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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of ``discrete differential forms'' built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes.
Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes. difference forms; lattice variety; cohomology; difference chains; local exactness; local difference potentials Mansfield, EL; Hydon, PE, Difference forms, Foundations of Computational Mathematics, 8, 427-467, (2008) Discrete version of topics in analysis, de Rham cohomology and algebraic geometry, Čech types, Exterior algebra, Grassmann algebras Difference forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article under review is a continuation of [\textit{E. Elmanto} et. al. ``Motivic infinite loop spaces'', Preprint, \url{arXiv:1711.05248}]. The main result is a concrete description of the \(\infty\)-category of modules over Voevodsky's algebraic cobordism spectrum \(\operatorname{MGL}\) as motivic spectra with finite syntomic transfers
\[
\mathbf{Mod}_{\operatorname{MGL}}(\mathbf{SH}(S)) \simeq \mathbf{SH}^{\operatorname{fsyn}}(S)
\]
for an arbitrary base scheme \(S\) (Theorem 4.1.3). The proof of this result strongly relies on the flexible theory of framed correspondences developed in [loc. cit.]. One important result about these is the reconstruction theorem (Theorem 18 in [\textit{M. Hoyois}, Compos. Math. 157, No. 1, 1--11 (2021; Zbl 1455.14042)]), which says that there is an equivalence of \(\infty\)-categories between motivic spectra and framed motivic spectra
\[
\mathbf{SH}(S) \simeq \mathbf{SH}^\mathrm{fr}(S)
\]
over any scheme \(S\). This may be viewed as the ``sphere spectrum version'' of the above equivalence. The rough idea is that certain motivic spectra of interest may have a more geometric description as motivic spectra with framed correspondences. The second main result of this paper is an example of such a description. Namely for a so called smooth stable tangential structure (Definitions 3.3.1 and 3.3.8) \(\beta\) there is an equivalence of motivic spectra
\[
M\beta \simeq \Sigma^\infty_{\mathbf{T},\operatorname{fr}} \mathcal{FQ}\operatorname{Sm}_S^\beta
\]
where the left hand side is the motivic Thom spectrum associated to \(\beta\) and the right hand side is the framed suspension spectrum of the moduli stack of finite quasi-smooth derived \(S\)-schemes with \(\beta\)-structure (Theorem 3.3.10). This general result has many concrete incarnations, a special case for example is the equivalence of motivic spectra
\[
\operatorname{MGL}_S \simeq \Sigma^\infty_{T,\operatorname{fr}}\mathcal{FS}\mathrm{yn}_S,
\]
where \(\mathcal{FS}\mathrm{yn}_S\) is the moduli stack of finite syntomic \(S\)-schemes (Theorem 3.4.1). From this equivalence the authors then deduce the above description of the \(\infty\)-category of \(\operatorname{MGL}_S\)-modules. Furthermore the motivic recognition principle (Theorem 3.5.14 in [\textit{E. Elmanto} et. al., ``Motivic infinite loop spaces'', Preprint, \url{arXiv:1711.05248}]) ensures that all of the results mentioned above have a more concrete form when \(S\) is a perfect field \(k\). For example one gets an equivalence of \(\infty\)-categories
\[
\mathbf{Mod}_{\operatorname{MGL}}(\mathbf{SH}^\mathrm{veff}(k)) \simeq \mathbf{H}^\mathrm{fsyn}(k)^\mathrm{gp}
\]
between very effective \(\operatorname{MGL}\)-modules and grouplike motivic spaces with finite syntomic transfers (Theorem 4.1.4). It is also noteworthy that for the proof of Theorem 4.1.3 the language of derived schemes, which is used throughout the paper, is essential even though the statement itself does not involve derived schemes. algebraic cobordism; framed correspondences Motivic cohomology; motivic homotopy theory, Stacks and moduli problems Modules over algebraic cobordism | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 explore an abstract algebraic geometry background of multi-operator functional calculus. Results in this paper consisted of arguments from commutative algebra and algebraic geometry combined with the results of the joint spectral theory. First result provides a necessary condition when a functional calculus does exist for the module \(M\) over an open subset \(U\subseteq X\). After a spectrum of an algebraic variety over an algebraically closed field is introduced. Further, the author generalizes the concept of a spectrum introduced above to schemes. Noetherian schemes; quasi-coherent sheaf; spectrum of a module; sheaf cohomology Schemes and morphisms, Sheaves in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Functional calculus in topological algebras, Homological dimension and commutative rings The spectrum of a module along scheme morphism and multi-operator functional calculus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 generalize the classical Lefschetz fixed point theorem in algebraic topology and the Lefschetz formula for the cohomology of coherent sheaf \(\mathcal{F}\) on a projective algebraic variety \(X\). Instead of a single endomorphism they consider a family of automorphisms of \(X\) parametrized by a one-dimensional torus \({\mathbb{G}}_m\). In the case when \(\mathcal{F}\) is a sheaf corresponding to a vector bundle \(F\) and the set of fixed points \(Z\) is finite, each term in the Lefschetz formula can be regarded as the trace of the action of \(f\in{\mathbb{G}}_m\) on the infinite dimensional space \({\widehat{\mathcal{O}}}_{X,x}\otimes F_x\) where \({\widehat{\mathcal{O}}}_{X,x}\) is the local ring at the point \(x\in Z\), and \(F_x\) is the fiber of the vector bundle over \(x\). The authors prove the Lefschetz formula using the cohomology theory of the adelic complexes \({\mathbb{A}}_X(\mathcal{F})^{\bullet}\) of coherent sheaves \(\mathcal{F}\) and the notion of traces of the action of the group \({\mathbb{G}}_m\) on the components of the adelic complex. The adelic Lefschetz formula for a locally free sheaf on a nonsingular projective variety of arbitrary dimension has the form
\[
Tr( {\mathbb{G}}_m,H^{\bullet}(X,\mathcal{F}))=Tr({\mathbb{G}}_m, {\mathbb{A}}^{fix}_X(\mathcal{F}) ),
\]
where \({\mathbb{A}}^{fix}_X(\mathcal{F}) \) is the ``fixed'' part of the adelic complex connected with the set of fixed points \(X^{{\mathbb{G}}_m}\). Since the complex \({\mathbb{A}}^{fix}_X(\mathcal{F}) \) is defined for any scheme, they conjecture that the adelic Lefschetz formula holds for an arbitrary proper scheme over a field \(k\). Lefschetz formula; Lefschetz number; Adelic complexes; one-dimensional torus С. О. Горчинский, А. Н. Паршин, ``Адельная формула Лефшеца для действия одномерного тора'', Тр. Санкт-Петербургского математического общества, 11, 2005, 37 -- 57 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group actions on varieties or schemes (quotients), Cycles and subschemes, Surfaces and higher-dimensional varieties Adelic Lefschetz formula for the action of a one-dimensional torus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{N. Schwartz} [in: Ordered algebraic structures, The 1991 Conrad Conf., 169-202 (1993; Zbl 0827.13008)] gave the following definitions: Let \(A\) be a ring or \(\mathbb{Q}\)-algebra contained in its real closure \(R(A)\), the so-called ring of abstract semialgebraic functions. For \(\alpha\) in the real spectrum \(\text{Sper}(A)\), we denote by \(\rho(\alpha)\) the real closure of the quotient field \(qf(A/\text{supp}(\alpha))\) with respect to the total order specified by \(\alpha\). This makes \(\prod \rho(\alpha)\) a lattice ordered ring with respect to the pointwise lattice order containing \(R(A)\). An \(a \in R(A)\) is piecewise polynomial if the real spectrum \(\text{Sper}(A)\) of \(A\) can be covered by finitely many closed constructible sets \(C_1, \ldots, C_r\) such that for certain \(a_1, \ldots, a_r\in A\) there holds \(a |C_i = a_i\) for all \(i\). Denoting the collection of piecewise polynomial functions by \(PW(A)\) and the collection of elements generated in the sense below by the elements of \(A\), the so-called sup-inf definable functions, by \(L(A)\), one has a chain \(L(A)\subseteq PW(A) \subseteq R(A) \subseteq \prod \rho (\alpha)\) of lattice ordered (sub)rings. A ring \(A\) is Pierce-Birkhoff if \(L(A) = PW(A)\). Defining inductively \(L_0(A) = A\), \(L_{n+1}(A) = L_n(A) [|a |: a \in L_n(A)]\), one has \(L(A) = \bigcup_n L_n (A)\).
The unresolved conjecture these authors made in 1956 [cf. \textit{G. Birkhoff} and \textit{R. S. Pierce}, Anais Acad. Bras. Cic. 28, 41-69 (1956; Zbl 0070.26602)] is that this equality holds for the case \(A=\mathbb{Q}[X_1,\dots,X_n]\), in which indeed \(L(A)\) and \(PW(A)\) assume the meaning the terminology suggests [see \textit{J. J. Madden}, Arch. Math. 53, No. 6, 565-570 (1989; Zbl 0691.14012) or \textit{C. N. Delzell}, Rocky Mt. J. Math. 53, No. 3, 651-668 (1989; Zbl 0715.14047)]. In the paper under review a construction corresponding to the inductive algebraic construction of \(L(A)\) is provided for \(PW(A)\). Interesting corollaries for understanding the Pierce-Birkhoff conjecture are derived. Assume (real) intermediate rings \(A \subset B\subset C\subset PW(A)\). Let \((\text{Spec}B)_{\text{re}}=\{\text{real primes of }B\) Pierce-Birkhoff conjecture; lattice ordered ring; semialgebraic function; piecewise polynomial functions; sup-inf definable; constructible sets Semialgebraic sets and related spaces, Real algebra An algebraic construction of the ring of piecewise polynomial functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper presents and compares different approaches to geometry over the field \({\mathbb F}_1\) of one element and proposes a definition of analytic functions through functions on roots of unity. It is widely accepted that roots of unity should play the role of algebraic integers over \({\mathbb F}_1\) and the author imposes the philosophy that whenever there appear naturally some functions with values in the roots of unity, one should consider them as cyclotomic coordinates of an underlying \({\mathbb F}_1\)-scheme. The author gives some examples of this instance. The philosophy is extended to define analytic functions on \({\mathbb F}_1\)-schemes via the construction of analytic functions by \textit{K. Habiro} [Publ. Res. Inst. Math. Sci. 40, No. 4, 1127--1146 (2004; Zbl 1098.13032)], in which paper it is shown that the evaluation map on
\[
\widehat {\mathbb Z}_n=\lim_{^\leftarrow_N}{\mathbb Z}[x_1,\dots,x_n]/\langle \phi_N(x_1),\dots,\phi_N(x_n)\rangle,
\]
where \(\phi_N(x)=(x^N-1)(x^{N-1}-1)\dots (x-1)\), gives an injective map
\[
\widehat{\mathbb Z}_n\hookrightarrow \text{Map}(\mu^n,{\mathbb Z}[\mu]),
\]
where \(\mu\) is the set of roots of unity. In this way, elements of \(\hat {\mathbb Z}_n\) are to be interpreted as analytic functions on roots of unity. The author proposes to establish analytic geometry over \({\mathbb F}_1\) using these functions. Manin, Yu. I., Cyclotomy and analytic geometry over \(\mathbb{F}_1\), (Quanta of Maths, Clay Math. Proc., vol. 11, (2010), Amer. Math. Soc.), 385-408 Finite ground fields in algebraic geometry, Relations with noncommutative geometry, Polynomials over commutative rings, Completion of commutative rings, Generalizations (algebraic spaces, stacks) Cyclotomy and analytic geometry over \(\mathbb{F}_1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 analogy with the situation in classical stable homotopy theory, the author constructs localization functors in stable motivic homotopy theory. The aim is to set up a ``motivic chromatic spectral sequence'' converging to the \(E_2\) term of the \(\mathbf{BP}\)-based motivic Adams-Novikov spectral sequence. This aim is reached provided a certain chain complex is exact.
The paper is full of interesting questions, e.g. does the chromatic convergence theorem hold in the motivic setting? However, the author cautions the reader that even if the questions listed in the current paper are answered satisfactorily, one is still far away from the structured state of affairs in classical stable homotopy theory.
The first part of the paper is devoted to setting up Hirschhorn's framework for localizations in motivic homotopy theory. The second part explores the possible application of these techniques in motivic homotopy, with the guiding aim of eventually being able to understand \([S^j,{\mathbf{G}_m}^{\wedge m}]_{\mathcal{SH}(k)}\) via an Adams-Novikov spectral sequence. In order to get information about the \(E_2\)-term the author attempts to set up a chromatic spectral sequence. Due to the lack of knowledge about the relevant spectra in motivic stable homotopy theory, it is not quite clear what the correct definition should be, nor that the ones chosen actually will work, but the author carefully points at many of the questions and problems that are associated with his choices.
The paper ends with an appendix showing that Bousfield localization for classical spectra is a Hirschhorn localization. motivic chromatic spectral sequence DOI: 10.1017/S030500410500890X Adams spectral sequences, Motivic cohomology; motivic homotopy theory, Homotopy theory and fundamental groups in algebraic geometry, Abstract and axiomatic homotopy theory in algebraic topology Localizations in motivic homotopy theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field of arbitrary characteristic \(p\geq 0\) and \(C\) a curve defined over \(k\) with only one branch at infinity. The only branch at infinity corresponds to a valuation \(v\) of \(K(C)\), the field of rational functions over \(k\). Let \(\Gamma_1(P) = \{-v(g) \mid g\in R\}\), where \(R\) is the affine \(k\)-algebra of \(C\) and \(P\) the branch point at infinity and let \(\{x, y\}\) be coordinates of \(R\) and \(f(x,y)=0\) the expression of \(C\) in the affine part. If \(p\) does not divide \(n= -v(x)\) and \(m= -v(y)\) at the same time, then it can be proved that there exists a positive integer \(h\) and a sequence of positive integers \(\delta_0, \dots, \delta_h\) such that they generate \(\Gamma_1(P)\) and:
(1) If \(d_i= \text{gcd} (\delta_0, \dots, \delta_{i-1})\), for \(1\leq i\leq h+1\) and \(n_i= d_i/d_{i+1}\), \(1\leq i\leq h\), then \(d_{h+1} =1\) and \(n_i>1\) for \(1\leq i\leq h\).
(2) For \(1\leq i\leq h\), \(n_i \delta_i\) belongs to the semigroup generated by \(\delta_0, \dots, \delta_{i-1}\).
(3) \(\delta_i < \delta_{i-1} n_{i-1}\) for \(i=2, \dots, h\).
The author considers the converse problem, that is, given a semigroup \(\Gamma\) generated by a sequence of natural numbers \(\delta_0,\delta_1,\dots,\delta_h\) satisfying (1), (2) and (3) and fixed a field \(k\), try to find curves with only one branch at infinity and such that \(\Gamma_1(P) = \Gamma\). In this paper the author studies curves with the above property, that is, only one branch at infinity and such that \(\Gamma_1(P) = \langle \delta_0, \delta_1, \dots, \delta_h \rangle\), using as main tool Newton polygons. The concept of cluster allows the author to classify the set of curves with only one branch at infinity and such that \(\Gamma_1(P)\) is the semigroup \(\Gamma = \langle \delta_0, \delta_1, \dots, \delta_h \rangle\). curve with only one branch at infinity; characteristic \(p\); Newton polygons Reguera López, A.: Semigroups and clusters at infinitiy. Algebraic geometry and singularities (La Rábida, 1991), Progr. Math., vol. 134, pp. 339-374. Birkhäuser, Basel (1996) Algebraic functions and function fields in algebraic geometry, Ramification problems in algebraic geometry Semigroups and clusters at infinity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and is virtually compact; a generalization to higher dimensions is also given. Among the one-dimensional sets which are not virtually compact, we show that at least a large subclass supports indeterminate moment sequences. Moreover, we prove that the determinateness of a moment sequence is implied by the same condition (in general easier to verify) of the push-forward sequence via finite morphisms. Putinar, M., Scheiderer, C.: Multivariate moment problems: geometry and indeterminateness. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) \textbf{5}(2), 137-157 (2006) Moment problems, Real algebraic sets Multivariate moment problems: geometry and indeterminateness | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is one of the best surveys the reviewer has ever read. It is written in Polish, but, fortunately, its even longer and more detailed version is to be published under the title ``On continuous rational functions'' in Proceedings of FJV 2017, World Scientific Publishing House.
The author reviews here a domain that he himself started in 2009 and that attracted many French (Fichou, Hulsman, Mangolte, Monnier\dots), Polish (Bilski, Kurdyka, A. and G. Valette\dots), American (see [Math. Ann. 370, No. 1--2, 39--69 (2018; Zbl 1407.14056)] by \textit{J. Kollár} et al.] and other mathematicians.
This new relatively domain deals with regulous functions.
Definition. A \(k\)-regulous function is a real function \(f\) with the \(k\)-th derivative continuous, that is equal to a rational function \(p/q\) outside the zero set of the function \(q\).
The 0-regular functions (only the continuity demanded) are also called hereditary rational or stratification regular.
Chapter 2 introduces all the notions needed in the sequel (like, for instance, algebraic sets, the Zariski topology) and contains several examples.
Chapter 3 presents 0-regulous functions. Three equivalent conditions are given for a function \(f\) defined and continuous on an algebraic set \(X\) are given.
Theorem 3.2 states the link between 0-regulous functions and the functions that have a rational representation (the latter are also defined in Chapter 1).
Theorems 3.3 and 3.4 speak about the functions that are 0-regulous on algebraic arcs or algebraic curves. Namely, under some assumptions, some of them are arc analytic (the definition first introduced by Kurdyka).
A clear proof and several interesting examples are given.
In Chapter 4, the author deals with \(k\)-regulous functions, \(k>0\). The ring of such functions is studied. This is not a noetherian ring but has a property that is an analogue to Hilbert Nullstellensatz. Namely, in Conclusions 4.9 and 4.10 we learn that for any ideal I of the ring of \(k\)-regulous functions there is the equality
\[
J(Z(I)) = \mathrm{Rad}(I)
\]
and, moreover, there exists a function \(f\) in the ring such that \(\mathrm{Rad}(f) = \mathrm{Rad}(I)\).
Conclusion 4.11 states that if \(f\) and \(g\) are two \(k\)-regulous functions, then
\[
\mathrm{Rad}(f) = \mathrm{Rad}(g)\quad\text{iff }Z(f) = Z(g)
\]
Chapters 5 and 6 contain the approximation theorems and the results concerning vector bundles.
It was a real pleasure to read this text. Not only does it present this new domain, but provides the reader with the history and the motivation of this research together with many pertinent examples. Real algebraic sets, History of algebraic geometry, Development of contemporary mathematics New trends in real algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper provides a generalization of the result of \textit{A. Bayer} and \textit{E. Macrì} [J. Am. Math. Soc. 27, No. 3, 707--752 (2014; Zbl 1314.14020)] relating the Bridgeland stability manifold and the movable cone of the moduli space of stable objects on smooth projective varieties to the setting of arbitrary separated schemes of finite type.
One of the key ideas is, as stated in the section 1.2, that one should introduce a proper definition of the stability function. For a separated scheme \(Y\) of finite type, the authors define a variant \(K^{\text{num}}_c(Y)\) of the numerical Grothendieck group as the quotient of \(K(D_c(Y))\) by the radical of the Euler pairing with perfect complexes on \(Y\), where \(D_c(Y)\) is the full subcategory of objects with proper support in the bounded derived category. Then they introduce the notion of numerical Bridgeland stability condition for compact support on \(Y\) as a pair \(\sigma=(Z_{\sigma}, \mathcal{P}_{\sigma})\) of a group homomorphism \(Z_{\sigma}:K^{\text{num}}_c(Y) \to \mathbb{C}\) and a slicing \(\mathcal{P}_{\sigma}\) of \(D_c(Y)\).
The main statement is Theorem 1.2.1, where a family of nef divisors on moduli spaces of stable objects in the above sense is constructed. In the proof, the authors construct a family of t-structures with left-compact support, which is introduced in Definition 2.1.3 as a replacement of the ordinary compact support property so that it behaves well under the derived restriction.
This paper also constructs stability conditions in the case when \(Y\) is a smooth scheme, projective over an affine scheme, equipped with a tilting bundle \(E\). In this setting, denoting by \(A\) the endomorphism algebra of \(E^{\vee}\), we have stability conditions \(\sigma_\theta\) on the derived category of finite \(A\)-modules parametrized by stability parameter \(\theta\) for \(A\)-modules in the sense of King. By the tilting equivalence one can construct stability conditions on \(D_c(Y)\) from \(\sigma_\theta\). Then Theorem 1.4.1 says that the numerical divisor class constructed by Theorem 1.2.1 is equivalent to the polarizing ample line bundle on the moduli space of stable \(A\)-modules.
The main result of this paper seems to have a wide range of applications as indicated in the section 1.5. The reviewer recommends this paper especially for those interested in some application of Bridgeland stability conditions to birational geometry of moduli spaces and geometric representation theory. Bridgeland stability conditions; derived categories; t-structures; moduli spaces of sheaves and complexes; nef divisors Bayer, A., Craw, A., Zhang, Z.: Nef divisors for moduli spaces of complexes with compact support. Sel. Math. (N.S.) \textbf{23}(2), 1507-1561 (2017) Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K3\) surfaces and Enriques surfaces, Derived categories, triangulated categories Nef divisors for moduli spaces of complexes with compact support | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article is devoted to the study of linear systems \({\mathcal L}_d(m^n)\) of plane curves of degree \(d\) passing through \(n\) points in general position with the same multiplicity \(m\) at each of them. These linear systems are known as homogeneous linear systems. Let \(v= [d(d+ 3)-nm(m+ 1)]/2\) and \(e= \max\{-1,v\}\) be respectively the virtual and the expected dimension of \({\mathcal L}_d(m^n)\). Since the base points are chosen in general position \(\ell= \dim{\mathcal L}_d(m^n)\) achieves its minimum value. A system \({\mathcal L}_d(m^n)\) is said to be non-special if \(\ell=e\) and special otherwise. Let \(A(n)= \sqrt{1+ 4nm(m+1)}\). Then the main result of the paper can be announced as follows:
Suppose that all homogeneous linear systems, of all degrees \(d\) and multiplicity \(m\), through \(n_1\) and \(n_2\) points in general position \({\mathcal L}_d(m^{n_1})\), and \({\mathcal L}_d(m^{n_2})\) respectively, are non-special, and that for an integer \(k\) such that \((A(n_2)- 3)\leq k\leq (A(n_2)- 1)/2\) the linear systems \({\mathcal L}_d(k^{n_1})\), \({\mathcal L}_{k-1}(m^{n_2})\) and \({\mathcal L}_k(m^{n_2})\) are non-special, then \({\mathcal L}_d(m^{n_1n_2})\) is also non-special.
This is a generalization of \textit{L. Evain}'s result [J. Algebra. Geom. 8, 787--796 (1999; Zbl 0953.14027)], who proved the theorem for \(n= 4^h\). In order to achieve their result the authors use a degeneration technique developed by \textit{C. Ciliberto} and \textit{R. Miranda} [J. Reine Angew. Math. 501, 191--220 (1998; Zbl 0943.14002)]. Buckley, A.; Zompatori, M.: Linear systems of plane curves with a composite number of base points of equal multiplicity. Trans. amer. Math. soc. 355, 539-549 (2003) Special divisors on curves (gonality, Brill-Noether theory), Plane and space curves, Divisors, linear systems, invertible sheaves Linear systems of plane curves with a composite number of base points of equal multiplicity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In 1960's, M. Sato introduced the notion of prehomogeneous vector spaces and proved the functional equations of zeta functions associated with prehomogeneous vector spaces defined over \(\mathbb{R}\) or \(\mathbb{C}\):
\[
\biggl( \bigl | f(x) \bigr |^{\mathfrak s}_ i \biggr)^ \wedge = \sum_ j \gamma_{ij} ({\mathfrak s}) \bigl | f(x) \bigr |_ j^{-n/d - {\mathfrak s}}.
\]
Similar functional equations associated with regular prehomogeneous vector spaces defined over \({\mathfrak p}\)-adic fields have been proved by J. Igusa and generalized to multivariables by F. Sato under certain assumption on their singular sets.
But even when a prehomogeneous vector space \((G,\rho,V)\) satisfies the sufficient conditions of F. Sato which assure the functional equations of zeta functions, the prehomogeneous vector space \((\widetilde G, \widetilde \rho, \widetilde V)\), which is obtained by the castling transformation of \((G, \rho, V)\), does not necessarily satisfy them. Thus even if the functional equations of zeta functions of \((G, \rho, V)\) hold, we do not know whether the functional equations of zeta functions of \((\widetilde G, \widetilde \rho, \widetilde V)\) hold or not. Since the castling transformation is a standard procedure of constructing new prehomogeneous vector spaces, it is natural to ask the existence of the functional equations of \((\widetilde G, \widetilde \rho, \widetilde V)\) when the functional equations of \((G, \rho, V)\) hold.
In this paper we prove the following: Theorem. If the functional equations of zeta functions of \((G, \rho, V)\) hold, then the functional equations of zeta functions of \((\widetilde G, \widetilde \rho, \widetilde V)\) hold and vice versa. functional equations of zeta functions; prehomogeneous vector space; castling transformation Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Homogeneous spaces and generalizations On functional equations of prehomogeneous vector spaces obtained from castling transforms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R=k[X_1,\dots,X_d]\) be the polynomial ring over a field \(K\) of characteristic 0, \(m\) the graded maximal ideal and \(I\) a proper homogeneous ideal. The authors investigate the asymptotic growth of \(\lambda(\text{ Ext}^d_R(R/I^n),R)\) as a function of \(n\), where \(\lambda(M)\) denotes the length of an \(R\)-module \(M\). When \(R\) is a local Gorenstein ring and \(I\) is an \(m\)-primary ideal, then this is easily seen to be equal to \(\lambda(R/I^n)\) and hence a polynomial in \(n\). It is also previously known that this extends to \(m\)-primary ideals in local Cohen-Macaulay rings \(R\). The authors consider homogeneous ideals in polynomial rings which are not \(m\)-primary and show that this limit exists asymptotically. The authors show that by local duality
\[
\lambda(\text{ Ext}^d_R(R/I^n,R(-d)))=\lambda(H_m^0(R/I^n))
\]
and the limit
\[
\lim_{n\rightarrow\infty}\frac{\lambda(H_m^0(R/I^n))}{n^d}=\lim_{n\rightarrow\infty}\frac{\lambda(\text{ Ext}^d_R(R/I^n,R(-d)))}{n^d}.
\]
The authors also give an example of a regular ring \(S\) of dimension 4 (which is essentially of finite type over the field of complex numbers) and an ideal \(J\) of \(S\) such that the limit
\[
\lim_{n\rightarrow \infty}\frac{\lambda(\text{ Ext}^d_S(S/J^n,S))}{n^d}
\]
is an irrational number. In particular \(\lambda(\text{ Ext}^d_S(S/J^n,S))\) is not a polynomial nor a quasi-polynomial for large \(n\). The authors' work uses ideas and techniques from [\textit{D. Kirby}, Math. Proc. Cambridge Philos. Soc. (3)105, 441--446 (1989; Zbl 0689.13004), \textit{V. Kodiyalam}, Proc. Am. Math. Soc. (3) 118, 757--764 (1993; Zbl 0780.13007), \textit{E. Theodorescu}, Math. Proc. Camb. Philos. Soc. 132, 75--88 (2002; Zbl 1054.13007)] on polynomial growth of the length of extension functors. powers of ideals; local cohomology; Hilbert function Cutkosky, Steven Dale; Hà, Huy Tài; Srinivasan, Hema; Theodorescu, Emanoil, Asymptotic behavior of the length of local cohomology, Canad. J. Math., 57, 6, 1178-1192, (2005) Local cohomology and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Local cohomology and algebraic geometry Asymptotic behavior of the length of local cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review provides the closing chapter to the story of arithmetic equidistribution begun in 1997 by \textit{L. Szpiro}, \textit{E. Ullmo}, and \textit{S. Zhang} [Invent. Math. 127, No. 2, 337--347 (1997; Zbl 0991.11035)]. Its author has written a lucid introduction in which he --- with remarkable clarity --- states the goals of his work, his main technical results on arithmetically big line bundles, the consequences for dynamical and arithmetic equidistribution, the context into which these results fit in the literature, and all of the requisite definitions. The interested researcher would be best served by perusing the extended introduction to the paper.
For the reader who would like a more hasty description of the results in the article, I offer the following summary (which assumes some familiarity with the objects of Arakelov theory).
Theorem. (Generic Equidistribution of Small Points) Suppose \(X\) is a projective variety of dimension \(n-1\) over a number field \(K\), and \(\overline{{\mathcal L}}\) is an adelic metrized line bundle over \(X\) such that \(\mathcal{L}\) is ample and the metric is semipositive. Let \(\{x_m\}\) be an infinite sequence of algebraic points in \(X(\overline{K})\) that is generic and small. Then for any place \(v\) of \(K\), the Galois orbits of the sequence \(\{x_m\}\) are equidistributed in the analytic space \(X_{\mathbb{C}_v}^{\text{an}}\) with respect to the probability measure \(d\mu_v = c_1\left(\overline{\mathcal{L}}\right)_v^{n-1} / \text{deg}_{\mathcal{L}}(X)\).
The seminal work of Szpiro/Ullmo/Zhang mentioned above proves this result in the case where \(v\) is an archimedean place of \(K\), and the curvature form \(c_1\left(\overline{\mathcal{L}}\right)_v^{n-1}\) is strictly positive. \textit{A. Chambert-Loir} [J. Reine Angew. Math. 595, 215--235 (2006; Zbl 1112.14022)] constructed the nonarchimedean analogue of this curvature form (supported on a \(v\)-adic analytic space as defined by Berkovich). He then proved the above result for \(v\) nonarchimedean under the added hypothesis that \(X\) is a curve, or that the metric on \(\overline{\mathcal{L}}\) is defined by an ample model. The contribution of the paper under review is to remove these positivity hypotheses by introducing the notion of an arithmetically big line bundle. Some large tensor power of such a line bundle has many small sections, and this is exactly the key to extending the proof of Szpiro/Ullmo/Zhang. The author's main result is an Arakelov-theoretic generalization of Siu's theorem. Essentially, it gives a numerical criterion for the existence of small sections of the difference of two ample line bundles. Here is a simplified version of it:
Theorem. (Arithmetic Siu's Theorem) Let \(\overline{\mathcal{L}}\) and \(\overline{\mathcal{M}}\) be two Hermitian line bundles over an arithmetic variety \(X\) of dimension \(n\). Assume that \(\overline{\mathcal{L}}\) and \(\overline{\mathcal{M}}\) are ample. Then
\[
\begin{aligned} h^0\left( \left(\overline{\mathcal{L}} \otimes \overline{\mathcal{M}}^{\vee}\right)^{\otimes N}\right) &:= \log \#\left\{s \in \Gamma\left(X, \left(\mathcal{L} \otimes \mathcal{M}^{\vee}\right)^{\otimes N} \right): \|s \|_{\text{sup}} < 1 \right\} \\ &\geq \frac{\widehat{c}_1\left(\overline{\mathcal{L}}\right)^n - n \cdot \widehat{c}_1\left(\overline{\mathcal{L}}\right)^{n-1} \widehat{c}_1\left(\overline{\mathcal{M}}\right)}{n!} N^n + o(N^n). \end{aligned}
\]
In particular, \(\overline{\mathcal{L}} \otimes \overline{\mathcal{M}}^{\vee}\) is arithmetically big if \ \(\widehat{c}_1\left(\overline{\mathcal{L}}\right)^n - n \cdot \widehat{c}_1\left(\overline{\mathcal{L}}\right)^{n-1} \widehat{c}_1\left(\overline{\mathcal{M}}\right) > 0\).
The paper is divided into three sections: Introduction, Arithmetic bigness, and Equidistribution theory. The second section forms the technical heart of the paper; it is quite difficult, although carefully written. The third section deals with applications of the bigness result, including how to deduce equidistribution from the arithmetic version of Siu's Theorem. It can be read independently from the second section. Arakelov theory; big line bundle; arithmetic variety; equidistribution; Siu's Theorem; arithmetic volume Yuan, X., \textit{big line bundles over arithmetic varieties}, Invent. Math., 173, 603-649, (2008) Arithmetic varieties and schemes; Arakelov theory; heights Big line bundles over arithmetic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors elucidate the cohomology meaning of the Kadomtsev-Petviashvili (KP) hierarchy of integrable equations which was considered for the first time by Mulase (1984). The KP flows are known to describe a point movement in the universal Grassmannian. The authors show that the Faà di Bruno polynomials, which are defined recursively, form a basis in a subspace of the universal Grassmannian associated to the KP hierarchy. For the algebraic geometrical solutions of the KP equations a point in the universal Grassmannian is associated via the Krichever map to the datum of a smooth algebraic spectral curve, a point on it with an appropriate local coordinate, a line bundle and a local trivialization in a neighborhood of the point. In this situation the Faà di Bruno recursion relations appear to be the cocycle condition for the Welters hypercohomology group which describes the deformations of the line bundle over the spectral curve. The authors illustrate the general theory by the example of an elliptic spectral curve. integrable systems; Kadomtsev-Petviashvili hierarchy; elliptic spectral curve; hypercohomology groups Falqui, G.; Reins, C.; Zampa, A.: Krichever maps, fa à di bruno polynomials, and cohomology in KP theories. Lett. math. Phys. 42, No. 4, 349-361 (1997) KdV equations (Korteweg-de Vries equations), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Other completely integrable equations [See also 58F07], Curves in algebraic geometry Krichever maps, Faà di Bruno polynomials, and cohomology in KP theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A set valued function \(f:X\longrightarrow 2^{Y}\) into the nonempty closed subsets of \(Y\) is upper semi-continuous (usc) if for each \(x \in X\) and each open set \(V \subset Y\) containing the set \(f (x)\), there exists an open set \(U\subset X\) such that \(x \in U\) and \(\bigcup_{u\in U} f (u) \subset V\). In the present paper the authors consider inverse limits \(\displaystyle{\lim_{\longleftarrow} f}\), obtained by usc set valued functions \(f:X\longrightarrow 2^{X}\) and then investigate the fundamental group of these inverse limits. They provide well chosen examples of bonding functions which demonstrate the behavior of loops in the inverse limit. The examples realize groups which are most likely to appear as the fundamental groups of one-dimensional spaces: free groups and the fundamental group of the Hawaiian Earring.
The left shift is an obvious construction, obtained by removing the first coordinate in the inverse limit. More precisely, the left shift map \(S : X^{\infty}\longrightarrow X^{\infty}\) defined by \(S(x_1, x_2, x_3, \ldots) = (x_2, x_3, \ldots)\) induces the map \(S :\displaystyle{\lim_{\longleftarrow} f}\longrightarrow \displaystyle{\lim_{\longleftarrow} f}\). In the present paper, the authors define right shifts of loops \(T:\displaystyle{\lim_{\longleftarrow} f}\longrightarrow \displaystyle{\lim_{\longleftarrow} f}\), for some inverse limits. Using the shift constructions they prove that the fundamental group of an inverse limit is often trivial or uncountable.
Finally, for every set \(S\) of cardinality \(|S|\leq \aleph_1\), they construct a surjective \(f:I\longrightarrow 2^{I}\) , such that \(\displaystyle{\lim_{\longleftarrow} f}\) is path connected and its fundamental group is the free group on the set \(S\). inverse limits; set valued functions; fundamental group Set-valued maps in general topology, Special constructions of topological spaces (spaces of ultrafilters, etc.), Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.), Homotopy theory and fundamental groups in algebraic geometry On the fundamental group of inverse limits | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors consider an affine hypersurface \(X\subset\mathbb A^n\) defined by a polynomial \(f\) in the ring \(R=k[x_1,\dots,x_n]\) over an algebraically closed field \(k\) of characteristic \(0\). The singular locus of \(X\) is defined by the vanishing of the Jacobian ideal \(I(f)\), which is generated by \(f\) and the \(n\) derivatives of \(f\). The ideal \(J(f)\) generated by the derivatives alone is the gradient ideal of \(f\). The authors compare the schemes defined by \(I(f)\) and by \(J(f)\) around a singular point \(p\) of \(X\), assuming that \(X\) has only isolated singularities (so that, in particular, it is reduced). The Milnor number of \(X\) at \(p\) is the dimension of the localization \((R/J(f))_p\), while the Tjurina number of \(X\) at \(p\) is defined as the dimension of the localization \((R/I(f))_p\). \(X\) is called `locally Eulerian' if the Milnor number is equal to the Tjurina number at each singular point. The authors prove that \(X\) is locally Eulerian if and only if the symmetric and the Rees algebras of \(I(f)\) in \(R\) are isomorphic. In turn, the condition is equivalent to say that the Tjurina algebra \(R/I(f)\) is artinian Gorenstein. Then, the authors extend their study to the case of reduced projective plane curves \(C\), defining \(C\) to be `of gradient type' if the restriction of \(C\) to each affine chart associated to the singular points is locally Eulerian. The authors prove that curves \(C\) with only simple singularities are of gradient type. Furthermore, they prove that (reduced) quartics are of gradient type, while there are examples of quintics and sextics which are not of gradient type. singularities of hypersurfaces; plane curves Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Hypersurfaces and algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of surfaces or higher-dimensional varieties, Plane and space curves Hypersurfaces with linear type singular loci | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review concers itself with constructing a pure Hodge structure on the \(M\)-equivariant topological \(K\)-theory of the underlying analytic space of a large class of smooth quasiprojective \(M\)-schemes equipped with a suitable generalization of the canonical stratification, where \(M\) is a compact Lie group. Note that in this case, the approach of Deligne to equivariant Hodge theory does not generalize to equivariant topological \(K\)-theory since the latter does not simply arise as the \(K\)-theory of a simplicial scheme.
Let \(G\) be a reductive algebraic group and let \(X\) be a smooth quasiprojective \(G\)-scheme which admits a complete Kirwan-Ness stratification. An example of such a \(G\) is given by the complexification of a compact Lie group. The Kirwan-Ness stratification was introduced by [\textit{C. Teleman}, Ann. Math. (2) 152, No. 1, 1--43 (2000; Zbl 0980.53102)] as a generalization of the canonical stratification of a \(G\)-variety considered among others by Hesselink, Ness, and Kirwan. In this case, the authors show that the non-commutative Hodge-de Rham spectral sequence for the periodic cyclic homology of the derived category \(\mathrm{Perf}(X/G)\) of \(G\)-equivariant perfect complexes of coherent sheaves on \(X\) degenerates on the first page. Previous work [\textit{D. Kaledin}, Prog. Math. 324, 99--129 (2017; Zbl 1390.14011)] establishes that the non-commutative Hodge-de Rham spectral sequence degenerates for dg-categories which are smooth and proper. In the situation of the paper, the scheme is in general not proper and \(\mathrm{Perf}(X/G)\) need not be smooth even though \(X\) is assumed to be smooth. The main technical tool, is the observation that \(\mathrm{Perf}(X/G)\) can be obtained as a retract of an infinite semi-orthogonal decomposition of smooth and proper Deligne-Mumford stacks together with the fact that Hochschild homology is an additive invariant in the sense that it takes semi-orthogonal decompositions of dg-categories to direct sums.
The authors use the degeneration of the Hodge-de Rham spectral sequence to construct a pure Hodge structure on the equivariant topological \(K\)-theory. Recent work of [\textit{A. Blanc}, Compos. Math. 152, No. 3, 489--555 (2016; Zbl 1343.14003)] provides a construction of a topological \(K\)-theory spectrum \(K^{\mathrm{top}}(\mathcal{A})\) for any dg-category \(\mathcal{A}\) over \(\mathbb{C}\), and additionally constructs a natural Chern character map \(K^{\mathrm{top}}(\mathcal{A}) \to \mathrm{HP}(\mathcal{A})\) to the periodic cyclic homology of \(\mathcal{A}\). The authors construct a topologization map \(K^{\mathrm{top}}(\mathcal{A}) \to K_M(X^{\mathrm{an}})\) for any smooth \(G\)-quasiprojective scheme \(X\), where \(G\) is the complexification of \(M\), and \(K_M(X^{\mathrm{an}})\) denotes the \(M\)-equivariant topological \(K\)-theory spectrum of the underlying analytic space of \(X\). Finally, the authors show that if \(X\) is a smooth quasiprojective \(G\)-scheme which admits a semicomplete Kirwan--Ness stratification, then the topologization map and the Chern character map provide equivalences
\[
K_M(X^{\mathrm{an}}) \otimes \mathbb{C} \xleftarrow{\simeq} K^{\mathrm{top}}(\mathrm{Perf}(X/G)) \otimes \mathbb{C} \xrightarrow{\simeq} C_{\bullet}^{\mathrm{per}}(\mathrm{Perf}(X/G)),
\]
where \(C_{\bullet}^{\mathrm{per}}(\mathrm{Perf}(X/G))\) denotes the complex whose homology is given by the periodic cyclic homology. Consequently, the equivariant topological \(K\)-theory \(K^n_M(X^{\mathrm{an}})\) admits a pure Hodge structure of weight \(n\) coming from the degeneration of the Hodge--de Rham spectral sequence for the periodic cyclic homology. In fact, the \(M\)-equivariant topological \(K\)-theory is a module over the representation ring \(\mathrm{Rep}(M)\) and the Hodge structure is \(\mathrm{Rep}(M)\)-linear. Furthermore, the authors identify the graded pieces in terms of the cohomology of the derived inertia stack which is sometimes also referred to as the derived loop stack.
Additionally, the authors establish degeneration of the Hodge-de Rham spectral sequence for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models. \(K\)-theory; Hochschild homology; Hodge structures; non-commutative geometry; equivariant Hodge theory; equivariant topological \(K\)-theory; Hodge-de Rham spectral sequence Noncommutative algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), \(K\)-theory and homology; cyclic homology and cohomology, Equivariant \(K\)-theory Equivariant Hodge theory and noncommutative 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 \(k\) be an algebraically closed field of characteristic 0, and let \(T\) denote the trace ring of \(m\) generic \(n\times n\) matrices over \(k\). The paper under review gives a description of the Brauer-Severi scheme for \(T\) (i.e., for the fibers of \(T\)). It is shown that the Brauer-Severi scheme for \(T\) is isomorphic to \(\text{Proj}(Q)\) for an appropriate (and described with its generators) graded ring \(Q\). Another important contribution in this paper is the study of the local properties of \(T\) considered as a sheaf over \(\text{Spec}(C)\) where \(C\) is the centre of \(T\). Sufficient conditions are found for the Brauer-Severi scheme of the fiber of \(T\) having smooth irreducible components that meet transversally. In addition, the case \(m=n=2\) is dealt with as an important example.
The paper contains new interesting results; it is organized in an appropriate way, and provides sources for further research. Brauer-Severi schemes; generic matrices; matrix invariants; trace rings; associated graded rings; sheaves; fibers Seelinger, G.: A description of the Brauer--Severi scheme of trace rings of generic matrices. J. Algebra 184, 852--880 (1996) Trace rings and invariant theory (associative rings and algebras), Brauer groups of schemes, Associative rings of functions, subdirect products, sheaves of rings A description of the Brauer-Severi scheme of trace rings of generic matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\pi:X\to Y\) be an algebraic map between algebraic sets. \(D_h(\pi)\) is defined to be the closure of the set of points \(x\in X\) such that \(\dim_x\pi^{-1}(\pi(x))=h\). Let \(Z\) be an irreducible algebraic set and \(\pi:Z\to Y\) be an algebraic map from \(Z\) to an algebraic set \(Y\). Let \(U\subset Z\) be Zariski open and dense such that the points of \(U\) are smooth, the points of \(\pi(U)\) are smooth, \(\pi|_U\) is of maximal rank, \(\pi|_U=s\circ r\), where \(r:U\to V\) is an algebraic map onto a quasiprojective manifold \(V\) with connected fibres and \(s:V\to\pi (U)\) is a covering map. The inclusion \(i:U\to Z\) induces an inclusion map of the \(k\)--fold product \(U \times_V \times\ldots \times_V U \) into \(Z \times_Y \times\ldots \times_Y Z\).
The main component \(Z^k_\pi\) of \(Z\) in \(Z \times_Y \times\ldots \times_Y Z\) is defined to be the closure of the image of this map.
It is proved that given an irreducible component \(Z\subset D_h(\pi)\) for an algebraic map \(\pi:X\to Y\) between irreducible algebraic sets satisfying \(\overline{\pi(X)}=Z\), the main component \(Z^k_\pi\) of \(Z\) in \(X \times_Y \times\ldots \times_Y X\) is an irreducible component for \(k\) sufficiently large. Numerical algorithms for identifying these components are given. The methods are illustrated by finding the rulings of a general quadric in \(\mathbb C^3\). components of solutions; exceptional loci of algebraic maps; fibre products; homotopy continuation; numerical algebraic geometry DOI: 10.1007/s10208-007-0230-5 Computational aspects in algebraic geometry, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation Exceptional sets and fiber products | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We estimate the number \(\mathcal A_{\lambda}\) of elements on a linear family \(\mathcal A\) of monic polynomials of \(\mathbb F_q[T]\) of degree \(n\) having factorization pattern \(\lambda=1^{\lambda_1}2^{\lambda_2}\cdots n^{\lambda_n}\). We show that \(| \mathcal A_{\lambda}| =\mathcal T(\lambda)q^{n-m}+\mathcal O(q^{n-m-1/2})\) where \(\mathcal T(\lambda)\) is the proportion of elements of the symmetric group of \(n\) elements with cycle pattern \(\lambda\) and \(m\) is the codimension of \(\mathcal A\). Furthermore, if the family \(\mathcal A\) under consideration is ``sparse'', then \(| \mathcal A_{\lambda}| =\mathcal T(\lambda)q^{n-m}+\mathcal O(q^{n-m-1})\). Our estimates hold for fields \(\mathbb F_q\) of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the \(\mathcal O\)-notation in terms of \(\lambda\) and A with ``good'' behavior. Our approach reduces the question to estimate the number of \(\mathbb F_q\)-rational points of certain families of complete intersections defined over \(\mathbb F_q\). Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of \(\mathbb F_q\)-rational points are established. factorization patterns; linear families of polynomials Polynomials over finite fields, Symmetric functions and generalizations, Combinatorial aspects of commutative algebra, Rational points, Finite ground fields in algebraic geometry The distribution of factorization patterns on linear families of polynomials over a finite field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action and the first step is always to reduce the determinacy question to an ``infinitesimal determinacy'', i.e., to the tangent spaces at the orbits of the group action. In this work we formulate a universal, characteristic-free approach to finite determinacy, not necessarily over a field, and for a large class of group actions. We do not restrict to pro-algebraic or Lie groups, rather we introduce the notion of ``pairs of (weak) Lie type'', which are groups together with a substitute for the tangent space to the orbit such that the orbit is locally approximated by its tangent space, in a precise sense. This construction may be considered as a kind of replacement of the exponential resp. logarithmic maps. It is of independent interest as it provides a general method to pass from the tangent space to the orbit of a group action in any characteristic. In this generality we establish the ``determinacy versus infinitesimal determinacy'' criteria, a far reaching generalization of numerous classical and recent results, together with some new applications. finite determinacy; pairs of Lie-type; sufficiency of jets; infinitesimal stability; group actions on modules Classification; finite determinacy of map germs, Normal forms on manifolds, Singularities in algebraic geometry, Deformations of singularities, Filtered associative rings; filtrational and graded techniques Pairs of Lie-type and large orbits of group actions on filtered modules: a characteristic-free approach to finite determinacy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely, tensor rank, Waring, partially symmetric rank, and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterize the behavior of sequences in the join set converging to the latter's boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results. join set; join decomposition problem; condition number; tensor rank decomposition; CP decomposition; Waring decomposition; block term decomposition P. Breiding and N. Vannieuwenhoven, \textit{The condition number of join decompositions}, SIAM J. Matrix Anal. Appl., 39 (2018), pp. 287--309, . Sensitivity analysis for optimization problems on manifolds, Numerical computation of matrix norms, conditioning, scaling, Local Riemannian geometry, Methods of local Riemannian geometry, Semialgebraic sets and related spaces, Complexity and performance of numerical algorithms, Multilinear algebra, tensor calculus The condition number of join decompositions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) parametrizing closed subschemes of \(\mathbb P^n\) with Hilbert polynomial \(p(t)\) has been of great interest every since Grothendieck constructed it in the early 1960s. Early results include the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)] and smoothness of \(\mathrm{Hilb}^{p(t)} (\mathbb P^2)\) due to \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. \textit{A. Reeves} and \textit{M. Stillman} showed that every non-empty Hilbert scheme contains a smooth Borel-fixed point [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] and \textit{A. P. Staal} classified those with exactly one such fixed point, which are necessarily smooth and irreducible [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)].
The main result classifies Hilbert schemes with two Borel-fixed points over a field \(k\) of characteristic zero. To describe the result, express the Hilbert polynomial \(p(t)\) in the form used by \textit{Gotzmann}, namely
\[
p(t) = \sum_{i=1}^m \binom{t+\lambda_i-i}{\lambda_i-1}
\]
where \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_m \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. Writing \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\), the theorem lists for exactly which \(\mathbf{\lambda}\) the Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) has two Borel-fixed points and further determines when it is (a) smooth, (b) irreducible and singular or (c) a union of two components. In each case the irreducible components are normal and Cohen-Macaulay and the singularities of the Hilbert scheme appear as cones over certain Segre embeddings of \(\mathbb P^a \times \mathbb P^b\). Since the writing of his paper, (a) \textit{A. P. Staal} [``Hilbert schemes with two Borel-fixed points in arbitrary characteristic'', Preprint, \url{arXiv:2107.02204}] has shown that the theorem is valid in all characteristics with a small modification when char \(k=2\) and (b) \textit{R. Skjelnes} and \textit{G. G. Smith} [J. Reine Angew. Math. 794, 281--305 (2023; Zbl 07640144)] have classified the smooth Hilbert schemes are described their geometry.
Despite the difficulty of the content, the paper is readably written. Section 1 gives preliminaries on Borel-fixed (strongly stable) ideals and the resolution of \textit{S. Eliahou} and \textit{M. Kervaire} [J. Algebra 129, No. 1, 1--25 (1990; Zbl 0701.13006)], while Section 2 identifies the tuples \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\) corresponding to Hilbert schemes with two components. Section 3 uses the comparison theorem of \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] to compute the tangent space of the non-lexicographic Borel-fixed ideal \(I(\mathbf{\lambda})\) and give a partial basis for the second cohomology group of \(k[x_0,\dots,x_n]/I(\mathbf{\lambda})\). These are used in Section 4 where the main theorem is proved to describe the universal deformation space of \(I(\mathbf{\lambda})\) and hence the nature of singularities of the Hilbert schemes. Finally in Section 5 the author gives examples of Hilbert schemes with three Borel-fixed points. The last three examples relate to Hilbert schemes studied in the literature [\textit{S. Katz}, in: Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 231--242 (1994; Zbl 0839.14001); \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004); \textit{D. Chen} et al., Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)]. Hilbert scheme; singularities; Borel-fixed points; deformations of ideals Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Hilbert schemes with two Borel-fixed points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Any ordered set of distinct points \((P_1,\dots,P_s)\) in \(\mathbb{P}^r\) gives rise to a sequence of natural numbers \({\mathcal S}_{P_1,\dots,P_s}= (d_1,\dots,d_i, \dots, d_s)\) defined as follows: \(d_i\), \(i>0\), is the minimum degree of a hypersurface separating \(P_i\) from \(P_1,\dots, P_{i-1}\), while \(d_1=0\). There, the function \(\gamma_S:\mathbb{N} \to \mathbb{N}\), defined by \(\gamma_S(n)=\text{card}\{i:d_i=n\}\), is proved to coincide with the Castelnuovo function \(\Gamma\) of any 0-dimensional, projective, reduced scheme \({\mathbf X}=\{P_1,\dots,P_s\}\), having \({\mathcal S}\) as an allowed sequence.
Here the following problem is faced: For any given Castelnuovo function \(\Gamma\), find the set \(S_\Gamma\) of all sequences allowed for at least one scheme \({\mathbf X}\), having that Castelnuovo function. The solution of this problem will be essential in a next step of the investigation, consisting in a partition of \(S_\Gamma\) into equivalence classes, each of which corresponds to schemes with some characteristic geometric property.
The starting point is Macaulay's theorem, characterizing the functions \(f:\mathbb{N}\to \mathbb{N}\) that can be the Castelnuovo function of some scheme. Such a set \(S_\Gamma\) is completely determined by its maximum with respect to the lexicographic order, which is named \({\mathcal M}_\Gamma\).
The description of \({\mathcal M}_\Gamma\) is the main job of this paper. In fact, in the special case of \(n+1\) points on a line, \({\mathcal M}_\Gamma\) simply is what we call a 1-segment starting with 0, that is \((0,1,2,\dots,n)\). For a scheme \({\mathbf X}\) lying in \(\mathbb{P}^2\), \({\mathcal M}_\Gamma\) can be partitioned into a collection of 1-segments, say \({\mathcal M}_\Gamma= ({\mathcal S}_0,\dots,{\mathcal S}_t)\) where \({\mathcal S}_i=(i,i+1, \dots,n_i)\), \(n_0 \geq n_1\geq\cdots\geq n_t\) and \(t+1\) is the minimal degree of a curve containing \(X\); so, we are led to introduce the concept of 2-segment starting with 0. In this situation, \({\mathcal M}_\Gamma\) is realized by any 0-dimensional scheme of distinct points on \(t+1\) lines \(r_0,\dots, r_1\), such that the points on \(r_h\) are in \(1-1\) correspondence with the elements of \({\mathcal S}_h\) and the intersection of two lines does not belong to \({\mathbf X}\). In the more general situation of points in \(\mathbb{P}^2\), \({\mathcal M}_\Gamma\) appears as a generalization of a 2-segment to higher dimension and we are led to introduce the concept of \(r\)-segment: an \({\mathcal M}_r\) is an \(r\)-segment starting with 0. The schemes described in \(\S3\) as a realization of \({\mathcal M}_\Gamma\) are \(k\)-configurations.
From the structure of \({\mathcal M}_\Gamma\) a bound can be read for the maximum number of points of \({\mathbf X}\) lying on a hypersurface of given degree, or on a linear subspace of given dimension. \(r\)-segment; configurations; Castelnuovo function G. Beccari, C. Massaza, Realizable sequences linked to the Hilbert function of a zero dimensional projective scheme, in: J. Herzog, G. Restuccia (Eds.), Geometric and Combinatorial Aspects of Commutative Algebra, Marcel Dekker, New York, 2001, pp. 21--41. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Realizable sequences linked to the Hilbert function of a 0-dimensional projective scheme. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is devoted to the study of some invariants under bi-Lipschitz transformations, called characteristic exponents, of germs of semialgebraic sets. Recall that a bi-Lipschitz transformation is a bijective map \(f\) satisfying \(ad(x,y)\leq d(f(x),f(y))\leq b d(x,y)\) where \(0<a\leq b\) and \(d\) denotes the relevant metric. Among the previous results about characteristic exponents, the following should be mentioned, due to the first author [Houston J. Math. 25, No.3, 453--472 (1999; Zbl 1007.32006)]:
Let \(X\subset {\mathbb R}^n\) be a two dimensional semialgebraic set and let \(x_0\) be an isolated singular point such that the link of \(X\) at \(x_0\) is connected. Then there exists a rational number \(\beta\geq 1\) such that the germ of \(X\) at \(x_0\) is bi-Lipschitz equivalent with respect to the intrinsic metric to a germ at \(0\) of a \(\beta\)-horn: \((x^2+y^2)^q=z^{2p},\;z\geq 0\) where \(\beta=\frac{p}{q}\). This number \(\beta\) is a complete bi-Lipschitz invariant, that is, if two \(\beta_i\)-horns, \(i=1,2\), are bi-Lipschitz equivalent, then \(\beta_1=\beta_2\).
On the other hand, the invariant \(\beta\) can be defined in terms of measure theory, using \(1\)-dimensional cycles in \(X\setminus\{x_0\}\). This kind of definition works not only for the \(2\)-dimensional case, and the main purpose of the authors in this paper is to establish a generalization and to show the existence of characteristic exponents for all dimensions and all kind of semialgebraic singularities. For that, they see an isolated singularity of a semialgebraic set as a collapsing family of \((n-1)\)-dimensional nonsingular Riemannian manifolds and use a result of \textit{H. Federer} and \textit{W.H. Fleming} [Ann. Math. (2), 72, 458--520 (1960; Zbl 0187.31301)], which claims that in a Riemannian compact manifold the cycles of dimension \(k\) and volume smaller than a certain constant are homologically trivial, to define the \(k\)-dimensional characteristic exponents as the vanishing rates of the previous constants \(C(k)\). More precisely, the volume growth number \(\mu(X,x_0)\) is the first exponent appearing in the asymptotic expansion of \(\text{ vol}(X \cap B(x_0,r))\). For \(k=0,1,\dots,\dim(X)\), the characteristic exponent \(\mu_k(X,x_0)\) is defined to be the minimal number \(\mu\) satisfying the following condition:
Let \(V^{x_0}\) be an small enough neighbourhood of \(x_0\), \(c\) a semialgebraic cycle of dimension \(k\) with support contained in \(V^{x_0} \setminus \{x_0\}\). If \(\eta\) is a semialgebraic cycle of dimension \(k+1\) with \(\partial \eta=c\), \(\text{ supp} (\eta)\subset V^{x_0}\) and \(\mu(\text{ supp}(\eta),x_0)>\mu\), then \(c\) is homologically trivial in \(V^{x_0}\setminus\{x_0\}\). semialgebraic sets; singular points; bi-Lipschitz invariants Birbrair, Characteristic exponents of semialgebraic singularities, Math Nachr 276 pp 23-- (2004) Semialgebraic sets and related spaces, Topology of real algebraic varieties Characteristic exponents of semialgebraic 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 A map \(f:\mathbb R^n\to\mathbb R^m\) is regular if there exist \(f_1,\dots,f_m, g\in\mathbb R[{\mathtt x}_1,\dots,{\mathtt x}_n]\) such that \(g^{-1}(0)\) is empty and for every point \(x\in\mathbb R^n\),
\[
f(x)=\Big(\frac{f_1(x)}{g(x)},\dots,\frac{f_m(x)}{g(x)}\Big).
\]
The map \(f\) is polynomial if \(g\) can be chosen to be constant. We are far from achieving a geometric characterization of the semialgebraic subsets \(\mathcal S\subset\mathbb R^m\) that are either a polynomial or a regular image of some \(\mathbb R^n\), that is, that can be represented as \(\mathcal S=f(\mathbb R^n)\) for some polynomial or regular map \(f:\mathbb R^n\to\mathbb R^m\). Motivation for such a characterization comes from the fact that important problems in real algebraic geometry involving semialgebraic sets \(\mathcal S\) (say, optimization, Positivstellensatz or computation of trajectories) can be reduced somehow to the case \(\mathcal S=\mathbb R^n\) if \(\mathcal S\) is either a polynomial or a regular image of an Euclidean space.
The authors of the work under review have contributed significantly in the last years to answer this question for a subclass of the class of semialgebraic sets with piecewise linear boundary. Let \(\mathcal P\subset\mathbb R^n\) be a convex polyhedron and denote \({\text{Int}}(\mathcal P)\) its interior as a manifold with boundary. In a previous work by the second author, [\textit{C. Ueno}, J. Pure Appl. Algebra 216, No. 11, 2436--2448 (2012; Zbl 1283.14024)] it is proved that for every convex polygon \(\mathcal P\subset\mathbb R^2\) that is neither a line nor a band, the semialgebraic sets \(\mathbb R^2\setminus\mathcal P\) and \(\mathbb R^2\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^2\). A band is the closed convex polygon determined by two parallel lines. In addition, it is shown that both \(\mathcal P\) and \({\text{Int}}(\mathcal P)\) are regular images of \(\mathbb R^2\).
Later on it was proved in [\textit{J. F. Fernando} and \textit{C. Ueno}, Int. J. Math. 25, No. 7, Article ID 1450071, 18 p. (2014; Zbl 1328.14088)] that for every convex polyhedron \(\mathcal P\subset\mathbb R^3\) which is neither a plane nor a layer both \(\mathbb R^3\setminus\mathcal P\) and \(\mathbb R^3\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^3\). A layer is the closed convex polyhedron determined by two parallel planes. In addition, they showed that for \(n\geq4\) the involved constructions do not work further to represent either \(\mathbb R^n\setminus\mathcal P\) or \(\mathbb R^n\setminus{\text{Int}}(\mathcal P)\) as polynomial images of \(\mathbb R^n\) for a general convex polyhedron \(\mathcal P\subset\mathbb R^n\).
In the article under review the authors go a step further. The main result is Theorem 1.1 where they prove that for arbitrary \(n\geq2\) and for every convex polyhedron \(\mathcal P\subset\mathbb R^n\) that is neither a hyperplane nor a layer the semialgebraic sets \(\mathbb R^n\setminus\mathcal P\) and \(\mathbb R^n\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^n\) in case \(\mathcal P\) is bounded and otherwise they are regular images of \(\mathbb R^n\). As far as the reviewer knows it remains open to characterize geometrically for \(n\geq 4\) the unbounded polyhedra \(\mathcal P\subset\mathbb R^n\) for which either \(\mathbb R^n\setminus\mathcal P\) or \(\mathbb R^n\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^n\).
The proof of the main result of the work under review is easier if \(\dim(\mathcal P)<n\). Indeed, it follows straightforwardly from a more general result, which is Theorem 3.1 in the article and has its own interest: for every proper basic semialgebraic set \(\mathcal S\subsetneq\mathbb R^n\), the set \(\mathbb R^{n+1}\setminus(\mathcal S\times\{0\})\) is a polynomial image of \(\mathbb R^{n+1}\). The hardest part is to deal with \(n\)-dimensional polyhedra \(\mathcal P\subset\mathbb R^n\). To attack it the authors use successfully a technique previously introduced in their joint work quoted above: to place the polyhedron in what they call first or second trimming position and they work by induction on the number of facets of the polyhedron if it is bounded and by double induction on the dimension and the number of facets of \(\mathcal P\) if it is unbounded.
In the last section of this work the authors substitute \(\mathcal P\) by the closed ball \(\mathcal B_n\subset\mathbb R^n\) (which can be understood as the convex polyhedron with infinitely many facets). It was proved in [\textit{J. F. Fernando} and \textit{J. M. Gamboa}, Isr. J. Math. 153, 61--92 (2006; Zbl 1213.14109)] that \(\mathbb R^2\setminus\mathcal B_2\) is a polynomial image of \(\mathbb R^3\) but it is not a polynomial image of \(\mathbb R^2\). Moreover, \(\mathbb R^2\setminus{\text{Int}}(\mathcal B_2)\) is a polynomial image of \(\mathbb R^2\). The proofs of the latter results are specific of the two-dimensional case. The authors have developed a completely different approach to show that for arbitrary \(n\geq2\) the set \(\mathbb R^n\setminus\mathcal B_n\) is a polynomial image of \(\mathbb R^{n+1}\) but it is not a polynomial image of \(\mathbb R^n\). Moreover, they show that \(\mathbb R^n\setminus{\text{Int}}(\mathcal B_n)\) is a polynomial image of \(\mathbb R^n\).
All proofs in this work are clearly written and have a strong geometrical flavor, illustrated with enlightening pictures. polynomial and regular image; convex polyhedron; first and second trimming positions Fernando, J. F.; Ueno, C., On complements of convex polyhedra as polynomial and regular images of \(\mathbb{R}^n\), Int. Math. Res. Not. IMRN, 2014, 18, 5084-5123, (2014) Semialgebraic sets and related spaces, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Three-dimensional polytopes On complements of convex polyhedra as polynomial and regular images of \(\mathbb{R}^{n}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Artin's approximation theorem states that a germ of an analytic solution \(y(x)\) of a polynomial system \(P(x,y)=0\) can be approximated by algebraic solutions in the Krull topology. In this paper the author proves a CR version of Artin's result. Suppose that \(M \subset {\mathbb{C}}^N\) is a real-algebraic CR submanifold with CR orbits of constant dimension, and let \(S' \subset {\mathbb{C}}^N \times {\mathbb{C}}^{N'}\) be a real-algebraic subset. Then for any \(\ell \in {\mathbb{N}}\) and any germ of a holomorphic mapping \(f : ({\mathbb{C}}^N,p) \to {\mathbb{C}}^{N'}\) at some \(p \in M\) whose graph over \(M\) lies in \(S'\), there exists a complex-algebraic mapping \(f^\ell\) whose graph over \(M\) lies in \(S'\), and \(f^\ell\) agrees with \(f\) up to order \(\ell\) at \(p\). CR submanifolds \(M\) with this approximation property are said to possess the Nash-Artin approximation property.
The theorem was proved for \(M\) that are minimal at some point in a previous paper by \textit{F. Meylan, D. Zaitsev} and the author [Int. Math. Res. Not. 2003, No. 4, 211--242 (2003; Zbl 1016.32017); in: Proceedings of the conference on partial differential equations, Forges-les-Eaux, France, 2003. Exp. Nos. I--XV. Nantes: Université de Nantes. Exp. No. XII, 20 p. (2003; Zbl 1054.32021)]. A corollary of these results is that any smooth real-algebraic hypersurface has the Nash-Artin approximation property. An important application of the present theorem is the following: If \(M\) and \(M'\) are two connected real-algebraic CR submanifolds of constant orbit dimension and are holomorphically equivalent as germs at some points \(p \in M\) and \(p' \in M'\), then they are algebraically equivalent. This theorem was known previously by work of \textit{M. S. Baouendi, L. Rothschild} and \textit{D. Zaitsev} [in: Complex analysis and geometry. Proceedings of a conference at the Ohio State University, Columbia, OH, USA, 1999. Berlin: de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 9, 1--20 (2001; Zbl 1023.32023)] and \textit{B. Lamel} and the author [Commun. Anal. Geom. 18, No. 5, 891--925 (2010; Zbl 1239.32029)] for \(p\) on a certain Zariski open subset of \(M\). This present work extends the result to all \(p \in M\). algebraic map; CR manifold; CR orbits Mir, N., Algebraic approximation in CR geometry, J. Math. Pures Appl. (9), 98, 72-88, (2012) Analysis on CR manifolds, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Real submanifolds in complex manifolds, Real-analytic sets, complex Nash functions, Real algebraic sets, Nash functions and manifolds Algebraic approximation in CR 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 \(A\) be an integer-valued \(d \times n\) matrix of rank \(d\) and \(\beta \in \mathbb C^d\). The \(A\)-hypergeometric system is a \(D\)-module defined by \(A\) and \(\beta\). Here \(D = \mathbb C[x_1, \dots, x_n] \langle \partial_1, \dots, \partial_n \rangle\) is the Weyl algebra. In the present paper, the authors studies the irregularity of \(M_A(\beta)\). To do this, they introduce a filtration \(L = \{L_k D\}_k\) of \(D\) and use the associated graded ring \(\text{gr}^L D\). Let \((l_1, \dots, l_{2n}) \in \mathbb Q^{2n}\) such that \(l_1 + l_{n+1}\), \dots, \(l_n + l_{2n} \geq 0\). Then let \(L_k D\) be the vector space spanned by monomials \(x_1^{e_1} \cdots x_n^{e_n} \partial_1^{f_1} \cdots \partial_n^{f_n}\) such that \(\sum e_i l_i + \sum f_i e_{i+n} \leq k\). For such a filtration, they define an \((A, L)\)-umbrella, which is a cell complex, and the characteristic variety \(\text{Ch}^L(M_A(\beta))\), which is a subvariety of \(\text{Spec} \text{gr}^L D\). Furthermore, they define slopes of \(M_A(\beta)\) by using a family of filtrations parameterized by rational numbers. For an irreducible subvariety \(C \subset \text{Spec} \text{gr}^L D\), let \(\mu^{L,C}_{A,0}(\beta)\) be the multiplicity of \(\text{gr}^L M_A(\beta)\). The main result of this paper is the computation of \(\mu^{L,C}_{A,0}(\beta)\). As its consequence, they prove the converse of Hotta's theorem. That is, any \(A\)-hypergeometric system is homogeneous with respect to some filtration.
The authors, moreover, define the Euler-Koszul \(L\)-characteristic along \(C\): \(\mu_{A,i}^{L,C}(-; \beta)\) (\(i = 0\), \(1\), \dots), which is a generalized notion of \(\mu_{A,0}^{L, C}(\beta)\). They also compute it. \((A, L)\)-umbrella; irregularity sheaf; Euler-Koszul characteristic M. Schulze and U. Walther, Irregularity of hypergeometric systems via slopes along coordinate subspaces , preprint,\arxivmath/0608668v2[math.AG] Commutative rings of differential operators and their modules, Toric varieties, Newton polyhedra, Okounkov bodies, Filtered associative rings; filtrational and graded techniques Irregularity of hypergeometric systems via slopes along coordinate subspaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results \[\Sigma \Omega \Sigma X \simeq \Sigma X \vee (X\wedge \Sigma\Omega \Sigma X) \] and \[ \Omega(X \vee Y) \simeq \Omega X\times \Omega Y\times \Omega \Sigma(\Omega X \wedge \Omega Y) \] in the maximal generality of an \(\infty \)-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary \(\infty \)-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting results in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field [\textit{K. Wickelgren} and \textit{B. Williams}, Geom. Topol. 23, No. 4, 1691--1777 (2019; Zbl 1428.14032)]. We also give two proofs of the metastable EHP sequence in the setting of \(\infty \)-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof. infinity categories Loop spaces, Suspensions, Homotopy theory, Homotopy groups of wedges, joins, and simple spaces, \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories, Motivic cohomology; motivic homotopy theory On the James and Hilton-Milnor splittings, and the metastable EHP sequence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 study of systems of differential equations with irregular singularities, it is well known that asymptotic analysis had been playing a deep role. In recent years cohomological methods are developed and being used in the study, recasting old results in precise cohomological terms and getting new results which seem to be out of reach by the classical methods. The author of the paper under review studies differential equations on complex manifolds with irregular singularities on divisors with normal crossings. He introduces the notion of strong asymptotic developability of functions defined in polysectors. This notion seems to be identical with Whitney's notion of infinite differentiability on boundaries of polysectors for functions holomorphic inside the polysectors. In this and a subsequent paper (see reference below), the author obtains 'vanishing theorems' for cohomology with coefficients in the sheaf of germs of such functions. The notations and preliminaries needed to state these theorems are lengthy and so we do not state these theorems here. The details of proofs with some simplifications can be found in the author's book [''Asymptotic analysis for integrable connections with irregular singular points'', Lecture Notes Math. 1075 (1984; Zbl 0546.58003)]. In the latter book, the author using these 'vanishing theorems' solves a version of the Riemann-Hilbert- Birkhoff problem for integrable connections with irregular singularities. This latter book is better written than the present and the subsequent papers (see the following review) and hence it is advisable to read this book which contains a wealth of materials and techniques. differential equations with irregular singularities; asymptotic analysis; strong asymptotic developability; vanishing theorems; cohomology with coefficients in the sheaf of germs Majima, H.: Vanishing theorems in asymptotic analysis. Proc. Japan Acad., 59A, 146-149 (1983). Vanishing theorems, Differential complexes, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Vanishing theorems in asymptotic analysis | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 his influential book [Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (AMS) (1990; Zbl 0715.14013)], \textit{V. G. Berkovich} defined analytic spaces over arbitrary Banach rings. The theory of these spaces was developed only over non-archimedean fields, where it was further generalized by Berkovich [Publ. Math., Inst. Hautes Étud. Sci. 78, 5--161 (1993; Zbl 0804.32019)] in order to better approximate classical rigid-analytic geometry. In the monograph under review, Poineau lays foundations for a theory of Berkovich analytic spaces over the rings of integers of number fields, spaces which encompass, at the same time, aspects of archimedean and non-archimedean analytic geometry. The analytic affine line over the ring of integers of a number field is studied in detail, both in terms of its topology and on the level of its analytic functions. Throughout the text, rings of integers of number fields are viewed as Banach rings via the maximum of their archimedean norms, and more general Banach base rings \((A,\|\cdot\|)\) are assumed to be uniform, a condition which guarantees that the polynomial ring in \(n\) variables over \(A\) is a subring of the ring of global functions on the analytic affine \(n\)-space over \(A\).
In the first chapter, the author recalls Berkovich's definition of analytic spaces over a uniform Banach ring \(A\), and he gives a description of analytic affine lines over Banach fields. He introduces spectrally convex compact subsets of the analytic affine \(n\)-space \(\mathbb A^n_{A}\) over \(A\), subsets which can naturally be viewed as spectra of uniform Banach rings and which are used for inductive arguments. It is shown that intersections of rational compact subsets provide examples for spectrally convex sets and that closed discs over spectrally convex sets are spectrally convex. The author then discusses the flow given by raising semi-norms to powers by positive real numbers, and he explains how to compare properties of points lying on the same path of that flow.
In the second chapter, Poineau defines rings of convergent power series on discs and annuli; he shows that their Banach spectra are naturally homeomorphic to the underlying discs and annuli, respectively.
He then considers inductive limits of these rings, limits which arise as stalks in certain points. He establishes Weierstrass division and preparation theorems by adapting methods used by Grauert and Remmert in the complex-analytic context, thereby avoiding both reduction ``mod p'' techniques and the Cauchy formula. He deduces that the stalks in points of analytic affine \(n\)-space which are split in their fibers are regular noetherian local rings. The author then proves that all stalks of Berkovich analytic spaces over uniform Banach rings are henselian, and he shows how this property can be used to establish local isomorphisms between analytic spaces in specific situations.
In Chapter 3, the author first uses Ostrowski's theorem to discuss the valuation spectrum of the ring of integers \(A\) of a number field \(K\); for this spectrum, he describes set-theoretical and topological properties, he computes rings of functions on open subsets as well as the stalks of the structural sheaf, and he describes the Shilov boundaries of compact and connected subsets. He then turns towards analytic affine spaces over the valuation spectrum of \(A\), for which he first specializes the results of the preceding chapter. As an application of the henselian property of the stalks, he reproves a theorem of Eisenstein stating that every element of \(K[[T]]\) that is integral over \(K[T]\) has strictly positive radius of convergence at all places. He then gives an explicit description of the rings of global sections on compact discs and annuli over compact connected subsets of the base. Emphasis is afterwards placed on points which are classical rigid points in their fibers or which lie over internal points of the base; for these points, Poineau describes a fundamental system of neighborhoods, and he deduces algebraic properties of their stalks, such as noetherianity, regularity and completeness. (Here a point in the base is called internal if it is neither the trivial valuation nor the trivial valuation on the residue class field of a finite place. To study points lying over internal points, Poineau uses the flow introduced in the first chapter.) Unfortunately, the author was not yet able to extend these results to the remaining points of analytic affine spaces over \(A\) -- except for the case of the analytic affine line, which is treated in Chapter 4. At the end of the third chapter, the author computes that the topological dimension of the analytic affine \(n\)-space \(\mathbb A^n_A\) over \(A\) equals \(2n+1\).
In Chapter 4, Poineau discusses the affine line over the ring of integers \(A\) of a number field, for which he extends the desired algebraic properties (noetherianity, regularity, excellence) to stalks in all points. Moreover, he shows that the structural sheaf of the analytic affine line over \(A\) is coherent.
In the fifth chapter, Poineau studies finite morphisms between Berkovich analytic spaces over an arbitrary uniform Banach ring \(A\). He gives conditions under which a unitary polynomial \(P\in A[T]\) yields a Banach norm on the quotient \(A[T]/(P)\) such that the natural homomorphism \(A\rightarrow A[T]/(P)\) is bounded and such that the induced morphism \(M(A[T]/(P))\rightarrow M(A)\) is finite with finite fibers and coherent direct image of the structural sheaf. Along the way, he obtains a Weierstrass division theorem for points which are classically rigid in their fibers, as well as a Weierstrass division theorem where radii of convergence are preserved. He applies these results to endomorphisms of the analytic line, first over a general uniform Banach ring and then over the ring of integers of a number field. The author suggests that the techniques of Chapter 5 should allow to investigate general analytic curves over rings of integers of number fields.
Chapter 6 is devoted to the study of Stein spaces in the analytic affine line over the ring of integers in a number field, where a subset of the analytic affine line is called Stein if the stalks of its coherent sheaves are generated by global sections and if its coherent sheaves have vanishing higher cohomology. Poineau exhibits various Stein subsets of the analytic affine line, like for instance relative discs and annuli; to do so, he transfers the lemmas of Cousin and Cartan to an arithmetic setting, using strong approximation, finiteness of the class number and his results on finite morphisms developed in the preceding chapter.
In Chapter 7, the author presents various applications of his theory. While complex analytic geometry yields statements on holomorphic functions, Poineau deduces properties of convergent arithmetic series. First, he shows the existence of power series \(f\) and \(g\) in \(\mathbb Z[[T]]\) converging everywhere on the complex open unit disc such that \(f/g\) has prescribed poles and prescribed lower bounds on vanishing orders. Furthermore, he establishes an analog of a theorem of Frisch: he shows that the ring of functions in the analytic affine line over the ring of integers of a number field around a compact apportionable Stein subset is noetherian. He thus obtains noetherianity of the ring of arithmetic series with prescribed minimal radii of convergence at \(\infty\) and at finitely many finite places, thereby giving a more conceptual proof of a result of \textit{D. Harbater} [Am. J. Math. 106, 801--846 (1984; Zbl 0577.13017), Theorem 1.8] and generalizing this result to any number field. Finally, the author presents an application to the inverse Galois problem: he reproves the fact, first shown by \textit{D. Harbater} [Am. J. Math. 110, No. 5, 849--885 (1988; Zbl 0683.14004), Corollary 3.8], that every finite group is a Galois group over the fraction field of the subring \(\mathbb Z_{1^-}[[T]]\) of \(\mathbb Z[[T]]\) given by the series with \(\mathbb Z\)-coefficients \(a_k\) such that \(\|a_k\| r^k\rightarrow 0\) with \(k\rightarrow\infty\) for all \(r<1\). The idea of Poineau's proof is the following: the ring \(\mathbb Z_{1^-}[[T]]\) is the ring of sections on the relative open disc of radius \(1\) around the zero section of the affine line over \(\mathbb Z\). One constructs Galois coverings of this disc by constructing cyclic coverings locally and glueing them. To algebraize the resulting analytic coverings, Poineau uses the fact, shown in Chapter 6, that the aforementioned open disc is a Stein space. Poineau thus gives a more geometric and conceptual proof of Harbater's result, and his methods work over any number field.
The exposition throughout Poineau's monograph is very clear and thorough. The author includes reminders on basic notions of Berkovich geometry, and he explains his arguments in detail; his text is thus pleasant to read, despite the numerous technicalities. Berkovich analytic spaces; ring of integers of a number field; global analytic geometry; convergent arithmetic series; noetherianity; Galois inverse problem Poineau, J., La droite de berkovich sur Z, Astérisque, 334, (2010) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Rigid analytic geometry, Global ground fields in algebraic geometry, Power series (including lacunary series) in one complex variable, Commutative Noetherian rings and modules, Inseparable field extensions The Berkovich line over \(\mathbf Z\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 matrix calculations such as diagonalization of quadratic forms under the aspect of additive complexity and relate these complexities to the complexity of matrix multiplication. While in \textit{P. Bürgisser}, \textit{M. Karpinski} and \textit{T. Lickteig} [Comput. Complexity 1, No. 2, 131-155 (1991; Zbl 0774.68056)] for multiplicative complexity the customary ``thick path existence'' argument was sufficient, here for additive complexity we need the more delicate finess of the real spectrum to obtain a complexity relativization. After its outstanding success in semi-algebraic geometry the power of the real spectrum method in complexity theory becomes more and more apparent. Our discussions substantiate once more the signification and future rôle of this concept in the mathematical evolution of the field of real algebraic algorithmic complexity.
A further technical tool concerning additive complexity is the structural transport metamorphosis from \textit{T. Lickteig}, On semi-algebraic decision complexity [Habilitationsschrift, Univ Tübingen, and Tech. Rep. TR-90-052 Int. Comp. Sci. Inst., Berkeley (1990)] which constitutes another use of the exponential and the logarithm as it appears in the work on additive complexity by \textit{D. Yu. Grigoryev} [Notes of scientific seminars of LOMI Vol. 118, Sankt Petersburg, 25-82 (1982)] and \textit{J. J. Risler} [SIAM J. Comput. 14, 178-183 (1985; Zbl 0562.12020)] through the use of \textit{A. G. Khovanskij} [SSSR 255, 804-807 (1980; Zbl 0569.32004)].
We confine ourselves here to diagonalization of quadratic forms. In the forthcoming paper \textit{T. Lickteig} and \textit{K. Meer} [Semi-algebraic complexity -- Additive complexity of matrix computational tasks, J. Complexity 13, No. 1, 83-107, Art. No. CM970430 (1997)] further such relativizations of additive complexity will be given for a series of matrix computational tasks. matrix calculations; multiplicative complexity Lickteig, T.; Meer, K.: Semi-algebraic complexity--additive complexity of diagonalization of quadratic forms. Revista matemática de la universidad complutense de Madrid 10 (1997, to appear) , Analysis of algorithms and problem complexity, Symbolic computation and algebraic computation, Semialgebraic sets and related spaces, Nash functions and manifolds Semi-algebraic complexity-additive complexity of diagonalization of quadratic forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 fan \(\Delta \) and a cone \(\sigma \in \Delta \), let \(\mathrm{star}^1(\sigma)\) be the set of cones that contain \(\sigma \) and are one dimension bigger than \(\sigma \). In this paper, we study two cones of piecewise linear functions defined on \(\Delta \): the cone of functions which are convex on \(\mathrm{star}^1(\sigma)\) for all cones, and the cone of functions which are convex on \(\mathrm{star}^1(\sigma)\) for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph \(K_5\), we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same. tropical linear space; functions convex on a fan; matroid; nef cone Combinatorial aspects of matroids and geometric lattices, Trees, Tropical geometry \(\mathrm{star}^{1}\)-convex functions on tropical linear spaces of complete graphs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, the authors study the dimension of the vector space of splines, or piecewise polynimial functions, of a fixed maximun polynomial degree and which are defined on planar polygonal partitions. The term ``mixed smoothness'' refers to the choice of different orders of smoothness across various edges of the partition. The approach follows the homological methods introduced to the study of splines in [\textit{L. J. Billera}, Trans. Am. Math. Soc. 310, No. 1, 325--340 (1988; Zbl 0718.41017)].
Starting from a spline space whose lower homology modules vanish, the authors present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness conditions on certain edges of the partition.
The results can be used to compute the dimensions of T-meshes with holes, therefore extending the results on mixed non-uniform bi-degree T-meshes presented in [\textit{D. Toshniwal} et al., Adv. Comput. Math. 47, No. 1, Paper No. 16, 42 p. (2021; Zbl 1473.13026)] and [\textit{D. Toshniwal} and \textit{N. Villamizar}, Comput. Aided Geom. Des. 80, Article ID 101880, 9 p. (2020; Zbl 07207288)]. splines; polygonal meshes; spline dimension formulas; mixed smoothness Syzygies, resolutions, complexes and commutative rings, Spline approximation, Numerical computation using splines, Geometric aspects of numerical algebraic geometry Counting the dimension of splines of mixed smoothness. A general recipe, and its application to planar meshes of arbitrary topologies | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the germ of a holomorphic function defining an isolated hypersurface singularity of multiplicity \(\mu\) in \({\mathbb{C}}^ n\). Furthermore, let F: (\({\mathbb{C}}^ n\times {\mathbb{C}}^{\mu},0\times 0)\to ({\mathbb{C}},0)\) be a (mini-)versal deformation of f, \(F_{\lambda}:=F(-,\lambda)\), and \(V_{\lambda}:=F_{\lambda}^{-1}(0)\) for any \(\lambda\in {\mathbb{C}}\). The discriminant of the versal deformation \((V_{\lambda})_{\lambda \in {\mathbb{C}}^{\mu}}\) of the given hypersurface singularity is defined to be the locus \(\Delta \subset {\mathbb{C}}^{\mu}\) of the parameters \(\lambda\) for which the hypersurface \(V_{\lambda}\) is singular. Then one has a natural representation of the fundamental group \(\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)\) as a linear transformation group of the intermediate homology groups \(H_{n-1}(V_{\lambda},{\mathbb{Z}})\cong {\mathbb{Z}}^{\mu}\), \(\lambda \in {\mathbb{C}}^{\mu}\), just given by displacement of cycles along closed paths. The image of this representation of \(\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)\) in \(GL_{\mu}({\mathbb{Z}})\) is called the monodromy group of the versal family. The monodromy group can be thought of as a subgroup of the isotropy group of the intersection form on the intermediate homology \(H_{n- 1}(V_{\lambda},Z)\). The classical Picard-Lefschetz theory describes the generators of the monodromy group of a versal deformation (of isolated hypersurface singularities) in terms of a particular basis in \(H_{n- 1}(V_{\lambda},{\mathbb{Z}})\), the so-called basis of vanishing cycles.
In the present paper, the author extends the classical Picard-Lefschetz theory, in that he investigates the action of the fundamental group \(\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)\) on the homology groups \(H_ n({\mathbb{C}}^ n\setminus V_{\lambda},{\mathcal Z}(q))\) of the complement of a hypersurface \(V_{\lambda}\) with values in a certain non-trivial locally constant sheaf \({\mathcal Z}(q)\). This ``twisted'' sheaf, with stalk \({\mathbb{Z}}[q,q^{-1}]\), involves the complex variable q and is characterized by the property that a closed path around \(V_{\lambda}\) induces the multiplication by q on its stalks. The main result of the paper provides an explicit set of generators of the monodromy group of the given versal deformation in \(H_ n({\mathbb{C}}^ n\setminus V_{\lambda},{\mathcal Z}(q))\), and that in terms of a suitable homology basis. In the concluding section of his paper, the author discusses examples, applications, and possible generalizations of his approach to the monodromy group. He shows how the Burau representation of the braid group \(\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)\) can be obtained, how the twisted cohomology of the complement of a non-singular hypersurface V in \({\mathbb{C}}^ n\) can be interpreted as the De Rham cohomology of a certain complex of singular differential forms on \({\mathbb{C}}^ n\setminus V\), and how the classical monodromy operator and the signature of the intersection form are calculated from his twisted Picard-Lefschetz formulae. cohomology groups; vanishing cycles; versal deformation; fundamental group; monodromy; Picard-Lefschetz theory; isolated hypersurface singularities; twisted cohomology Givental A.B.: Twisted Picard--Lefschetz formulas. Funct. Anal. Appl. 22, 10--18 (1988) Deformations of complex singularities; vanishing cycles, Complex singularities, Structure of families (Picard-Lefschetz, monodromy, etc.), Sheaves and cohomology of sections of holomorphic vector bundles, general results Twisted Picard-Lefschetz formulas | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present book deals with analytic spaces over \(\mathbb{R}\). Many decades back, the theory of complex analytic functions and their sets of zeroes was established. This led to the notion of an analytic space. Its local model is the zeroset of finitely many analytic functions on an open subset of \(\mathbb{C}^n\) with a sheaf of continuous functions.
Then, the question of formulating the notion of an analytic space over \(\mathbb{R}\) arose. A number of difficulties appeared, since the real analytic spaces did not behave as in the algebraic case. However, Henri Cartan proved that every analytic subset of \(\mathbb{R}^n\), defined as the zeroset of global analytic functions, is the support of a coherent sheaf of ideals, which defines a complex analytic subset of a Stein open neighborhood of \(\mathbb{R}^n\) in \(\mathbb{C}^n\). In this way he found a class of real analytic spaces globally defined in \(\mathbb{R}^n\), that have a good complexification. As Cartan himself wrote, the notion of analytic real subset has a global character, in contrast with the complex case, which is local.
The complexification and the existence of a coherent structure turned to be relevant in the study of real analytic spaces. The authors of the book obtain results on the so-called C-analytic spaces, by using properties of their complexification. We summarize in what follows the contents of the six chapters of the volume. It is worth to point out that many of the results are due to the work of the authors throughout the last 50 years. Note also that a guiding thread of the book is the comparison with the results in the complex case in contrast with the real one, and with the results in the algebraic case in contrast with the analytic.
The first two chapters recall classical results. In Chapter 1 the real analytic spaces, and C-analytic spaces, are defined. Let \(X\) be a topological space-Hausdorff, paracompact and second-countable- and \(\mathcal{O}_X\) a subsheaf of the sheaf of germs of continuous functions on \(X\). The pair \((X, \mathcal{O}_X)\) will be called a ringed space. Then, a C-analytic space is a ringed space \((X, \mathcal{O}_X)\) locally isomorphic to a local model \((Y, \mathcal{O}_Y)\) where \(Y \subset U \subset \mathbb{R}^n\) is the zeroset of finitely many analytic functions \(f_1, \dots, f_s \in \mathcal{O}(U)\), the ring of real analytic functions on \(U\), and \(\mathcal{O}_Y\) is the quotient sheaf of the sheaf \(\mathcal{O}_U\) by the ideal sheaf generated by \(f_1, \dots, f_s\). It is important to note that it is shown that there exist real analytic spaces not admitting a coherent structure, and hence not being C-analytic.
In Chapter 2, properties of the real and complex analytic sets are studied. Section 2.1 is devoted to the irreducible components of an analytic set, for what the authors are forced to restrict themselves to C-analytic sets. In Section 2.2 they consider the normalization in the case of real C-analytic spaces. The chapter ends with the study of the divisors in a C-analytic space \(X\), in order to decide when a given divisor is the divisor of a meromorphic function on \(X\).
Chapter 3 deals with the Nullstellensätze. The authors give the proof of the Nullstellensatz for closed ideals in \(\mathcal{O}(X)\) when \(X\) is a Stein space, and also for \(X\) a C-analytic space, using the Łojasiewicz radical, which is the radical of the convex hull of the ideal, generally larger than the real radical.
Hilbert's 17th Problem is considered in Chapter 4. The classical version questions whether a positive semidefinite polynomial in \(\mathbb{R}^n\) is a sum of squares in the field \(\mathbb{R}(x)\) of rational functions. In the present setting the authors consider the problem for analytic functions on \(\mathbb{R}^n\). Here the problem can be studied not only for finite sums, but for non-finite convergent sums of squares. The analytic Hilbert's 17th Problem in the global case is not completely solved, but only for some particular cases. The authors point out the present state-of-the-art. It is important to note that they show in Section 4.7.4 the relation between the Hilbert's 17th Problem and the estimation of the Pythagoras number of the field of meromorphic functions on \(\mathbb{R}^n\). This fact is in contrast with the independence of both questions in the algebraic setting.
Analytic inequalities are studied in Chapter 5. They lead to the concept of semianalytic set. The aim of this chapter is to determine the existence of a class of semianalytic sets defined only by global analytic functions, but behaving well with respect to boolean and topological operations, and by images of analytic maps. In particular, this last condition forced to define the concept of subanalytic sets. The chapter ends with a long and important Section 5.4, devoted to construct a theory of irreducibility and irreducible components in C-semianalytic sets. According to the authors, this theory should include that irreducible objects be connected, and that the Zariski closure of an irreducible object in an open neighborhood of it should be irreducible too.
A final Chapter 6 considers properties of the class of semianalytic sets of \(\mathbb{R}^n\), taken as constructible sets in the algebra \(\mathcal{E}(\mathbb{R}^n)\) of smooth functions.
All in all, this noteworthy book fulfills the goal of giving an excellently well written account of the present state of a number of relevant topics in the field of Real Analytic Geometry. real analytic geometry; analytic spaces; semianalytic spaces; C-analytic spaces; subanalytic spaces; Hilbert's 17 problem Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Real-analytic manifolds, real-analytic spaces, Normal analytic spaces, Real algebraic and real-analytic geometry Topics in global real 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 We show that every exponential-polynomial set may be realized as the return set for an action of \(\mathbb N^ n\) on a power of the multiplicative group via algebraic endomorphisms with an algebraic variety as the target set. Here, by an exponential-polynomial set we mean a subset of \(\mathbb N^ m\) defined by finitely many equations of the form \(f(\ell_1,\dots , \ell_m; \alpha^{\ell_1}_1, \dots , \alpha^{\ell_n}_1 , \dots , \alpha^{\ell_1}_t,\dots , \alpha^{\ell_m}_t) = 0\) where \(f\) is a polynomial with integral coefficients and each \(\alpha_j\) is an algebraic integer. An action of \(\mathbb N^ n\) on \(\mathbb G^ N_m\) is given by a sequence of \(n\) commuting algebraic group endomorphisms \(\psi _j: \mathbb G^ N_m\to \mathbb G^ N_m\) via the rule \((\ell_1,\dots , \ell_n)\cdot a:= \psi^{\circ\ell_1}_1\circ\cdots\circ \psi^{\circ\ell_n}_n(a)\). Given such an action, a point \(a \in \mathbb G^ N_m(\mathbb C)\) and a subset \(Y\subseteq \mathbb G^ N_m (\mathbb C)\), the return set is \(E(a, Y):= \{(\ell_1,\dots , \ell_n)\in \mathbb N ^n: (\ell_1,\dots , \ell_n)\cdot a \in Y\}\). Our main theorem is that every exponential-polynomial set is a return set for \(Y\) the rational points on an algebraic variety. The converse that every such return set is polynomial-exponential is implicit in the work of \textit{D. Ghioca} et al. [Duke Math. J. 161, No. 7, 1379--1410 (2012; Zbl 1267.37043)]. Our result shows that natural questions about algebraic dynamics are undecidable. exponential-polynomial set; algebraic endomorphism; algebraic variety; algebraic dynamics Scanlon, T., Yasufuku, Y.: Exponential-polynomial equations and dynamical return sets. Int. Math. Res. Not. (16), 4357-4367 (2014) Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Polynomials in number theory, Global ground fields in algebraic geometry, Linear algebraic groups over global fields and their integers, Arithmetic dynamics on general algebraic varieties, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps Exponential-polynomial equations and dynamical return sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex quasi-projective variety and \(\text{Hilb}^n (X)\) its Hilbert scheme of zero dimensional subschemes of length \(n\). The author expresses the virtual Hodge polynomials of \(\text{Hilb}^n (X)\) -- defined by cohomology with compact support -- in terms of those of \(X\) and the Hilbert scheme of subschemes of length \(n\) supported at a point of \(X\). -- The proof proceeds by comparison with the \(n\)-fold symmetric product of \(X\) and related spaces and uses a lemma on point Hilbert schemes from \(L\). Göttsche's 1991 Bonn thesis [see \textit{L. Göttsche}, ``Hilbertschemata nulldimensionaler Unterschemata glatter Varietäten'', Bonner Math. Schr. 243 (1991; Zbl 0846.14002)]. The key properties of the virtual Hodge polynomial used in the proof are its additivity over stratifications and multiplicativity for fibrations. The results extend those found for the Poincaré and Hodge polynomials of surfaces in a paper by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007)]. Hilbert scheme; virtual Hodge polynomials; symmetric product J. Cheah, ''On the Cohomology of Hilbert Schemes of Points,'' J. Algebr. Geom. 5, 479--511 (1996). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the cohomology of Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{M. Baker} and \textit{S. Norine} [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)] introduced a theory of linear systems on graphs very similar to the theory of divisors on smooth complex varieties. \textit{M. Baker} [Algebra Number Theory 2, No. 6, 613--653 (2008; Zbl 1162.14018)] gave a specialization map from smooth complex curves to strongly semistable curves which explains the similarities. In the paper under review the author extends the theory (also for metrizable graphs and tropical curves) to the real case, i.e. the case of graphs with a real structure. This is not a pointless generalization, not only because it is useful, but also because among the possible definitions of real structure, this seems to be the right one.
If \(X\) is a smooth projective curve defined over \(\mathbb {R}\), then it has a genus \(g(X)\) measured by the topological space \(X(\mathbb {C})\), the number \(s(X)\) of connected components of \(X(\mathbb {R})\) and an integer \(a(X)\) measuring if \(X(\mathbb {C})\setminus X(\mathbb {R})\) is connected or not. For a graph with a real structure \(G\) Coppens introduces a real locus \(G(\mathbb {R})\) and integers \(g(G)\), \(s(G)\) and \(a(G)\); the integer \(s(G)\) involves also the genera of the components of \(G(\mathbb {R})\). For real divisors on \(G\) he extends the main results (like parity invariant) of real linear systems on a real curve. He also look at some extremal case (\(X\) an \(M\)-curve) in which differences occur. real curve; real projective curve; stable curve; divisors on curves; divisors on graphs; real linear systems on graph; metrizable graph; tropical curve; \(M\)-curve Coppens, M.: Linear systems on graphs with a real structure. Q. J. Math. (to appear) Real algebraic and real-analytic geometry, Special divisors on curves (gonality, Brill-Noether theory), Special algebraic curves and curves of low genus, Graphs and linear algebra (matrices, eigenvalues, etc.), Paths and cycles Linear systems on graphs with a real 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 Recall that a map \(f:\mathbb R^n\to\mathbb R^m\) is said to be regular if there exist polynomials \(f_1,\dots,f_m,g\in\mathbb R[{\mathtt x}_1,\dots,{\mathtt x}_n]\) such that \(g^{-1}(0)=\emptyset\) and for every point \(x\in\mathbb R^n\),
\[
g(x)f(x)=(f_1(x),\dots,f_m(x)).
\]
If \(g\) is a constant polynomial the map \(f\) is said to be polynomial. It is a challenge problem to characterize geometrically the semialgebraic subsets \(S\subset\mathbb R^m\) that are either polynomial or regular images of \(\mathbb R^n\) for some positive integer \(n\). Just the case of one dimensional \(S\) is completely understood, see [\textit{J. F. Fernando}, ``On the one dimensional polynomial and regular images of \(\mathbb R^n\)'', to appear in J. Pure Appl. Algebra (2014)].
After some pioneering work by \textit{J. F. Fernando} and the reviewer, [J. Pure Appl. Algebra 179, No. 3, 241--254 (2003; Zbl 1042.14035)] and [Isr. J. Math. 153, 61--92 (2006; Zbl 1213.14109)], the first conclusive results in the bidimensional case were obtained by Ueno in the article under review. Throughout this paper a convex polygon is the intersection of a finite family of closed half-planes with nonempty interior in \(\mathbb R^2\), and a band is a convex polygon in \(\mathbb R^2\) bounded by two parallel lines. The main results of this article are the following:
\noindent Theorem 1. Let \(K\subset\mathbb R^2\) be a convex polygon which is not a band. Then, \(\mathbb R^2\setminus K\) and \(\mathbb R^2\setminus\text{Int}(K)\) are polynomial images of \(\mathbb R^2\).
\noindent Theorem 2. Each convex polygon and its interior are regular images of \(\mathbb R^2\).
Both results are optimal, in the following sense. First, the complement of a band is not connected, so it cannot be a continuous image of \(\mathbb R^2\). Secondly, singletons are the unique bounded polynomial images of \(\mathbb R^n\). Hence, regular mappings are needed, at least to realize bounded polygons and their interiors as images of \(\mathbb R^2\).
The article is based on a part of the doctoral dissertation of the author, written under the supervision of J.F. Fernando. It is transparently written and introduces very original techniques. In fact, its results are interesting by themselves and they have an extra value because they have been a fundamental seminal work for further progress. In [\textit{J. F. Fernando} and \textit{C. Ueno}, ``On complements of convex polyhedra as polynomial and regular images of \(\mathbb R^n\)'', Int. Math. Res. Not. (2014) to appear] the authors extend Theorem 1 to convex polyhedra \(K\) of \(\mathbb R^n\). They prove that if \(K\) is moreover bounded, both \(\mathbb R^n\setminus K\) and \(\mathbb R^n\setminus\text{Int}(K)\) are polynomial images of \(\mathbb R^n\). Indeed, the boundedness condition can be removed for \(n=3\), as it is proved by Fernando and Ueno (preprint).
Concerning Theorem 2, it was extended to convex polyhedra of \(\mathbb R^n\) in [\textit{J. F. Fernando} et al., Proc. Lond. Math. Soc. (3) 103, No. 5, 847--878 (2011; Zbl 1282.14101)] generalizing the techniques introduced by Ueno in the article under review. polynomial image of \(\mathbb R^n\); regular image of \(\mathbb R^n\); convex polyhedra Ueno, C., On convex polygons and their complements as images of regular and polynomial maps of \(\mathbb{R}^2\), J. Pure Appl. Algebra, 216, 11, 2436-2448, (2012) Semialgebraic sets and related spaces, \(n\)-dimensional polytopes On convex polygons and their complements as images of regular and polynomial maps of \(\mathbb R^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies the embedded resolution of an algebroid surface over an algebraically closed field of characteristic zero, that is the spectrum of a ring \( K[[X, Y,Z]]/(F)\).
The main combinatorial object associated to \(F\) is Hironaka's characteristic polygon \(\Delta(F)\). The original motivation of this work is: can the combinatorics bound, in some effective sense, the resolution process?
The paper studies in detail the resolution process for prepared equations, that \(F\) is a generic Weierstrass-Tchirnhausen equation, of the form \(Z^n +\sum_{k=0}^{n-2}a_k(X,Y)Z^k\) with \(a_k\) regular in \(X\) of order \(\nu_k=\nu(a_k)\geq n-k\).
The resolution strategy used is the following: (1) if \((Z,X)\) or \((Z, Y )\) are permissible curves, a monoidal transformation centered at them is performed, (2) otherwise, a quadratic transformation. For prepared equations bounds are given for the number of blow-ups needed before the multiplicity drops.
The paper contains many examples. resolution of surface singularities; Newton-Hironaka polygon; equimultiple locus; blowing-up Singularities of curves, local rings, Complex surface and hypersurface singularities Combinatorics and their evolution in resolution of embedded algebroid 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 show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives \(DM\) over a field \(k\). In the context of the Morel-Voevodsky \(\mathbb{A}^1\)-stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the \(E_1\)-term of Voevodsky's slice spectral sequence. Motivic cohomology; motivic homotopy theory, Algebraic cycles, Derived categories, triangulated categories, Higher algebraic \(K\)-theory, Stable homotopy theory, spectra On the convergence of the orthogonal spectral sequence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper is as follows: given a definable subset of \({\mathbb R}^{m+n}\) over an o-minimal structure over the reals (for example, a semi-algebraic or a restricted sub-Pfaffian set), the number of homotopy types of its projection to \({\mathbb R}^n\) admits a single exponential in \(mn\) upper bound. On the other hand, the only known upper bound for the number of topological types of such a projection is double exponential in \(mn\).
For the proof, the authors define a Whitney stratification of the given set by sign-invariant subsets of a family of real polynomials, then bound the number of homotopy types of fibres by the number of connected components of the complement in \({\mathbb R}^n\) of the projections of strata of dimension \(< n\) united with the critical values of the projection over the strata of dimension \(\geq n\), and finally use the upper bounds to the Betti numbers of projections of semi-algebraic sets established by \textit{A. Gabrielov, N. Vorobjov} and \textit{T. Zell} [J. Lond. Math. Soc. 69, 27--43 (2004; Zbl 1087.14038)]. Among applications of the main result are single exponential upper bounds to the number of homotopy types of semi-algebraic sets defined by fewnomials and for the radii of balls guaranteeing the local contractibility for semi-algebraic sets defined by integral polynomials. semi-algebraic sets; ssub-Pfaffian sets; Pfaffian maps; o-minimal structure; Whitney stratification; Betti numbers S. Basu and N. Vorobjov. On the number of homotopy types of fibres of a definable map. \textit{J. Lond. Math. Soc. (2)}, 76(3):757-776, 2007. Semialgebraic sets and related spaces, Topology of real algebraic varieties, Semi-analytic sets, subanalytic sets, and generalizations On the number of homotopy types of fibres of a definable map | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present an algorithm for determining the existence of the limit of a real multivariate rational function \(q\) at a given point which is an isolated zero of the denominator of \(q\). When the limit exists, the algorithm computes it, without making any assumption on the number of variables.
A process, which extends the work of \textit{C. Cadavid} et al. [J. Symb. Comput. 50, 197--207 (2013; Zbl 1263.68176)], reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components. limits of multivariate rational functions; real algebraic sets; regular chains P. Alvandi, M. Kazemi, and M. Moreno Maza. Computing limits of real multivariate rational functions. InProceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC 2016).ACM, 2016. To appear. Symbolic computation and algebraic computation, Polynomials in real and complex fields: factorization, Computational aspects of field theory and polynomials, Real algebraic sets, Real-analytic and semi-analytic sets, Real rational functions Computing limits of real multivariate rational functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be the function field of a geometrically integral curve over a field \(k\), and let \(S\) be a non-empty set of primes of \(K\). Let \(X\) be a scheme over \(\mathcal{O}_S= \{f\in K\,:\, v(f)\geq 0 \text{ for }v\in S\}\). This article describes, in certain situations, the image of the diagonal map
\[
X(\mathcal{O_S}) \to \prod_{v\in S} X(\mathcal{O}_v) \times \prod_{v\notin S} X(K_v)
\]
(where \(K_v\) is the completion of \(K\) at \(v\), \(\mathcal{O}_v\) its ring of integers) in terms of certain ``differential obstructions''.
To define these, one first realizes \(K\) as a finite separable extension of \(k(t)\). Let \(\delta:K\to K\) be the unique extension of the derivation with respect to \(t\), and let \(\Lambda\) be a polynomial in \(\delta\) (i.e., a linear differential operator). For any field \(L/K\) together with an extension of \(\delta\), let \(H^1(L, \Lambda) = L/\Lambda L\). Any \(f\in \Gamma(X, \mathcal O_X)\) defines maps \(X(L)\to L\to H^1(L, \Lambda)\), and hence an \((x_v)\in\prod_v X(K_v)\) defines an element of \(\prod_v H^1(K_v, \Lambda)\), which must lie in the image of \(H^1(K, \Lambda)\) if \((x_v)\) is in the image of \(X(K)\).
The author shows that this necessary condition suffices to describe the image of the diagonal map in case \(\mathrm{char} k = 0\) and \(X\) is affine and of finite type over \(\mathcal{O}_S\) (Theorem~2.2), or if \(S\) contains all primes of \(K\) and \(X\) is a smooth proper curve over \(\mathcal{O}_S\) with nonzero Kodaira-Spencer class (Theorem~3.1). differential obstruction; rational points Varieties over global fields, Positive characteristic ground fields in algebraic geometry Differential descent obstructions over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:X \rightarrow \Delta\) be a one-parameter degeneration of reduced compact complex analytic spaces of dimension \(n\). It is an important problem to compute the limit mixed Hodge structure on the cohomology of the fiber \(X_t\) with \(t \in \Delta^*\) from the geometry and cohomology of the central fiber \(X_0\) (e.g. compare geometric compactifications and Hodge-theoretical compactifications of moduli spaces). When \(f\) is a semistable degeneration, the limit mixed Hodge structure can be determined using the Clemens-Schmid sequence. In the case where \(X\) is smooth (and \(X_0\) has only rational singularities or more generally Du Bois singularities), several generalizations of the Clemens-Schmid sequence have been derived by the authors in [Sel. Math., New Ser. 27, No. 4, Paper No. 71, 48 p. (2021; Zbl 1478.14024)]. In the paper under review, the authors prove the invariance of the frontier Hodge numbers \(h^{p,q}\) with \(pq(n-p)(n-q)=0\) for the intersection cohomology of the fibers \(h^{p,q}_{\mathrm{IH}}(X_0)=h^{p,q}_{\mathrm{IH}}(X_t)\) and also for the cohomology of their desingularizations \(h^{p,q}(\widetilde{X}_0)=h^{p,q}(\widetilde{X}_t)\), under the assumption that the central fiber \(X_0\) is reduced, projective, and has only rational singularities. As a corollary (and assuming further smoothness of \(X_t\)), the the order of nilpontence of the logarithm monodromy \(N\) on \(H^j_{\lim}(X_t)\) is \(\max\{1,\min\{j-1,2n-j-1\}\}\) which is smaller than in the general singularity case by \(2\). The proof relies on the theory of mixed Hodge modules of the third named author, especially the results in [\textit{M. Saito}, Math. Ann. 295, No. 1, 51--74 (1993; Zbl 0788.32025)]. rational singularities; limit mixed Hodge structure; frontier Hodge numbers Variation of Hodge structures (algebro-geometric aspects), Deformations of singularities, Deformations of complex singularities; vanishing cycles Deformation of rational singularities and Hodge 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 The authors of this paper consider the following structured low-rank approximation problem. Let \(U\in \mathbb R^{m\times n}\) be a matrix, \(r\) an integer, \(\mathcal L\) a class of matrices that is a linear subspace of \(\mathbb R^{m\times n}\), and \(\parallel \cdot \parallel^2_{\Lambda}\) a weighted Frobenius distance, where the weight matrix \(\Lambda=(\lambda_{ij})\) has positive real entries. Then, the problem consists in finding one of the nearest structured low-rank matrices with respect to the given norm, i.e.~in finding \(X\in \mathcal L\) with \(\mathrm{rank}(X) \leq r\) such that
\[
\parallel X-U \parallel^2_{\Lambda}=\sum_{i=1}^m \sum_{j=1}^n \lambda_{ij}(x_{ij}-u_{ij})^2 = \min_{A\in \mathcal L, \;\mathrm{rank}(A)\leq r} \parallel A-U \parallel^2_{\Lambda}.
\]
This problem is difficult even when \(\Lambda\) is the all-one matrix and the authors consider also generic weights, distinguishing the cases in which \(\mathcal L\) is given either by homogeneous linear equations (linear case) or by inhomogeneous linear equations (affine case).
Among the possible approaches, the authors decide to use Gröbner bases methods to identify the critical points on the complex field. Then, the real solutions are singled out.
For generic data matrices \(U\) (the word \textit{generic} has here the usual meaning of algebraic geometry), the complex critical points are counted by the \(\Lambda\)-weighted Euclidean distance degree of the determinantal variety consisting of the matrices of \(\mathcal L\) of rank \(\leq r\). This gives also an estimate of the computational cost of the problem. Thus, under the hypothesis of genericity of \(\mathcal L\), some explicit formulas for the \(\Lambda\)-weighted Euclidean distance degree are exhibited by means of techniques from algebraic geometry. Finally, the authors focus on Henkel matrices and Sylvester matrices. Some examples support the description of the results. low-rank approximation; Gröbner bases; Euclidean distance degree Ottaviani, G., Spaenlehauer, P.-J., and Sturmfels, B., \textit{Exact solutions in structured low-rank} \textit{approximation}, SIAM J. Matrix Anal. Appl. 35 (2014), no. 4, 1521--1542. Computational aspects of higher-dimensional varieties, Numerical optimization and variational techniques, Symbolic computation and algebraic computation, System structure simplification Exact solutions in structured low-rank approximation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Phi\;:\;X \longrightarrow S\) a dominant morphism of \(k\)-varieties, where \(k\) is an algebraically closed field of characteristic 0. By Hironaka's theorem, you can make both \(X\) and \(S\) regular. But you cannot make \(\Phi\) smooth, an approximation of smoothness is monomialization.
Definition 1.1. Let \(\Phi:X \to S\) be a dominant morphism of regular \(k\)-varieties. \(\Phi\) is named monomial if, for all \(p\in X\) there exists an étale neighbourhood \(U\ni p\) and uniformizing parameters \((x_1,...,x_n)\) of \(U\) (i.e. \(x_i \in \Gamma(V,{\mathcal O}_V)\), \(1\leq i \leq n=\dim(X)\), and the natural morphism \(U \to \text{Spec}\,k[ x_1,...,x_n]\) is étale) and regular parameters \((y_1,...,y_n)\) in \({\mathcal O}_{S,\Phi(p)}\) and a matrix \((a_{i,j})\) of nonnegative integers (which necessarily has rank \(m=\dim(S)\)) such that \(y_j=x_1^{a_{j,1}}...x_n^{a_{j,n}},\;1\leq j \leq m.\) This notion of monomial is the best notion below the notion of smoothness which is in general impossible to get.
In the paper under review, \textit{S. D. Cutkosky} gives the state of art in the theory of monomialization and announces the next result published in his book ``Monomialization of morphisms from 3-folds to surfaces'' [ Lect. Notes Math. 1786 (2002; Zbl 1057.14009)]{}
Theorem 1.3. Let \(\Phi:X\longrightarrow S\) a dominant morphism from a 3-fold \(X\) to a surface \(S\), then there exist sequences of blowing-ups of non singular subvarieties \(X_1 \to X\) , \(S_1 \to S\) such that the induced map \(\Phi_1: X_1 \to S_1\) is everywhere defined in \(X_1\) and monomial.
The cases where \(X\) and \(S\) are surfaces and where \(X\) is a \(k\)-variety and \(S\) a curve are stated and proven in this paper. regular varieties; monomialization; desingularization; approximation of smoothness; dominant morphism Schemes and morphisms, Local structure of morphisms in algebraic geometry: étale, flat, etc. Resolution of morphisms. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X \subset \mathbb{P}^N\) be a smooth complex quasiprojective variety. A line \(L \subset \mathbb{P}^N\) is said \(k\)-secant to \(X\) if the scheme \(L \cap X\) is finite of degree \(\geq k\). The locus of \(k\)-secant lines to \(X \subset \mathbb{P}^N\) is a classical object of study in projective geometry, naturally in relation with the study of the projections \(\pi:X \to X_1 \subset \mathbb{P}^{N-1}\) into \(\mathbb{P}^{N-1}\) from a general point of \(\mathbb{P}^N\). In particular, it is of interest to study the subscheme \(X_{\{k_1, \dots, k_r\}} \subset X_1\) formed by the points \(x \in X_1\) such that \(\pi^{-1}(x)\) contains \(r\) points \(\{x_1,\dots, x_r\}\) (possibly equal) with multiplicity greater than or equal to \(k_i\) in \(x_i\). In the paper under review it is proven (see \textit{General Projection Theorem}, Theorem 1.1) that this scheme is pure dimensional of dimension \(N-1-\sum (k_i(N-n)-1)\). Moreover its singular locus is \(X_{\{k_1, \dots, k_r,1\}}\) and its normalization is smooth.
In order to be precise about the scheme structure of \(X_{\{k_1, \dots, k_r\}}\), the General Projection Theorem is presented as a consequence of a result stated in the language of Hilbert schemes of aligned points, naturally equipped with a map to the Grassmannian of lines \(G(1,N)\). This is Theorem 1.3, named \textit{Aligned Hilbert Scheme Theorem}, where the schemes \(X_{\{k_1, \dots, k_r\}}\) can be related naturally with fibers of natural projections of incidence varieties constructed by means of the map from the Hilbert scheme to the Grassmannian (see Theorem 1.3 for details).
The Aligned Hilbert Scheme Theorem results to be a consequence of the \textit{Aligned ordered Hilbert Scheme Theorem} where Hilbert schemes are substituted by ordered Hilbert schemes, parameterizing finite aligned subschemes supported in an ordered set of pints \((x_1,\dots, x_r)\) and with multiplicity \(k_i\) at \(x_i\). Being this ordered Hilbert scheme finite and flat over the non-ordered one, its smoothness implies smoothness of the non-ordered one.
An interesting section (Section 5) of examples, questions and conjectures (specially about the irreducibility of the locus of points of order \(k\) of a general projection) is also provided. \(k\)-secants; general projections; Hilbert schemes Gruson, L.; Peskine, C., On the smooth locus of aligned Hilbert schemes. the \textit{k}-secant lemma and the general projection theorem, Duke Math. J., 162, 553-578, (2013) Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry On the smooth locus of aligned Hilbert schemes, the \(k\)-secant lemma and the general projection theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\boldsymbol{f} = ( f_1, \ldots, f_s)\) be a sequence of polynomials in \(\mathbb{Q} [ X_1, \ldots, X_n]\) of maximal degree \(D\) and \(V \subset \mathbb{C}^n\) be the algebraic set defined by \textbf{f} and \(r\) be its dimension. The real radical \(\sqrt[ r e]{ \langle \boldsymbol{f} \rangle}\) associated to \textbf{\(f\)} is the largest ideal which defines the real trace of \(V\). When \(V\) is smooth, we show that \(\sqrt[ r e]{ \langle \boldsymbol{f} \rangle} \), has a finite set of generators with degrees bounded by \(\deg V\). Moreover, we present a probabilistic algorithm of complexity \(( s n D^n )^{O ( 1 )}\) to compute the minimal primes of \(\sqrt[ r e]{ \langle \boldsymbol{f} \rangle} \). When \(V\) is not smooth, we give a probabilistic algorithm of complexity \(s^{O ( 1 )} ( n D )^{O ( n r 2^r )}\) to compute rational parametrizations for all irreducible components of the real algebraic set \(V \cap \mathbb{R}^n\).
Let \(( g_1, \ldots, g_p)\) in \(\mathbb{Q} [ X_1, \ldots, X_n]\) and \(S\) be the basic closed semi-algebraic set defined by \(g_1 \geq 0, \ldots, g_p \geq 0\). The \(S\)-radical of \(\langle \boldsymbol{f} \rangle \), which is denoted by \(\sqrt[ S]{ \langle \boldsymbol{f} \rangle} \), is the ideal associated to the Zariski closure of \(V \cap S\). We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding \(\sqrt[ S]{ \langle \boldsymbol{f} \rangle} \). Assuming now that \(D\) is the maximum of the degrees of the \(f_i\)'s and the \(g_i\)'s, this algorithm runs in time \(2^p ( s + p )^{O ( 1 )} ( n D )^{O ( r n 2^r )} \). Experiments are performed to illustrate and show the efficiency of our approaches on computing real radicals. polynomial system; real radical; \(S\)-radical ideal; semi-algebraic set; real algebraic geometry Computational real algebraic geometry, Semialgebraic sets and related spaces, Symbolic computation and algebraic computation Computing real radicals and \(S\)-radicals of polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex algebraic variety, Whitney stratified by locally closed connected smooth subvarieties \(X_\lambda\). To each such stratum and local system \(\mathcal{L}\) of free \(\mathbb{Z}\)-modules on \(X_\lambda\), one associates the intersection cohomology \(\mathcal{D}_X\)-module \(\mathcal{IC}(\overline{X}_\lambda, E_{\mathcal{L}})\) extending the associated flat vector bundle \(E_{\mathcal{L}}\). Its characteristic cycle is a \(\mathbb{Z}\)-linear combination \(CC(\mathcal{IC}(\overline{X}_\lambda, E_{\mathcal{L}})) = \sum_\mu m^{\mathcal{L}}_{\lambda \mu} [\overline{T^*_{X_\mu} X}],\) where the multiplicities \(m^{\mathcal{L}}_{\lambda \mu}\) are certain Euler characteristics.
Let \(p\) be a prime number and \(\mathcal{M}\) a local system of \(\mathbb{F}_p\)-vector spaces on \(X_\mu\). Then the authors prove that \(\text{rank}(\mathcal{M})\cdot d^p_{\lambda, \mathcal{L}, \mu, \mathcal{M}} \leq m^{\mathcal{L}}_{\lambda \mu},\) where \(d^p_{\lambda, \mathcal{L}, \mu, \mathcal{M}}\) is the \textit{decomposition number}
\[
d^p_{\lambda, \mathcal{L}, \mu, \mathcal{M}} = [\mathbf{IC}(\overline{X}_\lambda, \mathcal{L})\otimes^L_{\mathbb{Z}} \mathbb{F}_p : \mathbf{IC}(\overline{X}_\mu, \mathcal{M})].
\]
Here, \(\mathbf{IC}\) denotes the extension by zero to \(X\) of the intersection chain sheaf complex extending a given local system. \(D\)-modules; intersection cohomology K. Vilonen, G. Williamson, Characteristic cycles and decomposition numbers, Math. Res. Lett. 20 (2013), no. 2, 359--366. Algebraic cycles, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Representation theory for linear algebraic groups Characteristic cycles and decomposition numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field, let \(\gamma\) be an arc of a \(k\)-variety \(X\), and assume that \(\gamma\) does not factor through the singular locus of \(X\). The Drinfeld-Grinberg-Kazhdan Theorem states that the formal neighborhood of \(\gamma\) in the arc scheme of \(X\) is isomorphic to the product of an infinite dimensional formal disc and a formal neighborhood in a finite type \(k\)-scheme. The latter formal neighborhood has been interpreted as a finite dimensional model of the formal neighborhood of \(\gamma\). The authors prove a generalization of the Drinfeld-Grinberg-Kazhdan theorem where \(X\) is replaced with a topologically finite type formal \(k[[T]]\)-scheme \(\mathcal{X}\), and they study consequences related to singularity theory.
In the case where \(\mathcal{X}\) is the completion of an affine \(k\)-variety, their proof gives an algorithm for computing the resulting finite dimensional model. In addition, they prove that these finite dimensional models are compatible with base change by separable extensions of \(k\), and they also prove a factoring statement involving the truncation morphisms from arc schemes to jet schemes.
The authors also present a proof, communicated to them by O. Gabber, of a cancellation theorem, which in particular allows them to define unique minimal finite dimensional models for an arc's formal neighborhood. The authors also use finite dimensional models to define an invariant they call the absolute nilpotency index of a singularity, and they prove that this is a formal invariant of the singularity. arc scheme; formal neighborhood Bourqui D., The Drinfeld--Grinberg--Kazhdan Theorem for Formal Schemes and Singularity Theory (2015) Arcs and motivic integration, Singularities in algebraic geometry The Drinfeld-Grinberg-Kazhdan theorem for formal schemes and singularity theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article presents a very extended overview presented in detail of the integrability of the Hirota-Kimura (H-K) type discretization method subject to algebraically completely intregrable systems with quadratic vector fields. This discretization scheme was first introduced by \textit{W. Kahan} [Unconventional numerical methods for trajectory calculations. Unpublished lecture notes. University of California, Berkeley (1993)] and is applicable for any system of ordinary differential equations for a vector \(x:\mathbb{R}\rightarrow \mathbb{R}^{n}\) of the form:
\[
\dot{x}=Q(x)+Bx+c, \tag{1}
\]
where each component of \(Q:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is a quadratic form, \(B\in \mathrm{End}\)\text{\ }\(\mathbb{R}^{n}\) and \(c\in \mathbb{R}^{n}.\) The Kahan's discretization reads
\[
\frac{\tilde{x}-x}{\varepsilon }=Q(x,\tilde{x})+\frac{1}{2}B(x+\tilde{x})+c, \tag{2}
\]
where
\[
Q(x,\tilde{x}):=\frac{1}{2}[Q(x+\tilde{x})-Q(x)-Q(\tilde{x})] \tag{3}
\]
is the symmetric bilinear form in \(\mathbb{R}^{n},\) corresponding to the quadratic form \(Q:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}.\)
The obtained discretization possesses the important reversibility property:
\[
\tilde{x}:=f(x,\varepsilon )=f^{-1}(\tilde{x},-\varepsilon ), \tag{4}
\]
implying, in particular, its birationality.
Since the discretization was later applied by Hirota and Kimura to the algebraically integrable Euler and Lagrange tops, this special case was named by the authors as the Hirota-Kiumura discretization. It was before conjectured by the authors that ``algebraically integrable dynamical systems are also H-K discretization integrable too'', but now, the authors claim that there exist examples which overturn this claim.
Based on the notion of H-K basis, the authors analyze the related problem of existence of conservation laws for the H-K discretized systems of equations (2) and apply this notion to different dynamical systems as the Wierstrass system, the Euler top, the Zhukovski-Volterra system and Volterra chain, the three wave system, the Lagrange top and some others. The authors elaborate a very large experimental material \ by means of computer calculations and conjecture some interesting statements to be verified. In particular, they pose the problem of finding some additional analytical structures for these H-K discretized systems as Lax representation, bi-Hamiltonian formulation, etc. which are up to now still not obtained for any of the studied system. algebraic integrability; integrable systems; integrable discretizations; birational dynamics Petrera M, Pfadler A, Suris Yu B. On integrability of Hirota-Kimura type discretizations. Regul Chaotic Dyn, 2011, 16(3--4): 245--289 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Rational and birational maps, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests On integrability of Hirota-Kimura type discretizations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here the notion of a prehomogeneous vector space is generalized to that of a prehomogeneous affine space and the associated zeta functions are studied. More precisely, let \(\rho\) be an algebraic homomorphism of a linear algebraic group \(G\) into the affine transformation group \(\text{Aff} (V)\) of a finite dimensional vector space \(V\), and \(\alpha\) a regular rational function on \(V \times G\) satisfying the cocycle condition \(\alpha(x,gg')=\alpha(x \rho(g),g')+\alpha(x,g)\) \((x \in V,g,g' \in G)\). A quartet \({\mathbf D} = (G, V,\rho,\alpha)\) is called a prehomogeneous affine datum (PAD) if there exists a proper algebraic subset \(S\) of \(V\) such that \(V-S\) is a single \(\rho(G)\)-orbit. It is proved that the dual \({\mathbf D}^*=(G,V^*,\rho^*,\alpha^*)\) of \({\mathbf D}\) is a PAD if \({\mathbf D}\) is regular in a suitable sense.
Let \({\mathbf D}=(G,V,\rho,\alpha)\) be a regular PAD and \(L\) a lattice of \(V_ \mathbb{R}\). Let \(n\) be the number of independent relative invariants of \({\mathbf D}\). Then one can define the associated zeta functions \(\xi_ i(s,L)\) \((s \in \mathbb{C}^ n\), \(1 \leq i \leq \nu=\) the number of connected components of \(V_ \mathbb{R}-S_ \mathbb{R})\). Let \(\xi^*_ j(s,L^*)\) \((1 \leq j \leq \nu)\) be the zeta functions associated to the dual lattice \(L^*\) of \(L\). The main result of the paper is that, under certain assumptions on \({\mathbf D}\), the functional equations hold between \(\xi_ i (s,L)\) and \(\xi^*_ j(s,L^*)\). Hermitian form; prehomogeneous vector space; prehomogeneous affine space; zeta functions; functional equations Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Homogeneous spaces and generalizations Zeta functions of prehomogeneous affine spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The notion of radius of convergence of a differential equation with analytic coefficients is known to be tricky in the \(p\)-adic setting: it may be finite even when the equation shows no singularity (think about the exponential, for instance). In order to study it, Dwork and his collaborators have introduced a notion of generic radius of convergence: if the differential equation is defined over an annulus \(\alpha \leq |t| \leq \beta\), there is such a generic radius associated to any radius between \(\alpha\) and \(\beta\).
In this paper, the authors investigate the behaviour of the generic radius when the radius varies. One of the main results of the paper states that, for a differential equation over a polyannulus, the logarithm of the generic radius of convergence is continuous and piecewise linear (see theorem 3.3.9). Let us mention that it had already been done in the one-dimensional case in an article by \textit{G. Christol} and \textit{B. Dwork} [Ann. Inst. Fourier 44, No. 3, 663--701 (1994; Zbl 0859.12004)] and more recently in a book by \textit{K. Kedlaya} [Cambridge Studies in Advanced Mathematics 125 (2010; Zbl 1213.12009)].
The techniques the authors use are those of differential modules. The first and second section are devoted to differential modules over fields and one-dimensional spaces respectively and are close to what had already been done in Kedlaya's book (the proofs actually often refer to it). The third section is devoted to general polyannuli. It begins with a general study of polyhedral functions before going back to the study of differential modules, for which the afore-mentioned continuity result is proven as well as a statement of decomposition according to the radii. To finish with, let us mention that the results not only hold over the \(p\)-adics but over any complete nonarchimedean field of characteristic 0 (of any residual characteristic) with nontrivial valuation. differential modules; polyannuli; nonarchimedean field; generic radius of convergence Kiran S. Kedlaya and Liang Xiao, Differential modules on \?-adic polyannuli, J. Inst. Math. Jussieu 9 (2010), no. 1, 155 -- 201. \(p\)-adic differential equations, Rigid analytic geometry, Modules of differentials Differential modules on \(p\)-adic polyannuli | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one. This approach comes from applying a moment relaxation to a suitable polynomial optimization formulation. The hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by Helton, Klep, and McCullough as well as by the authors. Moreover, several exactness results for the solitary criterion can be brought forward to the hierarchical approach. The hierarchy also applies to the (equivalent) question of checking whether a map between matrix (sub-)spaces is positive. In this context, the solitary criterion checks whether the map is completely positive, and thus our results provide a hierarchy between positivity and complete positivity. spectrahedra; containment; moment relaxation; positive map; complete positivity; semidefinite programming K. Kellner, T. Theobald, and Ch. Trabandt, \textit{A semidefinite hierarchy for containment of spectrahedra}, SIAM J. Optim., 25 (2015), pp. 1013--1033, . Semidefinite programming, Semialgebraic sets and related spaces A semidefinite hierarchy for containment of spectrahedra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a finite directed graph \(F\) whose vertices \(x \in \mathcal V\) are labelled by finite dimensional vector spaces \(V_ x\) over \(\mathbb{C}\). Let \(\mathcal E\) be the set of edges and \({\mathcal H} = \oplus\{\text{Hom}(V_ x \to V_ y) : (x \to y) \in {\mathcal E}\}\). The group \(G = \prod\{\text{GL}(V_ x) : x \in {\mathcal V}\}\) acts naturally on \(\mathcal H\), i.e the result of the action of \(g = (g_ x)_{x \in {\mathcal V}}\) on \(f = (f_{xy})_{(x \to y)\in {\mathcal E}}\) is given by \((gf)_{xy} = g_{y} f_{xy}g_{x}^{-1}\).
The author states the problem of calculation of invariants of this action in the symmetric algebra \(S({\mathcal H})\). The theorem 1 asserts that if \(F\) is of type \(A_ r\) with one-way-directed edges then the relative invariants in \(S({\mathcal H})\) amount to the monomials of the ``primitive determinant invariants'' and the last are algebraically independent. In the case of an \(\widetilde{A}_ r\)-graph with one-way-directed edges the absolute and relative invariants are described.
The author proposes the following generalization of the problem: Let \(G\) be a semisimple Lie group, \(P\) a parabolic subgroup, \(P = \text{LU}\) a Levi decomposition of \(P\) and \(N\) the Lie algebra corresponding to \(U\). What are the relative invariants under the adjoint action of \(L\) on \(V = N/[N,N]\)? finite directed graph; symmetric algebra; relative invariants; semisimple Lie group; adjoint action Koike, K.: Relative invariants of the polynomial rings over the ar,Ãr quivers. Adv. math. 86, 235-262 (1991) Representations of quivers and partially ordered sets, Trace rings and invariant theory (associative rings and algebras), Group actions on varieties or schemes (quotients), Geometric invariant theory, Representation theory for linear algebraic groups Relative invariants of the polynomial rings over type \(A_ r\), \(\widetilde A_ r\) quivers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 call a function constructible if it has a globally subanalytic domain and it can be expressed as a sum of products of globally subanalytic functions and logarithms of positively valued globally subanalytic functions.
In the paper under review two main results about constructible functions are proved. Let \(q>0\) and let \(f,\mu\) be constructible functions on \(E \times \mathbb R^n\). The first result describes the structure of the set
\[
\mathrm{LC}(f,|\mu|^q,E) := \big\{(x,p) \in E \times (0,\infty] : f(x,\cdot) \in L^p(|\mu|^q_x)\big\},
\]
where \(L^p(|\mu|^q_x)\) is the Lebesgue space with respect to the positive measure \(|\mu|^q_x\) on \(\mathbb R^n\) given by
\[
|\mu|^q_x(Y) = \int_Y |\mu(x,y)|^q\, dy,
\]
where \(dy\) denotes the Lebesgue measure. It is shown that the set of all fibers of \(\mathrm{LC}(f,|\mu|^q,E)\) over \(E\) is a finite set of open subintervals of \((0,\infty]\), and that set of all fibers of \(\mathrm{LC}(f,|\mu|^q,E)\) over \((0,\infty]\) is a finite set of subsets of \(E\) each of which is the zero locus of a constructible function on \(E\).
The second main result (used in the proof of the first) concerns a preparation theorem that expresses \(f\) and \(\mu\) as finite sums of terms of a simple form that reflect the structure of \(\mathrm{LC}(f,|\mu|^q,E)\).
These results extend previous work by the authors [Duke Math. J. 156, No. 2, 311--348 (2011; Zbl 1216.26008); Int. Math. Res. Not. 2012, No. 14, 3182--3191 (2012; Zbl 1250.26014)] and by \textit{J.-M. Lion} and \textit{J.-P. Rolin} [Ann. Inst. Fourier 48, No. 3, 755--767 (1998; Zbl 0912.32007); Ann. Inst. Fourier 47, No. 3, 859--884 (1997; Zbl 0873.32004)] and \textit{G. Comte, J.-M. Lion} and \textit{J.-P. Rolin} [Ill. J. Math. 44, No. 4, 884--888 (2000; Zbl 0982.32009)]. subanalytic functions; constructible functions; integrability locus; preparation theorem; \(L^{p}\)-spaces Cluckers, R; Miller, DJ, Lebesgue classes and preparation of real constructible functions, J. Funct. Anal., 264, 1599-1642, (2013) Semi-analytic sets, subanalytic sets, and generalizations, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Real-analytic and semi-analytic sets Lebesgue classes and preparation of real constructible functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study random walks on \(\mathrm{GL}_d(\mathbb{R} )\) whose proximal dimension \(r\) is larger than 1 and whose limit set in the Grassmannian \(Gr_{r,d} (\mathbb{R})\) is not contained any Schubert variety. These random walks, without being proximal, behave in many ways like proximal ones. Among other results, we establish a Hölder-type regularity for the stationary measure on the Grassmannian associated to these random walks. Using this and a generalization of Bourgain's discretized projection theorem, we prove that the proximality assumption in the Bourgain-Furman-Lindenstrauss-Mozes theorem can be relaxed to this Schubert condition. Matrix valued random walk; Grassmannian, Schubert variety; random matrix Generalized stochastic processes, Grassmannians, Schubert varieties, flag manifolds, Random matrices (probabilistic aspects), Sums of independent random variables; random walks Random walks on linear groups satisfying a Schubert condition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 gives detailed computations on the Dimitrov-Haiden-Katzarkov-Kontsevich categorical entropy \(h_t(\Phi)\) [\textit{G. Dimitrov} et al., ``Dynamical systems and categories'', Preprint, \url{arXiv:1307.8418}] of Fourier-Mukai transforms \(\Phi\) on abelian surfaces of Picard rank one using the matrix description of cohomological Fourier-Mukai transforms given in [\textit{S. Yanagida} and \textit{K. Yoshioka}, J. Reine Angew. Math. 684, 31--86 (2013; Zbl 1401.14100)]. The paper also gives an affirmative answer to a Gromov-Yomdin type conjecture
\(h_0(\Phi)=\log \rho(\Phi)\) proposed by \textit{K. Kikuta} et al. [Nagoya Math. J. 238, 86--103 (2020; Zbl 1436.14038)], where \(\Phi\) is a triangulated functor \(\mathbf{D}(X) \to \mathbf{D}(X)\) on an arbitrary abelian surface \(X\), and \(\rho(\Phi)\) is the spectral radius of the action of \(\Phi\) on the cohomology group. categorical entropy; abelian surfaces Algebraic moduli problems, moduli of vector bundles Categorical entropy for Fourier-Mukai transforms on generic abelian 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 Given a list of \(n\) cells \(L= [(p_1, q_1),\dots,(p_n, q_n)]\), where \(p_i, q_j\in\mathbb{Z}_{\geq 0}\), we let j\(\Delta_L= \text{det}\|(p_j!)^{-1} (q_j!)^{-1} x^{p_j}_i y^{q_j}_i\|\). The space of diagonally alternating polynomials is spanned by \(\{\Delta_L\}\), where \(L\) varies among all lists with \(n\) cells. For \(a> 0\), the operators \(E_a= \sum^n_{i=1} y_i \partial^a_{x_i}\) act on diagonally alternating polynomials. \textit{M. Haiman} [Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)] has shown that the space \(A_n\) of diagonally alternating harmonic polynomials is spanned by \(\{E_\lambda\Delta_n\}\), where \(\lambda= (\lambda_1,\dots, \lambda_\ell)\) varies among all partitions, \(E_\lambda= E_{\lambda_1}\cdots E_{\lambda_\ell}\) and \(\Delta_n= \text{det}\|((n- j)!)^{-1} x^{n-j}_i\|\). For \(t= (t_m,\dots, t_1)\in\mathbb{Z}^m_{> 0}\) with \(t_m>\cdots> t_1> 0\), we consider here the operator \(F_t=\text{det}\| E_{t_{m-j+1}+ (j-i)}\|\).
Our first result is to show that \(F_t\Delta_L\) is a linear combination of \(\Delta_{L'}\), where \(L'\) is obtained by moving \(\ell(t)= m\) distinct cells of \(L\) in some determined fashion. This allows us to control the leading term of some elements of the form \(F_{t_{(1)}}\cdots F_{t_{(r)}}\Delta_n\). We use this to describe explicit bases of some of the bihomogeneous components of \(A_n= \bigoplus A^{k,l}_n\), where \(A^{k,l}_n= \text{Span}\{E_\lambda: \ell(\lambda)= l\), \(|\lambda|= k\}\).
More precisely, we give an explicit basis of \(A^{k,l}_n\) whenever \(k< n\). To this end, we introduce a new variation of Schensted insertion on a special class of tableaux. This produces a bijection between partitions and this new class of tableaux. The combinatorics of these tableaux \(T\) allow us to know exactly the leading term of \(F_T\Delta_n\), where \(F_T\) is the operator corresponding to the columns of \(T\), whenever \(n\) is greater than the weight of \(T\). alternating; diagonal harmonic; explicit basis; low degree; Jeu de Taquin; Schensted insertion; tableaux Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes) Bases for diagonally alternating harmonic polynomials of low 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 The Krichever construction in one variable, that is, for spectral curves, linearizes the KdV-hierarchy on the Jacobian of the curve. We carry out an appropriate generalization of the Krichever construction for an arbitrary projective variety \(X\) and determine the corresponding nonlinear dynamics, which are then linearized on an extension of \(\text{Pic}^0(X)\).
The outline of the paper is as follows. Section 2 gives the basic results connecting the Fourier-Mukai transform with the theory of \(\mathcal D\)-modules. Section 3 applies this theory, to obtain commuting matrix partial differential operators. Section 4 addresses the problem of choosing a good basis. Section 5 brings in microdifferential operators.
The dynamics are obtained in section 6. In section 7 we obtain the modified dynamics that descend to quotients of \(Y\), gotten by throwing away the trivializations at some of the points of \(\phi^{-1}(P)\). Examples in which the flow descends to \(\text{Pic}^0(X)\) itself are given in section 8. Rothstein, M., Dynamics of the Krichever construction in several variables, J. Reine Angew. Math., 2004, 572, 111-138, (2004) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between surfaces, higher-dimensional varieties, and physics, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Dynamics of the Krichever construction in several variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors consider a smooth, algebraic curve \(C\) of genus \(g \geq 2\) over \(\mathbb{C}\) and consider the moduli of coherent systems over \(C\). A \textit{coherent system} of type \((n,d,k)\) is a pair \((E,V)\), where \(E \to C\) is a rank-\(n\) vector bundle of degree \(d\) and \(V \subseteq H^0(C;E)\) is a \(k\)-dimensional linear subspace.
The notion of slope stability extends to coherent systems, depending on a parameter \(\alpha \in \mathbb{R}\), via the slope \(\mu_\alpha = (d+\alpha k)/n\). \(G(\alpha;n,d,k)\) denotes the moduli space of \(\alpha\)-stable coherent systems on \(C\), which is related to the \textit{Brill-Noether locus} \(B(n,d,k)\) of \(\mu\)-stable bundles \(E \to C\) of rank \(n\) and degree \(d\) with \(h^0(C; E) \geq k\) (considered inside the moduli \(M(n,d)\) of all \(\mu\)-stable rank-\(n\) bundles of degree \(d\)).
The authors had previously studied spaces of coherent systems [J. Reine Angew. Math. 551, 123--143 (2002; Zbl 1014.14012); Int. J. Math. 18, No. 4, 411--453 (2007; Zbl 1117.14034); Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)] in the case \(k \leq n\). The present paper presents a generalization to arbitrary \(k\), subject to the constraint \(d\leq2n\); the final Section~7 illustrates with an example why the case \(d>2n\) is more complicated.
The main results are these: \(G(\alpha;n,d,k)\) is irreducible whenever it is non-empty. With the exception of the case where \(C\) is hyperelliptic and \(d=2n\) and \(k>n\), the conditions for non-emptiness of \(G(\alpha;n,d,k)\) are the same as for non-emptiness of \(B(n,d,k)\). With the exception of the case where \(C\) is hyperelliptic and \((n,d,k) = (n,2n,n+1)\) and \(n<g-1\), the dimension of \(G(\alpha;n,d,k)\) is the \textit{Brill-Noether number}
\[
\beta(n,d,k) {{\mathop:}=} n^2(g-1)+1-k(k-d+n(g-1)) \text{ .}
\]
We always require \(\alpha>0\), \(d>0\) and \(\alpha(n-k)<d\) for non-emptiness. The authors prove sharp inequalities on \(n\), \(d\) and \(k\) for the non-emptiness of \(G(\alpha;n,d,k)\). For example, when \(C\) is not hyperelliptic, then \(G(\alpha;n,d,k) \neq \varnothing\) if and only if \(k\leq n+(d-n)/g\) and \((n,d,k)\neq(n,n,n)\), or if \((n,d,k) = (g-1,2g-2,g)\); otherwise \(G(\alpha;n,d,k) = \varnothing = B(n,d,k)\) for all \(\alpha\). Moreover, a coherent system \((E,V)\) may be presented as an exact sequence
\[
0 \longrightarrow D^* \longrightarrow V \otimes \mathcal{O}_C \longrightarrow E \longrightarrow F \oplus T \longrightarrow 0 \text{ ,}
\]
where \(D\) and \(F\) are vector bundles, \(T\) is a torsion sheaf and \(h^0(C;D^*)=0\). The authors prove that if \((E,V) \in G(\alpha;n,d,k)\) is a generic element then: \(T=0=D\) if \(k<n\), \(F=0=D\) if \(k=n\), and \(F=0=T\) if \(k>n\). The case when \(C\) is hyperelliptic is treated separately with different methods.
Section~2 of the paper is a nice presentation of the basic properties of coherent systems, and Section~3 proves existence results for \(\alpha\)-stable coherent systems, in analogy to the existence of \(\mu\)-stable vector bundles on curves; these sections may be of interest in their own right. Sections~4, 5 and 6 contain the technical work of the paper, proving respectively irreducibility of \(G(\alpha;n,d,k)\), non-emptiness for non-hyperelliptic \(C\) and for hyperelliptic \(C\). algebraic curves; Brill-Noether loci; coherent systems; moduli of vector bundles Bradlow, B.; García-Prada, O.; Mercat, V.; Muñoz, V.; Newstead, P., Moduli spaces of coherent systems of small slope on algebraic curves, Commun. algebra, 37, 8, 2649-2678, (2009) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Special divisors on curves (gonality, Brill-Noether theory) Moduli spaces of coherent systems of small slope 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 Let \(S\) be a normal, locally Noetherian scheme, \(G\) a Chevalley-Demazure group scheme with ``épinglage'' over \(S\). Fix a generic geometric point \(\overline\eta:\text{Spec} K\to S\), and let \(\rho_ K:G_ K \to Gl(V_ K) \subset \text{End}(V_ K)\) be a representation of \(G_ K\). An \(S\)- form of \(V_ K\) is a pair \(\bigl( V({\mathcal O}_ S),i \bigr)\), where \(V({\mathcal O}_ S)\) is a vector bundle over \(S\) equipped with a \(G\)-action and \(i:V({\mathcal O}_ S)_ K \to V_ K\) is an isomorphism respecting the action of the group scheme \(G\). The author classifies all \(S\)-forms of \(V_ K\) by constructing a mapping from the set of such forms to \(\text{Pic} S\) and then classifying the elements in the fibre of the map as locally free \({\mathcal O}_ S\)-modules equipped with a graded structure defined in terms of a Chevalley system of \(G_ K\). This characterization is extended to nonsplit group schemes \(G\) by using the rigidity of the épinglage structure to construct descent data. Here the fibres are classified by \(H^ 1\bigl(S_{\text{ét}},\Aut(\rho)\bigr)\). The author finishes by describing \(\Aut(\rho)\) as an extension of \(G_ m\) by \(H\), where \(H\) is a semidirect product of \(\text{ad}(G)\) by the discrete group of automorphisms of the épinglage structure. \(S\)-forms; automorphism group; Chevalley-Demazure group scheme; action of group scheme Group schemes, Representation theory for linear algebraic groups Representations of reductive group 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 A stratification of a mapping \(f:N\to P\) is a pair of stratifications \((\Sigma_N, \Sigma_P)\) of the source and target such that \(f\) restricts on each \(X\in \Sigma_N\) to a submersion to a \(Y\in\Sigma_P.\) Generic proper \(C^\infty\)-smooth mappings can be canonically stratified [\textit{J. Mather}, Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 195--232 (1973; Zbl 0286.58003)]. All proper subanalytic mappings can be stratified [\textit{H. Hironaka}, Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Y. Akizuki, 453--493 (1973; Zbl 0297.32008)].
The author defines a new regularity condition \(B_f\) for stratified maps; it implies the Whitney-regularity of the restriction of \(\Sigma_P\) to the image of \(f.\) A stratum \(X\in\Sigma_N\) is \(B_f\)-regular over a stratum \(Y\in\Sigma_P\) at a \(y\in Y\) if the following condition holds: Assume \(P=\mathbb{R}^p.\) Let \(x_i\in X\) be a sequence convergent to an \(x\in N\) such that \(f(x)=y,\) and let \(y_i\in Y\) be a sequence convergent to \(y\in Y\). Assume that the lines connecting \(f(x_i)\) and \(y_i\) converge to a line. Then there exist tangent vectors \(v_i\) to the strata \(X\) at \(x_i\) such that
\[
\parallel v_i \parallel \to 0\text{ and }{{df(v_i ) - (y_i - f(x_i ))} \over {\min \{ \parallel df(v_i )\parallel ,\;\parallel y_i - f(x_i )\parallel \} }} \to 0\quad \text{ as }i\to \infty.
\]
It is shown that all complex analytic or subanalytic proper mappings admit Whitney-regular and \(B_f\)-regular stratifications, and with such regularity are semi-triangulable. The author goes on to discuss such topics as the local mod 2 Euler characteristic condition [\textit{W. Fulton} and \textit{R. MacPherson}, Mem. Am. Math. Soc. 243 (1981; Zbl 0467.55005)], generalizations of Sullivan's formula relating Stiefel-Whitney homology classes [\textit{D. Sullivan}, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969--1970, Lect. Notes Math. 192, 165--168 (1971; Zbl 0227.32005)], mapping cylinder methods and stratifications of Morin mappings. stratified mappings; semi-triangulable; Morin mapping; constructible functions; local mod 2 Euler characteristic; Whitney regularity; generic smooth mapping I. Nakai, Elementary topology of stratified mappings, In: Singularities-Sapporo 1998, (eds. J. P. Brasselet and T. Suwa), Adv. Stud. Pure Math., 29 , Kinokuniya, Tokyo, 2000, pp.,221-243. Stratifications in topological manifolds, Singularities in algebraic geometry, Stratified sets, Differentiable maps on manifolds, Singularities of differentiable mappings in differential topology, Semi-analytic sets, subanalytic sets, and generalizations, Global theory of complex singularities; cohomological properties, Characteristic classes and numbers in differential topology Elementary topology of stratified mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The approximation problem is the following: \(\varphi\) being a differentiable map between compact (affine) algebraic varieties \(X\subset \mathbb{R}^n\) and \(Y\subset \mathbb{R}^r\), find for all \(\varepsilon\) an open Nash isomorphism \(\iota\) from \(X\) to an algebraic variety \(\widehat X\) and a regular map \(\widehat\varphi\) such that \(d(\widehat \varphi (\iota(x)), \varphi (x)) < \varepsilon\) whenever \(\iota(x) \in \text{Reg} (\widehat X)\). If \(Y\) is smooth as a Nash variety the approximation problem can be solved by approximating maps with values in \(\mathbb{R}^r\) by the existence of Nash tubular neighborhoods; when \(Y\) is regular, a sufficient condition in order that \(\varphi\) have an approximation is that \((X,\varphi)\) be cobordant to a regular pair. In this paper the authors give sufficient conditions to solve the approximation problem, in the real and complex cases, \(Y\) being singular or not. For instance, they give the following result in the real case:
Let \(X\) be a codimension 1 Nash submanifold of a locally closed Nash subvariety \(S\) of \(\mathbb{R}^n\) and let \(\varphi\) be a Nash map from \(X\) to an algebraic subvariety \(Y\subset \mathbb{R}^r\). If there exists a Nash map \(\theta\) from \(S\) to \(Y\) such that \(\theta|X = \varphi\) and \(X' \subset S\) is a regular algebraic subvariety of \(\mathbb{R}^n\) of codimension 1 in \(S\) such that \(\theta |X'\) is regular and \(X'\)-generic, then, if \(X\) is \(K\)-equivalent to \(X'\) in \(S\), \(\varphi\) has an algebraic approximation on \(K\), where \(K\) is compact in \(X\). Nash approximation; Nash isomorphism; Nash tubular neighborhoods Nash functions and manifolds, Real-analytic and Nash manifolds, Real-analytic sets, complex Nash functions On the algebraic approximation of Nash maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The long-term purpose initiated by this paper is the rigorous construction of robust algorithms for approximate polynomial computations. In this direction a crucial task is to generalize to the multivariate case the continuity of the set of roots of a univariate polynomial with respect to its coefficients. \textit{A. Ostrowski}'s result in the book: ``Solution of equations in Euclidean and Banach spaces'', 3rd edition (1973; Zbl 0304.65002) give sharp bounds for the modulus of continuity. Indeed, the problem ``compute the solutions of a system of equations'' is ill-posed if the set of solutions does not depend continuously on the parameters (that are assumed to vary inside a well-described set) describing the system. In fact, continuity results are only a first step since in a computational setting we need a precise modulus of continuity in order to get a completely explicit computation. We choose here to work in the projective space and in the complex setting.
Inspired by classical results in algebraic geometry, we study the continuity with respect to the coefficients of the zero set of a system of complex homogeneous polynomials with a given pattern and when the Hilbert polynomial of the generated ideal is fixed. We prove topological properties of some classifying spaces, e.g. the space of systems with given pattern, fixed Hilbert polynomial is locally compact, and we establish continuous parametrizations of Nullstellensatz formulae. In the general case we get local rational results but in the complex case we get global results using rational polynomials in the real and imaginary parts of the coefficients. In a second companion paper, we shall treat the continuity of zero sets for the Hausdorff distance, i.e., from a metric point of view. algorithms for approximate polynomial computations; roots of a univariate polynomial A. Galligo, L. Gonzalez-Vega, H. Lombardi, Continuity properties for flat families of polynomials (II) A metric point of view, in preparation. Polynomials, factorization in commutative rings, Real algebraic and real-analytic geometry, Relevant commutative algebra, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation, Computational aspects in algebraic geometry, Computational aspects of field theory and polynomials Continuity properties for flat families of polynomials. I: Continuous parametrizations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 bijective mappings of real affine or projective spaces over finite-dimensional vector spaces that preserve collinearity are affine-linear or projective-linear, respectively. In this text, the authors study relaxations of the hypotheses of these ``fundamental theorems''. In the affine setting, they require that only lines parallel to finitely many directions are mapped to lines. In the projective setting, lines incident with elements of certain finite point sets are required to be mapped to lines.
The main results for the affine space \(\mathbb{R}^n\) are as follows: If lines in \(n\) linearly independent directions are mapped onto lines, the mapping is of a special polynomial form. If another general direction is added, the polynomial becomes even more restricted and mild additional assumptions ensure its affine-additivity or affine-linearity. Injections of the projective space \(\mathbb{RP}^n\) that map every line through one of \(n+2\) generic points onto a line are projective-linear. The genericity assumption on the points may be weakened to accommodate for affine versions of this results.
While the statements in this paper are formulated for real affine and projective spaces, many results hold true for spaces over other fields or division rings. Hints towards possible generalization are given throughout this text. In Section~4 it is shown how continuity influences results. Often, the requirement that lines are mapped to lines can be weakened to ``lines are mapped into lines'' and bijectivity can be abandoned for just injectivity. The concluding Section~5 contains an account of the history of problems in the spirit of those investigated in this paper. fundamental theorem; collineation; affine-additive map Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), General theory of linear incidence geometry and projective geometries, Linear incidence geometric structures with parallelism The fundamental theorems of affine and projective geometry revisited | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 algebraic \(K\)-theory of a scheme is a fundamental and much studied invariant. One of the most important foundational results in the subject, established by \textit{R. W. Thomason} and \textit{T. Trobaugh} [The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88 (1990; Zbl 0731.14001)] is the localization cofiber sequence
\[
K(\text{\(X\) on \((X-U)\)}) \to K(X) \to K(U)
\]
where \(X\) is quasi-separated and quasi-compact and \(U\) a quasi-compact open subscheme.
Topological Hochschild homology \(THH\) and topological cyclic homology \(TC\) are among the most powerful computational tools for accessing algebraic \(K\)-theory. The main result of this paper establishes these fundamental localization cofiber sequences for \(THH\) and \(TC\) (as well as for the intermediate theory \(TR\)) of schemes. This result is a specialization of the authors' more general localization cofiber sequence for \(THH\) (resp.~\(TC\), \(TR\)) of spectral categories (i.e.~catgeories enriched over symmetric spectra). Their general localization theorem says (roughly) that if \(\mathcal{A}\to \mathcal{B}\to \mathcal{C}\) is a sequence of pretriangulated spectral categories such that the sequence of triangulated categories \(\pi_{0}\mathcal{A}\to \pi_{0}\mathcal{B}\to \pi_{0}\mathcal{C}\) is exact, then the sequence
\[
THH(\mathcal{A})\to THH(\mathcal{B})\to THH(\mathcal{C})
\]
is a homotopy cofiber sequence of spectra. This generalizes Keller's construction of a localization cofiber sequence for Hochschild and cyclic homology of \(dg\)-categories [\textit{B. Keller}, J.~Pure Appl.~Algebra 136, No. 1, 1--56 (1999; Zbl 0923.19004)]. In order to deduce their localization theorem for schemes from the general result, the authors need to reformulate the definition of the topological Hochschild and cyclic homologies of schemes. The authors verify that their new definition yields a theory equivalant to Geisser-Hesselholt's original definition.
Several important consequences of the localization theorem are deduced: a Mayer-Vietoris homotopy pushout square for open covers, a similar one for blow-ups along regularly embedded centers, and a projective bundle theorem. The Mayer-Vietoris homotopy pushout square was previously established by Geisser-Hesselholt; the other two consequences are new. topological Hochschild homology; topological cyclic homology; localization sequence; Mayer-Vietoris sequence; projective bundle theorem; blow-up formula Blumberg, A.; Mandell, M., \textit{localization theorems in topological Hochschild homology and topological cyclic homology}, Geom. Topol., 16, 1053-1120, (2012) \(K\)-theory and homology; cyclic homology and cohomology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Localization theorems in topological Hochschild homology and topological cyclic homology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 given a set of weights and degrees defining \(\mathbb{C}^\times\) actions on \(\mathbb{C}^n\) and \(\mathbb{C}^p\) with \(n\geq p\). Necessary and sufficient conditions are obtained for the existence of an equivariant map \(f:\mathbb{C}^n \to \mathbb{C}^p\) such that \(f^{-1}(0)\) has an isolated singularity at 0. These are somewhat complicated, but simplify if \(n-p=0\) or 1 or if \(p=1\). The former case gives conditions for (weighted) homogeneously generated ideals of finite codimension in the ring \({\mathcal O}_n\) of germs of holomorphic functions; these are generated submodules of finite codimension in free \({\mathcal O}_n\)-modules. For maps \(f\) as above, there are known formulas for the Poincaré series of the Jacobian algebra and the \(\mathcal K\)-cotangent space; we also have a corresponding formula for the quotient in the submodule case.
For the case of \(\mathcal A\)- (right-left-) equivalence of maps \(f\), the above results can be used to give an algorithm for the Poincaré series of the \(\mathcal A\)-cotangent space (of a finitely \(\mathcal A\)-determined germ) in terms of the weights and degrees. The method yields necessary conditions for existence of a finitely \(\mathcal A\)-determined germ which are, however, not sufficient.
To express the condition for \(\mathcal K\)-finite maps, write the source as \(\mathbb{C}^I\) with weights \(\{w_i\mid i \in I\}\) and the target as \(\mathbb{C}^J\) with degrees \(\{d_j\mid j \in J\}\). For \(A \subseteq I\) write \(\mathbb{N}(A)\) for the additive monoid generated by \(\{w_i\mid i \in A\}\) and \(J(A)= \{j \in J\mid d_j\in \mathbb{N}(A)\}\); write \(\#\) to denote the cardinality. The condition is that for all \(A \subseteq I\) such that \(\# A>\#J(A)\) and all non-empty \(B \subseteq J\setminus J(A)\),
\[
\# \{i \in I\setminus A\mid \exists j \in B \text{ with } d_j-w_i \in \mathbb{N}(A)\}\geq \# A+\# B-\#J(A)-1\text{''}.
\]
weighted homogeneous maps; complete intersection; Poincaré series of cotangent space; isolated singularity; Poincaré series Wall, C.T.C. : Weighted Homogeneous Complete Intersections (preprint). Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Complete intersections Weighted homogeneous complete intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a smooth projective surface and denote with \(S^{[n]}\) the Hilbert scheme that parameterizes length \(n\) subschemes of \(S\). There exists a natural map from \(S^{[n]}\) to the symmetric power \(S^{(n)}\), whose fibers over multiplicity \(n\) cycles define the \textit{punctual Hilbert scheme} \(P_n\). In other words, \(P_n\) parameterizes subschemes of length \(n\) supported at one point. The authors study the tangent space \(T_\xi\) to \(P_n\) at a scheme \(\xi\in P_n\). The dimension \(\dim(T_\xi) \) is bounded below by the corank of the normal map \(\alpha_{n,\xi}\) which sends \(H^0(S, T_{S|\xi})\) to Hom\((\mathcal I_\xi,\mathcal O_\xi)\). The authors prove that when the ideal \(I_\xi\) of \(\xi\) is a monomial ideal, then \(\dim(T_\xi) \) is indeed equal to the corank of \(\alpha_{n,\xi}\). They also show how, for schemes \(\xi\) defined by monomials, the corank of \(\alpha_{n,\xi}\) can be computed from the Young diagram associated to \(I_\xi\). Hilbert scheme Parametrization (Chow and Hilbert schemes) The tangent space of the punctual Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We extend the explorations in [the author, J. Pure Appl. Algebra 104, No. 2, 213--233 (1995; Zbl 0842.52009)] to include the fractional power series expansions of \(k\) equations in \(d\) variables, where \(d> k\). An analog of Newton's polygon construction which uses the Minkowski sum \(P\) of the Newton polytopes \(P_1,\dots, P_k\) of the \(k\) equations is given for computing such series expansions. If the Newton polytopes of these equations are the same, then the common domains of convergence for the solutions correspond to the vertices of a certain fiber polytope \(\Sigma(P)\).
In general, our results suggest the existence of a ``mixed fiber polytope'' of \(k\) polytopes. It is also indicated that there may be a relationship between these mixed fiber polytopes and a generalization of the discriminant, which we call the mixed discriminant. Newton polytopes; fiber polytope McDonald J (2002) Fractional power series solutions for systems of equations. Discrete Comput Geom 27:501--529 \(n\)-dimensional polytopes, Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) Fractional power series solutions for systems of 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 Let \(\xi = (E,p,B)\) denote a vector bundle of rank \(n\) on a complete metric space \(B\). Fix a Riemannian metric on \(\xi\). An automorphism of \(\xi\) is denoted by \((X,\chi)\) and \(X[b]\) denotes the restriction of \(X\) to the fibre over \(b \in B\). Any element \(h\) of \(\Hom ({\mathbb Z}, \Aut \xi)\) can be written as \((X^m, \chi ^m)\), \(m\in {\mathbb Z}\). Let \(k\) be an integer , \(1\leq k \leq n-1\). The element \(h\) is called exponentially dichotomous with index \(k\) at \(b\in B\) if there exists a \(k\)-dimensional subspace \(R_0\) of the fibre \(p^{-1}(b)\) such that for every complement \(R_1\) of \(R_0\) in the fibre there exist \(\alpha >0\), \(\beta > 0\) with
\[
|X^m u||X^{m'} u |^{-1} \geq \alpha |X^m v||X^{m'} v |^{-1} \exp (\beta (m-m'))
\]
for any \(u \in R_1, v \in R_0, u\neq 0, v\neq 0\) and for any integers \(m \geq m'\geq 0\). The element \(h\) is called uniformly strongly positive at \(b\) if the sequence of operators \(X[\chi^m b]\), \(m\in {\mathbb Z}\) is uniformly strongly positive.
Theorem. A homomorphism \(h \in \Hom({\mathbb Z}, \Aut \xi)\) is exponentially dichotomous with index \(k\in \{1,\cdots,n-1\}\) at a point \(b\in B\) if and only if the homomorphism \(h^{n-k} \in \Hom ({\mathbb Z},\Aut \bigwedge ^{n-k} \xi)\) is uniformly strongly positive at \(b\). vector bundle automorphism; exponential dichotomy; uniformly strongly positive Smooth dynamical systems: general theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Dichotomy, trichotomy of solutions to ordinary differential equations, Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) Positiveness and exponential dichotomy of a family of automorphisms of a vector bundle | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Nash submanifold \(M\) of \(R^n\), where \(R\) is a real closed field, is a semialgebraic subset of \(R^n\) which is also a \(C^\infty\) submanifold of \(R^n\). A Nash subset of \(M\) is the zero set of a \(C^\infty\) semialgebraic function on \(M\). A mapping \(f: X \to Y\), where \(X\) and \(Y\) are Nash subsets, is called a Nash mapping if \(f\) is a \(C^\infty\) semialgebraic mapping.
The main result proved in this paper is the following:
Let \(f : X \to Y\) be a Nash mapping of Nash subsets. Then there exist finite compositions of Nash blowing-ups \(\sigma_X : \tilde X \to X\) and \(\sigma_Y : \tilde Y \to Y\) such that the induced Nash mapping \(\tilde f : \tilde X \to \tilde Y\) with \(\sigma_Y \circ \tilde f = f \circ \sigma_X\) has constant dimensional fibers.
It is worth to note that this is a global result, in the sense that the function \(\tilde f\) is global, and that it works for any real closed field \(R\). It has been inspired by a local equidimensionality theorem for real analytic spaces proved by \textit{A. Parusinski} [Trans. Am. Math. Soc. 344, 583--595 (1994; Zbl 0819.32006)]. Nash sets; Nash mappings Nash functions and manifolds Nash equidimensionality theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\subseteq\mathbb{P}^N =\mathbb{P}^N_K\) be a subvariety of dimension \(n\) and \(\Lambda\subseteq\mathbb{P}^N\) be a generic linear subspace of dimension \(N-k-1\) with \(k\geq n\). Then the linear projection \(\pi_\Lambda: X\to\mathbb{P}^k\) is a finite map. Let \(R(\pi_\Lambda)\) be its ramification locus. In this paper we study the map from the Grassmannian \(G(N-k-1,N)\) of planes of dimension \(N-k-1\) in \(\mathbb{P}^N\) to the Hilbert moduli space given by \(\Lambda\mapsto R(\pi_\Lambda)\). We wish to compute in particular the dimension, say, \(\eta\) of the image of this map. The motivation of this question comes from the fact that these ramification cycles are closely related to the Stückrad-Vogel cycle. We show that \(\eta\) is just the transcendence degree of a certain part of this cycle. The main result is that, under some mild hypothesis, in case of a projection \(X\to\mathbb{P}^n\), i.e. \(k=n\), the map \(\Lambda\mapsto R(\pi_\Lambda)\) is generically finite and so \(\eta\) takes its maximal possible value. Moreover, we show that in the case of smooth surfaces \(X\subseteq \mathbb{P}^4\) and generic projections onto \(\mathbb{P}^3\) this map is again generically finite if the normal bundle of \(X\) in \(\mathbb{P}^4\) is sufficiently positive. generic projections; Stückrad-Vogel intersection cycle; ramification locus; Grassmannian; Hilbert moduli space H. Flenner, M. Manaresi, Variation of ramification loci of generic projections, Math. Nachr. 194, 79--92, (1998).DOI: 10.1002/mana.19981940107 Ramification problems in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Variation of ramification loci of generic projections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper concerns the asymptotic behavior of the regularity of symbolic powers of ideals of points in a weighted projective plane. The regularity of such powers behaves asymptotically like a linear function, which is deeply related to the Seshadri constant of a blowup. The authors, in this paper, study the difference between regularity of such powers and this linear function. Under some conditions they prove that this difference is bounded or eventually periodic.
More in detail, suppose that \(H\) is an ample \({\mathbb{Q}}-\)Cartier divisor on a normal projective variety \(V\) and \(\mathcal{I}\) is an ideal sheaf on \(V\). Let \(\nu:W \to V\) be the blowup of \(\mathcal{I}\). Let \(E\) be the effective Cartier divisor on W defined by \(\mathcal{O}_W(-E)=\mathcal{I}\mathcal{O}_W\). The \(s-\)invariant, \(s_{\mathcal{O}_V(H)}(\mathcal{I})\), is defined by
\[
s_{\mathcal{O}_V(H)}(\mathcal{I})=\inf\{s \in \mathbb{R} | \nu^*(sH)-E \text{ is an ample } \mathbb{R}-\text{divisor on } W \}.
\]
The reciprocal, \(1/s_{\mathcal{O}_V(H)}(\mathcal{I})\), is the Seshadri constant of \(\mathcal{I}\). Several examples show that \(s_{\mathcal{O}_V(H)}(\mathcal{I})\) can be irrational, even when \(\mathcal{O}_V(H) \cong \mathcal{O}_{\mathbb{P}^1}(1)\) on ordinary projective space.
Let \(S=K[ x_o, \dots, x_n]\) be a polynomial ring, over a field \(K\), graded by the weighting wt\((x_i)=a_i\), \(1=0, \dots, n\). Assume that gcd\((a_0, a_2, \dots, a_{i-1}, a_{i+1}, \dots a_n)=1\) for \(0 \leq i \leq n\) and consider the weighted projective space \(\mathbb{P}=\mathrm{proj}(S)\). Suppose \(I\) is an homogeneous ideal of \(S\) and let \(\mathcal{I}\) be the sheaf associated to \(I\). In the first part of the paper the authors develop the basic properties of regularity on weighted projective space and show that the theory of asymptotic regularity on ordinary projective space extends naturally to weighted projective space. In particular, they show that reg\((I^m)\) is a linear function for \(m\gg 0\) and establish the following
Theorem 1. We have
\[
\lim_{m \to \infty} \frac{\text{reg}((I^m)^{\text{sat}})}{m}=s_{{\mathcal{O}}_{\mathbb{P}}(1) }(\mathcal{I}).
\]
\vskip0.4cm Define a function \(\sigma_I: \mathbb{N} \to \mathbb{Z}\) by
\[
\text{reg}((I^m)^{\text{sat}}= \lfloor ms_{{\mathcal{O}}_{\mathbb{P}}(1) }(\mathcal{I})\rfloor + \sigma_I(m).
\]
A function \(\sigma: \mathbb{N} \to \mathbb{Z}\) is \textit{eventually periodic} if \(\sigma(m)\) is periodic for \(m\gg 0\).
The authors fix their attention in the case of an ideal \(I \subset S=K[ x,y,z ]\) of a set of nonsingular points with multiplicity in a weighted two-dimensional projective space \(\mathbb{P}=\mathbb{P}(a,b,c)\) with wt\((x)=a\), wt\((y)=b\), wt\((z)=c\). Suppose that the points \(P_1, \dots P_r\) are distinct with multiplicities \(e_1, \dots, e_r\) in such a way their ideal is given as \(I=\cap I_{P_I}^{e_i}\). Define \(s(I)\) as
\[
s(I)=s_{{\mathcal{O}}_{\mathbb{P}}(1)}(\mathcal{I})
\]
The authors prove the following results.
Theorem 2. Let \(u=\sum_{i=1}^r e_i^2\). We have \(s(I)\geq \sqrt{abcu}\). If \(s(I)> \sqrt{abcu}\), then \(s(I)\) is a rational number.
Theorem 3. There exists a bounded function \(\sigma_I: \mathbb{N} \to \mathbb{Z}\) such that
\[
\text{reg}((I^{(m}))=\lfloor s(I)m\rfloor + \sigma_I(m).
\]
for all \(m \in \mathbb{N}\).
Theorem 4. Suppose that \(s(I)> \sqrt{abcu}\), and suppose that \(K\) has characteristic zero or is an algebraic closure of a finite field. Then the function \(\sigma_I(m)\) of the previous Theorem is eventually periodic. \vskip0.5cm Finally, the authors show, as a corollary, that if there exists a negative curve, then the regularity of symbolic powers of a monomial space curve is eventually a periodic linear function. Moreover they give a criterion for the validity of Nagata's conjecture in terms of the lack of existence of negative curves. symbolic powers; weighted projective spaces Cutkosky, SD; Kurano, K., Asymptotic regularity of powers of ideals of points in a weighted projective plane, Kyoto J. Math., 51, 25-45, (2011) Divisors, linear systems, invertible sheaves, General commutative ring theory, Computational aspects of algebraic surfaces, Global theory and resolution of singularities (algebro-geometric aspects) Asymptotic regularity of powers of ideals of points in a weighted projective 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 \textit{F. Chen, J. Zheng} and \textit{T. W. Sederberg} introduced the notion of a \(\mu \)-basis for a rational ruled surface [Comput. Aided Geom. Des. 18, 61--72 (2001; Zbl 0972.68158)] and showed that its resultant is the implicit equation of the surface, if the parametrization is generically injective. We generalize this result to the case of an arbitrary parametrization of a rational ruled surface. We also give a new proof for the corresponding theorem in the curve case and treat the reparametrization problem for curves and ruled surfaces. In particular, we propose a partial solution to the problem of computing a proper reparametrization for a rational ruled surface. implicitization; \(\mu \)-basis; ruled surface; reparametrization Dohm, M, Implicitization of rational ruled surfaces with \(\mu\)-bases, Journal of Symbolic Computation, 44, 479-489, (2009) Computational aspects of algebraic surfaces, Rational and ruled surfaces Implicitization of rational ruled surfaces with \(\mu \)-bases | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review studies the irregularity of the hypergeometric \(\mathcal{D}\)-modules \(\mathcal{M}_A(\beta)\) introduced by I. M. Gel'fand, M. I. Graev, M. M. Kapranov and A. V. Zelevinsky using explicit construction of Gevrey series along coordinate subspaces of \(\mathbb{C}^n\). Here \(A\) denotes a full rank \(d\times n\)-matrix with integer entries and \(\beta=(\beta_1,\dots,\beta_d)\) are complex parameters. The module \(\mathcal{M}_A(\beta)\) is the quotient \(\mathcal{D}/H_A(\beta)\) where \(H_A(\beta)\) is made of toric operators and Euler operators.
After reviewing facts on Gevrey series and slopes of \(\mathcal{D}\)-modules, the author explicitly constructs Gevrey series solutions of \(\mathcal{M}_A(\beta)\) along coordinate subspaces. Using those series, ideas from \textit{M. Schulze} and \textit{U. Walther} [Duke Math. J. 142, No. 3, 465--509 (2008; Zbl 1144.13012)] and the Comparison Theorem of the slopes of \textit{Y. Laurent} and \textit{Z. Mebkhout} [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 1, 39--69 (1999; Zbl 0944.14007)], she then gives in Theorem 5.9 a combinatorial description of the set of slopes of \(\mathcal{M}_A(\beta)\) along coordinate hyperplanes. This generalizes the result of Schulze and Walther (in the case of coordinate hyperplanes) because there are no further assumptions on \(A\) (for instance, pointedness).
The author also gives lower bounds for the dimensions of the Gevrey solution spaces, which are the actual dimensions for very generic parameters. Under some conditions on \(A\) and \(\beta\), she finally obtains a basis of the stalk at a generic point of the irregularity sheaves of \(\mathcal{M}_A(\beta)\) along a coordinate hyperplane. hypergeometric \(\mathcal D\)-modules; Gevrey series; slopes María-Cruz Fernández-Fernández, Irregular hypergeometric \?-modules, Adv. Math. 224 (2010), no. 5, 1735 -- 1764. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, Commutative rings of differential operators and their modules, Hypergeometric functions, Toric varieties, Newton polyhedra, Okounkov bodies Irregular hypergeometric \(\mathcal D\)-modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors of this paper are motivated by cohomological rigidity of simplicial polytopes. A simplicial polytope \(P\) is cohomologically rigid if there exists a quasitoric manifold \(M\) over \(P\) (i.e., \(M\) is a real \(2d\)-dimensional closed smooth manifold with a locally standard \(d\)-torus action whose orbit space is \(P\)) and if, whenever there exists a quasitoric manifold \(N\) over another polytope \(Q\) with a graded ring isomorphism \(H^{*}(M) \cong H^{*}(N)\), then \(P=Q\). Quasitoric manifolds were introduced by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No.~2, 417--451 (1991; Zbl 0733.52006)]. The authors define a simplicial polytope \(P\) to be combinatorially rigid if \(P=P'\) whenever \(P'\) is a simplicial polytope with the same \((i,j)\)th graded Betti numbers. By Poincaré duality, a \(3\)-dimensional simplicial polytope is combinatorially rigid exactly when it is determined by its \((i-1,i)\)th graded Betti numbers (which the authors call the special graded Betti numbers). By previous results [\textit{S. Choi} and \textit{T. Panov}, J. Lond. Math. Soc. Soc., II. Ser. 82, No.~2, 343--360 (2010; Zbl 1229.52008)], a simplicial polytope that supports a quasitoric manifold and is combinatorially rigid is also cohomologically rigid (all \(3\)-dimensional simplicial polytopes support quasitoric manifolds). To this effect, the authors prove several results about combinatorially rigid reducible \(3\)-dimensional simplicial polytopes (a \(3\)-dimensional simplicial polytope is called reducible if it is formed by attaching two polytopes along facets; the sum of two polytopes \(P_{1}\) and \(P_{2}\) is denoted \(P_{1} \# P_{2}\)). The first result is that a reducible and combinatorially rigid \(3\)-dimensional simplicial polytope is either a sum of three copies of the tetrahedron or a sum of the form \(P_{1} \# P_{2}\), where \(P_{1}\) and \(P_{2}\) are each one of a finite number of simplicial polytopes. The second result provides a list of sums of \(3\)-dimensional simplicial polytopes that are combinatorially rigid. 3-dimensional simplicial polytopes; combinatorial rigidity; graded Betti number S. Choi and J. S. Kim, ''Combinatorial Rigidity of 3-Dimensional Simplicial Polytopes,'' Int. Math. Res. Not. 2011(8), 1935--1951 (2011). Three-dimensional polytopes, Rigidity and flexibility of structures (aspects of discrete geometry), Special varieties, Combinatorial aspects of simplicial complexes Combinatorial rigidity of 3-dimensional simplicial polytopes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with the asymptotic behavior of the number of fixed points of the iteration of an endomorphism on a complex torus of dimension 2. If \(f\) is an endomorphism that fixes the origin on a complex torus \(X=\mathbb{C}^g/\Lambda\), then by the Holomorphic Lefschetz Fixed-Point Formula, the number of fixed points of \(f\) is precisely the product
\[
\#\text{Fix}(f)=\prod_{i=1}^g|1-\lambda_i|^2,
\]
where \(\lambda_1,\ldots,\lambda_g\) are the eigenvalues of the analytic representation of \(f\). The question of how the function \(n\mapsto\#\text{Fix}(f^n)=:F(n)\) behaves therefore boils down to understanding eigenvalues of (analytic representations of) endomorphisms of complex tori.
The fundamental result in this paper that allows for a complete classification of the behavior of \(F(n)\) is that for \(g=2\), if \(\lambda\) is an eigenvalue of an endomorphism of \(X\) and \(|\lambda|=1\), then \(\lambda\) must be a root of unity. The authors then proceed to show that \(F(n)\) can exhibit one of the following three types of behavior: it is either exponential, periodic, or a product of the previous two behaviors.
Examples are given that show that each of the three cases can already be seen in the projective case, but the third type of behavior does not appear for simple abelian surfaces. This last statement is obtained through the study of fixed points of endomorphisms on simple abelian surfaces using Albert's classification of finite dimensional division algebras with an anti-involution. abelian variety; endomorphism; fixed point; complex torus Bauer, T.; Herrig, T., Fixed-points of endomorphisms on two-dimensional complex tori, J. Algebra, 458, 352-363, (2016) Varieties and morphisms, Complex multiplication and abelian varieties, Automorphisms of surfaces and higher-dimensional varieties Fixed points of endomorphisms on two-dimensional complex tori | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow ``moderately discontinuous'' chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group \({M D H}_{\bullet}^b\) for any \(b \in [1, +\infty]\) and homomorphisms \({M D H}_{\bullet}^b \rightarrow {M D H}_{\bullet}^{b^\prime}\)for any \(b \geq b^\prime\). Here \(b\) is a ``discontinuity rate''. The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms \({M D H}_{\bullet}^b \rightarrow {M D H}_{\bullet}^{b^\prime}\) are essential. For \(b=1\) Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for \(b=+\infty\) the \({M D}\)-Homology recovers the homology of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi-Lipschitz subanalytic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer-Vietoris long exact sequences. Moreover, fixed a discontinuity rate \(b\) we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are \(b\)-moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, and recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs. Lipschitz properties; metric homology theory; metric singular subanalytic germs Moderately discontinuous homology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 purpose of this paper is to give a new proof of the Monodromy Theorem [see, e.g., \textit{A. Grothendieck}, Groupes de monodromie en géométrie algébrique, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006)], in a way which provides considerable new insights and leads to various generalizations of known results on algebraically finite (AF) surface transformations.
First, an AF-notion is introduced for a diffeomorphism and an isotopy class of diffeomorphisms on a compact smooth manifold. It is shown that such a diffeomorphism f induces a quasiunipotent action \(f^*\) on cohomology, which is a generalization of Nielsen's theorem on AF surface transformations. The polynomial growth of forms under f is used to bound the size of the Jordan blocks of \(f^*\), thus leading to a generalization of known estimates for the Jordan blocks of the monodromy action and a new result for AF surface transformations. Further extensions are given to the case of cohomology with twisted coefficients. The monodromy theorem is proved using essentially an approximation of an AF-system by a Morse-Smale one, combined with Shub's result that the cohomology action of a Morse-Smale diffeomorphism is quasiunipotent.
As a by-product of the present approach, it follows that Nielsen's theorem has no converse. It is also shown that the sequence of Lefschetz numbers of an AF diffeomorphism has a simple geometric description. Finally, the author computes Reidemeister torsion from the closed orbits of an AF and Morse-Smale flow. algebraically finite surface transformations; Monodromy Theorem; Nielsen's theorem; Morse-Smale flow [Fr]D. Fried, Monodromy and dynamical systems, Topology 25 (1986), 443--453. Morse-Smale systems, Étale and other Grothendieck topologies and (co)homologies Monodromy and dynamical systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal P=\{p_1, \dots, p_d\}\) be a finite set of integer primes, \(\mathbb Z_{(\mathcal P)}\) the ring of rationals with denominators prime to the \(p_i\) and \(X\) an affine \(\mathbb Z_{(\mathcal P)}\) scheme. The authors define arithmetic jet spaces \(\mathcal J^r_{\mathcal P}(X)\) for orders \(r=(r_1, \dots r_d)\), and the ring of global functions, \(A\), on it. The jet spaces are defined relative to some operators \(\delta_i\), analogous to derivations. Let \(\hat{A}_i\) denote the \(p_i\)-adic completion of \(A\).
Let \(P \in X\) be a \(\mathbb Z_{(\mathcal P)}\) point. The authors describe what it means for a tuple \((f_i) \in \Pi \hat{A}_i\) to be analytically continued along \(P\); such families are called \(\delta_{\mathcal P}\) functions, and are viewed as arithmetic (non--linear) PDE's on \(X\). For the case \(X=G\), a group scheme, with \(P\) its identity, there are additive \(\delta_{\mathcal P}\) functions, or \(\delta_{\mathcal P}\) characters. The authors determine all of these for \(G=\mathbb G_a\) and \(G= \mathbb G_m\), as well as when \(G\) is an elliptic curve. This latter requires extending the theory to non--affine \(X\).
The operators \(\delta_i\) are described as follows: for any integer prime \(p\), let \(C_p(X,Y)=p^{-1}(X^p+Y^p-(X+Y)^p)\). If \(R\) is a ring and \(S\) an \(R\) algebra then a map \(\delta: R \to S\) is a \(p\) derivation if \(\delta(a+b)=\delta a+ \delta b + C_p(a,b)\) and \(\delta(ab)=a^p\delta b+b^p\delta a+ p\delta a \delta b\). For \(R=S=\mathbb Z\), the unique \(p\) derivation is \(\delta_p(a)=p^{-1}(a-a^p)\).
If \(p\) and \(q\) are distinct integer primes let \(C_{p,q}=p^{-1}C_q(X^p,pY)-q^{-1}C_p(X^q,qZ)-q^{-1}\delta_pqZ^p+p^{-1}\delta_qpY^q\)
A \(\delta_{\mathcal P}\) ring \(R\) has \(p_i\) derivations \(\delta_i\) such that for each pair \(p=p_i\) and \(q=q_j\), \(\delta_p\delta_qa-\delta_q\delta_pa=C_{p,q}(a,\delta_pa,\delta_qa)\). Prolongation systems of \(\delta_{\mathcal P}\) rings are inductive systems indexed by \(d\) tuples of non--negative integers where the \(p_i\) derivations go to higher terms in the sequence, subject to compatibility conditions. There are universal such for a given ring \(R\), where the \(r=(r_1, \dots , r_n)\) term is denoted \(R^r\). When \(X=\mathrm{Spec}(R)\), the \(r\) jet space is \(\mathrm{Spec}(R^r)\). jet spaces; arithmetic partial differential equation; group characters Buium A., Simanca S.R.: Arithmetic Laplacians. Adv. Math. 220(1), 246--277 (2009) Group schemes Arithmetic Laplacians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X\) and \(Y\) are defined over some base scheme \(S\) which is the spectrum of either the integers or some field \(k\). Let \(\mathcal{P}(X,Y)\) denote the full subcategory of the category of coherent sheaves on \(X \times_{S} Y\) consisting of those sheaves \(\mathcal{F}\) whose support maps finitely to \(X\) and such that the pushforward of \(\mathcal{F}\) to \(X\) is locally free. This is an exact category, and \(K(X,Y)\) denotes the associated spectrum.
The author defines a tower of maps of spectra
\[
\dots \rightarrow W^{2}(X) \rightarrow W^{1}(X) \rightarrow W^0(X) \simeq K(X)
\]
in which \(W^{t}(X)\) is the homotopy colimit of a simplicial hypercube \(K(X \times \Delta^{\bullet},{\mathbb{P}}^{\wedge t})\), where \({\mathbb{P}}^{\wedge t}\) refers to a hypercube of schemes made up of copies of the projective line. The author uses Grayson's general construction of Adams operations for categories with exterior powers to construct Adams operations
\[
\Psi^{k}: W^{t}(X) \rightarrow W^{t}(X)
\]
which are compatible with the transition maps in the tower above. There are two main results:
1)\enskip These operations \(\Psi^{k}\) are compatible with the transition map
\[
K(X \times \Delta^{\bullet},{\mathbb{G}}_{m}^{\wedge t}) \rightarrow \Omega^{t}K(X \times \Delta^{\bullet},{\mathbb{P}}_{m}^{\wedge t})
\]
from the Grayson filtration.
2) The filtration of \(K_{q}(X)_{\mathbb{Q}}\) which is induced by the \(W^{t}(X)\) is the gamma filtration when \(X\) is a smooth variety over a field \(k\) which admits resolution of singularities.
The proof of the second assertion involves showing that certain complexes of rationalized \(K\)-groups in a fixed weight are acyclic, and these complexes can be described as complexes of rationalized higher Chow groups. Grayson filtration; Adams operations; filtrations; motivic cohomology Mark E. Walker, Adams operations for bivariant \?-theory and a filtration using projective lines, \?-Theory 21 (2000), no. 2, 101 -- 140. Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Motivic cohomology; motivic homotopy theory Adams operations for bivariant \(K\)-theory and a filtration using projective 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 short note, the authors consider
\[
{\mathcal G}=\{A \, \circ \, g \, \circ \, A^{-1} : A(z)=az+b, a, b \in \overline{\mathbb{Q}}, a\neq 0, g \in \mathrm{XDBP} \}
\]
where XDBP is the set of extra-clean dynamical Belyi polynomials [the second author, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 5, 671--693 (2000; Zbl 1066.14503)].
The authors prove that given any continuum \(K\subset \mathbb{C}\) and any \(\epsilon >0\), there exists a polynomial \(g\in \mathcal{G}\) with \(d(J(g), K)<\epsilon\), where the Julia set satisfies \(J(g)=\partial K(g)\).
This result is a generalized form of the result given by \textit{K. A. Lindsey} on Jordan curves [Ergodic Theory Dyn. Syst. 35, No. 6, 1913--1924 (2015; Zbl 1343.37028)] and it is obtained using an approximation result of the first author in [Invent. Math. 197, No. 2, 433--452 (2014; Zbl 1304.30028)].
Finally, some technical results are derived to construct the proof of the main result. dessin; Julia set; Belyi polynomial Bishop, C. J.; Pilgrim, K. M., Dynamical dessins are dense, Rev. Mat. Iberoam., 31, 3, 1033-1040, (2015) Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Small divisors, rotation domains and linearization in holomorphic dynamics, Dessins d'enfants theory, Continua theory in dynamics Dynamical dessins are dense | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 orbit space \(X_n = G_{n,2}/T^n\) of the standard action of the compact torus \(T^n\) on the complex Grassmann manifold \(G_{n,2}\). We describe the structure of the set of critical points \(\operatorname{Crit}G_{n,2}\) of the generalized moment map \(\mu_n \colon\, G_{n,2}\to \mathbb{R}^n\) whose image is a hypersimplex \(\Delta_{n,2} \). The canonical projection \(G_{n,2}\to X_n\) maps the set \(\operatorname{Crit} G_{n,2}\) to the set \(\operatorname{Crit}X_n\), which by definition consists of the orbits \(x\in X_n\) with nontrivial stabilizer subgroup in \(T^{n-1}=T^n/S^1\), where \(S^1\subset T^n\) is the diagonal one-dimensional torus. Introducing the notion of a singular point \(x\in\operatorname{Sing}X_n \subset X_n\) in terms of the parameter spaces of the orbits, we prove that the set \(Y_n = X_n\setminus\operatorname{Sing}X_n\) is an open manifold and is dense in \(X_n\). We show that \(\operatorname{Crit}X_n \subset\operatorname{Sing}X_n\) for \(n>4\), but \(\operatorname{Sing}X_4\subset\operatorname{Crit}X_4\). Our central result is the construction of a projection \(p_n \colon\, U_n= \mathcal{F}_n\times\Delta_{n,2}\to X_n, \dim U_n = \dim X_n\), where \(\mathcal{F}_n\) is a universal parameter space. Earlier, we have proved that \(\mathcal{F}_n\) is a closed smooth manifold diffeomorphic to a known manifold \(\,\overline{\!\mathcal{M}}(0,n)\). We show that the map \(p_n \colon\, Z_n = p_n^{-1}(Y_n)\to Y_n\) is a diffeomorphism, and describe the structure of the sets \(p_n^{-1}(x)\) for \(x\in\operatorname{Sing}X_n\). Grassmann manifold; torus action; chamber decomposition of hypersimplex; orbit space; universal parameter space Grassmannians, Schubert varieties, flag manifolds Resolution of singularities of the orbit spaces \(G_{n,2}/T^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author presents a new approach to results of \textit{K. Kurdyka} and \textit{L. Paunescu} [Duke Math. J. 141, No. 1, 123--149 (2008; Zbl 1140.15006)] and the reviewer [Trans. Am. Math. Soc. 363, No. 9, 4945--4977 (2011; Zbl 1241.26024); ibid. 365, No. 10, 5545--5577 (2013; Zbl 1387.47009)] on the spectral decomposition of real analytic multiparameter families of normal finite-dimensional operators. These results generalize classical theorems of Rellich and Kato.
It is proved that, for any real analytic normal operator \(L : F \to F\) acting on a real analytic vector bundle of finite rank over a connected real analytic manifold \(N\), there exists a locally finite composition \(\sigma : \tilde N \to N\) of (geometrically admissible) blowings-up such that locally over \(\tilde N\) the bundle \(\sigma^* F\) decomposes into a direct orthogonal sum of real analytic vector sub-bundles \(R_j\) and either the restriction of \(\sigma^* L\) to \(R_j\) is a multiple of the identity or \(R_j\) is real of rank \(2\) and the restriction of \(\sigma^* L\) to \(R_j\) is fiberwise a similitude. The latter may occur for real non-symmetric operators. As a corollary the eigenvalues of \(\sigma^*L\) can locally be chosen in a real analytic manner.
The main difference to the previous approaches is that the author only works with the eigenspaces and obtains the regularity of the eigenvalues as a by-product. He proves that the eigenspaces form real analytic vector bundles on the complement of the discriminant locus \(D_L\) of the reduced characteristic polynomial of \(L\). Then he constructs an ideal \(\mathcal F_L\) with the property that, if \(\mathcal F_L\) is principal, then the eigenspaces extend to \(D_L\). Principalization of \(\mathcal F_L\) yields the resolution map \(\sigma\). real analytic families of normal operators; eigenvalues and eigenvectors Semi-analytic sets, subanalytic sets, and generalizations, Eigenvalues, singular values, and eigenvectors, Real-analytic and semi-analytic sets Re-parameterizing and reducing families of normal operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a simple complete multipolytope \(\mathcal {P}\) in \(\mathbb{R}^{n}\), Hattori and Masuda defined a locally constant function \(\mathrm{DH}_{\mathcal{P}}\) on \(\mathbb{R}^{n}\) minus the union of hyperplanes associated with \(\mathcal {P}\), which agrees with the density function of an equivariant complex line bundle over a Duistermaat-Heckman measure when \(\mathcal{P}\) arises from a moment map of a torus manifold. We improve the definition of \(\mathrm{DH}_{\mathcal{P}}\) and construct a convex chain \(\overline{\mathrm{DH}}_{\mathcal{P}}\) on \(\mathbb{R}^{n}\). The well-definiteness of this convex chain is equivalent to the semicompleteness of the multipolytope \(\mathcal{P}\). Generalizations of the Pukhlikov-Khovanskii formula and an Ehrhart polynomial for a simple lattice multipolytope are given as corollaries. The constructed correspondence simple \{semicomplete multipolytopes\} \(\to\) \{convex chains\} is surjective but not injective. We will study its ``kernel''. Momentum maps; symplectic reduction, Three-dimensional polytopes, Toric varieties, Newton polyhedra, Okounkov bodies, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Multipolytopes and convex chains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author shows how the multigraded Hilbert scheme construction of \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] can be used to construct a quasi-projective scheme which parametrize left homogeneous ideals in the Weyl algebra having fixed Hilbert function. Fixing an integral domain \(k\) of characteristic zero, the Weyl algebra \(W=k \langle x_1, \dots, x_n, \partial_1, \dots, \partial_n \rangle\) has a \(k\)-basis consisting of the set \(\mathcal B = \{ x^\alpha \partial^\beta | \alpha, \beta \in \mathbb N^n \}\). If \(A\) is an abelian group, then any \(A\)-grading \(\mathbb N^n \to A\) on the polynomial ring \(S = k[x_1,\dots,x_n]\) extends to an \(A\)-grading \(\mathbb N^{2n} \to A\) on \(W\) by \(\deg (x^\alpha \partial^\beta) = \deg \alpha - \deg \beta\), which induces a decomposition \(W = \bigoplus_{a \in A} W_a\). Given a Hilbert function \(h:A \to \mathbb N\), the corresponding Hilbert functor \(H^h_W\) takes a \(k\)-algebra \(R\) to the set of homogeneous ideals \(I \subset R \otimes_k W\) such that \((R \otimes _k W_a)/I_a\) is a locally free \(R\)-module of rank \(h(a)\) for each \(a \in A\). The main theorem says that \(H^h_W\) is representable by a quasi-projective scheme over \(k\).
The strategy of the proof is similar to that of Haiman and Sturmfels [loc. cit.], but there are some new behaviors regarding monomials in the Weyl algebra \(W\) not seen in the polynomial ring \(S\). An obvious difference is that a product of monomials in \(W\) need not be a monomial. Another difference is that \(W\) has infinite antichains of monomial ideals, unlike the polynomial case: see work of \textit{D. MacLagan} [Proc. Am. Math. Soc. 129, 1609--1615 (2001; Zbl 0984.13013)]. Moreover, the natural extension of Gröbner basis theory for \(S\) to \(W\) does not work well, so the author considers the initial ideal of a left ideal in the associated graded algebra \(\text{gr} W\) and uses Gröbner basis theory for \(W\) developed by \textit{M. Saito} et al. [Gröbner deformations of hypergeometric differential equations. Berlin: Springer (2000; Zbl 0946.13021)]. The arguments are well presented along with examples showing the novel points. Hilbert schemes; Weyl algebras Parametrization (Chow and Hilbert schemes), Rings of differential operators (associative algebraic aspects) Multigraded Hilbert schemes parametrizing ideals in the Weyl algebra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \((R,\mathfrak{m})\) be a Noetherian \(d\)-dimensional local ring such that
the nilradical \(N(\hat{R})\) of the \(\mathfrak{m}\)-adic completion \(\hat{R}\) of
\(R\) has dimension < \(d\). Let \(\mathcal{I}(1)=\{\mathcal{I}(1)_{i}
\},\ldots,\mathcal{I}(r)=\{\mathcal{I}(r)_{i}\}\) be a system of \(r\)
filtrations on \(R\) (not necessarily Noetherian i.e. \(\oplus_{n\geq
0}\mathcal{I}(j)_{n}\) may not be finitely generated \(R\)-algebras), where
\(\mathcal{I}(j)_{i}\) are \(\mathfrak{m}\)-primary ideals. The authors prove that
for a finitely generated \(R\)-module \(M\) the function
\[
P(n_{1},\ldots,n_{r}):=\lim_{m\rightarrow\infty}\frac{\lambda(M/\mathcal{I}
(1)_{mn_{1}}\cdots\mathcal{I}(r)_{mn_{r}}M)}{m^{d}}
\]
is defined and is a homogeneous polynomial of total degree \(d\) with real
coefficients. Appropriately normalized coefficients of \(P\) are, by definition,
mixed multiplicities of \(M\) with respect to the filtrations \(\mathcal{I}
(1),\ldots,\mathcal{I}(r).\) They show many of the classical properties of
mixed multiplicities known for \(\mathfrak{m}\)-primary ideals \(I_{1}
,\ldots,I_{r}\) (i.e. for filtrations \(\mathcal{I}(j)_{i}:=I_{j}^{i}\)), in
particular the Minkowski inequalities. local ring; multiplicity; filtration;mixed multiplicity; Minkowski inequality Multiplicity theory and related topics, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Mixed multiplicities of filtrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the group \(G\) of invertible \(n\times n\) matrices over a field \(F\) act on the space \(M\) of all \(n\times n\) matrices over \(F\) by conjugation. A map \({\mathbb{N}} \rightarrow {\mathbb{N}}\) which is non-increasing and satisfies a certain convexity condition is called a \textit{rank function.} The relative codimension of a subvariety of \(M\) is defined in terms of a given rank function. Earlier [\textit{M. Skrzyński}, ibid. 40, 167-174 (2000; see the preceding review Zbl 1020.14013)] the author gave a characterization of \(G\)-invariant subcones of \(M\) of relative codimension one in case \(F\) is algebraically closed and of characteristic zero. In the paper under review he improves this result by completely characterizing subcones of \(M\) of relative codimension one over any field \(F\). He provides necessary and sufficient conditions for a \(G\)-invariant irreducible subcone to be the closure of the cone over a single conjugacy class. linear algebraic groups; group action; cones; invariant algebraic cones Group actions on varieties or schemes (quotients), Linear algebraic groups over arbitrary fields, Basic linear algebra On \(\mathcal{GL}_n(\mathbb{F})\)-invariant algebraic cones with extreme values of relative codimension | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{S. P. Novikov} constructed a spectral sequence to study de Rham cohomology of ``deformed differentials'' of the type \(d+t\omega\), where \(t\) is a complex parameter and \(\omega\) is a closed 1-form [Sov. Math., Dokl. 33, 551-555 (1986); translation from Dokl. Akad. Nauk SSSR 287, 1321-1324 (1986; Zbl 0642.58016)]. He showed that this spectral sequence converges (for almost all \(t\)) to cohomology with values in a certain coefficient system, which depends on \(\omega\).
In the present paper the author shows how this can be generalized to apply to the de Rham complex with values in an associative matrix algebra. deformed differentials; de Rham cohomology Alaniya, L. A.: On cohomologies with coefficients in a local system close to a trivial one. Russian math. Surveys 52, No. 2, 390-391 (1997) Homology with local coefficients, equivariant cohomology, de Rham cohomology and algebraic geometry On cohomologies with coefficients in a local system close to a trivial one | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the 21st century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto-Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: (1) to use a different relaxation order for each constraint, and (2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries. To the best of our knowledge, the Lasserre hierarchy was previously limited to small scale problems, while we solve a large scale industrial problem with thousands of variables and constraints to global optimality. multi-ordered Lasserre hierarchy; Hermitian sum-of-squares; chordal sparsity; semidefinite programming; optimal power flow C. Josz and D. K. Molzahn, \textit{Lasserre hierarchy for large scale polynomial optimization in real and complex variables}, SIAM J. Optim., 28 (2018), pp. 1017--1048, . Semidefinite programming, Large-scale problems in mathematical programming, Nonconvex programming, global optimization, Classical measure theory, Computational aspects in algebraic geometry, Applications of operator theory in optimization, convex analysis, mathematical programming, economics Lasserre hierarchy for large scale polynomial optimization in real and complex variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 101, No. 2, 411--424 (1990; Zbl 0723.32005)] and \textit{A. Parusinski} [Trans. Am. Math. Soc. 344, No. 2, 583--595 (1994; Zbl 0819.32006)] proved the following important rectilinearization result for continuous subanalytic functions:
Let \(U\) be a real analytic manifold and \(f:U\to\mathbb{R}\) a continuous subanalytic function. Then there is a locally finite covering \((\pi_j:U_j\to U)_j\) such that
(i) each \(\pi_j\) is a composite of finitely many mappings each of which is either a local blowing-up with smooth center or a local power substitution;
(ii) each \(f\circ \pi_j\) is analytic and identically \(0\) or a normal crossing or the inverse of a normal crossing.
In the paper under review, the author proves a rectilinearization theorem for functions definable in an o-minimal structure generated by a convergent Weierstrass system. Convergent Weierstrass systems were introduced in [\textit{L. Van den Dries}, J. Symb. Log. 53, No. 3, 796--808 (1988; Zbl 0698.03023)]. They are induced by certain subrings of the ring of all restricted real analytic functions satisfying similar properties, notably being closed under Weierstrass division. Taking all restricted analytic functions, one obtains the o-minimal structure \(\mathbb{R}_{\mathrm{an}}\) and the above result for globally subanalytic functions in a global form. The proofs rely on the fact that functions definable in an o-minimal structure generated by a convergent Weierstrass system are piecewise given by terms in a reasonable language. rectilinearization; subanalytic functions; Weierstrass systems DOI: 10.4064/ap99-2-2 Semi-analytic sets, subanalytic sets, and generalizations, Modifications; resolution of singularities (complex-analytic aspects), Real-analytic and semi-analytic sets, Quantifier elimination, model completeness, and related topics Rectilinearization of functions definable by a Weierstrass system and its applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Gamma \subset \mathbb{P}^c_K\) denote a nondegenerate finite subscheme of length \(d\). Related to \(\Gamma\) the authors investigate the following numerical invariants:\begin{itemize} \item[(1)] \(\operatorname{reg} \Gamma\), the Castelnuovo-Mumford regularity. \item[(2)] \(m(\Gamma)\) the maximal degree in a minimal generating set of \(I_{\Gamma}\), the saturated defining ideal of \(\Gamma\). \item[(3)] \(\ell(\Gamma)\) the largest integer \(\ell\) such that \(\Gamma\) admits a proper \(\ell\)-secant line. Then the following inequalities hold \( (\star) : \ell(\Gamma) \leq m(\Gamma) \leq \operatorname{reg}(\Gamma)\).\end{itemize} In their main result, the authors prove that equalities in \((\star)\) hold provided \((\#) : \operatorname{reg} \Gamma \geq \frac{d-c+5}{2}\). In case the last bound \((\#)\) is not fulfilled, then there is no strong relation between \(\operatorname{reg} \Gamma\) and \(\ell(\Gamma)\) as shown by examples. Moreover, if \((\#)\) is satisfied the authors study the Betti numbers of \(\Gamma\). Let \(\mathbb{L}\) denote a \(\operatorname{reg} \Gamma\)-secant line of \(\Gamma\) and let \(X\) denote the scheme-theoretic union \(\Gamma \cap \mathbb{L}\). It is shown that the Betti table of \(\Gamma\) is determined by those of \(X\) and \(\mathbb{L}\). This turns out because \(\operatorname{reg} X\) is strictly smaller than \(\operatorname{reg} \Gamma\). finite schemes; Castelnuovo-Mumford regularity; secant line Projective techniques in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Questions of classical algebraic geometry Regularity and multisecant lines of finite 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 \(f:\mathbb{R}^m \to\mathbb{R}^k\), \(m\geq k\geq 1\) be a polynomial mapping with singular locus \(\sum_f\). It is assumed that \(\sum_f\) is compact and contained in the fibre of 0 (condition 1). Let \(\sum_f\) be the zero set of \(\theta:\mathbb{R}^m \to\mathbb{R}_+\), a positive function, and \(B_\varepsilon= \{x\in \mathbb{R}^m \mid \theta(x)\leq \varepsilon\}\). Furthermore, let \(S_\eta^{k-1}=\{u\in \mathbb{R}^k\mid \sum u^2_i= \eta^2\}\) the \(k-1\)-sphere.
It is proved that \(f_{ |B_\varepsilon \cap f^{-1}(S_\eta^{k-1})}\) is a \(C^\infty\)-locally divided fibration onto \(S_\eta^{k-1}\) for sufficiently small \(\varepsilon\) and \(\eta\). Consider, furthermore, \(\pi_k:\mathbb{R}^k \to\mathbb{R}^{k-1}\) the projection, \(\pi_k (x_1, \dots,x_k)= (x_1,\dots, x_{k-1})\) and assume that \(\sum_f= \sum_{\pi_k \circ f}\) (condition 2). Then \((\pi_k\circ f_k)^{-1}(\widetilde c)\cap B_\varepsilon\) is homeomorphic to \((f^{-1}(c)\cap B_\varepsilon)\times[0,1]\) for \(\widetilde c\in S_\eta^{k-2}\), \(c\in S_\eta^{k-1}\) and \(\varepsilon,\eta\) sufficiently small.
Effective criteria are given to check whether condition 1 and 2 hold. Note that for isolated singularities the condition 1 and 2 for \(\sum_f\) are satisfied. locally trivial fibration; polynomial mapping A. Jacquemard, Thèse \(3\)ème cycle , Université de Dijon, France, 1982. Milnor fibration; relations with knot theory, Topology of real algebraic varieties On the fiber of the compound of a real analytic function by a projection | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors study the fine structure of the linearly presented perfect ideals of codimension two in three variables with particular emphasis on the Cohen-Macaulay, normal and fiber type properties of the associated Rees algebra and the special fiber. There is a vast literature on codimension two perfect ideals in the case the ideal satisfies the usual generic properties or the so-called ($G_d$) condition. One first attempt to get away from the ($G_d$) condition or others of similar nature is a recent result of \textit{N. P. H. Lan} [J. Pure Appl. Algebra 221, No. 9, 2180--2191 (2017; Zbl 1453.13018)]. In the present paper, the authors recover and extend his work. More precisely, the authors introduce the so-called \textit{chaos invariant} associated to the Hilbert-Burch matrix of $I$. In the case the chaos invariant is one (minimal possible), they recover Lan's result with additional contents. They also give applications of these results to three important models: linearly presented ideals of plane fat points, reciprocal ideals of hyperplane arrangements and linearly presented monomial ideals. \par In the paper the authors strongly focus on the behavior of the ideals of minors of the corresponding Hilbert-Burch matrix and on conjugation features of the latter. linear presentation; Cohen-Macaulay ideal; special fiber; Rees algebra; reduction number Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Linkage, complete intersections and determinantal ideals, Syzygies, resolutions, complexes and commutative rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Graded rings, Determinantal varieties Linearly presented perfect ideals of codimension 2 in three variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study linear systems on \((\mathbb{P}^{1})^{n}\). It is a classical question to ask about the dimension of a linear system of hypersurfaces in \(\mathbb{P}^{n}\) of a given degree passing through finitely many very general points with prescribed multiplicities. It is well-known that for \(n=2\) the Segre-Harbourne-Gimiliano-Hirschowitz conjecture predicts the dimension of such linear systems. It is natural to ask about possible generalizations of the SHGH conjecture for other values of \(n\) or another varieties.
Denote by \(\mathcal{L} = \mathcal{L}_{(d_{1}, \dots, d_{n})}(m_{1}, \dots, m_{r})\) the linear system of hypersurfaces in \((\mathbb{P}^{1})^{n}\) of degree \((d_{1}, \dots, d_{n})\) passing through \(r\) very general points \(q_{1}, \dots, q_{r}\) with multiplicities \(m_{1}, \dots, m_{r}\). \textit{The virtual dimension} of \(\mathcal{L}\) is
\[
\text{vdim}(\mathcal{L}) = \prod_{i=1}^{n}(d_{i}+1) - \sum_{i=1}^{r}{ n + m_{i}-1 \choose n} - 1.
\]
Then the expected dimension of \(\mathcal{L}\) is defined as \(\text{edim}(\mathcal{L}) = \max \{\text{vdim}(\mathcal{L}), -1 \}\). The inequality \(\text{dim}(\mathcal{L}) \geq \text{edim}(\mathcal{L})\) always holds. If \(\text{dim}(\mathcal{L}) > \text{edim}(\mathcal{L})\), then we say that \(\mathcal{L}\) is special.
For a given subset \(I \subset\{1, \dots, n\}\) we denote
\[
P_{I}: (\mathbb{P}^{1})^{n} \ni ([x_{1}:y_{1}], \dots, [x_{n}:y_{n}]) \mapsto ([x_{i}:y_{i}] \, : \, i \in I)\in (\mathbb{P}^{1})^{|I|}.
\]
Moreover, we denote by \(F_{j,I}\) the fiber of \(P_{I}\) through the point \(q_{j}\) for any \(j\). For a given vector \((d_{1}, \dots, d_{n}) \in \mathbb{Z}_{\geq 0}^{n}\) we denote by
\[
s_{I}:= \sum_{i \in I}d_{i} \text{ and } S_{I}:=1+ |I| +s_{I}
\]
with \(I \subset \{1, \dots,n\}\). By the assumption that points are very general one has \(F_{i,I}\cap F_{j,I} = \emptyset\) for \(i\neq j\). Then \textit{the fiber dimension} of \(\mathcal{L}\) is defined as
\[
\text{fdim}(\mathcal{L}) := \prod_{i=1}^{n}(d_{i}+1) - \sum_{1\leq j \leq r; \, I\subset \{1, \dots, n\}; \, S_{I} \leq m_{j}} (-1)^{|I|}{ m_{j}-S_{I}+n \choose n} -1.
\]
The fiber expected dimension is defined as \(\text{efdim}(\mathcal{L}) = \max \{\text{fdim}( \mathcal{L} ), -1\}\) and we say that \(\mathcal{L}\) is \textit{fiber special} if \(\dim(\mathcal{L}) > \text{efdim}(\mathcal{L})\). The first main result of the note is the following.
Theorem 1. For any linear system \(\mathcal{L}\) we have the following inequalities
\[
\dim(\mathcal{L}) \geq \text{efdim}(\mathcal{L}) \geq \text{edim}(\mathcal{L}).
\]
Another result gives us the following characterization of fiber non-special systems.
\textit{Theorem 2.} A linear system through two points in \((\mathbb{P}^{1})^{n}\) is fiber non-special.
Also if there are more than two points, then then there are examples of fiber special systems (see Example 5.2). linear systems; birational map; toric varieties Divisors, linear systems, invertible sheaves, Toric varieties, Newton polyhedra, Okounkov bodies Linear systems on the blow-up of \((\mathbb{P}^1)^n\) | 0 |
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