text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 known that, if \(X\) is a surface of general type whose canonical map is composed with a pencil, then the irregularity \(q(X)\leq 2\). However when \(\dim X\geq 3\), simple examples show that \(q(X)\) can be arbitrarily large whenever \(\Phi_X\) is composed with a pencil.
The Albanese dimension \(a(X)\) of a manifold \(X\) is defined to be the dimension of the image of the Albanese map \(\text{alb} :X\to\text{Alb}(X)\), and \(X\) is said to be of maximal Albanese dimension or of Albanese general type if \(a(X)=\dim X\). Note that \(a(X)\) is a topological invariant of \(X\) [\textit{F. Catanese}, Invent. Math. 104, 263--289 (1991; Zbl 0743.32025)]. The main result is the following.
Theorem 1. Let \(X\) be a smooth projective 3-fold of general type over the complex number field. Assume that the canonical linear system is composed with a pencil, and the irregularity \(q(X)\geq 6\). Then \(X\) is not of maximal Albanese dimension.
The author then considers the behaviour of the canonical map under smooth deformations.
Theorem 2. Let \(X\) be a smooth projective 3-fold of general type whose canonical map is composed with a pencil. Let \(f:X\to C\) be the fiber space associated to the canonical map. If \(g(C)\geq 2\), then the canonical map of any (global) smooth deformation of \(X\) is composed with a pencil. irregularity; Albanese dimension; 3-fold of general type \(3\)-folds, Topological properties in algebraic geometry The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 relation between finite shtukas and strict finite flat commutative group schemes and the relation between divisible local Anderson modules and formal Lie groups. Let \(\mathrm{Nilp}_{{\mathbb F}_{q}[[\xi]]}\) be the category of \({\mathbb F}_{q}[[\xi]]\)-schemes on which \(\xi\) is locally nilpotent. Let \(S \in \mathrm{Nilp}_{{\mathbb F}_{q}[[\xi]]}\). The main result of this dissertation is the following (section 2.5) interesting result: it is possible to associate a formal Lie group to any \(z\)-divisible local Anderson module over \(S\) in the case when \(\xi\) is locally nilpotent on \(S\).
For related and more general results see the paper by \textit{U. Hartl} and \textit{E. Viehmann} [J. Reine Angew. Math. 656, 87--129 (2011; Zbl 1225.14036)].
A general framework for the dissertation is the descent theory by \textit{A. Grothendieck} [Catègories fibrèes et descente, Exposè VI in Revètements ètales et groupe fondemental (SGA 1), Troisième èdition, corrigè, Institut des Hautes Études Scientifiques, Paris (1963)] and his colleagues, its extensions and specializations to finite characteristics.
In Chapter 1 the author defines cotangent complexes as in papers by \textit{S. Lichtenbaum} and \textit{M. Schlessinger} [Trans. Am. Math. Soc. 128, 41--70 (1967; Zbl 0156.27201)], by \textit{W. Messing} [The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Varieties. Lect. Notes Math. 264 (1972; Zbl 0243.14013)], by \textit{V. Abrashkin} [Compos. Math. 142, No. 4, 867--888 (2006; Zbl 1102.14032)] and prove that they are homotopically equivalent.
More generally to any morphism \(f: A \to B\) of commutative ring objects in a topos is associated a cotangent complex \(L_{B/A}\) and to any morphism of commutative ring objects in a topos of finite and locally free \(\mathrm{Spec} (A)\)-group scheme \(G\) is associated a cotangent complex \(L_{G/\mathrm{Spec} (A)}\) as has presented in books by [\textit{L. Illusie}, Complexe cotangent et déformations. I. Lecture Notes in Mathematics. 239. Berlin-Heidelberg-New York: Springer-Verlag. (1971; Zbl 0224.13014), II. (1972; Zbl 0238.13017)].
In section 1.5 the author investigates the deformations of affine group schemes follow to the mentioned paper of Abrashkin and defines strict finite \(O-\)module schemes. Next section is devoted to relation between finite shtukas by \textit{V. Drinfeld} [Funct. Anal. Appl. 21, No. 1-3, 107--122 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 23--41 (1987; Zbl 0665.12013)] and strict finite flat commutative group schemes. The comparison between cotangent complex and Frobenius map of finite \({\mathbb F}_p\)-shtukas is given in section 1.7.
\(z-\)divisible local Anderson modules by \textit{U. Hartl} [Number fields and function fields -- two parallel worlds. Boston, MA: Birkhäuser. Prog. Math. 239, 167--222 (2005; Zbl 1137.11322)] and local schtukas are investigated in Chapter 2.
Sections 2.1, 2.2 and 2.3 on formal Lie groups, local shtukas and divisible local Anderson-modules define and illustrate notions for later use. Many of these, if not new, are set in a new form,
In Section 2.4 the equivalence between the category of effective local shtukas over \(S\) and the category of \(z\)-divisible local Anderson modules over \(S\) is treated.
In the last section the theorem about canonical \({\mathbb F}_{{q}[[z]]}\)-isomorphism of \(z\)-adic Tate-module of \(z\)-divisible local Anderson module \(G\) of rank \(r\) over \(S\) and Tate module of local shtuka over \(S\) associated to \(G\) is given. Drinfeld module; local Anderson-module; local shtuka; formal Lie group; cotangent complex; commutative group scheme; decent theory Research exposition (monographs, survey articles) pertaining to algebraic geometry, Formal groups, \(p\)-divisible groups, Group schemes, Drinfel'd modules; higher-dimensional motives, etc. Local shtukas and divisible local Anderson-modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex algebraic curve. Denote by \(G\) the finite cyclic group generated by an automorphism \(h\) of \(X\) of order \(n\). Let \(Y=X/G\) be the quotient smooth complex algebraic curve. The authors fix an appropriate set \({\mathcal L}\) of \(G\)-equivariant structures on a \(G\)-equivariant line bundle \(L\) on \(X\). Denote by \({\mathcal E}_{\mathcal L}\) the set of isomorphism classes of \(G\)-equivariant vector bundles \((E,\tilde{h})\) of rank \(2\) whose determinant is \(G\)-equivariantly isomorphic to one of the structures from \({\mathcal L}\). By a sequence of \(G\)-equivariant elementary transformations at special orbits of \(h\), the authors associate to \((E,\tilde{h})\) a parabolic vector bundle on \(Y\) belonging to a set \(\tilde{P_a}\) of isomorphism classes of `admissible parabolic bundles'. It is shown that there is a bijection \(\tilde{\phi}: \tilde{P_a} \to {\mathcal E}_{\mathcal L}\). The bijection preserves (semi)stability, \(S\)-equivalence and hence induces a surjective morphism \(\phi: P_a \to M_L^G\) with finite fibres, where \(P_a\) is the moduli space of admissible semistable parabolic bundles on \(Y\) and \(M_L^G\) is the \(G\)-fixed subvariety of the moduli space of semistable parabolic bundles of rank two with fixed determinant \(L\) on \(X\). If \(n\) is even, there is an involution \(i\) on \(P_a\) induced by \((E,\tilde{h})\mapsto (E,- \tilde{h})\) on \({\mathcal E}_{\mathcal L}\). If \(n\) is odd (resp. \(n\) even) then the morphism \(\phi\) (resp. \(\overline{\phi}:P_a/i \to M_L^G\)) is shown to be a birational equivalence. Moreover, its restriction to components of \(P_a\) (\(P_a/i\) if \(n\) even) is normalizing map onto the corresponding component of \(M_L^G\). vector bundles; parabolic bundles; smooth curve Andersen, JE; Grove, J., Automorphism fixed points in the moduli space of semi-stable bundles, Quart. J. Math., 57, 1-35, (2006) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Automorphisms of curves Automorphism fixed points in the moduli space of semi-stable bundles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 article is to study the Leray spectral sequence of the inclusion \(i: U \rightarrow X\), where \(X\) is a complex manifold and \(U\) the complement of an arrangement of submanifolds, computing the cohomology of \(U\).
Let \(X\) be a smooth compact complex algebraic variety and let \(s\) be a natural number. An arrangement of distinct submanifolds \(\mathcal{Z} = \{Z_{1},\dots,Z_{s}\}\) is admissible of codimension \(c\) in \(X\) if
i) each \(Z_{i}\) is smooth of codimension \(c\);
ii) for any set of indices \(I \subset \{1,\dots,s\}\) and \(j \notin I\) the set-theoretical intersection
\[
\bigg( \bigcup_{i \in I} Z_{i} \bigg)\cap Z_{j}
\]
is empty or it is the union of an admissible codimension \(c\) arrangement in each \(Z_{j}\).
The idea behind this definition boils down to the fact that for such arrangements one can apply the so-called \textit{deletation-restriction argument}.
The main result of the paper can be formulated as follows.
Main Theorem. Let \(\mathcal{Z} = \{Z_{1},\dots,Z_{s}\}\) be an admissible arrangement of codimension \(c\) submanifolds in \(X\). Then the Leray spectral sequence of the inclusion \(j: U \setminus \bigcup_{i\in \{1,\dots,s\}} Z_{i} \rightarrow X\)
\[
E^{p,q}_{2} = H^{p}(X; R^{q}j_{*}\mathbb{Q}_{U}) \Rightarrow H^{p+q}(U;\mathbb{Q})
\]
has all differentials \(d_{r}\), \(r\geq 2\) vanishing except from \(d_{2c}\). Deligne spectral sequence; Hodge structure; Leray spectral sequence; arrangements of submanifolds Configurations and arrangements of linear subspaces, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Hodge theory in global analysis Leray spectral sequence for complements of certain arrangements of smooth submanifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an integral projective flat scheme over \(\mathrm{Spec}(\mathbb Z)\) and \(\overline{L}\) be an invertible sheaf on \(X\) equipped with a continuous metric on \(L_{\mathbb C}\), invariant by the complex conjugation, the asymptotic behaviour of the spaces of small sections of \(\overline{L}^{\otimes n}\) (\(n\geqslant 1\)) is described by the arithmetic volume function \(\widehat{\mathrm{vol}}(\overline L)\). This function is the arithmetic analog of the volume function in projective algebraic geometry.
In the article under review, the author shows that the arithmetic volume function can be continuously extended to the arithmetic Picard group with coefficient in \(\mathbb R\). This generalizes his previous result [J. Algebr. Geom. 18, No. 3, 407--457 (2009; Zbl 1167.14014)] on the continuity of the function \(\widehat{\mathrm{vol}}\) on the arithmetic Picard group \(\widehat{\mathrm{Pic}}(X)\). He also explain by an explicit example that the continuous extension does not follow formally from the continuity of \(\widehat{\mathrm{vol}}\). The proof relies on a refinement of the main estimation on small sections established in [loc. cit.] by using the distorsion function. arithmetic volume function; continuous extension Moriwaki, Atsushi, Continuous extension of arithmetic volumes, Int. Math. Res. Not. IMRN, 1073-7928, 19, 3598-3638, (2009) Arithmetic varieties and schemes; Arakelov theory; heights Continuous extension of arithmetic volumes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Since the introduction of rigid analytic spaces by Tate in 1961, rigid analytic geometry (i.e. the analog of complex analytic geometry over complete non archimedean fields) has had a growing importance in algebraic geometry, number theory, \(p\)-adic analysis, etc. However, the basic results of the theory had remained scattered in various papers, sometimes difficult to find, and some fundamental notions have undergone various changes as the theory was making progress. Thus, the subject was not so easily accessible to non-specialists, and there was an increasing need for a systematic account of the theory of rigid analytic spaces in addition to the introductory book of \textit{J. Fresnel} and \textit{M. van der Put} [``Géométrie analytique rigide et applications'', Prog. Math. 18 (1981; Zbl 0479.14015)]. The present book should meet that need.
The book is divided in three parts. The first one, ``Linear ultrametric analysis and valuation theory'', is devoted to prerequisites and fundamentals in ultrametric analysis and analytic function theory. In chapter 1, the basic notions concerning ultrametric norms on abelian groups and rings are introduced. They cover such topics as the reduction functor, power multiplicative semi-norms and smoothing procedures, strictly convergent power series with coefficients in a semi-normed ring. Chapter 2 is devoted to the theory of normed modules and vector spaces. The bulk of the chapter is a discussion of various types of normed vector spaces characterized by the existence of (eventually topological) basis for which the norm of the projections of a vector are closely related to the norm of the vector itself. Of particular interest for rigid analytic geometry is the case of normed vector spaces of countable type; for these spaces, the chapter includes a proof of the ``lifting theorem'', which gives conditions under which an algebraic basis of the reduction of a normed space can be lifted as a topological orthonormal basis of the space. The third and last chapter of part A discusses the extensions of norms and valuations. In addition to the case of field extensions and classical valuation theory, the chapter introduces for \(K\)-algebras over a non-archimedean field \(K\) the notions of spectral norm and supremum norm; a special attention is given to the case of Banach algebras, and to the behaviour of the supremum norm under finite extensions, which will be of great importance in the theory of affinoid algebras (e.g. for the proof of the ``maximum modulus principle'').
The second part, ``Affinoid algebras'', develops the algebraic foundations of rigid analytic geometry. Chapter 4 is devoted to the study of the Tate algebra \(T_n=K\langle X_1,\dots,X_n\rangle\) of strictly convergent power series in n variables over a complete non archimedean field K. A particularly important result is the Weierstrass division and preparation theorem, from which are derived some basic properties: \(T_n\) is noetherian, factorial, normal, its ideals are closed, etc. Chapter 5 then proceeds with the theory of affinoid algebras, i.e. quotient algebras of algebras \(T_n\), which play the same role in rigid analytic geometry as the affine algebras in algebraic geometry. After establishing the analog of Noether's normalisation theorem, which is one of the most useful tools in the study of affinoid algebras, some classical properties of the supremum semi-norm, such as the maximum modulus principle, are proved. The chapter ends with properties of the sub-algebra Å of elements of semi-norm \(\leq 1\) in an affinoid algebra, including the important finiteness theorem for the reduction functor.
The third part of the book deals with the theory of rigid analytic spaces. Chapter 7 develops the local theory. To any affinoid algebra \(A\) is attached its maximal spectrum \(\mathrm{Sp} A\), and the basic problem in order to get a good notion of analytic space is to define a topology on \(\mathrm{Sp} A\) having reasonable properties from the point of view of analytic continuation. This is achieved thanks to the construction of a Grothendieck topology on \(\mathrm{Sp} A\). A subset \(U\) of \(\mathrm{Sp} A\) is called an affinoid subdomain if there exists an affinoid A-algebra A', such that any homomorphism of affinoid algebras \(A\to B\) for which \(\mathrm{Sp} B\) is sent in \(U\) factors uniquely through \(A'\). For example, the rational domains, defined by inequalities \(| f_i(x)| \leq | f_0(x)|\), \(i=1,\dots,r\), where \(f_0,f_1,\dots,f_r\) are elements of A generating the unit ideal, are affinoid subdomains, and are shown to have good stability properties. A theorem of Gerritzen and Grauert, proved at the end of chapter 7, implies in particular that any affinoid subdomain is a finite union of rational domains. Affinoid subdomains and finite coverings by affinoid subdomains can then be used as generators for a Grothendieck topology, as explained in the beginning of chapter 9. Thanks to Tate's acyclicity theorem, to which is devoted chapter 8, it is possible to define the sheaf of analytic functions on \(\mathrm{Sp} A\) as a sheaf for that topology, associating to any affinoid subdomain the corresponding algebra. - Grothendieck topologies allow then to glue these local models to develop the global theory of rigid analytic spaces. This is done in the last sections of chapter 9, which cover the theory of coherent modules, finite morphisms, closed analytic subvarities, separated and proper morphisms. Kiehl's finiteness theorem for direct images is discussed, although the proof is not included. The book ends with the example which was the starting point of the theory: Tate's uniformization for elliptic curves with bad reduction. rigid analytic geometry; rigid analytic spaces; ultrametric analysis; affinoid algebras; Tate algebra; coherent modules; finiteness theorem for direct images; uniformization for elliptic curves with bad reduction Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold, Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wissenschaften, vol. 261, (1984), Springer-Verlag: Springer-Verlag Berlin Local ground fields in algebraic geometry, Non-Archimedean analysis, Valued fields, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Analytic algebras and generalizations, preparation theorems, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis Non-Archimedean analysis. A systematic approach to rigid analytic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S=\mathbb C[\,x_0,\ldots,x_n\,]\) be the graded polynomial ring over the complex numbers; for \(r\in\mathbb N_0\) let \(S_r\) be the \(\mathbb C\)-vector space of homogeneous polynomials of degree \(r\). Let \(f\in S_d\), let \(J=J_f\) be the Jacobian ideal of \(f\), generated by the partial derivatives \(f_0,\ldots,f_n\) of \(f\), and set \(M(f)=S/J\), the Milnor algebra of \(f\). Let \(D=V_+(f)\subset\mathbb P^n\) be the projective hypersurface defined by \(f\) and let \(\Sigma_f=V_+(J)\subset\mathbb P^n\) be the singular locus scheme of \(D\). Let \(p\) be an isolated singularity of \(D\) with local equation \(g=0\); then \(\mathcal O_{\Sigma_f,p}=T(g)\) is the local Tjurina algebra of the analytic germ \(g\), and \(\tau(g)=\dim(T(g))\) is the Tjurina number of the isolated singularity \((D,p)\). If \(D\) has only isolated singularities, then the Tjurina number \(\tau(D)\) is defined as the sum of the Tjurina numbers of the singularities of \(D\). In this case let \(\widehat J\) be the saturation of \(J\), and for \(k\in\mathbb N\) set \({\text{def}}_k(\Sigma_f)=\tau(D)-\dim(S_k/\widehat J_k)\).
Let \(AR(f)\subset S^{n+1}\) be the graded \(S\)-submodule whose \(m\)-th homogeneous part consists of all relations involving the \(f_j\)'s, namely \(a=(a_0,\ldots,a_n)\in AR(f)\) iff \(a_0f_0+\cdots+a_nf_n=0\). Inside \(AR(f)\) there is the homogeneous \(S\)-submodule \(KR(f)\) of Koszul relations; the quotient module \(ER(f)=AR(f)/ KR(f)\) is called the module of essential relations. Let \(K^*(f)\) be the Koszul complex of \(f_0,\ldots,f_n\), equipped with its natural grading. The main result of the paper is Th.\ 1: Let \(D=V_+(f)\) be a hypersurface of degree \(d\) having only isolated singularities. Then the three integers \(\dim(ER(f)_{nd-2n-1-k})\), \(\dim(H^n(K^*(f))_{nd-n-1-k})\) and \({\text{def}}_k(\Sigma_f)\) are equal for \(0\leq k\leq nd-2n-1\), and \(\dim(H^n(K^*(f))_j)=\tau(D)\) for \(j\geq n(d-1)\). [\,Note that here, in line -2 and -3 of p.\ 196, and in line -1 of p.\ 197, the letter \(N\) has to be replaced by \(d\).\,] The proof of this result is based on the same idea as the proof of Theorem 1.5 in the paper of the author and Sticlaru, Koszul complexes and pole order filtrations. \url{arXiv:1108.3976}, where nodal hypersufaces are treated. [\,In that paper \(f\) is a homogeneous polynomial of degree \(N\).\,] However, the use of the Cayley-Bacharach theorem (\,cf. \textit{D. Eisenbud}, \textit{M. Green} and \textit{J. Harris} [Bull. Am. Math. Soc., New Ser. 33, No. 3, 295--324 (1996; Zbl 0871.14024)]) is more refined. Using invariants introduced in the arXiv paper, from Theorem 1 the author gets a lower bound for those integers \(k\) such that \(\widehat J_j=J_k\).
In Remark 4 and Remark 5, the author sketches other proofs of Theorem 1. The author also shows that Theorem 1 can be used to give a new proof of an earlier result of \textit{A. D. R. Choudry} and \textit{A. Dimca} [Proc. Am. Math. Soc. 121, No. 4, 1009--1016 (1994; Zbl 0814.14037)]. projective hypersurfaces; singularities; global Milnor algebra; syzygies; saturation of an ideal Dimca, A.: Syzygies of Jacobian ideals and defects of linear systems, Bull. Math. Soc. Sci. Math. Roumanie Tome \textbf{56}(104) No. 2, 191-203 (2013) Singularities in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Divisors, linear systems, invertible sheaves, Syzygies, resolutions, complexes and commutative rings Syzygies of Jacobian ideals and defects of linear systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F_\mathbb{R}=(f_1,\dots,f_n):\mathbb{R}^n\to \mathbb{R}^n\) be a polynomial mapping, let \(M:\mathbb{R}^n\to \mathbb{R}\) be a polynomial and let \(B=\{x\in \mathbb{R}^n:M(x)>0\}\). If \(B\) is bounded and \(\partial B\cap F^{-1}_\mathbb{R}(0)=\emptyset\) then \(\text{deg}(F_\mathbb{R},B,0)\) will denote the topological degree of \(F_\mathbb{R}\) with respect to \(B\) and \(0\in \mathbb{R}^n\). Let \(A=R[x_1,\dots,x_n]/I\), where \(I\) is the ideal in \(R[x_1,\dots,x_n]\) generated by \(f_1,\dots,f_n\). The main result of the paper is to construct bilinear forms \(\Phi\) and \(\Phi_M:A\times A\to \mathbb{R}\) in the case \(\dim A<\infty\) such that (if \(\Phi_M\) is non-degenerate and if \(B\) is bounded)
\[
\text{deg}(F_\mathbb{R},B,0)=\frac12\text{(signature }\Phi+\text{signature }\Phi_M).
\]
The method may be derived from the theory of bilinear forms on finite intersection algebras. In the case of the local topological degree a similar formula is known as the Eisenbud-Levine formula. See also the note ``Applications of the Eisenbud and Levine's theorem to real algebraic geometry'' of the same authors in [Computational algebraic geometry, Progr. Math. 109, 177-184 (1993; Zbl 0861.68109)]. An algorithm in terms of the Gröbner basis for the computation is given at the end of the present article. signature; polynomial mapping; topological degree; Gröbner basis Andrzej Łȩcki and Zbigniew Szafraniec, An algebraic method for calculating the topological degree, Topology in nonlinear analysis (Warsaw, 1994) Banach Center Publ., vol. 35, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 73 -- 83. Degree, winding number, Real algebraic sets An algebraic method for calculating the topological degree | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 deals with some properties of the location of poles of the so- called Igusa's zeta function. Let K be a field of characteristic 0. Consider the k-tuple polynomials \(F(x):=(f_ 1(x),...,f_ k(x))\) on \(x=(x_ 1,...,x_ n)\in K^ n\) and regard it as a morphism from \(K^ n\) to \(K^ k\). We suppose that each \(f_ i\) vanishes at the origin. We define the integral
\[
Z_ F(s_ 1,...,s_ k):=\int_{K^ n}| f_ 1(x)|_ K^{s_ 1}... | f_ k(x)|_ K^{s_ k} \phi (x) | dx|.
\]
Here, \(\phi\) (x) is a test function with compact support; \(s=(s_ 1,...,s_ k)\in {\mathbb{C}}^ k\); \(| |\) is the normalized valuation; \(| dx|\) is the Haar measure on \(K^ n\). \(Z_ F\) is convergent if the real parts of \(s_ i\) are all sufficiently large, and it can be continued to the whole set \({\mathbb{C}}^ k\) as a meromorphic function. We call it Igusa's zeta function if K is a p-adic field. Thus it has possible poles whose locations makes a set of hyperplanes in \({\mathbb{C}}^ k\). The set of the directions of the hyperplanes, we denote it by \({\mathcal P}(F)\), is called ``pents'' of the morphism F. The author investigates the relation between \({\mathcal P}(F)\) and \({\mathcal P}(\Delta_ F)\), where \(\Delta_ F\) is the discriminant of F. The principal result is that \({\mathcal P}(F)\) is contained in \({\mathcal P}(\Delta_ F)\) under the suitable condition that the author called ``bon''. location of poles; Igusa's zeta function; p-adic field Loeser, F., Fonctions zêta locales d'igusa à plusieurs variables, intégration dans LES fibres, et discriminants, Ann. Sci. Éc. Norm. Supér. (4), 22, 3, 435-471, (1989) Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Fonctions zêta locales d'Igusa à plusieurs variables, intégration dans les fibres, et discriminants. (Igusa's local zeta functions in several variables, fiber integration and discriminants) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper proves counterparts to various theorems in arithmetic dynamics, in which a morphism is replaced by an ``arithmetically bounded'' infinite sequence of morphisms.
The first theorem proves the existence of canonical heights of points under such sequences. Let \(X\) be a projective variety over a number field \(K\), let \(L\) be a line bundle on \(X\), and let \(h_L\:X(\overline K)\to\mathbb R\) be a height function corresponding to \(L\). Let \(\mathcal H\) denote the set of morphisms \(f\:X\to X\) such that \(f^{*}L\cong L^{\otimes d_f}\) for some integer \(d_f\) (strictly speaking, it should be the set of pairs \((f,d_f)\), since \(d_f\) may not be uniquely determined by \(f\)). For \(f\in\mathcal H\), let \(c(f)\) be the (finite) number \(\sup_{x\in X(\overline K)}\bigl|\frac1{d_f}h_L(f(x))-h_L(x)\bigr|\). A sequence \(\mathbf f=(f_i)_{i=1}^\infty\) of morphisms \(f_i\in\mathcal H\) is said to be \textit{arithmetically bounded} if \(c(\mathbf f):=\sup_i c(f_i)\) is finite. We also define a \textit{shift} function \(S(\mathbf f)=(f_{i+1})_{i=1}^\infty\).
The first theorem then states that if \(\mathbf f\) is arithmetically bounded, then there is a canonical height function \(\widehat h_{L,\mathbf f}\:X(K)\to\mathbb R\) that satisfies (1) \(\bigl|\widehat h_{L,\mathbf f}(x)-h_L(x)\bigr|\leq 2c(\mathbf f)\) and (2) \(\widehat h_{L,S(\mathbf f)}(f_1(x))=\widehat h_{L,\mathbf f}(x)\) for all \(x\in X(\overline K)\). A uniqueness property is also proved. In addition, if \(L\) is ample, then \(\widehat h_{L,\mathbf f}(x)\geq0\) for all \(x\), with equality if and only if the orbit \(O^{+}_{\mathbf f}(x):=\{f_1(x),f_2(f_1(x)),f_3(f_2(f_1(x))),\dots\}\) is finite. The paper also gives an example showing that the boundedness condition on \(\mathbf f\) is necessary.
Next, the author proves a result modeled on work of \textit{J. E. Fornaess} and \textit{B. Weickert} [Ergodic Theory Dyn. Syst. 20, 1091--1109 (2000; Zbl 0973.37032)]. Let \(g_1,\dots,g_k\) be elements of \(\mathcal H\), let \(J=\{1,\dots,k\}\), let \(W=\prod_{i=1}^\infty J\), and let \(\mathbf f_w=(g_{w_i})\) for \(w=(w_i)\) in \(W\). Let \(\nu\) be the measure on \(J\) that assigns \(j\in J\) the mass \(d_{g_j}/(d_{g_1}+\dots+d_{g_k})\), and let \(\mu=\prod\nu\) be the product measure on \(W\). Then for all \(x\in X(\overline K)\), the average
\[
\int_W\widehat h_{L,\mathbf f_w}(x)\,d\mu(w)
\]
equals the height \(\widehat h_{L,(g_1,\dots,g_k)}(x)\) defined by \textit{S. Kawaguchi} [J. Reine Angew. Math. 597, 135--173 (2006; Zbl 1109.14025)].
The paper also proves a variant of the first theorem, showing the existence of canonical heights \(h_{L,\mathbf f}(Y)\) for certain closed subvarieties \(Y\) of \(X\).
Following work of \textit{S. Zhang} [J. Algebr. Geom. 4, 281--300 (1995; Zbl 0861.14019)], the paper shows the existence of bounded continuous metrics at each place, and also studies canonical heights of (closed) subvarieties from the point of view of adelically metrized line bundles.
Finally, the paper proves an equidistribution result extending work of \textit{X. Yuan} [Invent. Math. 173, 603--649 (2008; Zbl 1146.14016)]. canonical height; adelic metric Kawaguchi, Shu, Canonical heights for random iterations in certain varieties, Int. Math. Res. Not. IMRN, 7, Art. ID rnm 023, 33 pp., (2007) Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Heights, Arithmetic varieties and schemes; Arakelov theory; heights Canonical heights for random iterations in certain 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 Given a proper cone \(K\subseteq\mathbb{R}^n\), a multivariate polynomial \(f\in\mathbb{C}[\mathbf{z}]=\mathbb{C}[z_1,\dots,z_n]\) is called \(K\)-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of \(K\). If \(K\) is the non-negative orthant, then \(K\)-stability specializes to the usual notion of stability of polynomials.
We study conditions and certificates for the \(K\)-stability of a given polynomial \(f\), especially for the case of determinantal polynomials as well as for quadratic polynomials. A particular focus is on psd-stability. For cones \(K\) with a spectrahedral representation, we construct a semidefinite feasibility problem, which, in the case of feasibility, certifies \(K\)-stability of \(f\). This reduction to a semidefinite problem builds upon techniques from the connection of containment of spectrahedra and positive maps.
In the case of psd-stability, if the criterion is satisfied, we can explicitly construct a determinantal representation of the given polynomial. We also show that under certain conditions, for a \(K\)-stable polynomial \(f\), the criterion is at least fulfilled for some scaled version of \(K\). Computational real algebraic geometry, Polynomials in real and complex fields: location of zeros (algebraic theorems), Semialgebraic sets and related spaces, Real polynomials: location of zeros, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Polynomials and rational functions of several complex variables, Semidefinite programming Conic stability of polynomials and positive 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 Let \( X_{\mathbb{R}}\) be the zero locus in \( \mathbb{R}\mathrm {P}^n\) of one or two independently and Kostlan distributed random real quadratic forms (this is equivalent to the corresponding symmetric matrices being in the Gaussian Orthogonal Ensemble). Denoting by \( b(X_{\mathbb{R}})\) the sum of the Betti numbers of \( X_{\mathbb{R}}\), we prove that
\[
\lim _{n\to \infty }\frac {\mathbb{E}b(X_{\mathbb{R}})}{n}=1.\tag{1}
\]
The methods we use combine random matrix theory, integral geometry and spectral sequences: for one quadric hypersurface it is simply a corollary of Wigner's semicircle law; for the intersection of two quadrics it is related to the (intrinsic) volume of the set of singular symmetric matrices of norm one. Lerario, A, Random matrices and the expected topology of quadric hypersurfaces, Proc. Am. Math. Soc., 143, 3239-3251, (2015) Random matrices (probabilistic aspects), Topology of real algebraic varieties, Random matrices (algebraic aspects), Integral geometry, Singular homology and cohomology theory, General theory of spectral sequences in algebraic topology, Geometric probability and stochastic geometry Random matrices and the average topology of the intersection of two quadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book on the Adams - and Adams-Novikov spectral sequence and their applications to the computation of the stable homotopy groups of spheres is the first which does not only treat the definition and construction but leads the reader to concrete computations. It contains an overwhelming amount of material, examples and machinery and collects more or less anything known on the application of the Adams-Novikov spectral sequence \((=:\) ANS Seq.) to the computation of \(\pi^ s_*\). (Many of the results are due to the author.) In some sense this book is a continuation of \textit{H. Toda}'s book [Composition methods in homotopy groups of spheres (1962; Zbl 0101.407)] although styled completely differently.
The book starts with a long and detailed expository chapter which introduces and illustrates the main topics of the book. Without being troubled with technical details the reader becomes acquainted with most of the contents of the rest of the book. Chapter 2 contains the construction and decription of some general properties of both the classical Adams spectral sequence \((=AS\) Seq.) and the Adams spectral sequence associated to a generalized homology theory \(E_*\) (for \(E_*=BP\), Brown-Peterson homology, this spectral sequence is called the ANS Seq.). The next chapter treats the classical AS Seq. in some detail (May spectral Seq., Lambda-algebra, \(E_ 2\)-term of the AS Seq. in low filtration, vanishing line and periodicity).
In chapter 4 the most important properties of complex bordism and Brown- Peterson homology \(BP_*\) are discussed and derived. The point of view here is the one of formal group law theory. A first application is the computation of the p-component of \(\pi^ s_*\) for \(n\leq 2p^ 3-2p-1\) (p\(\neq 2\), and \(n\leq 24\) for \(p=2)\) using the ANS Seq. Every element in \(\pi^ s_*\) (and in the \(E_ 2\)-term of the ANS Seq.) belongs in some sense to a periodic family and every type of periodicity is associated with a generator \(v_ i\) of \(BP_*\). The chromatic spectral sequence [introduced by \textit{H. R. Miller}, the author, and \textit{W. S. Wilson}, Ann. Math., II. Ser. 106, 469-516 (1977; Zbl 0374.55022)] organizes and systematizes these periodicity phenomena. Besides the construction and discussion of this spectral seq. chapter 5 contains the known results on periodic families in \(\pi^ s_*\). The chromatic spectral seq. is not only useful in describing periodicity but is also of use in concrete computations. This is exemplified in chapter 6 where also the connection to Morava K-theories is discussed. As an application the odd primary analogues of the Kervaire invariant 1 elements are dealt with.
Chapter 7 (called by the author the computational payoff of the rest of the book) introduces a relatively effective method for the computation of the \(E_ 2\)-term of the ANS Seq. in a fixed dimension range. This is then carried through explicitly for \(p=3\) and 5 leading to the up to now most far reaching computation of the 5-primary component of \(\pi^ s_ n\), namely up to \(n=1000\). Among the problems which had to be solved in this computation are the questions, which power of \(\beta_ 1\) vanishes and whether \(\gamma_ 3\) is permanent or not.
The book contains three appendices. Most of the homological algebra used in the book (e.g. Hopf-algebroids, change of ring spectral seq., Massey products and Steenrod powers in spectral seq.) is collected in appendix 1, and the necessary material on formal groups is contained in appendix 2. Extensive tables in appendix 3 collect most of the known values of \(_{(p)}\pi^ s_ n\) for \(p=2,3,5\) together with the Adams and Adams- Novikov \(E_ 2\)-term for \(p=2.\)
The whole book is very well structured and organized. The summary of contents, the long introduction, introductions to the sections, survey reading sections and the appendices are very well done and very useful to the reader. Many examples show how to use the methods introduced but there still remains a lot of work to be done by the reader who wants to redo the most important computations. The reader is assumed to have a working knowledge of algebraic topology (covering e.g. the book of \textit{R. M. Switzer} [Algebraic topology - homotopy and homology (1975; Zbl 0305.55001)] and \textit{J. F. Adams}' Chicago notes [Stable homotopy and generalized homology (1974; Zbl 0309.55016)]), necessary experience and practice. The style of writing is very fluent, pleasant to read and typical for the author, as everyone who has read a paper written by him will recognize.
On the negative side, occasionally some minor details are not handled with enough care (e.g. the use of \(E\wedge MU\) and MU\(\wedge E\) on p. 123) and more seriously some proofs are written in a too condensed fashion (e.g., p. 125, \(\ell.-16).\)
As the author says in his introduction, his intention is: (i) to make BP- theory and the ANS Seq. more accessible to nonexperts, (ii) to provide a convenient reference for workers in the field, and (iii) to demonstrate the computational potential of the indicated machinery for determining \(\pi^ s_*\). In my opinion he has reached his aims, maybe with the restriction that one may argue about the definition of nonexpert. In any case this is very welcome book which fills a need and which certainly was not easy to write. Adams-Novikov spectral sequence; stable homotopy groups of spheres; Adams spectral sequence; Brown-Peterson homology; Lambda-algebra; complex bordism; formal group law theory; periodic family; chromatic spectral sequence; Morava K-theories; Kervaire invariant 1 elements; Hopf- algebroids; Massey products; Steenrod powers D. C. Ravenel, \textit{Complex Cobordism and Stable Homotopy Groups of Spheres}, Pure and Applied Mathematics, Vol. 121, Academic Press, Orlando, FL, 1986. Research exposition (monographs, survey articles) pertaining to algebraic topology, Adams spectral sequences, Stable homotopy of spheres, Bordism and cobordism theories and formal group laws in algebraic topology, Generalized cohomology and spectral sequences in algebraic topology, \(J\)-morphism, Massey products, Applied homological algebra and category theory in algebraic topology, Formal groups, \(p\)-divisible groups Complex cobordism and stable homotopy groups of spheres | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The \(\mu\)-basis of a rational ruled surface \(P(s,t)=P_0(s)+t P_1(s)\) is defined by \textit{F. Chen, J. Zheng} and \textit{T. W. Sederberg} [Comput. Aided Geom. Des. 18, 61--72 (2001; Zbl 0972.68158)] to consist of two polynomials \(p(x,y,z,s)\) and \(q(x,y,z,s)\) that are linear in \(x, y, z\). It is shown there that the resultant of \(p\) and \(q\) with respect to \(s\) gives the implicit equation of the rational ruled surface; however, the parametric equation \(P(s,t)\) of the rational ruled surface cannot be recovered from \(p\) and \(q\). Furthermore, the \(\mu\)-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the \(\mu\)-basis of a rational planar curve [\textit{D. A. Cox, T. W. Sederberg} and \textit{F. Chen}, Comput. Aided Geom. Design 15, 803--827 (1998; Zbl 0908.68174)]. In the paper under review, we introduce another polynomial \(r(x,y,z,s,t)\) that is linear in \(x, y, z\) and \(t\) such that \(p, q, r\) can be used to recover the parametric equation \(P(s,t)\) of the rational ruled surface; hence, we redefine the \(\mu\)-basis to consist of the three polynomials \(p, q, r\). We present an efficient algorithm for computing the newly-defined \(\mu\)-basis, and derive some of its properties. In particular, we show that the new \(\mu\)-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface. \(\mu\)-basis; Moving plane; Implicitization; rational ruled surface Chen, Falai; Zheng, Jianmin; Sederberg, Thomas W., \textit{\({\mu}\)}-Basis of a rational ruled surface, J. Symb. Comput., 36, 5, 699-716, (2003) Rational and ruled surfaces Revisiting the \(\mu\)-basis of a rational ruled surface. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Genericity of a polynomial ideal is often needed in mathematical argumentation. Noether normalization and the recently introduced concept of strong Noether normalization provide useful characterizations of ideals in general position. The author relates these concepts to \(\delta\)-regularity, a notion in the theory of Pommaret bases. In a previous paper [Appl. Algebra Eng. Commun. Comput. 20, No. 3--4, 261--338 (2009; Zbl 1175.13011)], the author has already established the equivalence of strong Noether position and \(\delta\)-regularity. Here this equivalence is carried further and various known results on Castelnuovo-Mumford regularity and strong Noether regularity are derived from the theory of Pommaret bases. Castelnuovo-Mumford regularity; Hilbert regularity; Pommaret basis; quasi-stable ideal; satiety Seiler, W.M., Effective genericity, \textit{ {\(\delta\)}}-regularity and strong Noether position, Commun. algebra, 40, 3933-3949, (2012) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Effectivity, complexity and computational aspects of algebraic geometry, Symbolic computation and algebraic computation Effective genericity, \(\delta\)-regularity and strong Noether position | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is structured as follows: Section 1 is introduction and motivational examples. In Section 2, the authors introduce a natural compactification of the linear space \({\mathcal L}\) which they denote by \(\overline{\mathcal L}\). In order to prove the main theorem they show that one has a characteristic map \(\chi\) defined on a Zariski open set of the variety \(\overline{\mathcal L}\). As a consequence the number of solutions in the critical dimension is equal to \(\deg\,\overline{\mathcal L}\) when counted with multiplicities and when some possible ``infinite solutions'' are taken into account. The results in Section 2 generalize mathematical ideas which have been developed for the static pole placement problem by \textit{R.W. Brockett} and \textit{C.I. Byrnes} [IEEE Trans. Autom. Control 26, 271--284 (1981; Zbl 0462.93026)] and for the dynamic pole placement problem by \textit{M.S.
Ravi, J. Rosenthal} and \textit{X. Wang} [SIAM J. Control Optimization 32, No. 1, 279--296 (1994; Zbl 0797.93018); ibid. 34, No.3, 813--832 (1996; Zbl 0856.93043)].
In Section 3 the authors compute the degree of \(\overline{\mathcal L}\) in many special cases. As a corollary they rediscover several matrix completion results as they were derived earlier.
Section 4 is concerned with the value of the ``generic degree''; this is the largest possible degree of a variety \(\overline{\mathcal L}\) of fixed dimension can have. The authors determine an upper bound for the generic degree in the case when \(d= n\) and prove that this bound is reached when \(m= n< 5\). inverse eigenvalue problems; Grassmann varieties; degree of a projective variety Kim, M., Rosenthal, J., and Wang, X., Pole Placement and Matrix Extension Problems: a Common Point of View, SIAM J. Control Optim., 2004, vol. 42, no. 6, pp. 2078--2093. Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors, Pole and zero placement problems, Eigenvalue problems Pole placement and matrix extension problems: a common point of view | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification. polynomial optimization; non-compact semi-algebraic sets; semidefinite programming relaxations; positivstellensatzë V. Jeyakumar, J. B. Lasserre, and G. Li, \textit{On polynomial optimization over non-compact semi-algebraic sets}, J. Optim. Theory Appl., 163 (2014), pp. 707--718. Nonlinear programming, Nonconvex programming, global optimization, Semialgebraic sets and related spaces, Optimality conditions and duality in mathematical programming On polynomial optimization over non-compact semi-algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the paper under review is to develop the classical theory of infinitely near singular points for plane curves to the higher dimensional case. The basic notion of the general theory is formulated in terms of permissible diagrams of local sequence of smooth blowing-ups, idealistic exponents and certain graded algebras associated with them [see \textit{H. Hironaka}, in: Algebraic geometry, The Johns Hopkins Centen. Lect., Symp. Baltimore/Maryland 1976, 52--125 (1977; Zbl 0496.14011)]. In fact, the idealistic exponents correspond to the first characteristic exponents in the classical case. The author underlines that his work technically depends upon the use of partial differential operators whose basic properties are reviewed in the first chapter of the paper. Then three key theorems for singular data on an ambient regular scheme of finite type over any perfect field of any characteristic are proved; they refer as Differentiation, Numerical Exponent and Ambient Reduction Theorems. As an application the author states and proves in the last two chapters the Finite Presentation Theorem which is, on his opinion, ``an important milestone in the development of a general theory of infinitely near singular points, giving an algebraic presentation of finite type to the total aggregate of all the trees of infinitely near singular points, geometrically diverse and intricate.'' It should be remarked that the original proof of this theorem is given in [\textit{H. Hironaka}, J. Korean Math. Soc. 40, No. 5, 901--920 (2003; Zbl 1055.14013)]. characteristic exponents; idealistic exponents; smooth blowing-ups; permissible diagrams Hironaka, H., Three key theorems on infinitely near singularities, (Singularités Franco-Japonaises. Singularités Franco-Japonaises, Sémin. Congr., vol. 10, (2005), Soc. Math. France: Soc. Math. France Paris), 87-126 Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects) Three key theorems on infinitely near singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathbb{P}}^ 1\) denote the projective line over a given field K. If D is a subset of \({\mathbb{P}}^ 1\) and \(\Phi\) maps \(D^ 2\) into D then \(\Phi\) is a three-cycle on D iff \(\Phi (u,\Phi (x,u))=x\) for all x,u\(\in D\). A three-cycle \(\Phi\) on D such that \(| D| \geq 3\) is called bilinear iff there is a list of constants a,b,c,d,e,f,g,h in K such that \(\Phi ((x,y),(u,v))=(axu+bxv+cyu+dyv,exu+fxv+gyu+hyv)\) for all (x,y),(u,v)\(\in D\). Three types of bilinear three-cycles are considered and it is proved that: That of the first type are equivalent to
((x,y),(u,v))\(\mapsto (\alpha yv,xu)\), for some \(\alpha \in K^*\), a bilinear three-cycle on a subset of \({\mathbb{P}}^ 1\setminus \{(1,0),(0,1)\}\). That of the second type are equivalent either \(to\)
((x,y),(u,v))\(\mapsto (xv+yu-\beta yv,-yv)\), for some \(\beta\in K\), or to \(((x,y),(u,v))\mapsto (w^ 2xv+wyu,-yv)\), for some \(w\in K\) such that \(w\neq 1\) and \(w^ 3=1\), a bilinear three-cycle on a subset of \({\mathbb{P}}^ 1\setminus \{(1,0)\}\), depending on whether \(\Phi\) is symmetric or not. That of the third type induce group structures on \({\mathbb{P}}^ 1\) and they may be expressed via those structures. equivalence; projective line; bilinear three-cycles Functional equations for functions with more general domains and/or ranges, Iteration theory, iterative and composite equations, General theory of linear incidence geometry and projective geometries, Elliptic curves, Loops, quasigroups Bilinear three-cycles on the projective line | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the present paper is to consider some spectrum functors which have significance in algebraic geometry. The author sets up a spectrum functor \({\mathcal T}\)-spec from the category of commutative rings with unit into the category of topological spaces, where \({\mathcal T}\) is a theory of commutative rings with unit admitting elimination of quantifiers, and \({\mathcal T}\)-spec A consists of equivalence classes of homomorphisms. Closed formulae are defined as those equivalent to positive quantifier- free formulae and are used to define a topology on \({\mathcal T}\)-spec A making it a spectral space. Criteria for \({\mathcal T}\)-closed formulae and spectrally closed constructible subsets, respectively, are given. Some examples illustrate additional properties such as the existence of open projections and enough prime models for a theory \({\mathcal T}\). Using results of \textit{G. Cherlin} and \textit{M. Dickmann} [Ann. Pure Appl. Logic 25, 213-231 (1983; Zbl 0538.03028)], the author discusses the theory RCVR of real closed valuation rings. The constructed spectrum functor RCVR- spec is new, therefore more details are described, especially for real Prüfer rings. An affine space of dimension n over a fixed model R of a theory \({\mathcal T}\) with enough prime models is defined as a certain n-fold fibred product. Such affine spaces enjoy an ``Artin-Lang'' property. The projection from the affine space \({\mathcal A}^ m\) to the affine space \({\mathcal A}^ n\) is open if \({\mathcal T}\) has open projections (both spaces are endowed with the same type of topology). Sheaves of definable sections and sheaves of closed and bounded definable sections (denoted by \({\mathcal D}\) and \({\mathcal S}\), respectively), are considered, where a section over \(X\subset {\mathcal T}\)-spec A is a function s: \(X\to {\mathcal T}\)-spec A[T] such that \(pr(s(\alpha))=\alpha\). Closed and bounded sections are continuous and a closed section is bounded if and only if it can be extended in a certain way.
The sheaves \({\mathcal R}\) (sheaf of rational functions), \({\mathcal S}\), and \({\mathcal D}\) are examples for definable sheaves of functions. Another example is the canonical Nash sheaf on the real spectrum. The final result deals with definable statements concerning the \({\mathcal L}\)- structure of constructible sets in \({\mathcal T}\)-spec A. The notions are defined in a model-theoretic way. continuous section; spectrum functors; algebraic geometry; commutative rings; topological spaces; spectral space; open projections; prime models; real closed valuation rings; affine space; fibred product; sheaves DOI: 10.1016/0022-4049(90)90071-O Model-theoretic algebra, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Valuations and their generalizations for commutative rings, Quantifier elimination, model completeness, and related topics, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Model theory and spectra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to prove a Givental type decomposition
for partition functions that arise out of topological recursion
applied to spectral curves. Copies of the Konstevich-Witten
Korteweg-de Vries (KdV) tau function arise out of regular spectral
curves and copies of the Brezin-Gross-Witten KdV tau function
arise out of irregular spectral curves. The authors present the
example of this decomposition for the matrix model with two hard
edges and spectral curve \((x^2-4)y^2=1\).
This paper is organized
as follows: The first section is an introduction to the subject.
In the second section the authors recall the definitions of the
two KdV tau functions which form the fundamental pieces of the
decomposition. In the third section, they introduce the
decomposition without the differential operator
\(\mathcal{\widehat{R}}\) via the elementary topological part of the
correlators. The forth section deals with the main result of this
paper which is a generalization of the decomposition theorem to
allow irregular singularities. One application of the main result
applied to the curve above is a Givental type decomposition for
the partition function of the Legendre ensemble. The paper is
supported by an appendix where the authors first consider
quantization in finite dimensions which easily generalizes to
infinite dimensions. Givental decomposition; topological recursion; spectral curve; KdV tau function Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Topological recursion with hard edges | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Deflation is an important method of regularizing nonreduced solution sets of systems of polynomials over the complex numbers so that, for example, Newton's method can be applied to approximate solutions to arbitrary accuracy. The current article defines three new, equivalent methods of deflation, each having specific strengths and weaknesses. In particular, the method based on determinants (as in the Thom-Boardman singularity theory) avoids introducing extra variables, so that the deflated algebraic set is the same as the original. A second method, which does employ additional variables, reduces the number of polynomials and does not increase degrees, thus making it useful for numerical work. The third method more closely resembles previous definitions of deflation. All three have the advantage of building a complete basis for the null space of \(Jf(x)\) (where \(f: \mathbb{C}^N \rightarrow \mathbb{C}^n\) is a polynomial system, and \(J\) is the Jacobian), as opposed to just the one vector in the null space that previous definitions produced. Additionally, the approach taken leads to a finite stratification of the system's singularities such that each solution is a smooth point on a unique isosingular subset, allowing Newton's method to be applied to non-generic, as well as generic, solutions.
Deflation is defined in terms of the deflation operator, the repeated application of which is called a deflation sequence. The authors prove that ultimately this sequence stabilizes, and they give both numeric and symbolic methods for computing deflation sequences, as well as for determining when they have stabilized.
The article defines an isosingular set as the (nonempty, irreducible) closure of a set of points in an algebraic set with the same deflation sequence. The deflation sequence stabilizes to the dimension of the set. The number of such sets for a given polynomial system is shown to be finite, and an algorithm is given to compute all of them. It is shown that every isosingular set is generically isomorphic to an irreducible and generically reduced component of a polynomial system constructed using deflation.
There are several examples within the paper. It is shown that the isosingular sets of the Whitney umbrella are the irreducible surface, the ``handle'', and the origin. Additionally, a detailed description of the computation of the isosingular sets of a ``foldable Stewart-Gough platform'', an example arising from kinematics, is provided. irreducible algebraic set; deflation; deflation sequence; multiplicity; isosingular set; isosingular point; local dimension; numerical algebraic geometry; polynomial system; witness point; witness set; singularity structure; regularization; Thom-Boardman singularity theory; systems of polynomials; Newton's method; symbolic method; algorithm Hauenstein, J.D.; Wampler, C.W.; Isosingular sets and deflation; Found. Computat. Math.: 2013; Volume 13 ,371-403. Numerical computation of solutions to systems of equations, Computational aspects in algebraic geometry, Symbolic computation and algebraic computation, Numerical computation of solutions to single equations, Solving polynomial systems; resultants Isosingular sets and deflation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal X}\) be an open orientable surface with finite genus and finite number of boundary components, and let \({\mathcal Y}\) be a closed orientable surface. An open continuous function from \({\mathcal X}\) to \({\mathcal Y}\) is called a \((p, q)\)-map, \(0< q< p\), if it has a finite number of branch points and assumes every point in \({\mathcal Y}\) either \(p\) or \(q\) times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all nontrivial functions having two valences between \({\mathcal X}\) and \({\mathcal Y}\).
In this paper we generalize and prove in a unified manner almost all the earlier covering and existence results involving \((p, q)\)-maps between orientable surfaces. Our results are combinatorial in nature; they relate the branch orders and exceptional sets of \((p, q)\)-maps to the valences \(p\), \(q\) and the topological invariants of \({\mathcal X}\) and \({\mathcal Y}\). The paper ends with open questions. open surface; closed surface; covering surface; orientable surface Covering spaces and low-dimensional topology, Embeddings in algebraic geometry, Compact Riemann surfaces and uniformization Covering theorems for open continuous mappings having two valences between orientable 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 At first, we study the resolution procedure by blowing-up an \(n\)- dimensional linear system of two variable power series. The main application of this study is an algebraic characterization of the linear system geometry by means of Zariski's complete ideal theory.
In the second part, we study more specifically the desingularization of the one-dimensional linear systems (pencils). We explain the parallelism between the geometry of a pencil and the associated meromorphic foliation. Finally, we characterize the set of pencils associated to a given foliation with a non-constant first integral. We work on the formal power series ring over the complex field but all the results are true on a two-dimensional, local, regular, integrally closed ring over an algebraically closed field. blowing-up an \(n\)-dimensional linear system; resolution; meromorphic foliation; formal power series ring Global theory and resolution of singularities (algebro-geometric aspects), Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Formal power series rings, Divisors, linear systems, invertible sheaves Linear systems of power series, pencils and associated foliations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using a reduction to the Beauville systems, a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism, is obtained. The structure of the paper is as follows: In Section 2 the relation between the Jacobians of two curves which are linked by a ramified cyclic covering of prime order is studied. In Section 3 a space of polynomial matrices of size \(p\) is introduced and its automorphisms of order \(p\) are studied, with particular attention to the fixed point set of such an automorphism. In Section 4, both the space and its fixed point set are related to the corresponding spectral curves, upon using the momentum map and the results of Section 2. Beauville's result also used to describe the fibers of the algebraic completely integrable systems under construction. The Hamiltonian structure of the space of polynomial matrices, its fixed point sets and their quotients (by the adjoint action) are considered in Section 5. In particular a multi-Hamiltonian structure of the newly constructed phase spaces is also obtained. The algebraic integrability this system is proved in Section 6. integrable systems; Jacobians; algebraic integrability; curves with automorphisms Poisson manifolds; Poisson groupoids and algebroids, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties Algebraic integrable systems related to spectral curves with automorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 finite subspaces of orderings of the ring of regular functions on an algebraic set \(V\) and compares them with those of the ring of analytic function germs at a point of \(V\). He proves that two semialgebraic subsets \(A\) and \(B\) in a real affine algebraic set \(X\) are generically separable by a regular function if and only if in the birational model of \(X\) made by contracting the union of their borders to a point \(a\), their germs at \(a\) are generically separable by an analytic function germ. The paper is very well written and contains a detailed introduction to the theory of constructible sets in real geometry. semialgebraic set; space of orderings; real spectrum; separation property; analytic orderings Semialgebraic sets and related spaces, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Real algebra, Real-analytic and semi-analytic sets Algebraic and analytic finite spaces of orderings. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this work, we find several interesting examples of surface singularities in \({\mathbb{C}}^ 3\), for which the \(\mu\)-constant stratum \(S_{\mu}\), in the miniversal deformation is not smooth. We study surface singularities \((V,0)\subset ({\mathbb{C}}^ 3,0)\) that can be resolved by a quadratic transformation, called superisolated singularities. Let p:\({\mathcal B}\to T\) of a superisolated singularity with smooth base T be the \(\mu\)-constant deformation. Using the Perron's theorem and results of Neuman we can see that such a deformation \({\mathcal B}\) is equimultiple along \(\sigma\) (T) (section of p) and p has a strong simultaneous resolution, starting with the monoidal transformation with center \(\sigma\) (T). Also, we can see that if \(D\subset {\mathbb{P}}_ 2\) is the projectivized tangent cone of (V,0), then p induces a deformation \(\pi\) :\({\mathcal D}\to T\) of D, which is equisingular as a deformation of the projective plane curve \(D\subset {\mathbb{P}}_ 2\). In the second section, we study how to compute in the base of a miniversal deformation of (V,0), the stratum \(\mu\)-constant (resp. \(\mu^*\)-constant) which denoted by \(S_{\mu}\) (resp. \(S_{\mu^*})\). Next, we study in B the equimultiplicity stratum E and the deformation of D over E induced by the miniversal deformation, the stratum \(\Sigma_ D\subset E\) of the points where the corresponding plane curve has the same equisingularity type in its singularities as D. Following is the main result: If (V,0) is as above, then \(S_{\mu}=S_{\mu^*}=\Sigma_ D\). mu-constant deformation; equimultiple deformation; mu-constant stratum; superisolated singularity Luengo, I., The \textit{ {\(\mu\)}}-constant stratum is not smooth, Invent. Math., 90, 1, 139-152, (1987) Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties The \(\mu\)-constant stratum is not smooth | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Bresinsky-Angermüller semigroup theorem proves that given a pair of plane algebroid branches \(\gamma, \lambda\) where \(\lambda\) is nonsingular and \(n\) is the intersection number of the branches, the semigroup of \(\gamma\) is generated by a characteristic sequence with initial term \(n\). Reciprocally, any characteristic sequence with initial term \(n>0\) admits a pair of branches \(\gamma, \lambda\), such that the sequence generates the semigroup of \(\gamma\), \(\lambda\) is nonsingular and both branches have intersection number equal to \(n\).
In this paper the authors consider Abhyankar-Moh characteristic sequences which are characteristic sequences with two added properties, one of them a conductor formula. The paper is then devoted to prove an interesting Bresinsky-Angermüller semigroup theorem for these type of sequences. This result states that the semigroup of the branch at infinity \(\gamma\) of a coordinate line of degree \(n >1\) is generated by an Abhyankar-Moh characteristic sequence with initial term \(n\). Moreover if \(G\) is a semigroup generated by an Abhyankar-Moh characteristic sequence with initial term \(n >1\), then there exists a coordinate line of degree \(n\) with branch at infinity \(\gamma\) whose semigroup is \(G\).
We conclude by recalling that an affine curve in \(K^2\) (\(K\) being any field) \(\Gamma\) is named a coordinate line whenever there exists a polynomial automorphism \((f,g): K^2 \rightarrow K^2\) such that \(f=0\) is the minimal equation of \(\Gamma\). Notice that this lines have ever one branch at infinity. Branch at infinity; Abhyankar-Moh characteristic sequence; Semigroup Singularities in algebraic geometry Semigroups corresponding to branches at infinity of coordinate lines in the affine plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f(X)\) be a non-constant rational function in \(\overline{\mathbb{Q}}(X)\) whose critical values lie in the set \(\{0,1,\infty\}\). We call such an \(f(X)\) a Belyi function of genus 0 and \(f^{-1}(\{0,1,\infty\})\subset \mathbb{P}^1(\mathbb{C})\) the set of cuspidal points of \(f(X)\). In this paper, we study a geometric balance of the cuspidal points arising from a Belyi function. First main result:
Theorem A. Let \(A\) be a finite set of \(\mathbb{P}^1(\mathbb{C})=\mathbb{C}\cup\{\infty\}\) with \(\infty\in A\). Then, \(A\) is realized as the set of cuspidal points of a Belyi function \(f(X)\) such that \(f(\infty) =\infty\) if and only if there exist a function \(e : A\to\mathbb{N}\) \((a\mapsto e_a)\), a constant \(C(\neq: 0)\) and a disjoint decomposition \(A = A_0\coprod A_1\coprod A_\infty\coprod \{\infty\}\) such that
\[
\begin{aligned} |A|-2 & =\sum_{a\in A_0}e_a=\sum_{a\in A_1}e_a=\sum_{a\in A_\infty\cup\{\infty\}}e_a,\tag{1}\\ \frac{e_a\prod_{x\in A\setminus\{a,\infty\}}(a-x)}{\prod_{x\in A_\infty}(a-x)^{e_x}}& =\begin{cases} -C & \text{for any }a\in A_0,\\ \;\;C& \text{for any }a\in A_1.\end{cases} \tag{2}\\ \prod_{x\in A_0}(a-x)^{e_x}& =\prod_{x\in A_1}(a-x)^{e_x}\quad \text{for any }a\in A_{\infty}.\tag{3}\end{aligned}
\]
Moreover, if this is the case, then the Belyi function \(f(X)\) is given by
\[
f(X)=\frac{e_\infty\prod_{a\in A_0}(X-a)^{e_a}}{C\prod_{a\in A_\infty}(X - a)^{e_a}}.
\]
where the values \(e_a\) \((a\in A)\) coincide with the branch indices of \(f(X)\) at the cuspidal points \(a\in A\) respectively.
\textit{A. Grothendieck} [Sketch of a programme, Lond. Math. Soc. Lect. Note Ser. 242, 5-48, 243-283 (1997; Zbl 0901.14001)] defines a dessin to be the graph \(f^{-1}([0,1])\) for a Belyi function of any genus. When the shape of a dessin is a tree, the dessin is necessarily of genus 0 and the cuspidal points of the associated Belyi function are precisely equal to \(\{\text{vertices of the dessin}\}\cup \{\infty\}\), in particular, the Belyi function is a polynomial. Thus, we specialize theorem A to the following:
Theorem B. Let \(A\) be a finite set of complex numbers. Then, \(A\) is the set of vertices of a tree dessin if and only if there exist a function \(e : A \to \mathbb{N}\) \((a\mapsto e_a)\), a constant \(C(\neq 0)\) and a disjoint decomposition \(A = A_0\coprod A_1\) such that
\[
|A|-1 =\sum_{a\in A_0} e_a=\sum_{a\in A_1} e_a,\tag{1}
\]
\[
e_a\prod_{x\in A\setminus\{a\}} (a-x)=\begin{cases} -C & \text{for any }a\in A_0;\\ C & \text{for any } a\in A_1.\end{cases}\tag{2}
\]
Moreover, if this is the case, then the Belyi function \(f(X)\) is given by
\[
\frac{|A|}{C} \prod_{a\in A_0}(X-a)^{e_a}
\]
where the values \(e_a\) \((a\in A)\) coincide with the branch indices of \(f(X)\) at the vertices \(a\in A\). respectively. dessins d'enfants; Riemann sphere; Belyi function; cuspidal points; tree dessin Arithmetic problems in algebraic geometry; Diophantine geometry, Riemann surfaces; Weierstrass points; gap sequences Geometric balance of cuspidal points realizing dessins d'enfants on the Riemann sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 or complex polynomial map \(f: \mathbb K^n \to \mathbb K^p\) defines a locally trivial fibration over the complement of the set of atypicla values. The authors study the relations between several regularity conditions at infinity. The central object of the paper is \(t\)-regularity. It is shown that it is equivalent to the regularity condition of \textit{K. Kurdyka, P. Orro} and \textit{S. Simon} [J. Differ. Geom. 56, No.~1, 67--92 (2000; Zbl 1067.58031)], and also to the Malgrange condition of \textit{T. Gaffney} [Compos. Math. 119, No.~2, 157--167 (1999; Zbl 0945.32013)], here also formulated in the real case. Under a `fairness' condition there is also an interpretation in terms of integral closure. The authors show that \(t\)-regularity implies \(\rho_E\)-regularity (a Milnor-type condition of transversality to the Euclidean distance function), which in turn implies topological triviality at infinity. \(t\)-regularity; integral closure of modules Dias, L.R.G.; Ruas, M.A.S.; Tibăr, M., Regularity at infinity of real maps and a Morse-sard theorem, J. topol., 5, 2, 323-340, (2012) Fibrations, degenerations in algebraic geometry, Global theory of complex singularities; cohomological properties, Equisingularity (topological and analytic), Topological properties of mappings on manifolds Regularity at infinity of real mappings and a Morse-Sard 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 Drawing on the work of Mourtada, we show that a family of vector fields with a generic algebraic polycycle of four hyperbolic apices possesses a maximum capacity of four limit cycles. This cyclicity is attained in an opening connecting the parameters which the edge contains, in particular a generic line of singularities of dovetail type. We also give an asymptotic estimation of the volume of this opening, as well as an explicit example of a family of polynomial vector fields replicating the above-described conditions and possessing five limit cycles. The methods employed are very diverse: geometrical arguments (Thom's theory of catastrophes and the theory of algebraic singularities), developments from Puiseux, the number of major roots by Descartes' law and calculated exactly by Sturm series, and other specific methods for formal calculus, such as for example the cylindrical algebraic decomposition and the resolution of algebraic systems via the construction of Gröbner bases. The calculations have been executed formally, that is to say without making the least appeal to numerical approximation, in using the formal calculus system AXIOM. family of vector fields; algebraic polycycle; limit cycles; cyclicity A. Jacquemard, F. Z. Khechichine-Mourtada, and A. Mourtada, Algorithmes formels appliqués à l'étude de la cyclicité d'un polycycle algébrique générique à quatre sommets hyperboliques, Nonlinearity 10 (1997), no. 1, 19 -- 53 (French, with English and French summaries). Local and nonlocal bifurcation theory for dynamical systems, Polynomials, factorization in commutative rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Structure of families (Picard-Lefschetz, monodromy, etc.), Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Structural stability and analogous concepts of solutions to ordinary differential equations Formal algorithms applied to the study of the cyclicity of a generic algebraic polycycle with four hyperbolic crests | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An almost CR-structure on a smooth manifold \(M\) consists of a subbundle \(E\) of the tangent bundle together with a smooth endomorphism \(J:E\to E\) such that \(J^2=-\text{Id}\). If this almost CR-structure is integrable, it is called a CR-structure and determines a foliation of \(M\) by complex manifolds. In [Ann. Math. 94, 494--503 (1971; Zbl 0236.57014)], \textit{H. B. Lawson jun.} constructed a codimension-one foliation of \(S^5\) and in [Ann. Math. 156, 915--930 (2002; Zbl 1029.32019)], the authors of the present paper constructed an integrable CR-structure whose underlying smooth foliation \({\mathcal F}\) is a variation of Lawson's. The two foliations are topologically distinct. Neither foliation is unique in the differentiable sense and the present paper is concerned with CR-structures on a fixed smooth foliation; however the results are independent of the choice.
The main results are as follows. Theorem A states that the Lawson foliation can be endowed with a compatible almost CR-structure, but no such structure can be integrable. On the other hand \({\mathcal F}\) does (as already stated) admit integrable CR-structures and the central purpose of the paper is to study the variation of such structures (for fixed \({\mathcal F}\)). After proving a rigidity theorem (Theorem B), the authors show that the set of such structures can be identified with \({\mathbb C}^3\) (Theorem C); moreover, this identification makes \({\mathbb C}^3\) into a coarse moduli space (Theorem D). Due to the presence of non-trivial CR-automorphisms, this is not a fine moduli space. It is interesting to note that the proof of Theorem B involves bundles on elliptic curves and divisors on Hirzebruch surfaces and cubic surfaces. almost CR-structure; foliation; integrable CR-structure; coarse moduli space Meersseman, L; Verjovsky, A, On the moduli space of certain smooth codimension-one foliations of the 5-sphere by complex surfaces, J. Reine Angew. Math., 632, 143-202, (2009) CR structures, CR operators, and generalizations, Foliations (differential geometric aspects), Special surfaces, Fine and coarse moduli spaces On the moduli space of certain smooth codimension-one foliations of the 5-sphere by complex surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the adjoint linear system \(|K_S + L|\) on a complex surface \(S\) of general type. For a nef divisor \(L\) on \(S\), we give a numerical criterion for \(\phi_{K_{S}+L}\) to be generically finite. A question of \textit{M. Chen} and \textit{E. Viehweg} [Pac. J. Math. 219, No. 1, 83--95 (2005; Zbl 1093.14056)] is studied and we give some partial evidence for this question to be true. algebraic surfaces; adjoint linear systems Surfaces of general type, Divisors, linear systems, invertible sheaves, Adjunction problems, Families, moduli, classification: algebraic theory Adjoint linear systems on algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\pi:\Gamma(2)\to H_ 1(\Gamma(2))\cong\mathbb{Z}^ 2\) be the canonical map and let \(R_ N:\mathbb{Z}^ 2\to(\mathbb{Z}/N\mathbb{Z})^ 2\) denote the reduction \(\bmod N\) where \(N\geq 1\) is an integer. The group \(\Phi(N):=\text{ker}(R_ N\circ\pi)\) defines an abelian Galois covering \(\Phi(N)\backslash\mathbb{H}\) of \(\Gamma(2)\backslash\mathbb{H}\) such that \(\overline{\Phi(N)\backslash\mathbb{H}}=F_ N\) where \(F_ N:x^ N+y^ N=1\) is the Fermat curve. The paper under review studies the eigenvalue theory of the hyperbolic Laplacian on \(L^ 2(\Phi(N)\backslash\mathbb{H})\). Since \(\Phi(N)\) is a congruence group if and only if \(N=1,2,4,8\), these groups constitute an interesting class of examples for investigations on the distribution of eigenvalues of \(\Delta\).
Let \(m(\lambda,N)\) be the dimension of the eigenspace for the eigenvalue \(\lambda\) of \(-\Delta\) on \(\Phi(N)\) and denote by \(M(\lambda,N)\) the sum of the \(m(\mu,N)\) with \(0<\mu<\lambda\). The superscripts ``cusp'' and ``Eis'' indicate that only the corresponding cusp forms or residues of Eisenstein series are counted. In particular, \(M(1/4,N)\) is the number of exceptional eigenvalues of \(-\Delta\) on \(\Phi(N)\).
Theorem 1. (i) There are continuous strictly increasing functions \(F,G:[0,1/4]\to\mathbb{R}\) with \(F(0)=G(0)=0\) such that as \(N\to\infty\)
\[
M^{cusp}(\lambda,N)=F(\lambda)N^ 2-G(\lambda)N+o(N)\quad\text{ for } \lambda\in]0,1/4[,
\]
\[
M^{Eis}(\lambda,N)=G(\lambda)N^ 2+O(1)\quad\text{ for } \lambda\in]0,1/4].
\]
(ii) \(\lambda_ 1(N)\sim 2/N\), \(\lambda_ 1^{cusp}(N)\sim 4/N\), where \(\lambda_ 1\) is the first non-trivial eigenvalue.
(iii) For all but finitely many \(\lambda\in[0,1/4]\) we have \(m(\lambda,N)=O(N^{2/3})\).
A more refined information based on numerical calculations (without detailed error analysis) is given in Theorem \(1'\).
The authors' investigation of \(-\Delta\) on \(L^ 2(\Phi(N)\backslash\mathbb{H})\) rests upon an analysis of \(-\Delta\) on \(L^ 2(\Gamma(2)\backslash\mathbb{H},\chi)\) where \(\chi:\Gamma(2)\to U(1)\) is a character. Note that \(m(\lambda,N)=\sum_{\chi^ N=1}m(\lambda,\chi)\). A basic result is that \(\lambda_ 1(\chi)\geq 1/4\) for all \(\chi\). This reduces the study of the exceptional spectrum of \(\Phi(N)\) to the investigation of \(\lambda_ 0(\chi)\) and its level curves \(C_ \lambda=\{\theta:\lambda_ 0(\theta)=\lambda\}\) where \(\theta\in\mathbb{R}^ 2/\mathbb{Z}^ 2\) parametrizes the characters on \(\Gamma(2)\). (Part of the analysis of the \(C_ \lambda\) is postponed to a future paper of the authors.) The proof of Theorem 1 then follows from standard results about lattice points in convex regions and on curves.
The authors also examine the spectrum of \(\Phi(N)\) on \([0,\infty[:\)
Theorem 4. There is a strictly increasing function \(F(\lambda)\) on \([0,\infty[\) such that
\[
M^{cusp}(\lambda,N)\sim F(\lambda)N^ 2\quad\text{ as } N\to\infty.
\]
The theoretical basis for the authors' computer calculations stems from a result due to Selberg the proof of which is reproduced in an appendix. Fermat curve; hyperbolic Laplacian; distribution of eigenvalues; exceptional eigenvalues; level curves; spectrum; computer calculations R. S. Phillips and P. Sarnak, The spectrum of Fermat curves, Geom. Func. Anal. 1 (1991) 80--146. Spectral theory; trace formulas (e.g., that of Selberg), Special algebraic curves and curves of low genus, Spectral problems; spectral geometry; scattering theory on manifolds The spectrum of Fermat 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 this short note we discuss some aspects of what could be called algebraic spline geometry. We concentrate on the concept of generalized Stanley-Reisner rings, namely the rings \(C^r(\Delta)\) of piecewise polynomial \(r\)-smooth functions on a simplicial complex \(\Delta\) in \(\mathbb R^d\). It is well known that the geometric realization of the ordinary Stanley-Reisner ring (or face ring) \(C^0(\Delta)\) reflects the structure of the simplicial complex: the irreducible components, corresponding to the maximal faces, are linear and intersect each other transversally in the pattern of the simplicial complex. We believe that the geometrical realizations of the generalized Stanley-Reisner rings behave similarly, except that the irreducible components are no longer linear and that they intersect with the appropriate multiplicity. We formulate this as a conjecture, the local spline ring conjecture, and show that it indeed holds in two very simple cases. For more complex examples, where the results are still conjectural, see [\textit{N. Villamizar}, Algebraic geometry for splines. University of Oslo (PhD Thesis) (2012), Ch. 4] for the case \(d=2\), \(r=1\). algebraic spline geometry; Stanley-Reisner rings; simplicial complex; local spline ring conjecture Computer-aided design (modeling of curves and surfaces), Numerical computation using splines, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Computational aspects in algebraic geometry Algebraic spline geometry: some remarks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article considers noncommutative schemes due to Kapranov, by considering the topological space as the spectrum of an abelianization of the associative \(\mathbb C\)-algebra, and by replacing the local ring of regular functions by a particular completion of the \(\mathbb C\)-algebra.
Let \(A\) be an associative \(\mathbb C\)-algebra, and define the lower central series filtration by \(L_1(A)=A\), \(L_k(A)=[A,L_{k-1}(A)]\). The \textit{Lower central series ideals} are \(M_k(A)=AL_k(A)A=AL_k(A)\), and \(A\) is called \textit{NC-complete} if it is complete in the \(M_k(A)\)-topology.
Let \(A\) be NC-complete with abelianization \(\pi:A\rightarrow A_{\mathrm{ab}}\). Let \(S=\pi^{-1}(\overline S)\subset A\) be the multiplicative subset corresponding to a multiplicative subset \(\overline{S}\subset A_{\mathrm{ab}}\). The \textit{stalk} of \(A\) at \(x\in\mathrm{Spec}(A_{\mathrm{ab}})\) is the direct limit of all localizations \(A[S^{-1}]\) where \(\overline{S}\subset A_{\mathrm{ab}}\) runs through the multiplicative subsets of functions not vanishing at \(x\). An NC-complete algebra is called \textit{NC-smooth of dimension} \(n\) if all its completed stalks are isomorphic to \(\hat{A}_n=\langle\langle x_1,\dots,x_n\rangle\rangle.\) An \textit{affine NC-manifold} is a pair \((\mathrm{Spec}(A_{\mathrm{ab}}),A)\) of a smooth affine variety and a NC-smooth algebra \(A\) abelianizing to it.
An \textit{NC-manifold} is a smooth variety \(X\) of dimension \(n\) and a sheaf \(\mathcal A\) of associative \(\mathbb C\)-algebras which is locally an affine NC-manifold of dimension \(n\). Finally, an \textit{NC-thickening} of a smooth variety \(X\) is a sheaf of algebras \(\mathcal A\) such that \((X,\mathcal A)\) is an NC-manifold. This defines a category \(\mathrm{NC-Th}_X\).
This article builds on earlier work by Jordan, Kapranov, Polishchuk, Tu and the author which studied NC-manifolds via commutative algebraic geometry on the abelianization. Kapranov's construction of NC-thickenings is not functorial, and so does not give such on non-affine \(X\). In this paper, a new geometric criterion for the existence of NC-thickenings are given by developing an analogue of formal algebraic geometry for NC-manifolds.
The article defines \textit{the bundle of coordinate systems} \(\mathcal M\) of \(X\). It is defined by its fibre over \(x\in X\) as the space of isomorphisms between the formal neighbourhood of \(x\) and the abstract formal disk \(\mathrm{Spec }\hat{\mathcal O}_n=\mathbb C[[x_1,\dots,x_n]]\). This is a principal bundle for the pro-algebraic group \(G_n^+\) of augmented algebra automorphisms of \(\hat{\mathcal O}_n\). \(G_n\) denotes the full group of algebra automorphisms.
The authors reformulate the theory of NC-manifolds to \textit{noncommutative coordinate systems}. These are principal bundles for \(H_n^+=\mathrm{Aut}_{\mathrm{aug}}\hat{A}\), the augmented automorphisms of the formal noncommutative disk arising from NC-thickenings as frame bundles, and form a category \(\mathrm{NC-Coord}_X\). The \textit{Gelfand-Kazhdan structure} is a splitting of the Atiyah sequence on \(\mathcal M\), valued in \(\mathrm{Lie}(G_n)\), the full group of automorphisms of \(\hat{\mathcal O}_n\). This structure is essential in the classical formal geometry, and the authors define a \textit{noncommutative Gelfand-Kazhdan structure} on \(\mathcal M\).
It is proved that the categories \(\mathrm{NC-Th}_X\) and \(\mathrm{NC-Coord}_X\) is equivalent, \textit{NC-connections} is defined, and then it is proved that the category of NC-thickenings \(\mathrm{NC-Th}_X\) is equivalent to NC-connections \(\mathrm{NC-Conn}_X\). Also an equivalence of \(\mathrm{NC-Coord}_X\) and \(\mathrm{NC-Conn}_X\) is a main result making the different categories under consideration equivalent.
Now, to prove the above, the author studies the corresponding graded and formal associative algebras. Kapranov constructed an isomorphism between a natural enhancement of the formal neighbourhood of the diagonal in \(X\times X\) and a natural enhancement of the formal neighbourhood of the zero section in the tangent bundle, giving an \(L_\infty\)-structure on \(\mathcal T_X[-1]\) by the geometry of the bundle of coordinate systems on \(X\). In an earlier text, the authors recover the \textit{structure sheaf} \(\mathcal A\) of an NC-manifold from the kernel of a collection of structure maps on \(\hat{T}_{\mathcal O}\Omega^1\), and in the present article, he proves how to construct the structure maps from the bundle of NC-coordinate systems associated with \(\mathcal A\). It is noted that from the perspective of Koszul duality, the paper can be viewed as the case of manifolds over the associative operad. It is also reasonable that this theory leads to a connection between Gelfand-Kazhdan structures and Koszul duality of operads.
The first parts of the article gives the necessary definitions and ideas in a nice and readable way. The final parts of the article which contains the proofs of the main results mentioned above, depends heavily on a good understanding of the previous articles referred in the text, so is more demanding for the reader. However, the article is a big step forward in unifying the NC-geometry. central series filtration; formal geometry; non-affine variety; NC-thickening; NC-connections; bundle of coordinate systems; Gelfand-Kazhdan structure; de Rham space; noncommutative formal geometry; noncommutative disk Noncommutative algebraic geometry, Classical groups (algebro-geometric aspects), Relevant commutative algebra, Differential graded algebras and applications (associative algebraic aspects) Formal geometry for noncommutative 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 authors consider functions from homogeneous spaces of functions defined on a locally compact Abelian group. The notion of Beurling spectrum, or essential spectrum, of functions is introduced. If a continuous unitary character is an essential point of the spectrum of a function, then it is the c-limit of a linear combination of shifts of the function in question. The notion of a slowly varying function at infinity is introduced, and properties of such functions are considered. The authors consider the Cauchy problem
\[
\frac{\partial x}{\partial t}=\Delta x, \qquad x(0,s)=x_0(s), \;s\in {\mathbb R}^n,
\]
with initial function \(x_0\) from a homogeneous space. It is proved that the weak solution as a function of the first argument is a slowly varying function at infinity. Beurling spectrum of a function; locally compact Abelian group; parabolic equation; continuous unitary character; Banach space; Fourier transform; Banach module; directed set; Stepanov set Žikov, V.V., Tjurin, V.M.: The invertibility of the operator \(d/dt+A(t)\) in the space of bounded functions. Mat. Zametki, \textbf{19}, 99-104 (1976). English translation: Math. Notes \textbf{19}(1-2), 58-61 (1976) Analysis on specific locally compact and other abelian groups, Homogeneous spaces and generalizations, Initial value problems for second-order parabolic equations Beurling's theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe an algorithm that associates to each positive real number \(\epsilon\) and each finite collection \(C_\epsilon\) of planar pixels of size \(\epsilon\) a planar piecewise linear set \(S_\epsilon\) with the following property: If \(C_\epsilon\) is the collection of pixels of size \(\epsilon\) that touch a given compact semialgebraic set \(S\), then the normal cycle of \(S_\epsilon\) converges in the sense of currents to the normal cycle of \(S\). In particular, in the limit we can recover the homotopy type of \(S\) and its geometric invariants such as area, perimeter and curvature measures. At its core, this algorithm is a discretization of stratified Morse theory. semialgebraic sets; pixelations; normal cycle; total curvature; Morse theory Curves in Euclidean and related spaces, Integral geometry, Stratified sets, Semialgebraic sets and related spaces, Pattern recognition, speech recognition Pixelations of planar semialgebraic sets and shape recognition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems To make a desingularization of a scheme \(X\), the classical way now is to define an upper-semicontinuous function \(\Phi\) defined on the singular locus of \(X\), with value in an ordered set \(E\). \(\Phi\) needs to have three properties:
(1) The set \(Y\) of points where \(\Phi\) is maximal is regular.
(2) If you blow-up \(Y\) on \(X'\), the strict transform of \(X\),
\[
\sup\bigl\{\Phi(x'):x'\in X'\bigr\}<{\sup\bigl\{\Phi(x):x\in X\bigr\}}=\sup\bigl\{\Phi(x):x\in Y\bigr\}.
\]
(3) There is no infinite strictly decreasing sequence of elements of \(E\).
That is what the author did in a previous article [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], in the case of characteristic 0. There were still some open questions:
\(Q_ 1\): Does the algorithm commute under smooth morphisms?
\(Q_ 2\): Can he lift on his desingularization of \(X\) any action of group of isomorphisms acting on \(X\)?
In the article under review, the author introduces the notions of trees, which are concatenations of blowing-ups and smooth morphisms, and of groves which are approximatively ``sheaves of trees''. Groves generalize the idealistic exponents used to make desingularization. The main result in this paper is that the function \(\Phi\) used in the previous paper cited above can be defined only with the set of groves of \(X\) which are geometrically constructed to control the composition of blowing-ups and smooth morphisms. This is a very strong result: it implies that the function \(\Phi\) is canonical and commutes under any smooth morphism, i.e., if \(f:X_ 1\to X_ 2\) is a smooth morphism, for any point \(x\in\text{Sing}(X_ 1)\), \(\Phi(x)=\Phi\bigl(f(x)\bigr)\). As a corollary, the author's answers \(Q_ 1\) and \(Q_ 2\).
At the end of the paper, two examples are completely exposed. patching; desingularization of a scheme; trees; groves; blowing-ups Villamayor U., O. E., Patching local uniformizations, Ann. Sci. École Norm. Sup. (4), 25, 6, 629-677, (1992) Global theory and resolution of singularities (algebro-geometric aspects) Patching local uniformizations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a log scheme locally of finite type over \(\mathbb{C}\), a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over \(\mathbb{C}\), another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over \(\mathbb{C}\), these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite étale homotopy type of its infinite root stack. log scheme; Kato-Nakayama space; root stack; profinite spaces; infinity category; étale homotopy type; topological stack Carchedi, D.; Scherotzke, S.; Sibilla, N.; Talpo, M., Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes, Geom. Topol., 21, 5, 3093-3158, (2017) Homotopy theory and fundamental groups in algebraic geometry, Localization and completion in homotopy theory, Abstract and axiomatic homotopy theory in algebraic topology, Generalizations (algebraic spaces, stacks), Limits, profinite groups Kato-Nakayama spaces, infinite root stacks and the profinite homotopy type of log 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 O-minimal structures are a generalization of the category of semialgebraic sets and functions. As a consequence of the axioms of o-minimality, every set which is definable in an o-minimal structure expanding the field of real numbers, can be partitioned into finitely many \(C^{k}\) cells, where \(k \geq 0\) is arbitrary [see \textit{L. van den Dries}, Tame Topology and o-minimal structures. London Mathematical Society Lecture Note Series. 248. (Cambridge): Cambridge University Press. (1998; Zbl 0953.03045)]. For a long time all known examples of o-minimal structures on the real field allowed actually analytic cell decomposition. Then the first author of the article under review together with \textit{P. Speissegger} and \textit{A. J. Wilkie} [J. Am. Math. Soc. 16, No. 4, 751--777 (2003; Zbl 1095.26018)] established an o-minimal structure on the real field which admits \(C^{\infty}\) but not \(C^{\omega}\) cell decomposition.
In the present nice short article the authors show now the following
Theorem. There is an o-minimal structure expanding the real field which does not admit \(C^{\infty}\) cell decomposition.
This theorem shows once more that o-minimal structures, although inspired by semialgebraic geometry and sharing many nice toplogical and geometric properties with it, go far beyound the category of semialgebraic sets and functions. To obtain the theorem the authors construct a function \(H\) on the real line which is \(C^{\infty}\) on the real line but not on any neighbourhood of 0 and which is piecewise given on the complement of every neighbourhood of 0 by finitely many polynomials. Moreover, the function \(H\) has the following additional important property: the smallest algebras of functions which contain the polynomials and in dimension 1 the function \(H\) and which are (roughly spoken) closed under taking partial derivatives in 0, composition and implicit functions, are quasianalytic; i.e. the map given by applying the Taylor expansion in 0, is injective. The authors realize this property with a nice trick. They construct the function \(H\) in such a way that the sequence of coefficients of the Taylor expansion of \(H\) has arbitrary high transcendence degree. Finally, the methods of the above paper of Rolin, Speissegger and Wilkie are applied to show that the function \(H\) generates an o-minimal structure on the real field and the theorem is proven. o-minimal structures; cell decomposition; quasianalyticity \beginbarticle \bauthor\binitsO. \bparticleLe \bsnmGal and \bauthor\binitsJ.-P. \bsnmRolin, \batitleUne structure o-minimale sans décomposition cellulaire \(C^\infty\), \bjtitleC. R. Math. Acad. Sci. Paris \bvolume346 (\byear2008), page 309-\blpage312. \endbarticle \endbibitem Real-analytic and semi-analytic sets, Model theory of ordered structures; o-minimality, Constructive real analysis An o-minimal structure which does not admit \(\mathcal C^{\infty}\) cellular decomposition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a Noetherian scheme. F. Morel and V. Voevodsky have introduced the homotopy category \({\mathcal H}(S)\) and the stable homotopy category \({\mathcal S}{\mathcal H}(S)\). On \({\mathcal S}{\mathcal H}(S)\) a symmetric monoidal structure can be introduced by using symmetric spectra as in [\textit{J. F. Jardine}, Doc. Math., J. DMV 5, 445--553 (2000; Zbl 0969.19004)]. Due to the fact that the above construction uses some particular properties of Nisnevich topology it is not a priori clear how to apply the method to the étale situation.
In this article the author gives a construction of the stable homotopy category \({\mathcal S}{\mathcal H}^T({\mathcal S},I)\), where \({\mathcal S}\) is a site, \(I\) a simplicial pre-sheaf on the site and \(T\) a simplicial pointed presheaf. If \(T\) is a suspension then \({\mathcal S}{\mathcal H}^T({\mathcal S},I)\) is naturally a triangulated category. The construction given in this article allows to introduce an étale version of \({\mathcal S}{\mathcal H}(S)\) defined as
\[
{\mathcal S}{\mathcal H}_{\text{ét}}(S)={\mathcal S}{\mathcal H}^{{\mathbb{P}}^1}(Sm/S_{\text{ét}},{\mathbb{A}}^1),
\]
where \(Sm/S_{\text{ét}}(Sm/S_{Nis})\) is the site given by the category \(Sm/S\) of smooth separated schemes of finite type on \(S\) equipped with the étale (Nisnevich) topology. Here \({\mathbb{P}}^1\) is pointed by \(\infty\). The morphism of sites \(\alpha: Sm/S_{\text{ét}}\to Sm/S_{Nis}\) induces adjoint triangulated functors \(R\alpha_*:{\mathcal S}{\mathcal H}_{\text{ét}}(S)\to{\mathcal S}{\mathcal H}(S)\) and \(L\alpha^*:{\mathcal S}{\mathcal H}(S)\to{\mathcal S}{\mathcal H}_{\text{ét}}(S)\). In particular \(R\alpha_*\) identifies \({\mathcal S}{\mathcal H}_{\text{ét}}(S)\) with a triangulated sub-category of \({\mathcal S}{\mathcal H}(S)\).
In this context some of the results given by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] can be formulated in terms of distinguished triangles. In particular one has the following
Theorem 1. Let \(S\) be a noetherian scheme \(X\in Sm/S\) and let \(U\) be an open subset of \(X\), \(p: V\to X\) an étale morphism inducing an isomorphim over the closed subset \(X- U\). Then there i.s a Mayer-Vietoris distinguished triangle
\[
p^{-1}(U)_+\to U_+\oplus V_+\to X_+\to p^{-1}(U)_+[1].
\]
If \(i: Z\to X\) is a closed immersion in \(Sm/S\) and \(p: X_Z\to X\) is the blow-up of \(X\) at \(Z\) then there is a distinguished triangle
\[
p^{-1}(Z)_+\to(X_Z)_+\oplus Z_+\to X_+\to p^{-1}(Z)_+[1].
\]
\(\mathbb{A}^1\)-homotopy theory; simplicial sheaves; spectra; triangulated category Riou, J., Catégorie homotopique stable d'un site suspendu avec intervalle, Bull. Soc. Math. France, 135, 495-547, (2007) Étale and other Grothendieck topologies and (co)homologies, Motivic cohomology; motivic homotopy theory, Grothendieck topologies and Grothendieck topoi, Simplicial sets, simplicial objects (in a category) [See also 55U10], Nonabelian homotopical algebra, Stable homotopy theory, spectra The stable homotopy category of a hanging site with interval | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From the introduction: These notes are an expanded version of the lectures given in the frame of the I.C.T.P. School held at Alexandria in Egypt from 12 to 24 November 2007.
Our purpose in this course was to give a survey of the various aspects -- algebraic, analytic and formal -- of the functional equations which are satisfied by the powers \(f^s\) of a function \(f\) and involve a polynomial in one variable \(b_f(s)\), called the Bernstein-Sato polynomial of \(f\). Since this course is intended to be useful for newcomers to the subject, we give enough significant details and examples in the most basic sections, which are the Sections 1, 2, and also 4. The latter is devoted to the calculation of the Bernstein-Sato polynomial or the basic example of quasi-homogeneous polynomials with isolated singularities. This case undoubtedly serve as a guide in the first developments of the theory.
We particularly focus our attention on the problem of the meromorphic continuation of the distribution \(f^s_+\) in the real case, which in turn motivates the problem of the existence of these polynomials, without forgetting related questions like the Mellin transform and the division of distributions. See the content of Section 3. The question of the analytic continuation property was brought up as early as 1954 at the congress of Amsterdam by I. M. Gelfand. The meromorphic continuation was proved 15 years later independently by Atiyah, and I. N. Bernstein and S. I. Gelfand, who used the resolution of singularities. The existence of the functional equations proved by I. N. Bernstein in the polynomial case allowed him to give a simpler proof which does not use the resolution of singularities. His proof establishes at the same time a relationship between the poles of the continuation and the zeros of the Bernstein-Sato polynomial. The already known rationality of the poles gives a strong reason for conjecturing the famous result about the rationality of the zeros of the \(b\)-function which was proved by Kashiwara and Malgrange.
We also want to mention another source of interest for studying functional equations due to Mikio Sato. It concerns the case of the semi-invariants of prehomogeneous actions of an algebraic group, and especially of a reductive group. In the latter case the functional equation is of a very particular type and the name \(b\)-function, frequently employed as a shortcut for the Bernstein-Sato polynomial, comes from this theory. We give the central step of the proof of the existence theorem in the reductive case in Subsection 2.4. functional equations; Bernstein-Sato polynomials; \(b\)-function; isolated singularities M. Granger, Bernstein-Sato polynomials and functional equations, in Algebraic approach to differential equations, World Sci., Hackensack, NJ, 2010, 225-291. Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local complex singularities Bernstein-Sato polynomials and functional equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper concerns a Nullstellensatz for ideals in the ring \(\mathcal{E}(M)\) of real-valued smooth functions on a smooth manifold \(M.\) An ideal \( \mathfrak{a}\subset \mathcal{E}(M)\) is a Łojasiewicz ideal if \(\mathfrak{a} \) is finitely generated and there exists \(f\in \mathfrak{a}\) such that for any compact \(K\subset M\) there exist \(C>0\) and an integer \(m\) such that \( \left| f(x)\right| \geq C\mathrm{d}(x,\mathcal{Z}(\mathfrak{a}))^{m}\) where \( \mathcal{Z}(\mathfrak{a})\) is the zero-set of \(\mathfrak{a}\) and \(\mathrm{d}\) is the distance function. The main result is that if \(\mathfrak{a\subset }\mathcal{E }(M)\) is a Łojasiewicz ideal then for the ideal \(\mathcal{I(Z}(\mathfrak{a} ))\) of functions in \(\mathcal{E}(M)\) vanishing on \(\mathcal{Z}(\mathfrak{a})\) we have
\[
\mathcal{I(Z}(\mathfrak{a}))=\overline{^{L}\!\!\!\sqrt{\mathfrak{a}}},
\]
where the Łojasiewicz radical \({}^{L}\!\!\!\sqrt{\mathfrak{a}}\) is defined as \(^{L}\!\!\!\sqrt{\mathfrak{a}}:=\{g\in \mathcal{E}(M):\exists _{f\in \mathfrak{a}}\exists _{m\geq 1}f>g^{2m}\) on \(M\}\) and the closure is taken in the compact-open topology. As a corollary the authors obtain known results by \textit{J. Bochnak} [Topology 12, 417--424 (1973; Zbl 0282.58003)], \textit{W. A. Adkins} and \textit{J. V. Leahy} [Duke Math. J. 42, 707--716 (1975; Zbl 0357.46032); Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 60, 90--94 (1976; Zbl 0367.14003)] concerning the case \(\mathfrak{a}\) is generated by analytic functions. Nullstellensatz; Łojasiewicz ideal; smooth function; smooth manifold Acquistapace, Francesca; Broglia, Fabrizio; Nicoara, Andreea, A Nullstellensatz for Łojasiewicz ideals, Rev. Mat. Iberoam., 30, 4, 1479-1487, (2014) Real-analytic and semi-analytic sets, Local complex singularities, Real-analytic functions A Nullstellensatz for Łojasiewicz 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 authors consider a class of finite-dimensional integral systems having the weak Kowalevski-Painlevé property, which generalizes the notion of algebraic complete integrability. The authors allow that their real invariant \(N\)-dimensional Liouville tori are extended to \(N\)-dimensional complex analytic nonlinear subvarieties of Abelian tori of dimension \(g\geq n\), so-called strata. Here, the nonlinearity means that such subvarieties are not complex Abelian groups. Moreover, the authors require that on universal coverings of the Abelian tori there exists coordinates \(\varphi_1,\dots, \varphi_g\) such that under the Hamiltonian flow, \(\varphi_1,\dots, \varphi_N\) change linearly in time \(t\), whereas the rest of the coordinates are analytic functions of the former. Some model examples are considered, such as the hierarchy of integrable generalizations of the Hénon-Heiles and the Neumann systems. Kowalevski-Painlevé property; complete integrability; Liouville tori; Hamiltonian flow Abenda, S.; Fedorov, Y., On the weak Kowalevski--Painlevé property for hyperelliptically separable systems, Acta Appl. Math., 60, 137-178, (2000) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Jacobians, Prym varieties, Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion, Relationships between algebraic curves and integrable systems On the weak Kowalevski-Painlevé property for hyperelliptically separable 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 It is well known that if \( \Delta \subset {\mathbb R}^{n} \) is an \( n \)-dimensional lattice polytope, then \( \sum_{k\geq 0} |{k \Delta \cap \mathbb Z^{n}}| t^{k} = h^{*}_{\Delta}(t) / (1-t)^{d+1} \) for some polynomial \( h^{*}_{\Delta}(t) \). This polynomial is called the \( h^{*} \)-polynomial of \( \Delta \). The author proves that the number of \( n \)-dimensional lattice polytopes (up to unimodular transformation) with a given \( h^{*} \)-polynomial is a monotonically increasing function of \( n \) that eventually becomes constant.
We now describe the main theorem of the paper, which the author uses to prove the result above. Given an \( n \)-dimensional lattice polytope \( \Delta \subset \mathbb R^{n} \), define the \textit{pyramid} \( \Pi(\Delta) \subset \mathbb R^{n+1} \) over \( \Delta \) to be the convex hull of \( \Delta \times \{0\} \) and \( (0, \dotsc, 0, 1) \in \mathbb R^{n+1} \). Let \({\text{Vol}_{\mathbb N}}(\Delta) \) = \( n! {\text{vol}}(\Delta) \) be the normalized volume of \( \Delta \), where \( {\text{vol}} \) denotes the usual euclidean volume. The \textit{codegree} of \( \Delta \) is \( \min \{k \in \mathbb Z_{\geq 0} : {\text{Int}}(k \Delta) \cap \mathbb Z^{n} \neq \emptyset\} \), and the \textit{degree} of \( \Delta \) is defined by \( \deg \Delta = n+1 - {\text{codeg}} \Delta \).
The author proves that if the dimension \( n \) of a lattice polytope exceeds a certain bound depending only on its degree and normalized volume, then that polytope must be a pyramid over an \( (n-1) \)-dimensional polytope. More precisely, if \( \deg \Delta = d \), \( {\text{Vol}_{\mathbb N}}(\Delta) = V \), and
\[
\dim \Delta = n \geq 4d \binom{ 2d + V - 1 }{ 2d },
\]
then \( \Delta = \Pi(\Delta') \) for some \( (n-1) \)-dimensional lattice polytope \( \Delta' \). The proof of this result uses techniques from commutative algebra.
The author concludes by conjecturing that there is a bound on the normalized volume of a degree-\( d \) lattice polytope that depends only on the leading coefficient of its \( h^{*} \)-polynomial. This is known to be true in dimension \( n=2 \) by an inequality due to \textit{P. R. Scott} [Bull. Aust. Math. Soc. 15, 395--399 (1976; Zbl 0333.52002)]. The author further conjectures that Scott's inequality (expressed in terms of the \( h^{*} \)-polynomial) applies whenever the \( h^{*} \)-polynomial is quadratic. Lattice polytopes; Cohen-Macaulay rings Batyrev, Victor V.: Lattice polytopes with a given h\ast-polynomial. Contemp. math. 423, 1-10 (2006) Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Lattice polytopes with a given \(h^*\)-polynomial | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mu = (\mu_1,\ldots,\mu_k)\) be a strictly increasing sequence of integers with \(\mu_1 \in \{0,1\}\). Consider the set \(X_{n,\mu}\) of all points \((x_1,\ldots,x_n) \in F^n\), with \(n \geq k\) and \(F\) an algebraically closed field of characteristic~0, such that the matrix \((x_i^{\mu_j})\) has rank less than~\(k\). Clearly \(X_{n,\mu}\) is the variety given by the \(k \times k\) minors of the matrix \((x_i^{\mu_j})\). The author proves that the ideal \(I_{n,\mu}\) of polynomials vanishing on \(X_{n,\mu}\) is actually generated by these minors. In fact, he shows that the minors form a universal Gröbner basis for \(I_{n,\mu}\), i.e., a Gröbner basis with respect to any monomial ordering on \(F[x_1,\ldots,x_n]\).
The proof uses the concept of critical ideals. A homogeneous ideal \(I\) in a polynomial ring is said to be critical if any homogeneous ideal \(I'\) strictly containing \(I\) is of dimension or degree strictly less than the corresponding number of \(I\). It is shown that the initial ideal of the \(k \times k\) minors is critical, and that it has the same dimension and degree as the initial ideal of \(I_{n,\mu}\). In the final section of the paper the author points out some connections to hyperplane arrangements given by irreducible pseudo-reflection groups.
The paper is clearly written and contains many interesting ideas. Gröbner basis; determinantal ideal; hyperplane arrangements; critical ideals M. Domokos, ''Gröbner bases of certain determinantal ideals,'' Beiträge Algebra Geom., 40, No. 2, 479--493 (1999). Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Linkage, complete intersections and determinantal ideals, Determinantal varieties Gröbner bases of certain 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 The authors consider smooth, irreducible, real, projective, algebraic curves. Such a curve has three topological invariants, the number of connected components \(k\), the algebraic genus \(g\), and the separability character \(\epsilon\). The complexification maps the curves of genus \(g\) into the moduli space \(\mathcal{M}_g\). The real locus, which is the image of the map, \(\mathcal{M}_g^\mathbb{R}\), is covered by the subsets \(\mathcal{M}_g^{k,\epsilon}\) proceeding from curves with given \(g\), \(k\) and \(\epsilon\). The subsets \(\mathcal{M}_g^{k,\epsilon}\) and \(\mathcal{M}_g^{k',\epsilon'}\) overlap if there is a curve with two real forms of types \((k,\epsilon)\) and \((k',\epsilon')\).
Using the functorial equivalence between these curves and the connected compact Riemann surfaces, the paper is written under this latter language. Then, a symmetry of a Riemann surface \(X\) is an antiholomorphic involution \(\sigma\), which is separating if \(X\backslash \text{Fix}(\sigma)\) is disconnected, and non-separating otherwise. The set of fixed points of \(\sigma\) consists of \(k\) curves called ovals. Then \((k,\epsilon)\) is called the topological type of \(\sigma\), where \(\epsilon=+1\) or \(-1\) according to \(\sigma\) be separating or not.
The goal of the paper is to study the real nerve \(\mathcal{N}(g)\). This is a simplicial complex whose vertices are the topological types \((k,\epsilon)\). A sequence \(((k_0,\epsilon_0),\dots,(k_n,\epsilon_n))\) is an \(n\)-simplex in \(\mathcal{N}(g)\) if there exists a surface having \(n+1\) symmetries of types \((k_0,\epsilon_0),\dots ,(k_n,\epsilon_n)\). Some results are already known for \(\mathcal{N}(g)\), for instance it has \([(3g+4)/2]\) vertices, and it is connected.
The main results of the paper under review concern the dimension of \(\mathcal{N}(g)\) for even values of \(g\). They are obtained using the theory of NEC groups, which are discrete co-compact subgroups of the group of all isometries (allowing orientation-reversing) of the hyperbolic plane. First, concerning geometrical dimension, Theorem 3.5 proves that for even \(g \geq 2\), the geometrical dimension of \(\mathcal{N}(g)\) is \(3\). And second, the authors deal with the homological dimension, that is to say, the greatest \(n\) such that \(\text{H}_n(\mathcal{N}(g), \mathbb{Z}) \neq 0\). In Theorem 4.4 they prove that for even \(g \geq 6\), the homological dimnesion of \(\mathcal{N}(g)\) is \(3\). For the remaining even numbers, the authors obtain \(\mathcal{N}(2) = 0\), and they announce that \(\mathcal{N}(4) = 1\).
The final Section 5 of the paper is devoted to global geometric properties of \(\mathcal{N}(g)\). Necessary and sufficient conditions for the existence of a 3-simplex in \(\mathcal{N}(g)\) whose vertices come from non-separating commuting symmetries are obtained. Those conditions are determined in terms of the number of ovals of the symmetries.
All in all, the paper is appealing and very well-written, and the usage of NEC groups is made in a very transparent way. complex algebraic curve; real locus; real nerve Gromadzki, G; Kozłowska-Walania, E, On the real nerve of the moduli space of complex algebraic curves of even genus, Ill. J. Math., 55, 479-494, (2011) Compact Riemann surfaces and uniformization, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences On the real nerve of the moduli space of complex algebraic curves of even genus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathbb K\) be an algebraically closed field. Fix integers \(r,m,n\) with \(1 < r \leq m \leq n\). Let \(A\) be an \(m \times n\) matrix of indeterminates. Let \(D_{m,n}^r\) be the subscheme of \(\mathbb P^{mm-1}_{\mathbb K}\) defined by the \(r \times r\) determinants of \(A\), and let \(P_{m,n}^r\) be the subscheme defined by the \(r \times r\) permanents of \(A\). Let \(F_k(D_{m,n}^r)\) be the subscheme of the Grassmannian \(\mathrm{Gr}(k~+~1,mn)\) parametrizing those \(k\)-dimensional planes in \(\mathbb P^{mn-1}_{\mathbb K}\) that are contained in \(D_{m,n}^r\), and analogously define \(F_k(P_{m,n}^r)\). \(F_k(D_{m,n}^r)\) and \(F_k(P_{m,n}^r)\) are examples of Fano schemes. The authors characterize when these Fano schemes are smooth, irreducible and connected. They also give examples to show that the schemes are not necessarily reduced. They show that \(F_1(D_{m,n}^r)\) always has the expected dimension, and they describe its components. Finally, they use their results to give a detailed study of the case \(r = m = n = 3\) and \(1 \leq k \leq 5\). Fano schemes; determinantal varieties; permanent M. Chan and N. Ilten, \textit{Fano schemes of determinants and permanents}, Algebra Number Theory, 9 (2015), pp. 629--679, . Configurations and arrangements of linear subspaces, Parametrization (Chow and Hilbert schemes), Determinants, permanents, traces, other special matrix functions, Infinitesimal methods in algebraic geometry, Determinantal varieties, Fano varieties Fano schemes of determinants and permanents | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is a survey paper focussing on compactification of parametrization spaces of convex projective structures on closed aspherical \(n\)-manifolds with (Gromov) hyperbolic fundamental group and covered by Euclidean \(n\)-space.
The action of the mapping class group of \(M\) on the parametrization space of convex projective structures on \(M\) extends continuously to an action on the compactification. The construction is reminiscent of Morgan-Shalen's compactification of Teichmüller space.
The main tools are inspired by tropical geometry, in particular, the tropical semifield and Maslov dequantization. The parameter space turns out to be a real semi-algebraic set and Maslov dequantization is applied to it. The limiting object resulting from dequantization represents the behaviour of the semi-algebraic set near infinity. This is the so-called logarithmic limit set which can be glued to the semi-algebraic set, compactifying it.
Alternately, in tropical geometry, algebraic varieties degenerate to tropical varieties by Maslov dequantization. Thus elements of the logarithmic limit set of the parametrization space of convex projective structures on \(M\) can be interpreted as tropical projective structures. Tropical geometry; compactification; convex projective structure; parametrization space; Maslov dequantization General geometric structures on low-dimensional manifolds, Dequantization of real convex 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 This paper presents the computation of the Betti numbers of the coarse moduli spaces \(\overline M_{0,n}(\mathbb P^r, d)\) of genus \(0\), degree \(d\), \(n\)-marked stable maps to \(\mathbb P^r\).
In fact, the following more general situation is considered. First, form the \(q\)-generating series
\[
\mu=\sum_{n=0}^{\infty} \sum_{d=0}^{\infty} q^d \left[M_{0,n}(\mathbb P^r, d)\right], \text{ and } \bar \mu=\sum_{n=0}^{\infty} \sum_{d=0}^{\infty} q^d \left[\overline M_{0,n}(\mathbb P^r, d)\right]
\]
where the coefficients, denoted by brackets, are classes in the Grothendieck group of complex quasiprojective varieties with symmetric group actions. An explicit relationship between \(\mu\) and \(\bar \mu\) is determined, by means of a transformation generalizing the Legendre transform in the theory of symmetric functions. The Betti/Hodge numbers are obtained by applying the Serre characteristics to the formulas relating \(\mu\) and \(\bar \mu\). The argument requires knowledge of the Serre characteristics of the open moduli spaces \(M_{0,n}(\mathbb P^r, d)\) of maps with smooth domains; these Serre characteristics are also computed in the paper.
Without being comprehensive, let me also mention some related work. \textit{E. Getzler}'s paper [in: The moduli space of curves, Prog. Math. 129, 199--230 (1995; Zbl 0851.18005)] presents the similar computation of the \(S_n\)-equivariant Poincare polynomials of the moduli space \(\overline M_{0,n}\). Similarly, \textit{E. Getzler}'s preprint ``Mixed Hodge structures on configuration spaces'' [\url{arxiv:math.AG/9510018}] gives the details for the case of the Fulton-MacPherson spaces. See also \textit{Yu. I. Manin} [in: The moduli space of curves, Prog. Math. 129, 401--417 (1995; Zbl 0871.14022)]. Betti numbers; stable maps Getzler, E.; Pandharipande, R., \textit{the Betti numbers of \(\overline{\mbox{\mathcalM}}\)_{0, \textit{n}}(\textit{r}, \textit{d})}, J. Algebraic Geom., 15, 709-732, (2006) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Fine and coarse moduli spaces, Applications of methods of algebraic \(K\)-theory in algebraic geometry The Betti numbers of \(\overline{\mathcal M}_{0,n}(r,d)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathbb{R}^\times:=\mathbb{R}\setminus\{0\}\) and let \(F:(\mathbb{R}^\times)^n\to\mathbb{R}\) be a map defined by Laurent polynomials having \(n+\ell\) distinct monomials, none of which is constant. In the paper under review a method to approximate the real solutions of the fewnomial system \(F(x)=b\), for a given \(b\in\mathbb{R}^n\), is described.
Denote by \(S_F\subset\mathbb{R}^n\) the set of real solutions of the system \(F(x)=b\), and suppose that it is finite. To find approximations to all the points of \(S_F\), the authors rely on the notion of Gale duality for polynomial systems, which is a correspondence between systems of polynomials defining complete intersections in the torus and systems of master functions defining complete intersections on a complement of certain hyperplane arrangements (see [\textit{F. Bihan} and \textit{F. Sottile}, Ann. Inst. Fourier 58, No. 3, 877--891 (2008; Zbl 1243.14044)]).
Applying Gale duality, the authors obtain a dual Gale system \(G(y)=1\), defined by certain rational functions. Then approximations \(S_G^*\) of the real solutions \(S_G\) of the system \(G(y)=1\) in the positive chamber \(\Delta\subset\mathbb{R}^\ell\) of the hyperplane arrangement under consideration are computed using a Khovanskii-Rolle continuation algorithm ([\textit{D. J. Bates} and \textit{F. Sottile}, Found. Comput. Math. 11, No. 5, 563--587 (2011; Zbl 1231.14047)]). This method traces real curves connecting the real solutions of certain start system to the points of \(S_G^*\). Finally, these approximations are used to obtain approximations \(S_F^*\) of the points of the set \(S_F\).
As the number of curves traced by the Khovanskii-Rolle method is essentially bounded above by any fewnomial bound for real solutions, the authors obtain an improved fewnomial bound for the kind of systems under consideration. Then they describe the first and third steps of the method above, and comment on an implementation in the software package \texttt{galeDuality}. fewnomial; Gale duality; Descartes' rule; Khovanskii-Rolle continuation; polynomial system; real algebraic geometry Real algebraic sets, Toric varieties, Newton polyhedra, Okounkov bodies, Computational aspects in algebraic geometry, Numerical computation of solutions to systems of equations, Global methods, including homotopy approaches to the numerical solution of nonlinear equations Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \((X,D)\) be a proper semistable variety over a mixed characteristic char \(V\) (that is, étale locally \(X\) is étale over \(\mathrm{Spec}(V[x_1, \ldots, x_n, y_1, \ldots, y_m]/(x_1\ldots x_r - \pi)\) with \(\pi\) the uniformizer, and with \(D\) given by \(y_1\ldots y_s = 0\)). There is a variety of notions of ''\(p\)-adic local systems'' on \(U:=X - D\) with regularity conditions along \(D\) that are considered in this paper:
(1) \(\mathrm{MIC}(U_K/K)^{reg}\), the category of modules with integrable connection on \(U_K\), where \(K\) is the fraction field of \(V\), with regular singularities along \(D_K\),
(2) \(\mathrm{MIC}((X_K, U_K)/K)^{lf, \Sigma}\), the category of locally free modules with integrable connection on \(X_K\) with logarithmic poles along \(D_K\) and exponents in \(\Sigma\) (see below),
(3) \(I_{\text{conv}}((X_k, M_k)/(\mathrm{Spf}(V), N))^{lf, \Sigma}\), the category of locally free convergent log isocrystals on \(X_k\), where \(k\) is the residue field of \(V\), \(M\) is the log structure on \(X\) induced by \(U_K\), and \(N\) is the standard log structure on \(\mathrm{Spf}(V)\), with exponents in \(\Sigma\),
(4) \(I^\dagger((X_k, U_k)/\mathrm{Spf}(V))^{\log, \Sigma}\), the category of overconvergent log isocrystals with \(\Sigma\)-unipotent monodromy along \(D_k\).
In (2)-(3), \(\Sigma\) is a collection of sets of \(p\)-adic integers assigned to the irreducible components of \(D_k\) with non-integer, non-\(p\)-adically Liouville differences. The notion of \(\Sigma\)-unipotent monodromy has been defined by \textit{A. Shiho} [Math. Ann. 348, No. 2, 467--512 (2010; Zbl 1268.12005)].
The main result (Theorem 1) of the paper is the construction of a fully faithful ``algebrization'' functor \((4)\to (1)\), which is the composition of three functors \((4)\to (3)\to (2)\to (1)\). First, the author proves that the natural restriction functor \((3)\to (4)\) is an equivalence; \((4)\to (3)\) will be its inverse. This generalizes the main result of Shiho [loc.cit.] which deals with the case of good reduction. Second, she constructs the functor \((3)\to (2)\) using the infinitesimal site of the \(p\)-adic completion of \(X\). The functor \((2)\to (1)\) is the restriction functor, well-defined thanks to the theory of algebraic logarithmic extension of André and Baldassarri.
A functor \((1')\to (4)\), where \((1')\) is a certain full subcategory of (1), has been defined by \textit{G. Christol} and \textit{Z. Mebkhout} [Invent. Math. 143, No. 3, 629--672 (2001; Zbl 1078.12501)], in case \(X\) is a smooth curve. Theorem 1 can be thought of as a vast generalization of this result, although the article lacks an intrinsic definition of the essential image \((1')\) of \((1)\to (4)\).
The paper is well-written and recalls most of the relevant definitions rather than referring the reader to other sources, which definitely makes it easier to read. logarithmic extension; log overconvergent isocrystals; semistable reduction; modules with integrable connection Di Proietto, V., On \(p\)-adic differential equations on semistable varieties, Math. Z., 274, 1047-1091, (2013) \(p\)-adic differential equations, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry On \(p\)-adic differential equations on semistable 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 Throughout the paper the base field is the filed of complex numbers \({\mathbb{C}}\).
An affine homology \(n\)-cell is a smooth affine \(n\)-fold \(U\) such that \(H^i(U, {\mathbb{Z}}) = 0\) for \( i > 0\). In this paper, the author considers the following problem:
Problem 1.1. Let \(f : X \rightarrow C\) be an extremal contraction of relative Picard number one from a smooth projective \( n\)-fold \(X\) to a smooth projective curve \(C\). Let \(U \subset X\) be an open subscheme.
(1) If \(U\) is an affine homology \(n\)-cell, then is it isomorphic to \({\mathbb{A}}^n\)?
(2) If \(U\) is isomorphic to \({\mathbb{A}}^n\), then can we construct an explicit birational map from \(X\) to a compactification of \({\mathbb{A}}^n\) with Picard number one preserving \(U \cong {\mathbb{A}}^n\)?
The answer to Problem 1.1 (1) is negative even when \(n=2\). So more condition should be added to it.
Definition 1.1. Let \(f : X \rightarrow C\) and \(U\) be as in Problem 1.1. Let \(D := X \backslash U\) be the boundary divisor. \( (X, D, f)\) is a compactification of \(U\) compatible with \(f\) if \(D\) contains a \(f\)-fiber. When \(D_f \subset D\) is a \(f\)-fiber and \( D_h \subset D\) is the other components, then \((X, D_h, D_f )\) is called a compactification of \(U\) compatible with \(f\).
A quadric fibration \(f : X \rightarrow C\) is an extremal contraction onto a curve \( C\) of relative Picard number one and of degree eight.
Problem 1.1 (1) is answered by the following theorem:
Theorem 1.3. Let \(Q\) be a smooth projective 3-fold and \( q : Q \rightarrow C\) a quadric fibration. Let \(D_h\) be a reduced effective divisor on \(Q\), and \(D_f\) a \(q\)-fiber. Then the following are equivalent.
(1) The complement \(Q \backslash (D_h \cup D_f )\) is an affine homology \(3\)-cell.
(2) It holds that \(C \cong {\mathbb{P}}^1\) and \(Q \backslash (D_h \cup D_f )\cong {\mathbb{A}}^3\).
In other two main theorems, the author proves that all such compactifications of affine homology 3-cells into quadric fibrations such that the boundary divisors contain fibers can be connected by explicit elementary links preserving \({\mathbb{A}}^3\) to the projective \(3\)-space \({\mathbb{P}}^3\). The main technique used in the proofs of theorems is the technique of elementary links. affine homology 3-cells; compactifications; quadric fibrations Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Minimal model program (Mori theory, extremal rays), \(3\)-folds On compactifications of affine homology 3-cells into quadric fibrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of \(\mathrm{SL}_r\) webs, and is built upon the \(r\)-fold dimer model on the network. When \(r\) is 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When \(r\) is 2 or 3, it is a reformulation of the \(\mathrm{SL}_2\) and \(\mathrm{SL}_3\) web immanants defined by the second author [J. Lond. Math. Soc., II. Ser. 92, No. 3, 633--656 (2015; Zbl 1328.05196)]. The basic result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of \(\mathrm{SL}_r\) webs, reproving a result of \textit{S. Cautis} et al. [Math. Ann. 360, No. 1-2, 351--390 (2014; Zbl 1387.17027)]. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity. dimer; web; boundary measurement; positroid; Grassmannian Fraser, Chris; Lam, Thomas; Le, Ian, From dimers to webs, (2017), preprint Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds From dimers to tensor invariants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the present note the authors study geometric criteria for the finite generation of the Cox rings of rational projective surfaces. Let us recall the following notions which are now specify to the case of surfaces. Let \(X\) be a smooth projective surface over an algebraically closed field \(k\). Then the Cox ring of \(X\) is defined as the \(k\)-algebra
\[
\text{Cox}(X) = \bigoplus_{(n_{1},\dots, n_{r}) \in \mathbb{Z}^{r}} H^{0}(X, \mathcal{O}(L_{1}^{n_{1}}) \otimes \cdots \otimes \mathcal{O}(L_{r}^{n_{r}})),
\]
where \(L_{1},\dots,L_{r}\) is a basis of the \(\mathbb{Z}\)-module \(\mathrm{Pic}(X)\) of classes of invertible sheaves on \(X\) modulo isomorphisms under the tensor product. Moreover, we assume that the linear and numerical equivalences on the group of Cartier divisors on \(X\) are the same (this is for instance the case of smooth rational projective surfaces). Denote by \(K_{X}\) the canonical divisor of \(X\) and by \(NS(X)\) the Neron-Severi group. The set of all effective elements in \(NS(X)\) is denoted by \(M(X)\). Since \(M(X)\) has an algebraic structure of a monoid, thus \(M(X)\) is called the effective monoid of \(X\).
The main result of the paper is the following criterion.
{ Theorem 1.}
Let \(X\) be a smooth projective rational surface defined over an algebraically closed field \(k\) of arbitrary characteristic such that the invertible sheaf associated to \(-K_{X}\) has a non-zero global section. Then the following conditions are equivalent: {\parindent=6mm \begin{itemize} \item[a)] \(\text{Cox}(X)\) is finitely generated, \item [b)] \(M(X)\) is finitely generated, \item [c)] \(X\) has only a finite number of \((-1)\)-curves and only a finite number of \((-2)\)-curves.
\end{itemize}} The second result of the paper provides another criterion.
{Theorem 2.}
Let \(X\) be a smooth projective rational surface defined over an algebraically closed filed \(k\) of arbitrary characteristic such that the invertible sheaf associated to the divisor \(-rK_{X}\) has at least two linearly independent sections for a certain positive integer \(r\). The following conditions are equivalent: {\parindent=6mm \begin{itemize} \item[a)] \(\text{Cox}(X)\) is finitely generated, \item [b)] the set of smooth projective rational curves of self-intersection \(-1\) on \(X\) is finite.
\end{itemize}} Cox rings; rational surfaces; effective monoids De La Rosa Navarro, B.L., Frías Medina, J.B., Lahyane, M., Moreno Mejía, I., Osuna Castro, O.: A geometric criterion for the finite generation of the Cox ring of projective surfaces. Rev. Mat. Iberoam. 31(4), 1131-1140 (2015) Divisors, linear systems, invertible sheaves, Picard groups, Riemann-Roch theorems A geometric criterion for the finite generation of the Cox rings of projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(H^N\) be the Hilbert scheme of \(N\) points in the projective plane. \textit{G. Gotzmann} [Math. Z. 199, 539--547 (1988; Zbl 0637.14003)] observed that \(H^N\) is naturally stratified by the Hilbert function, and he showed the irreducibility and dimension of the strata. The current paper, instead, studies the nested Hilbert scheme \(H_{N-i,N}\) parameterizing pairs \((Z,W)\) such that \(Z \in H^{N-i}, W \in H^{N}\) and \(Z \subset W\). In the case \(i=1\) this is known to be smooth and irreducible. In this paper, \(H_{N-1,N}\) is shown to again be stratified by irreducible subvarieties of the type \(H_{\phi, \psi} = \{ (Z,W) \in H_{N-1,N} | Z \in H_\phi, W \in H_\psi \}\), where \(H_\phi\) is the locally closed subscheme of the Hilbert scheme parameterizing zero-dimensional schemes with Hilbert function \(\phi\). The dimension of these strata is also computed.
On the other hand, when \(i>1\), it is shown that \(H_{N-i,N}\) is never smooth, and the corresponding strata may be reducible. However, if the Hilbert functions \(\phi\) and \(\psi\) are very close to each other (in a sense made precise in the paper), then the strata are irreducible, and the dimensions are computed. The authors also show that \(H_{N-2,N}\) is irreducible.
An important ingredient in the proof is the classical notion that algebraic liaison can be used (for codimension two arithmetically Cohen-Macaulay subschemes) to pass from any scheme to a simpler one. Here the authors reduce a statement about a specific nested scheme \(H_{\phi,\psi}\) to a corresponding one about a ``simpler'' nested scheme \(H_{\psi^*,\phi^*}\), via liaison. As a byproduct they find a new proof of Gotzmann's results. The results of this paper are motivated by the authors' study of globally generated and very ample Hilbert functions [Math. Z. 245, 155--181 (2003; Zbl 1079.14057)], and has applications to the classification of globally generated line bundles on the blow-up of \(\mathbb P^2\) at points, and to the study of Cayley-Bacharach schemes in \(\mathbb P^2\). globally generated line bundles; Cayley-Bacharach schemes; Hilbert function Parametrization (Chow and Hilbert schemes), Linkage, Projective techniques in algebraic geometry On the stratification of nested Hilbert schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be a smooth complex projective ruled surface and let L be a very ample line bundle on X, with genus \(g=g(L)\). The paper under review is concerned with the problem of classifying pairs (X,L) as above with a fixed g, when X is irregular. The authors look at the expression of L with respect to a suitable free basis of Num(X) and provide good estimates for the coefficients. This allows them to improve the classification obtained by the second author for \(g\leq 6\) and extend it to \(g=7.\)
Reviewer's remark. At the time when the paper was written the technique of iterating the adjunction process, developed by the authors for studying rational surfaces [Indiana Univ. Math. J. 36, 167-188 (1987; Zbl 0618.14015)], was not suitable for the irrational case. Subsequently, after complete results on the adjunction process have been allowable, the authors obtained more refined results [see Manuscr. Math. 64, No.1, 35-54 (1989)] by using the iteration technique also in the irrational case. sectional genus; classification of irregular ruled surfaces; very ample line bundle; iterating the adjunction process Biancofiore A., Livorni E.L.:Algebraic ruled surfaces with low sectional genus.Ricerche di Matematica.Vol.XXXVI,fasc.1{\(\deg\)} (1987) Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Moduli, classification: analytic theory; relations with modular forms Algebraic ruled surfaces with low sectional genus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be a \(d \times n\) integer matrix. Gel'fand et al. [4] proved that most \(A\)-hypergeometric systems have an interpretation as a Fourier-Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther [21] as the set of not strongly resonant parameters of \(A\). A similar statement relating \(A\)-hypergeometric systems to exceptional direct images was proved by Reichelt [16]. In this article, we consider a hybrid approach involving neighborhoods \(U\) of the torus of \(A\) and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated \(A\)-hypergeometric system is the inverse Fourier-Laplace transform of such a ``mixed Gauss-Manin'' system.
In order to describe which \(U\) work for such a parameter, we introduce the notions of fiber support and cofiber support of a \(D\)-module.
If the semigroup ring \(\mathbb{C} [\mathbb{N} A]\) is normal, we show that every \(A\)-hypergeometric system is ``mixed Gauss-Manin''. We also give an explicit description of the neighborhoods \(U\) which work for each parameter in terms of primitive integral support functions. algebraic geometry; commutative algebra; D-modules; local cohomology; affine semigroup rings; toric varieties; GKZ systems Structure of families (Picard-Lefschetz, monodromy, etc.), Toric varieties, Newton polyhedra, Okounkov bodies, Fibrations, degenerations in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials \(A\)-hypergeometric modules and Gauss-Manin systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over \(\mathbb{C}\) attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain \emph{motivic random variables}, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In the sequel, we find similar stability results in spaces of smooth hypersurface sections, providing new examples to be investigated through the lens of representation stability for symmetric, symplectic and orthogonal groups. representation stability; motivic stabilization; arithmetic statistics; configuration spaces; cohomological stability Applications of methods of algebraic \(K\)-theory in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Grothendieck groups (category-theoretic aspects), Discriminantal varieties and configuration spaces in algebraic topology Motivic random variables and representation stability. I: Configuration 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 \({\mathcal A}=\{H_1,\ldots H_n\}\) be an arrangement of complex hyperplanes, with complement \(M=M({\mathcal A})\). Let \(\mathcal L\) be a rank-one complex local system over \(M\). The local system \(\mathcal L\) is determined by its monodromy \(H^1(M) \to {\mathbb C}^*\), hence by a point \(t=(t_1,\ldots, t_n)\in ({\mathbb C}^*)^n.\)
This paper concerns the effect of isotopies of the arrangement \({\mathcal A}\) on the local system cohomology \(H^*(M,{\mathcal L}_t),\) in particular its dependence on \(t\).
Let \( B\) be a smooth component of the realization space of a simple matroid \(G\) on \(n\) points. Points \(b \in B\) give rise to arrangements of \(n\) labelled hyperplanes with the same underlying matroid \(G\), and with diffeomorphic complements. Fix \(t\in ({\mathbb C}^*)^n\) and let \((\lambda_1,\ldots,\lambda_n)\in {\mathbb C}^n\) with \(t_j=\text{exp}(-2\pi i \lambda_j)\). One obtains for each \(q\) a bundle of vector spaces over \( B\), whose fiber over \({\mathcal A}_b\) is the local system cohomology \(H^q(M({\mathcal A}_b),{\mathcal L}_t)\). The authors compute the monodromy of this bundle, and its infinitesimal generator, the Gauss-Manin connection. The main result is that the eigenvalues of the Gauss-Manin connection are integral linear combinations of the weights \(\lambda_j,\) confirming a conjecture of H.~Terao. The classical example arises from the family of discriminantal (Selberg-type) arrangements, parametrized by the configuration space \(b_k\) of \(k\) labelled points in the plane. In this case the Gauss-Manin connection was described by \textit{K. Aomoto} [J. Math. Soc. Japan 39, 191--208 (1987; Zbl 0619.32010)]. (N.B., In general, the realization space of \(G\) need not have any smooth components.)
Let \(\Lambda={\mathbb C}[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]\) denote the ring of Laurent polynomials. In previous work Cohen and Orlik used stratified Morse theory to construct a combinatorial cochain complex \((K^\bullet({\mathcal A}),\Delta(x))\) of \(\Lambda\)-modules and \(\Lambda\)-homomorphisms, such that specialization \({ x_j} \mapsto { t_j}\) yields a complex whose cohomology is \(H^*(M,{\mathcal L}_t).\) Here the authors develop a parametrized version of this construction. The universal \(\Lambda\)-complexes \((K^\bullet({\mathcal A}_b),\Delta(x))\) form a bundle over \( B\), with a universal monodromy operator \(\Phi^\bullet(x)\) acting by chain automorphisms on \((K^\bullet({\mathcal A}_{ b}),\Delta(x))\). The linearization at \(t={ 1}\) of \((K^\bullet({\mathcal A}_b),\Delta(x))\) is the Aomoto complex of \({\mathcal A}_b\), with ground ring \({\mathbb C}[y_1,\ldots, y_n].\) The linearization of \(\Phi^\bullet(x)\) acts on the Aomoto complex, inducing the Gauss-Manin connection upon specialization \(y_j\mapsto \lambda_j.\) Fixing \(\gamma\in \pi_1(b)\), an eigenvalue \(r(x)\) of \(\Phi^q(x)(\gamma)\) determines a nowhere zero analytic function \(({\mathbb C}^*)^n\to {\mathbb C}\) and thus is given by \(r(x)=x_1^{m_1}\cdots x_n^{m_n}\) for some \((m_1,\ldots, m_n)\in {\mathbb Z}^n.\) It follows that the eigenvalues of the Gauss-Manin connection are integral linear combinations of the weights \(\lambda_i\).
The constructions are illustrated with an extended example. Gauss-Manin connection; hyperplane arrangement; local system Cohen D, Orlik P. Gauss-Manin connections for arrangements, I. Eigenvalues. Compositio Math, 136:299--316 (2003) Relations with arrangements of hyperplanes, Structure of families (Picard-Lefschetz, monodromy, etc.), Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Homology with local coefficients, equivariant cohomology Gauss-Manin connections for arrangements. I: Eigenvalues | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a survey of the refinements of index theory associated with the local index formulas that arise from heat equation techniques. Local index formulas are expressed in terms of differential forms. Different perspectives on the same index problem may yield different representatives of the associated index class. Thus there are secondary invariants that transgress the index theorem to measure the difference between representative differential forms. Much of this paper is devoted to the eta forms that transgress the families index theorem of a fibration and to the (holomorphic) analytic torsion forms that refine the eta forms for a Kähler fibration. One perspective on the families index theorem involves the index bundle while the other perspective draws information from the geometry of the fibration's total space. Hence this approach to the families index theorem requires the full power of Quillen superconnections in an infinite-dimensional setting. Among the other topics discussed in this article are: the Atiyah-Patodi-Singer index theorem (including the approaches of J. Cheeger and R. Melrose); the families version of the Atiyah-Patodi-Singer theorem; adiabatic limits of secondary invariants; the geometry of determinant line bundles; analytic torsion currents associated with immersions; and the functoriality of analytic torsion forms and currents.
This article does not discuss the role of heat equation techniques in noncommutative geometry. It does mention the relation of the surveyed material with Arakelov geometry. Much of the material discussed in this paper first appeared in long papers by the author (some with co-authors). The author has also written other surveys with somewhat different emphases.
The present paper would be an excellent resource for a reader who has some familiarity with the field and who wants to learn how an expert views local index theory (with particular attention to its analytic aspects).
This paper is clearly written, informative, and self-contained in the sense that it contains definitions, theorems, sketches of proofs, and motivating comments.
The organization of the paper clearly communicates the logical structure of the subject. However a reader unfamiliar with the field might be overwhelmed by the quantity of information in this paper. Such a reader might get a gentler introduction to eta forms and (non-holomorphic) analytic torsion forms by reading \textit{S. Rosenberg} [Bull. Am. Math. Soc., New Ser. 34, No. 4, 423-433 (1997; Zbl 0886.58110)] and \textit{J. Lott} [in Connes, A. (ed.) et al., Quantum symmetries/ Symétries quantiques, pp. 947-955 (North-Holland, Amsterdam) (1998; Zbl 0941.58015)]. index theory; secondary invariants; eta forms; holomorphic torsion; analytic torsion; superconnection; determinant line bundle; Riemann-Roch theorem; survey Bismut, J.-M.: Local index theory, eta invariants and holomorphic torsion: a survey. Surveys in Differential Geometry, Vol. III, International Press, Boston, MA, pp 1--76 (1998) Index theory and related fixed-point theorems on manifolds, Determinants and determinant bundles, analytic torsion, Research exposition (monographs, survey articles) pertaining to global analysis, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Eta-invariants, Chern-Simons invariants, Heat and other parabolic equation methods for PDEs on manifolds, Characteristic classes and numbers in differential topology, Riemann-Roch theorems Local index theory, eta invariants and holomorphic torsion: A survey | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to determine the mean value of the Euler characteristic of an algebraic hypersurface defined by a random polynomial.
The vector space of homogeneous polynomials of degree \(m\) in \(n+1\) variables is denoted as \(H_m({\mathbb R}^{n+1})\). A random polynomial \(F \in H_m({\mathbb R}^{n+1})\) is a polynomial whose coefficients are random variables.
The orthogonal group, \(O(n+1)\), has a natural action on \(H_m({\mathbb R}^{n+1})\). A random polynomial \(F\) is said to be satisfactory if it has nontrivial \(O(n+1)\)-invariant normal distribution with zero mean. It is proved that the set \(V_F := \{F=0\}\) is a smooth hypersurface with probability 1 if \(F\) is satisfactory.
It can be shown that the following inequalities hold for a satisfactory random polynomial \(F \in H_m({\mathbb R}^{n+1})\):
\[
{1-(-1)^m \over 2} \leq r \leq {m(m+n-1) \over n}
\]
where \(r\) is the parameter of \(F\) defined as \(r= E({\partial F \over \partial x_1}(0))^2/EF(0)^2.\) -- The main result of the paper is the following:
If \(n\) is odd and \(F \in H_m({\mathbb R}^{n+1})\) is a satisfactory random polynomial of parameter \(r\) then the mean value of the Euler characteristic of \(V_F\) is given by \(M_n(r)\), where \(M_n(r) = {I_n(r^{1/2}) \over I_n(1)}\) and \(I_n(s) = \int_0^s(1-t^2)^{(n-1)/2} dt\). random polynomial; Euler characteristic; hypersurface; normal distribution S. Podkorytov, ''The mean value of the Euler characteristic of a random algebraic hypersurface,'' Algebra Analiz, 11, 185--193 (1999). Real algebraic sets, Group actions on varieties or schemes (quotients), Real-analytic and semi-analytic sets, Hypersurfaces and algebraic geometry The mean value of the Euler characteristic of an algebraic hypersurface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 reduced connected scheme and \(\mathcal{F}\) a vector bundle of rank \(m-1\) on \(X\). A multiple structure or more precisely a \(m\)-rope on \(X\) with conormal bundle \(\mathcal{F}\) is a scheme \(\overset{\sim}{X}\) with \(\overset{\sim}{X}_{red}=X\) such that \(\mathcal{I}^{2}_{X/\overset{\sim}{X}}=0\) and \(\mathcal{I}_{X/\overset{\sim}{X}}=\mathcal{F}\) as \(\mathcal{O}_{X}\)-modules, where \(\mathcal{I}_{X/\overset{\sim}{X}}\) is the ideal sheaf of the embedding of \(X\) in \( \overset{\sim}{X}\). A smoothing of multiple structures \(D\) is a flat family of schemes over a curve such that the central fibre is \(D\). It has been related to the deformations of morphisms in [\textit{F. J. Gallego} et al., Rev. Mat. Complut. 26, No. 1, 253--269 (2013; Zbl 1312.14009)] where Gallego et al. give a sufficient condition for a finite morphism to be deformed to an embedding. \newline In this paper the authors investigate the deformations of the composite of the universal covering map for an Enriques manifold \(Y\) of index \(d\) with an embedding of \(Y\) in a projective space \(\mathbb{P}^{N}\) using the infinitesimal condition described by Gallego et al.. The definition of an Enriques manifold used corresponds either to the one in the sense of [\textit{S. Boissière} et al., J. Math. Pures Appl. (9) 95, No. 5, 553--563 (2011; Zbl 1215.14046)] or [\textit{K. Oguiso} and \textit{S. Schröer}, J. Reine Angew. Math. 661, 215--235 (2011; Zbl 1272.14026)] depending on the index \(d\) and whether the universal cover of \(Y\) is a Calabi-Yau or a hyperkähler manifold. Given an embedding of an Enriques manifold \(Y\) of index \(d\) and dimension \(2n\) with canonical bundle \(K_{Y}\) in a large enough projective space, embedded \(d\)-ropes with conormal bundle \(\mathcal{E}=\displaystyle\bigoplus_{i=1}^{d-1}K_{Y}^{\otimes i}\) on \(Y\) are thoroughly studied and the dimension of the quasi-projective space parametrizing such \(d\)-ropes is computed. It turns out that \(d\)-ropes with conormal bundle \(\mathcal{E}\) appear naturally on Enriques manifolds of index \(d\) as flat limits of hyperkähler or Calabi-Yau manifolds. The main results on smoothing of such multiple structures are proven with the deformation results obtained in [\textit{F. J. Gallego} et al., Rev. Mat. Complut. 26, No. 1, 253--269 (2013; Zbl 1312.14009)]. These \(d\)-ropes on Enriques manifold are points in the Hilbert scheme of embedded Calabi-Yau or hyperkähler manifolds whose smoothness is proven for ropes with \(d=2\) as a consequence of smoothing. Moreover, in the Appendix the authors show that if an Enriques manifold \(Y\) of index \(d= 2\) whose universal cover is one of the known examples of hyperkähler six-folds is embedded inside \(\mathbb{P}^{N}\) then \(N\geq 13\).
Reviewer's remark: Small typos on the pages 1250 and 1253: ``will will'' should be ``we will'', and ``upto'' should be ``up to'' respectively. ropes; Enriques manifolds; hyper-Kähler manifolds; Calabi-Yau manifolds; smoothing of multiple structures; Hilbert scheme Holomorphic symplectic varieties, hyper-Kähler varieties, Calabi-Yau manifolds (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, Formal methods and deformations in algebraic geometry, Coverings in algebraic geometry, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Smoothing of multiple structures on embedded Enriques manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove the following theorem:
Let \(X\) be an \(n\)-dimensional algebraic variety and \(x\in X\) be a smooth point on \(X\). Then there is a Zariski open neighborhood \(U_x\subset X\) of \(x\) which is isomorphic to a closed smooth hypersurface in \(\mathbb{C}^{n+1}\).
In particular it implies that every \(n\)-dimensional smooth algebraic variety \(X\) can be covered by Zariski open subsets \(U_i\) which are isomorphic to closed smooth hypersurfaces in \(\mathbb{C}^{n+1}\).
As an application of the theorem above we give a characterization of components of the set \(S_f\) of points at which a polynomial mapping \(f:\mathbb{C}^n \to\mathbb{C}^m\) is not proper. Let us recall that \(f\) is not proper at a point \(y\) if there is no neighborhood \(U\) of \(y\) such that \(f^{-1} (\text{cl} (U))\) is compact. We showed [\textit{Z. Jelonek}, Math. Ann. 315, No. 1, 1-35 (1999; Zbl 0946.14039)] that the set \(S_f\) (if non-empty) has pure dimension \(n-1\) and it is \(\mathbb{C}\)-uniruled, i.e., for every point \(x\in S_f\) there is an affine parametric curve through this point. In this paper we show that, conversely, for every \(\mathbb{C}\)-uniruled \((n-1)\)-dimensional variety \(X \subset\mathbb{C}^m\) (where \(2\leq n\leq m)\), there is a generically-finite (even quasi-finite) polynomial mapping \(F:\mathbb{C}^n \to\mathbb{C}^m\) such that \(X\subset S_F\). This gives (together with the cited paper) a full characterization of irreducible components of the set \(S_f\). polynomial mapping Z. Jelonek. \textit{Local characterization of algebraic manifolds and characterization of components of the set \(S_f\).} Ann. Polon. Math. \textbf{75} (2000) 7--13 Local structure of morphisms in algebraic geometry: étale, flat, etc., Jacobian problem, Rational and birational maps, Coverings in algebraic geometry, Birational automorphisms, Cremona group and generalizations Local characterization of algebraic manifolds and characterization of components of the set \(S_f\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f : S^\prime \rightarrow S\) be a finite and faithfully flat morphism of locally noetherian schemes of constant rank \(n\) and let \(G\) be a smooth, commutative and quasi-projective \(S\)-group scheme with connected fibers. For every \(r \geq 1\), let \(\operatorname{Res}_G^{(r)} : H^r(S_{\text{ét}}, G) \rightarrow H^r(S_{\text{ét}}^\prime, G)\) and \(\operatorname{Cores}_G^{(r)} : H^r(S_{\text{ét}}^\prime, G) \rightarrow H^r(S_{\text{ét}}, G)\) be, respectively, the restriction and corestriction maps in étale cohomology induced by \(f\). For certain pairs \((f, G)\), we construct maps \(\alpha_r : \operatorname{Ker} \operatorname{Cores}_G^{(r)} \rightarrow \operatorname{Coker} \operatorname{Res}_G^{(r)}\) and \(\beta_r : \operatorname{Coker} \operatorname{Res}_G^{(r)} \rightarrow \operatorname{Ker} \operatorname{Cores}_G^{(r)}\) such that \(\alpha_r \circ \beta_r = \beta_r \circ \alpha_r = n\). In the simplest nontrivial case, namely when \(f\) is a quadratic Galois covering, we identify the kernel and cokernel of \(\beta_r\) with the kernel and cokernel of another map \(\operatorname{Coker} \operatorname{Cores}_G^{(r - 1)} \rightarrow \operatorname{Ker} \operatorname{Res}_G^{(r + 1)}\). We then discuss several applications, for example to the problem of comparing the (cohomological) Brauer group of a scheme \(S\) to that of a quadratic Galois cover \(S^\prime\) of \(S\). restriction map; Norm map; quadratic Galois cover; relative ideal class group; Tate-Shafarevich group; relative Brauer group Étale and other Grothendieck topologies and (co)homologies, Class numbers, class groups, discriminants, Arithmetic ground fields for abelian varieties, Brauer groups of schemes Cokernels of restriction maps and subgroups of norm one, with applications to quadratic Galois 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 Traditionally, uniformization theorems in geometry arise as an attempt to classify complicated structures by recasting them in terms of group actions on a small number of uniform geometries. For instance, every Riemann surface arises as the quotient by a discrete group of either the complex plane, the unit disc, or the Riemann sphere; or (for an example more closely related to the Langlands program) the moduli of bundles for a semisimple group on an algebraic curve can be realised as a quotient of the affine Grassmannian for the group.
Let \(E\) denote an complex elliptic curve. In this paper the authors construct non-abelian versions of the classical abelian analytic uniformizations \(E\simeq \mathbb{C}^\times / q^{\mathbb{Z}}\), \(E\simeq \mathbb{C}/(\mathbb{Z}\oplus\mathbb{Z}\tau)\) and \(\mathbb{C}^\times \simeq \mathbb{C}/\mathbb{Z}\). Specifically, let \(G\) be a semisimple simply-connected group, and denote by \(G_E\) the connected component of the trivial bundle in the stack of semistable \(G\)-bundles on \(E\). Then the authors show that as a complex analytic stack, \(G_E\) can be glued together from certain ``twisted'' adjoint quotients \(\mathfrak{g}_J^{se}/'G_J\), where the \(\mathfrak{g}_J\) are subalgebras of the holomorphic loop Lie algebra \(L_{hol}\mathfrak{g}\) and we take the quotient of an invariant subset of elements with ``small eigenvalues''. Similarly they show that as a complex analytic stack the adjoint quotient \(G/G\) can be glued together from twisted adjoint quotients \(G_J^{se}/'G_J\).
Using these descriptions, the authors prove that the \(\infty\)-categories of sheaves with nilpotent singular support \(\text{Sh}_{\mathcal{N}}(G/G)\) (i.e.\ character sheaves) and \(\text{Sh}_{\mathcal{N}}(G_E)\) can be obtained as limits of the corresponding categories of sheaves with nilpotent support on \(\mathfrak{g}_J/G_J\) and \(G_J/G_J\), respectively. In order to aid and orient the reader, the cases \(G=\mathrm{SL}_2\) and \(G=\mathrm{SL}_3\) are explicitly presented and clearly explained.
The results of this paper are strongly related to two areas of geometric representation theory. Firstly, they are related to Springer theory: there is a Fourier transform that decomposes any character sheaf into a system of compatible nilpotent orbital sheaves, i.e.\ we have that \(\text{Sh}_{\mathcal{N}}(G/G)\) is the limit of categories \(\text{Sh}(\mathcal{N}_{\mathfrak{g}_J} / G_J)\). The categories \(\text{Sh}(\mathcal{N}_{\mathfrak{g}_J} / G_J)\) are described by the generalized Springer correspondence, and allow the authors to construct a group \(\widehat{G}\) that realises a spectral description of character sheaves, in the sense that \(\text{Sh}_{\mathcal{N}}(G/G) \simeq \text{QCoh}(\widehat{G})\).
Secondly, they are related to the Betti Geometric Langlands program of Ben-Zvi and Nadler. The Betti Geometric Langlands conjecture proposes an equivalence of dg-categories \(\text{Sh}_{\mathcal{N}}(\text{Bun}_G(\Sigma)) \simeq \text{IndCoh}_{\check{\mathcal{N}}}(\text{LocSys}_{\check{G}}(\Sigma))\) for \(\Sigma\) a Riemann surface. The authors provide a recipe (some parts known, some parts conjectural) to construct an embedding of \(\text{Sh}_{\mathcal{N}}(G_E)\) into \(\text{IndCoh}_{\check{\mathcal{N}}}(\text{LocSys}_{\check{G}}(E))\), and thus prove a ``semistable part'' of the Betti Geometric Langlands conjecture for \(\Sigma = E\) an genus 1 curve. Moreover, the authors demonstrate the topological nature of the category \(\text{Sh}_{\mathcal{N}}(G_E)\) -- necessary if it is to play a role in the Betti Langlands conjecture -- by showing that as \(E\) varies one obtains a local system of \(\infty\)-categories over the moduli space of elliptic curves.
In order to prove the main results of this paper, the authors have the following key insights: (1) that they can construct analytic charts out of subspaces of small eigenvalues, invariant under a \textit{twisted} conjugation action; (2) that from a sheaf theoretic point of view, the small eigenvalue condition can be discarded, and (3) that the conjugation action can then be untwisted.
Finally, as is often the case with results that push the boundaries of the geometric Langlands program, the proofs in this paper involve a great deal of \(\infty\)-category theory, derived geometry, and Lie theoretic combinatorics. The authors treat these topics well, and provide clear exposition on a number of potentially forbidding topics (e.g.\ singular support conditions, the relation between the geometry of hyperplane arrangements and loop groups). character sheaves; singular support; Betti geometric Langlands; elliptic curve Geometric Langlands program (algebro-geometric aspects), Geometric Langlands program: representation-theoretic aspects Uniformization of semistable bundles on elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we present an expository account of the work done in the last few years in understanding a matrix Lax equation which arises in the study of scalar hyperbolic conservation laws with spectrally negative pure-jump Markov initial data. We begin with its extension to general \(N\times N\) matrices, which is Liouville integrable on generic coadjoint orbits of a matrix Lie group. In the probabilistically interesting case in which the Lax operator is the generator of a pure-jump Markov process, the spectral curve is generically a fully reducible nodal curve. In this case, the equation is not Liouville integrable, but we can show that the flow is still conjugate to a straight line motion, and the equation is exactly solvable. En route, we establish a dictionary between an open, dense set of lower triangular generator matrices and algebro-geometric data which plays an important role in our analysis. Markov initial data; Liouville integrability; algebro-geometric data; matrix Lie group Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems Nodal curves and a class of solutions of the Lax equation for shock clustering and Burgers turbulence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies rigidity of overconvergent isocrystals on an open subset \(U\) of the projective line. Under some natural and necessary conditions, the author proves that rigid overconvergent isocrystals over \(U\) admits Frobenius structure. This can be viewed as certain \(p\)-adic analogue of \textit{N. M. Katz}'s work on rigid local systems [Rigid local systems. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0864.14013)].
We summarize the main results as follows. Let \(K\) be a complete discrete valuation field of mixed characteristic \((0,p)\) with residue field \(k\). Let \(U \subseteq \mathbb P^1_k\) denote an open subset with finite complement \(S\). The first theorem of the paper says that, for an irreducible overconvergent isocrystal \(M^\dagger\) on \(U\) such that \(\mathrm{End}(M^\dagger)\) satisfies the non-Liouville condition of \textit{G. Christol} and \textit{Z. Mebkhout} [Ann. Math. (2) 146, No. 2, 345--410 (1997; Zbl 0929.12003)] at points in \(S\), if the Euler characteristic \(\chi(\mathrm{End}(M^\dagger)) = 2\), then \(M^\dagger\) is \textit{\(p\)-adically rigid} in the sense that \(M^\dagger\) is the unique isocrystal with the given local monodromy at \(S\). The proof is essentially the same as Katz's except using Christol and Mebhout's characteristic formula.
Next, the author points out that if \(M^\dagger\) comes from an algebraic local system \(M\) with regular singularity, satisfying some mild hypotheses including the convergence of the connection and the non-Liouville condition, the rigidity of the algebraic local system \(M\) implies that the \(p\)-adic rigidity of \(M^\dagger\).
Finally, the author proves that if in addition that the exponents of \(M^\dagger\) are rational, and if \(M^\dagger\) is irreducible and rigid, then \(M^\dagger\) admits a \(q\)-Frobenius structure for some \(p\)-power \(q\). By the rigidity result above, the proof of this result follows from that the restriction of \(M^\dagger\) at points in \(S\) admits Frobenius structure, which comes down to a local computation.
The paper is well-organized and well-written. \(p\)-adic differential equations; overconvergent isocrystals; rigidity \(p\)-adic cohomology, crystalline cohomology, \(p\)-adic differential equations Rigidity and Frobenius structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The \(k\)-th MD homotopy group is a group \(MDH^b_{\bullet}\) for any \(b\in [1,\infty]\) together with homomorphisms \(MD\pi^b\to MD\pi^{b'}\) for any \(b\geq b'\). We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For \(b=1\), it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for \(b=\infty\), the \(MD\)-homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case. Semialgebraic sets and related spaces, Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions, Other homology theories in algebraic topology Moderately discontinuous homotopy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 expand the known standard definitions of bi-Lipschitz \(\mathcal{R }\), \(\mathcal{A}\), \(\mathcal{K}\), \(\mathcal{K}^{\ast }\) equivalence of functions germs and bi-Lipschitz equivalence of analytic subsets in \(( \mathbb{C}^{n},0)\) to bi-Lipschitz equivalence of ideals in \(\mathcal{O}^{n}\) (\(\mathcal{O}^{n}\) is the ring of holomorphic function germs at \(0\in \mathbb{C}^{n}).\) Ideals \(I,J\subset \mathcal{O}^{n}\) are bi-Lipschitz equivalent if there exist \(f_{1},\ldots ,f_{p}\in I\) and \(g_{1},\ldots ,g_{q}\in J\) such that:
1. \(\overline{\left( f_{1},\ldots ,f_{p}\right) }=\overline{I},\;\;\) \( \overline{\left( g_{1},\ldots ,g_{q}\right) }=\overline{J}\) \ (\(\overline{I}\) denotes the integral closure of \(I\) in \(\mathcal{O}^{n}\)),
2. there is a bi-Lipschitz homeomorphism \(\phi :(\mathbb{C} ^{n},0)\rightarrow (\mathbb{C}^{n},0)\) such that
\[
\left| \left| \left( f_{1}(x),\ldots ,f_{p}(x)\right) \right| \right| \sim \left| \left| \left( g_{1}(\phi (x)),\ldots ,g_{q}(\phi (x))\right) \right| \right| \text{ for }x\text{ near }0.
\]
The authors study various numerical invariants for such equivalence: order of ideals, log canonical threshold of ideals, some types of Łojasiewicz exponents and others. They apply the results and methods to deformations \(f_{t}:( \mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)\) and show that they are not bi-Lipschitz trivial, specially focusing on several known examples of non-\( \mu ^{\ast }\)-constant deformations. bi-Lipschitz equivalence; Lojasiewicz exponent; log canonical threshold; deformation Local complex singularities, Equisingularity (topological and analytic), Deformations of singularities Invariants for bi-Lipschitz equivalence 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 goal of this paper is to generalize the results of [\textit{L. Clozel} and \textit{E. Ullmo}, Ann. Math. (2) 161, No. 3, 1571--1588 (2005; Zbl 1099.11031)] concerning the equi-distribution of special subvarieties of Shimura varieties.
Given a connected Shimura variety \(S\), one has the notion of special points, and of special subvarieties. Roughly speaking, a special subvariety is a sub-Shimura variety. Every special subvariety \(Z\subseteq S\) carries a probability measure \(\mu_Z\) with support \(Z\).
Whenever \(Z\) is not of the form
\[
Z = M_1 \times \{ x \} \subseteq M = M_1 \times M_2 \subseteq S
\]
for a special subvariety \(M\), then it is called ``NF'' (non facteur).
The author shows the following theorem: Let \((M_n)_n\) be a sequence of NF special subvarieties of \(S\). Then there exists an NF special subvariety \(Z\subseteq S\) and a subsequence \((\mu_{M_n})_{n\in\mathbb N}\), \(N\subseteq \mathbb N\), of the sequence of corresponding measures which converges weakly to \(\mu_Z\). Furthermore, for \(n\in\mathbb N\) sufficiently large, \(M_n\subseteq Z\).
It is clear that in the theorem one cannot replace ``NF special'' by ``special''. For instance, the special points (i.e.~the \(0\)-dimensional special subvarieties) are dense in \(S\) for the complex topology.
From the theorem, one can deduce the following result: Let \(Y\) be a subvariety of \(S\). There exists a finite set \(M_i\) of NF special subvarieties of \(S\), such that \(M_i\subseteq Y\) for all \(i\), and such that every NF special subvariety contained in \(Y\) is in fact contained in the union \(\bigcup M_i\). This result should be seen in the light of the André-Oort conjecture which states that the same should be true with ``NF special'' replaced by ``special'' everywhere. Compare also the work of Edixhoven and Yafaev in this direction.
As for loc.~cit., the proofs of the paper at hand rely on results in the ergodic theory of unipotent flows by Ratner and by Mozes and Shah. special subvarieties of Shimura varieties; André-Oort conjecture; ergodic theory of unipotent flows Ullmo, E, Equidistribution des sous-variétés spéciales II, J. Reine Angew. Math., 606, 193-216, (2006) Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Equidistribution of special subvarieties. 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 This paper studies the numerical semigroups, i.e. subsets H of \({\mathbb{N}}\) such that \(0\in H\), \(H+H\subseteq H\) and \({\mathbb{N}}-H\) is finite. It is well known that the study of numerical semigroups is closely related to the study of monomial curve singularities. In fact, for every numerical semigroup H one can associate the semigroup K-algebra \(K[[H]]=\{\sum_{h\in H}a_ ht^ h| \quad a_ h\in K\}\) (which is a K-subalgebra of K[[t]], where K is a fixed field). Many numerical invariants of the singularity K[[H]] such as the multiplicity, the embedding-dimension, the Hilbert function, etc. can be expressed in the language of semigroups. This point of view has been intensively studied by several authors e.g. Brezinski, Delorme, Herzog, Waldi, etc.
The author considers the set \(G_ m\) of numerical semigroups which contain the fixed number \(m\geq 3\), and associates to \(G_ m\) a convex polyhedral cone \(P_ m\subset {\mathbb{R}}^{m-1}\) in such a way that there is a bijection between \(G_ m\) and the set of points of \(P_ m\) with integral coordinates. This method has the advantage of translating the problem of classification of the semigroups of \(G_ m\) into a purely geometric language. In particular, one defines and studies the numerical invariant \(s_ m(H)\) which is defined as the dimension of the open face of \(P_ m\) containing the corresponding integral point of \(P_ m\) associated to H. This invariant is important because it gives a hierarchy of the semigroups of \(G_ m\). One also studies the Hilbert function of the integral points of a fixed face of \(P_ m.\)
In an appendix one gives the complete lists concerning the classification of the semigroups of \(G_ m\) with \(m\leq 7\) (by using the invariants studied before and a computer program). numerical semigroups; monomial curve singularities; multiplicity; embedding-dimension; Hilbert function; computer program E. Kunz, \textit{Über die Klassifikation Numerischer Halbgruppen}, Regensburger Matematische Schriften. Vol. 11 1987, https://books.google.com/books/about/%C3%9Cber_die_Klassifikation_numerischer_Hal.html?id=KfHuAAAAMAAJ. Singularities of curves, local rings, Multiplicity theory and related topics, General structure theory for semigroups, Software, source code, etc. for problems pertaining to algebraic geometry Über die Klassifikation numerischer Halbgruppen. (On the classification of numerical semigroups) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a regular local ring with maximal ideal \(\mathcal{M}\) and \(\hat R\) its \(\mathcal{M}\)-adic completion. Then \(R\subset \hat R\). An element \(f\in R\) can be expanded as a finite sum \(\displaystyle f=\sum_{\alpha\in\mathbb{Z}^n_{\geq 0}}a_{\alpha}x^{\alpha}\), where \(a_{\alpha}\in U(R)\cup\{0\}\), \(x=(x_1,\ldots,x_n)\) is a regular system of parameters of \(R\), \(\alpha=(\alpha_1,\ldots,\alpha_n)\) and \(x^{\alpha}=x_1^{\alpha_1}\ldots x_n^{\alpha_n}\). The main results of this paper are factorization theorems for elements in \(R\) and polynomials with coefficients in \(R\), using convex geometry. We need some preliminaries to formulate the first theorem. For \(\displaystyle f=\sum_{\alpha\in\mathbb{Z}^n_{\geq 0}}a_{\alpha}x^{\alpha}\in R\) a nonzero element, the Newton polyhedron \(\Delta(f)\) of \(f\) is the convex hull of the set \(\{\alpha;~a_{\alpha}\not=0\}+\mathbb{R}^n_{\geq 0}\). A set \(\Delta\subset\mathbb{R}^n_{\geq 0}\) is called a Newton polyhedron if \(\Delta=\Delta(f)\) for some \(0\not=f\in R\). For \(\xi\in\mathbb{R}^n_{\geq 0}\) and \(\Delta\subset\mathbb{R}^n_{\geq 0}\) a Newton polyhedron, we call the set \(\Delta^{\xi}=\{a\in \Delta;~<\xi,a>=min_{b\in\Delta}<\xi,b>\}\) a face of \(\Delta\). Here \(<.,.>\) denotes the standard scalar product. A face \(\Delta^{\xi}\) is compact if and only if \(\xi\in\mathbb{R}^n_>\). A face of dimension \(1\) is called an adge and a compact edge of a Newton polyhedron is called a loose adge if it is not contained in any compact face of dimension \(\geq 2\). Now, we are ready to give the first main result of the paper. Let \(R\) be a regular local ring and \(0\not=f\in R\). Assume that the Newton polyhedron \(\Delta(f)\) has a loose edge \(E\). If \(f_{/E}\) is a product of two relatively prime polynomials \(G\) and \(H\), where \(G\) is not divided by any variable, then there exists \(g,h\in\hat R\) such that \(f=gh\) and \(g_{/E_1}=G\), \(h_{/E_2}=H\) for some \(E_1\) and \(E_2\) such that \(E\) is the Minkowski sum of \(E_1\) and \(E_2\), \(E=E_1+E_2\). irreducibility; formal power series; Newton polyhedron Formal power series rings, Polynomials in general fields (irreducibility, etc.), Singularities in algebraic geometry Loose edges and factorization 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 Suppose that \(X\subset\mathbb{R}^n\) is a variety and \(E\subset X\) a finite subset. The article presents conditions on \(E\) in terms of real valued invariants of \(X\) such that the homology of \(X\) is uniquely determined by \(E\). The methods are constructive and translate into an efficient algorithm.
Of particular interest is when \(E\) is equal to \(S_\delta\) for some \(\delta\in\mathbb{R}_{>0}\), where \(S_\delta\subset X\) be obtained by intersecting \(X\) with a grid of linear varieties of complementary dimension such that the grid size is equal to \(\delta\). For example, if \(n=2\) and \(X\) is a curve, then the grid is defined by horizontal lines \(\{(k\,\delta,y)\in\mathbb{R}^2: k\in\mathbb{Z}, y\in\mathbb{R}\}\) and vertical lines \(\{(x,k\,\delta)\in\mathbb{R}^2: k\in\mathbb{Z}, x\in\mathbb{R}\}\). The finite subset \(E\) is called an \(\epsilon\)-\emph{sample} of \(X\) for some \(\epsilon\in\mathbb{R}_{>0}\), if for all \(x\in X\) there exists \(e\in E\) such that \(\|x-e\|<\epsilon\).
The authors show that if \(\delta<n^{-\frac{1}{2}}\min\{\epsilon,2\,b_2\}\), then \(S_\delta\) is an \(\epsilon\)-sample of~\(X\). The invariant \(b_2\in\mathbb{R}_{>0}\) is the \emph{radius of narrowest bottleneck} of~\(X\) and is defined in terms of \emph{bottlenecks} of \(X\) which are pairs of points \(x,y\in X\) that span a line that is orthogonal to \(X\) at both points.
If \(E\subset X\) is an \(\epsilon\)-sample such that \(\epsilon<\text{wfs}(X)\), then the authors show that 0th and 1st homology groups of \(X\) can be uniquely recovered from \(E\). Here, the invariant wfs\((X)\in\mathbb{R}_{>0}\) is called the \emph{weak feature size} of \(X\) and is again defined in terms of bottlenecks. If, moreover, \(\epsilon\) is bounded above in terms of an invariant called \emph{local reach}, then the remaining homology groups of \(X\) can be recovered as well. An algorithm is presented with as input implicit equations for \(X\), and the output is an \(\epsilon\)-sample of \(X\) such that \(\epsilon\in\mathbb{R}_{>0}\) satifies the hypothesis any of the above statements. bottleneck; homology; topological data analysis; real algebraic varieties; dense samples Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Effectivity, complexity and computational aspects of algebraic geometry, Topology of real algebraic varieties Sampling and homology via bottlenecks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a finite-dimensional Euclidean vector space \(V\) and a symmetric tensor \(f\in \text{Sym}^d V,\) a non-zero vector \(x\in V^\mathbb{C}\) such that \(||x||=1\) is called a \(E\)-eigenvector of \(f\) if there exists \(\lambda \in \mathbb{C}\) such that \(x\) is solution of the equation
\[
\dfrac{1}{d}\nabla f(x)=\lambda x.
\]
The scalar \(\lambda\) is called an \(E\)-eigenvalue of \(f\).
The author gives a formula for the product of all \(E\)-eigenvalues of \(f\) when \(f\) admits the maximum number of \(E\)-eigenvalues. The formula
\[
\lambda_1\cdots\lambda_N=\pm \frac{\text{Res}\left(\frac{1}{d}\nabla (f)\right)} {{\Delta_{\widetilde{Q}}(f)}^{\frac{d-2}{2}}}
\]
is given in terms of the resultant of \(\frac{1}{d}\nabla(f)\) and of the \(\widetilde{Q}\)-discriminant of \(f\).
In order to do this, the author studies the \(E\)-characteristic polynomial of \(f\), which generalizes the characteristic polynomial \(\psi_A(\lambda)=\text{det}(A-\lambda I)\) of a symmetric matrix \(A.\) symmetric tensor; \(E\)-eigenvalue; \(E\)-characteristic polynomial; isotropic quadric Multilinear algebra, tensor calculus, Rational and unirational varieties, Eigenvalues, singular values, and eigenvectors, Vector and tensor algebra, theory of invariants, Numerical solution of nonlinear eigenvalue and eigenvector problems The product of the eigenvalues of a symmetric tensor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Pencil-based algorithms are widely used to compute of the rank and to determine a Canonical Polyadic Decomposition (CDP) of a multidimensional matrix (tensor) \(T\). The basic idea behind these algorithms is the reduction, by means of an orthogonal projection, to tensors of type \(n_1\times n_2\times 2\), which can be treated as pencils of ordinary matrices, so that the original problem boils down to a problem of finding common eigenvectors of a pencil. The methods are particularly useful when the CDP of \(T\) is unique (identifiable tensors).
The authors study the reliability of these methods, in terms of the \textit{excess number} of
the procedure, defined as follows. Let \(\kappa\) be the condition number of the map
which sends a CDP \(A_1,\dots,A_n\) to the tensor \(T=A_1+\dots+A_r\). The excess number is the ratio between the maximal norm \(||A'-A||\) and \(\kappa(A)\epsilon\), where \(A\) is the CDP of \(T\) and \(A'\) is the decomposition that the algorithm returns for a tensor \(T'\) (floating point approximation of \(T\)) whose distance from \(T\) is less than \(\epsilon\).
The authors prove that the general condition number for the CDP of tensor of type \(n_1\times n_2\times 2\) is bigger than the condition number for tensors of type \(n_1\times n_2\times n_3\), \(n_3>2\), and the difference increases when \(n_3\) grows. As a consequence, it turns out that the excess number of the procedure can be arbitrarily big. Thus, the authors conclude that pencil-based algorithms for the computation of the CDP can be considered as unstable. canonical polyadic decomposition; numerical instability Geometric aspects of numerical algebraic geometry, Multilinear algebra, tensor calculus, Numerical computation of matrix norms, conditioning, scaling, Sensitivity analysis for optimization problems on manifolds Pencil-based algorithms for tensor rank decomposition are not stable | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 demonstrates the following results: Let \(A\) be the ring of formal polynomials in \(n>1\) variables over an algebraically closed field \(k\), and \(M\) a torsion-free \(A\)-module whose rank be \(r>1)\). Then there exists a monomorphism \(i\colon M\to A^r\) such that \(i(M)\) includes some elements whose set of zeros over \(k\) has \(\text{cdm} >1\) in \(k^n\). This property is true also if \(A\) is any unique factorization domain finitely generated over \(A\), which allows one to affirm that: any coherent torsion-free algebraic sheaf \(\mathcal M^{(r)}\), of rank \(r>1\), over an affine variety with unique factorization algebra variety \((V, \mathcal A_V\) of dimension \(n>1\), is isomorphic to a subsheaf \(\mathcal M'^{(r)}\) of \(\mathcal A_V^r\) which has some sections whose set of zeros has \(\text{cdm} >1\) in \(V\).
From this theorem it follows that for any coherent torsion-free algebraic sheaf \(\mathcal M^{(r)}\) over \((V, \mathcal A_V\) there exist \(r-1\) short exact sequences:
\[ 0\longrightarrow \mathcal A_V^h\longrightarrow \mathcal M^{(r)} \longrightarrow \mathcal M^{(r-h)} \longrightarrow 0 \]
\((1\le h\le r-1)\) of torsion-free sheaves, such that, for all \(h\), it is
\[ \text{Supp}(\mathcal A_V^r)/ \mathcal M'^{(r)}) \subseteq \text{Supp}(\mathcal A_V^{(r-h)}/\mathcal M'^{(r-h)}). \] algebraic geometry S. Baldassarri-Ghezzo, ?Proprietà di fasci algebrici coerenti e lisci su varietà algebriche affini ad algebra fattoriale,? Rend. Semin. Mat. Univ. Padova, 1968,41, 12?30 (1969). Algebraic geometry Proprietà di fasci algebrici coerenti e lisci su varietà algebriche ad algebra fattoriale | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathbb{K}\) be a field and \(M \simeq \mathbb{Z}^n\) a lattice. The classical Bernštein-Kušnirenko Theorem, also known as the BKK Theorem, states that given a system of Laurent polynomials \(f_1, \dotsc, f_n \in \mathbb{K}[M]\), the number of isolated solutions in the torus \(\mathbb{T}_M = \operatorname{Spec}\left(\mathbb{K}[M]\right)\) is bounded by the mixed volume \(\operatorname{MV}\left(\Delta_1, \dotsc, \Delta_n \right)\) of the corresponding Newton polytopes \(\Delta_i \subseteq M_{\mathbb{R}}\).
The goal of this article is to show an arithmetic analogue of the BKK Theorem. Assume that \(\mathbb{K}\) is an adelic field satisfying the product formula and write \(Z = Z\left(f_1, \dotsc, f_n\right)\) for the \(0\)-cycle on \(\mathbb{T}_M\) given by the isolated solutions of the system of equations
\[f_1 = \dotsc = f_n.
\]
Let \(X\) be a toric compactification of \(\mathbb{T}_M\) and \(\overline{D_0} = \left(D_0, \| \cdot \| \right)\) a metrized nef toric divisor on \(X\). This data induces the notion of a height \(h_{\overline{D_0}}\) of a \(0\)-cycle on \(X\), which is a non-negative real number. The main result of this article is an upperbound for \(h_{\overline{D_0}}(Z)\) in terms of mixed integrals of the concave local roof functions associated to the metrized divisor and to the \(f_i\)'s.
The authors also describe some families of examples in which their bound is close to optimal.
Their proof is based on the arithmetic intersection theory on toric varieties developed in [\textit{J. I. Burgos Gil} et al., Arithmetic geometry of toric varieties. Metrics, measures and heights. Paris: Société Mathématique de France (SMF) (2014; Zbl 1311.14050)].
Other attempts towards an arithmetic BKK Theorem have already been proposed by \textit{V. Maillot} [Mém. Soc. Math. Fr., Nouv. Sér. 80, 129 p. (2000; Zbl 0963.14009)] and by the second author in [J. Reine Angew. Math. 586, 207--233 (2005; Zbl 1080.14060)]. The result of this article improves these previously known upperbounds and generalizes them to any adelic filed satisfying the product formula and to height functions associated to arbitrary metrized nef toric divisors.
As a side result of their method of proof, the authors introduce a well-defined notion of global height for cycles with respect to metrized line bundles which are generated by small sections.
Finally, as an application of the main result, an upperbound for the size of the coefficients of the \(u\)-resultant of the direct image of the solution set of a systen of Laurent polynomial equations under a monomial map is given. height of points; Laurent polynomials; mixed integrals; toric varieties; \(u\)-resultants Arithmetic varieties and schemes; Arakelov theory; heights, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Convex functions and convex programs in convex geometry An arithmetic Bernštein-Kušnirenko inequality | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A scheme \(X\) is of characteristic \(p>0\) if the canonical morphism \(X \to \mathrm{Spec} (\mathbb{Z})\) factors through \(\mathrm{Spec} (\mathbb{Z}/p)\). For a scheme \(X\) of characteristic \(p>0\), there exists an absolute Frobenius morphism \(F_{X} : X \to X\), which is identity morphism on the base topological space and is the \(p\)-power morphism on the structure sheaf. A scheme \(X\) of characteristic \(p>0\) is \(F\)-finite is the Frobenius morphism \(F_{X}\) is a finite morphism of scheme.
Assume \(X\) is a \(F\)-finite normal scheme of characteristic \(p>0\) and \(\Delta\) is an effective \(\mathbb{R}\)-divisor on \(X\).
1. The pair \((X, \delta)\) is globally \(F\)-split if for every \(e \in \mathbb{Z}_{>0}\), the natural map \[ \mathcal{O}_{X} \to F_{X*}^{e}\mathcal{O}_{X} \hookrightarrow F_{X*}^{e}\mathcal{O}_{X} (\llcorner (p^{e}-1)\Delta \lrcorner) \] splits.
2. The pair \((X,\delta)\) is \(F\)-pure if \((\mathrm{Spec} \mathcal{O}_{X,x} , \Delta | _{\mathrm{Spec} \mathcal{O}_{X,x}})\) is globally \(F\)-split for every point \(x \in X\).
The main results of the paper under review is following two theorems:
Theorem 1.1. Over any algebraically closed filed \(k\) of characteristic \(p>0\), there exists a projective Kawamata log terminal surface \(X\) such that \(-K_{X}\) is ample and \(X\) is not globally \(F\)-split.
Theorem 1.2. Over any algebraically closed field \(k\) of characteristic \(p>0\), there exists a three-dimensional canonical variety \(X\), which is not \(F\)-pure. globally F-split; F-pure Positive characteristic ground fields in algebraic geometry, Minimal model program (Mori theory, extremal rays), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Klt del Pezzo surfaces which are not globally F-split | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:(\mathbb C^n,0) \to \mathbb C \) be an isolated (\(n\)-dimensional) isolated singularity, that is a germ of an analytic function such that \(\nabla f := (f_{z_1}, \ldots, f_{z_n})\) vanishes at \(0\) but not for \(z\) near \(0\), \(z \not= 0\). An important number attached to \(f\) is its Łojasiewicz exponent \(\L _0(f)\). This can be defined as follows. Let \(B_0(f)\) be the set of positive real numbers \(\alpha\) such that \(\parallel \nabla (f) \parallel \leq C_{\alpha}{\parallel z \parallel}^{\alpha}\), for a suitable real number \(C_{\alpha} > 0\) and \(z\) close enough to \(0\). Then \({\L} _0(f) = \inf B_0(f)\). There are many equivalent definitions of this exponent. This article establishes an upper bound for \(\L _0(f)\), for singularities satisfying certain conditions. This bound uses elements that can be obtained from \(\Gamma (f)\), the set of compact faces of \({\Gamma} _{+}(f)\), the Newton diagram, or polyhedron, of \(f\).
The main result of this paper has to do with isolated singularities which are nondegenerate in the sense of Kouchnirenko (K-nondegenerate). This concept is defined in terms of solutions to certain auxiliary polynomial equations associated to each compact face \(S \in \Gamma (f)\). Moreover, the main theorem involves the notion of exceptional faces, namely faces in \(\Gamma (f)\) whose intersections with coordinate planes contain certain special segments. Let \(N(f)\) be the set of faces in \(\Gamma (f)\) which are not exceptional. Finally we need, for \(S\) as above, the real number \(\alpha(S)\), defined in terms of the intersections of the hyperplane spanned by \(S \in \Gamma (f)\) with the coordinate axes. The main theorem precisely says:
If \(f:(\mathbb C^n,0) \to \mathbb C \) is an isolated singularity, K-nondedgenerate, where \(N(f) \not= \emptyset\), then
\[
\L _0(f) \leq \max\big\{m(S): S \in N(f)\big\}\, .
\]
This result improves, when it can be applied, a similar inequality found by \textit{T. Fukui} [Proc. Am. Math. Soc. 112, No. 4, 1169--1183 (1991; Zbl 0737.58001)]; although Fukui's assumptions are weaker than Oleksik's. An example shows that sometimes the estimate of this paper is better than that of Fukui. Another example is given where in Oleksik's formula we have an equality, so that the given estimate cannot be improved. The paper concludes with a problem: can we give a suitable notion of an ``exceptional face'' so that, using it, the inequality of the main theorem becomes an equality? Łojasiewicz exponent; isolated singularity; Kouchnirenko's nondegeneracy; Newton diagram; exceptional face Oleksik, The Łojasiewicz exponent of nondegenerate singularities, Univ. Iagel. Acta Math. 47 pp 301-- (2009) Invariants of analytic local rings, Singularities in algebraic geometry The Łojasiewicz exponent of nondegenerate singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth subscheme of a projective space over a finite field. In the case of a single hypersurface of large degree, the probability that its intersection with \(X\) is smooth of the correct dimension was computed by \textit{B. Poonen} [Ann. Math. 160, 1099--1127 (2003; Zbl 1084.14026)]. The authors of the paper under review generalize Poonen's theorem to the case of complete intersections. They then relate this result to the probabilistic model of rational points of such complete intersection.
These and some another author's results, although difficult to present in a short review, are well summarized in the introductory section. Briefly, the general idea of the proof of the authors' theorem is a sieve over the set of closed points of \(X\) similar to Poonen's proof of Theorem 1.1., in which the authors separately analyze the contribution of points of low degree, medium degree and high degree.
The authors also give an interesting corollary that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.
Finally, they indicate some other directions in which one can probably generalize Poonen's results from hypersurfaces to complete intersections and make the conjecture, generalizing the conditional Poonen's Theorem 5.5. Bertini theorem; smooth subscheme of a projective space; finite field; hypersurface of large degree; complete intersection; random smooth intersection; probabilistic model A. Bucur and K. S. Kedlaya, The probability that a complete intersection is smooth , J. Théor. Nombres Bordeaux 24 (2012), 541-556. Finite ground fields in algebraic geometry, Zeta and \(L\)-functions in characteristic \(p\) The probability that a complete intersection is smooth | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 give a cohomological definition of the \(n\)-dimensional symbol of \textit{C. Contou-Carrère} [C. R. Acad. Sci. Paris, Sér. I 318, No. 8, 743--746 (1994; Zbl 0840.14031)] over an artinian local ring as the distinguished element in the cohomology class \([\{\cdot , \ldots , \cdot\}_{n,A}] \in H^{n+1}(A((t_1)) \cdots A((t_n))^*, A^*)\), and to present an explicit expression for this symbol. The offered definition generalizes the author's recent definition of the two-dimensional Contou-Carrère symbol [\textit{F. Pablos Romo}, in: The geometry of Riemann surfaces and abelian varieties. Contemp. Math. 397, 163--175 (2006; Zbl 1105.19005)], and in the case where \(A=k\) is a field, it generalizes the \(n\)-dimensional tame symbol introduced by \textit{A. N. Parshin} [Proc. Steklov Inst. Math. 183, 191-201 (1991; Zbl 0731.11064)] and \textit{S. V. Vostokov} and \textit{I. Fesenko} [in: Selected papers in \(K\)-theory. Transl., Ser. 2, Am. Math. Soc. 154, 25-35 (1992); translation from Rings and modules. Limit theorems of probability theory, Leningrad 1, 75-87 (1986; Zbl 0796.11053)]. Further, the author shows that the proposed definition of an \(n\)-dimensional Contou-Carrère symbol allows to deduce the definition of an \(n\)-dimensional Witt residue extending the Witt-Parshin pairing which was introduced by Parshin to describe abelian \(p\)-extensions of \(n\)-dimensional local fields.
The final theorem gives the reciprocity law for the \(n\)-dimensional Contou-Carrère symbol associated with a flag of normal irreducible varieties \(x \subset X_1 \subset \cdots \subset X_{n-1} \subset X\), where \(X_1\) is a complete curve over an algebraically closed field. As a corollary the author obtains an \(n\)--dimensional Witt residue theorem as a particular case of this reciprocity law. Contou-Carrère symbol; tame symbol; Parshin symbol; Witt residue; reciprocity law Symbols and arithmetic (\(K\)-theoretic aspects), Commutator calculus, Higher symbols, Milnor \(K\)-theory, \(K\)-theory of local fields, Local ground fields in algebraic geometry An \(n\)-dimensional Contou-Carrère symbol over an Artinian local ring | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let f: (\({\mathbb{C}}^ n,0)\to ({\mathbb{C}},0)\) be a quasihomogeneous polynomial with weights \((w_ 1,...,w_ n)\) and w-degree d, and \(F:({\mathbb{C}}^ n\times {\mathbb{C}},0\times {\mathbb{C}})\to ({\mathbb{C}},0)\) be the deformation with the expansion
\[
F(x,t)= f(x)+ tf_ 1(x)+ t^ 2f_ 2(x)+...
\]
of F in powers of the deformation parameter t whose each term \(f_ i(x)\) is a linear combination of monomials of w-degree greater than or equal to d.
Let \(m_ t\) denote the multiplicity and \(\mu_ i\) the Milnor number of \(F_ t(x):=F(x,t)\) at the origin. The deformation F is equimultiple (resp., \(\mu\)-constant) if \(m_ 0=m_ t\) (resp., \(\mu_ 0=\mu_ t)\) for all t sufficiently close to 0.
Theorem. If f is a quasihomogeneous polynomial with an isolated singularity at the origin, then any deformation as above of f is equimultiple.
Theorem. Any \(\mu\)-constant deformation of a quasihomogeneous polynomial with an isolated singularity is the deformation as above.
The purpose of this paper is to show that particular families of singularities are equimultiple. This class includes all known examples of \(\mu\)-constant families which are not \(\mu^*\)-constant. upper deformations; mu-constant deformation; equimultiple deformation; quasihomogeneous polynomial; isolated singularity Teissier, B.: Cycles évanescents, section planes, et conditions de Whitney, Singularites a' Cargèse 1972, Astérisque No 7-8, Soc. Math. Fr. 285-362 (1973) Deformations of complex singularities; vanishing cycles, Complex singularities, Deformations of singularities Topologically trivial deformations of isolated quasihomogeneous hypersurface singularities are equimultiple | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 characterization of the sums of signs of global analytic functions on a real analytic manifold \(M\) of dimension 2. They prove that for dimension \(2\) the set of these sums of signs of global analytic functions is the intersection between two classes of constructible functions: the semianalytically constructible ones and the analytically constructible ones. Recall that a function \(\varphi:M\to{\mathbb Z}\) is semianalytically constructible if it is constant on each element of a finite partition of \(M\) into global semianalytic sets. On the other hand, an analytically constructible function is a linear combination, over the integers, of Euler's characteristics of fibers of proper analytic morphisms.
Even if \(M\) is a compact \(2\)-dimensional manifold the previous classes are not contained one in the other, and as it was pointed above their intersection is the set of the sum of signs of global analytic functions.
Similar results were known previously for the algebraic case [\textit{M. Coste} and \textit{K. Kurdyka}, Topology 37, 393--399 (1998; Zbl 0942.14031); \textit{A. Parusiński} and \textit{Z. Szafraniec}, in: Singularities symposium -- Lojasiewicz 70. Banach Center Publ. 44, 175--182 (1998; Zbl 0915.14032)] and also for the Nash compact case [\textit{I. Bonnard}, Manuscr. Math. 112, 55--75 (2003; Zbl 1025.14010)].
The authors also improve this last case avoiding the compactness asumption for dimension \(2\). In fact, they prove that unlike the algebraic case, obstructions at infinity are not relevant for analytic and Nash manifolds of dimension \(2\): {a function is a sum of signs (of functions of the suitable type) on a \(2\)-dimensional manifold \(M\) if and only if this is true on each compact subset of \(M\)}. sum of signs of global analytic functions; principal open sets Real-analytic and semi-analytic sets Constructible functions on 2-dimensional analytic 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 For a lattice \(\Lambda\) of determinant 1 in the Euclidean space \({\mathbb E}^n\) the Hermite invariant is defined by \(\mu(\Lambda)=\text{ inf} \left\{\langle x,x\rangle; x\in\Lambda,x\neq 0\right\}\). The systole of a compact Riemannian manifold \(M\), defined as the minimal length of a non-contractible curve in \(M\), is a far reaching generalization of this concept, because \(\mu(\Lambda)\) appears as the systole of the flat torus \({\mathbb E}^n/\Lambda\).
The author introduces a new unifying geometric framework for certain generalisations of the notion of the Hermite invariant, namely, flat tori, principally polarized abelian varieties of dimension \(g\), Riemannian surfaces of genus \(g\) endowed with the Poincaré metric. All these objects can be considered as points of certain Riemannian manifolds, namely of the space of \(P_n\) of Gram matrices (i.e., positive symmetric matrices of determinant 1), of the Siegel domain \({\mathcal H}_g\), and of the Teichmüller space \(T_g\), endowed with the Weil-Petersson metric, respectively.
The generalizations of the Hermite invariant appear as the minimum of a family of length functions on the corresponding parameter space. This permits to develop for each of these cases the theory in analogy to the case of lattices, which serves as guide. In the first chapter a geometric study of Hermite's invariant is given. The classical notions of the theory of lattices are transferred to geometrical notions on the space \(P_n\). A geometric interpretation of the length function of lattices as exponentials of the Busemann function is given.
In chapter II the key notions 'perfection' and 'eutaxy', due to Voronoï, are defined with the help of gradients of length functions and used to characterise local maxima of the Hermite invariant. For the case of Riemannian surfaces the following analogue of Voronoï's theorem is obtained: A Riemannian surface is a local maximum of the systole if and only if it is perfect and eutactic. In the last part it is observed that for principally polarized abelian varieties the situation changes: Eutactic principally polarized abelian varieties are not isolated and the Hermite invariant is not a Morse function on \({\mathcal H}_g\). systole; Hermite invariant; length function Busemann function; perfection; eutaxie; Riemannian surfaces; principally polarized abelian variety C. Bavard, ''Systole et invariant d'Hermite,'' J. Reine Angew. Math. 482, 93--120 (1997). Global Riemannian geometry, including pinching, Quadratic forms (reduction theory, extreme forms, etc.), Abelian varieties and schemes, Geodesics in global differential geometry, Differential geometry of symmetric spaces, Conformal metrics (hyperbolic, Poincaré, distance functions) Systole and Hermite invariant | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 subset \(Y\) of \(\mathbb{R}^m\) is called subanalytic if there exist a number \(d\in \mathbb{N}\) and a seminanalytic subset \(Z\) of the torus \(\mathbb{P}^{m+d}\) such that \(Y =\pi(Z)\cap \mathbb{R}^m\) by which \(\pi\) denotes the natural projection. Let \(D\) be a subanalytic subset of \(\mathbb{R}^m\). A function \(f : D\to \mathbb{R}\) is called globally subanalytic if its graph is globally subanalytic.
The authors prove the following Theorem: Let \(Y\) be a globally subanalytic subset of \(\mathbb{R}^{n+m}\) with fibers \(Y_x = Y\cap (\{x\}\times \mathbb{R}^m)\) of dimension at most \(k\). Let \(B\) be the set of points \(x\in \mathbb{R}^n\) such that the \(k\)-dimensional volume \(v(x)\) of \(Y_x\) is finite. Then \(B\) is a globally subanalytic subset of \(\mathbb{R}^n\) and \(v|B\) is of the form \(v = P(A_1,\dots,A_r,\log A_1,\dots,\log A_r)\) by which \(P\) is a polynomial and the \(A_i\) are globally subanalytic functions.
From this result the authors deduce a corollary about the log-analytic nature of the \(k\)-dimensional density of globally subanalytic subsets of \(\mathbb{R}^m\) with dimension at most \(k\).
If in the theorem \(Y\) is semialgebraic then \(B\) is semialgebraic, too.
Important for the proof of the theorem are a preparation theorem for subanalytic functions and Lipschitz stratification for compact subanalytic sets. density; preparation theorem; subanalytic sets COMTE G. , LION J.-M. , ROLIN J.-P. , Nature Log-analytique du volume des sous-analytiques , Illinois J. Math. (à paraître). Article | Zbl 0982.32009 Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets, Semialgebraic sets and related spaces Log-analytic nature of the volume of subanalytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the theory of functions of one complex variable, capacity is a measure of size for compact sets. This concept of capacity plays an essential role in the study of harmonic functions, equilibrium distribution, and general potential theory within the theory of one complex variable. On the other hand, it provides significant applications to algebraic number theory, too. By a classical theorem of \textit{M. Fekete} and \textit{G. Szegö} [cf. Math. Z. 63, 158--172 (1955; Zbl 0066.27002)], the distribution of complete conjugate sets of algebraic integers with respect to a compact set in the complex plane can be measured by the logarithmic capacity of such a set.
The naturally arising question of whether there is an analogous theorem for arbitrary number fields (rather than \({\mathbb{Q}})\) and more general sets at the finite primes has been partially answered by D. Cantor (1967, 1980) who defined a generalized capacity measure for adelic sets in the projective line, and proved, together with P. Roquette in 1984, a generalization of the Fekete-Szegö theorem, and a general existence theorem on rational points in affine varieties defined over arbitrary number fields.
In the present monograph, the author now extends capacity theory to arbitrary curves (of any genus) with an arbitrary global field as ground field. This pretentious enterprise requires to lay fairly new foundations, as on curves of higher genus (instead of \({\mathbb{P}}^ 1)\) one can only lean upon local methods. The author's far-reaching work to found a generalized capacity theory for algebraic curves and to apply it to arithmetical problems is based upon an ingenious combination of the classical methods, on the one hand, and several essentially new ingredients, such as local parametrizations by power series, a certain global maximum modulus principle, an existence theorem for algebraic functions with near-prescribed divisors, the Deligne-Mumford theorem on stable reduction, and Néron's canonical local height pairing on curves, on the other hand.
After a very enlightening and profound introduction to the classical background, D. Cantor's pioneering work, and the author's goals and results in this monograph, section \(1\) is devoted to some basic techniques needed in the sequel. In section \(2\) several constructions for canonical distance functions on algebraic curves are given. They form a crucial, and qualitatively new ingredient for defining capacity measures and Green functions on curves. The developed capacity theory is, for the present, a local one, and respectively dependent on the assumption of whether the ground field is archimedean or non-archimedean. The difference consists in the methodical approach, whereas the basic properties and results of the generalized (local) capacity theory for curves turn out to be the same. All this is comprehensively developed in section \(3\) (``Local capacity theory - Archimedean case'') and section \(4\) (``Local capacity -- non-archimedean case''), which form the crucial technical part of the monograph.
\(Section 5\) deals with the global capacity theory for curves, i.e., for adelic sets in complete smooth curves over global fields. This provides a suitable generalization of D. Cantor's theory for \({\mathbb{P}}^1\), where the results are, indeed, adapted from that special case, nevertheless heavily based on the author's new general local approach. A number of examples of capacities and Green functions for (archimedean and non-archimedean) local sets, as well as methods for the computation of the global capacity illustrate and verify the usefulness of this extended theory of capacity.
Finally, in the concluding section \(6\) the author discusses important arithmetical applications of his generalized capacity theory. After reproving the theorems of Fekete and Fekete-Szegö in their classical form, just for the convenience of the reader, and in order to prepare him for the following generalizations, the author provides a sagacious construction of rational functions on curves over a global field, which admit special properties needed for an adelic generalization of those classical theorems. This is certainly the ultimate goal, and the most significant result of the whole impressing treatise. The following proofs of the adelic generalizations of the theorems of Fekete and Fekete-Szegö are then rather formal consequences of the foregoing deep results. The book concludes with some examples showing that for the proofs of the Fekete theorem and the Fekete-Szegö theorem really different concepts of capacity are needed, and therefore generally justified and useful.
Altogether, the author has not only written a very deep-going research monograph, but also - at the same time -- an excellent textbook on classical and generalized capacity theory. All the proofs are carefully and detailedly carried out, and the ubiquitous motivating discussions and examples make this very advanced topic a real pleasure to the reader. A list of indications and conjectures concerning the possibility of strengthening, vaster applying, and extending (to higher dimensions) capacity theory serves as a good hint to current research. capacity; distribution of complete conjugate sets of algebraic integers; local parametizations by power series; global maximum modulus principle; stable reduction; local height pairing on curves; canonical distance functions on algebraic curves; capacity measures; Green functions R.S. Rumely, \textit{Capacity Theory on Algebraic Curves}, Lecture Notes in Math., Vol. 1378, Springer, New York, 1989. Arithmetic ground fields for curves, Harmonic functions on Riemann surfaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Global ground fields in algebraic geometry, Local ground fields in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to functions of a complex variable Capacity theory on algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The behavior near the boundary in the Deligne-Mumford compactification of many functions on \(\mathcal{M}_{h,n}\) can be conveniently expressed using the notion of ``point-like limit'' that we adopt from the string theory literature. In this note we study a function on \(\mathcal{M}_h\) that has been introduced by \textit{N. Kawazumi} [in: Handbook of Teichmüller theory. Volume II. Zürich: European Mathematical Society (EMS). 217--237 (2009; Zbl 1170.30001); ``Johnson's homomorphisms and the Arakelov-Green function'', Preprint, \url{arXiv:0801.4218}] and \textit{S.-W. Zhang} [Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031)], independently. We show that the point-like limit of the Zhang-Kawazumi invariant in a family of hyperelliptic Riemann surfaces in the direction of any hyperelliptic stable curve exists, and is given by evaluating a combinatorial analogue of the Zhang-Kawazumi invariant, also introduced by Zhang, on the dual graph of that stable curve. Arakelov-Green's function; hyperelliptic curve; point-like limit; stable curve; Zhang-Kawazumi invariant Arithmetic varieties and schemes; Arakelov theory; heights, Teichmüller theory for Riemann surfaces Point-like limit of the hyperelliptic Zhang-Kawazumi invariant | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 performs extensive calculations in the context of 2-complete stable motivic homotopy theory over \(\mathbb{C}\). To begin, the motivic May spectral sequence is used to compute the cohomology of the motivic Steenrod algebra over \(\mathbb{C}\) through the 70-stem. These computations serve as the input to the motivic Adams spectral sequence [\textit{D. Dugger} and \textit{D. C. Isaksen}, Geom. Topol. 14, No. 2, 967--1014 (2010; Zbl 1206.14041)], which allows the author to compute motivic stable homotopy groups through the 59-stem. Extensions of these calculations to the 65-stem are outlined, with details in [\textit{D. C. Isaksen}, \textit{G. Wang} and \textit{Z. Xu}, ``More stable stems'', Preprint, \url{arXiv:2001.04511}].
Traditionally, progress in motivic homotopy theory is often inspired by theorems in classical homotopy theory. While these results are interesting in their own right, it is perhaps more remarkable that the author uses motivic computations to prove new results about classical stable homotopy theory. For example, some hidden extensions can be more easily found or obstructed in the motivic setting, and some conjectural Adams differentials can be obstructed by motivic information. See e.g. [\textit{D. C. Isaksen} and \textit{Z. Xu}, Topology Appl. 190, 31--34 (2015; Zbl 1327.55007)]. The author also uses the motivic stable homotopy groups of the motivic element \(\tau\) to obtain new differentials and hidden extensions in the classical Adams-Novikov spectral sequence in a range beyond the prior state-of-the-art.
The organization and exposition of the book are especially noteworthy. The introduction provides a clear outline of the necessary foundations in motivic homotopy theory, as well as an instructive roadmap for the program of computing stable homotopy groups in the motivic and classical settings. To close, the author gives several carefully-prepared tables (with references) and charts to summarize the book's computational results. The charts are quite detailed and should be viewed on the digital version of the book, rather than in print. stable homotopy group; stable motivic homotopy theory; May spectral sequence; Adams spectral sequence; cohomology of the Steenrod algebra; Adams-Novikov spectral sequence Research exposition (monographs, survey articles) pertaining to algebraic topology, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Motivic cohomology; motivic homotopy theory, Stable homotopy of spheres, Steenrod algebra, Adams spectral sequences, Hopf algebras and their applications, Stable homotopy theory, spectra, Stable homotopy groups, Massey products Stable stems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Both the Picard group of a quasi-affine scheme and the class group of a normal domain may be described in a module theoretic setting as special cases of the so-called relative Picard group of a ring R associated to an idempotent kernel functor on R-mod. This paper aims to show that similar methods apply in a much more general context. Let us give some examples. It is well known that reflexive Modules (i.e., sheaves of modules) over an integral normal scheme X are locally free sheaves of rank one over \(X^{(1)}\), the subspace of points of codimension 1 of X. Moreover the class group of X is just the Picard group of \(X^{(1)}\). Another example is obtained by considering the local scheme at \(x\in X\) for some scheme X; i.e. consider the scheme Spe\(c({\mathcal O}_{X,x})\) canonically embedded in X. As a set this local scheme at x consists of all \(y\in X\) such that x is in the closure of \(\{\) \(y\}\) in X. - The general problem is the following: Given some subspace \(Y\subset X\), for example, an open subspace or one of the above mentioned, classify the extensions of a quasi-coherent sheaf on Y with structure sheaf \({\mathcal O}_ x| Y\) to a quasi-coherent sheaf on X. This problem arises in the description of relative invariants with respect to Y. If Y is open in X and \({\mathcal M}\) is quasi-coherent on Y then \(i_*{\mathcal M}\) is quasi-coherent on X, where i:\(Y\to X\) is the canonical inclusion. If X is Noetherian (we will assume this throughout) then a result of A. Grothendieck states that any coherent sheaf on Y may be extended to a coherent sheaf on X (note that \(i_*{\mathcal M}\) is not necessarily coherent itself). However, \(X^{(1)}\) is not even a subscheme of X, so in the situations we want to deal with Grothendieck's result cannot be applied. The subspaces Y we consider will be geometrically stable (generically closed), i.e. \(y\in Y\) and \(x\in X\), with \(y\in \{\bar x\}\), the closure of \(\{\) \(x\}\) in X, imply \(x\in Y\). The advantage of these spaces is that for a Noetherian ring the generically closed subsets of Spec(R) correspond bijectively to idempotent kernel functors in R-mod. We develop this correspondence further in a general geometric context, and we derive results similar to the extension result of A. Grothendieck. Along the way we show how some results of \textit{G. Horrocks} [Proc. Lond. Math. Soc., III. Ser. 14, 689- 713 (1964; Zbl 0126.168)] and \textit{R. Treger} [J. Algebra 54, 444-466 (1978; Zbl 0406.14001)] appear as applications of our constructions. generically closed subspace; geometrically stable subspace; Picard group of a quasi-affine scheme; class group of a normal domain; extensions of a quasi-coherent sheaf Van Oystaeyen F., J. Algebra 89 (1984) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard groups, Grothendieck groups, \(K\)-theory and commutative rings Extending coherent and quasi-coherent sheaves on generically closed spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The goal of the work is to show that a plane quartic over an algebraically closed field is uniquely determined by its configuration of inflection lines.
Let the inflection scheme \(\mathrm{Fl}(C)\) of a plane quartic \(C\) consist exactly of the lines having a point of intersection multiplicity at least 3 with \(C\). For describing the inflection scheme of a plane quartic, invariant theory of binary and ternary quartic forms is used. If the characteristics of the field is not 3, the inflection scheme \(\mathrm{Fl}(C)\) for \(C\) is the zero set of two associated contravariants. If the characteristics of the field is 3, the scheme defined by the contravariants is larger than the inflection scheme, so the invariant theory is not useful here and this case is not addressed by the paper.
Each smooth quartic has 24 inflection lines (counted with multiplicity), so its inflection scheme consists of 24 points. The only exception are curves projectively equivalent to the Fermat quartic \(x^4+y^4+z^4 = 0\) over algebraically closed field of char 3, as their inflection schemes are infinite.
So we can define \(\mathcal{F}:\ C\mapsto [\mathrm{Fl}(C)]\) as an element of a Hilbert scheme. Via the closure of the graph of \(\mathcal{F}\) we can study configurations of inflection lines associated to a singular quartic, though in case the curve is not reduced or contains a singular point of multipilicity at least 3, the inflection scheme is infinite. It turns out that each such configuration is different from a configuration associated to any regular curve, so it is possible to distinguish between smooth and singular curves.
As a result the authors conclude that if the characteristic of the field is comprime to 6, the plane quartic curve is uniquely determined by its inflection scheme, except a case in characteristic 13 where the uniqueness is up to projective equivalence.
The last part of the paper deals with the reconstrucion of a general plane quartic from its configuration of inflection lines, again in cases the characteristics of the field is coprime to 6. inflection lines of plane curves; invariants of quartic forms Computational aspects of algebraic curves, Special algebraic curves and curves of low genus, Plane and space curves, Configurations and arrangements of linear subspaces Reconstructing general plane quartics from their inflection lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we study flat deformations of real subschemes of \(\mathbb{P}^n\), hyperbolic with respect to a fixed linear subspace, i.e., admitting a finite surjective and real fibered linear projection. We show that the subset of the corresponding Hilbert scheme consisting of such subschemes is closed and connected in the classical topology. Every smooth variety in this set lies in the interior of this set. Furthermore, we provide sufficient conditions for a hyperbolic subscheme to admit a flat deformation to a smooth hyperbolic subscheme. This leads to new examples of smooth hyperbolic varieties. hyperbolic variety; Hilbert scheme; deformations Topology of real algebraic varieties, Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry On deformations of hyperbolic 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 Spectrahedral sets (or sets admitting an LMI representation) are linear sections of the cone of positive semidefinite matrices. A spectrahedral shadow is a (closed convex semialgebraic) set \(S\subset \mathbb{R}^{d}\) that can be represented as a projection to the first \(d\) coordinates of some spectrahedral subset of \(\mathbb{R}^{d+p}\), that is,
\[
S=\left\{ x\in \mathbb{R}^{d}\mid \exists y\in \mathbb{R}^{p}: \,\sum_{i=1}^{d}x_{i}A_{i}+\sum_{j=1}^{p}y_{j}B_{j}+C\succeq 0\right\},
\]
where \(A_i,B_j,C\) are real symmetric \(n\times n\) matrices. The question of whether or not every closed convex semialgebraic set is a spectrahedral shadow remains widely open. In this work, the authors study the structure of the boundary of spectrahedral shadows of type \((n,d,p)\) (referring to the dimension of matrices, and respectively the Euclidean spaces involved in the definition of \(S\)), under the assumption that both the linear section (defining the spectrahedral shadow in \(\mathbb{R}^{d+p}\)) and the projection to \(\mathbb{R}^{d}\) are generic. In this case, they characterize polynomials vanishing in the boundary of \(S\) (Theorem~1.1). They also discuss several examples of spectrahedral shadows in dimensions~2 (mostly) and~3. The authors fairly mention in the last section that the genericity assumption made on the spectahedral shadows compromises applications to optimization, at least in the current stage. spectahedral shadows; algebraic boundary, algebraic degree of semidefinite programming; extended formulations R. Sinn and B. Sturmfels, \textit{Generic spectrahedral shadows}, SIAM J. Optim., 25 (2015), pp. 1209--1220, . Semialgebraic sets and related spaces, Semidefinite programming, Projective techniques in algebraic geometry Generic spectrahedral shadows | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(B_m=\langle\sigma_1,\dots,\sigma_{m-1}\mid\sigma_j\sigma_{j+1}\sigma_j=\sigma_{j+1}\sigma_j\sigma_{j+1}\), \(\sigma_j\sigma_k=\sigma_k\sigma_j\) for \(|j-k|>1\rangle\). We call the elements of \(B_m\) \(m\)-braids or braids. An \(m\)-braid \(b\) is called quasipositive if \(b=\prod^k_{j=1}a_j\sigma_1a_j^{-1}\) for some \(a_j\in B_m\).
In a series of previous papers we exploited the observation that the quasipositivity of a certain braid provides a necessary condition for the realizability of a given isotopy type by a plane real algebraic curve of a given degree. As a test for quasipositivity we used the Murasugi-Tristram signature inequality, elementary arguments based on linking numbers, or the Garside normal form for braids with three strings. Here we propose a new simple test for quasipositivity and give an example when it gives some new restrictions for real algebraic curves of 7th degree.
The test is based on the following elementary observation. Suppose we are interested in whether a given braid \(b\) is quasipositive. If it is, then the number \(k\) of the factors in any quasipositive presentation is just the image of \(b\) under the Abelianization \(B_m\to\mathbb{Z}\). Let \(\rho\colon B_m\to\text{SU}(n)\) be any unitary representation. Then the matrix \(\rho(b)\) is a product of \(k\) matrices, each of which is conjugated to \(\rho(\sigma_1)\). A necessary and sufficient condition for a given matrix to be presented as a product of matrices from given conjugacy classes was obtained by \textit{S. Agnihotri} and \textit{C. Woodward} [Math. Res. Lett. 5, No. 6, 817-836 (1998; Zbl 1004.14013)]. quasipositivity; unitary representations; braid groups; real algebraic curves Stepan Yu. Orevkov , '' Quasipositivity test via unitary representations of braid groups and its applications to real algebraic curves '', J. Knot Theory Ramifications 10 (2001) no. 7, p. 1005-1023 Braid groups; Artin groups, Real algebraic sets, Knots and links in the 3-sphere, Ordinary representations and characters Quasipositivity test via unitary representations of braid groups and its applications to real algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper cited in the title by \textit{J. V. Leahy} and the author [Nagoya Math. J. 82, 27-56 (1981; Zbl 0528.14001)] the notion of a c- regular function on an algebraic variety over a closed field k of characteristic zero was introduced. The intention was to describe those k-valued functions on a variety that become regular functions when lifted to the normalization but without any reference to the normalization in the definition. Equivalently, the authors aspired to identify the c- regular functions on a given variety with the regular functions on the weak normalization of that variety. - Originally, a c-regular function was defined to be a continuous k-valued function that is regular at each nonsingular point. This does not provide the desired characterization as is evidenced by the following observation. By deleting finitely many points from the normalization of a weakly normal singular curve X, one obtains a curve X' that is homeomorphic to X via a birational morphism, but is not isomorphic to X as varieties. A regular function on X' that is not the restriction of a regular function on the normalization of X provides an example of a continuous function on X that is regular off the singular locus but does not lift to a regular function on the normalization. This is due to the very special nature of the Zariski topology in dimension one.
For varieties without one-dimensional components, the author offers a new characterization of those functions on a variety that lift to regular functions on the normalization. This characterization makes no reference to the normalization and hinges on the fact that a homeomorphic morphism onto a weakly normal variety without one-dimensional components is an isomorphism (the assumption that the ground field have characteristic zero is necessary here). In order to provide a unified treatment, the author now defines a c-regular function on a variety in terms of its normalization and subsequently offers alternative characterizations. weakly normal variety; homeomorphic morphism of varieties; c-regular function; functions on a variety that lift to regular functions on the normalization Vitulli, M. A.: Corrections to ''seminormal rings and weakly normal varieties''. Nagoya math. J. 107, 147-157 (1987) Varieties and morphisms, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), General commutative ring theory, , Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Corrections to ''Seminormal rings and weakly normal 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 Semi-algebraic decision complexity introduces a quantitative finiteness aspect into semi-algebraic geometry. In this paper, we combine methods from abstract real algebraic geometry and complexity theory in order to show lower bounds on the arithmetical cost of semi-algebraic decision trees. In contrast to the topological combinatorial methods the approach is local and based on the relations computed along paths distinguished by certain well defined points in the real spectrum of the polynomial ring \(R[X_1,\dots, X_n]\).
We describe the theme of semi-algebraic decision trees entirely from the point of view of the concept of the real spectrum which extracts the local ``quintessence'' of the behavior of decision trees. Together with the degree argument -- introduced into complexity theory by Strassen -- we obtain bounds that apply to concrete natural problems, and their range of application complements the one of topologically based lower bounds. Various new applications to test problems around interpolation (solvability of overdetermined interpolation tasks) and Chinese remaindering are included.
Having a lower bound on decision complexity of a semi-algebraic subset \(E\subset \mathbb{R}^n\) a further question naturally arises: Is the set of inputs from \(\mathbb{R}^n\) producing a long path in a decision tree ``significant'', or is it only an unspecified exceptional set of possibly very low dimension? Unlike the topological combinatorial methods the real spectrum approach provides such information. For instance, if \(E\) is an irreducible algebraic set then the subset of points in \(E\) producing a short path has dimension strictly less than the dimension of \(E\).
We discuss complexity questions throughout from the variable and relative standpoint. real spectrum; semi-algebraic; decision complexity T. Lickteig, On semialgebraic decision complexity, Technical Report TR-90-052, International Computer Science Institute, Berkeley, and Universität Tübingen, Habilitationsschrift, to appear. , Analysis of algorithms and problem complexity, Symbolic computation and algebraic computation, Semialgebraic sets and related spaces Semi-algebraic decision complexity, the real spectrum, and degree | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(P\) is a point in \(\mathbb{P}^n\) whose ideal is \(I_P\subset k[x_0,\dots,x_n]\), with \(k\) algebraically closed of characteristic zero, the scheme supported on \(P\) defined by \((I_P)^{m}\) is called an \(m\)-multiple point supported on \(P\).
In this paper, the authors completely solve the study of determining the Hilbert function of schemes \(X\subset \mathbb{P}^n\) which are the generic union of \(s\) lines and one \(m\)-multiple point for any \(s\) and \(m\) when \(n\geq 4\) (Theorem 3.2).
Section 4 is devoted to the case \(n=3\) (see Theorem 4.2). Hilbert function; multiple points; varieties of lines Carlini, E.; Catalisano, M. V.; Geramita, A. V., On the Hilbert function of lines union one non-reduced point, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), XV, 69-84, (2016) Configurations and arrangements of linear subspaces, Divisors, linear systems, invertible sheaves, Multiplicity theory and related topics On the Hilbert function of lines union one non-reduced point | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 geometric technique to study a singular point \(x\) of a complex analytic space \(X\) (embedded in affine space) is to consider the limits of tangent hyperplanes, i.e. hyperplanes \(H_u\) containing the tangent space \(T_{X,u}\) to \(X\) at a variable regular point \(u\in X\) near \(x\). Work in this direction was done by Lê, Teissier, Gonzalez-Sprinberg, Lejeune-Jalabert, Snoussi, etc. The case best understood is that of surfaces. This article is a survey of results about limits of tangent hyperplanes of a normal surface singularity.
The first part of the paper reviews results in the case where the surface is contained in \(\mathbb{C}^3\). Milnor numbers are introduced and criteria to conclude that a plane is not a limit of tangents, based on these numbers, are discussed. Other results, relating these limits to certain generatrices of the tangent cone to \(X\) at \(x\) are also explained.
Then the author moves to the case of a normal surface, not necessarily embeddable in three-space. He explains how many of the previous results can be extended to this situation. Much of this was done by Jawad Snoussi (dissertation, University of Marseille).
Finally, the theory of resolution of normal surface singularities is reviewed. Some connections of the previous notions to this theory are explained and, at the end, the author suggests the use of these techniques to attack certain unsolved problems. For instance, that of getting effective bounds for the number of normalized blowing ups (with zero-dimensional centers) necessary to resolve a normal surface singularity, or the similar one when one uses normalized Nash modifications.
This paper is written with great clarity and precision and it should be an eczellent introduction to this subject. modification; survey; tangent hyperplanes; normal surface singularity; Milnor numbers; tangent cone; resolution of normal surface singularities Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Invariants of analytic local rings, Singularities of surfaces or higher-dimensional varieties Geometry of complex surface singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An element \([\Phi] \in Gr_n \left(\mathcal{H}_+, \mathbf{F}\right)\) of the Grassmannian of \(n\)-dimensional subspaces of the Hardy space \(\mathcal{H}_+ = H^2\), extended over the field \(\mathbf{F} = \mathbf{C} (x_{1},\dots, x_{n})\), may be associated to any polynomial basis \(\phi = \{\phi_0, \phi_1,\dots\}\) for \(\mathbf{C}(x)\). The Plücker coordinates \(S_{\lambda, n}^{\phi}(x_1, \ldots, x_n)\; \text{of}\; [\Phi]\), labeled by partitions \(\lambda\), provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system \(\phi\) to the analog \(\{h_i^{(0)} \}\) of the complete symmetric functions generates a doubly infinite matrix \(h_i^{(j)}\) of symmetric polynomials that determine an element \([H] \in \mathrm{Gr}_n(\mathcal{H}_+, \mathbf{F})\). This is shown to coincide with \([\Phi]\), implying a set of generalized Jacobi identities, extending a result obtained by \textit{A. N. Sergeev} and \textit{A. P. Veselov} [Mosc. Math. J. 14, No. 1, 161--168 (2014; Zbl 1297.05244)] for the case of orthogonal polynomials. The symmetric polynomials \(S_{\lambda, n}^{\phi}(x_1, \ldots, x_n)\) are shown to be KP (Kadomtsev-Petviashvili) \(\tau\)-functions in terms of the power sums \([x]\) of the \(x_{a}\)'s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums \(\sum_{\lambda} S_{\lambda, n}^{\phi}([x]) S_{\lambda, n}^{\theta}(\mathbf{t})\) associated to any pair of polynomial bases \((\phi, \theta)\), which are shown to be 2D Toda lattice \(\tau\)-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.{
\copyright 2018 American Institute of Physics} Harnad, J.; Lee, E., Symmetric polynomials, generalized Jacobi-trudi identities and \textit{ {\(\tau\)}}-functions, J. Math. Phys., 59, 091411, (2018) Symmetric functions and generalizations, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Polynomials and rational functions of one complex variable, Hardy spaces, Grassmannians, Schubert varieties, flag manifolds, Schur and \(q\)-Schur algebras, KdV equations (Korteweg-de Vries equations) Symmetric polynomials, generalized Jacobi-Trudi identities and \(\tau\)-functions | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.