cohomology' (as stated in the cited paper) hold for this cohomology theory. In addition, we prove a (quasi-compact) base change theorem for rigid étale cohomology and a comparison theorem comparing rigid and algebraic étale cohomology of algebraic varieties. The main tools in this paper are analytic (resp. étale) points and rigid (resp. étale) overconvergent sheaves. In section 2 we (re)introduce some basic notations concerning analytic points and rigid overconvergent sheaves, which are needed later on. We (re)prove a number of folklore results most importantly: (1) Rigid cohomology agrees with Čech cohomology on quasi-compact spaces. (2) The cohomological dimension of a paracompact space is at most its dimension. (3) A base change theorem.
The rest of the paper deals with étale sites and étale cohomology. Étale points and étale overconvergent sheaves are introduced. A key point is the introduction of special étale morphisms of affinoids \(U\to X\), analogues to rational subdomains in the rigid case. Included in the paper is the proof by \textit{R. Huber} that any étale morphism of affinoids is special étale. This simplifies the original exposition somewhat. A structure theorem for étale morphisms (3.1.2) allows us to give a proof of the étale base change theorem following closely the proof in the rigid case. We calculate the cohomology groups of one dimensional spaces in section 4. This allows us to prove the basic results mentioned at the beginning of this introduction (sections 5, 6 and 7).
This paper may serve as an introduction to rigid and étale cohomology of rigid analytic spaces. étale cohomology for rigid analytic spaces; rigid overconvergent sheaves; Čech cohomology; cohomological dimension de Jong, Johan; van der Put, Marius, Étale cohomology of rigid analytic spaces, Doc. Math., 1, 01, 1-56, (1996) Local ground fields in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Non-Archimedean analysis Étale cohomology of rigid analytic 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 notions of multiplicity and log canonical threshold are the fundamental notions of
complex hypersurface \(H = \{f = 0\}\) defined by polynomials \(f\) on \(\mathbb C^{n+1}\). The former one is rather classical
which is defined by
\[
\mu_P(f) = \dim_{\mathbb C} \mathcal O_{\mathbb C^n,P}/(\delta f/\delta x_1, \ldots, \delta f/\delta x_n)
\]
at \(P \in H\). The latter notion of log canonical threshold is relative new which is given by
\[
\text{lct}_P(f) = \min \{(E_j) + 1/ \text{ord}_f(E_j): j \in S\}
\]
at \(P \in H\), where \((E_j)_{j \in S}\) are the \emph{resolution divisors} of a log resolution at \(0 \in H\) of the pair
\((\mathbb C^{n+1},H)\), whose precise current definition is
given by \textit{V. V. Shokurov} in birational geometry [Russ. Acad. Sci., Izv., Math. 40, No. 1, 95--202 (1992; Zbl 0785.14023); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105--201, Appendix 201--203 (1992)]. Both are algebraic invariants of hypersurface singularities
in complex algebraic geometry. The main result of the paper in review proves that these are indeed symplectic invariants of
the hypersurface in that when \(f, \, g: \mathbb C^{n+1} \to \mathbb C\) are two polynomials with isolated singular points at \(0\) with
embedded contactomorphic links, the multiplicity and the log canonical threshold of \(f\) and \(g\) are equal.
The main technical ingredient used to prove this result is to find formulas for the multiplicity and log canonical threshold
in terms of a sequence of fixed-point Floer cohomology groups in symplectic topology. The author does this by constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose
\(E^1\) page is explicitly described in terms of a log resolution of \(f\). This spectral sequence is a generalization
of a forumla by \textit{N. A'Campo} [Comment. Math. Helv. 50, 233--248 (1975; Zbl 0333.14008)]. The author first carries out a rather detailed technical
symplectic massaging, called \(\omega\)-regularization, of a germ of the neighborhood of intersections of
the symplectic crossing divisor \((V_i)_{i\in S}\) which are transversally intersecting codimension 2 symplectic submanifolds.
Then he applies the geometric notions of Liouville domains and open-books to construct a contact open book that is
well-behaved such that the mapping torus of the Milnor monodromy map is isotopic to the mapping torus of a
symplectomorphism arising from the open book. The paper provides much details of basic constructions in symplectic topology
that is expected to be useful for other similar future applications of symplectic machinery to complex algebraic geometry. log canonical threshold; Floer cohomology; singularity; Zariski conjecture; multiplicity; symplectic geometry; contact geometry Singularities of surfaces or higher-dimensional varieties, Milnor fibration; relations with knot theory, Contact manifolds (general theory), Symplectic aspects of Floer homology and cohomology Floer cohomology, multiplicity and the log canonical threshold | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 nonsingular polynomial maps \(F = (P,Q):\mathbb R^2 \to \mathbb R^2\) under the following regularity condition at infinity \((J_\infty)\): There does not exist a sequence \(\{(p_k,q_k)\} \subset \mathbb C^2\) of complex singular points of \(F\) such that the imaginary parts \((\mathrm{Im}(p_k), \mathrm{Im}(q_k))\) tend to \((0,0)\), the real parts \((\mathrm{Re}(p_k), \mathrm{Re}(q_k))\) tend to \(\infty\) and \(F(\mathrm{Re}(p_k),\mathrm{Re}(q_k))) \rightarrow a \in \mathbb R^2\). It is shown that \(F\) is a global diffeomorphism of \(\mathbb R^2\) if it satisfies Condition \((J_\infty)\) and if, in addition, the restriction of \(F\) to every real level set \(P^{-1}(c)\) is proper for values of \(| c|\) large enough. Jacobian conjecture; polynomial diffeomorphism Gutierrez E Nguyen Van Chau C.: On nonsingular polynomial maps of \({\mathbb{R}^2}\) . Annales Polonici Mathematici 88, 193--204 (2006) Jacobian problem On nonsingular polynomial maps of \(\mathbb R^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In a former paper [cf. Part I, Trans. Am. Math. Soc. 359, No. 3, 937--964 (2007; Zbl 1119.14021)] the authors introduced 1-forms of Wronskian type to the study of uniqueness polynomials and unique range sets. This article extends the result of the mentioned paper to a more general family of polynomials by using \(m\)-fold symmetric products of 1-forms of Wronskian type. One of the new main ingredients is the uniformization theorem for curves via the existence of \(m\)-fold symmetric product of 1-forms of Wronskian type. The classical uniformization theorem, used in Part I, is the special case \(m = 1\). Moreover, our method applies to all known characterization of uniqueness polynomials which were previously established by other means. Our results are more general in the case of positive characteristic than what was previously known and our method provides a uniform approach to the problem regardless of the characteristic of the underlying field. T. T. H. An, J. T. Y. Wang and P. M. Wong, ''Unique range sets and uniqueness polynomials in positive characteristic II,'' Acta Arithm. 115--143 (2005). Polynomials in general fields (irreducibility, etc.), Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.), Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Singularities of curves, local rings, Value distribution of meromorphic functions of one complex variable, Nevanlinna theory Unique range sets and uniqueness polynomials in positive characteristic. 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 From the abstract: ``The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a \(3\times3\) matrix spectral problem is derived by
using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD
hierarchy, we introduce a trigonal curve \(\mathcal{K}_{m-2}\) of arithmetic genus \(m-2\), from which the corresponding Baker-Akhiezer
function and meromorphic functions on \(\mathcal{K}_{m-2}\) are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel
differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the
meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.''
This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the authors obtain the CCIRD hierarchy related to a \(3\times3\) matrix spectral problem based on the Lenard recursion equations. In Section 3, a trigonal curve \(\mathcal{K}_{m-2}\) of arithmetic genus \(m-2\) with three infinite points is introduced by the use of the characteristic polynomial of the Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on \(\mathcal{K}_{m-2}\). Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, they present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. In Section 5, the authors extend all the Baker-Akhiezer functions, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case. Section 6 is devoted to some conclusions. Lenard recursion gradients; Baker-Akhiezer function; meromorphic functions; trigonal curve; Abel differentials; algebro-geometric solutions Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems, Reaction-diffusion equations Algebro-geometric solutions of the coupled Chaffee-Infante reaction diffusion hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book describes geometry and analysis of dynamical systems by looking at their scale transforms via \(\log_t\), where \(t > 1\). Of particular interest is the tropical transform, also known as Maslov dequantization [\textit{G. L. Litvinov} and \textit{V. P. Maslov}, in: Idempotency. Based on a workshop, Bristol, UK, October 3--7, 1994. Cambridge: Cambridge University Press. 420--443 (1998; Zbl 0897.68050)], obtained by \(t \to \infty\).
The author applies this strategy to three subjects:
Projective geometry: Chapters 2--5 are dedicated to the theory of iterated dynamics and pentagram maps. In Chapter 2, the author reduces real rational dynamics to tropical dynamics and shows that quasi-recursive rational dynamics is equivalent to recursive \((\text{max}, +)\)-dynamics. Theorem 5.6 proves, from the viewpoint of scale transform, that the continuous limit of the pentagram map induces the Boussinesq equation.
Infinite groups: Chapters 6--9 mostly treat (stable) state dynamics, automata groups, and Burnside groups. The idea is to study whether geometric and analytic properties of rational dynamics pass to automata groups in tropical geometry via scale transform.
Mathematical physics: Chapters 10--16 provide a classification of partial differential equations using tropical geometry. Chapter 14 introduces some analytic relation among PDEs in two variables which is based on asymptotic estimates of all positive solutions; then it is shown that two PDEs obtained from tropically equivalent rational functions are related. Chapter 15 and 16 are devoted to the basic analysis of hyperbolic Mealy systems, in particular, the existence of solutions involves the interplay of estimates between piecewise linear and differentiable dynamics.
Throughout the book, the author also shows that tropical geometry links different subject in mathematics. For instance, Korteweg-de Vries equations and lamplighter groups share structural similarities.
The book is self-contained and includes basic concepts in each subject such as iterated dynamics, pentagram map, lamplighter group, Burnside problem, Korteweg-de Vries equations and hyperbolic Mealy systems. tropical transform; tropical geometry; automata group; partial differential equation Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory, Topological dynamics, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Dynamical scale transform in tropical 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 Let \(Z=m_1p_1+\ldots +m_np_n\) be a fat point scheme in \(\mathbb{P}^N\) and let \(I=I(Z)\) be its saturated ideal in \(R=\mathbb{K}[x_0,\ldots , x_N]\).
The least \(i\) such that the Hilbert function and the Hilbert polynomial of \(I\) are equal (\(H(I,t)=P(I,t)\)) for all \(t\geq i\) is \(\mathrm{reg}(I)-1\), where \(\mathrm{reg}(I)\) is the Castelnouvo-Mumford regularity of \(I\). The iequality \[\mathrm{reg}(I)\geq \alpha(I)\] always holds, where \(\alpha(I)=\min\{d: I_d\neq 0\}\). The equality \(\mathrm{reg}(I)= \alpha(I)\) holds if and only if \(I\) admits a linear minimal free resolution.
In order to the containment problem Bocci and Harbourne introduced the resurgence of \(I\). It is defined as \[\rho(I)=\sup\{\frac{m}{r} : I^{(m)}\nsubseteq I^r\}.\] They also proved that \[\frac{\alpha(I)}{\widehat{\alpha}(I)}\leq \rho(I)\leq \frac{\mathrm{reg}(I)}{\widehat{\alpha}(I)},\] where \(\widehat{\alpha}(I)=\inf_{m\geq 1}\frac{\alpha(I^{(m)})}{m}\) is the Waldschmidt constant of \(I\). In particular if \(I\) has a linear minimal free resolution, then \[\rho(I)=\frac{\alpha(I)}{\widehat{\alpha}(I)}.\] Main results of paper under review are two theorems.\\
Theorem A. Let \(Z=m_1p_1+\ldots +m_np_n\) be a fat point subscheme of \(\mathbb{P}^N\) and let \(I(Z)\) be its saturated ideal in \(R\). Let for some positive integer \(s\), \(\mathrm{reg}(I(Z))=s\), and let \(r\) be a nonnegative integer. Let \(Z_r\) be a reduced scheme of \(\binom{s+N+r-1}{N}-\deg Z\) general points in \(\mathbb{P}^N\), where \(Z\cap Z_r=\emptyset\). Then for the fat point subscheme \(X_r=Z+Z_r\), we have \(\mathrm{reg}(I(X_r))=\alpha(I(X_r))=s+r\). In particular, \(I(X_r)\) has a linear minimal free resolution.\\
As a special case of above theorem we obtain the following theorem.\\
Theorem B. Let \(I\) be the saturated ideal of fat subscheme \(Z=(s-2)p_1+p_2+\ldots +p_s\) of \(\mathbb{P}^2\), supported on general points. Then
\[
\rho(I)= \begin{cases} (s+1)/s & \text{ if \(s\) is odd},\\
s(s-1)/((s-1)^2+1) & \text{ if \(s\) is even}. \end{cases}
\] containment problem; linear minimal free resolution; resurgence; symbolic power Ideals and multiplicative ideal theory in commutative rings, Configurations and arrangements of linear subspaces, Polynomial rings and ideals; rings of integer-valued polynomials, Projective techniques in algebraic geometry Fat point ideals in \(\mathbb{K}[\mathbb{P}^N]\) with linear minimal free resolutions and their resurgences | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 aim of the paper is to investigate the asymptotic behavior of the length of local cohomology modules of powers of an ideal $I$. More precisely, if $I$ is an ideal in a $d$-dimensional local noetherian ring $(R, \mathfrak{m}, k)$ or a polynomial ring over a field $k$ and maximal homogeneous ideal $\mathfrak{m}$, the authors study situations when the length $\lambda(H_{\mathfrak{m}}^i (R/I^n))$ is finite for $n \gg 0$ and consider the existence of the limit \[ \lim_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\tag{*} \] for $i > 0$. The paper is motivated by results of \textit{S. D. Cutkosky} [Adv. Math. 264, 55--113 (2014; Zbl 1350.13032)] who proved that this limit exists if $i=0$ and $R$ is analytically unramified. The most concrete results of the paper are obtained for monomial ideals in a polynomial ring. In this case, it is proved that $\lambda(H_{\mathfrak{m}}^i (R/I^n))< \infty $ for $n \gg 0$ and the sequence $\{\lambda(H_{\mathfrak{m}}^i (R/I^n))\}_{n \geq 0}$ agrees with a quasi-polynomial of degree $d$ for $n \gg 0$. Moreover, the generating function of this sequence is rational. Even though the limit (*) might not exist, a better behavior is obtained by replacing the filtration of powers of $I$ with the filtration of integral closures $\{\overline{I^n}\}_{n \geq 1}$, in which case $\lim_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/\overline{I^n}))}{n^d}$ exists and is a rational number. In the more general case when $I$ is a homogeneous ideal, the authors prove that \[ \limsup_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n)_{\geq -\alpha n})}{n^d} < \infty \] for every $\alpha \in \mathbb{Z}$ and leave open the question of whether or not $\limsup_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}$ is always finite. The authors also raise questions about situations when the limit (*) is rational and about finding interpretations of it in terms of volumes of geometric bodies associated with the ideal $I$. The paper also discusses several classes of ideals for which $\liminf_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}$ is positive. local cohomology; homogeneous ideals; Presburger arithmetic; singularities Local cohomology and commutative rings, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Singularities in algebraic geometry, Combinatorial aspects of commutative algebra Length of local cohomology of powers 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 paper under review contributes to the foundations of affinoid pre-adic spaces and, hence, perfectoid spaces. Most notably, the authors show that the structure presheaf \(\mathcal{O}_X\) of the affinoid pre-adic space \(X=\text{Spa}(R,R^+)\) associated with a Tate affinoid ring \((R,R^+)\) is a sheaf of complete topological rings whenever \((R,R^+)\) is stable uniform, i.e. whenever for each rational subset \(U\subseteq X\), the ring of power-bounded functions \(\mathcal{O}_X(U)^o\) on \(U\) is bounded. This criterion provides an independent proof of Scholze's result (see [\textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-archimedean fields. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)] Theorem (6.3 (iii)) that \(\mathcal{O}_X\) is a sheaf when \(X\) is perfectoid, avoiding the machinery of almost mathematics and tilting. The authors also provide various explicit examples where \(\mathcal{O}_X\) is not a sheaf.
The paper under review is carefully and lucidly written; its content is summarised and contextualised in the introductory first section. The authors point towards recent applications of their results in the context of Scholze's theory of diamonds, and they provide pedagogical asides.
In the preliminary second section of the paper, the authors review the definition of (affinoid) Tate rings, and they establish an elementary Lemma which provides an explicit description of the topology of the completion of a Tate ring. For the cover \(X=U\cup V\), \(U=\{|t|\ge 1\}\) and \(V=\{|t|\le 1\}\), associated with an element \(t\in R\), they then provide a detailed description of the natural restriction short sequence \(0\to\mathcal{O}_X (X)\to\mathcal{O}_X(U)\oplus\mathcal{O}_X(V)\to\mathcal{O}_X(U\cap V)\), and they explicit show that this sequence is exact if and only if the map \(\mathcal{O}_X(X)\to\mathcal{O}_X(U)\oplus\mathcal{O}_X(V)\) is strict at the level of non-complete topological localizations of \(R\).
In the third section of the paper, the authors proceed to establish their main result. They define a Tate ring \(R\) to be uniform if and only if its subring of power-bounded elements is bounded, and they explicitly show that if \(R\) is uniform, then the restriction short sequence from Section 2 is exact. Using Laurent covers, they infer that \(\mathcal{O}_X\) is at sheaf whenever \(R\) is stably uniform. They use this result to prove the fact, previously established by Scholze via different means, that \(\mathcal{O}_X\) is a sheaf if \((R,R^+)\) is a perfectoid algebra over a perfectoid field \(k\). They also infer a new criterion for checking the perfectoid property, showing that over a perfectoid field \(k\) of positive characteristic, a stably uniform complete Tate affinoid \(k\)-algebra is perfectoid if and only if this holds locally w.r.t. any rational covering. (The authors state that it is not known whether the corresponding statement holds in characteristic zero or whether stable uniformity can be weakened to uniformity. In general, e.g. without uniformity assumption, the perfectoid property cannot be checked locally; see the second example discussed below.)
In the fourth and final section of the paper, the authors provide 6 different examples, working over a complete non-archimedean field \(k\) and calling an affinoid \(k\)-algebra sheafy if and only if the associated presheaf \(\mathcal{O}_X\) of complete topological rings is a sheaf. First, they establish an elementary Lemma, showing that a grading on a Tate \(k\)-algebra w.r.t. a torsion-free abelian group induces a grading on the ring of power-bounded elements. They then provide a first explicit example, of a finitely generated \(k\)-algebra with a non-zero nilpotent element for which \(\mathcal{O}_X\) is not a sheaf due to failure of the first sheaf property. Next, they provide an second example, of a locally perfectoid pre-adic space whichis not perfectoid, by `perfectifying' the first example. The authors then proceed to provide an interesting explicit third example, of a pre-adic space with a locally zero function that is globally non-nilpotent. Next, they give an explicit fourth example, where the second sheaf property fails, i.e. where glueing fails. They then give a fifth example, of a uniform space that is not stably uniform. Finally, they give a sixth example, of a uniform non-sheafy space. Rigid analytic geometry, Varieties over finite and local fields Stably uniform affinoids are sheafy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 origin of the paper under review is the theory of prehomogeneous vector spaces due to \textit{M. Sato} [Sugaku 15, 85-157 (1970)], see also \textit{M. Sato} and \textit{T. Kimura} [Nagoya Math. J. 65, 1-155 (1977; Zbl 0321.14030)]. (A vector space is called prehomogeneous if it contains a Zariski dense orbit under an action of a reductive group.) The author puts the theory into a somewhat more general context including \(D\)- modules and perverse sheaves in the complex case, hyperfunctions in the real case, \(l\)-adic étale perverse sheaves in the finite field case. The exposition is concentrated around the notion of the Fourier transformation, also viewed in a very general sense including geometric Fourier transformations introduced by \textit{J.-L. Brylinski}, \textit{B. Malgrange}, and \textit{J.-L. Verdier} [C. R. Acad. Sci., Paris, Sér. I 297, 55-58 (1983; Zbl 0553.14005)] in the complex case, and geometric Fourier transformations of Deligne in the finite field case [see \textit{N. M. Katz} and \textit{G. Laumon}, Publ. Math., Inst. Hautes Étud. Sci. 62, 145-202 (1985; Zbl 0603.14015)].
The author uses what he calls the Lefschetz principle in a systematic way to translate some geometric statements into the arithmetic situation. He states some conjectures arising after such a translation, and give some heuristic arguments in their support. \(D\)-modules; prehomogeneous vector spaces; perverse sheaves; geometric Fourier transformation; Lefschetz principle Gyoja, A.: Lefschetz principle in the theory of prehomogeneous vector spaces. Adv. studies pure math. 21, 87-99 (1992) Homogeneous spaces and generalizations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, Relations of PDEs on manifolds with hyperfunctions, Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs Lefschetz principle in the theory of prehomogeneous vector spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field of any characteristic, and let \(X\) be a \(k\)-scheme, i.e. a separated scheme of finite type over \(k\). Denote by \(H^ \bullet(X)\) (and \(H^ \bullet_ c(X))\) the \(\ell\)-adic étale cohomology (with compact support) of \(X\). Thus if \(k\) has characteristic zero and is a subfield of \(\mathbb{C}\) one may also take rational singular cohomology. \(H^ \bullet(X)\) and \(H^ \bullet_ c(X)\) carry a weight filtration \(W_ \bullet\), and one defines the (pure) Euler characteristics of \(X\) by \(\chi_ m(X)=\sum_ i(-1)^ i\dim Gr^ W_ mH^ i(X)\), and similarly \(\chi^ m_ c(X)\) for \(Gr^ W_ mH^ i_ c(X)\). The (pure) Poincaré polynomial of \(X\) is defined as \(P_{\text{pur}}(X,t)=\sum_ m\chi_ m(X)t^ m\), and similarly \(P^ c_{\text{pur}}(X,t)=\sum_ m\chi^ m_ c(X)t^ m\). For a \(k\)-scheme \(X\) one can define a stratification or, a little more generally, a (filterable) decomposition. Such a stratification (or filterable decomposition) \(X=\bigcup X_ \alpha\) is called perfect if \(X\) and also the cells \(X_ \alpha\) are smooth and connected and one has the following relation between the Betti numbers of \(X\) and the \(X_ \alpha:b_ i(X)=\sum_ \alpha b_{i-2d_ \alpha}(X_ \alpha)\), where \(d_ \alpha\) is the codimension of \(X_ \alpha\) in \(X\). As a first result it is shown that for a smooth, connected, complete \(k\)-scheme \(X\) with Białynicki-Birula decomposition \(X=\bigcup X_ \alpha\) and with an action of an algebraic torus \(T\), one has \(b_ i(X)-\sum_ \alpha b_{i-2d_ \alpha}(X^ T_ \alpha)\) for all \(i\), where \(X^ T=\bigcup X^ T_ \alpha\) is the decomposition of the \(T\)-fixed part \(X^ T\) of \(X\) into connected components.
Next it is shown that one can define, for a linear \(k\)-algebraic group acting on a \(k\)-scheme \(X\), pure Euler characteristics for the equivariant cohomology
\[
H^ i_ G(X): \chi^ m_ G(X)=\sum_ i(- 1)^ i\dim Gr^ W_ mH^ i_ G(X).
\]
Similarly for the equivariant cohomology with compact support \(H^ i_{G,c}(X)\) one defines \(\chi^ m_{G,c}(X)\). Also, the pure equivariant Poincaré series of \(X\) is defined: \(P^ G_{\text{pur}}(X,t)=\sum_ m\chi^ m_ G(X)t^ m\). One can define (filterable) \(G\)-decompositions and \(G\)-stratifications of the \(G\)-scheme \(X\) in a straightforward way.
The final result says that for a smooth connected \(G\)-scheme \(X\) with \(G\)-decomposition \(X=\bigcup X_ \alpha\) such that the \(X_ \alpha\) are smooth, equidimensional of codimension \(d_ \alpha\) in \(X\), one has \(\chi^ m_ G(X)=\sum_ \alpha\chi_ G^{m-2d_ \alpha}(X_ \alpha)\) and \(P^ G_{\text{pur}}(X,T)=\sum_ \alpha t^{2d_ \alpha}P^ G_{\text{pur}}(X_ \alpha,t)\). This is applied to obtain inductively the Betti numbers of a symplectic or a geometric quotient of a variety. This leads to formulas obtained before by \textit{F. Kirwan}. A second application concerns the calculation of the Betti numbers of geometric quotients as done by Białynicki-Birula and Sommese.
The last section of the paper gives a motivic version of the preceding results. A motivic Euler characteristic \(\chi_{\text{mot}}(X)\) is defined by means of the system of mixed realizations \(SRM^ \bullet(X)\) defined by \(H^ \bullet(X)\). Similarly for compact support. In the situation of the Białynick-Birula decomposition of \(X\) with torus action as sketched above, one finds a relation for the Hodge numbers of \(X\) and the \(X^ T_ \alpha:h^{p,q}(X)=\sum_ \alpha h^{p-d_ \alpha,q-d_ \alpha}(X^ T_ \alpha)\). Also, for the \(G\)-scheme \(X\) with \(G\)-decomposition \(X=\bigcup X_ \alpha\) one finds for the motivic Poincaré series, \(P^ G_{\text{mot,pur}}(X,t)=\sum_ \alpha t^{2d_ \alpha}P^ G_{\text{mot,pur}}(X_ \alpha,t)(-d_ \alpha)\), where \((-d_ \alpha)\) means Tate twist. One deduces a formula for Hodge numbers. stratification; filterable decomposition; Euler characteristics; equivariant cohomology; Betti numbers; quotient of a variety; motivic Euler characteristic; Hodge numbers Navarro Aznar, V.: Stratifications parfaites et théorie des poids. Publ. Mat. 36 (1992), no. 2B, 807--825 (1993) Group actions on varieties or schemes (quotients), Topological properties in algebraic geometry, Stratifications in topological manifolds Perfect stratifications and weight 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 This article starts with a survey of basic contructions and results in transformation groups over algebraic and semialgebraic sets (such as realizability results), and then goes on to discuss a theory of transformation groups over larger categories.
From model theory one defines a structure \(\mathcal M\) by defining functions, relations, and constants on an underlying set. The larger categories are expansions of the standard structure \(\mathcal R\) consisting of the set of real numbers, multiplication, addition and the order relation \(<\,\). One defines formulas, then \(X \subset \mathbb R^n\) is definable (in \(\mathcal M\)) if it is defined by a formula, and continuous functions between definable sets are definable if their graphs are definable. An order-minimal (o-minimal) structure \(\mathcal M\) is one in which every definable subset of \(\mathbb R\) is a finite union of points and open intervals and this yields category properties that are needed. If \(\mathcal M=\mathcal R\), then it is order-minimal and every definable set is a semialgebraic one and every definable map is semialgebraic [\textit{A. Tarski}, A decision method for elementary algebra and geometry (1951; Zbl 0044.25102)].
The author presents several examples of o-minimal structures expanding \(\mathcal R\), and it is in this context that results are surveyed which appear in the literature [\textit{L. van den Dries}, Tame topology and o-minimal structures. Cambridge: Cambridge Univ. Press. Lond. Math. Soc. Lect. Note Ser. 248 (1998; Zbl 0953.03045); \textit{T. Kawakami}, Bull. Korean Math. Soc. 36, 183--201 (1999; Zbl 0939.14032) and Topology Appl. 123, 323--349 (2002; Zbl 1028.57034)]. G-manifolds; semialgebraic sets; definable sets; transformation groups; o-minimal structures Model theory of ordered structures; o-minimality, Compact Lie groups of differentiable transformations, Semialgebraic sets and related spaces, Differentiable mappings in differential topology, Real-analytic and Nash manifolds \(G\)-manifolds and \(G\)-vector bundles in algebraic, semialgebraic, and definable categories. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathcal{V}\) be a discrete valuation ring with fraction field \(K\) of characteristic 0 and residue field \(k\) of characteristic \(p>0\). Let \(f : X \to S\) be a morphism of \(k\)-varieties. Without going into the details, let us recall that an isocrystal \(E\) over \(X\) is a module with an integrable connection which is defined on some \(K\)-analytic rigid space associated to \(X\) (a strict neighbourhood of its tube in a certain formal scheme). The article under review gives sufficient conditions in order that the direct images \(R^i f_{\mathrm{rig}*}\) of an overconvergent isocrystal on \(X\) remain overconvergent. This will be the case when \(S\) is smooth over \(k\), \(f\) is smooth and projective and either \(X\) lifts to a flat scheme over \(\mathcal{V}\) or \(X\) is a relative complete intersection inside projective spaces over \(S\).
The first section of the text is devoted to base change theorems for proper morphisms, which will be a crucial ingredient in the proof. The author first handle the case of formal schemes, then of rigid analytic spaces. The second section gathers a few technical results on strict neighbourhoods, in particular conditions to ensure that a fundamental system of strict neighbourhoods remains so after pull-back. The third section contains a base change theorem for overconvergent modules by a proper morphism and ends with the proofs of the results concerning direct images of overconvergent isocrystals.
To finish, let us mention that the author announces two articles to follow, in which he will consider isocrystals endowed with a Frobenius structure in the convergent and overconvergent settings. rigid analytic spaces; formals schemes; overconvergent isocrystals; proper base change; rigid cohomology Étesse, Jean-Yves: Images directes I: Espaces rigides analytiques et images directes. J. théor. Nombres Bordeaux 24, No. 1, 101-151 (2012) Rigid analytic geometry, \(p\)-adic cohomology, crystalline cohomology Direct images. I: Rigid analytic spaces and direct images | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V\subset \mathbb{R}^{n}\) be an unbounded algebraic set and let \(F: \mathbb{R}^{n}\rightarrow \mathbb{R}^{m},\) \(m\geq n\geq 2,\) be a polynomial mapping. By the Łojasiewicz exponent at infinity \(\mathcal{L}_{\infty }(F|V)\) of \(F\) on \(V\) we mean the supremum of \(\nu \in \mathbb{R}\) for which
\[
\left| F(x)\right| \geq C\left| x\right| ^{\nu }\text{ \;\;\;as }x\in V\text{ and }\left| x\right| \rightarrow \infty .
\]
Generalizing an result by \textit{B. Osińska} [Bull. Sci. Math. 135, No. 2, 215--229 (2011; Zbl 1225.32034)] in complex case the authors prove the following.
Theorem. Let us assume \(F^{-1}(0)\) is a compact set. Then there exists a polynomial mapping \(G:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}\) such that:
(i) \(G\) is an extension of \(F|V\) to \(\mathbb{R}^{n},\) i.e. \(F|V=G|V,\)
(ii) \(\mathcal{L}_{\infty }(G|\mathbb{R}^{n})=\mathcal{L}_{\infty }(F|V),\)
(iii) \(\deg G\leq -[-\delta (V)(6\delta (V)-3)^{n-1}(\deg F+2-\mathcal{L} _{\infty }(F|V))-\mathcal{L}_{\infty }(F|V)]+\delta (V),\) where \(\delta (V)\) is the total degree of complexification \(V_{\mathbb{C}}\subset \mathbb{C}^{n} \) of \(V.\) polynomial mapping; Łojasiewicz exponent; real algebraic set [19]B. Osińska-Ulrych, G. Skalski and S. Spodzieja, Extensions of real regular mappings and the Łojasiewicz exponent at infinity, Bull. Sci. Math. 137 (2013), 718--729. Real algebraic sets, Complex singularities Extensions of real regular mappings and the Łojasiewicz exponent at infinity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Henselian couple \((A,a)\) has the approximation property if the solutions in \(A\) of every system of polynomial equations \(f\) over \(A\) are dense in the set of the solutions of \(f\) in the completion \(\widehat A_a\) of \(A\) with respect to \(a\). If \(A = \mathbb{C} \{X_1, \ldots, X_n\}\) is the complex convergent power series ring and \(a = (X)\) then \((A,a)\) has the approximation property. With this result started the Artin approximation theory almost thirty years ago. Now it is known that a Henselian couple \((A,a)\) with the completion map \(A \to \widehat A_a\) regular (that is, its fibers are geometrically regular) has the approximation property. This is a consequence of the following theorem:
A regular morphism of noetherian rings is a filtred inductive limit of smooth morphisms.
Proofs of this theorem were given by the reviewer [cf. Nagoya Math. J. 100, 97-126 (1985; Zbl 0575.14010), 104, 85-115 (1986; Zbl 0598.14012) and 118, 45-53 (1990; Zbl 0699.14018)], \textit{M. André} [``Cinq exposés sur la désingularisation'' (preprint 1991)], \textit{T. Ogoma} [J. Algebra 167, No. 1, 57-84 (1994; Zbl 0821.13003)]and \textit{M. Spivakovsky} [``Smoothing of ring homomorphisms approximation theorems, and the Bass-Quillen conjecture'' (preprint 1993)]. The above theorem gives also a partial positive answer to the Bass-Quillen conjecture concerning the projective modules over polynomial algebra over regular rings [see the papers of the reviewer, in Nagoya Math. J. 113, 121-128 (1989; Zbl 0663.13006) and \textit{M. Spivakovsky} (loc. cit.)]. approximation property; Henselian couple; Bass-Quillen conjecture; projective modules Teissier, B., \textit{Résultats récents sur l'approximation des morphismes en algèbre commutative (d'après André, Artin, Popescu, et spivakovsky)}, Séminaire Bourbaki, exposé n{\(\deg\)} 784, March 1994, 259-282, (1995), Société Mathématique de France, Paris Étale and flat extensions; Henselization; Artin approximation, Henselian rings, Projective and free modules and ideals in commutative rings, Morphisms of commutative rings, Local deformation theory, Artin approximation, etc., Regular local rings Recent results on the approximation of morphisms in commutative algebra (after André, Artin, Popescu and Spivakovsky) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(G\subseteq \text{GL}_n(\mathbb R)\) be a finite matrix group and \(X\subseteq \mathbb R^n\) be a \(G\)-variety. We propose a new approach for computing a stratification of \(X\) with respect to the orbit type of \(\mathbb R^n\) respectively of the quotient \(X/G\) and we present new algorithms for this task. For \(X = \mathbb R^n\) these algorithms yield an optimal description of each stratum and of the orbit space in terms of polynomial equations and inequalities (optimal with respect to the number of inequalities). Moreover we show that the dimension \(d\) of a stratum \(\widehat\Sigma_d\) of \(\mathbb R^n/G\) is an upper and lower bound for the number of inequalities needed for a description of \(\widehat\Sigma_d\) and its closure, which improves the upper bound \(d(d+1)/2\), which holds for general basic closed semialgebraic sets of dimension \(d\). Additionally, our algorithms allow us to compute strata of particular interest of \(X/G\), which demands less computational resources. By performing computations as long as possible in \(\mathbb R^n\) (and not in \(\mathbb R^n/G\)) and by refining results of \textit{C. Procesi} and \textit{G. Schwarz} [Invent. Math. 81, 539--554 (1985; Zbl 0578.14010)], it seems that our algorithms are more efficient than the present approach. We conclude by giving an application of our algorithms to the problem of constructing a potential for Nickel-Titanium alloys and compare the runtime with other algorithms. symbolic computation; stratification Symbolic computation and algebraic computation, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory Optimal descriptions of orbit spaces and strata of finite groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is divided into two sections. In the first section, certain explicit families of ``almost Belyi'' coverings of the projective line (over the algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\)) are constructed. In this case, ``almost Belyi'' means that the covering is branched at \(0\), \(1\), and \(\infty\), as well as a fourth point, above which the ramification is simple (meaning that there is only one ramification point above the branch point, and its ramification index is 2). The coverings constructed are of degrees 12, 11, and 20, and they all have genus zero, thus are defined by a single rational function. The strategy is to fix the ramification type, and then give an ansatz for what form the rational function should take. One can then take the logarithmic derivative of the ansatz form directly, and also realize (independently) that the logarithmic derivative must have certain poles and zeros. This leads to a set of equations for the coefficients of the rational form that can be solved by computer. In the case of the degree 12 covering, this is straightforward. For the other two coverings, further tricks (including solving the undetermined coefficient equations in various finite fields) are required to make the computation feasible.
In the second section, these almost Belyi coverings are applied to give solutions to Painlevé VI equations. This is possible because pulling back via these coverings can transform hypergeometric differential equations into \(2 \times 2\) matrix Fuchsian systems with 4 regular singular points. The paper computes an example for each of the coverings from the first section. Belyi map; dessin d'enfant; the Painlevé VI equation F. Klein, \textit{Vorlesungen über das Ikosaedar}, B. G. Teubner, Leipzig (1884). Dessins d'enfants theory, Low-dimensional topology of special (e.g., branched) coverings, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Painlevé-type functions Computation of highly ramified coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E=E_ 0\supset E_ 1\supset...\supset E_ t\) and \(F=F_ 0\twoheadrightarrow F_ 1\twoheadrightarrow...\twoheadrightarrow F_ t\) be flags of vector spaces. Call a complete collineation from E to F to be a sequence of linear maps \(v_ i:E_ i\to F_ i (i=0,...,t)\) up to scalar multiple such that \(E_ i=\ker v_{i-1}\), \(F_ i=co\ker v_{i- 1}\) and \(v_ t\) is injective \(\neq 0\). The sequence \((n+1\)-rank \(E_ i)\) where \(n+1=rank E\), is called the type of \((v_ i)\). Let \(S_ r\) denote the projective space of linear maps \(\bigwedge^{r+1}E\to \bigwedge^{r+1}F.\) Let \(S^ t\) be the closure of the set of all \((v,\bigwedge^{2}v,...,\bigwedge^{t+1}v)\) in \(S_ 0\times...\times S_ t\). It is shown that \(S^ t\) is smooth and equal to the successive blow-up of \(S_ 0\) along the loci of maps of rank \(\leq t\). The set of complete collineations of a given type is shown to be naturally parametrized by an orbit of \(S^ n\) under GL(E)\(\times GL(F)\) and the closure of any such is a transversal intersection of divisors. The Chow ring of \(S^ n\) is described and some applications to the enumerative geometry of collineations are given. It is shown that the coefficients appearing in a formula of A. Lascoux for the Segre classes of a tensor product of vector bundles are the solutions of a simple enumerative problem for collineations as stated by \textit{D. Laksov} in Enumerative Geometry and Classical Algebraic Geometry, Prog. Math. 24, 107-132 (1982; Zbl 0501.14031); this volume contains also the author's article ''Schubert calculus for complete quadrics'' [Prog. Math. 24, 199-235 (1982; Zbl 0501.14032)], where similar results are obtained for symmetric linear maps. Related material may be found in the following papers: \textit{C. De Concini} and \textit{C. Procesi}, ''Complete symmetric varieties'' (Prepr. Univ. Roma, 1982) and \textit{D. Luna} and \textit{T. Vust}, ''Plongements d'espaces homogènes'', Comm. Math. Helv. 58, 186-245 (1983). determinantal ideals; flags of vector spaces; Chow ring; enumerative geometry of collineations Vainsencher, I., \textit{complete collineations and blowing up determinantal ideals}, Math. Ann., 267, 417-432, (1984) Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Enumerative problems (combinatorial problems) in algebraic geometry Complete collineations and blowing up determinantal 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 \(Y,Z\) be quasi-projective algebraic varieties and let \(\Omega\) be an open subset of \(Y\). A map \(F:\Omega \to Z\) is said to be Nash algebraic if \(f\) is holomorphic and the graph of \(f\) is contained in an algebraic subvariety of \(Y \times Z\) of dimension equal to \(\dim Y\).
One of the main results in the paper is the following theorem concerning the approximation of holomorphic maps by Nash algebraic maps:
Theorem 1.1. Let \(\Omega\) be a Runge domain in an affine algebraic variety \(S\) and let \(f:\Omega \to X\) be a holomorphic map into a quasi-projective algebraic manifold \(X\). Then for every relatively compact domain \(\Omega_0 \Subset\Omega\), there is a sequence of Nash algebraic maps \(f_\nu: \Omega_0\to X\) such that \(f_\nu \to f\) uniformly on \(\Omega_0\).
As important applications of Theorem 1.1 the authors obtain that the Kobayashi-Royden pseudometric and the Kobayashi pseudodistance on projective algebraic manifolds can be approximated in terms of algebraic curves. It is proved that a type of algebraic approximation is also possible in the case of locally free sheaves.
Using the methods developed in the paper the authors give a more precise form of a result concerning the description of equivalent Nash algebraic vector bundle, obtained by \textit{T. Tancredi} and \textit{A. Tognoli} [Bull. Sci. Math., II. Ser. 117, No. 2, 173-183 (1993; Zbl 0798.32010)]. A result of \textit{E. L. Stout} [Contemp. Math. 32, 259-266 (1984; Zbl 0584.32027)] on the exhaustion of Stein manifolds by Runge domains in affine algebraic manifolds is proved by substantially different methods. approximation of holomorphic maps; Nash algebraic maps; quasi-projective algebraic manifold; Stein manifolds; Runge domains Demailly, J. -P.; Lempert, L.; Shiffman, B.: Algebraic approximation of holomorphic maps from Stein domains to projective manifolds, Duke math. J. 76, 333-363 (1994) Stein spaces, Nash functions and manifolds, Invariant metrics and pseudodistances in several complex variables Algebraic approximations of holomorphic maps from Stein domains to projective manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry. In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre's relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation scheme is governed by flatness, i.e., a rank-preserving property for associated dual SDPs. If flatness is observed, then optimizers can be extracted using the Gelfand-Naimark-Segal construction and the Artin-Wedderburn theory verifying exactness of the relaxation. To enforce flatness we employ a noncommutative version of the randomization technique championed by Nie. The implementation of these procedures in our computer algebra system \texttt{NCSOStools} is presented and several examples are given to illustrate our results. noncommutative polynomial; optimization; sum of squares; semidefinite programming; moment problem; Hankel matrix; flat extension; Matlab toolbox; real algebraic geometry; free positivity I. Klep and J. Povh, \textit{Constrained trace-optimization of polynomials in freely noncommuting variables}, J. Global Optim., 64 (2016), pp. 325--348. Semidefinite programming, Semialgebraic sets and related spaces, Real algebra, Linear operator methods in interpolation, moment and extension problems Constrained trace-optimization of polynomials in freely noncommuting variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Author's abstract: Let \(m\) and \(n\) be positive integers such that \(n\geqslant m\) and let \(B\) be a polynomial ring in \(m+n+1\) variables over a field \(k\) of characteristic 0. We give a bijective correspondence between the equivalence classes of embeddings \(\mathbb A^m \to \mathbb A\) and the equivalence classes of sequences of mutually commuting locally nilpotent derivations \(\delta _i\) (1\(\leqslant i \leqslant m\)) on \(B\) in some form, which are homogeneous with respect to a \(\mathbb Z\)-grading on \(B\) and have slices. The intersection \(A\) of the kernels of \(\delta i\) for \(1\leqslant i\leqslant m\) inherits the \(\mathbb Z\)-grading on \(B\). We show that \(A\) is a polynomial ring with homogeneous coordinates if and only if the corresponding embedding is rectifiable. locally nilpotent derivations; slice; embeddings problem; torus action Group actions on affine varieties, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Affine fibrations Homogeneous locally nilpotent derivations having slices and embeddings of affine spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{B C_n}(\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of \textit{G. I. Lehrer} [Bull. Lond. Math. Soc. 24, No. 1, 76--82 (1992; Zbl 0770.14013)] and \textit{M. Kim} [Proc. Am. Math. Soc. 120, No. 3, 697--703 (1994; Zbl 0816.14008)]. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^\ast(\mathcal{M}_{B C_n}(\mathbb{C}); \mathbb{C})$, and an asymptotic stability result for certain \textit{polynomial} statistics on monic squarefree polynomials over $\mathbb{F}_q$ with non-zero constant term. This result is the type B/C analogue of a theorem due to \textit{T. Church} et al. [Contemp. Math. 620, 1--54 (2014; Zbl 1388.14148)] in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as $\mathit{FI}_{\mathcal{W}}$-algebras finitely generated in $\mathit{FI}_{\mathcal{W}}$-degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators. complement of a union of hyperplanes; weights; mixed Hodge structure Combinatorial aspects of representation theory, \(p\)-adic cohomology, crystalline cohomology, Topological properties in algebraic geometry, Configurations and arrangements of linear subspaces Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article the author generalizes the notions of smooth, unramified and étale morphisms to the category of fine logarithmic schemes and gives criteria which are analogs of criteria in the category of schemes.
First he recalls the definition of logarithmic schemes, which have been introduced by \textit{K. Kato}: a log structure \((M_X , \alpha)\) on a scheme \(X\) is a sheaf of monoids \(M_X\) on the étale site on \(X\) and a homomorphism \(\alpha : M_X \to {\mathcal O}_X\) such that \(\alpha ^{-1} ({\mathcal O} _X ^*) \buildrel {\scriptstyle \simeq} \over \longrightarrow {\mathcal O} _X ^*\); a chart \((P \to M_X)\) of \(M_X\) is a homomorphism \(P_X \to M_X\), where \(P_X\) is the constant sheaf of monoids of value \(P\), with \(P\) finitely generated and integral.
Then, as in the category of schemes, we can give the following definitions: A morphism \(f : X \to Y\) of fine log schemes is formally smooth (resp. formally unramified, resp. formally étale) if for any strict closed immersion of affine fine log schemes \(Z_0 \buildrel {\scriptstyle i} \over \hookrightarrow Z\) and any morphism \(Z \to Y\) the map \(\text{Hom}_Y (Z,X) \to \text{Hom}_Y (Z_0 ,X)\) induced by \(i\) is surjective (resp. injective, resp. bijective); and we get the same basic properties for these morphisms of fine log schemes when we consider composition, base change, etc... A morphism \(f : X \to Y\) of fine log schemes is called smooth (resp. unramified, resp. étale) if it is formally smooth (resp. formally unramified, resp. formally étale) and the underlying morphism of schemes is locally of finite presentation. We have again the same properties as in the category of schemes, and the author gives the following characterization:
Theorem: Let \(f : X \to Y\) be a morphism of fine log schemes such that the underlying morphism of schemes is of finite presentation; the following properties are equivalent:
(a) \(f\) is unramified.
(b) The diagonal \(\Delta : X \to X \times _Y X\) is étale.
(c) The differential module \(\Omega _{X/Y} ^1\) is zero.
(d) For any \(y \in Y\) the fibre \(X_y = f^{-1} (y)\) provided with the induced log structure is unramified over Spec\(k(y)\).
(e) Étale locally on \(X\) there exists a chart \( (Q \to M_X , P \to M_Y, P \to Q)\) extending a given chart on \(P \to M_Y\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective with finite cokernel of order invertible on \(X\), (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}(\mathbb Z [P ])} \text{Spec}(\mathbb Z [Q ])\) is unramified.
In the last part the author gives criteria for a morphism to be smooth, flat or étale. First he shows that the morphism \(f\) is smooth if and only if locally it can be factorized over an étale map into the standard log affine space over \(Y\). Then he recalls the following theorem of Kato:
Theorem: Let \(f : X \to Y\) be a morphism of fine log schemes; the following properties are equivalent:
(a) \(f\) is smooth (resp. étale).
(b) Étale locally on \(X\) there exists a chart \( (Q \to M_X , P \to M_Y, P \to Q)\) of \(f\) extending a given chart on \(P \to M_Y\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective and the torsion part of its cokernel (resp. its cokernel) is finite of order invertible on \(X\), (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}({\mathbb Z} [P ])} \text{Spec}(\mathbb Z [Q ])\) is smooth (resp. étale).
The author gives the following definition: A morphism of fine log schemes \(f : X \to Y\) is called ``flat'' if fppf locally on \(X\) and \(Y\), there exist a chart \( (Q \to M_X ,P \to M_Y, P \to Q) \) of \(f\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective, (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}(\mathbb Z [P ])} \text{Spec}(\mathbb Z [Q ])\) is flat.
With this definition he can generalize the usual fibre criterion for flatness: Let \(f : X \to Y\) be an \(S\)-morphism of fine log schemes, with \(X/S\) flat, then \(f\) is flat if and only if the induced maps on the fibres \(f_s : X_s \to Y_s\) are flat.
He can also generalize criteria for étale and smooth morphisms: \(f : X \to Y\) is étale if and only if \(f\) is flat and unramified; \(f : X \to Y\) is smooth if and only if \(f\) is flat and the induced maps on the fibres \(f_y : X_y \to y\) are smooth. category of fine logarithmic schemes; log structure; morphism of fine log schemes; fibre criterion for flatness Werner Bauer, On smooth, unramified étale and flat morphisms of fine logarithmic schemes, Math. Nachr. 176 (1995), 5 -- 16. Local structure of morphisms in algebraic geometry: étale, flat, etc., Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies On smooth, unramified, étale and flat morphisms of fine logarithmic 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 authors study what they call piecewise algebraic (P. A.) sets and functions. The P. A. sets are simply the constructible subsets of \({\mathbb{C}}^ n\) defined over \({\mathbb{Z}}\) (given by finite Boolean combination of polynomial equations and inequalities with coefficients in \({\mathbb{Z}})\), and P. A. functions are multivalued functions whose graph is P. A. The main tool is a decomposition of P. A. sets into a finite disjoint union of ``normal'' P. A. sets, which are roughly finite sheeted coverings of complements of algebraic hypersurfaces in affine spaces corresponding to a subset of the coordinates. This gives closure properties of P. A. sets and functions with respect to several constructions, and a nice notion of dimension. The authors care that the constructions are performed inside ``certainly computable'' P. A. sets. Then they turn to some applications: dependence of the minimal complexity of the computation of a polynomial upon its coefficients, dependence of the Jordan canonical form of a matrix upon its coefficients, dependence of formal solutions of meromorphic differential equations upon the coefficients of the equation.
It is a pity that there is no mention of quantifier elimination theory, concerning the properties of constructible subsets defined over \({\mathbb{Z}}\). For instance the fact that ``the projection of a certainly computable P. A. set is a certainly computable P. A. set'' is just the effective quantifier elimination for algebraically closed field, and this has now been well studied from the point of view of complexity [\textit{J. Heintz}, Theor. Comput. Sci. 24, 239--277 (1983; Zbl 0546.03017)]. Also, the decomposition into normal P. A. sets plays a role very similar to the cylindrical algebraic decomposition of \textit{G. E. Collins} [see Autom. Theor. Formal Lang., 2nd GI Conf., Kaiserslautern 1975, Lect. Notes Comput. Sci. 33, 134--183 (1975; Zbl 0318.02051)] in the real case. piecewise algebraic functions; piecewise algebraic sets; P. A. sets; solutions of meromorphic differential equations; quantifier elimination; constructible subsets Effectivity, complexity and computational aspects of algebraic geometry, Explicit solutions, first integrals of ordinary differential equations, Abstract differential equations Piecewise algebraic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Binomial \(\mathcal D\)-modules are a generalization of classical Horn hypergeometric systems, introduced by \textit{A. Dickenstein} et al. [Duke Math. J. 151, No. 3, 385--429 (2010; Zbl 1205.13031)] and problems related to the holonomicity of this kind of systems are essential in the study of these objects.
The authors give in this paper an important contribution to this subject, focusing on the study of the irregularity of binomial \(\mathcal D\)-modules.
In particular, they are able to improve the characterization of regular holonomicity of a binomial \(\mathcal D\)-module \(M_A(I, \beta)\) presented in [loc. cit.].
Moreover, the authors present a description of the slopes of \(M_A(I, \beta)\) along coordinate subspaces in terms of known slopes of some related hypergeometric \(\mathcal D\)-modules depending on \(\beta\).
For a generic parameter \(\beta\), they also compute the dimension of the generic stalk of the irregularity of binomial \(\mathcal D\)-modules and finish the paper with ``a procedure to compute Gevrey solutions of binomial \(\mathcal D\)-modules by using known results in the hypergeometric case''.
These interesting results are presented in a well written article, carefully organized and containing illuminating examples. binomial \(\mathcal D\)-module; slope; Gevrey solution Fernández-Fernández, María-Cruz; Castro-Jiménez, Francisco-Jesús: On irregular binomial D-modules. Math. Z. 272, No. 3-4, 1321-1337 (2012) Sheaves of differential operators and their modules, \(D\)-modules, Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings of differential operators and their modules, Other hypergeometric functions and integrals in several variables On irregular binomial \(D\)-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 Given a simply connected bounded semianalytic subset \(\Omega \) of \(\mathbb{R}^{2}\) with an analytical smooth boundary \(\partial \Omega \) and two bounded continuous subanalytic functions \(f:\Omega \rightarrow \mathbb{R}\) and \(g:\partial \Omega \rightarrow \mathbb{R}\) the author shows that the (unique) solution \(u:\overline{\Omega}\rightarrow \mathbb{R} \) to the Poisson equation \(\triangle u=f\) in \(\Omega \) subject to the boundary condition \(u=g\) on \(\partial \Omega \) is definable in the o-minimal structure \(\mathbb{R}_{\text{an},\exp }\) (that is, the o-minimal structure generated by the globally subanalytic sets and the exponential mapping). Existence of solutions is based on the fact that the function \(f\) belonging to the polynomially bounded structure \(\mathbb{R}_{\text{an}}\) of globally subanalytic sets is Hölder continuous, while uniqueness is a consequence of the maximum principle.) Using a result of \textit{J.-M. Lion} and \textit{J.-P. Rolin} [Ann. Inst. Fourier 48, 755--767 (1998; Zbl 0912.32007)], the author establishes at first the result in the particular case when \(\Omega \) is the unit ball \(B(0,1)\), and concludes by the definability (in \(\mathbb{R}_{\text{an}}\)) of the Riemann mapping that maps \(\overline{\Omega}\) to \(\overline{B}(0,1).\) In particular case \(f=0\) (Laplace equation) it is known that the solution \(u\) is analytic in \(\Omega \), but eventually without analytic extension at the boundary \(\partial \Omega \). Under necessary assumptions on the boundary function \(g\) the author gives a dichotomy result stating that either \(u\) is analytic at \(x\in\partial \Omega \) (thus \(u\) is definable in \(\mathbb{R}_{\text{an}}\)) or \(u\) is definable in \(\mathbb{R}_{\text{an},\exp}\) but not in \(\mathbb{R}_{\text{an}}\). Poisson equation; Laplace equation; subanalytic set; Pfaffian closure; o-minimal structure Kaiser, T.: Definability results for the Poisson equation, Adv. geom. 6, 627-644 (2006) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Real-analytic and semi-analytic sets, Semi-analytic sets, subanalytic sets, and generalizations Definability results for the Poisson equation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(P_1,\dots,P_r\) be distinct points in \(\mathbb{P}^n\) with associated primes \(p_1,\dots,p_r\) in \(k[X_0, \dots,X_n]\) (\(k\) an algebraically closed field of characteristic zero); the subscheme \(Z=\{P_1,\dots,P_r;m_1,\dots,m_r\}\) associated to the ideal \(I=p^{m_1}_1\cap\dots\cap p^{m_r}_r\) is called a subscheme of fat points; if \(m_i = m\) for all \(i=1,\dots, r\), then \(Z\) is called a homogeneous scheme of fat points.
The Hilbert function of \(I\) has been studied by many authors, especially when \(n=2\) and in cases of fat points whose support lies in some special configuration.
The authors focus their attention to the homogeneous scheme of fat points
\[
X_{\text{grid}}=\{\text{CI}_{\text{grid}}(a,b);m\}
\]
whose support is a complete intersection given by the intersection of two set of lines in \(\mathbb{P}^2\) as an \(a\times b\) grid; they study the connections between the above fat point schemes and particular varieties of simple points called partial intersection (p.i. for short). Their main result is that the Hilbert function of a homogeneous fat point scheme whose underlying CI is an \(a \times b\) grid in \(\mathbb{P} ^2\) or an \(a \times b\) grid minus a point does not depend on the forms of degree \(a\) and \(b\) that generate the CI, but only on the numbers \(a, b\) and \(m\).
The authors also describe an alternative approach to the problem by considering the Gröbner basis of the ideal associated to a homogeneous ideal whose support is constructed on a grid. This paper is continued [cf. \textit{M. Buckels, E. Guardo} and \textit{A. Van Tuyl}, Matematiche 55, No. 1, 191-202 (2000; see the following review Zbl 1009.14010)]. Hilbert functions; Betti numbers; fat points; partial intersection Buckles, M.; Guardo, E.; Van Tuyl, A.: Fat points on a grid in P2, Matematiche (Catania) 55, 169-189 (2000) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Syzygies, resolutions, complexes and commutative rings Fat points on a grid in \(\mathbb{P}^2\) | 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 Let \(G\) be a compact group of linear transformations of a Euclidean space \(V\). The \(G\)-invariant \(C^{\infty}\) functions can be expressed as \(C^{\infty}\) functions of a finite basic set of \(G\)-invariant homogeneous polynomials, sometimes called an integrity basis. The mathematical description of the orbit space \(V/G\) depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semidefiniteness conditions of the \(\hat{P}\)-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of \(G\)-invariant homogeneous polynomials forming an integrity basis is not unique, so it is not unique the mathematical description of the orbit space too. If \(G\) is an irreducible finite reflection group, \textit{K. Saito} et al. [Commun. Algebra 8, 373--408 (1980; Zbl 0428.14020)] characterized some special basic sets of \(G\)-invariant homogeneous polynomials that they called \textit{flat}. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups \(E_7\) and \(E_8\). In this paper the flat basic sets of invariant homogeneous polynomials of \(E_7\) and \(E_8\) and the corresponding \(\hat{P}\)-matrices are determined explicitly. Using the results here reported one is able to determine easily the \(\hat{P}\)-matrices corresponding to any other integrity basis of \(E_7\) and \(E_8\). From the \(\hat{P}\)-matrices one may then write down the equations and inequalities defining the orbit spaces of \(E_7\) and \(E_8\) relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups \(E_7\) and \(E_8\) or one of the Lie groups \(E_7\) and \(E_8\) in their adjoint representations.{
\copyright 2010 American Institute of Physics} Talamini, V., Flat bases of invariant polynomials and \(\hat{P}\)-matrices of \(E_7\) and \(E_8\), J. Math. Phys., 51, (2010), 20 pp Group actions on varieties or schemes (quotients), Geometric invariant theory, Actions of groups on commutative rings; invariant theory, Trace rings and invariant theory (associative rings and algebras), Actions of groups and semigroups; invariant theory (associative rings and algebras) Flat bases of invariant polynomials and \(\hat{P}\)-matrices of \(E_7\) and \(E_8\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 brings a computational method for analyzing singularities of a parametrically given projective plane curve. More precisely, the authors consider a regular map $\mathbb{A}^1\to\mathcal{C}$ of degree 1 giving a proper parametrization $\mathcal{P}(t)$ of a plane projective curve.
For each point $P$ of the curve, the \textit{fibre function} is defined. It is a polynomial and its roots are the preimages (with proper multiplicities) of $P$ in the parametrization $\mathcal{P}(t)$. The exception is the case when the point $P$ is the image of the point obtained by the projective completion of the parameter space $\mathbb{A}^1$ (in this case the point $P$ is called the \textit{limit point}), and this issue is addressed by the paper.
Based on the parametrization $\mathcal{P}(t)$ also a \textit{$T$-function} is defined and it gives information about all singularities of the given curve $\mathcal{C}$, provided the singularities are ordinary. Again, the authors deal with the problematic case when the limit point is singular.
At the end, the algorithm is compared to other known methods in the literature.
The paper refers to the previous work of the authors [\textit{A. Blasco} and \textit{S. Pérez-Díaz}, ``Resultants and singularities of parametric curves'', \url{arXiv:1706.08430}], where the case of the limit point being a regular point of $\mathcal{C}$ is solved. algebraic parametric curve; rational parametrization; singularities; limit point; T-function; fibre function Computational aspects of algebraic curves, Singularities of curves, local rings, Solving polynomial systems; resultants, Symbolic computation and algebraic computation The limit point and the T-function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves a series of properties for the Hodge structure of local systems, generalizing earlier results of himself and other authors. For example: Let \(X\) be a quasi-projective complex manifold and \(\rho:\pi_1(X)\rightarrow U(N,\mathbb C)\) a character inducing a rank \(N\) local unitary system \(V_{\rho}\) on \(X\). The cohomology groups \(H^i(X,V_{\rho})\) support a canonical mixed Hodge structure and a Hodge filtration \(\dots\subset F^{p+1}H^n(V_\rho)\subset F^{p}H^n(V_\rho)\subset\dots\). Consider the family \({\mathcal W}=\{V_\rho\otimes L_\chi\,|\,\chi\in \text{ Hom}(\pi_1(X),S^1)\,\}\), where \(L_\chi\) is the rank one unitary local system induced by \(\chi\). The author describes for \(l\in\mathbb N\) the structure of the family \(S^{n,p}_l=\{W\in {\mathcal W}\,|\,\text{dim}F^pH^n(W)/F^{p+1}H^n(W)\geq l\,\}\) under the additional assumption that \(H^1(\overline{X},\mathbb C^*)=0\) for some non-singular compactification \(\overline{X}\) of \(X\). Assuming that \(H_1(X,\mathbb Z)\) is torsionfree and identifying the universal cover of the group Hom \((\pi_1(X),S^1)\) with the tangent space \(T\) at the neutral element , the universal covering map with the exponential map and the fundamental domain \({\mathcal U}\) of the \(\text{ Hom }(\pi_1(X),S^1)\)-action on \(T\) with the unit cube in \(T\) he proves that the family \(S^{n,p}_l\) is a finite union of polytopes in \({\mathcal U}\). The proof is based on a study of Deligne extensions of local systems on \(X\) to bundles on \(\overline{X}\) [\textit{P. Deligne}, Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics. 163. (Berlin-Heidelberg-New York): Springer-Verlag. (1970; Zbl 0244.14004)].
Another main result of the article deals with a local version of this setting: Let \({\mathcal X}\) be a germ of a complex space with isolated normal singularity such that the intersection of \({\mathcal X}\) with a sufficiently small sphere around the singularity is simply connected , and let \({\mathcal D}\) be a divisor on \({\mathcal X}\) with \(r\) irreducible components and \(X:={\mathcal X}\backslash {\mathcal D}\). The family of rank one local systems is then parametrized by the affine torus \(H^1(X,\mathbb C^*)=\mathbb C^{*r}\). The author proves that the characteristic varieties
\[
S^n_l=\{\chi\in H^1({\mathcal X}\backslash{\mathcal D},\mathbb C^*)\,|\,\text{ dim }{\mathcal X}\backslash{\mathcal D},L_\chi)\geq l\,\}, 1\leq n\leq\text{dim }X,
\]
of \(({\mathcal X},{\mathcal D})\) have a decomposition as a finite union of subtori of \(H^1({\mathcal X}\backslash{\mathcal D},\mathbb C^*)\), translated by points of finite order. For the proof he uses a Mayer-Vietoris spectral sequence for the union of tori bundles on quasi-projective manifolds which degenerates in the term \(E_2\), and also the results of \textit{D. Arapura} [J. Algebr. Geom. 6, No. 3, 563--597 (1997; Zbl 0923.14010)] in the global Kähler setting. The author also introduces twisted characteristic varieties as a multivariable generalization of twisted Alexander polynomials. He uses these varieties to obtain information about the homology of non unitary local systems. Finally explicit calculations of the graded components of the Hodge filtration are given for several examples where \(X\) is the complement of an arrangement of hypersurfaces in \(\mathbb P^n, n\leq 3\). unitary local system; Hodge filtration; Hodge number; polytope; Alexander polynomial; Deligne extension Libgober, A.: Alexander invariants of plane algebraic curves. Singularities, Part 2 (Arcata, Calif., 1981), pp. 135-143. Proceedings of Symposia in Pure Mathematics, vol. 40. Amer. Math. Soc., Providence (1983) Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), de Rham cohomology and algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Knots and links in the 3-sphere Non vanishing loci of Hodge numbers 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 \(\mathbb{P}^{n}_{k}\) denote a projective space of dimension \(n\) over an algebrically closed field \(k\). If \(\Omega_{{\mathbb{P}}^{n}_{k}}^{s}\) is the sheaf of differential \(s\)-forms, where \(1 \leq s \leq n-1\), a Pfaff field \(\eta\) of rank \(s\) is a map \(\eta:\Omega_{{\mathbb{P}}^{n}_{k}}^{s} \to\mathcal{L}\) where \(\mathcal{L}\) is an invertible sheaf. A closed subscheme \(X\subset \mathbb{P}^{n}_{k}\) is invariant under \(\eta\) if \(\eta\) induces a Pfaff field \(\Omega_{X}^{s} \to\mathcal{L}_{\mid X}\) on \(X\). The closed scheme \(\mathcal{S}\) supported on the set of points where \(\eta\) is not surjective is the singular locus of \(\eta\). The most interesting case appears when \(\eta\) is obtained taking determinants of a Pfaff system, which can be seen as a distribution outside \(\mathcal{S}\).
The main contribution of the paper under review (Theorem 3.1) states that if \(X\) is a connected reduced subscheme of pure dimension \(s>0\) which moreover is Cohen-Macaulay and subcanonical (for instance a complete intersection), and the singular locus of \(X\) has regularity bounded by the regularity of \(X\) minus \(2\), then \(h^{s}({\Omega}_{X}^{s})=1\). This theorem extends the results of the second author and \textit{S. Kleiman} in [Contemp. Math. 354, 57--67 (2004; Zbl 1065.37035)] for \(n>2\) and any \(s\). As application the authors study an \(X\) invariant by a Pfaff field of rank \(s\), obtaining bounds of regularity of \(X\) in terms of \(s\), regularity of the singular locus of \(X\) and degree of \(\mathcal{L}\) (Theorems 4.1 and 4.3) when the support of \(\mathcal{S}\) does not contain irreducible components of \(X\).
The paper gives interesting results concerning the the Poincaré problem. Pfaff systems; projective spaces; invariant schemes; regularity J. D. A. S. Cruz and E. Esteves, Bounding the regularity of subschemes invariant under Pfaff fields on projective spaces , Comment. Math. Helv. 86 (2011), 947-965. Dynamical aspects of holomorphic foliations and vector fields, Singularities of holomorphic vector fields and foliations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Bounding the regularity of subschemes invariant under Pfaff fields on projective 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 \(S\) be a fixed noetherian scheme. In the category of algebraic stacks (or Artin stacks) over \(S\), a morphism \(F:{\mathcal X}\to{\mathcal X}'\) of objects is called proper if \(F\) is of finite type, separated and universally closed. Like in the theory of schemes, properness criteria for morphisms of algebraic stacks are of particular practical importance, and in this respect the so far most complete account is given in the fundamental monoraph ``Champs algébriques'' by \textit{G. Laumon} and \textit{L. Moret-Bailly} [Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 39 (2000; Zbl 0945.14005)]. In Proposition 7.12 of their book, these authors prove a properness criterion for a morphism of algebraic stacks, which holds under a certain additional assumption (``condition \((*)\)'') on \(F\), and they derive a number of important consequences from that special criterion of properness. The paper under review substantially clarifies the situation concerning the condition \((*)\) in the properness criterion of Laumon and Moret-Bailly. Roughly speaking, the author proves that condition \((*)\) always holds, and that it can be omitted from the assumptions of this properness criterion. More precisely, the first main theorem of the present paper states that for any separated Artin stack \({\mathcal X}/S\) of finite type, there exists a proper surjective morphism \(p: X\to{\mathcal X}\) from a quasi-projective \(S\)-scheme \(X\) onto \({\mathcal X}\).
As the author points out, the approach to proving this theorem was suggested by O. Gabber, who also gave a first (unpublished) proof of it. Anyway, the present paper contains the first elaborated, complete and published proof of this theorem simplifying the properness criterion of Laumon and Moret-Bailly for morphisms of Artin stacks.
The second part of the paper is devoted to applications of this result. Among them is the second main theorem of the author's present work, which basically provides a version of the so-called Grothendieck existence theorem [\textit{A. Grothendieck}, EGA III, première partie, Publ. Math., Inst. Hautes Étud. Sci. 11, 349--511 (1962; Zbl 0118.36206)] for algebraic stacks. This theorem generalizes previous results, in this direction, obtained in the special cases of algebraic spaces [\textit{D. Knutson}, ``Algebraic spaces'', Lect. Notes Math. 203 (1971; Zbl 0221.14001)], tame Deligne-Mumford stacks [\textit{D. Abramovich} and \textit{A. Vistoli}, J. Am. Math. Soc. 15, No. 1, 27--75 (2002; Zbl 0991.14007)], and separated Deligne-Mumford stacks [\textit{M. Olsson} and \textit{J. Starr}, Commun. Algebra 31, No. 8, 4069--4096 (2003; Zbl 1071.14002)], respectively. morphisms; Chow's lemma; Grothendieck existence theorem; coherent sheaves Martin Olsson, ``On proper coverings of Artin stacks'', Adv. Math.198 (2005) no. 1, p. 93-106 Generalizations (algebraic spaces, stacks), Étale and other Grothendieck topologies and (co)homologies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Formal methods and deformations in algebraic geometry On proper coverings of Artin 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 The global uniformization problem for an analytic correspondence \(f(z,w)=0\) is the problem of finding a way to pass from the implicit description \(f(z,w)=0\) to an equivalent parametric description \(z=\varphi (t)\), \(w=\psi (t)\), where \(\varphi\) and \(\psi\) are single-valued meromorphic functions in a parameter \(t\). The article is devoted to the problem of explicit construction of a global uniformization.
The authors consider the simplest case of a uniformization of an algebraic correspondence \(f(z,w)=0\). They start with an algorithm for decomposing the polynomial \(f(z,w)\) into irreducible factors. In particular, the question of irreducibility of a given polynomial is solved. The content of the corresponding section is at least of methodological interest as a new application of the Carleman ``damping function.'' The further exposition mainly follows \textit{Eh.~I.~Zverovich} [Vestn. Beloruss. Gos. Univ. Im. V. I. Lenina, Ser. I 1991, No. 1, 36--39 (1991; Zbl 0773.30043)] and \textit{O.~B.~Dolgopolova, È.~I. Zverovich} [Boundary Value Problems, Spectral Functions and Fractional Calculus (in Russian), BGU, Minsk, 76-80 (1996)] but in more detail. In the final section, various examples of an explicit construction of a global uniformization are considered. global uniformization Compact Riemann surfaces and uniformization, Algebraic functions and function fields in algebraic geometry Explicit construction of a global uniformization for an algebraic correspondence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 smooth projective curve of genus \(g\) and let \(E\) be a vector bundle over \(C\). The Segre invariant, semistability and cohomological semistability for \(E\) are closely related and have been studied from different points of view for many years. The author consider \(S:=\mathbb{P}(E)\) the associated projective bundle and he describes the inflectional loci of projective models \(\psi: S\dashrightarrow \mathbb{P}^n\) in terms of Quot Scheme of \(E\). The author gives a characterization of the Segre invariant \(s_1(E)\) in terms of the osculating space and the inflectional locus. This allows to give an equivalence between the semistability of bundle \(E\) with slope \(\mu<1-2g \) and the osculating space. In the same direction, the author gives a natural characterization of cohomological stability of \(E\) via inflectional loci. Finally, he proved that for general \(S\), the inflectional loci are the expected dimension. curve; scroll; quot scheme; inflectional locus Vector bundles on curves and their moduli, Projective techniques in algebraic geometry Quot schemes, Segre invariants, and inflectional loci of scrolls over curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\subset A\) be factorial domains containing \(\mathbb Q\). In the paper under review the authors give an equivalent condition in terms of locally nilpotent derivations for \(A\) to be \(R\)-isomorphic to \(R[v,w]/(cw-h(v))\), where \(0\not=c\in R\) is not a unit and \(h(v)\in R[v]\) is nonconstant modulo every prime factor of \(c\): There exists an irreducible locally nilpotent \(R\)-derivation \(\xi\) of \(A\) with ring of constants \(A^{\xi}\) equal to \(R\) and such that there exists \(c=\xi(s)\in{\mathfrak p}{\mathfrak l}=\xi(A)\cap A^{\xi}\) with the property that the ideal \(R[s]\cap cA\) of \(R[s]\) is generated by \(c\) and a polynomial \(h(s)\in R[s]\). The result implies the isomorphism of the differential rings \((A,\xi)\) and \((R[v,w]/(cw-h(v)),\delta)\) where \(\delta(v)=c\) and \(\delta(w)=\partial_vh(v)\). The authors show that an example from a paper by Daigle gives that the result does not hold if \(A\) in not factorial. In the particular case when \(R\) is a polynomial ring in one variable, the result yields a natural generalization of of a result in Masuda characterizing Danielewski hypersurfaces whose coordinate ring is factorial. Finally, the authors apply their result to the study of triangularizable locally nilpotent \(R\)-derivations of the polynomial ring in two variables over \(R\). Danielewski hypersurface; locally nilpotent derivation; triangularizable derivation Group actions on affine varieties, Derivations and commutative rings Locally nilpotent derivations of factorial 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 \(f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C}^{m},0)\) be the germ of a holomorphic mapping with an isolated zero at \(0\) and \(e(f)\) its algebraic multiplicity (for instance the Hilbert-Samuel multiplicity) and \(\mathcal{L} _{0}(f)\) be the Łojasiewicz exponent of \(f.\) The authors give an effective formula for \(e(f)\) in the case \(f\) is a polynomial mapping and using this result they prove that for a given holomorphic deformation \(f_{s}\) of \(f\) there is a stratification of the parameter space such that \(\mathcal{L}_{0}(f_{s})\) is lower semicontinuous on each stratum and after a refinement of the stratification \(\mathcal{L}_{0}(f_{s})\) is constant on each new stratum. This generalizes the results by B. Teissier and A. Płoski on lower semicontinuous of the Łojasiewicz exponent in holomorphic families of mappings with constant multiplicity. germs of holomorphic mappings with an isolated zero; Łojasiewicz exponent; effective formula for the algebraic multiplicity Invariants of analytic local rings, Singularities in algebraic geometry, Computational aspects in algebraic geometry, Computational aspects of field theory and polynomials Multiplicity and semicontinuity of the Łojasiewicz exponent | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present paper is an enlightening work on motivic stable homotopy theory. The authors study the fundamental question how the motivic stable homotopy category behaves under base change. The main theorem is that the base change along an extension of algebraically closed fields whose exponential characteristic is prime to \(\ell\) induces a full and faithful functor between the mod-\(\ell\)-motivic stable homotopy categories over the fields. Such a behavior under base change is generally known as \textit{rigidity}. \textit{A. Suslin} has shown a similar property for algebraic \(K\)-theory in his celebrated paper [Invent. Math. 73, 241--245 (1983; Zbl 0514.18008)]. Since then several generalizations and applications in other theories have been achieved. The strategy of the proof is almost always based on Suslin's ingenious idea in [loc. cit.]. Suslin showed that certain maps in algebraic \(K\)-theory induced by different rational points on a smooth projective curve agree by proving that the maps factor through the group of divisors of degree zero of the curve and using the divisibility of this group. He called this result the Rigidity Theorem in loc. cit. which turned out to be the mother of all rigidity results. The authors of the present paper also use Suslin's idea. But on their way to prove the main result, they also show several other interesting properties of the motivic stable homotopy category and the category of motivic symmetric spectra which are needed to generalize Suslin's input, but which are of independent interest as well. The second main ingredient is the construction of suitable transfer maps. The authors construct them over general base schemes in two different ways for finite étale maps and for what they call linear maps between smooth schemes. The third main point is a very interesting discussion of motivic Moore spectra and an explicit fibrant replacement functor in the mod-\(n\)-localized category of motivic symmetric spectra. The authors conclude the paper with an Appendix on Homological Localization giving an alternative foundation for a localization theory of motivic symmetric spectra. motivic homotopy theory; rigidity; transfer maps; motivic symmetric spectra; localization Röndigs, Oliver; Østvær, Paul Arne, Rigidity in motivic homotopy theory, Math. Ann.. Mathematische Annalen, 341, 651-675, (2008) Motivic cohomology; motivic homotopy theory, Stable homotopy theory, spectra, Abstract and axiomatic homotopy theory in algebraic topology Rigidity in motivic homotopy theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review the authors provide an interesting generalization of the classical Riemann-Hurwitz and Pluecker formulas. Here is the setting of the paper under review. Let \(X\) be a smooth projective variety defined over an algebraically closed field \(\mathbb{K}\) of arbitrary characteristic, \(S\) a projective variety of dimension \(n\) and let \(\mathcal{D} \subset X \times S\) be a flat family of divisors on \(X\).
The key definition that the authors introduce is a generalization of the well-known definition of inflection points.
Definition. If we are given a smooth projective curve \(C\) and a morphism \(f : C \rightarrow X\), we say that a \(\mathbb{K}\)-point \(p \in C\) is an inflection point relative to \(\mathcal{D}\) if there exists a \(z \in S\) with the corresponding divisor \(D = \mathcal{D}_{z} \subseteq X\) such that \(f^{*}D\) has multiplicity at least \(n+1\) at \(p\).
The formula for the total inflection of such a morphism will be expressed in terms of two quantities, namely \(N(X,\mathcal{D}) \in \mathbb{Z}_{\geq 0}\) and \(\mathcal{H}(X,\mathcal{D}) \in \mathrm{Pic}(X)\) which arise in that setting. More precisely, for any positive integer \(m\) there is a natural evaluation morphism on the space of \(m\)-tuples of points on divisors in \(\mathcal{D}\), which is denoted by
\[
\mathrm{ev}_{m}\,:\, \mathcal{D}\times_{S}\mathcal{D} \times_{S} \cdots \times_{S} \mathcal{D} \rightarrow X^{m},
\]
where we have \(m\) copies of \(\mathcal{D}\) on the lefthand side .
Since both the source and the target of \(\mathrm{ev}_{n}\) have dimension \(n\cdot\dim X\) and the target is in addition irreducible, one defines
\[
N(X,\mathcal{D})=\deg \, \mathrm{ev}_{n}.
\]
Thus \(N(X,\mathcal{D})\) is simply the number of divisors in \(\mathcal{D}\) passing through \(n\) general points in \(X\). Secondly, if \(\alpha \in A_{(n+1)\dim X - 1}(X^{n+1}) = \mathrm{Pic}(X^{n+1})\) is the pushforward of the fundamental class of the source of \(\mathrm{ev}_{m+1}\) to its target, and \(\delta : X \rightarrow X^{n+1}\) is the small diagonal immersion, one defines
\[
\mathcal{H}(X,\mathcal{D}) = \delta^{*}\alpha.
\]
Thus \(\mathcal{H}(X,\mathcal{D})\) is the pullback to \(X\) under the small diagonal immersion of the class of the divisor consisting of \((n+1)\)-tuples of points on \(X\) contained in some divisor in \(\mathcal{D}\).
The main result of the paper under review can be formulated as follows.
Main Result. In the setting presented above, if we are given any smooth proper curve \(C\) and any morphism \(f : C \rightarrow X\) such that the set \(I\subset C\) of inflection points relative to \(\mathcal{D}\) is finite, then it is possible to assign non-negative integer multiplicities \(m_{p}\) for all \(p \in I\) such that
\[
\sum_{p \in I} m_{p}p \sim f^{*}\mathcal{H}(X,\mathcal{D}) + \binom{n+1}{2}N(X,\mathcal{D})K_{C},\tag{\(\star\)}
\]
where \(\sim\) denotes linear/rational equivalence. In fact, one has \(m_{p} > 0\) for all \(p \in I\) if the induced morphism \(S \rightarrow \mathrm{Hilb}(X)\) to the Hilbert scheme is geometrically finite, respectively \(m_{p}=0\) for all \(p \in I\) otherwise.
Note that in fact the lefthand side of \((\star)\) is a divisor while the righthand side is a divisor class, so it would be more precise to say that the lefthand side is an element of the righthand side.
For the completeness of the picture, the authors show the following.
Theorem. The righthand side of \((\star)\) specializes to the classical Riemann-Hurwitz and Pluecker formulas in the special cases that \(X\) is a curve, or that \(X\) is a projective space, respectively. Moreover, \(m_{p}\)'s constructed in the paper recover the usual multiplicities occurring in these formulas. Riemann-Hurwitz formula; Plücker formula; ramification point Coverings of curves, fundamental group, Divisors, linear systems, invertible sheaves A Riemann-Hurwitz-Plücker formula | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth variety and \(\mathfrak{a}_1, \dots, \mathfrak{a}_n\) be a set of ideals on \(X\). Inspired by the ACC property for log canonical thresholds, see for example [\textit{T. de Fernex} et al., Duke Math. J. 152, No. 1, 93--114 (2010; Zbl 1189.14044)], [\textit{J. Kollár}, ``Which powers of holomorphic functions are integrable?'', \url{arXiv:0805.0756}] and [\textit{C. Hacon, J. McKernan} and \textit{Ch. Xu}, ``ACC for log canonical thresholds'', \url{arXiv:1208.4150}], the authors consider the following generalization. Let \(\{ (t_1, \dots, t_n) \in \mathbb{R}^n \;|\; (X, \mathfrak{a}_1^{t_1} \cdots \mathfrak{a}_n^{t_n}) \text{ is log canonical}\}\). Such a set is called a LCT-polytope, see also [\textit{A. Libgober}, Manuscr. Math. 107, No. 2, 251--269 (2002; Zbl 1056.57005)].
The authors show that any ascending sequence of LCT-polytopes \(P_1 \subseteq P_2 \subseteq \dots \) eventually stabilizes, which should already be viewed as a substantial generalization of ACC statements for log canonical thresholds. However, they show the following even stronger statement. Consider the LCT- polytopes as elements of the space \(\mathcal{H}_n\) of compact subsets of \(\mathbb{R}^n\), with the Hausdorff metric. The main result, Theorem 3.3, says that if the limit of a sequence of LCT-polytopes \(\{P_m\}\) converges to a compact set \(Q\) in the Hausdorff metric, then \(Q = \bigcap_{m \gg 0} P_m\), \(Q\) is a rational convex polytope and, after possibly enlarging the base field, \(Q\) is the LCT-polytope of a set of ideals in a power series ring. The paper also contains many illuminating examples. log canonical threshold; ascending chain condition; polytope Multiplier ideals, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Minimal model program (Mori theory, extremal rays) Sequences of LCT-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 The author considers a special subalgebra, denoted by \(C_n(M)\), of the algebra of germs of \(C^\infty\) functions at the origin in \(\mathbb R^n\). This subalgebra is defined in Section 2.1. The objective of the paper is to study germs at the origin of sets defined as finite unions of sets of the form
\[
\{x: \varphi_0(x)=0, \varphi_1(x)>0,\dots,\varphi_q(x)>0\},
\]
where \(\varphi_0,\varphi_1,\dots,\varphi_q\) are germs of \(C_n(M)\). This kind of set germs are called by the author quasi semianalytic germs. Moreover, the author calls quasi subanalytic germs the projections of quasi semianalytic germs.
It is proven in the paper that, by a finite number of blowings-up of \(\mathbb R^n\) with smooth center, it is possible to transform any \(f\in C_n(M)\), modulo a product by an invertible element in \(C_n(M)\), to a monomial (this is shown in proposition 7). This implies that \(C_n(M)\) is topologically noetherian, that is, every decreasing sequence of germs is stationary. This property allows the author to extend some well known properties of semianalytic germs to quasi semianalytic germs. The author also proves that the closure and each connected component of a quasi semianalytic germ is also quasi semianalytic. The main results are theorem 7, which gives a uniform bound of the number of connected components of the fibers of a projection restricted to a bounded quasi subanalytic set, and lemma 7, which shows that the dimension of quasi semianalytic germs is well behaved. The author proves also the complement theorem for quasi subanalytic germs and the Łojasiewicz inequalities for functions in the class of quasi semianalytic germs. quasianalytic functions; subanalytic and semianalytic sets; Gabrielov's theorem Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets The theorem of the complement for a quasi subanalytic set | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 central concern of this paper is the development of a higher dimensional analogue of the remarkable fact that the existence of a pencil of genus \(\geq 2\) on a compact complex surface \(S\) is a topological property of \(S\). More precisely, let \(X\) be an irregular compact Kähler manifold of dimension n and \(\alpha: X\to\text{Alb}(X)\) the Albanese morphism. The author defines \(X\) to be of Albanese general type if the image of \(\alpha\) has dimension n but \(\alpha\) is not surjective; it is easy to see that this property depends only on the complex cohomology of \(X\). The main result concerns morphisms \(f: X\to Y\), where \(X\) is an irregular compact Kähler manifold and \(Y\) is normal, of dimension \(k<n\), and has a smooth model which is of Albanese general type; it states that there is a bijection between the set of such morphisms and a certain set of real subspaces of \(H^ 1(X;\mathbb{C})\) (namely the saturated \(2k\)-wedge subspaces).
In later sections, the author gives applications to moduli of algebraic surfaces and considers the problem of stability under deformations of the existence of irrational pencils and higher dimensional analogues.
\{For a fuller discussion of the motivation, ideas and methods of this interesting paper, see the author's excellent introduction.\}. Hodge theory; Albanese variety; Kähler manifold; moduli of algebraic surfaces; deformations; irrational pencils Catanese, F, Moduli and classification of irregular Kähler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math., 104, 263-289, (1991) Compact Kähler manifolds: generalizations, classification, Complex-analytic moduli problems, Transcendental methods, Hodge theory (algebro-geometric aspects), Global differential geometry of Hermitian and Kählerian manifolds, Algebraic moduli problems, moduli of vector bundles, Moduli, classification: analytic theory; relations with modular forms Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations. Appendix by Arnaud Beauville | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the present article the author constructs a \(p\)-adic analogue of the Picard-Lefschetz formula which describes the monodromy action in terms of its vanishing cycles. Let \(V\) be a discrete valuation ring of mixed characteristic with field of fractions \(K\) and residue class field \(k\). Furthermore let \(X\) denote an \(n\)-dimensional proper \(V\)-scheme whose generic fibre \(X_\eta\) is smooth and whose special fibre \(X_s\) has at most ordinary double points as singularities. Then for every \(q\) there exists a homomorphism \({\mathrm{sp}}^*\) mapping from the rigid cohomology of \(X_s\) to the algebraic de Rham cohomology of \(X_\eta\), which is an isomorphism for \(q\neq n,n+1\). For \(q=n,n+1\) there is a canonical exact sequence relating \(H^q_{\mathrm{rig}}(X_s)\), \(H^q_{\mathrm{dR}}(X_\eta)\) and a so-called ``vanishing cycle module'' \(\Phi_\Sigma = \bigoplus_{\sigma\in\Sigma} \Phi_\sigma\) whose dimension equals the number of ordinary double points on \(X_s\). On \(H^q_{\mathrm{dR}}(X_\eta)\) there exists the notion of a monodromy operator \(N\). In the case that \(X_\eta\) has a semistable model over \(V\) this \(N\) is defined via the Hyodo-Kato isomorphism by a corresponding operator on logarithmic crystalline cohomology.
The author generalizes this notion to our non-semistable setting. Furthermore, he shows that the action of \(N\) on \(H^n_{\mathrm{dR}}(X_\eta)\) corresponds to a ``variation map'' \(\mathrm{Var}_\Sigma = \bigoplus_{\sigma\in\Sigma} \mathrm{Var}_\sigma\) on \(\Phi_\Sigma\) which can be described by means of a vanishing cycle, in analogy to the classical situation. The construction of \({\mathrm{sp}}^*\), \(\Phi_\Sigma\) and \(\mathrm{Var}_\Sigma\) can be outlined as follows: Let \(\tilde{X},\tilde{Y}\) denote the blowing-up of \(X\), \(Y\) at \(\Sigma\), with \(Y:=X_s\). After a quadratic base extension the exceptional divisor \(D_\sigma\) of \(\tilde{X}\rightarrow X\) at every \(\sigma\in\Sigma\) is a quadric in projective space, and \(C_\sigma:=D_\sigma \cap \tilde{Y}\) coincides with a hyperplane section. We define \(\Phi^q_\sigma := H^q_{rig}(D_\sigma\setminus C_\sigma)\), and our task is to construct a long exact cohomology sequence relating \(H^q_{\mathrm{rig}}(Y)\), \(H^q_{\mathrm{dR}}(X_\eta)\) and \(\bigoplus_{\sigma\in\Sigma} H^q_{\mathrm{rig}}(D_\sigma\setminus C_\sigma)\). In order to obtain such a sequence the author uses two different generalizations of Illusie's de Rham-Witt complex that were studied by \textit{O. Hyodo} and \textit{K. Kato} [in: Périodes \(p\)-adiques, Astérisque 223,
221--268 (1994; Zbl 0852.14004)]. The first one denoted by \(W_n \Omega^\bullet_Z(\log D)\) computes the logarithmic crystalline cohomology of a pair \((Z,D)\), where \(Z\) is a smooth \(k\)-scheme and \(D\) is a normal crossing divisor on \(Z\). The second one computes the logarithmic crystalline cohomology of a \(k\)-variety \(Y\) whose singularities are at most ordinary double points. These two complexes are combined in order to construct a short exact sequence \(0 \rightarrow K_Y \rightarrow WA^\cdot \rightarrow \bigoplus_{\sigma\in\Sigma} W \Omega^\bullet_{D_\sigma}(\log C_\sigma) \rightarrow 0\) which yields the desired cohomology sequence. Also the variation map \(\mathrm{Var}_\Sigma\) is constructed on the level of the complex \(W\Omega^\bullet_{D_\sigma}(\log C_\sigma)\). Elementary results on the rigid cohomology of quadric hypersurfaces are used to deduce the final result. \(p\)-adic cohomology, crystalline cohomology, Structure of families (Picard-Lefschetz, monodromy, etc.) The Picard-Lefschetz formula for \(p\)-adic cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an \(r\)-dimensional smooth complex quasiprojective variety and denote the Hilbert scheme parametrizing zero-dimensional subschemes of length \(n\) of \(X\) by \(\text{Hilb}^nX\). A nested Hilbert scheme on \(X\) is defined to be a scheme of the form
\[
Z_{\mathbf n}(X):=\{(Z_1, Z_2,\dots, Z_m) : Z_i \in \text{Hilb}^{n_i}X
\]
and \(Z_i\) is a subscheme of \(Z_j\) if \(i < j\},\) where the symbol \(\mathbf n\) is used as a shorthand for the \(m\)-tuple \((n_1, n_2,\dots,n_m)\). If \(({\mathcal U}_1, {\mathcal U}_2,\dots, {\mathcal U}_m)\) is the universal element over the nested Hilbert scheme \(Z_{\mathbf n}(X)\), we call the scheme \({\mathcal U}_1\times_{Z_{\mathbf n}(X)}{\mathcal U}_2 \times_{Z_{\mathbf n}(X)}\cdots \times_{Z_{\mathbf n}(X)} {\mathcal U}_m\) the universal family over \(Z_{\mathbf n}(X)\). \textit{J. Cheah} [J. Algebr. Geom. 5, No. 3, 479-511 (1996)], expressed the virtual Hodge polynomials of the smooth Hilbert schemes \(\text{Hilb}^n X\) in terms of that of \(X\). In the paper under review we indicate how the arguments of the cited paper can be modified to express the virtual Hodge polynomials of all the smooth nested Hilbert schemes (and those of their universal families when \(r\geq 2\)) in terms of that of \(X\). More generally, we obtain the virtual Hodge polynomials of the schemes
\[
\begin{cases} \text{Hilb}^nX,\\ Z_{n-1,n}(X),\\ {\mathcal F}_n(X), \\ {\mathcal F}_{n-1,n}(X),\\ \{(P,Z_1,Z_2)\in X\times \text{Hilb}^{n-1}X\times \text{Hilb}^n X: \\ \qquad P \text{ lies in the support of }Z_2, Z_1 \text{ is a subscheme of }Z_2\} \\ \{(P, Q, Z) \in X \times X\times \text{Hilb}^nX: P \text{ and }Q\text{ lie in the support of }Z\}\end{cases}\tag{1}
\]
in terms of the virtual Hodge polynomial of \(X\) and those of the reduced schemes
\[
\text{Hilb}^k (\mathbb{A}^r,0) = \{Z \in \text{Hilb}^k\mathbb{A}^r : Z\text{ is supported at the origin\}}
\]
and
\[
{\mathcal Z}_{k-1,k}(\mathbb{A}^r, 0)=\{(Z_1, Z_2) \in \text{Hilb}^{k-1}(\mathbb{A}^r,0)\times \text{Hilb}^k(\mathbb{A}^r,0): Z_1\text{ is a subscheme of }Z_2\}.
\]
When \(r= 2\) or \(n\geq 3\), the virtual Hodge polynomials of the spaces listed in (1) can be given purely in terms of that of \(X\). Note that when \(r\geq 2\), this includes all the smooth nested Hilbert schemes \(Z_{\mathbf n}(X)\) and their universal families. If \(r = 1\), the schemes \(Z_{\mathbf n}(X)\) are products of symmetric powers of \(X\) and their virtual Hodge polynomials are easily determined using the formula for the virtual Hodge polynomials of symmetric powers given in the paper cited above.
From the equations of virtual Hodge polynomials, we obtain for free analogous equations of virtual Poincaré polynomials and Euler characteristics. In fact, since the Euler characteristics of the spaces \(\text{Hilb}^k(\mathbb{A}^r,0)\) and \({\mathcal Z}_{k-1,k}(\mathbb{A}^r,0)\) can be expressed in terms of the numbers of certain higher dimensional partitions, the Euler characteristics of the schemes listed in (1) are expressible in terms of these numbers and the Euler characteristic of \(X\). If \(X\) is projective, we also obtain formulae giving the Hodge (resp. Poincaré) polynomials of the smooth nested Hilbert schemes in terms of that of \(X\). virtual Hodge polynomials; nested Hilbert schemes; Poincaré polynomials; Euler characteristic J. Cheah, ''The Virtual Hodge Polynomials of Nested Hilbert Schemes and Related Varieties,'' Math. Z. 227, 479--504 (1998). Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The virtual Hodge polynomials of nested Hilbert schemes and related 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 \textit{D. Chen} et al. [Commun. Algebra 39, 3021--3043 (2011; Zbl 1238.14012] studied the Hilbert scheme component \(H_n \subset \text{Hilb} (\mathbb P^n)\) whose general point corresponds to the union of a pair of codimension two linear subspaces meeting transversely, showing that \(H_n\) is smooth, isomorphic to the blow up of the symmetric square of \(\mathbb G (n-2,n)\) along the diagonal, meets only one other component of the Hilbert scheme, and is a Mori dream space. Extending this work, the author offers a very complete study of the component \(\mathcal H_{c,d}^n \subset \text{Hilb} (\mathbb P^n)\) whose general point parametrizes a union of two linear subspaces of codimensions \(2 \leq c \leq d\) which meet transversely. He shows that \(\mathcal H_{c,d}^n\) is smooth, constructing it as (a) an iterated blow up over the symmetric square of \(\mathbb G (n-c,n)\) if \(c=d\) or (b) an iterated blow up over \(\mathbb G (n-c,n) \times \mathbb G (n-d,n)\) if \(d > c\). To show these isomorphisms, he constructs a Gröbner basis for the ideals of schemes parametrized by \(\mathcal H_{c,d}^n\), which leads to a classification of those schemes and the conclusion that \(\mathcal H_{c,d}^n\) has a unique Borel fixed point. In particular, there are exactly \(2^c\) schemes parametrized by \(\mathcal H_{c,d}^n\) up to projective equivalence. He goes on to determine the effective and Nef cones of \(\mathcal H_{c,d}^n\), concluding that \(\mathcal H_{c,d}^n\) is a Mori dream space. Furthermore \(\mathcal H_{c,d}^n\) is Fano if and only if \(c=3\) and \(n=5\) or \(c \neq 3\) and \(n=2c-1\) or \(2c\). Hilbert schemes; Grassmann varieties; Borel fixed points; Mori dream spaces; effective cones; Fano varieties Parametrization (Chow and Hilbert schemes), Rational and birational maps, Minimal model program (Mori theory, extremal rays), Syzygies, resolutions, complexes and commutative rings, Grassmannians, Schubert varieties, flag manifolds, Deformations and infinitesimal methods in commutative ring theory The Hilbert scheme of a pair of linear 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 present the linearized metrizability problem in the context of parabolic geometries and sub-Riemannian geometry, generalizing the metrizability problem in projective geometry studied by \textit{R. Liouville} [J. de l'Éc. Polyt. cah. LIX. 7--76 (1889; JFM 21.0317.02)].
The paper under review is concerned with bracket-generating distributions arising in parabolic geometries, which are Cartan-Tanaka geometries modelled on homogeneous spaces \(G/P\), where \(G\) is a semisimple Lie group and \(P\) a parabolic subgroup of \(G\). On a manifold \(M\) equipped with such a parabolic geometry, each tangent space is modelled on the \(P\)-module \(g/p\).
First, motivating examples are provided:
1. Parabolic geometries and Weyl structures.
2. Projective parabolic geometries.
3. Parabolic geometries on filtered manifolds.
4. \(BGG\) operators, local metrizability of the homogeneous model, and normal solutions.
Next, a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies are given. These tools lead to natural sub-Riemannian metrics on generic distributions of interest in geometric control theory.
By using an algebraic linearization condition, the authors establish a linearization principle. The metrizability procedure is illustrated by showing how the well-known example of projective geometry fits into the general method. In addition, the metric tractor bundle is investigated.
Let \(p_0=p/{p^\perp}\) and \(h\) a \(p_0\)-module. The main result is a classification of metric parabolic geometries with irreducible \(h\).
Finally, examples of reducible cases where the linearized metrizability procedure works are given. Cartan geometry; parabolic geometry; Bernstein-Gelfand-Gelfand resolution; projective metrizability; sub-Riemannian metrizability; overdetermined linear PDE; Weyl connections Other connections, Sub-Riemannian geometry, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Semisimple Lie groups and their representations, General geometric structures on manifolds (almost complex, almost product structures, etc.), Differential geometry of homogeneous manifolds, Natural bundles, Invariance and symmetry properties for PDEs on manifolds, Nonlinear systems in control theory Subriemannian metrics and the metrizability of parabolic geometries | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 is concerned with the problem of simultaneous pre-periodicity in parameterised families of algebraic dynamical systems. Previous work by \textit{M. Baker} and \textit{L. DeMarco} [Duke Math. J. 159, No. 1, 1--29 (2011; Zbl 1242.37062)] and by the authors [Algebra Number Theory 7, No. 3, 701--732 (2013; Zbl 1323.37056)] had established that if in a parametrised family of polynomial maps there are two points which are simultaneously pre-periodic for infinitely many parameter values, then these points must have the same image for all parameters.
The present work represents a considerable generalisation of the above result. We now have two parameterised families of rational maps of the projective line, indexed by the points of a smooth algebraic curve, and such that the degree of the numerator exceeds that of the denominator by at least two. If there are two points which are simultaneously pre-periodic for the corresponding maps for infinitely many parameter values, then these points must be simultaneously pre-periodic for all parameter values. A similar result is also established in two dimensions, for a specific family of endomorphisms. algebraic dynamics; quadratic family; polynomial orbits; preperiodic points D. Ghioca, L.-C. H'sia and T. Tucker, \textit{Preperiodic points for families of rational maps}, Proceedings of the London Mathematical Society, to appear. Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Heights, Arithmetic varieties and schemes; Arakelov theory; heights Preperiodic points for families of rational maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In an earlier paper [Ann. Inst. Fourier 56, No.~4, 1165--1205 (2006; Zbl 1118.14037)], the author presented a new construction of limit linear series which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The goal of the present paper is to obtain sufficiently sharp upper bounds for the dimensions of spaces of crude limit series that the author can apply the theoretical machinery of the cited paper to the loci of crude limit series in addition to refined limit series. The obtained estimates allow him to prove the following theorem.
Theorem 1.1. Fix integers \(r\), \(d\), and let \(X\) be a general curve of compact type of genus \(g\) over \(\text{Spec}\,k\) having no more than two components, with \(\text{char}\,k=0\), and general marked points. Then the space of limit linear series on \(X\) of degree \(d\) and dimension \(r\), with prescribed ramification sequences \(\alpha^i\) at the marked points, is proper and pure of exactly the expected dimension \(\rho=(r+1)(d-r)-rg-\sum_{i, j}\alpha^i_j\). If no ramification is specified, this space is nonempty, and if further \(\rho>0\), the space is connected. linear series; Brill-Noether theorem Osserman, B.: Linked Grassmannians and crude limit linear series. Int. Math. Res. Not. \textbf{2006}(25), 1-27 (2006b) \textbf{(Article ID 25782)} Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Linked Grassmannians and crude limit linear series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field, and let \(X\) be a separated, regular, Noetherian \(k\)-scheme of finite Krull dimension. Let \(\pi: \widetilde{X} \to X\) be the blowing-up of \(X\) along a regular equi-codimensional closed subscheme \(Y\), and set \(r\) to be the codimension of \(Y\) in \(X\). Assume \(r \geq 2\). There is a well-known theorem which relates the bounded derived category \(D^b(\text{Coh}(\widetilde{X}))\) to \(D^b(\text{Coh}(X))\) and \(D^b(\text{Coh}(Y))\): roughly, \(D^b(\text{Coh}(\widetilde{X}))\) is a semi-orthogonal decomposition of \(D^b(\text{Coh}(X))\) and \(r - 1\) copies of \(D^b(\text{Coh}(Y))\) (see Theorem 3.4 of the article under review for the precise statement). The main goal of this article is to formulate and prove an analogue of this theorem for categories of matrix factorizations.
Let \(W \in \Gamma(X, \mathcal{O}_X)\), and denote by \(\text{MF}(X, W)\) the triangulated category of \textit{matrix factorizations} of \(W\) over \(X\). In Section 2 of the article under review, the authors provide a comprehensive background on matrix factorization categories in the global setting, including detailed discussions of their dg enhancements and derived functors between them. Section 3 is devoted to a proof of their main theorem, which states, roughly, that \(\text{MF}(\widetilde{X}, W)\) is a semi-orthogonal decomposition of \(\text{MF}(X, W)\) and \(r-1\) copies of \(\text{MF}(Y, W)\) (here, we abuse notation slightly by denoting the pullbacks of \(W\) to \(\widetilde{X}\) and \(Y\) also by \(W\)); see Theorem 3.5 of the article for the precise statement of the main theorem.
The authors also obtain a semi-orthogonal decomposition of the category \(\text{MF}(E, W)\) of matrix factorizations associated to a projective bundle \(E\) over \(X\) (Theorem 3.2), and they discuss several applications of the main theorem (Section 3.3). matrix factorization; semi-orthogonal decomposition; blowing-up; projective space bundle; dg enhancement Lunts, V. A.; Schnürer, O. M., Matrix factorizations and semi-orthogonal decompositions for blowing-ups, J. Noncommut. Geom., 10, 907-979, (2016) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differential graded algebras and applications (associative algebraic aspects) Matrix factorizations and semi-orthogonal decompositions for blowing-ups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 various textbooks of algebraic geometry, the notion of smoothness is defined in different ways under different hypotheses. The aim of the paper is to reconcile these definitions and to give direct proofs and some new formulations of standard results about smoothness under weaker hypotheses. The starting point is the following definition:
A morphism \(R\to B\) of rings (commutative, with unit) is called quasi-smooth if, endowing the rings with the discrete topology, it is formally smooth in the sense of A. Grothendieck, i.e., for every \(R\)-algebra \(C\) and every ideal \(J\subset C\) with \(J^2= 0\), every morphism of \(R\)-algebras \(B\to C/J\) can be lifted to a morphism of \(R\)-algebras \(B\to C\). One defines, similarly, using \(\text{Spec\,}C\) instead of \(C\) and reversing arrows, quasi-smooth morphisms of schemes. Now, if \(B= A/I\) for a quasi-smooth algebra \(A\) over \(R\) (for example, a polynomial algebra in possibly infinitely many indeterminates) then \(B\) is quasi-smooth iff the surjection \(A/I^2\to A/I\) has a right inverse in the category of \(R\)-algebras. Moreover, if \(R\to B\) is quasi-smooth, the module \(\Omega_{B/R}\) of differentials is a projective \(B\)-module.
Interesting problems occur when one investigates the ``local nature'' of quasi-smoothness.
The author introduces the following definition: A \(B\)-module \(M\) is ``sum finitely presented'' if there exists a \(B\)-module \(M'\) such that \(M\oplus M'\) is a (possibly infinite) direct sum of finitely presented \(B\)-modules. For example, the quotient of a projective module by a finitely generated submodule is sum finitely presented. The author shows that if \(\Omega_{B/R}\) is sum finitely presented (for example, if \(B= A/I\) with \(A\) quasi-smooth over \(R\), and \(I/I^2\) is finitely generated) then \(R\to B\) is quasi-smooth iff, for every maximal \(m\) of \(B\) with \(p= m\cap R\), \(R_p\to B_m\) is quasi-smooth and gives an example showing that this is no longer true without any hypothesis on \(\Omega_{B/R}\). He also shows that if \(\Omega_{B/R}\) is a direct sum of finitely generated \(B\)-modules, if \(b_1,\dots, b_n\) generate the unit ideal of \(B\) and if all \(R\to B_{b_i}\) are quasi-smooth, then \(R\to B\) is quasi-smooth. It is an open question whether this remains true without assuming anything about \(\Omega_{B/R}\). This is related to the following question: If \(M\) is a \(B\)-module such that every \(M_{b_i}\) is a projective \(B_{b_i}\)-module, does it follow that \(M\) is a projective \(B\)-module?
Finally, the author proves the Jacobian criterion and shows that a finitely presented morphism \(R\to B\) is quasi-smooth iff it is flat with regular geometric fibres. In this case, the morphism is called smooth. formal smoothness; simple points; Jacobian criterion; sum finitely presented Local structure of morphisms in algebraic geometry: étale, flat, etc., Regular local rings, Relevant commutative algebra, Morphisms of commutative rings, Modules of differentials A new look at smoothness. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 build on the recent techniques of Codogni and Patakfalvi (2021, \textit{Inventiones Mathematicae} 223, 811-894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of \(\mathrm{K}\)-semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the \textit{Harder and Narasimhan polygons}. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, \textit{Inventiones Mathematicae} 223, 811-894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, \textit{Inventiones Mathematicae} 116, 109-138). One source of inspiration for our abstract study of \textit{Harder and Narasimhan data}, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, \textit{Commentarii Mathematici Helvetic} 59, 600-634). Another is the work of Faltings and Wüstholz (1994, \textit{Inventiones Mathematicae} 116, 109-138), and Evertse and Ferretti (2013, \textit{Annals of Mathematics} 177, 513-590), which is within the context of Diophantine approximation for projective varieties. K-stability; Diophantine approximation; Harder and Narasimhan filtrations Sheaves in algebraic geometry, Approximation to algebraic numbers, Families, moduli, classification: algebraic theory Vertices of the Harder and Narasimhan polygons and the laws of large numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider a Riemann surface \(X\) defined by a polynomial \(f(x,y)\) of degree \(d\), whose coefficients are chosen randomly. Hence, we can suppose that \(X\) is smooth, that the discriminant \(\delta (x)\) of \(f\) has \(d(d - 1)\) simple roots, \(\Delta \), and that \(\delta (0)\neq 0\), i.e. the corresponding fiber has \(d\) distinct points \(\{y_{1}, \dots ,y_d\}\). When we lift a loop \(0 \in \gamma \subset \mathbb C - \Delta \) by a continuation method, we get \(d\) paths in \(X\) connecting \(\{y_{1},\dots ,y_d\}\), hence defining a permutation of that set. This is called monodromy.
Here we present experimentations in \texttt{Maple} to get statistics on the distribution of transpositions corresponding to loops around each point of \(\Delta \). Multiplying families of ``neighbor'' transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.
Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions. bivariate polynomial; plane curve; random Riemann surface; absolute factorization; algebraic geometry; continuation methods; monodromy; symmetric group; algorithms; Maple code Galligo, A.; Poteaux, A., Computing monodromy via continuation methods on random Riemann surfaces, Theoret. Comput. Sci., 412, 1492-1507, (2011) Computational aspects of algebraic curves, Symbolic computation and algebraic computation, Polynomials, factorization in commutative rings, Numerical problems in dynamical systems Computing monodromy via continuation methods on random 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 In this paper, we study the intrinsic relations between the global injectivity of the differentiable local homeomorphism map \(F\) and the rate of the Spec \((F)\) tending to zero, where Spec \((F)\) denotes the set of all (complex) eigenvalues of Jacobian matrix \(JF(x)\), for all \(x\in \mathbb{R}^2\). They depend deeply on the \(W\)-condition which extends the \(*\)-condition and the \(B\)-condition. The \(W\)-condition reveals the rate that tends to zero of the real eigenvalues of \(JF\), which can not exceed \(\displaystyle O\Big(x\ln x(\ln \frac{\ln x}{\ln\ln x})^2\Big)^{-1}\) by the half-Reeb component method. This improves the theorems of
\textit{C. Gutierrez} and \textit{Nguyen Van Chau} [Discrete Contin. Dyn. Syst. 17, No. 2, 397--402 (2007; Zbl 1118.37024)] and
\textit{R. Rabanal} [Bull. Braz. Math. Soc. (N.S.) 41, No. 1, 73--82 (2010; Zbl 1191.14074)].
The \(W\)-condition is optimal for the half-Reeb component method in this paper setting. This work is related to the Jacobian conjecture. global injectivity; \(W\)-condition; half-Reeb component; Jacobian conjecture Jacobian problem, Implicit function theorems, Jacobians, transformations with several variables Global injectivity of differentiable maps via \(W\)-condition in \(\mathbb{R}^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors introduce symmetric obstruction theories, and compute the weighted Euler characteristic for schemes \(X\) that admit a \(\mathrm{G}_m\)-action and a compatible symmetric obstruction theory. They show that the weighted Euler characteristic is given by the formula
\[
\tilde{\chi} (X) = \sum (-1)^{\mathrm{dim}T_{X\mid P}},
\]
where the sum runs over the fixed points for the \(\mathrm{G}_m\)-action, where the fixed points are assumed finite and isolated. If the scheme \(X\) furthermore is projective, their formula above expresses the virtual count; the degree of the associated virtual fundamental class of \(X\).
A scheme which is locally the critical locus of a regular function on a smooth manifold, have a canonical symmetric obstruction theory. An example of such a scheme is the Hilbert scheme \(\mathrm{Hilb}^nY\) of \(n\)-points on a smooth three-fold \(Y\).
In the article it is shown that the weighted Euler characteristic of \(\mathrm{Hilb}^nY\), where \(Y\) is a smooth three-fold, is up to a sign, the ordinary Euler characteristic of \(\mathrm{Hilb}^nY\). For projective \(Y\) they then obtain the formula for the virtual count of \(\mathrm{Hilb}^nY\), as conjectured in [\textit{D. Maulik, N. Nekrasov, A. Okounkov} and \textit{R. Pandharipande}, Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)] symmetric obstruction theories; Hilbert schemes; Calabi-Yau threefolds; \(C^*\) actions; Donaldson-Thomas invariants; MNOP conjecture K. Behrend and B. Fantechi, 'Symmetric obstruction theories and Hilbert schemes of points on threefolds', \textit{Algebra Number Theory}2 (2008) 313-345. Parametrization (Chow and Hilbert schemes), Calabi-Yau manifolds (algebro-geometric aspects), \(3\)-folds Symmetric obstruction theories and Hilbert schemes of points on threefolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(C\subseteq \mathbb{P}^ N\) be a complete irreducible smooth curve over a field \(k\) with \(k=\overline k\), and let \({\mathcal G}\) be a linear system on \(C\) of projective dimension \(N\). To every point \(P \in C\) one can associate a sequence of integers \(b_ 0(P) <b_ 1(P)<\cdots<b_ N(P)\) that describes the possible intersection multiplicities at \(P\) of \(C\) with hyperplanes of \(\mathbb{P}^ N\). -- For all but finitely many points on the curve this sequence is the same and it is called the order sequence of \({\mathcal G}\). The points \(P\) having a nongeneric order sequence are called Weierstrass points of \({\mathcal G}\).
In a previous paper, the author [Bol. Soc. Bras. Mat. Nova Ser. 23, No. 1-2, 93-108 (1992; Zbl 0780.14019)] proved the inequality: \(b_ j+b_{N-j} \leq b_ N\). Such an inequality has been generalized by \textit{E. Esteves}, who proved that for each \(P \in C\) with order sequence \(\mu_ 0 (P),\dots,\mu_ N(P)\) and for all \(i,j\) with \(i+j=N\), we have: \(\mu_ i(P)+b_ j \leq \mu_{i+j}(P)\). -- In the present paper a more direct proof of the above inequality can be found, and an application of it to the study of linear systems with only one Weierstrass point. Namely what is proved in the paper is:
Theorem: Let \(W({\mathcal G})\) be the Wronskian (also called ramification) divisor of \({\mathcal G}\). Then \(\deg W({\mathcal G})=1\) if and only if \(C=\mathbb{P}^ 1\), \(N\) is even, \(j \neq N/2\) implies \(b_ j+b_{N-j}=\deg {\mathcal G}=2b_{N/2}+1\). In this case, the unique Weierstrass point \(Q\) is such that:
\[
\mu_ j(Q)=b_ j \text{ for }j \neq N/2\text{ and } \mu_{N/2} (Q)=b_{N/2}+1.
\]
In particular, \({\mathcal G}\) is without base points and \(\varphi_{\mathcal G}:\mathbb{P}^ 1 \to \mathbb{P}^ N\) can be given (with an appropriate choice of coordinates) by \(\varphi_{\mathcal G}(t)=(t^{b_ 0},\dots,t^{b_ N})\). intersection multiplicities; order sequence; Weierstrass points Riemann surfaces; Weierstrass points; gap sequences On Esteves' inequality of order sequences of 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 aim of the paper is to study the eigenscheme of order three partially symmetric and symmetric tensors. They also show that a subvariety of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenscheme of \(4 \times 4 \times 4\) symmetric tensors.
The spectral theory of tensors is a multi-linear generalization of the study of eigenvalues and eigenvectors in the case of matrices. The eigenscheme \(E(\mathcal{T})\) of a tensor \(\mathcal{T}\) can be roughly thought as the set of eigenpoints of the tensor, i.e. eigenvectors of \(\mathbb{C}^{n+1}\) of a particular contraction of the tensor. In the case of partial symmetric or symmetric tensors of order \(3\) a contraction may be the following. A partial symmetric tensor \(\mathcal{T} \in \operatorname{Sym}^2 \mathbb{C}^{n+1} \otimes \mathbb{C}^{n+1}\) can be seen as an \((n+1)\)-tuple of quadratic forms \((q_0,\dots,q_n)\) in the variables \(x_i\). Analogously, given a a symmetric tensor \(f \in \operatorname{Sym}^3 \mathbb{C}^{n+1}\), i.e. a homogeneous cubic polynomial, one can associate to it an \((n+1)\)-tuple of quadratic forms given by its derivatives \(\frac{\partial f}{\partial x_i}\). The authors investigates the eigenscheme and some its particular subschemes of the aforementioned tensors with those contractions.
At first they recall some basic notions regarding the theory. In particular they introduce the irregular eigenscheme \(\operatorname{Irr}(\mathcal{T})\) and the regular eigenscheme \(\operatorname{Reg}(\mathcal{T})\). The first can be thought as the subscheme of \(E(\mathcal{T})\) given by points with zero eigenvalue, while the second is the residue of \(E(\mathcal{T})\) with respect to \(\operatorname{Irr}(\mathcal{T})\). After that they focus on the case of order \(3\) symmetric tensors providing bounds on the dimensions and geometric properties of the irregular and regular eigenschemes. Numerous examples of symmetric tensors satisfying all the described properties are provided. As they observe, if the regular eigenscheme of a cubic polynomial is \(0\) dimensional, then it consists of at most \(2^{n+1}-1\) points. Therefore they investigate in the ternary and quaternary case whether there exists a cubic polynomial with a prescribed number of regular eigenpoints. Eventually they show that a open subvariety of a linear subspace of the Grassmannian \(Gr(3,\mathbb{P}^{14})\) parametrizes the eigenschemes of order \(3\) quaternary symmetric tensors. eigenpoints of tensors; cubic surfaces; Grassmannians Rational and ruled surfaces, Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors, Multilinear algebra, tensor calculus On the eigenpoints of cubic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(J\subset S= K[x_0,\dots, x_n]\) be a monomial strongly stable ideal. The collection \({\mathcal M}f(J)\) of the homogeneous polynomial ideals \(I\), such that the monomials outside \(J\) form a \(K\)-vector basis of \(S/I\), is called a \(J\)-marked family. It can be endowed with a structure of affine scheme, called a \(J\)-marked scheme. For special ideals \(J\), \(J\)-marked schemes provide an open cover of the Hilbert scheme \({\mathcal H}\text{ilb}^n_{p(t)}\), where \(p(t)\) is the Hilbert polynomial of \(S/J\). Those ideals more suitable to this aim are the \(m\)-truncation ideals \(\underline J_{\geq m}\) generated by the monomials of degree \(\geq m\) in a saturated strongly stable monomial ideal \(\underline J\).
Exploiting a characterization of the ideals in \({\mathcal M}f(\underline J_{\geq m})\) in terms of a Buchberger-like criterion, we compute the equations defining the \(\underline J_{\geq m}\)-marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of \({\mathcal M}f(\underline J_{\geq m})\) in an affine space of low dimension.
In this setting, explicit computations are achievable in many nontrivial cases. Moreover, for every \(m\), we give a closed embedding \(\phi_m:{\mathcal M}f(\underline J_{\geq m})\hookrightarrow{\mathcal M}f(\underline J_{\geq m+1})\), characterize those \(\phi_m\) that are isomorphisms in terms of the monomial basis of \(\underline J\), especially we characterize the minimum integer \(m_0\) such that \(\phi_m\) is an isomorphism for every \(m\geq m_0\). Hilbert scheme; strongly stable ideal; polynomial reduction relation Bertone, C.; Cioffi, F.; Lella, P.; Roggero, M., Upgraded methods for the effective computation of marked schemes on a strongly stable ideal, J. Symbolic Comput., 50, 263-290, (2013) Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Upgraded methods for the effective computation of marked schemes on a strongly stable ideal | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is well known that, in generic coordinates and given a term order, every polynomial ideal \(I\) can be deformed to a Borel-fixed ideal by a flat family, with the consequence that this Borel-fixed ideal preserves several properties of \(I\). Thus, there are some contexts in which Borel-fixed ideals have a special role, for example in the study of the Hilbert scheme or of minimal free resolutions. Over a field of characteristic zero, the set of Borel-fixed ideals coincides with the set of the strongly stable ideals, which have very nice combinatorial properties.
In this paper, the authors describe an algorithm to compute all the saturated strongly stable ideals defining projective schemes with a given Hilbert polynomial (and in a given projective space) (Section 4). Moreover, they exploit and specialize this algorithm obtaining two variants: the first one computes the almost lexsegment ideals with a given Hilbert polynomial (Definition 5.1 and Algorithm 5.8), with some consequences about the existence of ideals with minimal or maximal Hilbert function among the ideals with maximal Betti numbers, and the second one computes the saturated strongly stable ideals with a given Hilbert function (Algorithm 6.1). They also provide results in order to give an estimation of the complexity of their algorithms (Theorem 4.4 and Section 7). The algorithms have been implemented in Macaulay 2 and the implementations are available.
The idea behind the main algorithm described in this paper can be summarized in two essential steps: induction on the number of variables, as well as on the degree of the Hilbert polynomial, due to the successive hyperplane sections that are easily computable because the last variable is naturally a generic element for a strongly stable ideal; and manipulation of the ideals in construction by the identification of special minimal generators to be ``cancelled'', by means of the combinatorial properties of the strongly stable ideals.
As the authors of this paper note, algorithms computing all the saturated strongly stable ideals with a given Hilbert polynomial were already known: the algorithm in the appendix of the Ph.D thesis of \textit{A. A. Reeves} [``Combinatorial structure on the Hilbert scheme'', Thesis (Ph.D) Cornell University (1992)] and the algorithm in a paper co-authored by the reviewer [\textit{F. Cioffi} et al., Discrete Math. 311, No. 20, 2238--2252 (2011; Zbl 1243.14007)], which has been successfully improved in [\textit{P. Lella}, ``An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme'', in: ISSAC 2012-Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ACM, New York 242--248 (2012)]. The authors discuss relations of their algorithm with the others in Remark 7.2.
For an algorithm computing Borel-fixed ideals on a field of any characteristic, see [\textit{C. Bertone}, ``Quasi stable ideals and Borel-fixed ideals with a given Hilbert polynomial'', preprint, \url{arXiv:1409.5569}]. Hilbert polynomial; strongly stable ideal; Betti number; Castelnuovo-Mumford regularity; lexsegment ideal Bertone, C., Cioffi, F., Roggero, M.: A division algorithm in an affine framework for flat families covering Hilbert schemes, available at arXiv:1211.7264 [math.AC], (2013) Effectivity, complexity and computational aspects of algebraic geometry, Computational aspects and applications of commutative rings Algorithms for strongly stable 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 A subscheme \(Y\subset\mathbb P^ n\) is called arithmetically Buchsbaum if \(H^ p({\mathcal I}_{Y\cap M}(*))\) has trivial module structure for any linear subspace \(M\) of dimension \(m\) and \(p=1,...,m-2\). The main purpose of this article is to prove the following theorem: Let \(Y\) be a codimension-2 subscheme of \(\mathbb P^ n\) for \(n\geq 3\). Then \(Y\) is arithmetically Buchsbaum if and only if there exists an exact sequence
\[
0\to \oplus {\mathcal O}(-a_ i)\to \oplus \ell_ j\Omega^{p_ j}(-k_ j)\oplus {\mathcal O}(-c_ s)\to {\mathcal I}_ Y\to 0
\]
where \(P_ j\neq 0\).
The theorem, coupled with a general result [see the review of the author's paper, J. Reine Angew. Math. 397, 214--219 (1989; Zbl 0663.14008)] concerning smoothness or reducedness of the dependency locus of a map between vector bundles, allows us to determine the admissible \((a_ i,\ell_ j,p_ j,k_ j,c_ s)\), hence to give a classification of codimension-2 arithmetically Buchsbaum subschemes in \(\mathbb P^ n\) [the author, J. Reine Angew. Math. 401, 101--112 (1989; Zbl 0672.14026)].
Other applications of the theorem include bounding the degree and the number of minimal generators of \(\oplus H^ 0({\mathcal I}_ Y(k))\) and the regularity. classification of codimension-2 arithmetically Buchsbaum subschemes; degree; number of minimal generators; regularity Chang, MC, Characterization of arithmetically Buchsbaum subschemes of codimension 2 in \({\mathbb{P}}^n\), J. Differ. Geom., 31, 323-341, (1990) Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Characterization of arithmetically Buchsbaum subschemes of codimension 2 in \(\mathbb P^ n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A map \(f:{\mathbb R}^n\to{\mathbb R}^m\) is called a polynomial map if each of its components is a polynomial. A subset \(S\) of \({\mathbb R}^m\) is a polynomial image of \({\mathbb R}^n\) if \(S=f({\mathbb R}^n)\) for such an \(f\). For \(S\subset{\mathbb R}^m\), one defines \(p(S)\) as the smallest \(p\geq1\) such that \(S=f({\mathbb R}^p),\) with \(f\) a polynomial map. If this is not the case, then \(p(S)=+\infty.\)
Analogously, one has that a map \(f:{\mathbb R}^n\to{\mathbb R}^m\) is called regular if each of its components is a rational function which does is well-defined everywhere in \({\mathbb R}^n\). A subset \(S\) of \({\mathbb R}^m\) is a regular image of \({\mathbb R}^n\) if \(S=f({\mathbb R}^n)\) for such an \(f\). For \(S\subset{\mathbb R}^m\), one defines \(r(S)\) as the smallest \(r\geq1\) such that \(S=f({\mathbb R}^r),\) with \(f\) regular. If this is not the case, then \(r(S)=+\infty.\)
From these definitions, one has clearly \(\dim (S)\leq r(S)\leq p(S).\) The paper under review characterizes all the possible values of \(p(S)\) and \(r(S)\) in the case where \(\dim(S)=1.\)
The classification is as follows:
\[
\begin{matrix} r(S)& p(S)& S\\ 1&1&{\mathbb R}\,\text{or}\, [0,+\infty)\\ 1&2&\text{cannot happen}\\ 1&+\infty& [0,1)\\ 2&2& (0,+\infty)\\ 2&+\infty& (0,1)\\ +\infty& +\infty& \text{any non-rational algebraic curve} \end{matrix}
\]
polynomial maps; regular maps; semialgebraic sets; regularity Fernando, J. F., On the one dimensional polynomial and regular images of \(\mathbb{R}^n\), J. Pure Appl. Algebra, 218, 9, 1745-1753, (2014) Semialgebraic sets and related spaces, Real algebraic sets, Topology of real algebraic varieties, Computational aspects of algebraic curves On the one dimensional polynomial and regular images of \(\mathbb R^n\) | 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 The author studies the local behaviour of negative plurisubharmonic (psh) functions near their isolated singularities (\(-\infty\) points), say at the point \(0\) in \(\mathbb C ^n\) . It is assumed that the Monge-Ampère measure of those functions is zero except for the origin. For such a function \(\psi\) consider the approximation \(\mathcal{D}_k \psi \to \psi\) introduced by \textit{J.-P. Demailly} [J. Algebr. Geom. 1, No. 3, 361--409 (1992; Zbl 0777.32016)]. Then the convergence is pointwise, in \(L^1 _{loc}\), and the Lelong numbers of approximants converge to the Lelong number of \(\psi\) at any point. If \(\psi\) is of the form \(const. \log |F|\), for a holomorphic mapping \(F\) with \(F(0)=0\), then the we say thet \(\psi\) has an \textit analytic singularity. \textrm A general \(\psi\) has an \textit asymptotically analytic singularity \textrm if for any small \(\epsilon\) there is \(\psi _{\epsilon}\) with analytic singularity and such that \((1+\epsilon ) \psi _{\epsilon} + O(1) \leq \psi \leq (1-\epsilon ) \psi _{\epsilon} + O(1).\) The main theorems of the paper characterize the classes of functions with asymptotically analytic singularities, and those that can be represented as the limit of a decreasing sequence of functions with analytic singularities, in terms of the limiting behaviour of Demailly's approximants. For instance, \(\psi\) belongs to the latter class if and only if \((dd^c \psi )^n (\{ 0\} ) = \inf _k (dd^c \mathcal{D}_k \psi )^n (\{ 0\} )\). plurisubharmonic singularity; Monge-Ampère operator; multiplier ideal DOI: 10.1007/s00209-013-1179-0 Plurisubharmonic functions and generalizations, Lelong numbers, Plurisubharmonic extremal functions, pluricomplex Green functions, Singularities in algebraic geometry Analytic approximations of plurisubharmonic 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 authors present a unified approach to the study of \(F\)-signature and Hilbert Kunz multiplicity of local rings of positive characteristic. They provide proofs of existence of these limits, their semicontinuity, and their positivity using quite general ``uniform convergence'' techniques. In particular, they give new and simplified proofs of positivity of \(F\)-signature characterizes strongly \(F\)-regular rings, and that two ideals \(I\subseteq J\) have the same Hilbert-Kunz multiplicity if and only if \(J\subseteq I^*\). Furthemore, many results are generalized to the context of pairs and Cartier modules.
More importantly, using the uniform convergence technique the authors answer a question of \textit{K.-i. Watanabe} and \textit{K.-i. Yoshida} [J. Algebra 230, No. 1, 295--317 (2000; Zbl 0964.13008)]: the \(F\)-signature is the infimum of relative differences in the Hilbert-Kunz multiplicities of the cofinite ideals in a local ring.
One big open question is that, whether the infimum of the relative differences of Hilbert-Kunz multiplicities is actually a minimum. A positive answer to this question will establish that weakly \(F\)-regular and strongly \(F\)-regular are equivalent. The authors provide many partial results to this question. \(F\)-signature; Hilbert-Kunz multiplicity Polstra, Thomas; Tucker, Kevin, \textit{F}-signature and Hilbert-Kunz multiplicity: a combined approach and comparison, Algebra Number Theory, 12, 1, 61-97, (2018) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry \(F\)-signature and Hilbert-Kunz multiplicity: a combined approach and comparison | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 basic closed semialgebraic subset \(S\) of \(\mathbb R^n\) is defined by simultaneous polynomial inequalities \(g_1\geq 0,\dotsc,g_m\geq 0\). We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set \(S\) of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum \(f^\ast\) of \(f\) on \(S\) which is constructive and elementary. In the case where f possesses a unique minimizer \(x^\ast\), we prove that every sequence of ``nearly'' optimal solutions of the successive relaxations gives rise to a sequence of points in \(\mathbb R^n\) converging to \(x^\ast\). nonconvex optimization; positive polynomial; sum of squares; moment problem; Positivstellensatz; semidefinite programming M. Schweighofer, \textit{Optimization of polynomials on compact semialgebraic sets}, SIAM J. Optim., 15 (2005), pp. 805--825. Nonconvex programming, global optimization, Semidefinite programming, Semialgebraic sets and related spaces Optimization of polynomials on compact semialgebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a smooth affine \(\mathbb C\)-scheme, and let \(X = \text{Spec}(A)\) be an affine \(S\)-scheme. Consider a projective \({\mathcal O}_X\)-module \(\mathcal E\) of finite rank, and an integrable \(\mathbb C\)-connection \(\nabla : {\mathcal E} \rightarrow \Omega^1_{X/{\mathbb C}} \otimes_{{\mathcal O}_X}{\mathcal E}.\) Let \(f_1, \ldots, f_r \in A\) be a regular sequence defining the smooth complete intersection \(Y = \text{Spec}(A/(f_1, \ldots, f_r)),\) and let \(j : Y \rightarrow X\) be the inclusion of \(S\)-schemes. Set \({\mathbb A}_X^r = \text{Spec}(A[T_1, \ldots, T_r])\) so that \(\pi : {\mathbb A}_X^r \rightarrow X\) is the projection. It is proved that for any \(n \in {\mathbb N}\) there is an isomorphism of \({\mathcal O}_S\)-modules with \(\mathbb C\)-connection: \(H^n_{DR}(Y/S, (j^\ast({\mathcal E}), \nabla_Y)) \cong H^{n+2r}_{DR}({\mathbb A}_X^r/S, (\pi^\ast({\mathcal E}), \nabla_F)),\) where \(\nabla_Y\) is the pullback of \(\nabla\) to a connection on \(j^\ast({\mathcal E}),\) \(F = T_1f_1 + \ldots + T_rf_r,\) and \(\nabla_F\) is the corresponding twisted connection. The cohomology groups on the right hand side can be interpreted as analogues of Dwork cohomology groups in the case of characteristic zero. This result can be considered as a generalization to the complete intersections case of a result by \textit{N. Katz} [Publ. Math., Inst. Hautes Étud. Sci. 39, 175-232 (1970; Zbl 0221.14007)] which states that for smooth projective hypersurfaces the Dwork cohomology and the primitive part of the de Rham cohomology coincide. As an application the authors show how one can compute the Picard-Fuchs equations for smooth complete intersections. integrable connection; Gauss-Manin connection; complete intersection; de Rham cohomology; Dwork cohomology; hypergeometric equations; Picard-Fuchs equations de Rham cohomology and algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Structure of families (Picard-Lefschetz, monodromy, etc.), Connections (general theory) Dwork cohomology, de Rham cohomology, and hypergeometric functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The book under review is the third and last volume of a treatise on projective spaces over a finite field. This trilogy is the most complete work to date on this subject, and is an indispensable reference for anyone working in the area. The first book in the series [the first author, `Projective geometries over finite fields', Oxford Univ. Press, New York (1979; Zbl 0418.51002)] dealt mainly with projective planes over the finite field \(GF (q)\), although some general introductory material was also presented. The second book in the series [the first author, `Finite projective spaces of three dimensions', Oxford Univ. Press, New York (1985; Zbl 0574.51001)] dealt primarily with finite projective 3- space as its title implies. The present book deals with \(PG(n,q)\) for arbitrary dimension \(n\). In all cases the approach taken is one that might reasonably be called ``finite algebraic geometry''. That is, the group theoretic point of view is not emphasized, but rather a combinatorial approach is taken to characterize various curves and collections of subspaces in finite projective space. The main proof techniques thus involve algebraic manipulations of coordinates over finite fields and various counting strategies. It should be noted that complete proofs are given for almost all results in the three volumes. For a more group theoretic approach to finite geometry one is referred to the classic book by \textit{P. Dembowski}, `Finite geometries', Springer, Berlin (1968; Zbl 0159.500), although many results in the latter reference are not proven.
The main topics discussed in this third volume are quadrics, various varieties (Hermitian, Grassmann, Veronese, Segre), polar spaces, generalized quadrangles, partial geometries, arcs and caps. The chapter on quadrics extends some work done in the previous two volumes, where the properties of quadrics in two, three and five dimensions were carefully developed. The chapters characterizing Hermitian and Grassmannian varieties over finite fields are quite different than what one would see in the classical setting, where, for instance, a Hermitian manifold over the complex numbers is not an algebraic variety. However, the chapter developing the properties of Veronese and Segre varieties closely follows the classical model. The chapter on polar spaces and generalized quadrangles is one of the few places where a number of results are stated without proof. The volume concludes with an appendix listing the known results for the existence of ovoids and spreads in the finite classical polar spaces.
There are a few topics in finite geometry that are intentionally omitted in this treatise. For instance, nondesarguesian planes are not at all discussed, and the interested reader is referred to a book such as by \textit{H. Lüneburg}, `Translation planes', Springer, Berlin (1980; Zbl 0446.51003) for an account of this important subject. Other topics of interest today that are not discussed in this treatise include flocks of quadrics in \(PG(3,q)\), generalized \(n\)-gons for \(n>4\), and Buekenhout diagram geometries. Nonetheless, this is a very encompassing piece of work, and the topics covered are discussed with incredible detail. This is certainly true for the volume under review.
The compilation of the bibliography in and of itself is a tremendous accomplishment. There are over 2000 references given in the three volumes with the most recent publications cited appearing in print in 1991. finite geometry; Hermitian; quadrics; varieties; Grassmann; Veronese; Segre; partial geometries; arcs; caps; polar spaces; generalized quadrangles; ovoids; spreads; bibliography J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford Math. Monogr., Oxford University Press, New York, 1991. Research exposition (monographs, survey articles) pertaining to geometry, Other finite linear geometries, Generalized quadrangles and generalized polygons in finite geometry, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Blocking sets, ovals, \(k\)-arcs, Finite partial geometries (general), nets, partial spreads, Grassmannians, Schubert varieties, flag manifolds, Combinatorial structures in finite projective spaces General Galois geometries | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 generalized étale cohomology theory is a representable cohomology theory for presheaves of spectra on an étale site of an algebraic variety. These cohomology theories simultaneously generalize the homotopy-theoretic cohomologies of algebraic topology and the algebraic theories (for example: étale and crystalline) of Grothendieck.
Consequently this volume, in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. This is an extremely delicate development, obstructed by the need for coherent constructions involving very ``large'' objects such as limits of Čech constructions involving presheaves of spectra. The development of an adequate theory, particularly in respect of its applications to algebraic \(K\)-theory, was held up by difficulties with smash-products of spectra and with transfer constructions. This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in \(K\)-theory applications [i.e., the closed model structure of \textit{A. K. Bousfield} and \textit{E. M. Friedlander}, Lect. Notes Math. 658, 80-130 (1978; Zbl 0405.55021)].
As an application of the techniques the author gives proofs of the descent theorems of \textit{R. W. Thomason} and \textit{Y. A. Nisnevich}. In particular, this implies the celebrated result of \textit{R. W. Thomason}, ``Algebraic \(K\)-theory and étale cohomology'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)] which identifies \(\text{mod }p\) \(K\)-theory, after being inflicted with Bott periodicity in the manner introduced by the reviewer, with \(\text{mod }p\) étale \(K\)-theory.
The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason's theorem without Bott periodicity). The recent proof of this conjecture, by \textit{V. Voevodsky}, when \(p=2\) for fields of characteristic zero makes this volume compulsory reading for all who want to be au fait with current trends in algebraic \(K\)-theory! étale site of algebraic variety; generalized étale cohomology; presheaves of spectra; closed model category; Lichtenbaum-Quillen conjecture Jardine, J. F., Generalized Étale Cohomology Theories, Progress in Mathematics, vol. 146, (1997), Birkhäuser Verlag: Birkhäuser Verlag Basel Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Stable homotopy theory, spectra, Research exposition (monographs, survey articles) pertaining to category theory, Research exposition (monographs, survey articles) pertaining to algebraic topology, Étale and other Grothendieck topologies and (co)homologies, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Generalized étale cohomology theories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathcal C\) denote any smoothness category of real valued functions in \(n\) variables. Let \(\mathcal F\) be any filter of subsets of \(\overline{\mathbb R}^n\) with connected basis converging to \(0\), constituted by open subsets of \(\mathbb R^n\). Let \(\mathcal C(\mathcal F)\) be the ring of germs in \(0\) following \(\mathcal F\) of \(\mathcal C\)-functions with the pointwise defined operations. A subfield \(K\subset\mathcal C(\mathcal F)\) is said to be a \(\mathcal C\)-Hardy field in \(n\) variables if \(K\) is closed with respect to partial differentiation for all variables. Let \(\mathcal F_1\) be the filter of \(\mathbb R\) with the basis \(\mathcal B_1=\{(0,1/n: n\in\mathbb N\}\). Denote by \(K_1\) the Hardy field of germs of \(1\)-variable rational functions in \(0\) for \(\mathcal F_1\). The algebraic closure of \(K_1\) is the Nash Hardy field in \(1\) variable. Starting from this field, the authors give an inductive construction of Nash Hardy fields in several variables. signed places; valuations; ordered fields; Hardy field; Nash Hardy fields in several variables Pasini, L.; Marchiò, C.: Nash Hardy fields in several variables. Note di matematica 8, 231-240 (1988) Ordered fields, Nash functions and manifolds, Valued fields Nash Hardy fields in several variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group \(G\), a parabolic subgroup \(P\), and a topological surface \(M\), the (enhanced) spectral Eisenstein series category of \(M\) is the factorization homology over \(M\) of the \(\mathsf{E}_2\)-Hecke category \(\mathsf{H}_{G ,P} = \mathsf{IndCoh}(\mathsf{LS}_{G ,P}(D^2, S^1))\), where \(\mathsf{LS}_{G ,P}(D^2, S^1)\) denotes the moduli stack of \(G\)-local systems on a disk together with a \(P\)-reduction on the boundary circle.
More generally, for any pair of stacks \(\mathcal{Y} \to \mathcal{Z}\) satisfying some mild conditions and any map between topological spaces \(N \to M\), we define \((\mathcal{Y} ,\mathcal{Z})^{N ,M} = \mathcal{Y}^N \times_{\mathcal{Z}^N} \mathcal{Z}^M\) to be the space of maps from \(M\) to \(\mathcal{Z}\) along with a lift to \(\mathcal{Y}\) of its restriction to \(N\). Using the pair of pants construction, we define an \(\mathsf{E}_n\)-category \(\displaystyle \mathsf{H}_n(\mathcal{Y}, \mathcal{Z}) = \mathsf{IndCoh}_0 \Big(\big((\mathcal{Y} ,\mathcal{Z})^{S^{n - 1} ,D^n}\big)_{\mathcal{Y}}^\wedge\Big)\) and compute its factorization homology on any \(d\)-dimensional manifold \(M\) with \(d \leq n\),
\[
\int_M \mathsf{H}_n(\mathcal{Y}, \mathcal{Z}) \simeq \mathsf{IndCoh}_0 \Bigg(\bigg((\mathcal{Y} ,\mathcal{Z})^{\partial (M \times D^{n - d}) ,M}\bigg)_{\mathcal{Y}^M}^\wedge\Bigg),
\]
where \(\mathsf{IndCoh}_0\) is the sheaf theory introduced by Arinkin-Gaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi-Francis-Nadler and Beraldo. Betti Langlands program; factorization homology; Betti spectral gluing; Eisenstein series Geometric Langlands program (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Bordism and cobordism theories and formal group laws in algebraic topology Eisenstein series via factorization homology of Hecke categories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 real homogenoues polynomial \(f\) of degree \(d\) is called \textit{hyperbolic with respect to a point,} if every real line through the point intersects the algebraic curve defined by \(f\) in only real points. By the Helton-Vinnikov Theorem, every hyperbolic polynomial in three variables \(x\), \(y\), \(z\) admits a representation of the shape \(f = \det(xM_1+yM_2+zM_3)\) with real symmetric matrices \(M_1\), \(M_2\), \(M_3\) of dimension \(d \times d\). This article presents a new algorithm for computing determinantal representations of this form for a given point \((e_1,e_2,e_3)\) but simplifies the problem by requiring only \textit{Hermitian matrices.}
The proposed algorithm first computes the intersection points of the algebraic curves defined by \(f\) and the derivative of \(f\) in direction of \((e_1,e_2,e_3)\). This is the computationally most expensive step. After this computationlly expensive step, only basic tasks of linear algebra are required. Existence and uniqueness of the solution to an overdetermined system of linear equations have to be proved.
Experiments indicate that a numerical version of this algorithm is much more efficient than previously known methods. It allows to compute determinantal representations for polynomials of degree not greater than 15 with good accuracy in just a few minutes. The authors also discuss possible variants of their algorithm that, after further theoretical clarifications, may help to improve its symbolic aspects. hyperbolic polynomials; determinantal representations; interlacing; Hermitian matrices of linear forms D. Plaumann, R. Sinn, D. E. Speyer, and C. Vinzant, \textit{Computing Hermitian Determinantal Representations of Hyperbolic Curves}, arXiv:1504.06023 [math.AG], 2015. Real algebraic and real-analytic geometry, Matrices, determinants in number theory, Convex sets in \(2\) dimensions (including convex curves), Semidefinite programming Computing Hermitian determinantal representations of hyperbolic 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 author investigates the asymptotic expansion of the Hermitian matrix integral
\[
Z_n(t,m) = \int_{{\mathcal H}_n} \exp ( - (1/2) \cdot \text{ trace} (X^2) ) \exp ( \text{ trace } \sum_{j=3}^{2m} (t_j/j)\cdot X^j ) dX,
\]
where \({\mathcal H}_n\) is the space of \(n\times n\) Hermitian matrices, \(dX\) is the Lebesgue measure of \({\mathcal H}_n\) and \(t_j\) are parameters. This asymptotic expansion implies the one of the so-called Penner model which gives the orbifold Euler characteristic of the moduli space of pointed algebraic curves by the formula of Harer-Zagier and Penner.
\textit{R. C. Penner's} original calculation [J. Differ. Geom. 27, No. 1, 35-53 (1988; Zbl 0608.30046)] involves analytic continuation. However, the formula he suggested to calculate the matrix integral does not seem to hold in the holomorphic category.
The main result of the present paper is the rigorous proof of the formula of Harer-Zagier and Penner in the asymptotic category. To this aim, the author develops instruments of asymptotic analysis replacing the analytic part of Penner's proof. Hermitian matrix integral; Feynman diagram; Penner model; moduli space of pointed algebraic curves; asymptotic analysis; Euler characteristic M. Mulase, Asymptotic analysis of a Hermitian matrix integral, Internat. J. Math. 6 (1995), 881-892. Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Feynman diagrams Asymptotic analysis of a Hermitian matrix integral | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 generalize the \(m\)-Segre invariant of vector bundles on curves to coherent systems.
Let \(X\) be a non-singular irreducible complex curve of genus \(g\). Let \(G(\alpha;n,d,k)\) be the moduli space of \(\alpha\)-stable coherent systems of type \((n,d,k)\) on \(X\). The author first generalize the \(m\)-Segre invariant of vector bundles to the \((m,t)\)-Segre invariant of coherent systems as follows.
Recall that a coherent system \((E,V)\) on \(X\) of type \((n,d,k)\) consists of a vector bundle \(E\) on \(X\) of rank \(n\) and degree \(d\) and a subspace \(V \subset H^0(X, E)\) of dimension \(k\). Also, the \(\alpha\)-slope of \((E,V)\) is defined by
\[
\mu_\alpha (E,V) := \frac{d}{n} + \alpha \frac{k}{n}.
\]
A coherent subsystem \((F,W)\) of \((E,V)\) is called called \textit{principal} if \(W = V \cap H^0(X,F)\). For any pair of integers \((m,t), \ 0 < m < n, \ 0 \le t \le k\), the \((m,t)\)-Segre invariant is defined by
\[
S_{m,t}^\alpha (E,V) := mn \cdot \min \{ \mu_\alpha (E,V) - \mu_\alpha (F,W) \},
\]
where the minimum is taken over all principal subsystems \((F,W) \subset (E,V)\) with \(\mathrm{rk}(F) =m\) and \(\dim W = t\). The author porves:
Theorem 1.1. The \((m,t)\)-Segre function is lower semicontinuous.
As a consequence of this result, the \((m,t)\)-Segre invariant yields a stratification on the moduli space \(G(\alpha;n,d,k)\) into locally closed subvarieties according to the value \(s\) of the invariant:
\[
G(\alpha;n,d,k;m,t;s) := \{ (E,V) \in G(\alpha;n,d,k) : S_{m,t}^\alpha (E,V) =s \}.
\]
Also the following issues are discussed:
\begin{itemize}
\item An upper bound on \(S_{m,t}^\alpha\) (Proposition 3.1)
\item Non-emptyness of the strata \(G(\alpha;n,d,k;m,t;s)\) (Theorem 1.2 and 1.3)
\item Dimension of the strata (Proposition 4.1)
\item Application to the crossing of critical values (\S 5)
\end{itemize}
The non-emptyness of \(G(\alpha;n,d,k;m,t;s)\) is closely related to the non-emptyness of the moduli spaces \(G(\alpha;n,d,k)\). algebraic curves; moduli of vector bundles; coherent systems; Segre invariant; stratification of moduli space Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Segre invariant and a stratification of the moduli space of coherent 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 \(X\) denote either \(\mathbb {CP}^m\) or \(\mathbb {C}^m\). We study certain analytic properties of the space \(\mathcal {E}^n(X,gp)\) of ordered geometrically generic \(n\)-point configurations in \(X\). This space consists of all \(q=(q_{1},\dots ,q_{n}) \in X^{n}\) such that no \(m + 1\) of the points \(q_1,\dots ,q_n\) belong to a hyperplane in \(X\). In particular, we show that for large enough \(n\) any holomorphic map \(f : \mathcal {E}^n(\mathbb {CP}^m,gp) \to {\mathcal E}^n(\mathbb {CP}^m,gp)\) commuting with the natural action of the symmetric group \(S(n)\) in \({\mathcal E}^n(\mathbb {CP}^m,gp)\) is of the form \(f(q) = \tau(q)q = (\tau(q)q_{1},\dots ,\tau(q)q_{n})\), \(q \in {\mathcal E}^n(\mathbb {CP}^m,gp)\), where \(\tau:{\mathcal E}^n(\mathbb {CP}^m,gp) \to \text{PSL}(m+1,\mathbb C)\) is an S(\(n\))-invariant holomorphic map. A similar result holds true for mappings of the configuration space \({\mathcal E}^n(\mathbb {C}^m,gp)\). configuration space; geometrically generic configurations; vector braids; points in general position; holomorphic endomorphism Y. Feler, ''Spaces of geometrically generic configurations,'' J. Eur. Math. Soc., 10:3 (2008), 601--624. Picard-type theorems and generalizations for several complex variables, Automorphisms of surfaces and higher-dimensional varieties, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Complex spaces with a group of automorphisms Spaces of geometrically generic configurations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an arithmetic variety (i.e. a proper flat scheme over the ring of integers in a number field \(K\) whose generic fiber is smooth) of dimension \(d\). Let \(\overline L\) be a positive Hermitian line bundle on \(X\), i.e. a line bundle on \(X\) endowed with a smooth Hermitian metric on \(X(\mathbb{C})\) whose curvature form \(c_1(\overline L)\) is strictly positive. Using Arakelov geometry as developed by Gillet and Soulé, these data allow one to define height functions on the set of subvarieties of \(X_K\) which are actual functions, as opposed to functions defined up to a bounded one.
The main result of the paper is Theorem 3.1, which states: Suppose the height \(h_{\overline L}(X)=0\) and choose an embedding \(\sigma\) of \(K\) into the complex numbers. Then for any sequence \((x_n)\) of points in \(X(\overline K)\) such that (a) \(h_{\overline L}(x_n)\) converges to \(0\) and (b) any subsequence of \((x_n)\) is dense in \(X\) with respect to the Zariski topology, the sequence of probability measures \((\mu_n)\) on \(X_\sigma(\mathbb{C})\) given by
\[
\mu_n = {1\over \#O(x_n)} \sum_{x\in O(x_n)} \delta_{\sigma(x)}
\]
(\(O(x_n)\) being the orbit of \(x_n\) under the Galois group of \(\overline K/K\)) converges vaguely to the measure \(\mu= c_1(\overline L)^{d-1} / c_1(L)^{d-1}\).
Note that the result of \textit{S. Zhang} on arithmetic ampleness [J. Am. Math. Soc. 8, No. 1, 187-221 (1995; Zbl 0861.14018)] implies under hypotheses (a) and (b) that the height \(h_{\overline L}(X)\leq 0\). Granted the equality, the proof of the theorem goes by twisting the Hermitian metric by any small enough smooth function.
The authors also show (Proposition 4.1) how the theory of adelic metrics, as developed by \textit{S. Zhang} [J. Algebr. Geom. 4, No. 2, 281-300 (1995; Zbl 0861.14019)], allows one to extend the result stated to quite more general situations in the most interesting example of abelian varieties and Neron-Tate heights. The limit measure is in this case the normalized Haar measure on the complex tori underlying the abelian variety.
Under Bogomolov's conjecture and in the case of abelian varieties, Hypothesis (b) of Theorem 4.1 may be replaced by supposing that no subsequence of \((x_n)\) is contained in a strict algebraic subgroup of \(X\). Note, however, that this theorem is a cornerstone to the recent proof of Bogomolov's conjecture by \textit{E. Ullmo} [Ann. Math. (2) 147, 167-179 (1998; Zbl 0934.14013)] and \textit{S. Zhang} [Ann. Math. (2) 147, 159-165 (1998; Zbl 0991.11030)]. Note also that an analogue of this uniform distribution theorem has been given recently by \textit{Y. Bilu} [Duke Math. J. 89, 465-476 (1997; Zbl 0918.11035)] in the case of tori using entirely different methods. equidistribution theorem; abelian variety; Bogomolov's conjecture; Arakelov geometry Szpiro, L.; Ullmo, E.; Zhang, S., Équirépartition des petits points, Invent. Math., 127, 337-347, (1997) Heights, Arithmetic varieties and schemes; Arakelov theory; heights, Rational points, Modular and Shimura varieties, Abelian varieties of dimension \(> 1\) Equidistribution of small points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves the following refinement of the main theorem of part I of this paper [ibid. 22, No.2, 161-179 (1989; see the preceding review)] the notation of which we preserve. Assume that R is global, i.e. arises as a localization of a ring of algebraic numbers K or as an open affine set of a smooth algebraic curve over a finite field. Suppose we are given additionally a finite set \(\Sigma\) of valuations of K which do not come from maximal ideals of R, for each \(v\in \Sigma\) we fix a finite Galois extension \(L_ V\) of the completion \(K_ V\) of K and a non-empty open subset \(\Omega_ V\) of \(X(L_ V)\) whose points are smooth and invariant with respect to \(Gal(L_ V/K_ V)\). Then, assuming that \(f:X\to Spec(R)\) is surjective and the union of \(\Sigma\) and the maximal spectrum of R is not equal to the set of all places of K, one can find an irreducible closed subscheme Y of X finite over R such that for any \(v\in \Sigma\), \(Y\otimes_ RL_ V\) consists of \(L_ V\)-rational points contained in \(\Omega_ V.\)
The case \(\Sigma =\emptyset\) corresponds to a theorem of Rumely (see part I). The present geometric proof specializes in this case to the proof from part I. Skolem problems; solution of system of diophantine equations; ring of algebraic numbers; rational points Moret-Bailly, Laurent, Groupes de Picard et problèmes de Skolem. I, II, Ann. Sci. École Norm. Sup. (4), 0012-9593, 22, 2, 161-179, 181-194, (1989) Global ground fields in algebraic geometry, Algebraic numbers; rings of algebraic integers, Dedekind, Prüfer, Krull and Mori rings and their generalizations, Higher degree equations; Fermat's equation Groupes de Picard et problèmes de Skolem. II. (Picard groups and Skolem problems. 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 Let \(U\) be an open subset of \(\mathbb{R}^n\) and \(X\) a coherent real analytic subset of \(U\). Let \(\varphi : U \to \mathbb{R}\) be a \({\mathcal C}^\infty\) function such that \(\varphi_{|X}\) is analytic. The authors show that for each neighbourhood \(B_\varphi\) of \(\varphi\) in the Whitney \({\mathcal C}^\infty\)-topology and for each \(\sigma : U \to \mathbb{R}^+\) with some control of the growth such that \(\sigma_{|X} = 0\) one can find an analytic function \(f\) such that \(|f(x) - \varphi (x) |\leq \sigma (x)\) for \(x \in U\). Similar results are also obtained in the algebraic and the Nash cases.
The authors also prove that if \((V, \partial V)\) and \((W, \partial W)\) are two analytic manifolds with boundary and \(\varphi : V \to W\) is a smooth map such that \(\varphi (\partial V) \subset \partial W\), then \(\varphi\) can be approximated in the Whitney \({\mathcal C}^0\)-topology by an analytic map \(f : V \to W\) such that \(f (\partial V) \subset \partial W\). real analytic sets; Nash sets; Whitney topology; approximation of smooth maps Real-analytic sets, complex Nash functions, Real-analytic and semi-analytic sets, Nash functions and manifolds, Real-analytic and Nash manifolds, Real-analytic functions, \(C^\infty\)-functions, quasi-analytic functions Some improvements of 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 authors prove two main theorems. The Milnor algebra of a surface \(V(f)\subseteq \mathbb{P}^3\) for \(f\in S=\mathbb{C}[x_0,x_1,x_2,x_3]\) which is a homogeneous polynomial, is defined by \(M(f)=\bigoplus_kM(f)_k=S/J_f\) where \(J_f=\mathrm{ideal}\big(f_i=\partial_{x_i}f:0\leq i\leq 3\big)\). The Hilbert function \(H(M(f)):\mathbb{N}\longrightarrow \mathbb{N}\) of the graded \(S\)-module \(M(f)\) is defined by \(H(M(f))(k)=\dim(M(f)_k)\).
For a free surface, \(V(f)\subseteq \mathbb{P}^3\) and for a nearly free surface, also denoted by, \(V(f)\subseteq \mathbb{P}^3\), in two separate theorems they have expressed the coefficients of the Hilbert polynomial \(P(M(f))\) associated to the Milnor algebra \(S/J_f\) which here becomes a linear polynomial in terms of their exponents. This gives some definite information about the singular locus \(\Sigma_f\) of such surfaces \(V(f)\). The computation of the Hilbert polynomial of the Milnor algebra for a general hypersurface in \(\mathbb{P}^3\) is not straight forward.
The authors also mention that the coefficients of the Hilbert Polynomial can be calculated in two examples where it is a linear polynomial. The first example is, for transversal intersections of two smooth surfaces \(D,D'\subseteq \mathbb{P}^3\) giving a smooth complete intersection curve
\(C=D\cap D'\) as a singular locus of the surface \(D\cup D'\). The coefficients are calculated in terms of their degrees \(e,e'\) respectively. The second example, is for cones \(V\subseteq \mathbb{P}^3\) over hypersurfaces \(W\subseteq \mathbb{P}^2\) with isolated singularities. The coefficients are found in terms of stability threshold \(st(W)\) and total Tjurina number \(\tau(W)\). They also compute the same in a third scenario, for the variant of the cone construction given by the surface \(V: f(x_0,x_1,x_2,x_3)=x_0g(x_1,x_2,x_3)=0 \subseteq \mathbb{P}^3\) for the surface \(W:g(x_1,x_2,x_3)=0\subseteq \mathbb{P}^2\) having isolated singularities. Here the coefficients of the Hilbert polynomial are given in terms of \(st(W),\tau(W)\) and degree \(g\).
An analogue of Saito criterion is proved for nearly free surfaces which expresses the (unique) second order syzygy in terms of determinants constructed using the first order syzygies.
As examples, are given the infinite families of surfaces \(D_d:f_d=x^{d-1}z+y^d+x^{d-2}yw+x^4y^{d-4}=0,d\geq 4\), \(D_d':f_d=x^{d-1}z+y^d+x^{d-2}yw+x^{d-5}y^5=0,d\geq 6\), \(D_d'': f_d=x^{d-1}z+y^d+x^{d-2}yw=0, d\geq 4\) which have been classified as free and nearly free surfaces and also their exponents have been given. Jacobian ideal; Milnor algebra; free divisor; nearly free divisor; Saito's criterion A. Dimca G. Sticlaru Free and nearly free surfaces in P 3 Hypersurfaces and algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Computational homological algebra, Divisors, linear systems, invertible sheaves, Relations with arrangements of hyperplanes Free and nearly free surfaces in \(\mathbb{P}^3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and \(\tau\)-functions as operator-valued. Drawing from Part I, we produce a pre-determinant structure for a class of \(\tau\)-functions defined in the setting of the similarity class of projections of a certain Banach \(^\ast\)-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator \(\tau\)-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued \(\tau\)-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We directly link this cross-ratio with Fay's trisecant identity for the \(\tau\)-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall-Chaundy \(C^\ast\)-algebra (Part I) to provide a link to the study of the KP hierarchy.
For Part I, see [ibid. 7, No. 4, 739--763 (2013; Zbl 1276.19005)]. Hilbert module; polarization; tau-function; projective structure; cross-ratio; Schwarzian derivative; KP hierarchy; Fay trisecant identity; Cowen-Douglas theory Dupré, M.J., Glazebrook, J.F., Previato, E.: Differential algebras with Banach-algebra coefficients II: The operator cross-ratio Tau function and the Schwarzian derivative, Complex Anal. Oper. Theory. doi: 10.1007/s11785-012-0219-9 \(C^*\)-modules, Projective connections, Differential geometry of homogeneous manifolds, Relationships between algebraic curves and integrable systems Differential algebras with Banach-algebra coefficients. II: The operator cross-ratio tau-function and the Schwarzian derivative | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper takes into consideration sets of six points (also infinitely near) \(P_1,\dots ,P_6\) in \({\mathbb P}^2\), and studies how the Hilbert function and graded Betti numbers of schemes of ``fat points'' (i.e. schemes defined by ideals of type \(I_{P_1}^{m_1}\cap \dots \cap I_{P_6}^{m_6}\)) are related to the geometry of the surface \(X\) obtained by blowing up \({\mathbb P}^2\) at the points. The case treated here is when \(-K_X\) is effective and the points are allowed to be infinitely near (thus generalizing previous work by the authors which considered the case of distinct points).
It turns out that the graded Betti numbers (hence also the Hilbert function) only depend on multiplicities \(m_1,\dots ,m_6\) and the structure of effective divisor classes on \(X\) having negative self-intersection.
The structure of these divisor classes is determined via the ``configuration type'' \(T\) of the points, i.e. (given an order to the points) by using a matrix made of 1's and 0's, with 7 columns and a row for each line which contains three or more of the \(P_i\)'s. An explicit numerical procedure is given to get the Betti numbers of the fat points from \(T\) and \(m_1,\dots ,m_6\). Matrids; Fat points; Free Resolutions; Hilbert Function Guardo, E.; Harbourne, B.: Configuration types and cubic surfaces, Journal of algebra 320, 3519-3533 (2008) Special surfaces Configuration types and cubic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is twofold. The first is to apply the method introduced in the works of \textit{A. Nakayashiki} and \textit{F. A. Smirnov} [in: The Kowalevski property. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes 32, 239--246 (2002; Zbl 1064.14027)] on the Mumford system to its variants. The other is to establish a relation between the Mumford system and the isospectral limit \(\mathcal{Q}_g^{(I)}\) and \(\mathcal{Q}_g^{(II)}\) of the Noumi-Yamada system [\textit{M. Noumi} and \textit{Y. Yamada}, Funkc. Ekvacioj, Ser. Int. 41, No. 3, 483--503 (1998; Zbl 1140.34303)]. As a consequence, we prove the algebraically completely integrability of the systems \(\mathcal{Q}_g^{(I)}\) and \(\mathcal{Q}_g^{(II)}\), and get explicit descriptions of their solutions. Inoue, R.; Yamazaki, T.: Cohomological study on variants of the Mumford system, and integrability of the Noumi--Yamada system. Comm. math. Phys. 265, 699-719 (2006) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems Cohomological study on variants of the Mumford system, and integrability of the Noumi-Yamada system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author characterises the smooth \(n\)-dimensional hyperquadrics as Fano manifolds of length \(n\). Given a Fano manifold \(X\) (resp. a pair \((X,x_0)\) of a Fano manifold \(X\) and a closed point \(x_0\) on it), the (global) length \(l(X)\) of \(X\) (resp. the local length \(l(X,x_0)\) of \((X,x_0))\) is defined to be the positive integer \(\min_{C \subset X}\{(C,-K_X)\}\), where \(C\) runs through the set of the rational curves contained in \(X\) (resp. the set of the rational curves such that \(x_0\in C\subset X)\). The local length \(l(X,x_0)\) is a lower semicontinuous function in \(x_0\) and the global length \(l(X)\) is by definition equal to \(\inf_{x_0\in X}l(X,x_0)\). For a given closed point \(x_0\in X\), it is known that \(l(X,x_0)\leq\dim X+1\), the equality holding if and only if \(X\) is projective space. In terms of the notions above, the main result is the following
Theorem 1. Let \(X\) be a smooth Fano variety of dimension \(n\geq 3\) defined over an algebraically closed field \(k\) of characteristic zero. Then the following three conditions are equivalent:
(1) \(X\) is isomorphic to a smooth hyperquadric \(Q_n \subset \mathbb{P}^{n+1}\).
(2) The global length \(l(X)\) is \(n\).
(3) \(\rho(X)= 1\) and \(l(X,x_0)=n\) for a sufficiently general point \(x_0\in X\), where \(\rho (X)\) stands for the Picard number.
This simple numerical result involves the preceding characterisations due to \textit{E. Brieskorn} [Math. Ann. 155, 184--193 (1964; Zbl 0019.37703)], \textit{S. Kobayashi} and \textit{T. Ochiai} [J. Math. Kyoto Univ., 13, 31--47 (1973; Zbl 0261.32014)], \textit{K. Cho} and \textit{E. Sato} [Math. Z. 217, 553--565 (1994; Zbl 0815.14035)] and [J. Math. Kyoto Univ., 35, 1--33 (1995; Zbl 0832.14031)] as immediate corollaries. rational curves; local length; global length Miyaoka, Y.: Numerical characterisations of hyperquadrics. Proceedings of The Fano Conference, on 50th anniversary of death of Gino Fano, (ed.), Collino, Conte, Marchisio, Univ. Torino 2004, pp. 559--561 Hypersurfaces and algebraic geometry, Fano varieties Numerical characterisations of hyperquadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\Gamma\) be an integral plane curve of degree \(d>k\geq 1\) with \(\delta\) ordinary nodes and cusps as its singularities, and let p: \(C\to \Gamma\) be its normalization. Let \({\mathbb{P}}_ k\) be the projective space parametrizing effective divisors of degree k on \({\mathbb{P}}^ 2\). The authors generalize a result of \textit{M. Namba} [``Families of meromorphic functions on compact Riemann surfaces'', Lect. Notes Math. 767 (1979; Zbl 0417.32008)]) concerning linear systems on a smooth curve to the case with ordinary nodes and cusps as follows.
Let \(g^ 1_ n\) be a fixed point free linear system on C with \(n+\delta <k(d-k)\) for some integer \(k>0\). Then there exists a pencil \({\mathbb{P}}\subset {\mathbb{P}}_{k-1}\) including \(g^ 1_ n\) on C. From this main lemma the authors deduce several theorems, and give examples to show the sharpness of the theorems. The results are used by the first author in Math. Ann. 289, No.1, 89-93 (1991; Zbl 0697.14019).
The proof of the main lemma is corrected in Manuscr. Math. 71, No.3, 337- 338 (1991)]. integral plane curve; linear systems Marc Coppens and Takao Kato, The gonality of smooth curves with plane models, Manuscripta Math. 70 (1990), no. 1, 5 -- 25. , https://doi.org/10.1007/BF02568358 Marc Coppens and Takao Kato, Correction to: ''The gonality of smooth curves with plane models'', Manuscripta Math. 71 (1991), no. 3, 337 -- 338. Divisors, linear systems, invertible sheaves, Singularities of curves, local rings The gonality of smooth curves with plane models | 0 |
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