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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 In this paper the author considers the following situation generalizing results of Atiyah. Let $\pi : X \to S$ be an arbitrary morphism of schemes. We fix $I$ to be a quasi-coherent $\O_X$-module and set $d : \mathcal{O}_X \to I$ be an $\pi^{-1}(\mathcal{O}_S)$-linear derivation. Then for an arbitrary coherent $\mathcal{O}_X$-module $\mathcal{E}$ one can form $J_I^1(\mathcal{E} := I \otimes \mathcal{E} \oplus \mathcal{E}$. For $\mathcal{E} = \O_X$, $J_I^1(\mathcal{O}_X) = J_I^1$ can be given a ring structure $(x, a)(y,b) = (xb+ay,ab)$ and can easily make $J_I^1(\mathcal{E})$ into a left-$J_I^1$-module. The author then obtains a short exact sequence $0 \to I \otimes \mathcal{E} \to J^1_I(\mathcal{E}) \to \mathcal{E} \to 0$ (which the author calls the generalized Atiyah sequence). From this the author defines the Atiyah-class of $\mathcal{E}$ (with respect to $J_I^1$) as $c_J(\mathcal{E}) \in \text{Ext}_{J_I^1}(\mathcal{E}, I \otimes \mathcal{E})$ (likewise one can take the $\text{Ext}$ with respect to $\O_X$). The author then proves a number of results about this class (for instance that $c_J(\mathcal{E}) = 0$ if and only if $\mathcal{E}$ has an $(I, d_{\mathcal{J}})$-connection (see the discussion above 3.5 in the paper for precise definitions). In the case where $S = \mathbb{C}$, $I = \Omega_{X/S}$ and $\mathcal{E}$ is locally free, one obtains the classical Atiyah sequence see [\textit{M. F. Atiyah}, Trans. Am. Math. Soc. 85, 181--207 (1957; Zbl 0078.16002)] and corresponding extension class (with similar results). The paper concludes with a series of examples exhibiting this generalized theory. Atiyah sequence; jet bundle; characteristic class; square zero extension Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry On jets, extensions and characteristic classes. II	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For $n\geq 1$ and $\mu$ a Borel measure in $\mathbb{C}^n$, the weighted log canonical threshold of a holomorphic function $f$ defined in a neighborhood of the origin of $\mathbb{C}^n$ is defined as $c_{\mu}(f):=$sup$\{ c\geq 0$ : $\|f\|^{-2c}$ is $L^1(\mu)$ in a neighbourhood of $0~\}$. Set \par $\mathcal{C}(\mu):=\{ c_{\mu}(f)$ : $f$ is holomorphic in a neighbourhood of $0~\}$. \par The ACC conjecture for weight $\mu$ reads:\par $\mathcal{C}(\mu)$ satisfies the ascending chain condition every convergent increasing sequence in $\mathcal{C}(\mu)$ to be stationary.\par The authors show that the conjecture holds true for $n=2$ and $\mu =\\| z\\| ^{2t}dV_4$, $t\geq 0$. holomorphic function; log canonical threshold; ACC conjecture Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Lelong numbers ACC conjecture for weighted log canonical thresholds	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition, goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ``contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system $A_n$) introduced by \textit{T. K. Petersen} [Electron. J. Comb. 25, No. 4, Research Paper P4.64, 28 p. (2018; Zbl 1441.05243)]. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ``meta-matrix'' formed by the numbers of contingency matrices of various sizes, is totally positive. symmetric products; contingency matrices; stratifications; total positivity Exact enumeration problems, generating functions, General topology of complexes, Polyhedral manifolds, Contingency tables, Linear algebraic groups over the reals, the complexes, the quaternions, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Contingency tables with variable margins (with an appendix by Pavel Etingof)	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 $f_t : {\mathbb C}^n \to {\mathbb C}$, $t \in [0,1]$, be a family of polynomial maps, $f_t \in {\mathbb C}[t,x_1, \dots, x_n]$, $n \geq 2$, having only isolated singularities and $\mu(t)$ the (affine) Milnor number (the sum of the local Milnor numbers at critical points of $f_t$) and $\lambda(t)$ the Milnor number at infinity (the sum of the local Milnor numbers at critical points of $f_t$ at infinity). Suppose that $\mu(t)$, $\lambda (t)$, the number of critical points and the number of critical points at infinity for fixed $t$ do not depend on $t$. Moreover, suppose that the critical values at infinity depend continuously on $t$. Then the fibrations $f_0^{-1}({\mathbb C} \smallsetminus B\bigl(0)\bigr) \to {\mathbb C} \smallsetminus B(0)$ and $f_1^{-1}\bigl({\mathbb C} \smallsetminus B(1)\bigr) \to {\mathbb C} \smallsetminus B(1)$ are fibre homotopy equivalent and for $n \not= 3$ differentiably isomorphic. Here $B(0)$, resp.\ $B(1)$, is the set of critical points of $f_0$, resp.\ $f_1$. Under the additional assumption that $\lambda(t) = 0$ and $n \not= 3$, it is proved that $f_0$ and $f_1$ are topologically equivalent. In the case $n = 2$ the same holds without requiring $\lambda(t) = 0$ but $\deg(f_t)$ not depending on $t$. $\mu$-constant theorem; family of polynomials; singularities at infinity; Milnor number; Milnor number at infinity [3]A. Bodin, Invariance of Milnor numbers and topology of complex polynomials, Comment. Math. Helv. 78 (2003), 134--152. Equisingularity (topological and analytic), Singularities of curves, local rings Invariance of Milnor numbers and topology of complex polynomials	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A map $f : \mathbb{C}^n \to \mathbb{C}^n$ is called a square system of polynomial-exponential functions if it is polynomial in both the variables $x_1,\dots,x_n$ and finitely many exponentials of the form $e^{\beta x_i}$ where $\beta \in \mathbb{C}$. Systems of polynomial-exponential functions arise in many applications such as engineering, mathematical physics, control theory and so on. Smale's $\alpha$-theory states that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Given a system of polynomial-exponential equations, the authors consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. Furthermore, they discuss methods for generating numerical approximations to solutions of polynomial-exponential systems. Finally, based on these results, they describe an algorithm for certifying solutions to polynomial-exponential systems, and demonstrate their algorithm on several examples. certified solutions; alpha theory; polynomial system; polynomial-exponential systems; numerical algebraic geometry; alphacertified Hauenstein, J. D.; Levandovskyy, V.: Certifying solutions to square systems of polynomial-exponential equations. (Sept. 2011) Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation, Numerical computation of solutions to systems of equations Certifying solutions to square systems of polynomial-exponential 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 [For the entire collection see Zbl 0518.00005.] The paper under review deals with the following problem: given morphisms of affine schemes $s: \tilde X\to Y,$ $p: Y\to X$ with p finitely presented, find a morphism $f: Z\to Y$ such that s factors through f, pf is smooth and f is ''smooth except above the singular locus of p''. This problem is motivated by work on approximation property. The authors solve the problem in two cases: $\tilde X=X$ and in the isolated singularity case (i.e. $\tilde X,$ X are spectra of henselian local rings with the same completion and p is smooth at every point of s(X) except the closed point). Meanwhile a complete answer to the above problem was given by \textit{M. Cipu} and \textit{D. Popescu}, ''A desingularisation theorem of Néron type'' [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 30, 63-76 (1984)]; this paper in its turn is based on the papers: \textit{D. Popescu}, ''General Néron desingularisation'' [to appear in Nagoya Math. J. 100 (1985)] and \textit{D. Popescu}, ''General Néron desingularisation and approximation'' (to appear). factorization of smooth maps; morphisms of affine schemes; approximation property; isolated singularity 6. M. Artin and J. Denef , Smoothing of a ring homomorphism along a section, in Arithmetic and Geometry, Vol. 2 (Birkhäuser, Boston, 1983), pp. 5-32. genRefLink(16, 'S0219498817500724BIB006', '10.1007%252F978-1-4757-9286-7_2'); Local structure of morphisms in algebraic geometry: étale, flat, etc., Singularities in algebraic geometry, Local deformation theory, Artin approximation, etc. Smoothing of a ring homomorphism along a section	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 considers continuous actions of a non-compact real reductive Lie group $G$ on a Hausdorff topological space $\mathcal{M}$. The aim of the authors is to develop a geometric invariant theory for such actions, inspired by well-known results for actions on Kähler manifolds. More precisely, write $G=K\cdot \exp(\mathfrak{p})$ where $K$ is a maximal compact subgroup of $G$. In this general setting, in Section 2 the authors define Kempf-Ness functions as maps $\Psi: \mathcal{M} \times G\to \mathbb R$ satisfying certain conditions. Such a Kempf-Ness function gives rise to a map $\mathfrak{F}: \mathcal{M}\to \mathfrak{p}^*$, called gradient map. In Section 5, given a point $x\in \mathcal{M}$, the authors define when $x$ is polystable (resp. stable, semi-stable, unstable) in terms of the intersection of the $G$-orbit of $x$ with $\mathfrak{F}^{-1}(0)$. A feature is that these various notions of stability can be detected by a numerical criteria (see Sections 5 and 6), using the maximal weight function introduced in Section 4. The end of the paper is devoted to examples: Section 7 addresses the classical gradient map for Kähler manifolds extending a construction of \textit{I. Mundet i Riera} [Trans. Am. Math. Soc. 362, No. 10, 5169--5187 (2010; Zbl 1201.53086)], Section 8 addresses the action on the space of measures on $\mathcal{M}$, and Section 9 exemplifies explicitly the latter when $\mathcal{M}$ is real projective space. reductive Lie group; gradient maps; geometric invariant theory; stability; polystability; semi-stability L. Biliotti and M. Zedda. Stability with respect to actions of real reductive Lie groups. Ann. Mat. Pura Appl. (4), 196(6), 2185-2211, 2017. Geometric invariant theory, Momentum maps; symplectic reduction Stability with respect to actions of real reductive Lie groups	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Thanks to the the author's previous work, it is possible to reconstruct a scheme from its derived category of perfect complexes, provided we equip the latter with its derived tensor product. The scheme is recovered by means of the ``triangular spectrum'' construction, $\mathrm{Spc}(T)$, which can be applied to any abstract tensor triangulated category~$T$ to yield a nice (locally ringed) space in a functorial way. This is the starting point of ``tensor triangular geometry'', which is a geometric theory of tensor triangulated categories generalizing algebraic geometry (see [\textit{P. Balmer}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 85--112 (2011; Zbl 1235.18012)]). This very short paper addresses the question: How should we generalize the definition of the Chow group of a scheme, so that it still make good sense for more general tensor triangulated categories? Assuming that the tensor triangulated category~$T$ is rigid and its spectrum Noetherian (which is quite reasonable), the proposed definition reads as follows. The group of ``generalized $p$-cycles'' of~$T$ is the direct sum, over all $p$-dimensional points $P\in \mathrm{Spc}(T)$ in the spectrum, of the (triangulated) Grothendieck $K$-group $K_0(\mathrm{Min}(T_P))$, where $\mathrm{Min}(T_P)$ denotes the subcategory of objects with minimum support in $T_P$, the local category of $T$ at~$P$. This notion works for any reasonable dimension function on the spectrum. The Chow group of~$T$ is then obtained from the group of cycles by quotienting out a suitable subgroup of boundaries. Two different definitions of boundaries are proposed here, and at least one of them allows to recover the Chow group of a scheme in the regular case. Both definitions make geometric sense in view of the classical theory. Their precise relationship and several other questions are investigated by the author's PhD student, Sebastian Klein. Chow groups; triangulated categories; triangular spectrum Balmer, P, Tensor triangular Chow groups, J. Geom. Phys., 72, 3-6, (2013) Derived categories, triangulated categories, Applications of methods of algebraic $K$-theory in algebraic geometry Tensor triangular Chow groups	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $G$ be a finite group and $\Lambda(G)$ be the set of irreducible complex representations of $G$. Associated to a fixed finite-dimensional representation $V$, are the numbers $m_k^\lambda$, which are the multiplicity of $\lambda\in \Lambda(G)$ in $V^{\otimes k}$, $A=(a_{\lambda \, \mu})$, the adjacency matrix of the representation graph $R_V(G)$, and the Bratteli diagram which is the infinite graph with vertices labeled by the elements of $\Lambda(G)$ on level $k$, keeping tracks of finite steps walks on $R_V(G)$. The Poincaré series for $\lambda\in \Lambda(G)$ is defined by \[ m^{\lambda}(t)=\sum_{k=0}^\infty m_k^\lambda t^k. \] The main result of the paper under review is as follows: suppose that $G$ acts faithfully on $V$, and $V^* \cong V$ as $G$-representations. Let $M_\mu$ be the matrix $I - tA$ with the column indexed by $\mu$ replaced by $(1\; 0\; 0 \dots 0)^T$. Then \[ m^\mu(t)=\frac{\det M_\mu}{\det(I-tA)}. \] In the special case where $G$ is a finite subgroup of $\mathrm{SU}(2)$, and $V=\mathbb{C}^2$ is its natural representation, then \[ m^0(t)=\frac{\det(I-t\hat{A})}{\det(I-tA)}, \] where $0\in \Lambda(G)$ is the trivial representation, and $\hat{A}$ is the adjacency matrix of the finite Dynkin diagram obtained by removing the affine node. In the continuation, the paper under review finds further explicit formulas for $m^0(t)$ and express the results in terms of Chebyshev polynomials, and the Bratteli diagrams. tensor invariants; Poincaré series; McKay correspondence; Schur-Weyl duality Benkart, G., Poincaré series for tensor invariants and the mckay correspondence, Adv. Math., 290, 236-259, (2016) McKay correspondence, Combinatorial aspects of representation theory, Group rings of finite groups and their modules (group-theoretic aspects) Poincaré series for tensor invariants and the McKay correspondence	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [See also the review to the journal version in Invent. Math. 117, 165-180 (1994; Zbl 0822.58046).] In a previous paper [Adv. Sov. Math. 16, No. 1, 211-241 (1993; Zbl 0802.58051)] the authors proved a version of the classical Riemann-Roch theorem for solutions of general elliptic equations with point singularities. In this paper they extend the results to much more general singularities supported on arbitrary compact nowhere dense subsets. The only restriction is that the allowed singularities should be taken from a finite-dimensional space. Dually a finite set of conditions may be imposed on another nowhere dense compact set. This leads to a notion of rigged divisor which includes two nowhere dense compact sets with finite- dimensional distribution spaces supported on them. Then the allowed singularities on the first given set are described as singularities of solutions which may be extended as distributions to the whole given manifold so that after applying the given elliptic operator one gets into the first given space of distributions. The conditions on the second compact set are just orthogonality conditions to the second space of distributions. The main theorem of the authors then connects the dimension of the space of solutions having the allowed singularities and satisfying the imposed conditions, with another dimension defined in the same way from the dual (or inverse) divisor which is obtained by changing places of two given compact subsets and distribution spaces and replacing the given operator by the adjoint operator. In distinction to their previous paper, the authors here restrict to closed manifolds. general Riemann-Roch theorem; elliptic operators [GrS2]M. Gromov, M. Shubin, The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets, Preprint ETH Zürich, 1993. Elliptic equations on manifolds, general theory, Exotic index theories on manifolds, Higher-order elliptic equations, Riemann-Roch theorems The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 [Duke Math. J. 53, 457--502 (1986; Zbl 0679.14001)], \textit{G. W. Anderson} showed that the following are equivalent: (1) an Abelian $t$-module E is uniformisable, which roughly means that it permits an analytic description as a quotient $V/\Lambda$ of a finite dimensional $\mathbb C_\infty$-vector space $V$ by a free and discrete sub-$\mathbb F_q[t]$-module $\Lambda$, (2) the associated $t$-motive $M$ is analytically trivial, which roughly means that it admits a $\tau$-invariant basis after extending scalars from $\mathbb C_{\infty[t]}$ to the Tate algebra $\mathbb C\langle t\rangle$ (the reader who is unfamiliar with these notions is strongly encouraged to consult Anderson (loc. cit.) first). The paper under review studies how these notions behave in families. The locus of uniformisable $t$-modules in an algebraic family is usually not algebraic: it typically depends on inequalities between valuations of coordinates. (The reader who is unfamiliar with this phenomenon is encouraged to read section 7 of the paper under review first, it contains a detailed treatment of an example due to Pink.) It is therefore most natural to study uniformisability in families over a rigid analytic base, and this is exactly what the authors do. After carefully working out the notions of a rigid analytic family of $t$-motives (and of $t$-modules), and of analytic triviality (and uniformisability) of such families. The main result (Theorem 5.5) states that the locus of uniformisability in a rigid analytic family is Berkovich open (i.e., the set of analytic points over which the given family is uniformisable is open in the topological space of all analytic points of the base.) The exposition is excellent and the authors have done considerable effort to recall the necessary notions and results from rigid analytic geometry. Drinfeld modules; t-motives; uniformisability; rigid analytic geometry Gebhard Böckle & Urs Hartl , Uniformizable families of $t$ -motives , Trans. Am. Math. Soc. 359 (2007), p. 3933-3972 Drinfel'd modules; higher-dimensional motives, etc., Rigid analytic geometry Uniformizable families of $t$-motives	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Determining when a 0-dimensional scheme in projective space $\mathbb P^n_K$ (where $K$ is a field) which is the complete intersection of $n$ hypersurfaces has played an important role in both Algebraic Geometry and Commutative Algebra. For example, in projective 2-space, work of \textit{E. Davis} and \textit{P. Maroscia} [Lect. Notes Math. 1092, 253--269 (1984; Zbl 0556.14025)] provided a characterization of such schemes using the Hilbert function symmetry and the Cayley-Bacharach property. In [\textit{E. D. Davis} et al., Proc. Am. Math. Soc. 93, 593--597 (1985; Zbl 0575.14040)] and [\textit{M. Kreuzer}, Math. Ann. 292, No. 1, 43--58 (1992; Zbl 0741.14030)], it was then shown that this also provides a characterization for arithmetically Gorenstein schemes. There have also been characterizations of 0-dimensional local complete intersections using the Kähler different (see [\textit{G. Scheja} and \textit{U. Storch}, J. Reine Angew. Math. 278/279, 174--190 (1975; Zbl 0316.13003)]). In this paper the authors uses the Kähler and Dedekind differents of $R/K[x_0]$ and $R/K$ to further the characterizations of 0-dimensional complete intersections. A sample characterization is the following theorem: Theorem. Let $\mathbb X$ be a smooth 0-dimensional subscheme of $\mathbb P^n_K$. Then $\mathbb X$ is a complete intersection if and only if $\mathbb X$ is a Cayley-Bacharach scheme and the Hilbert function of the Kähler different of $\mathbb X$ in degree $r_{\mathbb X}$ is non-zero, where $r_{\mathbb X}$ is the regularity index of the Hilbert function of $\mathbb X$. Most of the characterizations can be found in Section 5 of the paper. Indeed, in this section the authors present three characterizations: one generalizing the work of Scheja and Storch; one using a value of the Hilbert function of the Kähler different of the scheme to separate the classes of complete intersections from arithmetically Gorenstein schemes; and one is the above stated theorem. Moreover, using the Dedekind different rather than the Kähler different, the authors characterize 0-dimensional arithmetically Gorenstein schemes. One particular gem of this paper is its useful exposition with many included necessary definitions and references to the literature. zero-dimensional scheme; complete intersection; Kähler different; Dedekind different; arithmetically Gorenstein scheme; Cayley-Bacharach scheme; Hilbert function Complete intersections, Modules of differentials, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Characterizations of zero-dimensional 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 This paper gives a useful generalization of the usual Castelnuovo-Mumford's regularity to the case of weighted projective space, a generalization different from the one of \textit{D. Maclagan} and \textit{G. Smith} [J. Reine Angew. Math. 571, 179--212 (2004; Zbl 1062.13004)] and the multigraded regularity of \textit{N. Botbol} and \textit{M. Chardin} [J. Algebra 474, 361--392 (2017; Zbl 1368.13019)]. With their definition (which obviously depends on the weights of the weighted projective space) the authors are able to prove a theorem for the generation the multigraded ring associated to a coherent sheaf $\mathcal{F}$ and a definition of weight regularity for $\mathcal{F}$ so that $m$-weight regularity implies $(m+1)$-weight regularity. Castelnuovo-Mumford regularity; weighted projective spaces; multigraded ring Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Syzygies, resolutions, complexes and commutative rings Weighted Castelnuovo-Mumford regularity and weighted global generation	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 $w=(w_ 0,...,w_ n)$ be a set of integer positive weights and let S be the polynomial ring ${\mathbb{C}}[x_ 0,...,x_ n]$ graded by the conditions $\deg (x_ i)=w_ i$. Let f be a weighted homogeneous polynomial of degree $ N.$ The Milnor fibration of f is the locally trivial fibration $f:\;{\mathbb{C}}^{n+1}\setminus f^{-1}(0)\to {\mathbb{C}}\setminus \{0\}$ with typical fiber F: $f^{-1}(1).''$ Spectral sequence techniques are used to arrive at a formula for the cohomology groups of F. In the case of isolated singularities more is known about F [see for example \textit{E. Brieskorn}, Manuscr. Math. 2, 103- 161 (1970; Zbl 0186.261)]. The results in this work apply to more general f with (possibly) nonisolated singularities. There is a good number of interesting examples of explicit calculations using the spectral sequences introduced which illustrate why they should be useful in many situations. - A theorem is stated relating a filtration induced by one of the spectral sequences introduced and the Hodge filtration, and it allows one to extend a result of \textit{P. A. Griffiths} [Ann. Math., II. Ser. 90, 460-495; 496-541 (1969; Zbl 0215.081)] on reductions of rational n-forms on (complex) projective spaces with a nonsingular polar locus to the singular case. The author points out in a note that the theorem is correct (although the proof given is flawed), consequence of a more general result in work done with \textit{P. Deligne} [``Hodge and order of the pole filtrations for singular hypersurfaces'' (preprint); see also Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, No.4, 645-656 (1990)]. nonisolated singularity; Milnor fibration; spectral sequences A. DIMCA, On the Milnor fibrations of weighted homogeneous polynomials, Composito Math., 76 (1990), pp. 19-47. Zbl0726.14002 MR1078856 Singularities in algebraic geometry, Families, fibrations in algebraic geometry, Complex singularities, Singularities of differentiable mappings in differential topology On the Milnor fibrations of weighted homogeneous polynomials	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $R_d=\mathbb R[x,y]_d$ be the space of degree-$d$ real binary forms. A rank $r$ is typical in $R_d$ if it is realized in an open (euclidean) subset of $R_d$, or equivalently if the interior $\mathcal R_{d,r}$ of the semi-algebraic set $\{f\in R_d \ \| \ rk_{\mathbb R}(f)=r\}$ is not empty. Typical ranks for $R_d$ are exactly the ranks $r$ such that $\frac{d+1}{2}\leq r \leq d$. Let $\partial(\mathcal R_{d,r})$ be the topological (euclidean) boundary in $R_d$. One defines the algebraic boundary $\partial_{alg}(\mathcal R_{d,r})$ to be the Zariski closure of $\partial(\mathcal R_{d,r})$ in the complex space $\mathbb P(\mathbb C[x,y]_d)$. The algebraic boundaries for minimum rank $r=\lceil\frac{d+1}{2}\rceil$ and maximum rank $r=d$ (for any $d$) have already been described, as well as the ones for any typical rank $\frac{d+1}{2}\leq r \leq d$ when $d\leq 8$. In this work the authors extend this study to any typical rank $r$ and any degree $d$. More precisely, given $\lambda=(\lambda_1\leq \ldots \leq \lambda_k)$ a partition of $d$ having length $k$, the associated coincident root locus is the $k$-dimensional variety \[ \Delta_{\lambda}=\left\{f \in \mathbb P^d=\mathbb P(\mathbb C[x,y]_d) \ \bigg\| \ \exists l_1,\ldots, l_k \in \mathbb C[x,y]_1 \ : \ f=\sum_{i=1}^kl_i^{\lambda_i}\right\} \subset \mathbb P^d \ ; \] then the authors' main results (Theorem 3.2, Theorem 3.3) describe the algebraic boundaries for any degree $d$ and any typical rank $r$ as the union of dual varieties to coincident root loci $\Delta_\lambda$, as $\lambda$ varies among certain partitions of $d$ whose length depends on $r$. The main tools used to reach the results are the $j$-th higher associated variety $C\!H_j(X)$ to a $k$-dimensional variety $X\subset \mathbb P^d$, defined as the closure of the subset in the grassmannian $\mathbb G(d-k-1+j,\mathbb P^d)$ \[ \left\{H\in \mathbb G(d-k-1+j,\mathbb P^d) \ \| \ H \cap X \neq \emptyset , \ \dim(H \cap T_xX)\geq j \ \text{for some } x \in H\cap X_{sm}\right\} \ , \] and the apolar map \[ \begin{matrix} \Psi_{d,r}: & \mathbb P^d=\mathbb P(\mathbb C[x,y]_d) & \dashrightarrow & \mathbb G(2r-d-1,\mathbb P^r)\\ & f & \mapsto & \mathbb P((f^\perp)_r) \end{matrix} \ , \] where, for $d-r\leq r\leq d$, $(f^\perp)_r\subset \mathbb C[\partial_x,\partial_y]_r$ is the degree-$r$ component of the apolar ideal $f^\perp$. In Section 1 the basic notions and results about coincident root loci, higher associated varieties and apolar maps are recalled. In Section 2 the authors give a detailed study of the pullbacks via apolar maps of the higher associated varieties to a coincident root locus.\\ In section 3 the main results are proven. typical rank; real rank boundary; algebraic boundary; binary form; multiple root locus; coincident root locus; Waring problem; apolar map Secant varieties, tensor rank, varieties of sums of powers, Semialgebraic sets and related spaces, Multilinear algebra, tensor calculus Algebraic boundaries among typical ranks for real binary forms of arbitrary 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 In this paper it is proved that there exist completely normal spaces that are not real spectra of rings. Real spectra of rings are known to be spectral spaces in the sense of Hochster and the problem of characterizing topologically real spectra has been considered from the beginning of this theory. A property which is quite easy to notice for real spectra, and closely linked to the ordered structures of fields considered in this theory, is the following, called complete normality: whenever points $x$ and $y$ are specializations of the same point $z$ (this means that they belong to the closure of the singleton $\{z\})$, then either $x$ is a specialization of $y$ or $y$ is a specialization of $x$. This paper proves that in order to find a topological description of real spectra, new topological properties of real spectra have to be identified. A completely normal spectral space is constructed which is not the real spectrum of a ring. The example presented is one dimensional which is the first dimension where such a result can be hoped. The construction relies on the Dedekind completion -- through cuts -- of completely dense totally ordered sets and to a construction due to Hausdorff of an $\eta_2$-set (where $\omega_2$ is the second infinite cardinal): it is a set $Y$ such that for every two subsets $A$ and $B$ of $Y$ such that $A < B$ with cardinality less than $\omega_2$, there exist $y$ in $Y$ such that $A < y < B$. completely normal spaces that are not real spectra of rings; spectral spaces Delzell, Charles N.; Madden, James J., A completely normal spectral space that is not a real spectrum, J. Algebra, 169, 1, 71-77, (1994) Real algebraic sets, Real and complex fields, Relevant commutative algebra A completely normal spectral space that is not a real spectrum	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 real analytic set in $\mathbb R^n$ and $x\in X$. For a relatively compact neighbourhood $U$ of $x$ in $\mathbb R^n$ and a function $f\in C^{\infty}(\overline{U})$, we define \[ \\|f\\|_{m,X}^U= \inf_h \|h\|_m^U \] over all $h\in C^\infty(\overline{U})$ such that $h\|X=f,$ where \[ \|h\|_m^U= \sum_{\|\gamma\|\leq m}\|D^\gamma h\|^U \] and $\|\cdot\|^U$ is the ordinary sup-norm on $U$. The author proves a generalization of the Gagliardo-Nirenberg inequality for analytic sets $X$ at a point $x\in X$: there exist $U\ni x$, a real number $s\geq1$ and constants $C_m>0,$ $m\geq0$, such that for all $f\in C^\infty(\overline{U})$ ($f\not \equiv 0$ on $X\cap U$) \[ \frac{\\|f\\|_{k,X}^U}{\\|f\\|_{0,X}^U}\leq C_0C_m^k\left( \frac{\\|f\\|_{m,X}^U} {\\|f\\|_{0,X}^U}\right)^{\frac{sk}{m}} \] (for $X=$ compact domain in $\mathbb R^n$ with ``good'' boundary and $s=1$, we obtain the classical Gagliardo-Nirenberg inequality). He gives also examples of algebraic sets for which $s>1$ at an isolated singularity. real analytic set; Gagliardo-Nirenberg inequality Local complex singularities, Real-analytic sets, complex Nash functions, Real-analytic and semi-analytic sets, Real algebraic sets A note on Gagliardo-Nirenberg type inequalities on analytic sets	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate an approximate projecting of the Banach space element in the neighborhood of still points of a given resolving operator onto the corresponding stable manifold. The projection operator is given by a basis, which describes an admissible displacement. Thus, the original problem is reduced to the solution of a special kind of a nonlinear equation. Solvability of this equation at standard conditions is proved. Local equivalence of the given method to known algorithms of the stable manifold approximation as well as high efficiency of the method is demonstrated. Numerical calculations are performed for the two-dimensional Chaffey-Enfant equation. approximate projecting; Hadamard-Perron theorem; stable manifold; Chaffey-Enfant equation Invariant manifold theory for dynamical systems, Projective techniques in algebraic geometry, Theoretical approximation of solutions to ordinary differential equations, Stability theory for smooth dynamical systems, Numerical nonlinear stabilities in dynamical systems On approximate projecting on a stable manifold	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a successor to \textit{B. Elazar} and \textit{A. Shaviv} [Can. J. Math. 70, No. 5, 1008--1037 (2018; Zbl 1445.14078)] in which the authors succeeded to extend the famous class of Schwartz functions on $\mathbb{R}^n$ to real algebraic varieties which might have singularities; it also uses many results from \textit{A. Aizenbud} and \textit{D. Gourevitch} [Int. Math. Res. Not. 2008, Article ID rnm155, 37 p. (2008; Zbl 1161.58002)] which was among the first to study more deeply Schwartz functions on Nash manifolds. In the current paper the categories of Nash varieties and real algebraic varieties are put under the umbrella of a new category, called Quasi-Nash of which the former are subcategories. After a very careful outline of context and main results, given in Section 1, Section 2 presents the technical preliminaries of the study. Here semialgebraic subsets are defined (with a subtle logical error), the concept of complete variety, the algebraic Alexandrov compactification of noncomplete varieties, various results on Fréchet spaces and versions of the open mapping theorem and the Hahn Banach theorem for such spaces are recounted. As background or complementary literature here serve the well known books by \textit{F. Trèves} [Topological vector spaces, distributions and kernels. New York-London: Academic Press (1967; Zbl 0171.10402)] and \textit{J. Bochnak} et al. [Real algebraic geometry. Transl. from the French. Rev. and updated ed. Berlin: Springer (1998; Zbl 0912.14023)], as well as the paper by Aizenbud and Gourevitch [loc. cit.]. The author often works with restricted topological spaces. These deviate from the usual concept by requiring of the family of `open' sets only that it is closed with respect to finite unions. Continuity is defined as usual. Among the central notions are these: an $ \mathbb{R}$-space is a pair $(M,\mathcal O_M)$ consisting of a restricted topological space $M$ and a sheaf $\mathcal O_M$ of $\mathbb{R}$-algebras over $M$ which is a subsheaf of the the $\mathbb{R}$-algebra of all continuous functions on $M.$ A continous map $\varphi: (M,\mathcal O_M)\rightarrow (N,\mathcal O_N) $ is a morphism of $\mathbb{R}-$spaces if for any open set $U\subset N$ and any $f\in \mathcal O_N(U)$ there holds $(f\circ \varphi)\|_{\varphi^{-1}(U)} \in \mathcal O_M(\varphi^{-1}(U)).$ A Nash submanifold of $\mathbb{R}^n$ is simply a semialgebraic subset of $ \mathbb{R}^n$ which is a smooth submanifold; a Nash function on $M$ is a smooth semialgebraic function; an affine Nash manifold is an $ \mathbb{R}$-space which is isomorphic to the $ \mathbb{R}$-space associated to a closed Nash submanifold; and a Nash manifold itself is an $\mathbb{R}$-space $(M,\mathcal N_M)$ where $M$ admits a finite open cover $\{M_i\}_{i=1}^n$ so that every $ \mathbb{R}$-space $(M_i, \mathcal N_{M_i})$ is an affine Nash manifold. Bochnak et. al. [loc. cit.] contains not all of these concepts or not in this form. The same is true of the concept of a Nash differential operator $D$ on an affine Nash manifold which is an element of the algebra with 1 generated by multiplication and derivations along Nash sections of the tangent bundle. The space of Schwartz functions on such an $M$ is $\mathcal S(M)=\{\phi\in C^\infty (M): D\phi \text{ is bounded for any Nash differential operator }\}. $ The norm used is defined by $\|\|\phi\|\|_D:=\sup_{x\in M} \|D\phi(x)\|.$ A function $t:\mathbb{R}^n\rightarrow \mathbb{R}$ is tempered if it is smooth and all its derivatives are polynomially bounded. Tempered functions on $M$ are naturally defined, then, via closed embeddings $i:M\hookrightarrow \mathbb{R}^n.$ Section 3, titled `Geometry', defines gradually the new category: Just as before happened with the concept of a Nash manifold, the concept of a `Quasi Nash variety' is also defined by a tower of notions. First the Naive Quasi Nash (NQN) category is defined: its objects are the locally closed semialgebraic subsets of $\mathbb{R}^n;$ and if $X\subset \mathbb{R}^n,$ $Y\subset \mathbb{R}^m$ are such sets, then $\varphi: X \rightarrow Y$ is a morphism in the category if there exists an open set $U\supset X$ and a Nash map $g: U\rightarrow \mathbb{R}^m$ such that $g_{\|X}=\varphi.$ Next an affine Quasi Nash (QN) variety is an $\mathbb{R}$-space which is isomorphic to an $\mathbb{R}$-space obtained from appropriate `sheafificaction' of some closed NQN set. At that point the author can prove already his Lemma 3.6: Affine Nash manifolds and affine real algebraic varieties form a subcategories of affine QN-varieties; the former subcategory is full; but not so the latter. Finally a QN-variety is an $\mathbb{R}$-space $(X,QN_X)$ so that $X$ is a restricted topological space which admits a finite open cover $\{X_i\}_{i=1}^m$ so that the $\mathbb{R}$-spaces $(X_i, QN_{X\|X_i})$ are affine QN varieties. Section 4, titled `Schwartz functions, tempered functions and distributions' , consists of three subsections aptly dubbed Schwartz functions on, respectively, Naive Quasi Nash sets (4.1); Affine Quasi Nash varieties (4.2); and Quasi Nash Varieties (4.3). This happens in parallelism with the gradual ascent to this general concept made in section 3. What concerns Section 4.1, if $X$ is a NQN set and $U\supset X$ open, let $I_{\mathrm{Sch}}^U(X)$ be the ideal of Schwartz functions on $U$ that vanish on $X.$ Since, as shown in Corollary 4.1.1, the quotient $\mathcal S(U)/ I_{\mathrm{Sch}}^U(X)$ does not depend on $U,$ one can define $\mathcal S(X)$ as being this quotient. It is also a Fréchet space. If $X_1\stackrel{\varphi}{\rightarrow} X_2$ is an NQN isomorphism then the pullback $\mathcal S(X_2) \stackrel{\varphi\|\mathcal S(X_2)}{\rightarrow} \mathcal S(X_1)$ is an isomorphism of Fréchet spaces. Like a few other theorems this result occurs (formulated for Schwartz functions on (general) Quasi Nash varieties) again in Section 4.3 as Lemma 4.3.5 and is one of the main results of the paper. In 4.2 first the concept of Schwartz function is extended for the case that $X$ is an affine QN variety: $s$ is a Schwartz function on $X$ if it is a pullback from the closed NQN set corresponding to $X.$ Another concept for laying down the other main result is defined: Let $X$ be an affine QN variety corresponding to the closed set $\tilde X\subset \mathbb{R}^n .$ Then a real valued function $f$ defined on $\tilde X$ is flat at $\tilde p$ if there exists an open neighbourhood $U$ of $\tilde p$ and an analytic function $F$ such that $f\|_{U\cap \tilde X}= F\|_{U\cap \tilde X}$ and the Taylor series of $F$ around $\tilde p$ is identically zero. For $X$ an affine QN variety the space of continuous linear functionals is called the space of tempered distributions, denoted $\mathcal S^ (X).$ Section 4.3: Let $\text{Func}(X,R)$ denote the set of real valued functions on a set $X.$ Let $X$ be a (general) QN variety with a finite open affine QN cover $X=\bigcup_{i=1}^m X_i.$ The natural map $\psi: \bigoplus_{i=1}^m \text{Func}(X_i,R) \rightarrow \text{Func}(X,R)$ is used to define the Schwartz functions on $X$ as $\mathcal S(X)= \psi \left( \bigoplus_{i=1}^m \mathcal S(X_i) \right).$ It is shown via an isomorphism that this definition is good (not dependent on the cover) and $\mathcal S(X)$ again Fréchet space. The notion of flatness is duely adapted and after further work the second main result is obtained. Theorem 4.3.15: Let $X$ be a QN variety and $Z\subset X$ a closed subset. Let $U=X\setminus Z$ and $W_Z:=\{\phi\in \mathcal S(X): \phi \text{ is flat on (the points of)} Z\}.$ Then $W_Z$ is a closed subspace of $\mathcal S(X)$ and hence Fréchet. One has a Fréchet space isomorphism $W_Z\cong \mathcal S(U)$ which is given by the extension by zero from $U$ to $X,$ and its inverse, the restriction from $X$ to $U.$ Th 4.3.17: Under similar conditions the extension defines a closed embedding $\mathcal S(U)\hookrightarrow S(X)$ and the restriction morphism $\mathcal S^(X) \rightarrow \mathcal S^(U)$ is onto. Concerning sections 5 and 6, it is proved that tempered functions and tempered distributions form sheaves and that Schwartz functions define a cosheaf. Further a characterization of Schwartz sections on open subsets for vector bundless are given. The proofs are said to be largely as in Elazar and Shaviv [loc. cit.] and Aizenbud and Gourevitch [loc. cit] with adjustments. An appendix prepares the reader with some definitions and facts to adapt a partition of unity theorem found again in Aizenbud and Gourevitch to the current case. The author makes a very honest attempt to make his topic accessible and by and large seems very careful with his definitions. The towers of notions building one upon the other make it nevertheless demanding reading for non-specialists. affine algebraic varieties; semi-algebraic sets; Nash maps; Nash manifolds; categories; tempered distributions; tempered functions Nash functions and manifolds, Real algebraic sets, Semialgebraic sets and related spaces, Linear function spaces and their duals Quasi-Nash varieties and Schwartz functions on them	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we introduce constructible analogs of the discrete complexity classes $\mathbf {VP}$ and $\mathbf {VNP}$ of sequences of functions. The functions in the new definitions are constructible functions on $\mathbb {R}^n$ or $\mathbb {C}^n$. We define a class of sequences of constructible functions that play a role analogous to that of $\mathbf {VP}$ in the more classical theory. The class analogous to $\mathbf {VNP}$ is defined using Euler integration. We discuss several examples, develop a theory of completeness, and pose a conjecture analogous to the $\mathbf {VP}$ versus $\mathbf {VNP}$ conjecture in the classical case. In the second part of the paper we extend the notions of complexity classes to sequences of constructible sheaves over $\mathbb {R}^n$ (or its one point compactification). We introduce a class of sequences of simple constructible sheaves, that could be seen as the sheaf-theoretic analog of the Blum-Shub-Smale class $\mathbf {P}_\mathbb {R}$. We also define a hierarchy of complexity classes of sheaves mirroring the polynomial hierarchy, $\mathbf {PH}_\mathbb {R}$, in the B-S-S theory. We prove a singly exponential upper bound on the topological complexity of the sheaves in this hierarchy mirroring a similar result in the B-S-S setting. We obtain as a result an algorithm with singly exponential complexity for a sheaf-theoretic variant of the real quantifier elimination problem. We pose the natural sheaf-theoretic analogs of the classical $\mathbf {P}$ versus $\mathbf {NP}$ question, and also discuss a connection with Toda's theorem from discrete complexity theory in the context of constructible sheaves. We also discuss possible generalizations of the questions in complexity theory related to separation of complexity classes to more general categories via sequences of adjoint pairs of functors. constructible functions; constructible sheaves; polynomial hierarchy; complexity classes; adjoint functors H. Edelsbrunner, J. Harer, \textit{Computational Topology. An Introduction}, (American Mathematical Society, 2010). Complexity classes (hierarchies, relations among complexity classes, etc.), Semialgebraic sets and related spaces, Topology of real algebraic varieties, Symbolic computation and algebraic computation A complexity theory of constructible functions and sheaves	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any point $x = (x_1, x_2) \in \mathbb{R}^2$ we let $G_x = \mathbb{Z} x_1 + \mathbb{Z} x_2 + \mathbb{Z}$ be the subgroup of the additive group $\mathbb{R}$ generated by $x_1, x_2, 1$. When $\operatorname{rank}(G_x) = 3$ we say that $x$ is a \textit{rank 3 point.} We prove the existence of an infinite set $\mathcal{I} \subseteq \mathbb{R}^2$ of rank 3 points having the following property: For every two-dimensional continued fraction expansion ${\mu}$ and $x \in \mathcal{I}$, letting $\mu(x) = T_0 \supseteq T_1 \supseteq \cdots$, it follows that infinitely many triangles $T_n$ have some angle $\leq \arcsin(23^{1 / 2} / 6)\approx \pi /(3.3921424) \approx 53^\circ$. Thus $\lim \inf_{n \rightarrow \infty} \operatorname{area}(T_n) / \operatorname{diam}(T_n)^2 \leq 23^{1 / 2} / 12$. \par At the opposite extreme, we construct a two-dimensional continued fraction expansion ${\mu}$ and a dense set $\mathcal{D} \subseteq \mathbb{R}^2$ of rank 3 points such that for each $x \in \mathcal{D}$ the sequence $T_0 \supseteq T_1 \supseteq \cdots$ of triangles of $\mu(x)$ has the following property: Letting $\omega_n$ denote the smallest angle of $T_n$, it follows that $\omega_0 < \omega_1 < \cdots$ and $\lim_{n \rightarrow \infty} \omega_n = \pi / 3$. Further, the other two angles of $T_n$ are $> \pi / 3$. Thus $\lim_{n \rightarrow \infty} \operatorname{area}(T_n) / \operatorname{diam}(T_n)^2 = 3^{1 / 2} / 4$, and the vertices of the triangles $T_n$ strongly converge to $x$. Our proofs combine a classical theorem of Davenport and Mahler with binary stellar operations of regular fans. regular cone; unimodular cone; Fan; continued fraction expansion; simultaneous Diophantine approximation; multidimensional continued fraction algorithm; stellar operation; starring; Farey mediant; Farey sum; Davenport-Mahler theorem Continued fractions, Farey sequences; the sequences $1^k, 2^k, \dots$, Continued fraction calculations (number-theoretic aspects), Continued fractions and generalizations, Diophantine approximation in probabilistic number theory, Toric varieties, Newton polyhedra, Okounkov bodies, General theory of group and pseudogroup actions, Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations, Dynamics induced by group actions other than $\mathbb{Z}$ and $\mathbb{R}$, and $\mathbb{C}$, Convergence and divergence of continued fractions Triangles in Diophantine 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 $R$ be a real closed field, $D$ an ordered subdomain in $R$ and $\mathcal{P}$ a family of symmetric polynomials in $k$ variables of degree $\leq d$ over $D$. The algorithmic problem of computing Betti numbers of arbitrary semi-algebraic subsets of $R^k$ is well-studied and has applications in the theory of computational complexity and in robot motion planning. Betti numbers of semi-algebraic subsets of $R^k$ satisfy a singly exponential (in $k$) complexity; singly exponential dependence on $k$ of the bound is unavoidable. The authors prove that for each fixed $\ell$, $d\geq 0$, there exists an algorithm that takes as input a quantifier-free first-order formula $\Phi$ with atoms $P=0$, $P>0$, $P<0$ with $P\in \mathcal{P}$, and computes the ranks of the first $\ell +1$ cohomology groups of the symmetric semi-algebraic set defined by $\Phi$. The complexity of this algorithm (measured by the number of arithmetic operations in $D$) is bounded by a polynomial in $k$ and card($\mathcal{P}$) (for fixed $d$ and $\ell$). This result contrasts with the PSPACE-hardness of the problem of computing just the zeroth Betti number (i.e., the number of semi-algebraically connected components) in the general case for $d\geq 2$ (taking the ordered domain $D$ to be equal to $\mathbb{Z}$). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. The authors prove that the Specht modules corresponding to partitions having long lengths cannot occur in the isotypic decompositions of low-dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks. symmetric semi-algebraic sets; isotypic decomposition; Specht module; Betti numbers; mirrored spaces; computational complexity Semialgebraic sets and related spaces, Topology of real algebraic varieties, Symmetric functions and generalizations, Effectivity, complexity and computational aspects of algebraic geometry, Symbolic computation and algebraic computation Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable \[ \zeta =\sum\limits_{k=0}^\infty\frac{2}{3^k}\varepsilon_{\eta_k}\equiv\Delta_{\eta_1\dots\eta_k\dots}, \] where the indices $\eta_k$ are independent random variables taking the values 0, 1, 2, 3, 4 with probabilities $p_{0k}$, $p_{1k}$, $p_{2k}$, $p_{3k}$, $p_{4k}$ respectively, $p_{ik}\geq 0$, $p_{ik} \geq 0$, $\sum_{i=0}^4 p_{ik}=1$, $\sum_{i=0}^4 p_{ik}=1$ for any $k \in \mathbb N$, $\varepsilon_m=e^{\frac{m\pi i}{2}}=i^m(m = 0,1,2,3)$ are 4th roots of unity, $\varepsilon_4=0$. We prove that the distribution of $\zeta$ is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of $\zeta$ to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found. probability measure; fractal curve; Vicsek fractal; discrete probability distribution; singularly continuous probability distribution; point spectrum; spectrum of probability distribution Probability measures on topological spaces, Fractals, Plane and space curves Probability measures on fractal curves (probability distributions on the Vicsek fractal)	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The problem of resolution of singularities in characteristic 0 was solved by Hironata at the beginning of the sixties of the last century. Much later algorithmic methods have been developed for solving this problem. The study of algorithmic equiresolution started about 15 years ago. Some basic results in this direction can be found in the article of \textit{S. Encinas, A. Nobile} and \textit{O. E. Villamayor} [Proc. Lond. Math. Soc., III. Ser. 86, No. 3, 607--648 (2003; Zbl 1076.14020)]. Here families of ideals or of embedded schemes, parametrized by smooth varieties are studied. The equiresolution proposed required that the centers for the transformations are smooth over the parameter variety (condition AE) or required the local constancy of a certain invariant associated to each fibre (condition $\tau$). In this paper a definition of equiresolution is proposed for families parametrized by not necessarily reduced schemes, called condition E. Other approaches are proposed, condition A, C and F. Condition A corresponds to AE mentioned above. The main objective of the paper is to prove that when the parameter space is regular all these conditions are quivalent. Assuming the properness of certain projections it is proved that they are also equivalent to $\tau$ mentioned above. resolution algorithm; embedded variety; coherent ideal; basic object Nobile, A.: Simultaneous algorithmic resolution of singularities, Geom. dedic. 163, 61-103 (2013) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, fibrations in algebraic geometry Simultaneous algorithmic resolution of singularities	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review gives a description of the set of all locally nilpotent derivations of the quotient ring $\mathcal{B} = K[X, Y, Z]/(f(X)Y-\phi(X, Z))$ constructed from the defining equation $f(X)Y=\phi(X, Z)$ of a generalized Danielewski surface in $\mathbb{K}^{3}$ in the case when $\mathbb{K}$ is an algebraically closed field of characteristic zero, $\phi(X, Z) = Z^{m} + b_{m-1}(X)Z^{m-1} +\dots + b_{1}(X)Z + b_{0}(X)$ ($m > 1$) and $\deg f > 1$. As a consequence of this description, the authors obtain that both $ML$- and $HD$- invariants of $\mathcal{B}$ introduced in [\textit{L. Makar-Limanov}, Isr. J. Math. 96, Part B, 419--429 (1996; Zbl 0896.14021)] and [\textit{H. Derksen}, Constructive invariant theory and the Linearisation problem. Ph.D. Thesis, Univ. of Basel (1997)], respectively, are equal to $\mathbb{K}[X]$. (Recall that the Makar-Limanov (or $ML$-) invariant of a ring $A$ is the intersection of the kernels of all locally nilpotent derivations of $A$. The Derksen (or $HD$-) invariant of $A$ is defined as the subring of $A$ generated by the union of kernels of all nontrivial locally nilpotent derivations of $A$.) The paper also presents a description of a set of generators for the group of $\mathbb{K}$-automorphisms of the ring $K[X, Y, Z]/(f(X)Y-\phi(Z))$ where $\phi(Z)\in\mathbb{K}[Z]$ and $\deg \phi > 1$. automorphisms; Danielewski surface; locally nilpotent derivations; ML-invariant Derivations and commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate the possibility of a finite characterization of multiple structures in projective space $\mathbb P^n_k$, where $k$ is an algebraically closed field, defined on linear subspaces of small codimension under the assumption of unmixedness. \textit{N. Manolache} gave such a characterization for scheme-theoretically Cohen-Macaulay multiple structures of degree at most 4 [Math. Z. 210, No. 4, 573--579 (1992; Zbl 0784.14020)] as well as for locally complete intersection multiple structures of degree at most 6 [\textit{N. Manolache}, Math. Z. 219, No. 3, 403--411 (1995; Zbl 0837.14039)]. In addition, \textit{B. Engheta} gave a characterization of unmixed ideals of height 2 and multiplicity 2 [J. Algebra 316, No. 2, 715--734 (2007; Zbl 1132.13006)]. In contrast to these characterizations, the main result of this paper shows that no such finite characterization is possible under the unmixedness assumption alone. In particular, the authors prove the following: Theorem. For any integers $h, e \geq 2$ with $(h,e) \not = (2,2)$ and integer $p \geq 5$, there exists a homogeneous ideal $I_{h,e,p}$ in a polynomial ring $R$ over the algebraically closed field $k$ such that: $I_{h,e,p}$ has height $h$; the Hilbert-Samuel multiplicity of $R/I_{h,e,p}$ is $e$; the projective dimension of $R/I_{h,e,p}$ is at least $p$; $I_{h,e,p}$ is primary to a linear prime $(x_1, \ldots, x_h)$. After reviewing necessary background, at the beginning of Section 3 the authors provide the following useful summary of their approach to the proof of the above theorem. For the ease of notation, as in the main theorem, denote by $L_{h,e,p}$ an ideal in a polynomial ring $R$ over an algebraically closed field $k$ such that $L_{h,e,p}$ has height $h$, $R/L_{h,e,p}$ has Hilbert-Samuel multiplicity $e$, and the canonical module for $R/L_{h,e,p}$ has projective dimension greater than or equal to $p$. Using this notation, the authors first define four key families of primary ideals with large projective dimension and nice resolutions and canonical modules. The four key families are $L_{2,5,p}, L_{2,6,p}, L_{2,20,p}$ and $L_{3,6,p}$. Then by linking via a complete intersection from one of these four key families, the authors present primary and radical linear ideals of height 2 and any multiplicity greater than or equal to 3 with arbitrarily large projective dimension. To obtain examples with arbitrary height, one adds extra linear generators. It is also noted that an additional construction is needed for the case of $L_{3,2,p}$ case since the case for $L_{2,2,p}$ is finite. Section 4 of the paper is dedicated to the details of the construction for the four key families. The key ideals were originally found with the use of Macaulay2 and have many generators (see Section 5 for an example demonstrating the minimal number of generators for $I_{2,4,p}$). The paper concludes with a discussion of classifying ideals with Serre's ($S_2$) property. The authors note that the ideals constructed in the paper satisfy Serre's ($S_1$) property but none are ($S_2$). The authors close with the following question: Question: Is there a finite classification of homogeneous unmixed or primary ideals of a given height and multiplicity that satisfy Serre's ($S_2$) condition? Is there a classification for such ideals that are ($S_2$) on the punctured spectrum? multiple structures; projective dimension; multiplicity; primary ideals; unmixed ideals; free resolution; linkage Huneke, C.; Mantero, P.; McCullough, J.; Seceleanu, A., Multiple structures with arbitrary large projective dimension supported on linear subspaces, J. Algebra, 447, 183-205, (2016) Homological dimension and commutative rings, Linkage, Low codimension problems in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Multiple structures with arbitrarily large projective dimension supported on linear 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 Consider the natural map $F$ between the space $E$ of ordinary differential equations on marked Riemann surfaces with prescribed local monodromies and the space $R$ of representation classes of $\pi_1$ (punctured surface) with prescribed local representations around the punctures. A one-parameter family of differential equations lying in the fiber $F^{-1}(r)$ is called isomonodromic. In the note an infinitesimal description of isomonodromic families is given in terms of a completely integrable system of partial differential equations on some local coordinate parameters of $E$. In geometric terms this amounts to describe the tangential directions to the fibers $F^{-1}(r)$ based on the existence of a natural symplectic structure on $R$. The method is applied to the Fuchsian system $d^2y/dz^2=q(z)y$ on the elliptic curve ${\mathbb C}/({\mathbb Z}.1+{\mathbb Z}.\tau)$, $\tau\in {\mathbb C}$, Im$\tau >0$. The function $q$ equals \[ \begin{aligned} k &+\sum _{i=0}^m[H_i\zeta (z-t_i,\tau)+ (1/4)(\theta _i^2-1)\wp (z-t_i,\tau)]\\ & +\sum _{\alpha =0}^m[-\mu _{\alpha } \zeta (z-\lambda _{\alpha },\tau)+(3/4)\wp (z-\lambda _{\alpha },\tau)],\end{aligned} \] with $\sum _{i=0}^mH_i-\sum _{\alpha =0}^m\mu _{\alpha }=0$ where $\zeta (z,\tau)$ and $\wp (z,\tau)$ denote the Weierstrass $\zeta$- and $\wp$-functions with fundamental periods 1 and $\tau$. monodromy representation; isomonodromic family; symplectic structure; marked Riemann surfaces S. Kawai,Isomonodromic Deformation of Fuchsian-type Projective Connections on Elliptic curves, RIMS Kokyuroku Vol. 1022 (RIMS, Kyoto University, Kyoto, 1997), pp. 53--57. Structure of families (Picard-Lefschetz, monodromy, etc.), Riemann surfaces; Weierstrass points; gap sequences, Infinitesimal methods in algebraic geometry, Ordinary differential equations on complex manifolds, Isomonodromic deformation of Fuchsian-type projective connections on elliptic curves	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to answer to a question of M. Gromov, formulated somewhere and transmitted to the author by a friend of him. More precisely, one consider families \[ X=\{X_{t}\}_{t\in T}\text{ \;\;}X_{t}\in\mathbb{R}_{t}^{n},t\in T,T\subset \mathbb{R}^{m} \] with an index set $T$ (called a base) which is a complex analytic set (or subanalytitic, or semianalytic) or complex algebraic set (or semialgebraic). All these conditions are expressed by the symbols $\mathcal{U}_{i}, T\in\mathcal{U}_{i},i=1,2,3,4,5.$ Lipschitz equivalence with respect to $X$ is defined as ordinary and the notion of the complexity of a curve $p(\mu),\mu>0,$ is recalled (the number $N$ of polynomial equations and inequalities that describe the graph of the curve $p$ determines the complexity of at most $N).$ Gromov's question is stated as follows: Is the set of Lipschitz equivalence classes with respect to $X$ of curves of complexity at most $N$ finite? The answer of the author is affirmative. The obtained proof deals with an appropriate stratification of $\mathbb{R}^{m}$, and respectively of $\mathbb{R}^{n}$ compatible with $T$ with skeleton in $\mathcal{U}_{i}.$ The exposition is highly sophisticated and it seems that the proof is correct, but difficult to be summarized shortly. Lipschitz equivalence; curve complexity Semi-analytic sets, subanalytic sets, and generalizations, Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions, Semialgebraic sets and related spaces, Real-analytic and semi-analytic sets Lipschitz stratifications and Lipschitz isotopies	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers semi-algebraic sets $K\subset\mathbb{R}^n$, that is, subsets of the form $K=\{{\mathbf x}\in\mathbb{R}^n:p_j({\mathbf x})\geq0,~ j=0,1,\ldots,m\}$, where $(g_j)_{j=0}^m$ is a (finite) family of real polynomials on $\mathbb{R}^n$ with $g_0=1$. Such a set is clearly closed but not necessarily compact. The aim of the paper is to solve the $K$-moment problem for certain continuous linear functionals. More precisely, the algebra of all polynomials $\mathbb{R}[{\mathbf x}]$ is endowed with a norm induced by the sequence ${\mathbf w}= (w_\alpha)_{\alpha\in\mathbb{N}^n}$, where $w_\alpha=2\lceil\|\alpha\|/2\rceil!$. If $f=\sum_{\alpha\in\mathbb{N}^n}f_\alpha{\mathbf x}^\alpha$ is an arbitrary polynomial, what the author calls the $\ell_{\mathbf w}$-norm of $f$ is the quantity given by $\sum_\alpha w_\alpha\| f_\alpha\|$. For a semi-algebraic set $K\subset\mathbb{R}^n$ defined by the family of polynomials $\{p_0,p_1,\break\ldots,p_m\}$, with $p_0=1$, and for a linear functional $L$ on $\mathbb{R}[{\mathbf x}]$ which is $\ell_{\mathbf w}$-continuous, there exists a finite positive Borel measure $\mu$ on $K$ such that $L(f)=\int_Kfd\mu$ for all polynomials $f\in\mathbb{R}[{\mathbf x}]$ if and only if $L(h^2g_j)\geq 0$ for all $h\in \mathbb{R}[{\mathbf x}]$, $j=0,1,\ldots,m$, and $\sup_{\alpha\in\mathbb{N}^n}\| L({\mathbf x}^\alpha)\|/w_\alpha<\infty$. The main ingredient of the proof is the use of Carleman's condition, insured by the choice of the sequence $\mathbf w$. Other related results are also obtained, exploiting the $\ell_{\mathbf w}$-continuity. In particular, with the notation from above, a polynomial $f$ is positive on the support of $\mu$ if and only if $\int h^2fd\mu\geq0$ for all $h\in \mathbb{R}[{\mathbf x}]$. moment problems; real algebraic geometry; positive polynomials; semi-algebraic sets Lasserre, J. B., The $K$-moment problem for continuous linear functionals, Trans. Amer. Math. Soc., 365, 5, 2489-2504, (2013) Moment problems, Polynomials over commutative rings, Semialgebraic sets and related spaces, Polynomials and rational functions of one complex variable The $K$-moment problem for continuous linear functionals	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a theory of ``limit linear series'' for rational ribbons -- that is, for schemes that are double structures on $\mathbb{P}^1$. This allows us to define a ``linear series Clifford index'' for ribbons. Our main theorem shows that this is the same as the Clifford index of ribbons studied by \textit{D. Bayer} and \textit{D. Eisenbud} in this same volume [Trans. Am. Math. Soc. 347, No. 3, 719-756 (1995; see the following review)]. This allows us to prove that the Clifford index is semicontinuous in degenerations from a smooth curve to a ribbon. A result of \textit{L.-Y. Fong} [J. Algebr. Geom. 2, No. 2, 295-307 (1995; Zbl 0788.14027)] then shows that ribbons may be deformed to smooth curves of the same Clifford index. Thus the canonical curve conjecture of \textit{M. L. Green} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] would follow, at least for a general smooth curve of each Clifford index, from the corresponding statement for ribbons. Green's conjecture; linear series Clifford index; rational ribbon; resolution Clifford index; canonical curve conjecture D. Eisenbud and M. Green, Clifford indices of ribbons, Trans. Amer. Math. Soc. 347 (1995), no. 3, 757--765. Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Complexes, Families, moduli of curves (algebraic) Clifford indices of ribbons	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $M\subset\mathbb R^{n}$ be a real-analytic submanifold and ${\mathcal H}(M)$ the algebra of real analytic functions on $M$. If $K\subset M$ is a compact subset we consider $S_{K}=\{f\in {\mathcal H}(M) \mid f(x)\neq 0$ for all $x\in K\}$; $S_{K}$ is a multiplicative subset of ${\mathcal H}(M)$. Let $S_{K}^{-1}{\mathcal H}(M)$ be the localization of ${\mathcal H}(M)$ with respect to $S_{K}$. In this paper we prove, first, that $S_{K}^{-1}{\mathcal H}(M)$ is a regular ring (hence Noetherian) and use this result in two situations: 1) For each open subset ${\Omega}\subset\mathbb R^{n}$, we denote by ${\mathcal O}({\Omega})$ the subalgebra of ${\mathcal H}({\Omega})$ defined as follows: $f\in {\mathcal O}({\Omega})$ if and only if for all $x\in{\Omega}$, the germ of $f$ at $x$, $f_{x}$, is algebraic on ${\mathcal H}(\mathbb R^{n})$. We prove that if ${\Omega}$ is a bounded subanalytic subset, then ${\mathcal O}({\Omega})$ is a regular ring (hence Noetherian). 2) Let $M\subset \mathbb R^{n}$ be a Nash submanifold and ${\mathcal N}(M)$ the ring of Nash functions on $M$; we have an injection ${\mathcal N}(M) \rightarrow {\mathcal H}(M)$. In a former paper it was proved that every prime ideal $\wp$ of ${\mathcal N}(M)$ generates a prime ideal of analytic functions $\wp{\mathcal H}(M)$ if $M$ or $V(\wp)$ is compact. We use our Theorem 1 to give another proof in the situation where $V(\wp)$ is compact. Finally we show that this result holds in some particular situation where $M$ and $V(\wp)$ are not assumed to be compact. analytic algebra; Nash functions; subanalytic sets; regular rings Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets, Real-analytic sets, complex Nash functions Noetherianness of certain analytic function algebras and applications	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The moduli spaces ${\mathcal M}_{0,n}^\delta$ are partial compactifications of the moduli space of $n$-pointed non-singular rational curves, indexed by the choice of a dihedral structure $\delta$ on the set $\{1,\dots,n\}$ of markings. They were introduced in [\textit{F. C. S. Brown}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 3, 371--489 (2009; Zbl 1216.11079)] because they give a symmetric set of smooth affine charts on the moduli space $\overline{\mathcal M}_{0,n}$ of $n$-pointed stable rational curves. In this note, the authors investigate the structure of the cohomology of ${\mathcal M}_{0,n}^\delta$. The Betti number of these spaces in the maximal degree $n-3$ is specifically interesting in view of connections with the Lie operad and with multiple zeta values. As in the case of ${\mathcal M}_{0,n}$, over the field of complex numbers the $k$th cohomology group of these partial compactifications carries a pure Hodge structures of weight equal to $2k$. This ensures that the cohomology with rational coefficients is completely determined by the Poincaré polynomial. Exploiting the combinatorial relationship between the structure of ${\mathcal M}_{0,n}^\delta$ and Stasheff polynomials, it is shown that the generating series for the Poincaré polynomials of ${\mathcal M}_{0,n}^\delta$ is the inverse of the corresponding generating series for ${\mathcal M}_{0,n}$. As an application of the result, recurrence relations for the Poincaré polynomials and explicit formulas for the Betti numbers of ${\mathcal M}_{0,n}^{\delta}$ in small degree are given. Note that the inverse of the \textit{exponential} generating series of the Poincaré polynomial of ${\mathcal M}_{0,n}$ is known to coincide with the corresponding series for the Poincaré polynomial of $\overline{\mathcal M}_{0,n}$ by work of \textit{V. Ginzburg} and {it M. Kapranov} [Duke Math. J. 76, No. 1, 203--272 (1994; Zbl 0855.18006)] and \textit{E. Getzler} [Prog. Math. 129, 199--230 (1995; Zbl 0851.18005)]. moduli spaces; curves of genus 0; Poincaré polynomial; generating series; inversion of series; Lie operad; multiple zeta values Bergström, Jonas; Brown, Francis: Inversion of series and the cohomology of the moduli spaces M0,n$\delta $. Clay math. Proc. 12, 119-126 (2010) Families, moduli of curves (algebraic), Inversion of series and the cohomology of the moduli spaces ${\mathcal M}^\delta_{0,n}$	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 For a given rational parametrization ${\mathcal P}(\bar{t})$, $\bar{t}=(t_1,\ldots,t_r)$, of a variety in ${\mathbb K}^n$, the base points are the values of $\bar{t}$ such that all the numerators and denominators in the parametrization simultaneously vanish. The presence of base points is typically a source of difficulties in a number of geometric problems usually attacked by means of elimination methods; therefore, understanding their behavior is important. Additionally, the \textit{fibre} of a point $P\in {\mathbb K}^n$ is the set of $\bar{t}$'s such that ${\mathcal P}(\bar{t})=P$. Studying the cardinality of the fibre of a generic point is also important, because it is related to the number of times that ${\mathcal P}(\bar{t})$ covers the variety, and therefore with whether or not the parametrization is ``redundant'', i.e. with the optimality, in a certain sense, of the parametrization. Both base points and fibres are addressed in the paper. The content itself is very well described in the abstract of the paper: Given a rational parametrization ${\mathcal P}(\bar{t})$, $\bar{t}=(t_1,\ldots,t_r)$, of an $r$-dimensional unirational variety, we analyze the behavior of the variety of the base points of ${\mathcal P}(\bar{t})$ in connection to its generic fibre, when successively eliminating the parameters $t_i$. For this purpose, we introduce a sequence of of generalized resultants whose primitive and content parts contain the different components of the projected variety of the base points and the fibre. In addition, when the dimension of the base points is strictly smaller than one (as in the well-known cases of curves and surfaces), we show that the last element in the sequence of resultants is the univariate polynomial in the corresponding Gröbner basis of the ideal associated to the fibre; assuming that the ideal is in $t_1$-general position and radical. rational parametrization; unirational variety; degree of a rational map; fiber of a rational map; base points; generalized resultants Pérez-Díaz, S.; Sendra, J. R.: Behavior of the fiber and the base points of parametrizations under projections, Math. comput. Sci. 7, No. 2, 167-184 (2013) Computational aspects of higher-dimensional varieties, Computational aspects of algebraic surfaces, Symbolic computation and algebraic computation, Rational and birational maps Behavior of the fiber and the base points of parametrizations under 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 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 $S\subset\mathbb R^m$ that are either a polynomial or a regular image of some $\mathbb R^n$, that is, that can be represented as $S=f(\mathbb R^n)$ 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 $S$ (say, optimization, Positivstellensatz or computation of trajectories) can be reduced somehow to the case $S=\mathbb R^n$ if $S$ 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 $X\subset\mathbb R^n$ be a convex polyhedron and denote $\mathrm{Int}(X)$ 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 was proved that for every convex polygon $X\subset\mathbb R^2$ that is neither a line nor a band, the semialgebraic sets $\mathbb R^2\setminus X$ and $\mathbb R^2\setminus\mathrm{Int}(X)$ 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 $X$ and $\mathrm{Int}(X)$ 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 $X\subset\mathbb R^3$ which is neither a plane nor a layer both $\mathbb R^3\setminus X$ and $\mathbb R^3\setminus\mathrm{Int}(X)$ 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 $\mathbb R^n\setminus X$ or $\mathbb R^n\setminus\mathrm{Int}(X)$ as polynomial images of $\mathbb R^n$ for a general convex polyhedron $X\subset\mathbb R^n$. 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 $X\subset\mathbb R^n$ that is neither a hyperplane nor a layer the semialgebraic sets $\mathbb R^n\setminus X$ and $\mathbb R^n\setminus\mathrm{Int}(X)$ are polynomial images of $\mathbb R^n$ in case $X$ 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 $X\subset\mathbb R^n$ for which either $\mathbb R^n\setminus X$ or $\mathbb R^n\setminus\mathrm{Int}(X)$ are polynomial images of $\mathbb R^n$. The proof of the main result of the work under review is easier if $\dim(X)<n$. 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 $S\subsetneq\mathbb R^n$, the set $\mathbb R^{n+1}\setminus(S\times\{0\})$ is a polynomial image of $\mathbb R^{n+1}$ . The hardest part is to deal with $n$-dimensional polyhedra $X\subset\mathbb R^n$. 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 $X$ if it is unbounded. In the last section of this work the authors substitute $X$ by the closed ball $\mathbb B_n\subset\mathbb R^n$ (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 $\mathbb R^2\setminus\mathbb B_2$ is a polynomial image of $\mathbb R^3$ but it is not a polynomial image of $\mathbb R^2$. Moreover, $\mathbb R^2\setminus\mathrm{Int}(\mathbb B_2)$ 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 $\mathbb R^n\setminus\mathbb B_n$ is a polynomial image of $\mathbb R^{n+1}$ but it is not a polynomial image of $\mathbb R^n$. Moreover, they show that $\mathbb R^n\setminus\mathrm{Int}(\mathbb B_n)$ 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 the set of points at infinity of a polynomial image of $\mathbb{R}^n$, Discrete Comput. Geom., 52, 4, 583-611, (2014) Semialgebraic sets and related spaces, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Three-dimensional polytopes On the set of points at infinity of a polynomial image of $\mathbb R^n$	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this article is to investigate a codimension-two analogue of Ueda's theory. Given a complex manifold $X$ and a smooth hypersurface $S$ of $X$, let $C$ be a smooth compact hypersurface of $S$. The author gives a sufficient condition for the line bundle ${\mathcal{O}}_X(S)$ to be flat in a neighborhood of $C$ in $X$. Precisely, under the assumption on the normal bundle $N_{S/X}$ to be flat around $C$, the author defines an obstruction class \[ u_{n,m}(C,S,X)\in H^1(C, {{N_{S/X}}\|}_C^{-n}\oplus N_{C/S}^{-m}) \] for each $n\geq 1$, $m\geq 0$ such that ${\mathcal{O}}_X(S)$ is not flat around $C$ if there exist $n\geq 1$, $m\geq 0$ with $u_{n,m}(C,S,X)\neq 0$. In particular if $C$ is a smooth compact Kähler hypersurface of $S$ such that ${{N_{S/X}}\|}V$ is flat, where $V$ is a small neighborhood of $C$ in $S$, under some additional assumption on $N_{C/S}$ and ${N_{S/X}}\|{}_C$, if $u_{n,m}(C,S,X)= 0$ for all $n\geq 1$ and $m\geq 0$ then there exists a neighborhood $U$ of $C$ in $X$ such that ${\mathcal{O}}_X(S)\| U $ is flat. As an application the author gives a sufficient condition for the anticanonical bundle of the blow-up of $\mathbb{CP}^3$ at $8$ points to be semi-ample. flat line bundles; Ueda's theory; blow-up of the three-dimensional projective space at eight points Koike, T, Toward a higher codimensional ueda theory, Math. Z., 281, 967-991, (2015) Transcendental methods of algebraic geometry (complex-analytic aspects), Divisors, linear systems, invertible sheaves Toward a higher codimensional Ueda theory	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [\textit{E. Mukhin} and \textit{A. Varchenko}, Commun. Contemp. Math. 6, No. 1, 111--163 (2004; Zbl 1050.17022)], some correspondences were defined between critical points of master functions associated to $sl_{N+1}$ and subspaces of $\mathbb{C}[x]$ with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus. Belkale P, Mukhin E, Varchenko A (2006) Multiplicity of critical points of master functions and Schubert calculus. In: Bertrand D, Enriquez B, Mitschi C, Sabbah C, Schäfke R (eds) Differential equations and quantum groups: Andrey A Bolibrukh memorial volume. IRMA Lect Math Theor Phys, vol 9. Europ Math Soc Publ, pp 59--84 Classical problems, Schubert calculus, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Exactly solvable models; Bethe ansatz Multiplicity of critical points of master functions and Schubert 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 study smooth manifolds defined by simple convex polytopes. With every $n$-dimensional simple polytope $P$ with $m$ facets is associated a smooth $(m+n)$-dimensional manifold ${\mathcal Z}_P$ that admits a canonical action of the compact $m$-torus $T^m$. Many manifolds that are important in topology, algebra and symplectic geometry arise in this context as quotients of manifolds ${\mathcal Z}_P$ by toric subgroups $T^k$ of $T^m$ which act freely on ${\mathcal Z}_P$. The class of quasitoric manifolds is obtained when $k=m-n$, which is the largest possible value for $k$; this class includes all smooth projective toric varieties (toric manifolds). The authors take the following approach to constructing these manifolds. The combinatorial structure of $P$ determines a certain collection of affine planes in $\mathbb{C}^m$, whose complement $U(P)$ in $\mathbb{C}^m$ admits an action by $(\mathbb{C}^)^m$. It is always possible to find certain subgroups $R$ of $(\mathbb{C}^)^m$ which are isomorphic to $(\mathbb{R}^*_+)^{m-n}$ and act freely on $U(P)$. The manifolds then are the quotients of $U(P)$ by such subgroups. In particular, the authors investigate the relationships between the combinatorics of simple polytopes and the topology of the manifolds associated with them. It is proved that the cohomology of ${\mathcal Z}_P$ possesses a natural structure as a bigraded algebra. This bigraded cohomology algebra carries all the information about the combinatorics of $P$. For example, there are appropriate interpretations of the Dehn-Sommerville equations and the upper-bound-theorem for simple convex polytopes. toric manifolds; simple convex polytopes; toric varieties; cohomology algebra Buchstaber V. M. and Panov T. E., ''Torus actions and combinatorics of polytopes,'' Proc. Steklov Inst. Math., 225, 87--120 (1999). Polyhedral manifolds, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies, Compact groups of homeomorphisms Torus actions and combinatorics of 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 It is a well-known fact that a connected component and the closure of a semialgebraic set are again semialgebraic sets and that a similar statement holds true for semianalytic sets. The author considers the problem of whether a union of connected components of a global semianalytic set is again global semianalytic. The answer to this problem was known to be positive for boundary bounded global semianalytic sets, see the book of \textit{C. Andradas, L. Bröcker} and \textit{J. M. Ruiz} [Constructible sets in real geometry. Ergebn. Math. Grenzgeb. 3, 33 (1996; Zbl 0873.14044)], and for global semianalytic subsets of a connected paracompact real manifold of dimension two, see [\textit{A. Castilla} and \textit{C. Andradas}, J. Reine Angew. Math. 475, 137--148 (1996; Zbl 0862.32003)]. In this paper it is proved that a union of connected components of a global semianalytic subset $X$ of a coherent analytic subset of dimension two with affine normalization is again a global semianalytic subset. It must be noticed that a different proof of the same statement has been given by \textit{C. Andradas} and the reviewer [Ill. J. Math. 48, No.~2, 519--537 (2004; Zbl 1077.14087)]. The author also proves partial results for connected components of a global semianalytic subset of a three-dimensional analytic manifold. In order to get the main results the author proves some useful technical lemmas to deal with semianalytic subsets and also an interesting generalization of Thom's lemma for convergent power series. Global semianalytic sets; coherent analytic surfaces Fujita, M, On the connected components of a global semianalytic subset of an analytic surface, Hokkaido Math. J., 35, 155-179, (2006) Real-analytic and semi-analytic sets, Real algebra On the connected components of a global semianalytic subset of an analytic surface	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $f:{\mathbb C}^2 \to \mathbb C$ be a nonconstant polynomial in two complex variables with a finite set of critical points; let $C^{ t}$ be the projective closure of the fiber $f^{-1}(t)$, let $L_{ \infty}$ in ${\mathbb P}^2 ({\mathbb C})$ be the line at infinity and let $C_{ \infty}= C^{ t} \cap L_{ \infty}$. The set $\Lambda(f) = \{ t \in {\mathbb C}:\mu_{p}^t > \mu_{p}^{min} \;\text{ for a } p \in C_{ \infty} \}$, where $\mu_{p}^t = \mu_{p}(C^t)$ is the Milnor number and $\mu_{p}^{min}= \inf_{t \in {\mathbb C}} \mu_{p}^t$, is called the set of irregular values of $f$ at infinity. In this note the authors characterize polynomials $f$ with no critical points and one irregular value at infinity: improving a recent result by \textit{A. Assi} [see Math. Z. 230, No. 1, 165--183 (1999; Zbl 0934.32017)], they give a description of the irregular fiber of such a polynomial. This result is applied to the estimation of the number of points at infinity of a polynomial with no critical points and at most one irregular value at infinity. The authors give also a discriminant criterion for polynomials to have one irregular value and present a list of open questions. affine curves; irregular value J. Gwozdziewicz and A. Ploski, On the singularities at infinity of plane algebraic curves, Rocky Mountain J. Math. 32 (2002), 139--148. Singularities of curves, local rings, Milnor fibration; relations with knot theory, Singularities in algebraic geometry On the singularities at infinity of plane 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 Nash functions are analytic functions which are algebraic over the polynomials. The paper under review is an excellent survey on some problems concerning Nash functions and the methods for their solutions. The problems consider here are: separation, factorization, global equations and extension. To be more precise, the separation problem asks whether a prime ideal of Nash functions on a Nash manifold $M \subset \mathbb R^n$ has a prime extension to the ring of global analytic functions on $M$. In the factorization problem the question is whether a Nash function $f$ with a factorization $f = f_1 f_2$ as a product of analytic functions has a similar factorization $f = g_1 g_2$ as a product of Nash functions. Finally, whether a finite ideal sheaf is generated by its global sections and whether every Nash function on $M$ is the restriction of a Nash function on $\mathbb R^n$ to $M$ are the global equation and the extension problems, respectively. In the case of a compact Nash manifold $M$, using the so-called Néron desingularization an important approximation theorem is proved and then these problems are solved in the affirmative [see the paper of \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota}, Am. J. Math. 117, No. 4, 905-927 (1995; Zbl 0873.32007)]. When $M$ is not necessarily compact \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [Compos. Math. 103, No. 1, 31-62 (1996; Zbl 0885.14029)] showed that these problems are equivalent. After that \textit{M. Coste} and \textit{M. Shiota} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, No. 1, 139-149 (2000; Zbl 0981.14027)] solved the global equation problem. The proof of \textit{M. Coste} and \textit{M. Shiota} relies in a semialgebraic version of Thom's first isotopy lemma proved by these two authors. The same problems have been considered in a more general setting, namely, replacing the field of real numbers by an arbitrary real closed field. Again, these problems have been answered in the affirmative by \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [J. Reine Angew. Math. 536, 209-235 (2001; Zbl 0981.14028)]. In that paper the authors, using the Tarski-Seidenberg theorem and proving uniform bounds on the complexities of Nash functions, obtain the solution to the already mentioned problems. Also, they prove the idempotency of the real spectrum and discuss conditions for the rings of abstract Nash functions to be noetherian. approximation theorem; Nash manifolds; separation problem; factorization problem; extension problem; global equations; complexity of Nash functions Nash functions and manifolds, Real-analytic sets, complex Nash functions, Real-analytic and Nash manifolds Global problems of Nash functions	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $K$, $K_\infty$, $A$, $C$ and $\Gamma$ be respectively a global field $K$ of positive characteristic, its completion $K_\infty$ at a fixed place $\infty$, the ring $A$ of elements of $K$ regular outside $\infty$, the completion $C$ of an algebraic closure of $K_\infty$ and an arithmetic subgroup $\Gamma$ of $GL_2 (K)$, i.e. a subgroup commensurable with $GL_2 (A)$. Set $\Omega= C-K_\infty$. This is a Drinfeld upper half-plane, $\Gamma$ acts by fractional linear transformations on $\Omega$ and the rigid analytic space $M_\Gamma= \Gamma\setminus \Omega$ is indeed an affine curve over $C$; this is a Drinfeld modular curve. These curves, for various $\Gamma$, are the substitutes in positive characteristic of the classical modular curves. Let $\overline{M}_\Gamma$ be the canonical completion of $M_\Gamma$. The author studies divisors on $\overline{M}_\Gamma$ with support in the set of cusps, i.e. in $\overline{M}_\Gamma- M_\Gamma$ (cuspidal divisors). To use analytic methods [as in \textit{E.-U. Gekeler} and \textit{M. Reversat}, J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)], in \S 2, the author generalizes the theory of theta functions developed in (loc. cit.) to ``degenerate parameters''. One knows that the period lattice (in the sense of Manin-Drinfeld) of $\overline{M}_\Gamma$ is described by a set of harmonic cochains or a set of theta functions (loc. cit.). The author adds to that interpretation the description of the links between new theta functions, cuspidal divisors and harmonic cochains (\S 3). In \S 4, where $\Gamma$ is assumed to be a congruence subgroup, the author studies, with the help of the preceding methods, the canonical map between the group ${\mathcal C}(\Gamma)$ generated in the Jacobian ${\mathcal J}_\Gamma$ of $\overline{M}_\Gamma$ by the cusps (the group ${\mathcal C}(\Gamma)$ is finite here), and the group $\Phi_\infty (\Gamma)$ of connected components of the Néron model of ${\mathcal J}_\Gamma$ at $\infty$. Finally (\S 5), in the case of $A= \mathbb{F}_q [T]$ and for a Hecke congruence subgroup, the author obtains more information about his new theta functions and the (eventually non-empty) kernel of the map ${\mathcal C}(\Gamma)\to \Phi_\infty(\Gamma)$. class group; Drinfeld upper half-plane; Drinfeld modular curve; theta functions; cuspidal divisors; harmonic cochains; congruence subgroup E.-U. Gekeler,On the cuspidal divisor class group of a Drinfeld modular curve, Documenta Mathematica. Journal der Deutschen Mathematiker-Vereinigung2 (1997), 351--374. Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties, Holomorphic modular forms of integral weight, Modular and Shimura varieties On the cuspidal divisor class group of a Drinfeld modular curve	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers the Dirichlet series $s\mapsto Z(P;s) =\sum\limits_{m\in{\mathbb N}^{n}}P(m)^{-s} (s\in{\mathbb C})$ where $P\in{\mathbb R}[X_1,...,X_n]$. Say that $Z(P;s)$ exists if this multiple series is absolutely convergent. In this paper he studies meromorphic continuations of such series, under the assumptions that there exists a constant $B\in]0,1[$ such that: i) $P(x)\to +\infty$ when $\|x\|\to + \infty$ and $x \in [B,+\infty[^n$ and ii) $d(Z(P),\;[B,+\infty[^n)>0$ where $Z(P)=\{ z \in{\mathbb C}^n\mid P(z)=0\}$. This assumption is probably optimal, and in any way strictly includes all classes of polynomials previously treated. Under this assumption, he proves the existence of meromorphic continuations of Dirichlet series and gives a set of candidate poles and an upper bound to the orders of these poles. Moreover the author obtains bounds for these meromorphic continuations on vertical bands. As an application, he shows the existence of a finite asymptotic expansion of the counting function: \[ N_P(t)=\#\{ m \in {\mathbb N}^{n}\mid P(m)\leq t \} \text{ when } t\to + \infty. \] Dirichlet series; meromorphic continuation; integral representation; resolution of singularities; semi-algebraic set Essouabri, D., Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres, Ann. inst. Fourier (Grenoble), 47, 429-484, (1996) Other Dirichlet series and zeta functions, Lattice points in specified regions, Semialgebraic sets and related spaces, Meromorphic functions of several complex variables, Integral representations; canonical kernels (Szegő, Bergman, etc.), Singularities in algebraic geometry Singularités de séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres. (Singularities of Dirichlet series associated to polynomials of several variables and applications to analytic number 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 biregular geometry of punctual Hilbert schemes in dimensions 2 and 3, i.e. of schemes parametrizing fixed-length zero-dimensionaI subschemes supported at a given point on a smooth surface or a smooth three-dimensional variety, is studied. A precise biregular description of these schemes has only been known for the trivial cases of lengths 3 and 4 in dimension 2. The next case of length 5 in dimension 2 and the two first nontrivial cases of lengths 3 and 4 in dimension 3 are considered. A detailed description of the biregular properties of punctual Hilbert schemes and of their natural desingularizations by varieties of complete punctual flags is given. Stein expansion; Briançon classification; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes) Punctual Hilbert schemes of small length in dimensions 2 and 3	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a $C^{\infty}$ function $\phi:\quad M\to {\mathbb{R}}$ (where M is a real algebraic manifold) the following problem is studied. If $\phi^{- 1}(0)$ is an algebraic subvariety of M, can $\phi$ be approximated by rational regular functions f such that $f^{-1}(0)=\phi^{-1}(0)?$ We find that this is possible if and only if there exists a rational regular function $g:\quad M\to {\mathbb{R}}$ such that $g^{-1}(0)=\phi^{- 1}(0)$ and g(x)$\cdot \phi (x)\geq 0$ for any x in ${\mathbb{R}}^ n$. Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus is also studied. approximation of $C^{\infty }$-functions by regular functions; algebraic manifold; zero set Broglia, Ann. Inst. Fourier, Grenoble 39 pp 611-- (1989) Real algebraic and real-analytic geometry Approximation of $C^{\infty}$-functions without changing their zero-set	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $X$ be a quasi-projective scheme over a field $k$, $M(X)$ the category of coherent sheaves on $X$ and $M_p(X)$ be subcategory of sheaves supported on a closed set of dimension at most $p$. The inclusion functor $M(X)\to M_p(X)$ induces a homomorphism $F_{n,p}: K_n(M_p(X))\to G_n(X)= K_n(M(X))$. The image $G_n(X)_{(p)}$ of $F_{n,p}$ is the $p$th term of the topological filtration on $G_n(X)$. Let's denote by $G_n(X)_{(p/p-1)}$ the subsequent factors of the filtration. The niveau spectral sequence \[ E^1_{p,q}= \coprod_{x\in X_p} K_{p+q}(k(X))\Rightarrow G_{p+q}(X)\tag{1} \] (where $X_p$ is the set of points of $X$ of dimension $p$) yields a surjective homomorphism \[ \phi_p: CH_p(X)= E^2_{p,-p}\to G_0(X)_{(p/p-1)} \] whose kernel is detected by the differentials of the spectral sequence arriving at $E^*_{p,-p}$. In this paper the author proves the following constraints for the order of differentials: Theorem 2. Let $\delta: E^s_{p,q}\to E^s_{p-s,q+s-1}$ be the differential in the spectral sequence (1) with $p+ q\leq 2$. The order $\text{ord}(\delta)$ is finite and, if $l$ is a prime divisor of $\text{ord}(\delta)$, then $l\leq p$ and $l-1$ divides $s-1$. Merkurjev, A., \textit{Adams operations and the Brown-Gersten-Quillen spectral sequence}, Quadratic forms, linear algebraic groups, and cohomology, vol. 18, 305-313, (2010), Springer, New York Applications of methods of algebraic $K$-theory in algebraic geometry, $K$-theory of schemes, $J$-homomorphism, Adams operations Adams operations and the Brown-Gersten-Quillen 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 Let $A\subset\mathbb{Z}^{n-1}$ be a finite subset, $\mathbb{C}^ A$ the linear $\mathbb{C}$-space of Laurent polynomials $f=\sum_{\omega\in A}a_ \omega X^ \omega$, $a_ \omega\in\mathbb{C}$ in some indeterminates $X$ and $\nabla_ 0\subset\mathbb{C}^ A$ the set of those $f$ for which there is $\kappa\in(\mathbb{C}^)^{n-1}$ such that $f(\kappa)=(\partial f/\partial X_ i)(\kappa)=0$ for all $i$. The closure $\nabla_ A$ of $\nabla_ 0$ is an irreducible variety defined in fact on $\mathbb{Z}$. When $\nabla_ A$ has codimension 1 then an irreducible polynomial $\Delta_ A\in\mathbb{Z}[a_ \omega;\omega\in A]$, which is zero on $\nabla_ A$, is unique up to the sign and it is called the $A$-discriminant. If $\text{codim}(\nabla_ A)>1$ then put $\Delta_ A=1$. The $A$- discriminant is homogeneous and satisfies the following quasi homogeneous $(n-1)$-conditions: ``$\sum_{\omega\in A}m(\omega)\cdot\omega\in\mathbb{Z}^{n-1}$ is constant for all monomials $\prod_{\omega\in A}a_ \omega^{m(\omega)}$ which enter in $\Delta_ A$''. This notion extends the classical notions of discriminant and resultant. --- Let $A=\{\omega_ 1,\ldots,\omega_ N\}$ and $Y_ A$ be the closure of the set $\{(\kappa^{\omega_ 1},\ldots,\kappa^{\omega_ N}\mid\kappa\in\mathbb{C}^{n-1}\}$ in $\mathbb{P}^{N-1}$. Then $\nabla_ A$ and $Y_ A$ are dual projective varieties and the description of $\Delta_ A$ follows if we can describe the equations of the dual projective variety of a given projective one $Y\subset\mathbb{P}^{N-1}$ [see the authors' previous paper in Sov. Math., Dokl. 39, No. 2, 385-389 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 6, 1294-1298 (1989; Zbl 0715.14042)]. Let $G$ be a free abelian group of rank $n$, $G_ \mathbb{C}:=\mathbb{C}\otimes_ \mathbb{Z} G$, $\lambda:G\to(\mathbb{Q},+)$ a nonzero group morphism, $S\subset G$ a finitely generated semigroup such that $o\in S$ and $\lambda(s)\geq 1$ for all $s\in S$, $S_ e=\{t\in S\mid\lambda(t)=e\}$ for $e\in\mathbb{Q}$ and $A\subset S_ 1$ a finite subset generating in $G_ \mathbb{R}=\mathbb{R}\otimes_ \mathbb{Z} G$ the same convex cone as $S$. For $k\in\mathbb{Z}_ +$, $e\in\mathbb{Q}$, $\omega\in A$ let $\bigwedge^ k(e)$ be the space of all maps $S_{k+e}\to\bigwedge^ kG_ \mathbb{C}$, $\partial_ \omega:\bigwedge^ k(e)\to\bigwedge^{k+1}(e)$ the map given by $\partial_ \omega(\gamma)(u)=\omega\wedge\gamma(u-\omega)$, if $u-\omega\in S_{k+e}$, otherwise $\partial_ \omega(\gamma)(u)=0$ and $\partial_ f=\sum_{\omega\in A}a_ \omega\partial_ \omega$ if $f=\sum a_ \omega X^ \omega\in\mathbb{C}^ A$. The complex $(\overset{.}\bigwedge (e),\partial_ f)$ is called the Cayley-Koszul complex. --- Choose a basis $u$ in terms of $\overset{.}\bigwedge(e)$ and let $E_ e(f)$ be the determinant of the complex $(\overset{.}\bigwedge (e),\partial_ f)$ with respect to $u$ [see \textit{F. Fischer}, Math. Z. 26, 497-550 (1927) or \textit{J.-K. Bismut} and \textit{D. S. Freed}, Commun. Math. Phys. 106, 159-176 (1986; Zbl 0657.58037)]. For $e$ sufficiently high $E_ A(f):=E_ e(f)$ is a polynomial of $(a_ \omega)$, $f=\sum a_ \omega X^ \omega$ which depends on $e$ only by a constant multiple. --- Let $Q_ A$ be the convex closure of $A$ in $G_ \mathbb{R}$. If $f=\sum a_ \omega X^ \omega$ we can express $E_ A(f)=\sum_ \varphi c_ \varphi\prod_{\omega\in A}a_ \omega^{\varphi(\omega)}$, where $\varphi$ runs in the set $\mathbb{Z}^ A_ +$ of the maps $A\to\mathbb{Z}_ +$. Let $M(E_ A)\subset\mathbb{R}^ A$ be the convex closure of those $\varphi\in\mathbb{Z}^ A_ +$ for which $c_ \varphi\neq 0$. Then there exists a nice correspondence between the vertices of $M(E_ A)$ and some special triangulations of $Q_ A$. The theory is applied to the following examples: the discriminant of a polynomial in two indeterminates, the resultant of two quadratric polynomials, the elliptic curve in Tate normal form\dots Laurent polynomials; Cayley-Koszul complex; determinant; discriminant of a polynomial in two indeterminates; elliptic curve Gel'fand, I.; Zelevinskiǐand, A.; Kapranov, M., Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz, 2, 1, (1990) Toric varieties, Newton polyhedra, Okounkov bodies, Polynomial rings and ideals; rings of integer-valued polynomials, Complexes, Determinantal varieties Discriminants of polynomials in several variables and triangulations of Newton polyhedra	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies linear systems of curves defined by fixed points of given multiplicities on Hirzebruch surfaces ${{\mathbb F}_n}$. If $\operatorname{Pic}{\mathbb {F}_n} = \langle H,F\rangle$, where $F$ is a fiber and $H^2=n$, $F\cdot H=1$, the linear systems under consideration are of type ${\mathcal L}(a,b,m_1,\dots,m_s)$, i.e. the subsystem of the complete linear system $\| aF+bH\|$ given by the curves having multiplicity at least $m_i$ at $P_i$, for $s$ generic points $P_1,\dots ,P_s$. In analogy with what happens in ${\mathbb P}^2$, the author conjectures that such linear systems are special (i.e. their dimension is bigger than expected) if and only if they are $(-1)$-special, i.e. there is a $(-1)$-curve $E$ such that the base locus of ${\mathcal L}(a,b,m_1,\dots,m_s)$ contains $\alpha \Gamma_n + tE$, $t\geq 2$, $\alpha \geq 0$. Here $\Gamma_n\in \| H-nF\|$ is the curve with $\Gamma ^2=-n$, while a $(-1)$-curve $E$ is an irreducible curve such that $E^2=-1=K_{{\mathbb F}_n}\cdot E$. In the paper the conjecture is proved for $m_1=\cdots =m_s\leq 3$; the main idea of the proof is to list all $(-1)$-special systems with multiplicities $m_i\leq 3$ via birational transformations ${\mathbb F}_n \rightarrow {\mathbb F}_{n-1}$ (a generalization of quadratic transformations in ${\mathbb P}^2$) and then to work with suitable deformations of ${\mathbb F}_n$. fat points; Hirzebruch surfaces; $(-1)$-special A. Laface: ''On linear systems of curves on rational scrolls'', Geom. Dedicata, Vol. 90, (2002), pp. 127--144. Divisors, linear systems, invertible sheaves, Birational automorphisms, Cremona group and generalizations, Rational and ruled surfaces On linear systems of curves on rational scrolls	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $V_{g}$ be a universal space of Jacobian varieties of flat hyperelliptic curves in the form \[ V(\mu,\nu)={(\mu,\nu)\in\mathbb{C}^{2}: \nu^{2}-f_{g}(\mu)=0}, \] where $f_{g}(\mu)=4\mu^{2g+1}+\sum_{i=0}^{2g-1}\mu^{i}\lambda_{i}, g=1,2,\dots$. Let us introduce on $V_{g}$ the coordinates $(u,\lambda)=(u_{1},u_{2},\dots,u_{g},\lambda_{0}, \lambda_{1},\dots,\lambda_{2g-1}),$ where the $u$ are coordinates on the Jacobian variety, related with the canonical basis of holomorphic differentials on curves $V(\mu,\nu)$. Consider the system of functional equations \[ \varphi(u,\omega d+\omega'c,\omega b+\omega'a)=\varphi(u,\omega,\omega'),\tag{1} \] \[ \begin{multlined} \varphi(u+\omega m+\omega'm',\omega,\omega') =\\ \exp[2(u+\omega m+\omega'm')(\eta m+\zeta'm')^{t}]\varphi(u,\omega,\omega'),\end{multlined}\tag{2} \] where $m,m'\in\mathbb{Z}^{g}$, $\left(\begin{smallmatrix} a&b\\ c&d\end{smallmatrix}\right)\in \text{Sp}(2g,\mathbb{Z})$, and the $(g\times g)$-matrices $\omega,\omega',\eta,\eta '$ are functions of $\lambda$ given with the help of curvilinear integrals on the curve $V(\mu,\nu)$. Then the hyperelliptic $\sigma$-function is called a continuation for special solutions $\varphi_{0}(u,\omega,\omega')$ for the system of equations (1)-(2) on all the spaces $V_g$. Let $U$ be an algebra of linear differential operators on $u_1,\dots,u_g$ and $\lambda_{2g-1},\dots,\lambda_0$, the $U(\sigma)$ annihilator ideal for the $\sigma$-function, that is, the subalgebra of operators $L\in U$ such that $L\sigma(u,\lambda)\equiv 0$. Lemma 1. (a) The ideal $U(\sigma)$ is generated by $2g$ generators $l_{j}$, $j = 0, 1,\dots , 2g -1$. (b) The operators $l_{j}$ are homogeneous for $\deg u_{i} = 2(g - i) + 1,$ $\deg \lambda_{k} = -2(2g - k) - 2,$ where $\deg l_{j} = -2j$. (c) The differential operators $l_{j}$, $j > 0,$ are of second order in $u$ and of first order in $\lambda,$ and the operator $l_{0}$ is up to multiplication by a constant, equal to \[ \sum_{i=1}^{g}(2(g - i) + 1)u_{i} \partial_{u_{i}} - \sum_{k=0}^{2g-1} 2(2g - k + 1) \lambda_{k}\partial_{\lambda_{k}} - g(g + 1)/2. \] (d) The coefficients of operators $l_{j}$ are polynomials. The main result of this paper is Theorem 2. Graded generators $l_{i}$ of the ideal $U(\sigma)$ and the operators of multiplication by $\lambda_{j}$ generate a graded Lie algebra. The authors note that (1) the hyperelliptic $\sigma$-functions were introduced by F. Klein, as a generalization of the elliptic $\sigma$-function of Weierstrass; (2) Annihilator's ideal of the Weierstrass $\sigma$-function was studied by K. Weierstrass himself; (3) H. F. Baker proposed and developed Klein's construction for the case $g=2$. Families, moduli of curves (analytic), Theta functions and curves; Schottky problem, Elliptic curves, Graded Lie (super)algebras Graded Lie algebras that define hyperelliptic sigma 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 In this article we introduce a stability concept for the coercivity of multivariate polynomials $f\in\mathbb R [x]$. In particular, we consider perturbations of $f$ by polynomials up to the so-called degree of stable coercivity, and we analyze this stability concept in terms of the corresponding Newton polytopes at infinity. For coercive polynomials $f\in\mathbb R [x]$ we also introduce the order of coercivity as a measure expressing the order of growth of $f$, and we identify a broad class of multivariate polynomials $f\in\mathbb R [x]$ for which the order of coercivity and the degree of stable coercivity coincide. For these polynomials we give a geometric interpretation of this phenomenon in terms of their Newton polytopes at infinity, which we call the degree of convenience. We relate our results to the existing literature and we illustrate them with some examples. As applications we show that the gradient maps corresponding to a broad class of polynomials are always subjective, we establish Hölder type global error bounds for such polynomials, and we link our results to the existence of solutions in the calculus of variations. Newton polytope; coercivity; stability; order of growth; error bound; convenient polynomials Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic sets, Real polynomials: analytic properties, etc., Newton-type methods Coercive polynomials: stability, order of growth, and Newton polytopes	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a method to desingularize every three-dimensional toric variety $X = X_ \Delta$, where $\Delta$ is a simplicial fan. Letting $\Delta^{(3)}$ denote the set of three-dimensional cones of $\Delta$, each cone $\sigma \in \Delta^{(3)}$ of multiplicity $\text{mult} (\sigma) > 1$ is starred at a nonzero primitive vector $p_ \sigma \in \sigma \cap \mathbb{Z}^ 3$ such that the sum of the multiplicities of the cones thus obtained is the smallest possible. In this way, after a finite number of steps one obtains a fan $\nabla (\Delta)$ yielding a nonsingular subdivision of $\Delta$. Our algorithm generalizes the well known construction of the coarsest nonsingular subdivision of a fan in $\mathbb{Z}^ 2$ as given by the Hirzebruch-Jung continued fraction algorithm. For each three-dimensional toric variety $X = X_ \Delta$, we thus obtain a desingularization $X' = X_{\nabla (\Delta)}$ of $X$ whose Euler characteristic $E(X')$ satisfies the inequality $E(X') \leq - E(X) + 2 \sum_{\sigma \in \Delta^{(3)}} \text{mult} (\sigma)$. It is shown that this upper bound is best possible. three-dimensional toric variety; subdivision of fan; desingularization; Euler characteristic Aguzzoli, S.; Mundici, D., An algorithmic desingularization of 3-dimensional toric varieties, \textit{Tohoku Math. J.}, 46, 4, 557-572, (1994) Computational aspects of higher-dimensional varieties, $3$-folds, Toric varieties, Newton polyhedra, Okounkov bodies An algorithmic desingularization of 3-dimensional toric varieties	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $S:=K[x_0,\dots,x_n]$ be a polynomial ring over an algebraically closed field $K$, and let $I$ be a homogeneous ideal of $S$ defining a subscheme ${\mathfrak X}$ of projective $n$-space $\mathbb{P}^n_K$. The Castelnuovo-Mumford regularity (or simply regularity) of $I$, $\text{reg} I$, is defined as follows: If \[ 0\to \bigoplus^{\beta_p}_{j=1} S(-e_{pj}) @>\varphi_p>> \cdots @>\varphi_1>> \bigoplus^{\beta_0}_{j=1}S(-e_{0 j}) @>\varphi_0>> I\to 0\tag{0.1} \] is a minimal graded free resolution of $I$, setting $e_i:=\max\{e_{ij};\;1\leq j\leq\beta_i\}$, then $\text{reg} I:= \max \{e_i-i;\;0\leq i\leq p\}$. In other words, $\text{reg} I$ is the smallest integer $m$ for which $I$ is $m$-regular, i.e., $e_{ij}\leq m+i$ for all $i,j$. When $I$ is saturated (i.e., when it is the largest ideal defining ${\mathfrak X})$, we call this the regularity of ${\mathfrak X}$. -- The regularity is a numerical invariant of the ideal $I$ and is, ``an important measure of how hard it will be to compute a free resolution''. We give an effective method to compute the regularity of a saturated ideal $I$ defining a projective curve that also determines in which step of a minimal graded free resolution of $I$ the regularity is attained. Buchberger's syzygy algorithm; polynomial ring; Castelnuovo-Mumford regularity; free resolution Bermejo, I.; Gimenez, P.: On Castelnuovo -- Mumford regularity of projective curves. Proc. amer. Math. soc. 128, 1293-1298 (2000) Syzygies, resolutions, complexes and commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects of algebraic curves On Castelnuovo-Mumford regularity of projective 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 $\mathcal{P} = \{P_{1},\dots, P_{s}\} \subset \mathbb{P}^{2}_{\mathbb{K}}$ be a finte set of points and denote by $I = m_{1}P_{1} + \dots +m_{s}P_{s}$ the associated fat point scheme, where $\mathbb{K}$ is an algebraically closed field of charcteristic $0$. For a homogeneous ideal $I$ we define the initial degree $\alpha(I)$ to be the least integer $t$ such that $I_{t} \neq 0$. The Waldschmidt constant of $I$ is defined as \[ \widehat{\alpha}(I) = \lim_{m \rightarrow \infty} \frac{\alpha(I^{(m)})}{m}, \] where $I^{(m)}$ is the $m$-th symbolic power of $I$. In the present paper the authors provide a complete classification of fat point schemes of the form $I = m_{1}P_{1} +\dots+ m_{s}P_{s}$ for which the Waldschmidt constant $\widehat{\alpha}(I)$ is strictly less than $\frac{5}{2}$. Let us define the following fat subschemes: i) $Z$ which consists of three simple points, plus one double point, such that the simple points lie on a line $L_1$ and the double point lies on another line $L_2$. Formally, we denote it as $Z = 2q + p_1 + p_2 + p_3$. ii) $W$ which consists of $r+s$ simple points, plus one double point, where $r,s \geq 1$, such that $r$ simple points out of them lie on a line $L_1$, and $s$ simple points out of them lie on another line $L_2$, and the single double point lies at the intersection of these two lines. We denote it as $W = 2q +q_1 + q_2 +\cdots+ q_s + p_1 + p_2 +\cdots+ p_r$. iii) $T$ which consists of $r$ double points, plus $s$ simple points, where $r \geq 1$ and $s \geq 0$, such that all of these points are contained in a single line. We denote it as $T = 2p_1 +\cdots+ 2p_r + q_1 + q_2 +\cdots+ q_s$. Main Theorem. Let $\mathcal{A}$ be a fat point subscheme of $\mathbb{P}^{2}_{\mathbb{K}}$ with defining ideal $I$, then $\widehat{\alpha}(I) < \frac{5}{2}$ if and only if $\mathcal{A}$ is one of the fat point subschemes $Z$, $W$, or $T$. fat points; Waldschmidt constant; symbolic power; star configuration; configuration of points Configurations and arrangements of linear subspaces, Graded rings, Projective techniques in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials All fat point subschemes in $\mathbb{P}^2$ with the Waldschmidt constant less than 5/2	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $\phi : \mathbb P^1 \to \mathbb P^1$ be a rational map defined over a field $K$, let $N$ be a positive integer. A point $P\in \mathbb P^1$ is called periodic of primitive period $N$ if $N$ is the smallest number such that $\phi^N(P)=P$. The notion of a point of formal period $N$ is a slight generalization of this notion. The points of formal period $N$ are by definition the roots of the so called $N$th dynatomic polynomial for $\phi$. Let $M_d$ denote the moduli space parameterizing the rational maps of degree $d$ up to coordinate change, i.~e., up to conjugacy by elements from $\mathrm{PGL}_2(\overline K)$. The author constructs the moduli space $M_d(N)$ parameterizing conjugacy classes of degree-$d$ maps together with a point of formal period $N$. One of the main results of the paper is Theorem~4.5 stating that $M_{2}(N)$ is geometrically irreducible for $N>1$. For a rational map $\phi$ an element $h \in \mathrm{PGL}_2(\overline K)$ is called an automorphism of $\phi$ if $h^{ - 1}\circ \phi \circ h=\phi$. Let $p$ be a prime number, let $\mathfrak C_p$ denote the cyclic group of order $p$. Then the author considers the moduli space $M_d(N, \mathfrak C_p)$ of rational maps of degree $d$ with a point of formal period $N$ having an automorphism of order $p$. Another important result of the paper is Theorem~7.1, whose essence is expressed by Corollary~7.10 saying that for $d\geq 2$ the moduli space $M_d(pN, \mathfrak C_p)$ is reducible for all but finitely many prime integers $N$. In the case of the field of zero characteristic it is reducible for all but finitely many positive integers $N$. In particular the curves $M_2(2N, \mathfrak C_2)$ are reducible for infinitely many $N$. The paper consists of 9 sections. Section~1 is an introduction. An overview of the paper and its main results are given here. Some important notions and objects ($N$th dynatomic polynomial, points of formal period $N$, moduli space $M_d$, etc.) are defined and described in Section~2. In Section~3 a construction of moduli spaces with level structures is presented. An algebraic proof of the irreducibility of $M_2(N)$ is given in Section~4 (Theorem~4.5). Section~5 deals with maps with automorphisms. In particular for a rational map $\phi$ on $\mathbb P^1$ and for its automorphism $h$ the notion of $h$-periodic points as well as the notion of $h$-tuned dynatomic polynomials are introduced. In Section~6 basic properties of $h$-tuned dynatomic polynomials are studied. Section~7 contains another main result of the paper, i.~e., Theorem~7.1 and Corollary~7.10. It turns out that $h$-tuned dynatomic polynomials, which are defined as rational functions, are in fact polynomials, which justifies their name. In generic situation they turn out to be non-trivial divisors of dynatomic polynomials. Hence the latter are reducible and this leads to a constructive proof of the statements mentioned above. Section~8 provides complete factorizations of the dynatomic polynomials for the cases $\phi(z)=z^d$ and $\phi(z)=z^{-d}$. In these cases the dehomogenized dynatomic polynomials turn out to be products of certain cyclotomic polynomials. In Section~9 the author shows in Proposition~9.2 that $M_2(N, \mathfrak{C}_2)$ is irreducible if $2^N-1$ is a prime number. rational map; moduli space; level structure; dynatomic polynomial; irreducibility; reducibility Manes, M., Moduli spaces for families of rational maps on $\mathbb{P}$1, J. Number Theory, 129, 1623-1663, (2009) Algebraic moduli problems, moduli of vector bundles, Families and moduli spaces in arithmetic and non-Archimedean dynamical systems Moduli spaces for families of rational maps on $\mathbb P^1$	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 this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial $\delta$-uniform permutations is obtained, for any $\delta$, as well as an upper bound for differentially two-valued functions. Concerning quadratic functions, significant relations between nonlinearity and differential uniformity are exhibited. In particular, we show that the quadratic differentially 4-uniform permutations should be differentially two-valued and possess the best known nonlinearity. vectorial Boolean function; quadratic function; nonlinearity; differential uniformity; differentially two-valued function Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry New links between nonlinearity and differential uniformity	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $F$ be a hypersurface of degree $m$ with an isolated singular point $p\in RP^ n$. Let $F_ t$, $t\in(0,1]$, be a continuous family of hypersurfaces of degree $m$, where $F_ 0=F$ and $F_ t$, $t\in(0,1]$, are nonsingular and isotropic in a closed spherical regular neighbourhood $D(p)$ of $p$. The sets $F_ t\cap D(p)$ are called smoothings of the singular point $p$ of $F$. The smoothings are called $q$-hyperbolic if for some continuous family $q_ t\in RP^ n$, $q_ 0=q$ for each real line $L\supset\{q_ t\}$ we have $L\cap F_ t\subset D(p)$. Author shows that all $q$-hyperbolic smoothings of a given singular point are rigidly isotopic, i.e. they are joined by continuous families of smoothings. He proved that the number of isotopic types of minimal smoothings of a solitary singular point of order $k$ equals $k/2+1$, and that the number of isotopic types of minimal smoothings of a nonsolitary singular point equals $(2r)!/(r!(r+1)!)$, where $r$ is the number of real branches. It is also proved that by a small perturbation of the coefficients of a plane algebraic curve, one can realize the smoothings of all its singular points, isotopic to their preassigned minimal smoothings. isotopic hypersurfaces; smoothings of singular point of hypersurface; hyperbolic smoothings; family of hypersurfaces; plane algebraic curve Shustin, E.: Hyperbolic and minimal smoothings of singular points. Selecta math. Sov. 10, 19-25 (1991) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Real algebraic sets, Hypersurfaces and algebraic geometry Hyperbolic and minimal smoothings of singular 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 For a scheme $X$ let $GW_{0}(X)$ denote the Grothendieck-Witt group of symmetric bilinear spaces over $X$. This is the abelian group generated by isometry classes $[\mathcal V,\phi ]$ of vector bundles $\mathcal V$ over $X$ with a nonsingular symmetric bilinear form $\phi: \mathcal V\otimes \mathcal V\to O_X$ subject to the relations $[(\mathcal V,\phi) \perp (\mathcal V ',\phi ' ]= [{\mathcal V},{\phi}]+[\mathcal V ',\phi ']$ and $[\mathcal M,\phi]=[\mathcal H(\mathcal N)]$ for every metabolic space $(\mathcal M, \phi)$ with Lagrangian subbundle $\mathcal N=\mathcal N^{\perp} \subset \mathcal M$ and associated hyperbolic space $\mathcal H(\mathcal N)$. The higher Grothendieck-Witt groups $GW_{i}(X), \, i\in {\mathbb N}$ were defined by the author [``Higher Grothendieck-Witt groups of exact categories'', J. K-theory (to appear)]. In the paper the author proves the Mayer-Vietoris sequence for open covers (Theorem 1). This main theorem of the paper (in fact Theorem 16 of the paper is more general and includes versions for skew symmetric forms and coefficients in line bundles different than $O_{X}$ ) is derived from the localization (Theorem 2) and Zariski excision (Theorem 3) theorems. The author proves also additivity, fibration and approximation theorems for the hermitian $K$-theory of exact categories with weak equivalences and duality. As the author noticed, \textit{P. Balmer} [K-Theory 23, No.~1, 15--30 (2001; Zbl 0987.19002)] and \textit{J. Hornbostel} [Topology 44, No.~3, 661--687 (2005; Zbl 1078.19004)] proved similar results to theorems 1--3. However, their assumptions are stricter than these of the author. Grothendieck-Witt groups; hermitian K-theory; Mayer-Vietoris sequence; localization; excision Berrick, J., Karoubi, M., Østvær Paul, A., Schlichting, M.: The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-Theory, arxiv:1011.4977v2 ( 2011) Hermitian $K$-theory, relations with $K$-theory of rings, $K$-theory of schemes, Witt groups of rings, Algebraic theory of quadratic forms; Witt groups and rings, Applications of methods of algebraic $K$-theory in algebraic geometry The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus $g$ over $\mathbb{F}_q$ $(q \equiv 1\pmod 3)$ as the curve varies in an irreducible component of the moduli space. They show that for $q$ fixed and $g$ increasing, the limiting distribution of the trace of Frobenius equals the sum of $q+1$ independent random variables taking the value $0$ with probability $2/(q+2)$ and $1,\,e^{2\pi i/3},\,e^{4\pi i/3}$ each with probability $q/(3(q+2))$. This extends the work of \textit{P. Kurlberg} and \textit{Z. Rudnick} [J. Number Theory 129, No. 3, 580--587 (2009; Zbl 1221.11141)] who considered the same limit for hyperelliptic curves (although not on the moduli space, which makes a slight difference explained in Theorem 1.1). They also show that when both $g$ and $q$ go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to $p$-fold covers of the projective line. Frobenius; traces of cyclic trigonal curves; Gaussian distribution Bucur, A.; David, C.; Feigon, B.; Lalın, M., Statistics for traces of cyclic trigonal curves over finite fields, Int. Math. Res. Not., 2010, 5, 932-967, (2009) Curves over finite and local fields, Finite ground fields in algebraic geometry Statistics for traces of cyclic trigonal curves over finite fields	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a log. smooth log. scheme $(X,M)$, defined over a discrete valuation ring $V$, with residue field $k$ of characteristic $p$. Fix a uniformizer $\pi$ of $V$. The author studies the extensions of automorphisms $\sigma$ of the special fiber $X_s$ of $X$ to the whole family $X$. Assuming that the smooth locus is dense in $X_s$, the author proves the existence of an integer $m$ (actually computable) such that $\sigma$ lifts to the log. structure $(X_s,M_s)$ over the special fiber if and only it lifts locally to the infinitesimal neighbourhood $X/\pi^m$. This explains in greater detail the known fact that sheaves of nearby cycles to the special fiber depend only on the completion of $X$. In the case of (relative) curves, the author shows a trace formula for the number of points collapsing to fixed points of $\sigma$, in its lifting to $X$. endomorphisms; valuation ring; logarithmic schemes Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry Endomorphisms of logarithmic schemes	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hilbert schemes $\mathrm{Hilb}^{p(t)} (\mathbb P^m)$ parametrizing closed subschemes $X \subset \mathbb P^m$ with Hilbert polynomial $p(t)$ have received much attention from algebraic geometers since their construction in the early 1960s. Although connected [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)], $\mathrm{Hilb}^{p(t)} (\mathbb P^m)$ exhibits many bad behaviors, for example it can have non-reduced components [\textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and every singularity type appears in some Hilbert scheme [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. In the paper under review, the authors classify all smooth Hilbert schemes. Recall that a Hilbert polynomial of a closed subscheme $X \subset \mathbb P^m$ can be uniquely written in the form $p(t) = \sum_{i=1}^r \binom{t+\lambda_i-i}{\lambda_i-1}$ where $\lambda = (\lambda_1, \dots, \lambda_r)$ is a partition of integers satisfying $\lambda_1 \geq \dots \geq \lambda_r \geq 1$ [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. The authors prove that the partitions corresponding to a smooth Hilbert scheme are precisely those in the following list: (1) $m = 2 \geq \lambda_1$. (2) $m \geq \lambda_1$ and $\lambda_r \geq 2$. (3) $\lambda = (1)$ or $\lambda = (m^{r-2}, \Lambda_{r-1},1)$, where $r \geq 2$ and $m \geq \lambda_{r-1} \geq 1$. (4) $\lambda = (m^{r-s-3}, \lambda_{r-s-s}^{s+2},1)$, where $r-3 \geq s \geq 0$ and $m-1 \geq \lambda_{r-s-2} \geq 3$. (5) $\lambda = (m^{r-s-5}, 2^{s+4},1)$, where $r-5 \geq s \geq 0$. (6) $\lambda = (m^{r-3},1^3)$, where $r \geq 3$. (7) $\lambda = (m+1)$ or $r=0$. Moreover, the authors describe the schemes parametrized by each family listed. For example, the general member of family (3) is a union of a hypersurface of degree $r-2$, a linear subspace of dimension $\lambda_{r-1}$ and a point while the general member of family (5) is a union of a hypersurface of degree $r-s-3$, a hypersurface of degree $s+2$ of a linear subspace of dimension $\lambda_{r-s-2}$ and a point. The families in (7) correspond to the one-point Hilbert scheme parametrizing $\mathbb P^m$ and the empty scheme. All families in the theorem were previously known to be smooth: smoothness for families (1) and (6) follows from work of \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], families (2) and (3) were shown smooth by \textit{A. P. Staal} [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)] and families (4) and (5) were shown smooth by \textit{R. Ramkumar} [J. Algebra 617, 17--47 (2023; Zbl 1503.13007)] in his work on Hilbert schemes with at most two Borel-fixed ideals. The main contribution here is that this list is complete. As to the strategy of the proof, it is known from work of \textit{A. Reeves} and \textit{M. Stillman} that the lexicographical point is always smooth on the Hilbert scheme and determines a unique irreducible component of $\mathrm{Hilb}^{p(t)} (\mathbb P^m)$ of computable dimension [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)]. In theory one could attempt to show smoothness by computing the dimension of the Zariski tangent space at the other Borel-fixed points, but this is unwieldy. Instead, the authors construct families of subschemes corresponding to points on Hilbert schemes that necessarily singular. To describe these points, define a \textit{residual inclusion} $X \subset Y \subset \mathbb P^m$ to be a closed immersion such that there is a linear subspace $\Lambda \subset \mathbb P^m$ containing $X$ and a hypersurface $D \subset \Lambda$ with $Y$ the residual scheme of $D \subset X$ in $\Lambda$. A \textit{residual flag} is a flag $\emptyset \subset X_e \subset X_{e-1} \subset \dots \subset X_1$ of residual inclusions. For most Hilbert polynomials not in the list, the authors produce such an $X_1$ near the lexicographic point which corresponds to a singular point on the Hilbert scheme. The others not on the list are handled with three other singular families. The proof is valid over $\mathrm{Spec}\,\mathbb Z$. The authors credit \texttt{Macaulay2} [\url{http://www.math.uiuc.edu/Macaulay2/}] for many experimental computations that were indispensable in discovering their results. Hilbert schemes; Borel fixed ideals; partial flag varieties Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, Geometric invariant theory, Actions of groups on commutative rings; invariant theory Smooth Hilbert schemes: their classification and geometry	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $X$ be an integral variety over a perfect field $k$, $\mathcal L_m(X)$ its jet scheme of level $m\in \mathbb N$ and $\mathcal L_{\infty}(X)$ its arc scheme. The general component $\mathcal G_m(X)$ of $\mathcal L_m(X)$ is the Zariski closure of $\mathcal L_m(\mathrm{Reg}(X))$. If $X$ is smooth on $k$ then the geometry and topology of $\mathcal L_m(X)$ are well understood. In this paper the authors consider the case $X$ is not smooth and study some properties of the general component $\mathcal G_m(X)$ by means of a smooth birational model of $X$. Indeed, under the further hypothesis that $X$ is affine embedded in $\mathbb A^N_k$, the authors prove that a birational model of $X$ provides a description of $\mathcal G_m(X)$ that gives rice to an algorithm which computes a Groebner basis of the defining ideal of $\mathcal G_m(X)$ in $\mathbb A^N_k$ as a subscheme of $\mathcal L_m(X)$ (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7. The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all $\mathcal G_m(X)$'' (Sections 8 and 9). computational aspects of algebraic geometry; derivation module; jet and arc scheme; singularities in algebraic geometry Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Computational aspects of algebraic curves, Computational aspects of higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry, Local complex singularities Two algorithms for computing the general component of jet scheme and 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 $G$ be a semisimple algebraic group defined over $\mathbb {Q}_p$, and let $\Gamma $ be a compact open subgroup of $G(\mathbb {Q}_p)$. We relate the asymptotic representation theory of $\Gamma $ and the singularities of the moduli space of $G$-local systems on a smooth projective curve, proving new theorems about both: \begin{itemize} \item[(1)] We prove that there is a constant $C$, independent of $G$, such that the number of $n$-dimensional representations of $\Gamma $ grows slower than $n^{C}$, confirming a conjecture of \textit{M. Larsen} and \textit{A. Lubotzky} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 351--390 (2008; Zbl 1142.22006)]. In fact, we can take $C=3\cdot {{\mathrm{dim}}}(E_8)+1=745$. We also prove the same bounds for groups over local fields of large enough characteristic. \item[(2)]We prove that the coarse moduli space of $G$-local systems on a smooth projective curve of genus at least $\lceil C/2\rceil +1=374$ has rational singularities. \end{itemize} For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. 1 A. Aizenbud and N. Avni, 'Representation growth and rational singularities of the moduli space of local systems', \textit{Invent. Math.}204 (2016) 245-316. MR 3480557. Algebraic moduli problems, moduli of vector bundles, Singularities in algebraic geometry, Asymptotic properties of groups, Linear algebraic groups over local fields and their integers, Deformations of singularities, Symplectic structures of moduli spaces Representation growth and rational singularities of the moduli space of local systems	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book systematically develops a theory of ``continuously varying families'' of Berkovich spaces. In algebraic geometry, the idea of ``continuously varying families'' of schemes are formalized by \textit{flat} morphisms between schemes: in the words of \textit{D. Mumford} [The red book of varieties and schemes. Includes the Michigan lectures (1974) on ``Curves and their Jacobians''. 2nd, expanded ed. with contributions by Enrico Arbarello. Berlin: Springer (1999; Zbl 0945.14001)], ``the concept of flatness is a riddle that comes out of algebra, but which technically is the answer to many prayers.'' A standard result in algebraic geometry and commutative algebra, exemplifying the kind of ``continuous variation'' that is imposed by flatness, states that all fibers of a flat morphism between two equidimensional noetherian schemes must have the same dimension. Note also that, like all reasonable classes of morphisms of schemes, flat morphisms are stable under base change and ground field extension. In Berkovich geometry, one could consider the class morphisms of Berkovich spaces which are flat as morphisms of locally ringed spaces, just as in the case of schemes. Ducros calls these \textit{naively flat} morphisms. As the addition of the adverb ``naively'' suggests, this class is ill-behaved: it turns out to not be stable under base change and ground field extension, and Ducros describes a counterexample due to Temkin in sections 1.3 and 4.4. Ducros then remedies this roughly by defining a \textit{flat} morphism between Berkovich spaces to be one which is naively flat and remains naively flat after any base change and ground field extension. Ducros proceeds to show that this class of flat morphisms of Berkovich spaces has many of the same properties that flat morphisms of schemes do in algebraic geometry; for example, the result mentioned above about the constancy of dimension is discussed in section 4.5. More importantly, Ducros uses the machinery of flat morphisms between Berkovich spaces to prove the following. Given a morphism $\phi : Y \to X$, we can consider the fibers $Y_x$ over points $x \in X$, which is a Berkovich space over the completed residue field $\mathscr{H}(x)$. Suppose also that $P$ assigns to each $x \in X$ some property $P(x)$ of Berkovich spaces over $\mathscr{H}(x)$. Ducros proves that the set of points $y \in Y$ for which $\mathscr{O}_{Y,y}$ is flat over $\mathscr{O}_{X,\phi(y)}$ and for which $Y_{\phi(y)}$ satisfies $P(\phi(y))$ is Zariski-open when $P$ is either empty or any of the following: geometrically regular, Gorenstein, complete intersection, Cohen-Macaulay, geometrically reduced (if $Y$ is relatively equidimensional), or geometrically normal (if $Y$ is relatively equidimensional). Ducros also proves that the set of points $y \in Y$ for which $Y_{\phi(y)}$ satisfies $P(\phi(y))$ is locally constructible when $P$ is any of the following: geometrically regular, Gorenstein, complete intersection, Cohen-Macaulay, geometrically reduced, or geometrically normal. In rough outline, the book is structured as follows. Chapters 1, 2, and 3 review some background information on Berkovich spaces. Flatness is defined formally in chapter 4, where its basic properties are also developed. Chapters 5 through 9 form the technical heart of the book, though they are very likely to be of independent interest. Their contents are summarized in the subsequent paragraphs. The results mentioned above, about open-ness and local constructibility of various loci, are described in chapter 10. Chapter 11 discusses descent of various properties along flat morphisms. Parts of the results of this book generalize earlier work of \textit{R. Kiehl} [Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl., 2. Abhdl. 25--49 (1968; Zbl 0177.06101)] in the setting of affinoid rigid spaces. Chapter 5 introduces quasi-smooth morphisms. These are morphisms which satisfy a property akin to the Jacobian criterion of smoothness for schemes. The main result of the section is proving that quasi-smooth morphisms of Berkovich spaces are precisely the ones which are flat and fiberwise geometrically regular. Ducros's approach is similar to that of \textit{S. Bosch} et al. [Néron models. Berlin etc.: Springer-Verlag (1990; Zbl 0705.14001)], and expands on the quasi-étale morphisms of \textit{V. G. Berkovich} [Invent. Math. 115, No. 3, 539--571 (1994; Zbl 0791.14008)]. Chapter 6 discusses a technique of ''spreading out from the generic fiber.'' Suppose $\phi : Y \to X$ is a morphism of Berkovich spaces and $x$ is a point of $X$ such that $\mathscr{O}_{X,x}$ is a field (such a point should be thought of as the generic point of an irreducible component of $X$), and $y \in Y_x$. There is a natural map of local rings $\mathscr{O}_{Y,y} \to \mathscr{O}_{Y_x,y}$. Were $\phi$ a morphism of schemes, this map would be an isomorphism. This is not true when $\phi$ is a morphism of analytic spaces, but Ducros proves that this map is often a flat map; this ensures that many algebraic properties can be descended along this map. Chapters 7 and 9 study images of morphisms of analytic spaces. The main result, which appears in chapter 9, states that if $\phi : Y \to X$ is a map of affinoid spaces and $Y$ is the suppert of a flat coherent sheaf, then $\phi(Y)$ is an analytic domain in $X$. This is a generalization of an earlier result by \textit{S. Bosch} and \textit{W. Lütkebohmert} [Math. Ann. 296, No. 3, 403--429 (1993; Zbl 0808.14018), cor 5.11]. The proof relies on an analysis of the local situation, which is conducted in chapter 7. Chapter 8 develops a theory of dévissage as in the work of \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)]. The author defines dévissages and proves that they always exist. Berkovich analytic spaces; flatness; dévissage Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, moduli of curves (analytic), Rigid analytic geometry, Foundations of algebraic geometry, Topology of analytic spaces Families of Berkovich spaces	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $G$ be a group of $n \times n$ matrices over the field $\mathbb{R}$ of real numbers acting on a finite dimensional vector space $M$ over $\mathbb{R}$. Assume there is a function $\phi:G\times M\to M$ with the following properties: $\phi(I,x)=x$ for all $x\in M$, and $\phi(A,\phi(B,x))=\phi(AB,x)$ for all $A, B \in G$ and all $x\in M$. An element $x\in M$ has the global [local] Lipschitz property if there is a constant $K > 0$ [are constants $K > 0$, $\varepsilon > 0$] depending on $x$ such that for every vector $y \in O(x)=\{z \in M: \phi(A,x) =z$ for some $A\in G\}$ [with $\|y -z\|_2 < \varepsilon]$ there is an $A\in G$ satisfying $y=\phi(A,x)$ and $\|I - A\|_1 \leq K \|y - x\|_2$ where $\|\;\|_1$ is a norm on $\mathbb{R}^{n \times n}$ and $\|\;\|_2$ is a norm on $M$. The authors study Lipschitz properties for matrix groups over the ground fields $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. Let $G$ be a matrix group with elements $(G_1, G_2)$ where $G_1 \in \mathbb{R}^{m \times m}$, $G_2 \in \mathbb{R}^{n \times n}$ are invertible, and assume the action $\phi$ of $G$ on $\mathbb{R}^{m \times n}$ is defined by $\phi((G_1, G_2),x)=G^{-1}_1 x G_2$, $x \in \mathbb{R}^{m \times n}$, $(G_1, G_2) \in G$. Then the action $\phi$ has the local Lipschitz property. This result can be extended to complex or quaternionic matrices. Further, this theorem encloses some important particular cases, namely simultaneous similarity, restricted simultaneous similarity, equivalence, and simultaneous equivalence for $m$-tuples of matrices. From the local Lipschitz property the authors deduce the global Lipschitz property for (restricted) simultaneous similarity and for (restricted) simultaneous equivalence over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. The authors also study simultaneous congruence. They consider a set of $m$-tuples $(A_1, \dots, A_m)$ of complex Hermitian $n \times n$ matrices viewed as a real vector space. Let $\phi$ be the action $\phi(S, (A_1, \dots, A_m))=(S^* A_1 S, \dots, S^* A_m S)$. Assume that the $n\times n$ complex Hermitian matrices $A_1, \dots, A_m$ have no common isotropic vector. Then the $m$-tuple $(A_1, \dots, A_m)$ has the global Lipschitz property with respect to the action $\phi$. The crucial point in the proof of this theorem is the proof of the local property which takes the most part of the paper. Moreover, a generalization for simultaneous congruence of not necessarily Hermitian matrices is stated. complex Hermitian matrices; matrix groups; actions; simultaneous similarity; simultaneous equivalence; local Lipschitz property; global Lipschitz property; simultaneous congruences DOI: 10.1016/0024-3795(94)90435-9 Linear algebraic groups over the reals, the complexes, the quaternions, Hermitian, skew-Hermitian, and related matrices, Group actions on varieties or schemes (quotients), Other matrix groups over fields, Matrices over special rings (quaternions, finite fields, etc.) Lipschitz properties of some actions of matrix groups	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author (along with G. Brumfiel) has defined abstract semialgebraic functions on the constructible subsets of the real spectrum of a ring A. The ring R of these functions on the real spectrum is called the real closure of A [the author, Mem. Am. Math. Soc. 397 (1989; Zbl 0697.14015)]. The abstract semialgebraic functions also form a sheaf for the ``inverse'' topology (for which the usual closed constructible subsets of the real spectrum form a basis of open sets), giving a ringed space $Spec^R$. These ringed spaces are the building blocks of inverse real closed spaces [the author, Illinois J. Math. (to appear)], an abstraction of weakly semialgebraic spaces [\textit{M. Knebusch}, ``Weakly semialgebraic spaces'', Lect. Notes Math. 1367 (1989; Zbl 0681.14008)]. The purpose of the paper is to characterize $Spec^R$ by a universal property. Let ${\mathcal D}$ be the category of ringed spaces (X,${\mathcal O}_ X)$ whose stalks are real closed (in the above sense) integral domains, with morphisms of ringed spaces $\phi: X\to Y$ such that $\phi^({\mathcal O}_ Y)\to {\mathcal O}_ X$ is injective. Then $Spec^R$ is in ${\mathcal D}$, and represents the functor $X\mapsto Hom(A,H^ 0(X,{\mathcal O}_ X)).$ The first section of the paper contains a characterization of real closed integral domains which gives a first order axiomatisation of these rings. - The second section is devoted to the ``inverse affine scheme'' $Spec^*A$ of any ring A. This is the same as the ``integral spectrum'' of \textit{P. T. Johnstone} [J. Algebra 49, 238-260 (1977; Zbl 0369.13019)], which was used to characterize those rings which are representable as rings of global sections of sheaves of integral domains [cf. \textit{J. F. Kennison}, Math. Z. 151, 35-56 (1976; Zbl 0346.18012)]. semialgebraic functions; real spectrum; inverse real closed spaces Schwartz, N.: Eine universelle eigenschaft reell abgeschlossener rǎume, Commun. algebra 18, No. 3, 755-774 (1990) Semialgebraic sets and related spaces, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Eine universelle Eigenschaft reell abgeschlossener Räume. (A universal property of real closed spaces)	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The first question raised in this paper, as suggested by its title, is whether one can define higher-dimensional formal loops in a scheme. Such a notion has been developed previously, generalizing earlier work of Contou-Carrère. Here, the author chooses instead to generalize an approach of Kapranov and Vasserot. Given a scheme $X$, one can take the space of maps from the formal neighborhood of 0 in affine $d$-space and likewise the space of maps from a punctured formal neighborhood. The definition given here is that the $d$-dimensional formal loops are given by the completion of the latter space in the former. Making such a definition precise is subtle, but this loop space is shown to be representable by a derived ind-pro-scheme. The author uses derived algebraic geometry as the appropriate context in which to make these definitions, and indeed allows for generalization from schemes to stacks. The paper begins with an introduction to the higher categorical methods and derived algebraic geometry results that are needed. The second question is whether, if $X$ has a symplectic structure, the higher loop space can also be endowed with a natural symplectic structure. While an initial answer might be that such a structure is impossible, since this loop space is typically infinite-dimensional, the author develops a notion of Tate stacks as the appropriate context in which to frame the question. If $X$ is symplectic, then its higher dimensional formal loop space has the structure of a Tate stack. A variant, called the bubble space of $X$, is also shown to have this structure. formal loops; derived algebraic geometry; shifted symplectic structures Categories in geometry and topology, Topological categories, foundations of homotopy theory, Abstract and axiomatic homotopy theory in algebraic topology, Schemes and morphisms, Generalizations (algebraic spaces, stacks) Higher dimensional formal loop 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 This paper is devoted to the Poisson geometry behind directed networks on surfaces. In previous works, the authors have already used Postnikov's construction in order to parametrize cells in Grassmannians and proved that the space of weights of networks in a disk can be naturally endowed with Poisson brackets. The purpose of this paper is to generalize the construction for directed weighted networks in an annulus. To this aim, the authors first associate with every network a matrix of boundary measurements. Then, they characterize all universal Poisson brackets on the space of edge weights. They show that the universal Poisson brackets induce a family of Poisson structures on rational matrix-valued functions and on the space of loops in the Grassmannian. Poisson geometry; network; annulus; weight; Grassmannian; R-matrix M. Gekhtman, M. Shapiro and A. Vainshtein, \textit{Poisson geometry of directed networks in an annulus}, arXiv:0901.0020. Poisson manifolds; Poisson groupoids and algebroids, Grassmannians, Schubert varieties, flag manifolds Poisson geometry of directed networks in an annulus	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 notion of regular separation for solutions of systems of ODEs $y^\prime = F (x, y)$, where $F$ is definable in a polynomially bounded o-minimal structure and $y = (y_1, y_2)$. Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author. solutions of ODEs; non-oscillating trajectories of vector fields; o-minimality; Hardy field; transcendental formal solutions Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations, Model theory of ordered structures; o-minimality, Asymptotic properties of solutions to ordinary differential equations, Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $M$ be a smooth manifold and $\mathfrak Z$ a subspace of the space of complex vector fields on $M$, which is closed under multiplication by complex valued smooth functions. Then one considers weak $L^2_{loc}$ solutions of the system of equations $Zu=0$ for all $Z\in\mathfrak Z$. The main aim of the article is to find conditions on $\mathfrak Z$ which imply that such solutions satisfy a maximum modulus principle and/or unique continuation. The basic conditions to be imposed are versions of hypoellipticity on one hand and conditions on the behavior of $\mathfrak Z$ under the Lie bracket on the other hand. The latter conditions are either expressed as a Hörmander condition or via the Sussmann leaf associated to $\mathfrak Z$. The authors first show that Lipschitz-hypoellipticity together with openness of the Sussmann leaf implies a local maximal modulus principle. If in addition weak unique continuation holds, one obtains a global maximal modulus principle. Second, the authors study weak unique continuation, deriving results in the real analytic case, and for embedded CR manifolds satsifying weakenings of pseudoconcavity. complex vector fields; maximum modulus principle; weak unique continuation; $CR$ manifold CR structures, CR operators, and generalizations, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, Homotopy groups of topological groups and homogeneous spaces Complex vector fields, unique continuation and the maximum modulus principle	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $(X, 0)$ be the germ of a complex normal surface singularity, with link $Sigma$. A smoothing of $(X, 0)$ is a morphism $f : (X , 0) \to (\mathbb{C}, 0)$, with $(X , 0)$ a threedimensional isolated Cohen-Macaulay singularity, equipped with an isomorphism $(f ^{1} (0), 0)\simeq(X, 0)$. The Milnor fibre $M$ is the general fibre $f^{-1} (\delta)$. The second Betti number of $M$ is called $\mu$, the Milnor number of the smoothing. We say that $f$ is a $QHD$ smoothing if $\mu = 0$. There are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a $QHD$ smoothing. The fundamental group of the Milnor fibre has been known for all except three exceptional families. In this work the authors settle these cases, they present a new explicit construction for one of the exceptional families, showing the fundamental group is non-abelian. They show that the fundamental groups for the remaining two exceptional families are abelian. In all three cases, they show that it is possible to use the plane curves and their blow-ups to compute the fundamental group of the complement. surface singularity; rational homology sphere; Milnor fibre Singularities of curves, local rings, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Fundamental group, presentations, free differential calculus Fundamental group of rational homology disk smoothings of surface singularities	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the following classical interpolation problem: Given a finite set of points $\{P_1,\dots, P_n\}$ in the projective plane and a set $\{m_1,\dots, m_n\}$ of non negative integers, determine the dimension of the linear system $\|L\|$ of plane curves of degree $d$ passing through each $P_i$ with multiplicity greater or equal than $m_i$. Since every $(m_i)$-uple point imposes $m_i(m_i-1)/2$ conditions to curves, it is natural to expect that $\|L\|$ has dimension either $(d(d-3)-\sum_1^n m_i(m_i-1))/2$ or is empty, when the previous number is negative. On the other hand it is easy to write down examples in which the true dimension is different from the expected one. Harbourne and Hirschowitz conjectured that the true and the expected dimensions differ exactly when the the proper transform of $\|L\|$ in the blow up of ${\mathbb P}^2$ at the $P_i$'s has a fixed rational component of self intersection $-1$. By now, only few particular cases of the conjecture are established. The authors examine here the \textit{homogeneous} case, in which all the multiplicities are the same (say equal to $m$). In this situation, they give an explicit description of all the cases in which the $(-1)$-curve quoted above actually exists. Then they use an inductive procedure, which relies on the degeneration of the plane to a union of rational surfaces, to attack the conjecture. The authors are able to control the inductive steps and prove the conjecture for all $m\leq 12$. plane curves; multiplicity; interpolation problem; linear system C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of egual multiplicity , Trans. Amer. Math. Soc. 352 (2000), 4037-4050. JSTOR: Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves Linear systems of plane curves with 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 Let $\kappa$ be a field and $K$ its algebraic closure. For every positive integer $m\in \mathbb{N}$, let $A^m(\kappa)$ be the $m$-dimensional affine space over $\kappa$. A subset $C \subset A^m(K)$ is called constructible if it is equal to the image of an algebraic variety under the projection map from $A^M(K)$ to $ A^m(K)$ with $M \ge m$. In the literature, the degree of a constructible set $C$ is defined to be the degree of its Zariski closure; however, as is shown in this paper, Bézout's inequality does not hold for this definition. In the first part of the paper, the authors present two different notions of degree for a constructible set and prove that they satisfy a Bézout inequality. Furthermore, the authors study several properties of the new notions. The second part of the paper is devoted to correct test sequences. Let $X$ be a set, $K$ a field and $m\in \mathbb{N}$ a positive integer. Let $\Omega$ be a set whose elements are sequences of length $m$ of functions from $X$ to $K$. Let $\Sigma \subset \Omega$. A correct test sequence of length $L$ for $\Omega$ with discriminant $\Sigma$ is a finite set of $L$ elements $\{x_1, \ldots , x_L \} \subseteq X$ such that for each $f\in \Omega$, if $f(x_1)=\cdots=f(x_L)=(0,\ldots ,0)$ then $f\in \Sigma$. Correct test sequences were introduced by Heintz and Schnorr in 1982 and they appeared in different places in mathematics. In this paper, by using the results of the first part, the authors prove that correct test sequences for lists of polynomials are densely distributed in any constructible set of accurate co-dimension and degree. Finally, they apply correct test sequences of asymptotically optimal length to describe a randomized algorithm to decide whether a list of polynomials forms a ``suite sécante'': a sequence of polynomials $f_1,\ldots,f_m$ in terms of $n$ variables is called suite sécante if the associated variety has dimension $n-m$. degree of constructible and locally closed affine sets; Bézout's inequality; correct test sequences; probability; existence; equality test of lists of functions; bounded error probability; polynomial-time algorithms Symbolic computation and algebraic computation, Polynomials in general fields (irreducibility, etc.), Polynomial rings and ideals; rings of integer-valued polynomials, Solving polynomial systems; resultants, Effectivity, complexity and computational aspects of algebraic geometry, Analysis of algorithms and problem complexity, Randomized algorithms, Analysis of algorithms A promenade through correct test sequences. I: Degree of constructible sets, Bézout's inequality and density	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to prove a criterion for projectively Cohen- Macaulay-two-codimensional subschemes $Y$ of $\mathbb{P}_ k^ N$ ($k$ algebraically closed of characteristic 0) to be smoothable. The considered schemes are determinantal, i.e. the homogeneous ideal $I(Y)$, defining $Y$, is generated by the $t\times t$-minors of a homogeneous $m\times n$-matrix $M$ with $\text{ht}(I(Y))= (m-t+1) (n-t+1)$. Determinantal schemes are equidimensional with a Cohen-Macaulay coordinate ring. It is easy to see that a closed subscheme $Y\subset \mathbb{P}_ k^ N$ of equicodimension two with a Cohen-Macaulay coordinate ring is necessarily determinantal. The author determines first the codimension of the singular locus of $Y$ (see propositions 1 and 2 in section 2.1) under certain conditions on the entries of $M$. Then she proves the existence of a deformation $f:X\to Z$ of $Y$ (determinantal and of codimension 2) over an irreducible $k$-scheme $Z$ (i.e. $Y=f^{- 1}(z_ 0)$ over a closed point $z_ 0$); and $Y$ is then smoothable iff $2\leq N\leq 5$ (see theorem in section 2.2). This result generalizes a corresponding criterion of \textit{T. Sauer} [Math. Ann. 272, 83-90 (1985; Zbl 0546.14023)] for curves in $\mathbb{P}_ k^ 3$. determinantal schemes; smoothability Frauke, S.:Generic determinantal schemes and the smoothability of determinantal schemes of codumension 2, Manuscripta Math.82 (1994) 417--431. Determinantal varieties, Linkage, complete intersections and determinantal ideals, Deformations and infinitesimal methods in commutative ring theory Generic determinantal schemes and the smoothability of determinantal schemes of codimension 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 provides a new proof of the result in [\textit{J.-P. Rolin} et al., J. Amer. Math. Soc. 16, 751--777 (2003; Zbl 1095.26018)] and a result similar to that in [\textit{J.-P. Rolin} et al., Proc. London Math. Soc. 95, 413--442 (2007; Zbl 1123.03031)] using the result of [\textit{J.-M. Lion}, J. Symbolic Logic 67, 1616--1622 (2002; Zbl 1042.03030)] on o-minimality of the structure generated by geometric, regular and $0$-regular family of functions. They provide examples of o-minimal structures without $\mathcal C^\infty$-cell decompositions. The first main result (Theorem 4.3) of this paper is as follows: Let $\mathcal A = \{\mathcal A_m\}_{m \in \mathbb N}$ be a family of algebras of restricted smooth functions closed under taking partial derivatives. If the geometric family $S(\mathcal A)$ generated by the germs of $\mathcal A$ at every point in $[-1,1]^m$ is quasi-analytic, then $\mathcal A$ is contained in a certain o-minimal structure. In his proof, the author introduces the notion of a geometric family of almost $\mathcal C^{\infty}$-germs and develops a normal crossings transformation theorem on their quasianalytic algebras, which is an adaptation of the classical reduction of analytic functions to normal crossings. Using this theorem together with Lion's result, the author gets another proof of the above result. Using Lion's result again, the author constructed an o-minimal structure generated by the sum of a function which is semialgebraic outside every neighborhood of zero and infinitely flat at zero and a strongly transcendental smooth function (Theorem 6.6). This result implies the existence of two functions such that each of their germs is contained in the Hardy field of an o-minimal structure, but both of them are not simultaneously contained in the Hardy field of any o-minimal structure (Proposition 7.6). o-minimality; theorem of Lion; Hardy field Model theory of ordered structures; o-minimality, Semialgebraic sets and related spaces Theorem of lion and o-minimal structures which do not admit $\mathcal{C}^{\infty}$-cell 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 purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings $f:C \to\mathbb{P}^1$ of the projective line by smooth curves. Thus, from the very outset, one sees that there are essentially two ways to approach the Hurwitz scheme: (1) We start with $\mathbb{P}^1$ and regard the objects of interest as coverings of $\mathbb{P}^1$. (2) We start with $C$ and regard the objects of interest as morphisms from $C$ to $\mathbb{P}^1$. One finds that one can obtain the most information about the Hurwitz scheme and its compactification by exploiting interchangeably these two points of view. Our first main result is the following theorem. Let $b,d$, and $g$ be integers such that $b=2d+ 2g-2$, $g\geq 5$ and $d>2g+4$. Let ${\mathcal H}$ be the Hurwitz scheme over $\mathbb{Z} [{1\over b!}]$ parametrizing coverings of the projective line of degree $d$ with $b$ points of ramification. Then $\text{Pic} ({\mathcal H})$ is finite. The number $g$ is the genus of the ``curve $C$ upstairs'' of the coverings in question. Note, however, that the Hurwitz scheme ${\mathcal H}$, and hence also the genus $g$, are completely determined by $b$ and $d$. -- Note that although in the statement of the theorem we spoke of ``the'' Hurwitz ``scheme,'' there are in fact several different Hurwitz schemes used in the literature, some of which are, in fact, not schemes, but stacks. The main idea of the proof is that by combinatorially analyzing the boundary of the compactification of the Hurwitz scheme, one realizes that there are essentially three kinds of divisors in the boundary, which we call excess divisors, which are ``more important'' than the other divisors in the boundary in the sense that the other divisors map to sets of codimension $\geq 2$ under various natural morphisms. On the other hand, we can also consider the moduli stack ${\mathcal G}$ of pairs consisting of a smooth curve of genus $g$, together with a linear system of degree $d$ and dimension 1. The subset of ${\mathcal G}$ consisting of those pairs that arise from Hurwitz coverings is open in ${\mathcal G}$, and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of ${\mathcal M}_g$, we show that these three divisors on ${\mathcal G}$ form a basis of $\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}$, and in fact, we even compute explicitly the matrix relating these three divisors on ${\mathcal G}$ to a certain standard basis of $\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}$. The above theorem then follows formally. Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack $({\mathcal A}, M)$ parametrizing such coverings: Fix non-negative integers $g,r,q,s,d$ such that $2g-2+r =d(2q-2+s) \geq 1$. Let ${\mathcal A}$ be the stack over $\mathbb{Z}$ defined as follows: For a scheme $S$, the objects of ${\mathcal A}(S)$ are admissible coverings $\pi:C\to D$ of degree $d$ from a symmetrically $r$-pointed stable curve $(f:C\to S$; $\mu_f \subseteq C)$ of genus $g$ to a symmetrically $s$-pointed stable curve $(h:D \to S$; $\mu_h \subseteq D)$ of genus $q$; and the morphisms of ${\mathcal A} (S)$ are pairs of $S$-isomorphisms $\alpha: C\to C$ and $\beta: D\to D$ that stabilize the divisors of marked points such that $\pi\circ \alpha= \beta\circ \pi$. Then ${\mathcal A}$ is a separated algebraic stack of finite type over $\mathbb{Z}$. Moreover, ${\mathcal A}$ is equipped with a canonical log structure $M_{\mathcal A} \to {\mathcal O}_{\mathcal A}$, together with a logarithmic morphism $({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}$ (obtained by mapping $(C;D;\pi) \mapsto D)$ which is log étale (always) and proper over $\mathbb{Z} [{1\over d!}]$. finite Picard group; Hurwitz scheme; Hurwitz coverings; admissible coverings S. Mochizuki, ''The geometry of the compactification of the Hurwitz scheme,'' Publ. Res. Inst. Math. Sci., vol. 31, iss. 3, pp. 355-441, 1995. Families, moduli of curves (algebraic), Picard groups, Coverings in algebraic geometry The geometry of the compactification of the Hurwitz 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 Let $S$ be a regular scheme and ${\mathcal S}m/S$ the category of smooth and separated schemes of finite type over $S$. \textit{F. Morel} and \textit{V. Voevodsky} proved [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] representability of the algebraic $K$-theory of regular schemes in the $\mathbb A^1$-homotopy theory: Theorem 0.1 [Morel, Voevodsky]: Let $S$ be a regular scheme. Then, for any $n\in\mathbb N$ and $X\in{\mathcal S}m/S$, there is a canonical isomorphism \[ \Hom_{{\mathcal H}_\bullet(S)}(S^n\wedge X_+,\mathbb Z\times\mathbb G\text{r})\cong K_n(X). \] $\mathbb G\text{r}$ is a colimit of the system $(\mathbb G\text{r}_{d,r})_{(d,r)\in\mathbb N^2}$ in the category of presheaves over ${\mathcal S}m/S_{\text{Nis}}$ and $\mathbb G\text{r}_{d,r}$ denotes the Grassmann scheme parametrizing subbundles of rank $d$ in the trivial bundle of rank $d+r$. From the isomorphism in the above theorem it is clear that the endomorphisms of $\mathbb Z\times\mathbb G\text{r}$ in ${\mathcal H}_\bullet(S)$ act on algebraic $K$-groups of schemes in ${\mathcal S}m/S$. The author proves the following theorem: Theorem 0.2: Let $S$ be a regular scheme. We let $K_{0}(-)$ be the presheaf of sets on ${\mathcal S}m/S$ which maps $X$ to $K_{0}(X)$. Then the map induced by Theorem 0.1 is a bijection: \[ \text{End}_{{\mathcal H}(S)}(\mathbb Z\times\mathbb G\text{r}) \overset\sim \rightarrow \text{End}_{{\mathcal S}m/S^{\text{opp}}{\mathcal S}ets}(K_{0}(-)). \] The theorem shows that operations defined on the level of $K_{0}$ [\textit{P. Berthelot, A. Grothendieck} and \textit{L. Illusie} (eds.), Séminaire de géométrie algébrique du Bois Marie 1966/67. Lect. Notes Math. 225 (1971; Zbl 0218.14001)] for example ${\lambda}^n$ or ${\Psi}^{n}$ lift uniquely to ${\mathcal H}(S)$ and therefore via Theorem 0.1 extend uniquely to operations on higher algebraic $K$-theory. This also works for operations which involve several operands, for example products. In section 2 the author studies the structures on $\mathbb Z\times\mathbb G\text{r}$ that come from from algebraic structures on $K_{0}$. In particular he shows that $\mathbb Z\times\mathbb G\text{r}$ is equipped with a structure of a special $\lambda$-ring with duality. In section 3 the author compares the constructions of products, $\lambda$-operations and Adams' operations by various authors [cf. \textit{J.-L. Loday}, Ann. Sci. Éc. Norm. Supér. (4) 9, 309--377 (1976; Zbl 0362.18014); \textit{F. Waldhausen}, Lect. Notes Math. 1126, 318--419 (1985; Zbl 0579.18006); \textit{Ch. Kratzer}, Comment. Math. Helv. 55, 233--254 (1980; Zbl 0444.18008); \textit{C. Soulé}, Can. J. Math. 37, 488--550 (1985; Zbl 0575.14015); \textit{D. R. Grayson}, K-Theory 6, No. 2, 97--111 (1992; Zbl 0776.19001); \textit{F. Lecomte}, Algebraic $K$-theory and its applications. Proceedings of the workshop and symposium, ICTP, Trieste, Italy, September 1-19, 1997. Singapore: World Scientific. 437--449 (1999; Zbl 0969.55008) and \textit{M. Levine}, Fields Inst. Commun. 16, 131--184 (1997; Zbl 0883.19001)] to his constuction. In section 4 the author relates his constructions to virtual categories of \textit{P. Deligne} [Contemp. Math. 67, 93--117 (1987; Zbl 0629.14008)]. In section 5 the author examines the operations ${\tau}: K_{0}(-)\rightarrow K_{0}(-)$ such that ${\tau}(x+y)={\tau}(x)+{\tau}(y).$ These correspond to $H$-group endomorphisms of $\mathbb Z\times\mathbb G\text{r}$. In section 6 a version of homotopical Riemann-Roch formulas is derived. operations in algebraic $K$-theory; Riemann-Roch theorems; Chern character Joël Riou, ``Algebraic $K$-theory, $A^1$-homotopy and Riemann-Roch theorems'', J. Topol.3 (2010) no. 2, p. 229-264 $Q$- and plus-constructions, Motivic cohomology; motivic homotopy theory, Homotopy theory Algebraic $K$-theory, $A^1$-homotopy and Riemann-Roch theorems	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The so-called Brenner-Hochster-Kollár problem about vector-valued continuous solutions of finite systems of linear equations was solved by \textit{C. Fefferman} and \textit{J. Kollár} [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)]. In the paper under review, the analytic method to settle this problem is extended to solve it for the space $C^m(\mathbb{R}^n)$ of real-valued functions whose derivatives up to order $m$ are continuous and bounded in $\mathbb{R}^n$, and for the space $C^{m,\omega}(\mathbb{R}^n)$ of real-valued functions whose $m$-th derivatives have modulus of continuity $\omega$. These cases had been left open. For $C^m(\mathbb{R}^n)$, the authors solve a more general problem that includes also a general Whitney extension problem for vector-valued functions and jets. The more general problem is formulated in terms of sections for bundles. A variant of this general problem for $C^{m,\omega}(\mathbb{R}^n,\mathbb{R}^d)$ is also solved by means of a theorem which is a type of ``finiteness principle''. The dependence of a certain ``finiteness constant'' with respect to $m$, $n$ and $d$ is also discussed. This paper continues recent, deep work by Bierstone-Milman-Pawlucki, Brudnyi-Shvartsman, Fefferman-Kollár, among others. generalized Whitney problems; extending vector-valued functions; continuous closures; differentiable closures; Brenner-Hochster-Kollár problem Fefferman, C.; Luli, G.K., The Brenner-hochster-Kollár and Whitney problems for vector-valued functions and jets, Revista Matematica Iberoamericana, 30, 875-892, (2014) Topological linear spaces of continuous, differentiable or analytic functions, Functions of several variables, Computational aspects in algebraic geometry The Brenner-Hochster-Kollár and Whitney problems for vector-valued functions and jets	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A real linear operator ${\mathcal M}$ acts on $\mathcal{C}^n$ according to $z\rightarrow {\mathcal M}z=Mz+M_{\#}\bar{z}$ for a pair of matrices $M,M_{\#} \in \mathcal{C}^{n\times n}$ called the linear and antilinear parts of ${\mathcal M}.$ If ${\mathcal M}_{\#}=0$ (resp. ${\mathcal M}=0$) then ${\mathcal M}$ is called $\mathcal{C}$-linear (resp. antilinear). The spectrum of ${\mathcal M}$ consists of those points $\lambda \in \mathcal{C}$ for which $\lambda I-{\mathcal M}$ is not invertible giving rise to a bounded, possibly empty, real algebraic plane curve of degree $2n.$ The authors presents results concerning the location of the eigenvalues of ${\mathcal M}$ and classify components of the spectrum. Path continuation techniques are implemented for computing components and subsets of the spectrum once an eigenvalue is available. Three numerical examples illustrating the aspect studied have been presented. real linear operator; spectrum; characteristic bivariate polynomial; path following techniques; real algebraic plane curve; location of eigenvalues; numerical examples Huhtanen M., von Pfaler J.: The real linear eigenvalue problem in \$\$\{$\backslash$mathbb C\^n\}\$\$ . Linear Algebra Appl. 394, 169--199 (2005) Eigenvalues, singular values, and eigenvectors, Linear transformations, semilinear transformations, Computational aspects of algebraic curves, Numerical computation of eigenvalues and eigenvectors of matrices The real linear eigenvalue problem in $\mathbb C^n$	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 In this paper, we study length categories using iterated extensions. We fix a field $k$, and for any family $\mathsf{S}$ of orthogonal $k$-rational points in an Abelian $k$-category $\mathcal{A} $, we consider the category $\mathbf{Ext}(\mathsf{S})$ of iterated extensions of $\mathsf{S}$ in $\mathcal{A} $, equipped with the natural forgetful functor $\mathbf{Ext}(\mathsf{S}) \to \mathcal{A}(\mathsf{S})$ into the length category $\mathcal{A}(\mathsf{S})$. There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel's criterion, we give a complete classification of the indecomposable objects in $\mathcal{A}(\mathsf{S})$ when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family $\mathsf{S}$ in $\mathcal{A}$. As an application, we classify all graded holonomic $D$-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when $D$ is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish. finite length categories; uniserial categories; iterated extensions; noncommutative deformations Abelian categories, Grothendieck categories, Representation theory of associative rings and algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Iterated extensions and uniserial length categories	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper extends previous work by \textit{R. M. Skjelnes} and \textit{D. Laksov} [Compos. Math. 126, 323-334 (2001; Zbl 1056.14500)] and gives a description of the Hilbert scheme of $n$ points on the affine scheme $C:=\text{Spec}(K[x]_U)$, where $K$ is a field and $K[x]_U$ is a fraction ring of the polynomial ring in one variable. The Hilbert functor of $n$ points on $C$, $\text{Hilb}^n$, associates to a $K$-algebra $A$ the set $\text{Hilb}^n(A)$, formed by the ideals $I$ of $A\otimes{_KK}[x]_U$ such that the residue class ring $A\otimes{_KK}[x]_U/I$ is locally free of finite rank $n$ (as $A$-module). The main result is that $\text{Hilb}^n$ is represented by the fraction ring $H=K[s_1,...,s_n]_{U(n)}$, where $s_1,...,s_n$ are the elementary symmetric functions in the variables $t_1,...,t_n$ and $U(n)=\{f(t_1)\dots f(t_n)\mid f\in U\}$. Moreover, the universal family of $n$-points on $C$ is isomorphic to $\text{Sym}_K^{n-1}(C)\times_K C$, as in the case where $C$ is a smooth curve. The result relies on a characterization of the characteristic polynomial of the multiplication by the residue of a polynomial in $A[x]/(F)$, where $A$ is any commutative ring and $F$ a monic polynomial in $A[x]$. Hilbert scheme; resultants; points on the line; characteristic polynomial of the multiplication Skjelnes, R. M.: Resultants and the Hilbert scheme of the line. Ark. mat. 40, No. 1, 189-200 (2002) Parametrization (Chow and Hilbert schemes), Algebraic functions and function fields in algebraic geometry, Fine and coarse moduli spaces, Polynomial rings and ideals; rings of integer-valued polynomials Resultants and the Hilbert scheme of points on the line	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a basic compact semialgebraic set $\mathbf{K}\subset\mathbb{R}^n$, we introduce a methodology that generates a sequence converging to the volume of $\mathbf{K}$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on $\mathbf{K}$ can be approximated as closely as desired, which permits the approximation of the integral on $\mathbf{K}$ of any given polynomial; the extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed. computational geometry; volume; integration; $\mathbf{K}$-moment problem; semidefinite programming D. Henrion, J. B. Lasserre, and C. Savorgnan, \textit{Approximate volume and integration for basic semialgebraic sets}, SIAM Rev., 51 (2009), pp. 722--743, . Semialgebraic sets and related spaces, Sums of squares and representations by other particular quadratic forms, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Convex programming Approximate volume and integration for basic semialgebraic sets	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``\textit{Appell's hypergeometric functions} $F_2$ and $F_3$ are the analytic continuations of Appell's hypergeometric series'' defined by the relations \[ {F_2}\left( {\alpha ,\beta ,\beta ',\gamma ,\gamma ';x,y} \right) = \sum_{m,n \geqslant 0} {\frac{{{(\alpha)_{m + n}}{(\beta)_m}{(\beta')_n}}} {{{(\gamma)_m}{(\gamma')_n}m!n!}}{x^m}{y^n}} ,\quad \left\| x \right\| + \left\| y \right\| < 1 \tag{1} \] \[ {F_3}\left(\alpha,\alpha',\beta,\beta',\gamma ;x,y \right) = \sum_{m,n \geqslant 0} {\frac{{{(\alpha)_m}{(\alpha')_n}{(\beta)_m}{(\beta')_n}}} {{{(\gamma)_{m + n}}m!n!}}{x^m}{y^n}} ,\quad \left\| x \right\| < 1,\left\| y \right\| < 1 \tag{2} \] where, ${(a)_n}$ is the Pochhammer symbol (shifted factorial) defined by \[ {(a)_n} = a(a+1)(a+2)\cdots (a+n-1) \tag{3} \] The functions $F_2$ and $F_3$ are known to satisfy the following systems of partial differential equations denoted by $E_2$ and $E_3$ (each of rank 4) respectively: \[ \left({E_2}\right)\begin{cases} {\left[ {x\left( {1 - x} \right)\frac{{{\partial ^2}}} {{\partial {x^2}}} - xy\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma - \left( {\alpha + \beta + 1} \right)x} \right\}\frac{\partial } {{\partial x}} - \beta y\frac{\partial } {{\partial y}} - \alpha \beta } \right]F = 0,} \\ {\left[ {y\left( {1 - y} \right)\frac{{{\partial ^2}}} {{\partial {y^2}}} - xy\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma ' - \left( {\alpha + \beta ' + 1} \right)y} \right\}\frac{\partial } {{\partial y}} - \beta 'x\frac{\partial } {{\partial x}} - \alpha \beta '} \right]F = 0} \end{cases} \tag{4} \] and \[ \left( {{E_3}} \right)\begin{cases} {\left[ {x\left( {1 - x} \right)\frac{{{\partial ^2}}} {{\partial {x^2}}} + y\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma - \left( {\alpha + \beta + 1} \right)x} \right\}\frac{\partial } {{\partial x}} - \alpha \beta } \right]F = 0,} \\ {\left[ {y\left( {1 - y} \right)\frac{{{\partial ^2}}} {{\partial {y^2}}} + x\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma - \left( {\alpha ' + \beta ' + 1} \right)y} \right\}\frac{\partial } {{\partial y}} - \alpha '\beta '} \right]F = 0} \end{cases} \tag{5} \] both on the space \[ {\mathbb{C}^2}\backslash \left\{ {\left\{ {x = 0} \right\} \cup \left\{ {x = 1} \right\} \cup \left\{ {y = 0} \right\} \cup \left\{ {y = 1} \right\} \cup \left\{ {x + y = 1} \right\}} \right\} \subset {\left( {{\mathbb{P}^1}} \right)^2}. \] E. Nakagiri had earlier in his Master's thesis [\textit{E. Nakagiri}, Monodromy representations of hypergeometric differential equations of two variables (Japanese). Master's Thesis. Kobe: Kobe University (1979)] investigated the monodromy representations associated with $F_2$ and $F_3$, where he gave ``explicit formulas about the matrix elements of the representations'' by imposing the following conditions \[ \begin{cases} {\text{Re} \left( {1 - \alpha } \right),\text{Re} \left( \beta \right),\text{Re} \left( {\beta '} \right),\text{Re} \left( {\gamma - \beta } \right),\text{Re} \left( {\gamma ' - \beta '} \right) > 0,} \\ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}{\text{ and }}\gamma ,\gamma ' \notin \mathbb{Z}}\end{cases} \tag{6} \] in the $E_2$ case and \[ \begin{cases} {\text{Re} \left( {1 - \alpha } \right),\text{Re} \left( {1 - \alpha '} \right),\text{Re} \left( \beta \right),\text{Re} \left( {\beta '} \right),\text{Re} \left( {\gamma - \beta - \beta '} \right) > 0,} \\ {\gamma - \alpha - \alpha ',\gamma - \alpha - \beta ',\gamma - \alpha ' - \beta \notin {\mathbb{Z}_{ \leqslant 0}}{\text{ and }}\beta - \alpha ,\beta ' - \alpha ' \notin \mathbb{Z}} \end{cases}\tag{7} \] in the $E_3$ case. The author of the present paper relaxes these two conditions (i.e. (6) and (7)) of Nakagiri [loc. cit.] to the following: \[ \begin{cases} { - \alpha ,\beta - 1,\beta ' - 1,\gamma - \beta - 1,\gamma ' - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}},} \\ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \end{cases} \tag{8} \] in the $E_2$ case and to \[ \begin{cases} { - \alpha , - \alpha ',\beta - 1,\beta ' - 1,\gamma - \beta - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}},} \\ {\gamma - \alpha - \beta ',\gamma - \alpha ' - \beta ,\gamma - \alpha - \alpha ' - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \end{cases} \tag{9} \] in the $E_3$ case and derives ``Nakagiri's matrix elements of the five generators of the monodromy representations associated with $F_2$ and $F_3$.'' He utilizes the integrals of multivalued functions and deforms the ``the chains on which the integrals are defined'' by showing ``each of the chains in the two dimensional complex space by the product of the paths in the complex plane''. He also gives an elegant derivation of the sufficient conditions for the systems of the partial differential equations $E_2$ and $E_3$ being irreducible. In the first section of the paper the author introduces his \textit{auxiliary integrals} as \[ \int {\int_C {t_1^{{\lambda _1}}{{\left( {{t_1} - {z_1}} \right)}^{{\lambda _2}}}t_2^{{\lambda _3}}{{\left( {{t_2} - {z_2}} \right)}^{{\lambda _4}}}{{\left( {{t_1} + {t_2} - 1} \right)}^{{\lambda _5}}}d{t_1}d{t_2}} } \tag{10} \] over a suitable chain $C$. The following abbreviated symbols \[ e(A) = \exp \left( {2\pi \sqrt { - 1} A} \right),\quad {\lambda _{ijk \ldots l}} = {\lambda _i} + {\lambda _j} + {\lambda _k} + \cdots + {\lambda _l},\quad {e_{ijk \ldots l}} = e\left( {{\lambda _{ijk \ldots l}}} \right) \] \[ {\partial _x} = \frac{\partial } {{\partial x}},{\theta _x} = x{\partial _x},\quad t = \left( {{t_1},{t_2}} \right),\quad dt = d{t_1}d{t_2} \] the author uses to formulate his results throughout this paper. By letting $z = \left( {{z_1},{z_2}} \right)$ to be a point of \[ Z = {\mathbb{C}^2}\backslash \left\{ {\left\{ {{z_1} = 0} \right\} \cup \left\{ {{z_1} = 1} \right\} \cup \left\{ {{z_2} = 0} \right\} \cup \left\{ {{z_2} = 1} \right\} \cup \left\{ {{z_1} + {z_2} = 1} \right\}} \right\} \subset {\left( {{\mathbb{P}^1}} \right)^2} \] and $u(t) = u\left( {z;t} \right),$ a function defined by \[ u\left( {z;t} \right) = \prod_i {{f_i}{{(t)}^{{\lambda _i}}}} = t_1^{{\lambda _1}}{\left( {{t_1} - {z_1}} \right)^{{\lambda _2}}}t_2^{{\lambda _3}}{\left( {{t_2} - {z_2}} \right)^{{\lambda _4}}}{\left( {{t_1} + {t_2} - 1} \right)^{{\lambda _5}}} \tag{11} \] on ${T_z} = {\mathbb{C}^2}\backslash \bigcup_{i,{\lambda _i} \notin {Z_{ \geqslant 0}}} {\left\{ {{f_i}(t) = 0} \right\}} $ where, \[ {f_1}(t) = {t_1},{f_2}(t) = \left( {{t_1} - {z_1}} \right), \dots ,{f_5}(t) = \left( {{t_1} + {t_2} - 1} \right) \] and for a pair of complex variables $\left( {{z_1},{z_2}} \right)$ which are real and sufficiently small positive numbers by allotting certain domains $D_j$, $1 \leqslant j \leqslant 4$, of the real manifold $T_{\mathbb{R}}$ and letting ${u_{{D_j}}}\left( {z;t} \right) = \prod_i {{{\left( {{\varepsilon _i}{f_i}(t)} \right)}^{{\lambda _i}}}} $ ``where ${\varepsilon _i} = \pm $ is so determined that each ${{\varepsilon _i}{f_i}(t)}$ is positive and the argument of ${{\varepsilon _i}{f_i}(t)}$ is assigned to be zero on $D_j$'', the author, by following \textit{M. Kita} [Jpn. J. Math., New Ser. 18, No. 1, 25--74 (1992; Zbl 0767.33009)] ``for a way of constructing the cycle $\text{reg} {D_j}$ when ${\lambda _i} \notin Z$ for all $1 \leqslant i \leqslant 5$'' and \textit{K. Mimachi} and \textit{T. Sasaki} [Kyushu J. Math. 66, No. 1, 35--60 (2012; Zbl 1259.33007)] ``for the regularization of the chain when some of $\lambda_i$ are integers'' states the following notable results: Theorem 1.1. If ${\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4},{\lambda _5} \notin {\mathbb{Z}_{ < 0}}$ then each integral \[ {I_j}\left( z \right) = \int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {x,y;t} \right)dt} \] for $1 \leqslant j \leqslant 4$ satisfies the system of differential equations \[ \begin{cases} {\left[ {{\theta _{{z_1}}}\left( {{\theta _{{z_1}}} - {\lambda _{12}} - 1} \right) - {z_1}\left( {{\theta _{{z_1}}} - {\lambda _2}} \right)\left( {{\theta _{{z_1}}} + {\theta _{{z_2}}} - 2 - {\lambda _{12345}}} \right)} \right]F = 0,} \\ {\left[ {{\theta _{{z_2}}}\left( {{\theta _{{z_2}}} - {\lambda _{34}} - 1} \right) - {z_2}\left( {{\theta _{{z_2}}} - {\lambda _4}} \right)\left( {{\theta _{{z_1}}} + {\theta _{{z_2}}} - 2 - {\lambda _{12345}}} \right)} \right]F = 0.} \end{cases} \tag{12} \] Theorem 1.2. If ${\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4},{\lambda _5} \notin {\mathbb{Z}_{ < 0}}$ and ${\lambda _{125}} + 2,{\lambda _{345}} + 2,{\lambda _{12345}} + 3 \notin {\mathbb{Z}_{ \leqslant 0}}$ then the integrals ${I_j}\left( z \right) = \int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {z;t} \right)dt} $ for $1 \leqslant j \leqslant 4$ are linearly independent. A combination of these two theorems is the important consequence: Corollary 1.3. If ${\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4},{\lambda _5} \notin {\mathbb{Z}_{ < 0}}$ and ${\lambda _{125}} + 2,{\lambda _{345}} + 2,{\lambda _{12345}} + 3 \notin {\mathbb{Z}_{ \leqslant 0}}$ then the integrals ${I_j}\left( z \right) = \int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {z;t} \right)dt} $ for $1 \leqslant j \leqslant 4$ constitute a fundamental set of solutions of (12). The lengthy proof of the Theorem 1.1 is given in Subsection 1.1 which is based on the statement and proof of a lemma, while the author derives a Wronskian formula for $\int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {t} \right)dt} ,1 \leqslant j \leqslant 4$ at the beginning of the Subsection 1.2 to ultimately prove the Theorem 1.2 in this subsection by stating and proving two more lemmas. Remarking that from Section 2 of the paper and onwards ``the superscript (2) is used to indicate the case of Appell's $F_2$ (similarly, the superscript (3) is used to indicate the case of Appell's $F_3$)'', the author gives fundamental sets of solutions of $E_2$ and $E_3$ in Section 2 of the paper. For a point $\left(x,y\right)$ of ${X^{(2)}} = {\mathbb{C}^2}\backslash \left\{ {\left\{ {x = 0} \right\} \cup \left\{ {x = 1} \right\} \cup \left\{ {y = 0} \right\} \cup \left\{ {y = 1} \right\} \cup \left\{ {x + y = 1} \right\}} \right\} \subset {\left( {{\mathbb{P}^1}} \right)^2}$ on $T_{\left( {x,y} \right)}^{(2)} = {\mathbb{C}^2}\backslash \bigcup\nolimits_{i,{\lambda _i} \notin {Z_{ \geqslant 0}}} {\left\{ {f_i^{(2)}(t) = 0} \right\}} $ where, $f_1^{(2)}(t) = {t_1},f_2^{(2)}(t) = {t_1} - 1, \ldots ,f_5^{(2)}(t) = 1 - x{t_1} - y{t_2}$ when $\left(x,y\right)$ are sufficiently small positive numbers, by assigning certain domains $D_j^{(2)}$ for $1 \leqslant j \leqslant 4$ of the real manifold $T_{\mathbb{R}}^{(2)}$ and by defining $u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right) = \prod_i {{{\left( {{\varepsilon _i}f_i^{(2)}(t)} \right)}^{{\lambda _i}}}} $ in which ${\lambda _1} = \beta - 1,{\lambda _2} = \gamma - \beta - 1,{\lambda _3} = \beta ' - 1,{\lambda _4} = \gamma ' - \beta ' - 1,{\lambda _5} = - \alpha $ and ${\varepsilon _i} = \pm $ is so chosen ``that each ${{\varepsilon _i}f_i^{(2)}(t)}$ is positive and the argument of ${{\varepsilon _i}f_i^{(2)}(t)}$ is assigned to be zero on $D_j^{(2)}$'' the author proves: Theorem 2.1. If $\beta - 1,\gamma - \beta - 1,\beta ' - 1,\gamma ' - \beta ' - 1, - \alpha \notin {\mathbb{Z}_{ < 0}}$ then each of the integrals $I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)} $ for $1 \leqslant j \leqslant 4$ satisfies the system of differential equations (4). Next he proves: Theorem 2.2. If $\beta - 1,\gamma - \beta - 1,\beta ' - 1,\gamma ' - \beta ' - 1, - \alpha \notin {\mathbb{Z}_{ < 0}}$ and \[ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \] the integrals $I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)}dt $ for $1 \leqslant j \leqslant 4$ are linearly independent. A combination of these two theorems is the result: Corollary 2.3. If $\beta - 1,\gamma - \beta - 1,\beta ' - 1,\gamma ' - \beta ' - 1, - \alpha \notin {\mathbb{Z}_{ < 0}}$ and \[ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \] the integrals $I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)}dt $ for $1 \leqslant j \leqslant 4$ constitute a fundamental set of solutions of $E_2$. The author proves parallel results for the Appell's function $F_3$ in the Theorems 2.4, 2.5 and Corollary 2.6 by showing ``that four integrals in the form \[ \int {\int_C {t_1^{\beta - 1}t_2^{\beta ' - 1}{{\left( {1 - {t_1} - {t_2}} \right)}^{\gamma - \beta - \beta ' - 1}}{{\left( {1 - x{t_1}} \right)}^{ - \alpha }}{{\left( {1 - y{t_2}} \right)}^{ - \alpha '}}d{t_1}d{t_2}} } \tag{13} \] satisfy (5) under the condition $ - \alpha , - \alpha ',\beta - 1,\beta ' - 1,\gamma - \beta - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}}$ and that the four integrals are linearly independent and constitute a fundamental set of solutions of $E_3$'' when the conditions stated in (9) hold. Returning to the \textit{auxiliary integrals}, introduced in the first section of the paper, the author presents monodromy representation associated with these integrals (10) in the third section of the paper where he computes the `` matrix elements of the five generators of the monodromy representation with respect to the four integrals by deforming the chains appropriately'' following the approach in [\textit{K. Mimachi}, Int. Math. Res. Not. 2005, No. 33, 2031--2057 (2005; Zbl 1129.32017); Funkc. Ekvacioj, Ser. Int. 51, No. 1, 107--133 (2008; Zbl 1157.33308)] and [\textit{K. Mimachi} and \textit{T. Sasaki}, Kyushu J. Math. 66, No. 1, 35--60 (2012; Zbl 1259.33007); ibid. 66, No. 1, 61--87 (2012; Zbl 1259.33025); ibid. 66, No. 1, 89--114 (2012; Zbl 1259.33028)]. The results of these investigations are reported in the Theorem 3.1 ``which gives rise to the matrix elements in the cases $E_2$ and $E_3$''. By considering the monodromy representation $\rho :{\pi _1}\left( {{X^{(2)}},\left( {{x^0},{y^0}} \right)} \right) \to GL\left( {4,\mathbb{C}} \right)$ around a base point ${\left( {{x^0},{y^0}} \right)}$ of real and sufficiently small positive numbers on the solution space $V = \sum_{1 \leqslant j \leqslant 4} {\mathbb{C}I_j^{(2)}\left( {x,y} \right)} $ of $E_2$ for $I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)dt}$, the author establishes ``Nakagiri's matrix elements in the $E_2$ case'' in Theorem 4.1 under the conditions ${ - \alpha ,\beta - 1,\beta ' - 1,\gamma - \beta - 1,\gamma ' - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}}}.$ Similarly, the author deduces Nakagiri's matrix elements for the $E_3$ case in the Theorem 4.2 under the conditions $- \alpha$, $-\alpha'$, $\beta - 1$, $\beta' - 1$, $\gamma - \beta - \beta ' - 1 \notin {\mathbb{Z}_{< 0}}.$ The reviewer feels that the Theorems 3.1, 4.1 and 4.2 are the very important results of this paper, only because their statements are quite lengthier, so, they are not quoted here for the reasons of brevity. In the fifth section of the paper the author utilizes the matrix elements developed in Theorem 3.1 to establish ``a sufficient condition for the monodromy representation associated with the auxiliary integrals being irreducible'' where he proves the notable result: Theorem 5.1. If none of $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_{125}, \lambda_{345}$ and $\lambda_{12345}$ are integers, the monodromy representation $\rho$ in Section 3 is irreducible. As an important corollary of this theorem the author deduces: Theorem 5.2. {\parindent=6mm\begin{itemize}\item[(1)] If none of $\alpha, \beta, \beta',\gamma - \beta, \gamma'- \beta', \gamma - \alpha, \gamma' - \alpha, \gamma + \gamma' - \alpha$ is an integer, then the monodromy representation for $E_2$ is irreducible. \item[(2)] If none of $\alpha, \alpha',\beta, \beta',\gamma-\beta-\beta',\gamma-\alpha-\beta',\gamma-\alpha'-\beta,\gamma-\alpha-\alpha'$ is an integer, then the monodromy representation for $E_3$ is irreducible. \end{itemize}} In an earlier work the author has proved (see [\textit{K. Mimachi} and \textit{T. Sasaki}, Kyushu J. Math. 69, No. 2, 429--435 (2015; Zbl 1329.33020)]) a complementary result of the Theorem 5.2 above which states that: Theorem 5.3. {\parindent=6mm\begin{itemize}\item[(1)] If at least one of $\alpha, \beta, \beta',\gamma - \beta, \gamma'- \beta', \gamma - \alpha, \gamma' - \alpha, \gamma + \gamma' - \alpha$ is an integer, then the monodromy representation of the system $E_2$ is reducible. \item[(2)] If at least one of $\alpha, \alpha',\beta, \beta',\gamma-\beta-\beta',\gamma-\alpha-\beta',\gamma-\alpha'-\beta,\gamma-\alpha-\alpha'$ is an integer, then the monodromy representation of the system $E_3$ is reducible. \end{itemize}} A combination of these two last theorems leads us to ``the necessary and sufficient conditions for the systems $E_2$ and $E_3$ being irreducible obtained by \textit{E. Bod} [J. Differ. Equations 252, No. 1, 541--566 (2012; Zbl 1236.33026)] and \textit{M. Kato} [Kyushu J. Math. 66, No. 2, 325--363 (2012; Zbl 1259.33027)]''. Finally in the sixth section (appendix) of the paper the author shows that if $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_{125}, \lambda_{345}, \lambda_{12345} \notin \mathbb{Z}$ then his monodromy representation $\rho$ of Theorem 3.1 may also be considered as the irreducible monodromy representations on the twisted homology groups. The reviewer has noticed a few misprints in the paper which are not listed here. The reviewer opines that this well organized and nicely written paper is a valuable addition to our existing knowledge of the monodromy representations associated with the Appell's hypergeometric functions in general, and the Appell's functions $F_2$ and $F_3$ in particular. hypergeometric integrals; Nakagiri's monodromy representations; irreducibility; auxiliary integrals; chains; cycles; twisted homology; Appell's hypergeometric functions; Appell's function $F_2$; Appell's function $F_3$ Appell, Horn and Lauricella functions, Monodromy; relations with differential equations and $D$-modules (complex-analytic aspects), Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Monodromy on manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.) Nakagiri's monodromy representations associated with Appell's hypergeometric functions $F_2$ and $F_3$	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)], \textit{B. Toën} and \textit{M. Vaquié} defined a scheme theory for a closed monoidal category $(\mathcal C, \otimes, 1)$. In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category $s$ of simplicial objects in a monoidal category and all the categories $sA$-mod, $sA$-$\mathrm{alg }(A \in \mathrm{sComm}(\mathcal C))$ with compatible model structures using the work of \textit{C. Rezk} [Topology Appl. 119, No. 1, 65--94 (2002; Zbl 0994.18008)]. We then give a general notion of smoothness in $\mathrm{sComm}(\mathcal C)$. We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in $\mathbb N$-alg and Comm(Set). symmetric monoidal category; simplicial category; model category; relative geometry; smoothness; cohomology DOI: 10.1017/is013010008jkt242 Monoidal categories (= multiplicative categories) [See also 19D23], Étale and other Grothendieck topologies and (co)homologies, Simplicial sets, simplicial objects (in a category) [See also 55U10] Smoothness in relative 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 F be a surface in ${\mathbb{C}}^ 3$ with an isolated singularity at the origin 0. 0 is called a quasi-ordinary singularity if there is a finite map p: (F,0)$\to ({\mathbb{C}}^ 2,0)$ such that the branched locus of p is $xy=0$ or $x=0$ in the degenerate case. It is known by \textit{J. Lipman}, ''Quasi-ordinary singularities of embedded surfaces''. Thesis, Harvard Univ. (1965) that there is a fractional power series $\zeta =H(x^{1/n},y^{1/n})$ with $H(0,0)=0$ so that $f(x,y,z)=\prod (z- \zeta_ i)$ where f is the defining function of F and $\zeta_ i$ has the form $H(\mu^ j x^{1/n},\mu^ k y^{1/n})$, $\mu =\exp (2\pi i/n)$. (u/n,v/n) is a distinguished pair if the monomial $x^{u/n} y^{v/n}$ appears as the leading term of some $\zeta_ i-\zeta_ j$. The normalized parametrization has the following property: The distinguished pairs are linearly ordered $(0,0)<(\lambda_ 1,\mu_ 1)<...<(\lambda_ s,\mu_ s)$. Monomial $x^{\alpha}y^{\beta}$ with non-zero coefficient in $\zeta$ satisfies $(\alpha,\beta)\geq (\lambda_ 1,\mu_ 1)$. $(\lambda_ 1,...,\lambda_ s)\geq (\mu_ 1,...,\mu_ s)$ (lexicographically) and $\mu_ 1=0$ if $\lambda_ 1\geq 1$. The main result in this paper is: Theorem (2.2.4). The topological type of a quasi-ordinary singularity determines and is determined by its distinguished pairs. isolated surface singularity; characteristic pairs; multiplicity; quasi- ordinary surface singularity; distinguished pair; normalized parametrization Gau, Topology of the quasi-ordinary singularities, Topology 25 (4) pp 495-- (1986) Complex singularities, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Topology of the quasi-ordinary surface singularities	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $p$ and $q$ be multiplicatively independent positive integers and consider the so-called Mahler operators $x\mapsto x^p$ and $x\mapsto x^q$. A power series $F(x)\in K[[x]]$ satisfies a $p$-Mahler fuctional equation if the vector space generated by $F(x), F(x^p), F(x^{p^2}), \ldots$ has finite dimension. \textit{B. Adamczewski} and \textit{J. P. Bell} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 4, 1301--1355 (2017; Zbl 1432.11086)] proved that a power series $F(x)$ satisfies both a $p$- and a $q$-Mahler functional equation if and only if it is a rational function. \textit{J.-P. Bézivin} and \textit{A. Boutabaa} [Collect. Math. 43, No. 2, 125--140 (1992; Zbl 0778.39009)] gave a similar result for dilations $x\mapsto px$ and $x\mapsto qx$. This paper considers similar objects called elliptic $(p,q)$-difference modules for an elliptic curve $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice, and the field of $\Lambda$-elliptic functions. These are finite-dimensional vector spaces with commuting automorphisms given by $\sigma f(z)=f(z/p), \tau f(z)=f(z/q)$ on the field of $\Lambda$-elliptic functions. The main theorem is a structure theorem for elliptic $(p,q)$-difference modules in the case when $p$ and $q$ are relatively prime. As a consequence, it is shown that a power series which satisfies simultaneously a $p$-difference equation and a $q$-difference euqation with coefficients Laurent expansions of $\Lambda$-elliptic functions lies in the ring generated over the field of $\Lambda'$-elliptic functions by $z^{\pm1}$ and $\zeta(z,\Lambda')$ for some lattice $\Lambda'\subset \Lambda$. Here, $\zeta(z,\Lambda)$ is the Weierstrass zeta function for the elliptic curve. Conversely, every function from this ring satisfies a $p$-difference equation and a $q$-difference equation with elliptic functions as coefficients. The derivation explains the appearance of $\zeta(z,\Lambda)$ in the elliptic case, unlike the cases for the simpler additive and multiplicative groups. In the 1930s, Mahler built a method for studying the transcendence of functions satisfying functional equations in the multiplicative case $x\mapsto x^p$ and its higher-dimensional generalisations. These ideas found applications in computability because the generating functions of sets recognised by finite automata are related to Mahler functions. In the 1960s, Cobham proved that if a set is recognised by finite automata in base $p$ and in base $q$ then it is a union of a finite set and a finite number of arithmetic progressions. However, the precise link between finite automata and Mahler functions was elusive. Difference algebra, as further developed in this paper, has delivered the recent progress in this area. difference equations; simultaneous difference equations; elliptic functions Difference algebra, Elliptic curves, Additive difference equations Elliptic $(p,q)$-difference modules	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $R$ be a complete discrete valuation ring with quotient field of characteristic zero and algebraically closed residue field $k$ of characteristic $p>0$. Let $\phi: \mathbb{R} \to \mathbb{R}$ be the unique lifting of the Frobenius of $k$, and define $\delta: \mathbb{R} \to \mathbb{R}$ by $\delta x =\frac{\phi (x) - x^p}{p}$. Let $G/R$ be a smooth group scheme of finite type. The author defines a $\delta$-formal function of order $\leq n$ on $G(R)$ to be an $R$ valued function which can locally (in the Zariski topology of $G$) be expressed as a polynomial in affine coordinate functions and ($n$ or fewer) iterates of $\delta$. These form a ring $O^n(G)$. For each $n$, these rings have comultiplications, coinverses, and counits coming from the group scheme structure of $G$. A coherent family $\mathcal J$ of ideals $J_n \subset O^n(G)$ respecting these cooperations determines a subgroup of $G(R): \mathcal J^{\text{sol}} = \{x \in G(R) \mid f(x)=0$ for all $f \in J_n$, $n \geq 0\}$. The author calls such subgroups $\delta$-subgroups. The author calls a subgroup $\Gamma \subset G(R)$ Zariski dense modulo $p$ if the image of $\Gamma$ in the closed fibre $G_0(k)$ of $G/R$ is Zariski dense. The main result of the paper is the following theorem: Let $G/R$ be such that $G_0/k$ is a simple algebraic group such that the adjoint representation of $G_0$ on $\text{Lie}(G_0)$ is irreducible. Let $\Gamma$ be a $\delta$-subgroup which is Zariski dense modulo $p$. Then $\Gamma = G(R)$. The proof depends heavily on the author's previous work on $p$ formal groups and arithmetic jet theory. complete discrete valuation ring; lifting of the Frobenius; characteristic $p$; smooth group scheme; $p$ formal groups; arithmetic jet theory Buium, A.: Differential subgroups of simple algebraic groups over p-adic fields. Amer. J. Math. 120, 1277-1287 (1998) Differential algebra, Linear algebraic groups over local fields and their integers, Derivations, actions of Lie algebras, Group schemes, Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure Differential subgroups of simple algebraic groups over $p$-adic 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 The purpose of this work is to construct a general semiregularity map for cycles on a complex analytic or algebraic manifold and to show that such semiregularity map can be obtained from the classical tool of the Atiyah-Chern character. The first part of the paper is fairly detailed, deducing the existence and explicit form of a generalized semiregularity map from known results and constructions. As a corollary we obtain in the second part as well a description of the infinitesimal Abel-Jacobi map for smooth cycles as the leading term of this generalized semiregularity map, indicate why for locally complete intersections the appropriate component of our semiregularity map coincides with the one constructed by \textit{S. Bloch} [Invent. Math. 17, 51--66 (1972; Zbl 0254.14011)], and give applications to embedded deformations and deformations of coherent modules. We restrict ourselves mainly to the case of a smooth ambient space, avoiding thus the formidable technical machinery of resolvents, powers of cotangent complexes, traces and so on. The underlying ``metaresult'' in its utmost generality is only stated at the end with details to be given in a later paper [\textit{R. O. Buchweitz} and \textit{H. Flenner}, Compos. Math. 137, 135--210 (2003; Zbl 1085.14503)]. semiregularity map; Atiyah-Chern character; Abel-Jacobi map; smooth cycles R.-O. Buchweitz, H. Flenner, The Atiyah-Chern character yields the semiregularity map as well as the infinitesimal Abel-Jacobi map, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proceeding Lecture Notes vol. 24, American Mathematical Society, Providence, RI, 2000, pp. 33-46. Algebraic cycles, Transcendental methods of algebraic geometry (complex-analytic aspects), Formal methods and deformations in algebraic geometry The Atiyah-Chern character yields the semiregularity map as well as the infinitesimal Abel-Jacobi 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 Consider a noetherian abelian hereditary Ext-finite category $\mathcal{H}$ over a field $k$, and assume that $\mathcal{H}$ is equipped with an equivalence $\tau$ which satisfies Serre duality. In the paper under review, two major invariants are attached to $\mathcal{H}$, namely its function field $k(\mathcal{H})$ and its Euler characteristic $\chi_{\mathcal{H}}$, and they are proved to determine the shape of $\mathcal{H}$ under many aspects. Several examples of these categories are outlined in the paper, the most relevant being that of coherent sheaves on a smooth projective curve $C$ over $k$, in which case the invariants of $\mathcal{H}$ equal those of $C$. Consider the full subcategory $\mathcal{H}_{0}$ consisting of objects of finite length. This is the union over a set $C(\mathcal{H})$ (called the set of points of $\mathcal{H}$) of categories whose Auslander-Reiten quiver can be of type $\mathbb{Z\,A}/\tau^{p}$ (a finite point) or of type $\mathbb{Z\,A}_{\infty}$ (an infinite point). The authors first prove that these alternatives cannot coexist, and that in the latter $\mathcal{H}$ can have at most two points. On the full subcategory $\mathcal{H}_{+}$ of objects (called bundles) with no indecomposable subobject, a convenient notion of rank takes positive value. Once chosen a line bundle on $\mathcal{H}$ the authors provide a definition of $\chi_{\mathcal{H}}$, a degree function, and a definition of semistable bundles in analogy with the case of smooth projective curves [see \textit{C. S. Seshadri}, Astérisque 96 (1982; Zbl 0517.14008)]. The case of $\chi_{\mathcal{H}}$ being positive ($\mathcal{H}$ is then called domestic) is studied in great detail. The condition $\chi_{\mathcal{H}} > 0$ takes place, for instance, if and only if the rank function is bounded on some component of $\mathcal{H}_{+}$, or if and only if each indecomposable is stable (or exceptional). The case of $\mathcal{H}$ having an infinite point is proved to be domestic too. Moreover, in analogy with the case of $\mathbb{P}^{1}$, it is proved that $\mathcal{H}$ is domestic if and only if $\mathcal{H}$ has a hereditary tilting class, which in turn amounts to $\mathcal{H}$ being derived equivalent to $\roman{mod}(\Lambda)$, where $\Lambda$ is a hereditary locally bounded category. This allows to complete the classification of domestic categories over an algebraically closed field, which turn out to be associated to path algebras of extended simply laced Dynkin graphs, or of an infinite quiver of type $\mathbb{A}_{\infty}^{\infty}$ or $\mathbb{D}_{\infty}$. This complements the classification appearing by \textit{I. Reiten} and \textit{M. Van den Bergh} [J. Am. Math. Soc. 15, 295--366 (2002; Zbl 0991.18009)]. The tubular case (that is, when $\chi_{\mathcal{H}}=0$) is also studied, and the Auslander-Reiten quiver of $\mathcal{H}$ must consist of infinitely many points of finite period. A number of open questions is listed at the end of the paper. hereditary noetherian categories; Euler characteristic; domestic categories; path algebras; infinite points; non-commutative curves H. Lenzing and I. Reiten, \emph{Hereditary {N}oetherian categories of positive {E}uler characteristic}, Math. Z. \textbf{254} (2006), no.~1, 133--171. \MR{2232010 (2007e:18008)} Categorical embedding theorems, Module categories in associative algebras, Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc., Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Families, moduli of curves (algebraic) Hereditary noetherian categories of positive Euler characteristic	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author gives a necessary and sufficient condition for the infinitesimal Torelli theorem to hold and identifies the $\mu =const$. stratum in the discriminant of the miniversal deformation S. After examining the geometric reasons causing Torelli to fail for certain deformations, the author defines a local moduli space and proves a local Torelli theorem for singularities. Definition: The subset ${\mathcal K}$ of S is the union of all analytic arcs through 0 in S corresponding to the deformations $f_ r=f+rg_ 1+r^ 2g_ 2+..$. with \[ g_ 1\in ''V^ 1{\mathcal O},\quad g_ 2=g^ 2_ 1\partial_ t,\quad g_ 3=2g_ 1g_ 2\partial_ t-g^ 3_ 1\partial^ 2_ t,\quad.... \] Theorem. The $\mu =const$. stratum $D_{\mu}={\mathcal K}.$ [For the notations and further explanations see the paper itself.] $\mu $ $=const$. deformation; mixed Hodge structure; Torelli theorem KARPISHPAN (Y.) . - Torelli theorems for singularities , Inv. Math., t. 100, 1990 , p. 97-141. MR 91b:32040 \| Zbl 0694.32013 Deformations of complex singularities; vanishing cycles, Local complex singularities, Deformations of singularities, Period matrices, variation of Hodge structure; degenerations Torelli theorems for singularities	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review proves existence of a parametrization of bounded definable subsets of real Euclidean space such that the number of parametrizing functions needed is explicitly bounded. More specifically, the sets under consideration are definable in some fixed o-minimal expansion of a real field and by a parametrization of such a set $X \subset {\mathbb R}^n$ the authors mean a finite collection of definable maps from $(0,1)^m$ to ${\mathbb R}^n$ where $m:=\dim(X)$, whose range covers $X$. The fact that such a parametrization exists is a consequence of the cell decomposition theorem, and in an earlier paper the second and the third author constructed parametrizing functions with certain differentiability conditions and bounds on derivatives. Further, in the same paper it was shown that if ${\mathcal X}=\{X_t: t\in T\}$ is a definable family of $m$-dimensional subsets of $(0,1)^n$ in the sense that relation $ ``t \in T \text{ and } x\in X_t$'' is definable in both $x$ and $t$ and the parametrizing functions have the $r$-th continuous derivatives, then there exists an integer $N_r$ such that for each $t$ at most $N_r$ functions are required to parametrize $X_t$ and each such function is definable. The earlier work however does not give an explicit bound on $N_r$, and one of the results in the paper under review in the case of the ambient o-minimal structure being the restricted analytic field ${\mathbb R}_{\mathrm{an}}$ where the bounded definable sets are precisely the bounded subanalytic sets, shows that $N_r$ can be taken to be polynomial in $r$. The authors use the explicit bound on the number of the parametrizing functions to achieve in some settings a better estimate of the number of rational points contained in definable sets. rational points of bounded height; $C^r$-parameterizations; quasi-parameterizations; pre-parameterizations; subanalytic sets; restricted Pfaffian functions; weakly mild functions; a-b-m functions; Gevrey functions; power maps; entropy; dynamical system theory; Weierstrass preparation Model theory of ordered structures; o-minimality, Counting solutions of Diophantine equations, Heights, Model theory (number-theoretic aspects), Real-analytic and semi-analytic sets Uniform parameterization of subanalytic sets and Diophantine 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 $R$ be a complete discrete valuation ring of mixed characeristic, let $K$ be its fraction field and let $k$ be its residue field. Let $X$ be a proper strictly semistable scheme over $R$. Then the generic fibre $X_K$ is smooth over $K$, whereas the special fibre $X_s$ is a strictly normal crossing divisor in $X$, in particular, it is the union of smooth proper $k$-schemes $Y_1,\ldots, Y_n$ intersecting transversally. Let ${\mathcal X}$ denote the formal completion of $X$; for any subscheme $E$ of $Y$, the preimage $]E[_{\mathcal X}$ of $E$ under the specialization map ${\mathcal X}_K\to Y$ is an admissible open subset in the rigid analytic $K$-space ${\mathcal X}_K$ associated with ${\mathcal X}$. For a non empty subset $I$ of $\{1,\ldots,n\}$ put $Y_I=\cap_{i\in I}Y_i$ and $U_I=Y_I-\cup_{I'\supsetneq I}Y_{I'}$. Let $\Omega^{\bullet}_{c,I;{\mathcal X}}$ be the total complex of the bicomplex \[ \Omega^{\bullet}_{]Y_I[_{{\mathcal X}}/K}\longrightarrow (\text{ push forward of }\Omega^{\bullet}_{]Y_I-U_I[_{{\mathcal X}}/K}). \] Theorem. If $\|I\|\geq2$ then there is a spectral sequence converging to $H^*(]Y_I[_{{\mathcal X}},\Omega^{\bullet}_{c,I;{\mathcal X}})$ with \[ E_2^{pq}=H_{c,\text{rig}}^p(U_I/K)\otimes_K\bigwedge(V'_I) \] for a certain $K$-vector space $V'_I$ of dimension $\|I\|-1$. As an application one obtains two formulae comparing the Euler Poincaré characteristic $\chi_{\text{rig}}(X_s)$ of the rigid cohomology of $X_s$ with the Euler Poincaré characteristic $\chi_{dR}(X_K)$ of the de Rham cohomology of $X_K$: Proposition. \[ \chi_{\text{rig}}(X_s)-\chi_{dR}(X_K)=\sum_{\|I\|\geq2}\chi_{c}(U_I), \] \[ \chi_{\text{rig}}(X_s)-\chi_{dR}(X_K)=\sum_{\|I\|\geq2}(-1)^I(\|I\|-1)(\Delta Y_I.\Delta Y_I) \] where $\chi_{c}(U_I)$ is the Euler Poincaré characteristic of the rigid cohomology with proper support of $U_I$, and $\Delta Y_I.\Delta Y_I$ is the self intersection number of $Y_I$. One important ingredient in the proof is to elaborate further certain local cohomology calculations in a semistable reduction situation occuring in [\textit{E. Grosse-Klönne}, Duke Math. J. 113, No. 1, 57--91 (2002; Zbl 1057.14023)]. rigid cohomology; strictly semistable reduction; Euler Poincare characteristic de Rham cohomology and algebraic geometry, Schemes and morphisms A spectral sequence for de Rham 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 In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a 2-$(6, 3, 78)_5$ design by computer, which corresponds to a halving $\operatorname{LS}_5 [2](2, 3, 6)$. The application of the new recursion method to this halving and an already known $\operatorname{LS}_3 [2](2, 3, 6)$ yields two infinite two-parameter series of halvings $\operatorname{LS}_3 [2](2, k, v)$ and $\operatorname{LS}_5 [2](2, k, v)$ with integers $v \geq 6$, $v \equiv 2\pmod 4$ and $3 \leq k \leq v - 3$, $k \equiv 3 \pmod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$. $q$-analog; combinatorial design; subspace design; large set; halving Braun, M; Kiermaier, M; Kohnert, A; Laue, R, Large sets of subspace designs, J. Comb. Theory Ser. A, 147, 155-185, (2014) Other designs, configurations, Combinatorial aspects of block designs, Grassmannians, Schubert varieties, flag manifolds Large sets of subspace designs	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 examines conditions under which the effective monoid $M(X)$ of a rational surface $X$ is finitely generated. The monoid $M(X)$ of a surface $X$ is the set of effective divisor classes on $X$ modulo algebraic equivalence. The author considers the case of surfaces $X$ obtained by blowing up the projective plane along a $0$-dimensional subscheme $Z \subset {\mathbb P}^ 2$. \textit{B. Harbourne} [Duke Math. J. 52, 129--148 (1985; Zbl 0577.14025)] studied the case when $Z$ is contained in a smooth plane cubic curve and gave a characterization of finite generation of $M(X)$ in terms of the $({-}2)$-curves lying on $X$. Here, the author assumes that $Z$ is contained in a degenerate cubic that is the union of an integral conic and a line. The subscheme $Z$ can contain infinitely near points and is not necessarily disjoint from the singular locus of the cubic curve. Whereas smooth projective rational surfaces with finitely generated effective monoids and small Picard number are well understood, little is known in the case of large Picard number. Note that $X$ may have arbitrarily large Picard number. It is known that the effective monoid $M(X)$ of a rational surface obtained by blowing up ${\mathbb P}^ 2$ along the points of a constellation with the origin a single point is not necessarily finitely generated. The article treats the case when the constellation has ``support'' on a degenerate cubic curve of the type above. The article determines a numerical criterion for the finite generation of $M(X)$. As a byproduct of this criterion, the author shows that nef divisors on $X$ are regular. finite generation of monoids; rational surfaces; constellations Lahyane, M.: On the finite generation of the effective monoid of rational surfaces. J. Pure Appl. Algebra. 214, 1217-1240 (2010) Rational and ruled surfaces, Divisors, linear systems, invertible sheaves, Riemann-Roch theorems, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps, Families, moduli, classification: algebraic theory, Special surfaces, Enumerative problems (combinatorial problems) in algebraic geometry On the finite generation of the effective monoid of rational 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 An infinite $(1+1)$-dimensional chain $u_{\alpha,t}=\phi_{\alpha,1}u_{1,x}+...+\phi_{\alpha,\alpha+1}u_{\alpha+1,x}$ is investigated. The authors introduce the notion of its hydrodynamical reduction in the following way. It is a chain of semi-Hamiltonian systems $r_t^i=p_i(r^1,\dots,r^N)r^i_x$ and a function $u(r^1,\dots,r^N)$ such that for every solution $r^1,\dots,r^N$ of the semi-Hamiltonian system the function $u(r^1,\dots,r^N)$ is a solution of the infinite chain. The notion of equivalence of chains is given. By integrality of the chain the authors mean the existence of an infinite set of hydrodynamical reductions (see details in Theor. Math. Phys. 161, No. 1, 1340--1352 (2009; Zbl 1180.37096)). The purpose of the paper is a classification of integrable $(1+1)$-dimensional chains of hydrodynamical type. Some examples of $(2+1)$-dimensional chains of hydrodynamical type are constructed. The problem of classification of chains is reduced to the classification of Gibbon-Tsarev systems (in the paper -- the system (1.5),(1.6)). The structure of the paper is the following: at first the main definitions are given. Only $N=3$ case of hydrodynamical reduction is considered since the existence of a reduction with $N>3$ gives nothing new. Then the statements of the previous results used in the paper are presented. In the next section the classification of admissible polynomial coefficients Gibbon-Tsarev systems is obtained. Then the constructions of integrable chains in general case and in some degenerate cases are given. Also some examples of $(2+1)$-dimensional integrable chains in the sense of this paper of hydrodynamical type are given. Infinitesimal symmetries of Gibbon-Tsarev system are studied. integrable chains of hydrodynamical type; hydrodynamical reduction Odesskii, A.V.; Sokolov, V.V., Classification of integrable hydrodynamic chains, J. Phys. A: Math. Gen., 43, 434027, (2010) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Applications of Lie algebras and superalgebras to integrable systems, Applications of holomorphic fiber spaces to the sciences, Relationships between algebraic curves and integrable systems Classification of integrable hydrodynamic 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 In this short article, given a smooth diagonalizable group scheme $G$ of finite type acting on a smooth quasi-compact separated scheme $X$, we prove that (after inverting some elements of representation ring of $G)$ all the information concerning the additive invariants of the quotient stack $[X/G]$ is ``concentrated'' in the subscheme of $G$-fixed points $X^G$. Moreover, in the particular case where $G$ is connected and the action has finite stabilizers, we compute the additive invariants of $[X/G]$ using solely the subgroups of roots of unity of $G$. As an application, we establish a Lefschtez-Riemann-Roch formula for the computation of the additive invariants of proper push-forwards. Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Stacks and moduli problems Motivic concentration 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 The author studies a function $k\mapsto N_ x(k)=\{y\in X:$ there is a sequence $(x_ 0,x_ 1,...,x_ k)$, $x_ 0=x$, $x_ k=y$; $(x_{i- 1},x_ i)\in \Gamma_{\phi}$, $1\leq i\leq k\}$ where $\Gamma_{\phi}$ is the graph of a multivalued map (correspondence) $\phi$ from a set X into itself. The author defines the notions of a symplectic, Liouville integrable, algebraic and abelian correspondence and gives several results concerning the estimation of the number $N_ x(k)$ for generic points x. The main result states that for an integrable m-valued correspondence there is a constant A such that $N_ x(k)=Ak^{m-1}$ contrary to the natural expectation that $N_ x(k)=m^ k$. The paper contains some examples and conjectures. No proofs are given. set-valued mapping; symplectic manifolds; Liouville integrability; correspondence Veselov, A. P.: On the growth of the numbers of images of the point under the iterations of multiple-valued mapping. Mat. Zametki,49 (2) (1991) (Russian) Set-valued and function-space-valued mappings on manifolds, Rational and birational maps Growth of the number of images of a point under iterates of a multivalued 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 The aim of this paper is to study the variation of mixed Hodge structure arising from a projective morphism $f: X\to D^$ near the puncture of the disc $D$ in ${\mathbb{C}}$. Namely to construct a limit mixed Hodge structure satisfying the condition given by \textit{P. Deligne} in his article in Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014). We use the results on filtered mixed Hodge complex giving rise to spectral sequences of mixed Hodge structures [previous note, C. R. Acad. Sci., Paris, Sér. I 295, 669-672 (1982; Zbl 0511.14004)]. First we construct such a complex in the case where $X$ is a normal crossing divisor in some ambiant space smooth over $D^$, and then we give the construction for a projective morphism. This paper contains the detailed proof with comments on previous work by W. Schmid, J. Steenbrink and H. Clemens in the case of smooth morphism f. [For a short announcement of this paper see the following title.] variation of mixed Hodge structure Fouad El Zein, Théorie de Hodge des cycles évanescents, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 107 -- 184 (French, with English summary). Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), de Rham cohomology and algebraic geometry Théorie de Hodge des cycles évanescents. (Hodge theory of vanishing cycles)	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article gives ``{evidence for a connection between the work of Kuo-Tsai Chen on iterated integrals and de Rham homotopy theory on the one hand, and the work of Wei-Liang Chow on algebraic cycles on the other.}'' It surveys the subject, presents new results, and prospects further developments. Sometimes certain constructions are presented in detail, whereas at others only a rough idea is given. The result is an overview for novices and experts about the role iterated integrals play in algebraic (and differential) geometry. In the opinion of the reviewer, this article merits being worked into a book on this subject. Below are short descriptions of the content of the 14 sections of this article. Section 1: The definition of differential forms on the path space $PX$ of a smooth manifold $X$ is given. Furthermore, there exists an exterior differential, making $E^\bullet(PX)$ into a differential graded algebra (dga). Basic properties (pull back, integration along compact fibers, Stokes' theorem) are given. Section 2: Iterated integrals on $PX$ which form the Chen complex $Ch^\bullet(PX) \subset E^\bullet(PX)$ are introduced. For $\omega_1,\ldots,\omega_r \in E^\bullet(X)$, the iterated integral $\int \omega_1 \ldots \omega_r$ on $PX$ is the push forward of the form $p_1^\omega_1 \wedge \ldots \wedge p_r^\omega_r$ on $\Delta^r \times PX$ to $PX$. Here, $\Delta^r$ is the $r$ dimensional simplex and $p_i$ maps $(t_1,t_2,\ldots,t_r, \alpha)$ to $\alpha(t_i)$. The Chen complex is filtered by length and again a dga. These constructions can be carried through for natural subspaces of $PX$, like the pointed loop space $P_{x,x}X$ or $P_{x,y}$. In the case of $Ch^\bullet(P_{x,x}X)$, a Hopf algebra structure is obtained. Section 3: The cohomology of this algebra is related to the cohomology and homotopy of $P_{x,x}X$ and $X$. Section 4: It is shown how iterated integrals of holomorphic one forms give multi valued holomorphic functions. For example, the iterated integral $\int \frac{dz}{1-z} \frac{dz}{z}$ on ${\mathbb P}^1 \setminus \{ 0, 1, \infty \}$ is Euler's dilogarithm function. Section 5: Bruno Harris' construction of the harmonic volume is presented. Iterated integrals are used to calculate the volume (mod ${\mathbb Z}$) of a submanifold with given boundary. Section 6: The notation of iterated integrals is extended to currents. Here certain transversality conditions are needed to define this product. Section 7: Presentation of Chen's reduced bar construction and its connection to the Chen complex $Ch(P_{x,y}X)$ and the (co)homology of $P_{x,y}X$. Section 8: Exact sequences are deduced from the above construction which relate homotopy and cohomology of a simply connected manifold $X$. Section 9: Hodge theory is taken into account. This way Hodge and weight filtrations on $Ch^\bullet(P_{x,y}X)$ are obtained. Section 10: Application to algebraic cycles, to intermediate Jacobians, and the theorem of \textit{J. Carlson}, \textit{H. Clemens} and \textit{J. Morgan} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 14, 323--338 (1981; Zbl 0511.14005)] on the image of homological trivial cycles of codimension two in the intermediate Jacobian of a projective manifold $X$. Section 11: The coordinate ring of the unipotent closure of $\pi_1(X,x)$ is identified with the ring $H^0(Ch^\bullet(P_{x,x}X))$. Section 12: The above construction is extended to the algebraic completion of $\pi_1(X,x)$ relative to a Zariski dense homomorphism $\rho:\pi_1(X,x) \to S$ to a reductive algebraic group. Section 13: Several approaches to extend iterated integrals to algebraic de Rham theory are presented. Section 14: \textit{J.F. Adams}' cobar construction [Proc. Natl. Acad. Sci. USA 42, 409--412 (1956; Zbl 0071.16404)] is explained. This gives a cosimplicial space which models the path space $PX$. The iterated integrals are its de Rham realization. path spaces; mixed Hodge stucture; intermediate Jacobian Richard Hain, Iterated integrals and algebraic cycles: examples and prospects, Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000) Nankai Tracts Math., vol. 5, World Sci. Publ., River Edge, NJ, 2002, pp. 55 -- 118. Algebraic cycles, Algebraic topology on manifolds and differential topology, Homotopy theory and fundamental groups in algebraic geometry, Homotopy and topological questions for infinite-dimensional manifolds Iterated integrals and algebraic cycles: examples and prospects.	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 variety ${\mathcal N}_n({\mathbb F})$ of all nilpotent $n\times n$ matrices over ${\mathbb F}$ are considered, where ${\mathbb F}$ is an algebraic closed field. A partition of $n$ is a nonincreasing sequence of positive integers $\underline{\lambda}=(\lambda_1,\lambda_2,\dots,\lambda_s)$ with $\sum_{i=1}^s=n$. ${\mathcal P}(n)$ denotes the set of all partitions of $n$. The nilpotent commutator of a matrix $B\in{\mathcal N}_n({\mathbb F})$ is the set ${\mathcal N}_B$ of all matrices $A\in{\mathcal N}_n({\mathbb F})$ such that $AB=BA$. If the Jordan canonical form of $B$ is given by a partition $\underline{\lambda}\in{\mathcal P}(n)$, then the orbit of $B$ under the conjugated action of $GL_n({\mathbb F})$ on ${\mathcal N}_n({\mathbb F})$ is denoted ${\mathcal O}_B={\mathcal O}_{\underline{\lambda}}$; therefore ${\mathcal O}_{\underline{\lambda}}$ is the set of all nilpotent matrices with the Jordan canonical form given by ${\mathcal O}_{\underline{\lambda}}$. The set of all partitions that are Jordan canonical forms of matrices in ${\mathcal N}_B$ is denoted by ${\mathcal P}({\mathcal N}_B)$. The structure of the nilpotent commutator ${\mathcal N}_B$ is studied. It is shown that for $n\geq 3$ ${\mathcal P}({\mathcal N}_B)={\mathcal P}(n)$, while for $n\geq 4$ ${\mathcal P}({\mathcal N}_B)={\mathcal P}(n)$ if and only if $B^2=0$. A characterization of the pairs of Jordan canonical forms of two commuting nilpotent matrices is provided in the case of matrices with two Jordan blocks. Some sufficient conditions and some necessary conditions are given in order for partitions to be Jordan canonical forms of matrices which have the dimension of the kernel equal to 2. The map ${\mathcal D}$ on ${\mathcal P}(n)$ is studied, where ${\mathcal D}(\underline{\lambda})=\underline{\mu}$ = the largest partition such that ${\mathcal O}_{\underline{\mu}}\cap{\mathcal N}_B\neq\emptyset$. The partitions $\underline{\lambda}$ such that the parts of ${\mathcal D}(\underline{\lambda})$ differ by 2 are characterized and the preimage of ${\mathcal D}$ for certain partitions is described. nilpotent matrices; commuting matrices; nilpotent commutator; nilpotent orbit; Jordan canonical form; maximal partition Oblak, P., On the nilpotent commutator of a nilpotent matrix, Linear Multilinear Algebra, 60, 599-612, (2012) Commutativity of matrices, Canonical forms, reductions, classification, Hermitian, skew-Hermitian, and related matrices, Group actions on varieties or schemes (quotients) On the nilpotent commutator of a nilpotent matrix	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Branched covers of the complex projective line ramified over $0,1,$ and $\infty $ (Grothen\-dieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over $\infty$ and given numbers of preimages of $0$ and $1$ are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers. dessins d'enfant; branched covers; Kadomtsev-Petviashvili hierarchy P. Zograf, \textit{Enumeration of Grothendieck's dessins and KP hierarchy}, arXiv:1312.2538 [INSPIRE]. Arithmetic aspects of dessins d'enfants, Belyĭ theory, Dessins d'enfants theory Enumeration of Grothendieck's dessins and KP hierarchy	0
The authors identify iterated function systems $\Phi$ on a metric space $X$ and regular Borel measures $\mu$ such that the matrix subdivision process relative to a finite family $\mathcal A$ converges. They give a characterization of the convergence of the subdivision process to a function in $L_p(\Omega,\mu)$ in terms of the spectral properties of $\mathcal A$. First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of $\mu$-uniformity of a family of contractions for any $\mu$-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study some actions on the set $M_{n,m}$ of matrices with coefficients in the ring of formal power series $R:=K[[x_1,\ldots, x_n]]$, where $K$ is a field. They study the action of $G:=Gl_n(R)$ on $M_{n,m}$ by multiplication from the left (or from the right, or from both sides). Then they give a condition to insure the finitely determinacy of a matrix $A$. One says that a matrix $A:=(a_{ij}(x))$ is finitely determined (resp. $G$-finitely determined) if, for every matrix $(b_{ij}(x))$ whose entries coincide with those of $A$ up to some high power of the maximal ideal of $R$, there exists an automorphism $\varphi$ of $R$ such that $(\phi(b_{ij}(x)))=(a_{ij}(x))$ (resp. such that $(\phi(b_{ij}(x)))$ belongs to the orbit of $A$ under the action of $G$). The condition given in this paper is expressed in terms of the tangent image of the orbit map. In characteristic zero, they prove that this condition is not only sufficient but also necessery. These results generalize previous classical results concerning the finite determinacy of power series or of vectors of power series. matrices with power series coefficients; finite determinacy Singularities in algebraic geometry, Formal power series rings, Matrices over function rings in one or more variables On finite determinacy for matrices of power series	0

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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 In this paper the author considers the following situation generalizing results of Atiyah. Let \(\pi : X \to S\) be an arbitrary morphism of schemes. We fix \(I\) to be a quasi-coherent \(\O_X\)-module and set \(d : \mathcal{O}_X \to I\) be an \(\pi^{-1}(\mathcal{O}_S)\)-linear derivation. Then for an arbitrary coherent \(\mathcal{O}_X\)-module \(\mathcal{E}\) one can form \(J_I^1(\mathcal{E} := I \otimes \mathcal{E} \oplus \mathcal{E}\). For \(\mathcal{E} = \O_X\), \(J_I^1(\mathcal{O}_X) = J_I^1\) can be given a ring structure \((x, a)(y,b) = (xb+ay,ab)\) and can easily make \(J_I^1(\mathcal{E})\) into a left-\(J_I^1\)-module. The author then obtains a short exact sequence \(0 \to I \otimes \mathcal{E} \to J^1_I(\mathcal{E}) \to \mathcal{E} \to 0\) (which the author calls the generalized Atiyah sequence). From this the author defines the Atiyah-class of \(\mathcal{E}\) (with respect to \(J_I^1\)) as \(c_J(\mathcal{E}) \in \text{Ext}_{J_I^1}(\mathcal{E}, I \otimes \mathcal{E})\) (likewise one can take the \(\text{Ext}\) with respect to \(\O_X\)). The author then proves a number of results about this class (for instance that \(c_J(\mathcal{E}) = 0\) if and only if \(\mathcal{E}\) has an \((I, d_{\mathcal{J}})\)-connection (see the discussion above 3.5 in the paper for precise definitions). In the case where \(S = \mathbb{C}\), \(I = \Omega_{X/S}\) and \(\mathcal{E}\) is locally free, one obtains the classical Atiyah sequence see [\textit{M. F. Atiyah}, Trans. Am. Math. Soc. 85, 181--207 (1957; Zbl 0078.16002)] and corresponding extension class (with similar results). The paper concludes with a series of examples exhibiting this generalized theory. Atiyah sequence; jet bundle; characteristic class; square zero extension Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry On jets, extensions and characteristic classes. II	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For \(n\geq 1\) and \(\mu\) a Borel measure in \(\mathbb{C}^n\), the weighted log canonical threshold of a holomorphic function \(f\) defined in a neighborhood of the origin of \(\mathbb{C}^n\) is defined as \(c_{\mu}(f):=\)sup\(\{ c\geq 0\) : \(\|f\|^{-2c}\) is \(L^1(\mu)\) in a neighbourhood of \(0~\}\). Set \par \(\mathcal{C}(\mu):=\{ c_{\mu}(f)\) : \(f\) is holomorphic in a neighbourhood of \(0~\}\). \par The ACC conjecture for weight \(\mu\) reads:\par \(\mathcal{C}(\mu)\) satisfies the ascending chain condition every convergent increasing sequence in \(\mathcal{C}(\mu)\) to be stationary.\par The authors show that the conjecture holds true for \(n=2\) and \(\mu =\\| z\\| ^{2t}dV_4\), \(t\geq 0\). holomorphic function; log canonical threshold; ACC conjecture Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Lelong numbers ACC conjecture for weighted log canonical thresholds	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the property of being quasi-regular. The other, more economical, decomposition, goes back to the work of Fox-Neuwirth and Fuchs on the cohomology of braid groups. We give a criterion for a sheaf constructible with respect to the ``contingency decomposition'' to be constructible with respect to the complex stratification. We also study a polyhedral ball which we call the stochastihedron and whose boundary is dual to the two-sided Coxeter complex (for the root system \(A_n\)) introduced by \textit{T. K. Petersen} [Electron. J. Comb. 25, No. 4, Research Paper P4.64, 28 p. (2018; Zbl 1441.05243)]. The Appendix by P. Etingof studies enumerative aspects of contingency matrices. In particular, it is proved that the ``meta-matrix'' formed by the numbers of contingency matrices of various sizes, is totally positive. symmetric products; contingency matrices; stratifications; total positivity Exact enumeration problems, generating functions, General topology of complexes, Polyhedral manifolds, Contingency tables, Linear algebraic groups over the reals, the complexes, the quaternions, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Contingency tables with variable margins (with an appendix by Pavel Etingof)	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(f_t : {\mathbb C}^n \to {\mathbb C}\), \(t \in [0,1]\), be a family of polynomial maps, \(f_t \in {\mathbb C}[t,x_1, \dots, x_n]\), \(n \geq 2\), having only isolated singularities and \(\mu(t)\) the (affine) Milnor number (the sum of the local Milnor numbers at critical points of \(f_t\)) and \(\lambda(t)\) the Milnor number at infinity (the sum of the local Milnor numbers at critical points of \(f_t\) at infinity). Suppose that \(\mu(t)\), \(\lambda (t)\), the number of critical points and the number of critical points at infinity for fixed \(t\) do not depend on \(t\). Moreover, suppose that the critical values at infinity depend continuously on \(t\). Then the fibrations \(f_0^{-1}({\mathbb C} \smallsetminus B\bigl(0)\bigr) \to {\mathbb C} \smallsetminus B(0)\) and \(f_1^{-1}\bigl({\mathbb C} \smallsetminus B(1)\bigr) \to {\mathbb C} \smallsetminus B(1)\) are fibre homotopy equivalent and for \(n \not= 3\) differentiably isomorphic. Here \(B(0)\), resp.\ \(B(1)\), is the set of critical points of \(f_0\), resp.\ \(f_1\). Under the additional assumption that \(\lambda(t) = 0\) and \(n \not= 3\), it is proved that \(f_0\) and \(f_1\) are topologically equivalent. In the case \(n = 2\) the same holds without requiring \(\lambda(t) = 0\) but \(\deg(f_t)\) not depending on \(t\). \(\mu\)-constant theorem; family of polynomials; singularities at infinity; Milnor number; Milnor number at infinity [3]A. Bodin, Invariance of Milnor numbers and topology of complex polynomials, Comment. Math. Helv. 78 (2003), 134--152. Equisingularity (topological and analytic), Singularities of curves, local rings Invariance of Milnor numbers and topology of complex polynomials	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A map \(f : \mathbb{C}^n \to \mathbb{C}^n\) is called a square system of polynomial-exponential functions if it is polynomial in both the variables \(x_1,\dots,x_n\) and finitely many exponentials of the form \(e^{\beta x_i}\) where \(\beta \in \mathbb{C}\). Systems of polynomial-exponential functions arise in many applications such as engineering, mathematical physics, control theory and so on. Smale's \(\alpha\)-theory states that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Given a system of polynomial-exponential equations, the authors consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. Furthermore, they discuss methods for generating numerical approximations to solutions of polynomial-exponential systems. Finally, based on these results, they describe an algorithm for certifying solutions to polynomial-exponential systems, and demonstrate their algorithm on several examples. certified solutions; alpha theory; polynomial system; polynomial-exponential systems; numerical algebraic geometry; alphacertified Hauenstein, J. D.; Levandovskyy, V.: Certifying solutions to square systems of polynomial-exponential equations. (Sept. 2011) Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation, Numerical computation of solutions to systems of equations Certifying solutions to square systems of polynomial-exponential 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 [For the entire collection see Zbl 0518.00005.] The paper under review deals with the following problem: given morphisms of affine schemes \(s: \tilde X\to Y,\) \(p: Y\to X\) with p finitely presented, find a morphism \(f: Z\to Y\) such that s factors through f, pf is smooth and f is ''smooth except above the singular locus of p''. This problem is motivated by work on approximation property. The authors solve the problem in two cases: \(\tilde X=X\) and in the isolated singularity case (i.e. \(\tilde X,\) X are spectra of henselian local rings with the same completion and p is smooth at every point of s(X) except the closed point). Meanwhile a complete answer to the above problem was given by \textit{M. Cipu} and \textit{D. Popescu}, ''A desingularisation theorem of Néron type'' [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 30, 63-76 (1984)]; this paper in its turn is based on the papers: \textit{D. Popescu}, ''General Néron desingularisation'' [to appear in Nagoya Math. J. 100 (1985)] and \textit{D. Popescu}, ''General Néron desingularisation and approximation'' (to appear). factorization of smooth maps; morphisms of affine schemes; approximation property; isolated singularity 6. M. Artin and J. Denef , Smoothing of a ring homomorphism along a section, in Arithmetic and Geometry, Vol. 2 (Birkhäuser, Boston, 1983), pp. 5-32. genRefLink(16, 'S0219498817500724BIB006', '10.1007%252F978-1-4757-9286-7_2'); Local structure of morphisms in algebraic geometry: étale, flat, etc., Singularities in algebraic geometry, Local deformation theory, Artin approximation, etc. Smoothing of a ring homomorphism along a section	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 considers continuous actions of a non-compact real reductive Lie group \(G\) on a Hausdorff topological space \(\mathcal{M}\). The aim of the authors is to develop a geometric invariant theory for such actions, inspired by well-known results for actions on Kähler manifolds. More precisely, write \(G=K\cdot \exp(\mathfrak{p})\) where \(K\) is a maximal compact subgroup of \(G\). In this general setting, in Section 2 the authors define Kempf-Ness functions as maps \(\Psi: \mathcal{M} \times G\to \mathbb R\) satisfying certain conditions. Such a Kempf-Ness function gives rise to a map \(\mathfrak{F}: \mathcal{M}\to \mathfrak{p}^*\), called gradient map. In Section 5, given a point \(x\in \mathcal{M}\), the authors define when \(x\) is polystable (resp. stable, semi-stable, unstable) in terms of the intersection of the \(G\)-orbit of \(x\) with \(\mathfrak{F}^{-1}(0)\). A feature is that these various notions of stability can be detected by a numerical criteria (see Sections 5 and 6), using the maximal weight function introduced in Section 4. The end of the paper is devoted to examples: Section 7 addresses the classical gradient map for Kähler manifolds extending a construction of \textit{I. Mundet i Riera} [Trans. Am. Math. Soc. 362, No. 10, 5169--5187 (2010; Zbl 1201.53086)], Section 8 addresses the action on the space of measures on \(\mathcal{M}\), and Section 9 exemplifies explicitly the latter when \(\mathcal{M}\) is real projective space. reductive Lie group; gradient maps; geometric invariant theory; stability; polystability; semi-stability L. Biliotti and M. Zedda. Stability with respect to actions of real reductive Lie groups. Ann. Mat. Pura Appl. (4), 196(6), 2185-2211, 2017. Geometric invariant theory, Momentum maps; symplectic reduction Stability with respect to actions of real reductive Lie groups	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Thanks to the the author's previous work, it is possible to reconstruct a scheme from its derived category of perfect complexes, provided we equip the latter with its derived tensor product. The scheme is recovered by means of the ``triangular spectrum'' construction, \(\mathrm{Spc}(T)\), which can be applied to any abstract tensor triangulated category~\(T\) to yield a nice (locally ringed) space in a functorial way. This is the starting point of ``tensor triangular geometry'', which is a geometric theory of tensor triangulated categories generalizing algebraic geometry (see [\textit{P. Balmer}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 85--112 (2011; Zbl 1235.18012)]). This very short paper addresses the question: How should we generalize the definition of the Chow group of a scheme, so that it still make good sense for more general tensor triangulated categories? Assuming that the tensor triangulated category~\(T\) is rigid and its spectrum Noetherian (which is quite reasonable), the proposed definition reads as follows. The group of ``generalized \(p\)-cycles'' of~\(T\) is the direct sum, over all \(p\)-dimensional points \(P\in \mathrm{Spc}(T)\) in the spectrum, of the (triangulated) Grothendieck \(K\)-group \(K_0(\mathrm{Min}(T_P))\), where \(\mathrm{Min}(T_P)\) denotes the subcategory of objects with minimum support in \(T_P\), the local category of \(T\) at~\(P\). This notion works for any reasonable dimension function on the spectrum. The Chow group of~\(T\) is then obtained from the group of cycles by quotienting out a suitable subgroup of boundaries. Two different definitions of boundaries are proposed here, and at least one of them allows to recover the Chow group of a scheme in the regular case. Both definitions make geometric sense in view of the classical theory. Their precise relationship and several other questions are investigated by the author's PhD student, Sebastian Klein. Chow groups; triangulated categories; triangular spectrum Balmer, P, Tensor triangular Chow groups, J. Geom. Phys., 72, 3-6, (2013) Derived categories, triangulated categories, Applications of methods of algebraic \(K\)-theory in algebraic geometry Tensor triangular Chow groups	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group and \(\Lambda(G)\) be the set of irreducible complex representations of \(G\). Associated to a fixed finite-dimensional representation \(V\), are the numbers \(m_k^\lambda\), which are the multiplicity of \(\lambda\in \Lambda(G)\) in \(V^{\otimes k}\), \(A=(a_{\lambda \, \mu})\), the adjacency matrix of the representation graph \(R_V(G)\), and the Bratteli diagram which is the infinite graph with vertices labeled by the elements of \(\Lambda(G)\) on level \(k\), keeping tracks of finite steps walks on \(R_V(G)\). The Poincaré series for \(\lambda\in \Lambda(G)\) is defined by \[ m^{\lambda}(t)=\sum_{k=0}^\infty m_k^\lambda t^k. \] The main result of the paper under review is as follows: suppose that \(G\) acts faithfully on \(V\), and \(V^* \cong V\) as \(G\)-representations. Let \(M_\mu\) be the matrix \(I - tA\) with the column indexed by \(\mu\) replaced by \((1\; 0\; 0 \dots 0)^T\). Then \[ m^\mu(t)=\frac{\det M_\mu}{\det(I-tA)}. \] In the special case where \(G\) is a finite subgroup of \(\mathrm{SU}(2)\), and \(V=\mathbb{C}^2\) is its natural representation, then \[ m^0(t)=\frac{\det(I-t\hat{A})}{\det(I-tA)}, \] where \(0\in \Lambda(G)\) is the trivial representation, and \(\hat{A}\) is the adjacency matrix of the finite Dynkin diagram obtained by removing the affine node. In the continuation, the paper under review finds further explicit formulas for \(m^0(t)\) and express the results in terms of Chebyshev polynomials, and the Bratteli diagrams. tensor invariants; Poincaré series; McKay correspondence; Schur-Weyl duality Benkart, G., Poincaré series for tensor invariants and the mckay correspondence, Adv. Math., 290, 236-259, (2016) McKay correspondence, Combinatorial aspects of representation theory, Group rings of finite groups and their modules (group-theoretic aspects) Poincaré series for tensor invariants and the McKay correspondence	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [See also the review to the journal version in Invent. Math. 117, 165-180 (1994; Zbl 0822.58046).] In a previous paper [Adv. Sov. Math. 16, No. 1, 211-241 (1993; Zbl 0802.58051)] the authors proved a version of the classical Riemann-Roch theorem for solutions of general elliptic equations with point singularities. In this paper they extend the results to much more general singularities supported on arbitrary compact nowhere dense subsets. The only restriction is that the allowed singularities should be taken from a finite-dimensional space. Dually a finite set of conditions may be imposed on another nowhere dense compact set. This leads to a notion of rigged divisor which includes two nowhere dense compact sets with finite- dimensional distribution spaces supported on them. Then the allowed singularities on the first given set are described as singularities of solutions which may be extended as distributions to the whole given manifold so that after applying the given elliptic operator one gets into the first given space of distributions. The conditions on the second compact set are just orthogonality conditions to the second space of distributions. The main theorem of the authors then connects the dimension of the space of solutions having the allowed singularities and satisfying the imposed conditions, with another dimension defined in the same way from the dual (or inverse) divisor which is obtained by changing places of two given compact subsets and distribution spaces and replacing the given operator by the adjoint operator. In distinction to their previous paper, the authors here restrict to closed manifolds. general Riemann-Roch theorem; elliptic operators [GrS2]M. Gromov, M. Shubin, The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets, Preprint ETH Zürich, 1993. Elliptic equations on manifolds, general theory, Exotic index theories on manifolds, Higher-order elliptic equations, Riemann-Roch theorems The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 [Duke Math. J. 53, 457--502 (1986; Zbl 0679.14001)], \textit{G. W. Anderson} showed that the following are equivalent: (1) an Abelian \(t\)-module E is uniformisable, which roughly means that it permits an analytic description as a quotient \(V/\Lambda\) of a finite dimensional \(\mathbb C_\infty\)-vector space \(V\) by a free and discrete sub-\(\mathbb F_q[t]\)-module \(\Lambda\), (2) the associated \(t\)-motive \(M\) is analytically trivial, which roughly means that it admits a \(\tau\)-invariant basis after extending scalars from \(\mathbb C_{\infty[t]}\) to the Tate algebra \(\mathbb C\langle t\rangle\) (the reader who is unfamiliar with these notions is strongly encouraged to consult Anderson (loc. cit.) first). The paper under review studies how these notions behave in families. The locus of uniformisable \(t\)-modules in an algebraic family is usually not algebraic: it typically depends on inequalities between valuations of coordinates. (The reader who is unfamiliar with this phenomenon is encouraged to read section 7 of the paper under review first, it contains a detailed treatment of an example due to Pink.) It is therefore most natural to study uniformisability in families over a rigid analytic base, and this is exactly what the authors do. After carefully working out the notions of a rigid analytic family of \(t\)-motives (and of \(t\)-modules), and of analytic triviality (and uniformisability) of such families. The main result (Theorem 5.5) states that the locus of uniformisability in a rigid analytic family is Berkovich open (i.e., the set of analytic points over which the given family is uniformisable is open in the topological space of all analytic points of the base.) The exposition is excellent and the authors have done considerable effort to recall the necessary notions and results from rigid analytic geometry. Drinfeld modules; t-motives; uniformisability; rigid analytic geometry Gebhard Böckle & Urs Hartl , Uniformizable families of \(t\) -motives , Trans. Am. Math. Soc. 359 (2007), p. 3933-3972 Drinfel'd modules; higher-dimensional motives, etc., Rigid analytic geometry Uniformizable families of \(t\)-motives	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Determining when a 0-dimensional scheme in projective space \(\mathbb P^n_K\) (where \(K\) is a field) which is the complete intersection of \(n\) hypersurfaces has played an important role in both Algebraic Geometry and Commutative Algebra. For example, in projective 2-space, work of \textit{E. Davis} and \textit{P. Maroscia} [Lect. Notes Math. 1092, 253--269 (1984; Zbl 0556.14025)] provided a characterization of such schemes using the Hilbert function symmetry and the Cayley-Bacharach property. In [\textit{E. D. Davis} et al., Proc. Am. Math. Soc. 93, 593--597 (1985; Zbl 0575.14040)] and [\textit{M. Kreuzer}, Math. Ann. 292, No. 1, 43--58 (1992; Zbl 0741.14030)], it was then shown that this also provides a characterization for arithmetically Gorenstein schemes. There have also been characterizations of 0-dimensional local complete intersections using the Kähler different (see [\textit{G. Scheja} and \textit{U. Storch}, J. Reine Angew. Math. 278/279, 174--190 (1975; Zbl 0316.13003)]). In this paper the authors uses the Kähler and Dedekind differents of \(R/K[x_0]\) and \(R/K\) to further the characterizations of 0-dimensional complete intersections. A sample characterization is the following theorem: Theorem. Let \(\mathbb X\) be a smooth 0-dimensional subscheme of \(\mathbb P^n_K\). Then \(\mathbb X\) is a complete intersection if and only if \(\mathbb X\) is a Cayley-Bacharach scheme and the Hilbert function of the Kähler different of \(\mathbb X\) in degree \(r_{\mathbb X}\) is non-zero, where \(r_{\mathbb X}\) is the regularity index of the Hilbert function of \(\mathbb X\). Most of the characterizations can be found in Section 5 of the paper. Indeed, in this section the authors present three characterizations: one generalizing the work of Scheja and Storch; one using a value of the Hilbert function of the Kähler different of the scheme to separate the classes of complete intersections from arithmetically Gorenstein schemes; and one is the above stated theorem. Moreover, using the Dedekind different rather than the Kähler different, the authors characterize 0-dimensional arithmetically Gorenstein schemes. One particular gem of this paper is its useful exposition with many included necessary definitions and references to the literature. zero-dimensional scheme; complete intersection; Kähler different; Dedekind different; arithmetically Gorenstein scheme; Cayley-Bacharach scheme; Hilbert function Complete intersections, Modules of differentials, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Characterizations of zero-dimensional 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 This paper gives a useful generalization of the usual Castelnuovo-Mumford's regularity to the case of weighted projective space, a generalization different from the one of \textit{D. Maclagan} and \textit{G. Smith} [J. Reine Angew. Math. 571, 179--212 (2004; Zbl 1062.13004)] and the multigraded regularity of \textit{N. Botbol} and \textit{M. Chardin} [J. Algebra 474, 361--392 (2017; Zbl 1368.13019)]. With their definition (which obviously depends on the weights of the weighted projective space) the authors are able to prove a theorem for the generation the multigraded ring associated to a coherent sheaf $\mathcal{F}$ and a definition of weight regularity for $\mathcal{F}$ so that $m$-weight regularity implies $(m+1)$-weight regularity. Castelnuovo-Mumford regularity; weighted projective spaces; multigraded ring Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Syzygies, resolutions, complexes and commutative rings Weighted Castelnuovo-Mumford regularity and weighted global generation	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(w=(w_ 0,...,w_ n)\) be a set of integer positive weights and let S be the polynomial ring \({\mathbb{C}}[x_ 0,...,x_ n]\) graded by the conditions \(\deg (x_ i)=w_ i\). Let f be a weighted homogeneous polynomial of degree \( N.\) The Milnor fibration of f is the locally trivial fibration \(f:\;{\mathbb{C}}^{n+1}\setminus f^{-1}(0)\to {\mathbb{C}}\setminus \{0\}\) with typical fiber F: \(f^{-1}(1).''\) Spectral sequence techniques are used to arrive at a formula for the cohomology groups of F. In the case of isolated singularities more is known about F [see for example \textit{E. Brieskorn}, Manuscr. Math. 2, 103- 161 (1970; Zbl 0186.261)]. The results in this work apply to more general f with (possibly) nonisolated singularities. There is a good number of interesting examples of explicit calculations using the spectral sequences introduced which illustrate why they should be useful in many situations. - A theorem is stated relating a filtration induced by one of the spectral sequences introduced and the Hodge filtration, and it allows one to extend a result of \textit{P. A. Griffiths} [Ann. Math., II. Ser. 90, 460-495; 496-541 (1969; Zbl 0215.081)] on reductions of rational n-forms on (complex) projective spaces with a nonsingular polar locus to the singular case. The author points out in a note that the theorem is correct (although the proof given is flawed), consequence of a more general result in work done with \textit{P. Deligne} [``Hodge and order of the pole filtrations for singular hypersurfaces'' (preprint); see also Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, No.4, 645-656 (1990)]. nonisolated singularity; Milnor fibration; spectral sequences A. DIMCA, On the Milnor fibrations of weighted homogeneous polynomials, Composito Math., 76 (1990), pp. 19-47. Zbl0726.14002 MR1078856 Singularities in algebraic geometry, Families, fibrations in algebraic geometry, Complex singularities, Singularities of differentiable mappings in differential topology On the Milnor fibrations of weighted homogeneous polynomials	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R_d=\mathbb R[x,y]_d\) be the space of degree-\(d\) real binary forms. A rank \(r\) is typical in \(R_d\) if it is realized in an open (euclidean) subset of \(R_d\), or equivalently if the interior \(\mathcal R_{d,r}\) of the semi-algebraic set \(\{f\in R_d \ \| \ rk_{\mathbb R}(f)=r\}\) is not empty. Typical ranks for \(R_d\) are exactly the ranks \(r\) such that \(\frac{d+1}{2}\leq r \leq d\). Let \(\partial(\mathcal R_{d,r})\) be the topological (euclidean) boundary in \(R_d\). One defines the algebraic boundary \(\partial_{alg}(\mathcal R_{d,r})\) to be the Zariski closure of \(\partial(\mathcal R_{d,r})\) in the complex space \(\mathbb P(\mathbb C[x,y]_d)\). The algebraic boundaries for minimum rank \(r=\lceil\frac{d+1}{2}\rceil\) and maximum rank \(r=d\) (for any \(d\)) have already been described, as well as the ones for any typical rank \(\frac{d+1}{2}\leq r \leq d\) when \(d\leq 8\). In this work the authors extend this study to any typical rank \(r\) and any degree \(d\). More precisely, given \(\lambda=(\lambda_1\leq \ldots \leq \lambda_k)\) a partition of \(d\) having length \(k\), the associated coincident root locus is the \(k\)-dimensional variety \[ \Delta_{\lambda}=\left\{f \in \mathbb P^d=\mathbb P(\mathbb C[x,y]_d) \ \bigg\| \ \exists l_1,\ldots, l_k \in \mathbb C[x,y]_1 \ : \ f=\sum_{i=1}^kl_i^{\lambda_i}\right\} \subset \mathbb P^d \ ; \] then the authors' main results (Theorem 3.2, Theorem 3.3) describe the algebraic boundaries for any degree \(d\) and any typical rank \(r\) as the union of dual varieties to coincident root loci \(\Delta_\lambda\), as \(\lambda\) varies among certain partitions of \(d\) whose length depends on \(r\). The main tools used to reach the results are the \(j\)-th higher associated variety \(C\!H_j(X)\) to a \(k\)-dimensional variety \(X\subset \mathbb P^d\), defined as the closure of the subset in the grassmannian \(\mathbb G(d-k-1+j,\mathbb P^d)\) \[ \left\{H\in \mathbb G(d-k-1+j,\mathbb P^d) \ \| \ H \cap X \neq \emptyset , \ \dim(H \cap T_xX)\geq j \ \text{for some } x \in H\cap X_{sm}\right\} \ , \] and the apolar map \[ \begin{matrix} \Psi_{d,r}: & \mathbb P^d=\mathbb P(\mathbb C[x,y]_d) & \dashrightarrow & \mathbb G(2r-d-1,\mathbb P^r)\\ & f & \mapsto & \mathbb P((f^\perp)_r) \end{matrix} \ , \] where, for \(d-r\leq r\leq d\), \((f^\perp)_r\subset \mathbb C[\partial_x,\partial_y]_r\) is the degree-\(r\) component of the apolar ideal \(f^\perp\). In Section 1 the basic notions and results about coincident root loci, higher associated varieties and apolar maps are recalled. In Section 2 the authors give a detailed study of the pullbacks via apolar maps of the higher associated varieties to a coincident root locus.\\ In section 3 the main results are proven. typical rank; real rank boundary; algebraic boundary; binary form; multiple root locus; coincident root locus; Waring problem; apolar map Secant varieties, tensor rank, varieties of sums of powers, Semialgebraic sets and related spaces, Multilinear algebra, tensor calculus Algebraic boundaries among typical ranks for real binary forms of arbitrary 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 In this paper it is proved that there exist completely normal spaces that are not real spectra of rings. Real spectra of rings are known to be spectral spaces in the sense of Hochster and the problem of characterizing topologically real spectra has been considered from the beginning of this theory. A property which is quite easy to notice for real spectra, and closely linked to the ordered structures of fields considered in this theory, is the following, called complete normality: whenever points \(x\) and \(y\) are specializations of the same point \(z\) (this means that they belong to the closure of the singleton \(\{z\})\), then either \(x\) is a specialization of \(y\) or \(y\) is a specialization of \(x\). This paper proves that in order to find a topological description of real spectra, new topological properties of real spectra have to be identified. A completely normal spectral space is constructed which is not the real spectrum of a ring. The example presented is one dimensional which is the first dimension where such a result can be hoped. The construction relies on the Dedekind completion -- through cuts -- of completely dense totally ordered sets and to a construction due to Hausdorff of an \(\eta_2\)-set (where \(\omega_2\) is the second infinite cardinal): it is a set \(Y\) such that for every two subsets \(A\) and \(B\) of \(Y\) such that \(A < B\) with cardinality less than \(\omega_2\), there exist \(y\) in \(Y\) such that \(A < y < B\). completely normal spaces that are not real spectra of rings; spectral spaces Delzell, Charles N.; Madden, James J., A completely normal spectral space that is not a real spectrum, J. Algebra, 169, 1, 71-77, (1994) Real algebraic sets, Real and complex fields, Relevant commutative algebra A completely normal spectral space that is not a real spectrum	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 real analytic set in \(\mathbb R^n\) and \(x\in X\). For a relatively compact neighbourhood \(U\) of \(x\) in \(\mathbb R^n\) and a function \(f\in C^{\infty}(\overline{U})\), we define \[ \\|f\\|_{m,X}^U= \inf_h \|h\|_m^U \] over all \(h\in C^\infty(\overline{U})\) such that \(h\|X=f,\) where \[ \|h\|_m^U= \sum_{\|\gamma\|\leq m}\|D^\gamma h\|^U \] and \(\|\cdot\|^U\) is the ordinary sup-norm on \(U\). The author proves a generalization of the Gagliardo-Nirenberg inequality for analytic sets \(X\) at a point \(x\in X\): there exist \(U\ni x\), a real number \(s\geq1\) and constants \(C_m>0,\) \(m\geq0\), such that for all \(f\in C^\infty(\overline{U})\) (\(f\not \equiv 0\) on \(X\cap U\)) \[ \frac{\\|f\\|_{k,X}^U}{\\|f\\|_{0,X}^U}\leq C_0C_m^k\left( \frac{\\|f\\|_{m,X}^U} {\\|f\\|_{0,X}^U}\right)^{\frac{sk}{m}} \] (for \(X=\) compact domain in \(\mathbb R^n\) with ``good'' boundary and \(s=1\), we obtain the classical Gagliardo-Nirenberg inequality). He gives also examples of algebraic sets for which \(s>1\) at an isolated singularity. real analytic set; Gagliardo-Nirenberg inequality Local complex singularities, Real-analytic sets, complex Nash functions, Real-analytic and semi-analytic sets, Real algebraic sets A note on Gagliardo-Nirenberg type inequalities on analytic sets	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate an approximate projecting of the Banach space element in the neighborhood of still points of a given resolving operator onto the corresponding stable manifold. The projection operator is given by a basis, which describes an admissible displacement. Thus, the original problem is reduced to the solution of a special kind of a nonlinear equation. Solvability of this equation at standard conditions is proved. Local equivalence of the given method to known algorithms of the stable manifold approximation as well as high efficiency of the method is demonstrated. Numerical calculations are performed for the two-dimensional Chaffey-Enfant equation. approximate projecting; Hadamard-Perron theorem; stable manifold; Chaffey-Enfant equation Invariant manifold theory for dynamical systems, Projective techniques in algebraic geometry, Theoretical approximation of solutions to ordinary differential equations, Stability theory for smooth dynamical systems, Numerical nonlinear stabilities in dynamical systems On approximate projecting on a stable manifold	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a successor to \textit{B. Elazar} and \textit{A. Shaviv} [Can. J. Math. 70, No. 5, 1008--1037 (2018; Zbl 1445.14078)] in which the authors succeeded to extend the famous class of Schwartz functions on \(\mathbb{R}^n\) to real algebraic varieties which might have singularities; it also uses many results from \textit{A. Aizenbud} and \textit{D. Gourevitch} [Int. Math. Res. Not. 2008, Article ID rnm155, 37 p. (2008; Zbl 1161.58002)] which was among the first to study more deeply Schwartz functions on Nash manifolds. In the current paper the categories of Nash varieties and real algebraic varieties are put under the umbrella of a new category, called Quasi-Nash of which the former are subcategories. After a very careful outline of context and main results, given in Section 1, Section 2 presents the technical preliminaries of the study. Here semialgebraic subsets are defined (with a subtle logical error), the concept of complete variety, the algebraic Alexandrov compactification of noncomplete varieties, various results on Fréchet spaces and versions of the open mapping theorem and the Hahn Banach theorem for such spaces are recounted. As background or complementary literature here serve the well known books by \textit{F. Trèves} [Topological vector spaces, distributions and kernels. New York-London: Academic Press (1967; Zbl 0171.10402)] and \textit{J. Bochnak} et al. [Real algebraic geometry. Transl. from the French. Rev. and updated ed. Berlin: Springer (1998; Zbl 0912.14023)], as well as the paper by Aizenbud and Gourevitch [loc. cit.]. The author often works with restricted topological spaces. These deviate from the usual concept by requiring of the family of `open' sets only that it is closed with respect to finite unions. Continuity is defined as usual. Among the central notions are these: an \( \mathbb{R}\)-space is a pair \((M,\mathcal O_M)\) consisting of a restricted topological space \(M\) and a sheaf \(\mathcal O_M\) of \(\mathbb{R}\)-algebras over \(M\) which is a subsheaf of the the \(\mathbb{R}\)-algebra of all continuous functions on \(M.\) A continous map \(\varphi: (M,\mathcal O_M)\rightarrow (N,\mathcal O_N) \) is a morphism of \(\mathbb{R}-\)spaces if for any open set \(U\subset N\) and any \(f\in \mathcal O_N(U)\) there holds \((f\circ \varphi)\|_{\varphi^{-1}(U)} \in \mathcal O_M(\varphi^{-1}(U)).\) A Nash submanifold of \(\mathbb{R}^n\) is simply a semialgebraic subset of \( \mathbb{R}^n\) which is a smooth submanifold; a Nash function on \(M\) is a smooth semialgebraic function; an affine Nash manifold is an \( \mathbb{R}\)-space which is isomorphic to the \( \mathbb{R}\)-space associated to a closed Nash submanifold; and a Nash manifold itself is an \(\mathbb{R}\)-space \((M,\mathcal N_M)\) where \(M\) admits a finite open cover \(\{M_i\}_{i=1}^n\) so that every \( \mathbb{R}\)-space \((M_i, \mathcal N_{M_i})\) is an affine Nash manifold. Bochnak et. al. [loc. cit.] contains not all of these concepts or not in this form. The same is true of the concept of a Nash differential operator \(D\) on an affine Nash manifold which is an element of the algebra with 1 generated by multiplication and derivations along Nash sections of the tangent bundle. The space of Schwartz functions on such an \(M\) is \(\mathcal S(M)=\{\phi\in C^\infty (M): D\phi \text{ is bounded for any Nash differential operator }\}. \) The norm used is defined by \(\|\|\phi\|\|_D:=\sup_{x\in M} \|D\phi(x)\|.\) A function \(t:\mathbb{R}^n\rightarrow \mathbb{R}\) is tempered if it is smooth and all its derivatives are polynomially bounded. Tempered functions on \(M\) are naturally defined, then, via closed embeddings \(i:M\hookrightarrow \mathbb{R}^n.\) Section 3, titled `Geometry', defines gradually the new category: Just as before happened with the concept of a Nash manifold, the concept of a `Quasi Nash variety' is also defined by a tower of notions. First the Naive Quasi Nash (NQN) category is defined: its objects are the locally closed semialgebraic subsets of \(\mathbb{R}^n;\) and if \(X\subset \mathbb{R}^n,\) \(Y\subset \mathbb{R}^m\) are such sets, then \(\varphi: X \rightarrow Y\) is a morphism in the category if there exists an open set \(U\supset X\) and a Nash map \(g: U\rightarrow \mathbb{R}^m\) such that \(g_{\|X}=\varphi.\) Next an affine Quasi Nash (QN) variety is an \(\mathbb{R}\)-space which is isomorphic to an \(\mathbb{R}\)-space obtained from appropriate `sheafificaction' of some closed NQN set. At that point the author can prove already his Lemma 3.6: Affine Nash manifolds and affine real algebraic varieties form a subcategories of affine QN-varieties; the former subcategory is full; but not so the latter. Finally a QN-variety is an \(\mathbb{R}\)-space \((X,QN_X)\) so that \(X\) is a restricted topological space which admits a finite open cover \(\{X_i\}_{i=1}^m\) so that the \(\mathbb{R}\)-spaces \((X_i, QN_{X\|X_i})\) are affine QN varieties. Section 4, titled `Schwartz functions, tempered functions and distributions' , consists of three subsections aptly dubbed Schwartz functions on, respectively, Naive Quasi Nash sets (4.1); Affine Quasi Nash varieties (4.2); and Quasi Nash Varieties (4.3). This happens in parallelism with the gradual ascent to this general concept made in section 3. What concerns Section 4.1, if \(X\) is a NQN set and \(U\supset X\) open, let \(I_{\mathrm{Sch}}^U(X)\) be the ideal of Schwartz functions on \(U\) that vanish on \(X.\) Since, as shown in Corollary 4.1.1, the quotient \(\mathcal S(U)/ I_{\mathrm{Sch}}^U(X)\) does not depend on \(U,\) one can define \(\mathcal S(X)\) as being this quotient. It is also a Fréchet space. If \(X_1\stackrel{\varphi}{\rightarrow} X_2\) is an NQN isomorphism then the pullback \(\mathcal S(X_2) \stackrel{\varphi\|\mathcal S(X_2)}{\rightarrow} \mathcal S(X_1)\) is an isomorphism of Fréchet spaces. Like a few other theorems this result occurs (formulated for Schwartz functions on (general) Quasi Nash varieties) again in Section 4.3 as Lemma 4.3.5 and is one of the main results of the paper. In 4.2 first the concept of Schwartz function is extended for the case that \(X\) is an affine QN variety: \(s\) is a Schwartz function on \(X\) if it is a pullback from the closed NQN set corresponding to \(X.\) Another concept for laying down the other main result is defined: Let \(X\) be an affine QN variety corresponding to the closed set \(\tilde X\subset \mathbb{R}^n .\) Then a real valued function \(f\) defined on \(\tilde X\) is flat at \(\tilde p\) if there exists an open neighbourhood \(U\) of \(\tilde p\) and an analytic function \(F\) such that \(f\|_{U\cap \tilde X}= F\|_{U\cap \tilde X}\) and the Taylor series of \(F\) around \(\tilde p\) is identically zero. For \(X\) an affine QN variety the space of continuous linear functionals is called the space of tempered distributions, denoted \(\mathcal S^ (X).\) Section 4.3: Let \(\text{Func}(X,R)\) denote the set of real valued functions on a set \(X.\) Let \(X\) be a (general) QN variety with a finite open affine QN cover \(X=\bigcup_{i=1}^m X_i.\) The natural map \(\psi: \bigoplus_{i=1}^m \text{Func}(X_i,R) \rightarrow \text{Func}(X,R)\) is used to define the Schwartz functions on \(X\) as \(\mathcal S(X)= \psi \left( \bigoplus_{i=1}^m \mathcal S(X_i) \right).\) It is shown via an isomorphism that this definition is good (not dependent on the cover) and \(\mathcal S(X)\) again Fréchet space. The notion of flatness is duely adapted and after further work the second main result is obtained. Theorem 4.3.15: Let \(X\) be a QN variety and \(Z\subset X\) a closed subset. Let \(U=X\setminus Z\) and \(W_Z:=\{\phi\in \mathcal S(X): \phi \text{ is flat on (the points of)} Z\}.\) Then \(W_Z\) is a closed subspace of \(\mathcal S(X)\) and hence Fréchet. One has a Fréchet space isomorphism \(W_Z\cong \mathcal S(U)\) which is given by the extension by zero from \(U\) to \(X,\) and its inverse, the restriction from \(X\) to \(U.\) Th 4.3.17: Under similar conditions the extension defines a closed embedding \(\mathcal S(U)\hookrightarrow S(X)\) and the restriction morphism \(\mathcal S^(X) \rightarrow \mathcal S^(U)\) is onto. Concerning sections 5 and 6, it is proved that tempered functions and tempered distributions form sheaves and that Schwartz functions define a cosheaf. Further a characterization of Schwartz sections on open subsets for vector bundless are given. The proofs are said to be largely as in Elazar and Shaviv [loc. cit.] and Aizenbud and Gourevitch [loc. cit] with adjustments. An appendix prepares the reader with some definitions and facts to adapt a partition of unity theorem found again in Aizenbud and Gourevitch to the current case. The author makes a very honest attempt to make his topic accessible and by and large seems very careful with his definitions. The towers of notions building one upon the other make it nevertheless demanding reading for non-specialists. affine algebraic varieties; semi-algebraic sets; Nash maps; Nash manifolds; categories; tempered distributions; tempered functions Nash functions and manifolds, Real algebraic sets, Semialgebraic sets and related spaces, Linear function spaces and their duals Quasi-Nash varieties and Schwartz functions on them	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we introduce constructible analogs of the discrete complexity classes \(\mathbf {VP}\) and \(\mathbf {VNP}\) of sequences of functions. The functions in the new definitions are constructible functions on \(\mathbb {R}^n\) or \(\mathbb {C}^n\). We define a class of sequences of constructible functions that play a role analogous to that of \(\mathbf {VP}\) in the more classical theory. The class analogous to \(\mathbf {VNP}\) is defined using Euler integration. We discuss several examples, develop a theory of completeness, and pose a conjecture analogous to the \(\mathbf {VP}\) versus \(\mathbf {VNP}\) conjecture in the classical case. In the second part of the paper we extend the notions of complexity classes to sequences of constructible sheaves over \(\mathbb {R}^n\) (or its one point compactification). We introduce a class of sequences of simple constructible sheaves, that could be seen as the sheaf-theoretic analog of the Blum-Shub-Smale class \(\mathbf {P}_\mathbb {R}\). We also define a hierarchy of complexity classes of sheaves mirroring the polynomial hierarchy, \(\mathbf {PH}_\mathbb {R}\), in the B-S-S theory. We prove a singly exponential upper bound on the topological complexity of the sheaves in this hierarchy mirroring a similar result in the B-S-S setting. We obtain as a result an algorithm with singly exponential complexity for a sheaf-theoretic variant of the real quantifier elimination problem. We pose the natural sheaf-theoretic analogs of the classical \(\mathbf {P}\) versus \(\mathbf {NP}\) question, and also discuss a connection with Toda's theorem from discrete complexity theory in the context of constructible sheaves. We also discuss possible generalizations of the questions in complexity theory related to separation of complexity classes to more general categories via sequences of adjoint pairs of functors. constructible functions; constructible sheaves; polynomial hierarchy; complexity classes; adjoint functors H. Edelsbrunner, J. Harer, \textit{Computational Topology. An Introduction}, (American Mathematical Society, 2010). Complexity classes (hierarchies, relations among complexity classes, etc.), Semialgebraic sets and related spaces, Topology of real algebraic varieties, Symbolic computation and algebraic computation A complexity theory of constructible functions and sheaves	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any point \(x = (x_1, x_2) \in \mathbb{R}^2\) we let \(G_x = \mathbb{Z} x_1 + \mathbb{Z} x_2 + \mathbb{Z}\) be the subgroup of the additive group \(\mathbb{R}\) generated by \(x_1, x_2, 1\). When \(\operatorname{rank}(G_x) = 3\) we say that \(x\) is a \textit{rank 3 point.} We prove the existence of an infinite set \(\mathcal{I} \subseteq \mathbb{R}^2\) of rank 3 points having the following property: For every two-dimensional continued fraction expansion \({\mu}\) and \(x \in \mathcal{I}\), letting \(\mu(x) = T_0 \supseteq T_1 \supseteq \cdots\), it follows that infinitely many triangles \(T_n\) have some angle \(\leq \arcsin(23^{1 / 2} / 6)\approx \pi /(3.3921424) \approx 53^\circ\). Thus \(\lim \inf_{n \rightarrow \infty} \operatorname{area}(T_n) / \operatorname{diam}(T_n)^2 \leq 23^{1 / 2} / 12\). \par At the opposite extreme, we construct a two-dimensional continued fraction expansion \({\mu}\) and a dense set \(\mathcal{D} \subseteq \mathbb{R}^2\) of rank 3 points such that for each \(x \in \mathcal{D}\) the sequence \(T_0 \supseteq T_1 \supseteq \cdots\) of triangles of \(\mu(x)\) has the following property: Letting \(\omega_n\) denote the smallest angle of \(T_n\), it follows that \(\omega_0 < \omega_1 < \cdots\) and \(\lim_{n \rightarrow \infty} \omega_n = \pi / 3\). Further, the other two angles of \(T_n\) are \(> \pi / 3\). Thus \(\lim_{n \rightarrow \infty} \operatorname{area}(T_n) / \operatorname{diam}(T_n)^2 = 3^{1 / 2} / 4\), and the vertices of the triangles \(T_n\) strongly converge to \(x\). Our proofs combine a classical theorem of Davenport and Mahler with binary stellar operations of regular fans. regular cone; unimodular cone; Fan; continued fraction expansion; simultaneous Diophantine approximation; multidimensional continued fraction algorithm; stellar operation; starring; Farey mediant; Farey sum; Davenport-Mahler theorem Continued fractions, Farey sequences; the sequences \(1^k, 2^k, \dots\), Continued fraction calculations (number-theoretic aspects), Continued fractions and generalizations, Diophantine approximation in probabilistic number theory, Toric varieties, Newton polyhedra, Okounkov bodies, General theory of group and pseudogroup actions, Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Convergence and divergence of continued fractions Triangles in Diophantine 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 \(R\) be a real closed field, \(D\) an ordered subdomain in \(R\) and \(\mathcal{P}\) a family of symmetric polynomials in \(k\) variables of degree \(\leq d\) over \(D\). The algorithmic problem of computing Betti numbers of arbitrary semi-algebraic subsets of \(R^k\) is well-studied and has applications in the theory of computational complexity and in robot motion planning. Betti numbers of semi-algebraic subsets of \(R^k\) satisfy a singly exponential (in \(k\)) complexity; singly exponential dependence on \(k\) of the bound is unavoidable. The authors prove that for each fixed \(\ell\), \(d\geq 0\), there exists an algorithm that takes as input a quantifier-free first-order formula \(\Phi\) with atoms \(P=0\), \(P>0\), \(P<0\) with \(P\in \mathcal{P}\), and computes the ranks of the first \(\ell +1\) cohomology groups of the symmetric semi-algebraic set defined by \(\Phi\). The complexity of this algorithm (measured by the number of arithmetic operations in \(D\)) is bounded by a polynomial in \(k\) and card(\(\mathcal{P}\)) (for fixed \(d\) and \(\ell\)). This result contrasts with the PSPACE-hardness of the problem of computing just the zeroth Betti number (i.e., the number of semi-algebraically connected components) in the general case for \(d\geq 2\) (taking the ordered domain \(D\) to be equal to \(\mathbb{Z}\)). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. The authors prove that the Specht modules corresponding to partitions having long lengths cannot occur in the isotypic decompositions of low-dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks. symmetric semi-algebraic sets; isotypic decomposition; Specht module; Betti numbers; mirrored spaces; computational complexity Semialgebraic sets and related spaces, Topology of real algebraic varieties, Symmetric functions and generalizations, Effectivity, complexity and computational aspects of algebraic geometry, Symbolic computation and algebraic computation Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable \[ \zeta =\sum\limits_{k=0}^\infty\frac{2}{3^k}\varepsilon_{\eta_k}\equiv\Delta_{\eta_1\dots\eta_k\dots}, \] where the indices \(\eta_k\) are independent random variables taking the values 0, 1, 2, 3, 4 with probabilities \(p_{0k}\), \(p_{1k}\), \(p_{2k}\), \(p_{3k}\), \(p_{4k}\) respectively, \(p_{ik}\geq 0\), \(p_{ik} \geq 0\), \(\sum_{i=0}^4 p_{ik}=1\), \(\sum_{i=0}^4 p_{ik}=1\) for any \(k \in \mathbb N\), \(\varepsilon_m=e^{\frac{m\pi i}{2}}=i^m(m = 0,1,2,3)\) are 4th roots of unity, \(\varepsilon_4=0\). We prove that the distribution of \(\zeta\) is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of \(\zeta\) to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found. probability measure; fractal curve; Vicsek fractal; discrete probability distribution; singularly continuous probability distribution; point spectrum; spectrum of probability distribution Probability measures on topological spaces, Fractals, Plane and space curves Probability measures on fractal curves (probability distributions on the Vicsek fractal)	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The problem of resolution of singularities in characteristic 0 was solved by Hironata at the beginning of the sixties of the last century. Much later algorithmic methods have been developed for solving this problem. The study of algorithmic equiresolution started about 15 years ago. Some basic results in this direction can be found in the article of \textit{S. Encinas, A. Nobile} and \textit{O. E. Villamayor} [Proc. Lond. Math. Soc., III. Ser. 86, No. 3, 607--648 (2003; Zbl 1076.14020)]. Here families of ideals or of embedded schemes, parametrized by smooth varieties are studied. The equiresolution proposed required that the centers for the transformations are smooth over the parameter variety (condition AE) or required the local constancy of a certain invariant associated to each fibre (condition \(\tau\)). In this paper a definition of equiresolution is proposed for families parametrized by not necessarily reduced schemes, called condition E. Other approaches are proposed, condition A, C and F. Condition A corresponds to AE mentioned above. The main objective of the paper is to prove that when the parameter space is regular all these conditions are quivalent. Assuming the properness of certain projections it is proved that they are also equivalent to \(\tau\) mentioned above. resolution algorithm; embedded variety; coherent ideal; basic object Nobile, A.: Simultaneous algorithmic resolution of singularities, Geom. dedic. 163, 61-103 (2013) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, fibrations in algebraic geometry Simultaneous algorithmic resolution of singularities	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review gives a description of the set of all locally nilpotent derivations of the quotient ring \(\mathcal{B} = K[X, Y, Z]/(f(X)Y-\phi(X, Z))\) constructed from the defining equation \(f(X)Y=\phi(X, Z)\) of a generalized Danielewski surface in \(\mathbb{K}^{3}\) in the case when \(\mathbb{K}\) is an algebraically closed field of characteristic zero, \(\phi(X, Z) = Z^{m} + b_{m-1}(X)Z^{m-1} +\dots + b_{1}(X)Z + b_{0}(X)\) (\(m > 1\)) and \(\deg f > 1\). As a consequence of this description, the authors obtain that both \(ML\)- and \(HD\)- invariants of \(\mathcal{B}\) introduced in [\textit{L. Makar-Limanov}, Isr. J. Math. 96, Part B, 419--429 (1996; Zbl 0896.14021)] and [\textit{H. Derksen}, Constructive invariant theory and the Linearisation problem. Ph.D. Thesis, Univ. of Basel (1997)], respectively, are equal to \(\mathbb{K}[X]\). (Recall that the Makar-Limanov (or \(ML\)-) invariant of a ring \(A\) is the intersection of the kernels of all locally nilpotent derivations of \(A\). The Derksen (or \(HD\)-) invariant of \(A\) is defined as the subring of \(A\) generated by the union of kernels of all nontrivial locally nilpotent derivations of \(A\).) The paper also presents a description of a set of generators for the group of \(\mathbb{K}\)-automorphisms of the ring \(K[X, Y, Z]/(f(X)Y-\phi(Z))\) where \(\phi(Z)\in\mathbb{K}[Z]\) and \(\deg \phi > 1\). automorphisms; Danielewski surface; locally nilpotent derivations; ML-invariant Derivations and commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Locally nilpotent derivations and automorphism groups of certain Danielewski surfaces	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate the possibility of a finite characterization of multiple structures in projective space \(\mathbb P^n_k\), where \(k\) is an algebraically closed field, defined on linear subspaces of small codimension under the assumption of unmixedness. \textit{N. Manolache} gave such a characterization for scheme-theoretically Cohen-Macaulay multiple structures of degree at most 4 [Math. Z. 210, No. 4, 573--579 (1992; Zbl 0784.14020)] as well as for locally complete intersection multiple structures of degree at most 6 [\textit{N. Manolache}, Math. Z. 219, No. 3, 403--411 (1995; Zbl 0837.14039)]. In addition, \textit{B. Engheta} gave a characterization of unmixed ideals of height 2 and multiplicity 2 [J. Algebra 316, No. 2, 715--734 (2007; Zbl 1132.13006)]. In contrast to these characterizations, the main result of this paper shows that no such finite characterization is possible under the unmixedness assumption alone. In particular, the authors prove the following: Theorem. For any integers \(h, e \geq 2\) with \((h,e) \not = (2,2)\) and integer \(p \geq 5\), there exists a homogeneous ideal \(I_{h,e,p}\) in a polynomial ring \(R\) over the algebraically closed field \(k\) such that: \(I_{h,e,p}\) has height \(h\); the Hilbert-Samuel multiplicity of \(R/I_{h,e,p}\) is \(e\); the projective dimension of \(R/I_{h,e,p}\) is at least \(p\); \(I_{h,e,p}\) is primary to a linear prime \((x_1, \ldots, x_h)\). After reviewing necessary background, at the beginning of Section 3 the authors provide the following useful summary of their approach to the proof of the above theorem. For the ease of notation, as in the main theorem, denote by \(L_{h,e,p}\) an ideal in a polynomial ring \(R\) over an algebraically closed field \(k\) such that \(L_{h,e,p}\) has height \(h\), \(R/L_{h,e,p}\) has Hilbert-Samuel multiplicity \(e\), and the canonical module for \(R/L_{h,e,p}\) has projective dimension greater than or equal to \(p\). Using this notation, the authors first define four key families of primary ideals with large projective dimension and nice resolutions and canonical modules. The four key families are \(L_{2,5,p}, L_{2,6,p}, L_{2,20,p}\) and \(L_{3,6,p}\). Then by linking via a complete intersection from one of these four key families, the authors present primary and radical linear ideals of height 2 and any multiplicity greater than or equal to 3 with arbitrarily large projective dimension. To obtain examples with arbitrary height, one adds extra linear generators. It is also noted that an additional construction is needed for the case of \(L_{3,2,p}\) case since the case for \(L_{2,2,p}\) is finite. Section 4 of the paper is dedicated to the details of the construction for the four key families. The key ideals were originally found with the use of Macaulay2 and have many generators (see Section 5 for an example demonstrating the minimal number of generators for \(I_{2,4,p}\)). The paper concludes with a discussion of classifying ideals with Serre's (\(S_2\)) property. The authors note that the ideals constructed in the paper satisfy Serre's (\(S_1\)) property but none are (\(S_2\)). The authors close with the following question: Question: Is there a finite classification of homogeneous unmixed or primary ideals of a given height and multiplicity that satisfy Serre's (\(S_2\)) condition? Is there a classification for such ideals that are (\(S_2\)) on the punctured spectrum? multiple structures; projective dimension; multiplicity; primary ideals; unmixed ideals; free resolution; linkage Huneke, C.; Mantero, P.; McCullough, J.; Seceleanu, A., Multiple structures with arbitrary large projective dimension supported on linear subspaces, J. Algebra, 447, 183-205, (2016) Homological dimension and commutative rings, Linkage, Low codimension problems in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Multiple structures with arbitrarily large projective dimension supported on linear 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 Consider the natural map \(F\) between the space \(E\) of ordinary differential equations on marked Riemann surfaces with prescribed local monodromies and the space \(R\) of representation classes of \(\pi_1\) (punctured surface) with prescribed local representations around the punctures. A one-parameter family of differential equations lying in the fiber \(F^{-1}(r)\) is called isomonodromic. In the note an infinitesimal description of isomonodromic families is given in terms of a completely integrable system of partial differential equations on some local coordinate parameters of \(E\). In geometric terms this amounts to describe the tangential directions to the fibers \(F^{-1}(r)\) based on the existence of a natural symplectic structure on \(R\). The method is applied to the Fuchsian system \(d^2y/dz^2=q(z)y\) on the elliptic curve \({\mathbb C}/({\mathbb Z}.1+{\mathbb Z}.\tau)\), \(\tau\in {\mathbb C}\), Im\(\tau >0\). The function \(q\) equals \[ \begin{aligned} k &+\sum _{i=0}^m[H_i\zeta (z-t_i,\tau)+ (1/4)(\theta _i^2-1)\wp (z-t_i,\tau)]\\ & +\sum _{\alpha =0}^m[-\mu _{\alpha } \zeta (z-\lambda _{\alpha },\tau)+(3/4)\wp (z-\lambda _{\alpha },\tau)],\end{aligned} \] with \(\sum _{i=0}^mH_i-\sum _{\alpha =0}^m\mu _{\alpha }=0\) where \(\zeta (z,\tau)\) and \(\wp (z,\tau)\) denote the Weierstrass \(\zeta\)- and \(\wp\)-functions with fundamental periods 1 and \(\tau\). monodromy representation; isomonodromic family; symplectic structure; marked Riemann surfaces S. Kawai,Isomonodromic Deformation of Fuchsian-type Projective Connections on Elliptic curves, RIMS Kokyuroku Vol. 1022 (RIMS, Kyoto University, Kyoto, 1997), pp. 53--57. Structure of families (Picard-Lefschetz, monodromy, etc.), Riemann surfaces; Weierstrass points; gap sequences, Infinitesimal methods in algebraic geometry, Ordinary differential equations on complex manifolds, Isomonodromic deformation of Fuchsian-type projective connections on elliptic curves	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to answer to a question of M. Gromov, formulated somewhere and transmitted to the author by a friend of him. More precisely, one consider families \[ X=\{X_{t}\}_{t\in T}\text{ \;\;}X_{t}\in\mathbb{R}_{t}^{n},t\in T,T\subset \mathbb{R}^{m} \] with an index set \(T\) (called a base) which is a complex analytic set (or subanalytitic, or semianalytic) or complex algebraic set (or semialgebraic). All these conditions are expressed by the symbols \(\mathcal{U}_{i}, T\in\mathcal{U}_{i},i=1,2,3,4,5.\) Lipschitz equivalence with respect to \(X\) is defined as ordinary and the notion of the complexity of a curve \(p(\mu),\mu>0,\) is recalled (the number \(N\) of polynomial equations and inequalities that describe the graph of the curve \(p\) determines the complexity of at most \(N).\) Gromov's question is stated as follows: Is the set of Lipschitz equivalence classes with respect to \(X\) of curves of complexity at most \(N\) finite? The answer of the author is affirmative. The obtained proof deals with an appropriate stratification of \(\mathbb{R}^{m}\), and respectively of \(\mathbb{R}^{n}\) compatible with \(T\) with skeleton in \(\mathcal{U}_{i}.\) The exposition is highly sophisticated and it seems that the proof is correct, but difficult to be summarized shortly. Lipschitz equivalence; curve complexity Semi-analytic sets, subanalytic sets, and generalizations, Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions, Semialgebraic sets and related spaces, Real-analytic and semi-analytic sets Lipschitz stratifications and Lipschitz isotopies	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers semi-algebraic sets \(K\subset\mathbb{R}^n\), that is, subsets of the form \(K=\{{\mathbf x}\in\mathbb{R}^n:p_j({\mathbf x})\geq0,~ j=0,1,\ldots,m\}\), where \((g_j)_{j=0}^m\) is a (finite) family of real polynomials on \(\mathbb{R}^n\) with \(g_0=1\). Such a set is clearly closed but not necessarily compact. The aim of the paper is to solve the \(K\)-moment problem for certain continuous linear functionals. More precisely, the algebra of all polynomials \(\mathbb{R}[{\mathbf x}]\) is endowed with a norm induced by the sequence \({\mathbf w}= (w_\alpha)_{\alpha\in\mathbb{N}^n}\), where \(w_\alpha=2\lceil\|\alpha\|/2\rceil!\). If \(f=\sum_{\alpha\in\mathbb{N}^n}f_\alpha{\mathbf x}^\alpha\) is an arbitrary polynomial, what the author calls the \(\ell_{\mathbf w}\)-norm of \(f\) is the quantity given by \(\sum_\alpha w_\alpha\| f_\alpha\|\). For a semi-algebraic set \(K\subset\mathbb{R}^n\) defined by the family of polynomials \(\{p_0,p_1,\break\ldots,p_m\}\), with \(p_0=1\), and for a linear functional \(L\) on \(\mathbb{R}[{\mathbf x}]\) which is \(\ell_{\mathbf w}\)-continuous, there exists a finite positive Borel measure \(\mu\) on \(K\) such that \(L(f)=\int_Kfd\mu\) for all polynomials \(f\in\mathbb{R}[{\mathbf x}]\) if and only if \(L(h^2g_j)\geq 0\) for all \(h\in \mathbb{R}[{\mathbf x}]\), \(j=0,1,\ldots,m\), and \(\sup_{\alpha\in\mathbb{N}^n}\| L({\mathbf x}^\alpha)\|/w_\alpha<\infty\). The main ingredient of the proof is the use of Carleman's condition, insured by the choice of the sequence \(\mathbf w\). Other related results are also obtained, exploiting the \(\ell_{\mathbf w}\)-continuity. In particular, with the notation from above, a polynomial \(f\) is positive on the support of \(\mu\) if and only if \(\int h^2fd\mu\geq0\) for all \(h\in \mathbb{R}[{\mathbf x}]\). moment problems; real algebraic geometry; positive polynomials; semi-algebraic sets Lasserre, J. B., The \(K\)-moment problem for continuous linear functionals, Trans. Amer. Math. Soc., 365, 5, 2489-2504, (2013) Moment problems, Polynomials over commutative rings, Semialgebraic sets and related spaces, Polynomials and rational functions of one complex variable The \(K\)-moment problem for continuous linear functionals	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a theory of ``limit linear series'' for rational ribbons -- that is, for schemes that are double structures on \(\mathbb{P}^1\). This allows us to define a ``linear series Clifford index'' for ribbons. Our main theorem shows that this is the same as the Clifford index of ribbons studied by \textit{D. Bayer} and \textit{D. Eisenbud} in this same volume [Trans. Am. Math. Soc. 347, No. 3, 719-756 (1995; see the following review)]. This allows us to prove that the Clifford index is semicontinuous in degenerations from a smooth curve to a ribbon. A result of \textit{L.-Y. Fong} [J. Algebr. Geom. 2, No. 2, 295-307 (1995; Zbl 0788.14027)] then shows that ribbons may be deformed to smooth curves of the same Clifford index. Thus the canonical curve conjecture of \textit{M. L. Green} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] would follow, at least for a general smooth curve of each Clifford index, from the corresponding statement for ribbons. Green's conjecture; linear series Clifford index; rational ribbon; resolution Clifford index; canonical curve conjecture D. Eisenbud and M. Green, Clifford indices of ribbons, Trans. Amer. Math. Soc. 347 (1995), no. 3, 757--765. Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Complexes, Families, moduli of curves (algebraic) Clifford indices of ribbons	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(M\subset\mathbb R^{n}\) be a real-analytic submanifold and \({\mathcal H}(M)\) the algebra of real analytic functions on \(M\). If \(K\subset M\) is a compact subset we consider \(S_{K}=\{f\in {\mathcal H}(M) \mid f(x)\neq 0\) for all \(x\in K\}\); \(S_{K}\) is a multiplicative subset of \({\mathcal H}(M)\). Let \(S_{K}^{-1}{\mathcal H}(M)\) be the localization of \({\mathcal H}(M)\) with respect to \(S_{K}\). In this paper we prove, first, that \(S_{K}^{-1}{\mathcal H}(M)\) is a regular ring (hence Noetherian) and use this result in two situations: 1) For each open subset \({\Omega}\subset\mathbb R^{n}\), we denote by \({\mathcal O}({\Omega})\) the subalgebra of \({\mathcal H}({\Omega})\) defined as follows: \(f\in {\mathcal O}({\Omega})\) if and only if for all \(x\in{\Omega}\), the germ of \(f\) at \(x\), \(f_{x}\), is algebraic on \({\mathcal H}(\mathbb R^{n})\). We prove that if \({\Omega}\) is a bounded subanalytic subset, then \({\mathcal O}({\Omega})\) is a regular ring (hence Noetherian). 2) Let \(M\subset \mathbb R^{n}\) be a Nash submanifold and \({\mathcal N}(M)\) the ring of Nash functions on \(M\); we have an injection \({\mathcal N}(M) \rightarrow {\mathcal H}(M)\). In a former paper it was proved that every prime ideal \(\wp\) of \({\mathcal N}(M)\) generates a prime ideal of analytic functions \(\wp{\mathcal H}(M)\) if \(M\) or \(V(\wp)\) is compact. We use our Theorem 1 to give another proof in the situation where \(V(\wp)\) is compact. Finally we show that this result holds in some particular situation where \(M\) and \(V(\wp)\) are not assumed to be compact. analytic algebra; Nash functions; subanalytic sets; regular rings Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets, Real-analytic sets, complex Nash functions Noetherianness of certain analytic function algebras and applications	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The moduli spaces \({\mathcal M}_{0,n}^\delta\) are partial compactifications of the moduli space of \(n\)-pointed non-singular rational curves, indexed by the choice of a dihedral structure \(\delta\) on the set \(\{1,\dots,n\}\) of markings. They were introduced in [\textit{F. C. S. Brown}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 3, 371--489 (2009; Zbl 1216.11079)] because they give a symmetric set of smooth affine charts on the moduli space \(\overline{\mathcal M}_{0,n}\) of \(n\)-pointed stable rational curves. In this note, the authors investigate the structure of the cohomology of \({\mathcal M}_{0,n}^\delta\). The Betti number of these spaces in the maximal degree \(n-3\) is specifically interesting in view of connections with the Lie operad and with multiple zeta values. As in the case of \({\mathcal M}_{0,n}\), over the field of complex numbers the \(k\)th cohomology group of these partial compactifications carries a pure Hodge structures of weight equal to \(2k\). This ensures that the cohomology with rational coefficients is completely determined by the Poincaré polynomial. Exploiting the combinatorial relationship between the structure of \({\mathcal M}_{0,n}^\delta\) and Stasheff polynomials, it is shown that the generating series for the Poincaré polynomials of \({\mathcal M}_{0,n}^\delta\) is the inverse of the corresponding generating series for \({\mathcal M}_{0,n}\). As an application of the result, recurrence relations for the Poincaré polynomials and explicit formulas for the Betti numbers of \({\mathcal M}_{0,n}^{\delta}\) in small degree are given. Note that the inverse of the \textit{exponential} generating series of the Poincaré polynomial of \({\mathcal M}_{0,n}\) is known to coincide with the corresponding series for the Poincaré polynomial of \(\overline{\mathcal M}_{0,n}\) by work of \textit{V. Ginzburg} and {it M. Kapranov} [Duke Math. J. 76, No. 1, 203--272 (1994; Zbl 0855.18006)] and \textit{E. Getzler} [Prog. Math. 129, 199--230 (1995; Zbl 0851.18005)]. moduli spaces; curves of genus 0; Poincaré polynomial; generating series; inversion of series; Lie operad; multiple zeta values Bergström, Jonas; Brown, Francis: Inversion of series and the cohomology of the moduli spaces M0,n\(\delta \). Clay math. Proc. 12, 119-126 (2010) Families, moduli of curves (algebraic), Inversion of series and the cohomology of the moduli spaces \({\mathcal M}^\delta_{0,n}\)	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 For a given rational parametrization \({\mathcal P}(\bar{t})\), \(\bar{t}=(t_1,\ldots,t_r)\), of a variety in \({\mathbb K}^n\), the base points are the values of \(\bar{t}\) such that all the numerators and denominators in the parametrization simultaneously vanish. The presence of base points is typically a source of difficulties in a number of geometric problems usually attacked by means of elimination methods; therefore, understanding their behavior is important. Additionally, the \textit{fibre} of a point \(P\in {\mathbb K}^n\) is the set of \(\bar{t}\)'s such that \({\mathcal P}(\bar{t})=P\). Studying the cardinality of the fibre of a generic point is also important, because it is related to the number of times that \({\mathcal P}(\bar{t})\) covers the variety, and therefore with whether or not the parametrization is ``redundant'', i.e. with the optimality, in a certain sense, of the parametrization. Both base points and fibres are addressed in the paper. The content itself is very well described in the abstract of the paper: Given a rational parametrization \({\mathcal P}(\bar{t})\), \(\bar{t}=(t_1,\ldots,t_r)\), of an \(r\)-dimensional unirational variety, we analyze the behavior of the variety of the base points of \({\mathcal P}(\bar{t})\) in connection to its generic fibre, when successively eliminating the parameters \(t_i\). For this purpose, we introduce a sequence of of generalized resultants whose primitive and content parts contain the different components of the projected variety of the base points and the fibre. In addition, when the dimension of the base points is strictly smaller than one (as in the well-known cases of curves and surfaces), we show that the last element in the sequence of resultants is the univariate polynomial in the corresponding Gröbner basis of the ideal associated to the fibre; assuming that the ideal is in \(t_1\)-general position and radical. rational parametrization; unirational variety; degree of a rational map; fiber of a rational map; base points; generalized resultants Pérez-Díaz, S.; Sendra, J. R.: Behavior of the fiber and the base points of parametrizations under projections, Math. comput. Sci. 7, No. 2, 167-184 (2013) Computational aspects of higher-dimensional varieties, Computational aspects of algebraic surfaces, Symbolic computation and algebraic computation, Rational and birational maps Behavior of the fiber and the base points of parametrizations under 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 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 \(S\subset\mathbb R^m\) that are either a polynomial or a regular image of some \(\mathbb R^n\), that is, that can be represented as \(S=f(\mathbb R^n)\) 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 \(S\) (say, optimization, Positivstellensatz or computation of trajectories) can be reduced somehow to the case \(S=\mathbb R^n\) if \(S\) 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 \(X\subset\mathbb R^n\) be a convex polyhedron and denote \(\mathrm{Int}(X)\) 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 was proved that for every convex polygon \(X\subset\mathbb R^2\) that is neither a line nor a band, the semialgebraic sets \(\mathbb R^2\setminus X\) and \(\mathbb R^2\setminus\mathrm{Int}(X)\) 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 \(X\) and \(\mathrm{Int}(X)\) 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 \(X\subset\mathbb R^3\) which is neither a plane nor a layer both \(\mathbb R^3\setminus X\) and \(\mathbb R^3\setminus\mathrm{Int}(X)\) 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 \(\mathbb R^n\setminus X\) or \(\mathbb R^n\setminus\mathrm{Int}(X)\) as polynomial images of \(\mathbb R^n\) for a general convex polyhedron \(X\subset\mathbb R^n\). 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 \(X\subset\mathbb R^n\) that is neither a hyperplane nor a layer the semialgebraic sets \(\mathbb R^n\setminus X\) and \(\mathbb R^n\setminus\mathrm{Int}(X)\) are polynomial images of \(\mathbb R^n\) in case \(X\) 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 \(X\subset\mathbb R^n\) for which either \(\mathbb R^n\setminus X\) or \(\mathbb R^n\setminus\mathrm{Int}(X)\) are polynomial images of \(\mathbb R^n\). The proof of the main result of the work under review is easier if \(\dim(X)<n\). 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 \(S\subsetneq\mathbb R^n\), the set \(\mathbb R^{n+1}\setminus(S\times\{0\})\) is a polynomial image of \(\mathbb R^{n+1}\) . The hardest part is to deal with \(n\)-dimensional polyhedra \(X\subset\mathbb R^n\). 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 \(X\) if it is unbounded. In the last section of this work the authors substitute \(X\) by the closed ball \(\mathbb B_n\subset\mathbb R^n\) (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 \(\mathbb R^2\setminus\mathbb B_2\) is a polynomial image of \(\mathbb R^3\) but it is not a polynomial image of \(\mathbb R^2\). Moreover, \(\mathbb R^2\setminus\mathrm{Int}(\mathbb B_2)\) 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 \(\mathbb R^n\setminus\mathbb B_n\) is a polynomial image of \(\mathbb R^{n+1}\) but it is not a polynomial image of \(\mathbb R^n\). Moreover, they show that \(\mathbb R^n\setminus\mathrm{Int}(\mathbb B_n)\) 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 the set of points at infinity of a polynomial image of \(\mathbb{R}^n\), Discrete Comput. Geom., 52, 4, 583-611, (2014) Semialgebraic sets and related spaces, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Three-dimensional polytopes On the set of points at infinity of a polynomial image of \(\mathbb R^n\)	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this article is to investigate a codimension-two analogue of Ueda's theory. Given a complex manifold \(X\) and a smooth hypersurface \(S\) of \(X\), let \(C\) be a smooth compact hypersurface of \(S\). The author gives a sufficient condition for the line bundle \({\mathcal{O}}_X(S)\) to be flat in a neighborhood of \(C\) in \(X\). Precisely, under the assumption on the normal bundle \(N_{S/X}\) to be flat around \(C\), the author defines an obstruction class \[ u_{n,m}(C,S,X)\in H^1(C, {{N_{S/X}}\|}_C^{-n}\oplus N_{C/S}^{-m}) \] for each \(n\geq 1\), \(m\geq 0\) such that \({\mathcal{O}}_X(S)\) is not flat around \(C\) if there exist \(n\geq 1\), \(m\geq 0\) with \(u_{n,m}(C,S,X)\neq 0\). In particular if \(C\) is a smooth compact Kähler hypersurface of \(S\) such that \({{N_{S/X}}\|}V\) is flat, where \(V\) is a small neighborhood of \(C\) in \(S\), under some additional assumption on \(N_{C/S}\) and \({N_{S/X}}\|{}_C\), if \(u_{n,m}(C,S,X)= 0\) for all \(n\geq 1\) and \(m\geq 0\) then there exists a neighborhood \(U\) of \(C\) in \(X\) such that \({\mathcal{O}}_X(S)\| U \) is flat. As an application the author gives a sufficient condition for the anticanonical bundle of the blow-up of \(\mathbb{CP}^3\) at \(8\) points to be semi-ample. flat line bundles; Ueda's theory; blow-up of the three-dimensional projective space at eight points Koike, T, Toward a higher codimensional ueda theory, Math. Z., 281, 967-991, (2015) Transcendental methods of algebraic geometry (complex-analytic aspects), Divisors, linear systems, invertible sheaves Toward a higher codimensional Ueda theory	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [\textit{E. Mukhin} and \textit{A. Varchenko}, Commun. Contemp. Math. 6, No. 1, 111--163 (2004; Zbl 1050.17022)], some correspondences were defined between critical points of master functions associated to \(sl_{N+1}\) and subspaces of \(\mathbb{C}[x]\) with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus. Belkale P, Mukhin E, Varchenko A (2006) Multiplicity of critical points of master functions and Schubert calculus. In: Bertrand D, Enriquez B, Mitschi C, Sabbah C, Schäfke R (eds) Differential equations and quantum groups: Andrey A Bolibrukh memorial volume. IRMA Lect Math Theor Phys, vol 9. Europ Math Soc Publ, pp 59--84 Classical problems, Schubert calculus, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Exactly solvable models; Bethe ansatz Multiplicity of critical points of master functions and Schubert 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 study smooth manifolds defined by simple convex polytopes. With every \(n\)-dimensional simple polytope \(P\) with \(m\) facets is associated a smooth \((m+n)\)-dimensional manifold \({\mathcal Z}_P\) that admits a canonical action of the compact \(m\)-torus \(T^m\). Many manifolds that are important in topology, algebra and symplectic geometry arise in this context as quotients of manifolds \({\mathcal Z}_P\) by toric subgroups \(T^k\) of \(T^m\) which act freely on \({\mathcal Z}_P\). The class of quasitoric manifolds is obtained when \(k=m-n\), which is the largest possible value for \(k\); this class includes all smooth projective toric varieties (toric manifolds). The authors take the following approach to constructing these manifolds. The combinatorial structure of \(P\) determines a certain collection of affine planes in \(\mathbb{C}^m\), whose complement \(U(P)\) in \(\mathbb{C}^m\) admits an action by \((\mathbb{C}^)^m\). It is always possible to find certain subgroups \(R\) of \((\mathbb{C}^)^m\) which are isomorphic to \((\mathbb{R}^*_+)^{m-n}\) and act freely on \(U(P)\). The manifolds then are the quotients of \(U(P)\) by such subgroups. In particular, the authors investigate the relationships between the combinatorics of simple polytopes and the topology of the manifolds associated with them. It is proved that the cohomology of \({\mathcal Z}_P\) possesses a natural structure as a bigraded algebra. This bigraded cohomology algebra carries all the information about the combinatorics of \(P\). For example, there are appropriate interpretations of the Dehn-Sommerville equations and the upper-bound-theorem for simple convex polytopes. toric manifolds; simple convex polytopes; toric varieties; cohomology algebra Buchstaber V. M. and Panov T. E., ''Torus actions and combinatorics of polytopes,'' Proc. Steklov Inst. Math., 225, 87--120 (1999). Polyhedral manifolds, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies, Compact groups of homeomorphisms Torus actions and combinatorics of 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 It is a well-known fact that a connected component and the closure of a semialgebraic set are again semialgebraic sets and that a similar statement holds true for semianalytic sets. The author considers the problem of whether a union of connected components of a global semianalytic set is again global semianalytic. The answer to this problem was known to be positive for boundary bounded global semianalytic sets, see the book of \textit{C. Andradas, L. Bröcker} and \textit{J. M. Ruiz} [Constructible sets in real geometry. Ergebn. Math. Grenzgeb. 3, 33 (1996; Zbl 0873.14044)], and for global semianalytic subsets of a connected paracompact real manifold of dimension two, see [\textit{A. Castilla} and \textit{C. Andradas}, J. Reine Angew. Math. 475, 137--148 (1996; Zbl 0862.32003)]. In this paper it is proved that a union of connected components of a global semianalytic subset \(X\) of a coherent analytic subset of dimension two with affine normalization is again a global semianalytic subset. It must be noticed that a different proof of the same statement has been given by \textit{C. Andradas} and the reviewer [Ill. J. Math. 48, No.~2, 519--537 (2004; Zbl 1077.14087)]. The author also proves partial results for connected components of a global semianalytic subset of a three-dimensional analytic manifold. In order to get the main results the author proves some useful technical lemmas to deal with semianalytic subsets and also an interesting generalization of Thom's lemma for convergent power series. Global semianalytic sets; coherent analytic surfaces Fujita, M, On the connected components of a global semianalytic subset of an analytic surface, Hokkaido Math. J., 35, 155-179, (2006) Real-analytic and semi-analytic sets, Real algebra On the connected components of a global semianalytic subset of an analytic surface	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:{\mathbb C}^2 \to \mathbb C\) be a nonconstant polynomial in two complex variables with a finite set of critical points; let \(C^{ t}\) be the projective closure of the fiber \(f^{-1}(t)\), let \(L_{ \infty}\) in \({\mathbb P}^2 ({\mathbb C})\) be the line at infinity and let \(C_{ \infty}= C^{ t} \cap L_{ \infty}\). The set \(\Lambda(f) = \{ t \in {\mathbb C}:\mu_{p}^t > \mu_{p}^{min} \;\text{ for a } p \in C_{ \infty} \}\), where \(\mu_{p}^t = \mu_{p}(C^t)\) is the Milnor number and \(\mu_{p}^{min}= \inf_{t \in {\mathbb C}} \mu_{p}^t\), is called the set of irregular values of \(f\) at infinity. In this note the authors characterize polynomials \(f\) with no critical points and one irregular value at infinity: improving a recent result by \textit{A. Assi} [see Math. Z. 230, No. 1, 165--183 (1999; Zbl 0934.32017)], they give a description of the irregular fiber of such a polynomial. This result is applied to the estimation of the number of points at infinity of a polynomial with no critical points and at most one irregular value at infinity. The authors give also a discriminant criterion for polynomials to have one irregular value and present a list of open questions. affine curves; irregular value J. Gwozdziewicz and A. Ploski, On the singularities at infinity of plane algebraic curves, Rocky Mountain J. Math. 32 (2002), 139--148. Singularities of curves, local rings, Milnor fibration; relations with knot theory, Singularities in algebraic geometry On the singularities at infinity of plane 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 Nash functions are analytic functions which are algebraic over the polynomials. The paper under review is an excellent survey on some problems concerning Nash functions and the methods for their solutions. The problems consider here are: separation, factorization, global equations and extension. To be more precise, the separation problem asks whether a prime ideal of Nash functions on a Nash manifold \(M \subset \mathbb R^n\) has a prime extension to the ring of global analytic functions on \(M\). In the factorization problem the question is whether a Nash function \(f\) with a factorization \(f = f_1 f_2\) as a product of analytic functions has a similar factorization \(f = g_1 g_2\) as a product of Nash functions. Finally, whether a finite ideal sheaf is generated by its global sections and whether every Nash function on \(M\) is the restriction of a Nash function on \(\mathbb R^n\) to \(M\) are the global equation and the extension problems, respectively. In the case of a compact Nash manifold \(M\), using the so-called Néron desingularization an important approximation theorem is proved and then these problems are solved in the affirmative [see the paper of \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota}, Am. J. Math. 117, No. 4, 905-927 (1995; Zbl 0873.32007)]. When \(M\) is not necessarily compact \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [Compos. Math. 103, No. 1, 31-62 (1996; Zbl 0885.14029)] showed that these problems are equivalent. After that \textit{M. Coste} and \textit{M. Shiota} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, No. 1, 139-149 (2000; Zbl 0981.14027)] solved the global equation problem. The proof of \textit{M. Coste} and \textit{M. Shiota} relies in a semialgebraic version of Thom's first isotopy lemma proved by these two authors. The same problems have been considered in a more general setting, namely, replacing the field of real numbers by an arbitrary real closed field. Again, these problems have been answered in the affirmative by \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [J. Reine Angew. Math. 536, 209-235 (2001; Zbl 0981.14028)]. In that paper the authors, using the Tarski-Seidenberg theorem and proving uniform bounds on the complexities of Nash functions, obtain the solution to the already mentioned problems. Also, they prove the idempotency of the real spectrum and discuss conditions for the rings of abstract Nash functions to be noetherian. approximation theorem; Nash manifolds; separation problem; factorization problem; extension problem; global equations; complexity of Nash functions Nash functions and manifolds, Real-analytic sets, complex Nash functions, Real-analytic and Nash manifolds Global problems of Nash functions	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\), \(K_\infty\), \(A\), \(C\) and \(\Gamma\) be respectively a global field \(K\) of positive characteristic, its completion \(K_\infty\) at a fixed place \(\infty\), the ring \(A\) of elements of \(K\) regular outside \(\infty\), the completion \(C\) of an algebraic closure of \(K_\infty\) and an arithmetic subgroup \(\Gamma\) of \(GL_2 (K)\), i.e. a subgroup commensurable with \(GL_2 (A)\). Set \(\Omega= C-K_\infty\). This is a Drinfeld upper half-plane, \(\Gamma\) acts by fractional linear transformations on \(\Omega\) and the rigid analytic space \(M_\Gamma= \Gamma\setminus \Omega\) is indeed an affine curve over \(C\); this is a Drinfeld modular curve. These curves, for various \(\Gamma\), are the substitutes in positive characteristic of the classical modular curves. Let \(\overline{M}_\Gamma\) be the canonical completion of \(M_\Gamma\). The author studies divisors on \(\overline{M}_\Gamma\) with support in the set of cusps, i.e. in \(\overline{M}_\Gamma- M_\Gamma\) (cuspidal divisors). To use analytic methods [as in \textit{E.-U. Gekeler} and \textit{M. Reversat}, J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)], in \S 2, the author generalizes the theory of theta functions developed in (loc. cit.) to ``degenerate parameters''. One knows that the period lattice (in the sense of Manin-Drinfeld) of \(\overline{M}_\Gamma\) is described by a set of harmonic cochains or a set of theta functions (loc. cit.). The author adds to that interpretation the description of the links between new theta functions, cuspidal divisors and harmonic cochains (\S 3). In \S 4, where \(\Gamma\) is assumed to be a congruence subgroup, the author studies, with the help of the preceding methods, the canonical map between the group \({\mathcal C}(\Gamma)\) generated in the Jacobian \({\mathcal J}_\Gamma\) of \(\overline{M}_\Gamma\) by the cusps (the group \({\mathcal C}(\Gamma)\) is finite here), and the group \(\Phi_\infty (\Gamma)\) of connected components of the Néron model of \({\mathcal J}_\Gamma\) at \(\infty\). Finally (\S 5), in the case of \(A= \mathbb{F}_q [T]\) and for a Hecke congruence subgroup, the author obtains more information about his new theta functions and the (eventually non-empty) kernel of the map \({\mathcal C}(\Gamma)\to \Phi_\infty(\Gamma)\). class group; Drinfeld upper half-plane; Drinfeld modular curve; theta functions; cuspidal divisors; harmonic cochains; congruence subgroup E.-U. Gekeler,On the cuspidal divisor class group of a Drinfeld modular curve, Documenta Mathematica. Journal der Deutschen Mathematiker-Vereinigung2 (1997), 351--374. Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties, Holomorphic modular forms of integral weight, Modular and Shimura varieties On the cuspidal divisor class group of a Drinfeld modular curve	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers the Dirichlet series \(s\mapsto Z(P;s) =\sum\limits_{m\in{\mathbb N}^{n}}P(m)^{-s} (s\in{\mathbb C})\) where \(P\in{\mathbb R}[X_1,...,X_n]\). Say that \(Z(P;s)\) exists if this multiple series is absolutely convergent. In this paper he studies meromorphic continuations of such series, under the assumptions that there exists a constant \(B\in]0,1[\) such that: i) \(P(x)\to +\infty\) when \(\|x\|\to + \infty\) and \(x \in [B,+\infty[^n\) and ii) \(d(Z(P),\;[B,+\infty[^n)>0\) where \(Z(P)=\{ z \in{\mathbb C}^n\mid P(z)=0\}\). This assumption is probably optimal, and in any way strictly includes all classes of polynomials previously treated. Under this assumption, he proves the existence of meromorphic continuations of Dirichlet series and gives a set of candidate poles and an upper bound to the orders of these poles. Moreover the author obtains bounds for these meromorphic continuations on vertical bands. As an application, he shows the existence of a finite asymptotic expansion of the counting function: \[ N_P(t)=\#\{ m \in {\mathbb N}^{n}\mid P(m)\leq t \} \text{ when } t\to + \infty. \] Dirichlet series; meromorphic continuation; integral representation; resolution of singularities; semi-algebraic set Essouabri, D., Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres, Ann. inst. Fourier (Grenoble), 47, 429-484, (1996) Other Dirichlet series and zeta functions, Lattice points in specified regions, Semialgebraic sets and related spaces, Meromorphic functions of several complex variables, Integral representations; canonical kernels (Szegő, Bergman, etc.), Singularities in algebraic geometry Singularités de séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres. (Singularities of Dirichlet series associated to polynomials of several variables and applications to analytic number 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 biregular geometry of punctual Hilbert schemes in dimensions 2 and 3, i.e. of schemes parametrizing fixed-length zero-dimensionaI subschemes supported at a given point on a smooth surface or a smooth three-dimensional variety, is studied. A precise biregular description of these schemes has only been known for the trivial cases of lengths 3 and 4 in dimension 2. The next case of length 5 in dimension 2 and the two first nontrivial cases of lengths 3 and 4 in dimension 3 are considered. A detailed description of the biregular properties of punctual Hilbert schemes and of their natural desingularizations by varieties of complete punctual flags is given. Stein expansion; Briançon classification; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes) Punctual Hilbert schemes of small length in dimensions 2 and 3	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a \(C^{\infty}\) function \(\phi:\quad M\to {\mathbb{R}}\) (where M is a real algebraic manifold) the following problem is studied. If \(\phi^{- 1}(0)\) is an algebraic subvariety of M, can \(\phi\) be approximated by rational regular functions f such that \(f^{-1}(0)=\phi^{-1}(0)?\) We find that this is possible if and only if there exists a rational regular function \(g:\quad M\to {\mathbb{R}}\) such that \(g^{-1}(0)=\phi^{- 1}(0)\) and g(x)\(\cdot \phi (x)\geq 0\) for any x in \({\mathbb{R}}^ n\). Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus is also studied. approximation of \(C^{\infty }\)-functions by regular functions; algebraic manifold; zero set Broglia, Ann. Inst. Fourier, Grenoble 39 pp 611-- (1989) Real algebraic and real-analytic geometry Approximation of \(C^{\infty}\)-functions without changing their zero-set	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a quasi-projective scheme over a field \(k\), \(M(X)\) the category of coherent sheaves on \(X\) and \(M_p(X)\) be subcategory of sheaves supported on a closed set of dimension at most \(p\). The inclusion functor \(M(X)\to M_p(X)\) induces a homomorphism \(F_{n,p}: K_n(M_p(X))\to G_n(X)= K_n(M(X))\). The image \(G_n(X)_{(p)}\) of \(F_{n,p}\) is the \(p\)th term of the topological filtration on \(G_n(X)\). Let's denote by \(G_n(X)_{(p/p-1)}\) the subsequent factors of the filtration. The niveau spectral sequence \[ E^1_{p,q}= \coprod_{x\in X_p} K_{p+q}(k(X))\Rightarrow G_{p+q}(X)\tag{1} \] (where \(X_p\) is the set of points of \(X\) of dimension \(p\)) yields a surjective homomorphism \[ \phi_p: CH_p(X)= E^2_{p,-p}\to G_0(X)_{(p/p-1)} \] whose kernel is detected by the differentials of the spectral sequence arriving at \(E^*_{p,-p}\). In this paper the author proves the following constraints for the order of differentials: Theorem 2. Let \(\delta: E^s_{p,q}\to E^s_{p-s,q+s-1}\) be the differential in the spectral sequence (1) with \(p+ q\leq 2\). The order \(\text{ord}(\delta)\) is finite and, if \(l\) is a prime divisor of \(\text{ord}(\delta)\), then \(l\leq p\) and \(l-1\) divides \(s-1\). Merkurjev, A., \textit{Adams operations and the Brown-Gersten-Quillen spectral sequence}, Quadratic forms, linear algebraic groups, and cohomology, vol. 18, 305-313, (2010), Springer, New York Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, \(J\)-homomorphism, Adams operations Adams operations and the Brown-Gersten-Quillen 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 Let \(A\subset\mathbb{Z}^{n-1}\) be a finite subset, \(\mathbb{C}^ A\) the linear \(\mathbb{C}\)-space of Laurent polynomials \(f=\sum_{\omega\in A}a_ \omega X^ \omega\), \(a_ \omega\in\mathbb{C}\) in some indeterminates \(X\) and \(\nabla_ 0\subset\mathbb{C}^ A\) the set of those \(f\) for which there is \(\kappa\in(\mathbb{C}^)^{n-1}\) such that \(f(\kappa)=(\partial f/\partial X_ i)(\kappa)=0\) for all \(i\). The closure \(\nabla_ A\) of \(\nabla_ 0\) is an irreducible variety defined in fact on \(\mathbb{Z}\). When \(\nabla_ A\) has codimension 1 then an irreducible polynomial \(\Delta_ A\in\mathbb{Z}[a_ \omega;\omega\in A]\), which is zero on \(\nabla_ A\), is unique up to the sign and it is called the \(A\)-discriminant. If \(\text{codim}(\nabla_ A)>1\) then put \(\Delta_ A=1\). The \(A\)- discriminant is homogeneous and satisfies the following quasi homogeneous \((n-1)\)-conditions: ``\(\sum_{\omega\in A}m(\omega)\cdot\omega\in\mathbb{Z}^{n-1}\) is constant for all monomials \(\prod_{\omega\in A}a_ \omega^{m(\omega)}\) which enter in \(\Delta_ A\)''. This notion extends the classical notions of discriminant and resultant. --- Let \(A=\{\omega_ 1,\ldots,\omega_ N\}\) and \(Y_ A\) be the closure of the set \(\{(\kappa^{\omega_ 1},\ldots,\kappa^{\omega_ N}\mid\kappa\in\mathbb{C}^{n-1}\}\) in \(\mathbb{P}^{N-1}\). Then \(\nabla_ A\) and \(Y_ A\) are dual projective varieties and the description of \(\Delta_ A\) follows if we can describe the equations of the dual projective variety of a given projective one \(Y\subset\mathbb{P}^{N-1}\) [see the authors' previous paper in Sov. Math., Dokl. 39, No. 2, 385-389 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 6, 1294-1298 (1989; Zbl 0715.14042)]. Let \(G\) be a free abelian group of rank \(n\), \(G_ \mathbb{C}:=\mathbb{C}\otimes_ \mathbb{Z} G\), \(\lambda:G\to(\mathbb{Q},+)\) a nonzero group morphism, \(S\subset G\) a finitely generated semigroup such that \(o\in S\) and \(\lambda(s)\geq 1\) for all \(s\in S\), \(S_ e=\{t\in S\mid\lambda(t)=e\}\) for \(e\in\mathbb{Q}\) and \(A\subset S_ 1\) a finite subset generating in \(G_ \mathbb{R}=\mathbb{R}\otimes_ \mathbb{Z} G\) the same convex cone as \(S\). For \(k\in\mathbb{Z}_ +\), \(e\in\mathbb{Q}\), \(\omega\in A\) let \(\bigwedge^ k(e)\) be the space of all maps \(S_{k+e}\to\bigwedge^ kG_ \mathbb{C}\), \(\partial_ \omega:\bigwedge^ k(e)\to\bigwedge^{k+1}(e)\) the map given by \(\partial_ \omega(\gamma)(u)=\omega\wedge\gamma(u-\omega)\), if \(u-\omega\in S_{k+e}\), otherwise \(\partial_ \omega(\gamma)(u)=0\) and \(\partial_ f=\sum_{\omega\in A}a_ \omega\partial_ \omega\) if \(f=\sum a_ \omega X^ \omega\in\mathbb{C}^ A\). The complex \((\overset{.}\bigwedge (e),\partial_ f)\) is called the Cayley-Koszul complex. --- Choose a basis \(u\) in terms of \(\overset{.}\bigwedge(e)\) and let \(E_ e(f)\) be the determinant of the complex \((\overset{.}\bigwedge (e),\partial_ f)\) with respect to \(u\) [see \textit{F. Fischer}, Math. Z. 26, 497-550 (1927) or \textit{J.-K. Bismut} and \textit{D. S. Freed}, Commun. Math. Phys. 106, 159-176 (1986; Zbl 0657.58037)]. For \(e\) sufficiently high \(E_ A(f):=E_ e(f)\) is a polynomial of \((a_ \omega)\), \(f=\sum a_ \omega X^ \omega\) which depends on \(e\) only by a constant multiple. --- Let \(Q_ A\) be the convex closure of \(A\) in \(G_ \mathbb{R}\). If \(f=\sum a_ \omega X^ \omega\) we can express \(E_ A(f)=\sum_ \varphi c_ \varphi\prod_{\omega\in A}a_ \omega^{\varphi(\omega)}\), where \(\varphi\) runs in the set \(\mathbb{Z}^ A_ +\) of the maps \(A\to\mathbb{Z}_ +\). Let \(M(E_ A)\subset\mathbb{R}^ A\) be the convex closure of those \(\varphi\in\mathbb{Z}^ A_ +\) for which \(c_ \varphi\neq 0\). Then there exists a nice correspondence between the vertices of \(M(E_ A)\) and some special triangulations of \(Q_ A\). The theory is applied to the following examples: the discriminant of a polynomial in two indeterminates, the resultant of two quadratric polynomials, the elliptic curve in Tate normal form\dots Laurent polynomials; Cayley-Koszul complex; determinant; discriminant of a polynomial in two indeterminates; elliptic curve Gel'fand, I.; Zelevinskiǐand, A.; Kapranov, M., Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz, 2, 1, (1990) Toric varieties, Newton polyhedra, Okounkov bodies, Polynomial rings and ideals; rings of integer-valued polynomials, Complexes, Determinantal varieties Discriminants of polynomials in several variables and triangulations of Newton polyhedra	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies linear systems of curves defined by fixed points of given multiplicities on Hirzebruch surfaces \({{\mathbb F}_n}\). If \(\operatorname{Pic}{\mathbb {F}_n} = \langle H,F\rangle\), where \(F\) is a fiber and \(H^2=n\), \(F\cdot H=1\), the linear systems under consideration are of type \({\mathcal L}(a,b,m_1,\dots,m_s)\), i.e. the subsystem of the complete linear system \(\| aF+bH\|\) given by the curves having multiplicity at least \(m_i\) at \(P_i\), for \(s\) generic points \(P_1,\dots ,P_s\). In analogy with what happens in \({\mathbb P}^2\), the author conjectures that such linear systems are special (i.e. their dimension is bigger than expected) if and only if they are \((-1)\)-special, i.e. there is a \((-1)\)-curve \(E\) such that the base locus of \({\mathcal L}(a,b,m_1,\dots,m_s)\) contains \(\alpha \Gamma_n + tE\), \(t\geq 2\), \(\alpha \geq 0\). Here \(\Gamma_n\in \| H-nF\|\) is the curve with \(\Gamma ^2=-n\), while a \((-1)\)-curve \(E\) is an irreducible curve such that \(E^2=-1=K_{{\mathbb F}_n}\cdot E\). In the paper the conjecture is proved for \(m_1=\cdots =m_s\leq 3\); the main idea of the proof is to list all \((-1)\)-special systems with multiplicities \(m_i\leq 3\) via birational transformations \({\mathbb F}_n \rightarrow {\mathbb F}_{n-1}\) (a generalization of quadratic transformations in \({\mathbb P}^2\)) and then to work with suitable deformations of \({\mathbb F}_n\). fat points; Hirzebruch surfaces; \((-1)\)-special A. Laface: ''On linear systems of curves on rational scrolls'', Geom. Dedicata, Vol. 90, (2002), pp. 127--144. Divisors, linear systems, invertible sheaves, Birational automorphisms, Cremona group and generalizations, Rational and ruled surfaces On linear systems of curves on rational scrolls	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V_{g}\) be a universal space of Jacobian varieties of flat hyperelliptic curves in the form \[ V(\mu,\nu)={(\mu,\nu)\in\mathbb{C}^{2}: \nu^{2}-f_{g}(\mu)=0}, \] where \(f_{g}(\mu)=4\mu^{2g+1}+\sum_{i=0}^{2g-1}\mu^{i}\lambda_{i}, g=1,2,\dots\). Let us introduce on \(V_{g}\) the coordinates \((u,\lambda)=(u_{1},u_{2},\dots,u_{g},\lambda_{0}, \lambda_{1},\dots,\lambda_{2g-1}),\) where the \(u\) are coordinates on the Jacobian variety, related with the canonical basis of holomorphic differentials on curves \(V(\mu,\nu)\). Consider the system of functional equations \[ \varphi(u,\omega d+\omega'c,\omega b+\omega'a)=\varphi(u,\omega,\omega'),\tag{1} \] \[ \begin{multlined} \varphi(u+\omega m+\omega'm',\omega,\omega') =\\ \exp[2(u+\omega m+\omega'm')(\eta m+\zeta'm')^{t}]\varphi(u,\omega,\omega'),\end{multlined}\tag{2} \] where \(m,m'\in\mathbb{Z}^{g}\), \(\left(\begin{smallmatrix} a&b\\ c&d\end{smallmatrix}\right)\in \text{Sp}(2g,\mathbb{Z})\), and the \((g\times g)\)-matrices \(\omega,\omega',\eta,\eta '\) are functions of \(\lambda\) given with the help of curvilinear integrals on the curve \(V(\mu,\nu)\). Then the hyperelliptic \(\sigma\)-function is called a continuation for special solutions \(\varphi_{0}(u,\omega,\omega')\) for the system of equations (1)-(2) on all the spaces \(V_g\). Let \(U\) be an algebra of linear differential operators on \(u_1,\dots,u_g\) and \(\lambda_{2g-1},\dots,\lambda_0\), the \(U(\sigma)\) annihilator ideal for the \(\sigma\)-function, that is, the subalgebra of operators \(L\in U\) such that \(L\sigma(u,\lambda)\equiv 0\). Lemma 1. (a) The ideal \(U(\sigma)\) is generated by \(2g\) generators \(l_{j}\), \(j = 0, 1,\dots , 2g -1\). (b) The operators \(l_{j}\) are homogeneous for \(\deg u_{i} = 2(g - i) + 1,\) \(\deg \lambda_{k} = -2(2g - k) - 2,\) where \(\deg l_{j} = -2j\). (c) The differential operators \(l_{j}\), \(j > 0,\) are of second order in \(u\) and of first order in \(\lambda,\) and the operator \(l_{0}\) is up to multiplication by a constant, equal to \[ \sum_{i=1}^{g}(2(g - i) + 1)u_{i} \partial_{u_{i}} - \sum_{k=0}^{2g-1} 2(2g - k + 1) \lambda_{k}\partial_{\lambda_{k}} - g(g + 1)/2. \] (d) The coefficients of operators \(l_{j}\) are polynomials. The main result of this paper is Theorem 2. Graded generators \(l_{i}\) of the ideal \(U(\sigma)\) and the operators of multiplication by \(\lambda_{j}\) generate a graded Lie algebra. The authors note that (1) the hyperelliptic \(\sigma\)-functions were introduced by F. Klein, as a generalization of the elliptic \(\sigma\)-function of Weierstrass; (2) Annihilator's ideal of the Weierstrass \(\sigma\)-function was studied by K. Weierstrass himself; (3) H. F. Baker proposed and developed Klein's construction for the case \(g=2\). Families, moduli of curves (analytic), Theta functions and curves; Schottky problem, Elliptic curves, Graded Lie (super)algebras Graded Lie algebras that define hyperelliptic sigma 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 In this article we introduce a stability concept for the coercivity of multivariate polynomials \(f\in\mathbb R [x]\). In particular, we consider perturbations of \(f\) by polynomials up to the so-called degree of stable coercivity, and we analyze this stability concept in terms of the corresponding Newton polytopes at infinity. For coercive polynomials \(f\in\mathbb R [x]\) we also introduce the order of coercivity as a measure expressing the order of growth of \(f\), and we identify a broad class of multivariate polynomials \(f\in\mathbb R [x]\) for which the order of coercivity and the degree of stable coercivity coincide. For these polynomials we give a geometric interpretation of this phenomenon in terms of their Newton polytopes at infinity, which we call the degree of convenience. We relate our results to the existing literature and we illustrate them with some examples. As applications we show that the gradient maps corresponding to a broad class of polynomials are always subjective, we establish Hölder type global error bounds for such polynomials, and we link our results to the existence of solutions in the calculus of variations. Newton polytope; coercivity; stability; order of growth; error bound; convenient polynomials Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic sets, Real polynomials: analytic properties, etc., Newton-type methods Coercive polynomials: stability, order of growth, and Newton polytopes	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a method to desingularize every three-dimensional toric variety \(X = X_ \Delta\), where \(\Delta\) is a simplicial fan. Letting \(\Delta^{(3)}\) denote the set of three-dimensional cones of \(\Delta\), each cone \(\sigma \in \Delta^{(3)}\) of multiplicity \(\text{mult} (\sigma) > 1\) is starred at a nonzero primitive vector \(p_ \sigma \in \sigma \cap \mathbb{Z}^ 3\) such that the sum of the multiplicities of the cones thus obtained is the smallest possible. In this way, after a finite number of steps one obtains a fan \(\nabla (\Delta)\) yielding a nonsingular subdivision of \(\Delta\). Our algorithm generalizes the well known construction of the coarsest nonsingular subdivision of a fan in \(\mathbb{Z}^ 2\) as given by the Hirzebruch-Jung continued fraction algorithm. For each three-dimensional toric variety \(X = X_ \Delta\), we thus obtain a desingularization \(X' = X_{\nabla (\Delta)}\) of \(X\) whose Euler characteristic \(E(X')\) satisfies the inequality \(E(X') \leq - E(X) + 2 \sum_{\sigma \in \Delta^{(3)}} \text{mult} (\sigma)\). It is shown that this upper bound is best possible. three-dimensional toric variety; subdivision of fan; desingularization; Euler characteristic Aguzzoli, S.; Mundici, D., An algorithmic desingularization of 3-dimensional toric varieties, \textit{Tohoku Math. J.}, 46, 4, 557-572, (1994) Computational aspects of higher-dimensional varieties, \(3\)-folds, Toric varieties, Newton polyhedra, Okounkov bodies An algorithmic desingularization of 3-dimensional toric varieties	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S:=K[x_0,\dots,x_n]\) be a polynomial ring over an algebraically closed field \(K\), and let \(I\) be a homogeneous ideal of \(S\) defining a subscheme \({\mathfrak X}\) of projective \(n\)-space \(\mathbb{P}^n_K\). The Castelnuovo-Mumford regularity (or simply regularity) of \(I\), \(\text{reg} I\), is defined as follows: If \[ 0\to \bigoplus^{\beta_p}_{j=1} S(-e_{pj}) @>\varphi_p>> \cdots @>\varphi_1>> \bigoplus^{\beta_0}_{j=1}S(-e_{0 j}) @>\varphi_0>> I\to 0\tag{0.1} \] is a minimal graded free resolution of \(I\), setting \(e_i:=\max\{e_{ij};\;1\leq j\leq\beta_i\}\), then \(\text{reg} I:= \max \{e_i-i;\;0\leq i\leq p\}\). In other words, \(\text{reg} I\) is the smallest integer \(m\) for which \(I\) is \(m\)-regular, i.e., \(e_{ij}\leq m+i\) for all \(i,j\). When \(I\) is saturated (i.e., when it is the largest ideal defining \({\mathfrak X})\), we call this the regularity of \({\mathfrak X}\). -- The regularity is a numerical invariant of the ideal \(I\) and is, ``an important measure of how hard it will be to compute a free resolution''. We give an effective method to compute the regularity of a saturated ideal \(I\) defining a projective curve that also determines in which step of a minimal graded free resolution of \(I\) the regularity is attained. Buchberger's syzygy algorithm; polynomial ring; Castelnuovo-Mumford regularity; free resolution Bermejo, I.; Gimenez, P.: On Castelnuovo -- Mumford regularity of projective curves. Proc. amer. Math. soc. 128, 1293-1298 (2000) Syzygies, resolutions, complexes and commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects of algebraic curves On Castelnuovo-Mumford regularity of projective 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 \(\mathcal{P} = \{P_{1},\dots, P_{s}\} \subset \mathbb{P}^{2}_{\mathbb{K}}\) be a finte set of points and denote by \(I = m_{1}P_{1} + \dots +m_{s}P_{s}\) the associated fat point scheme, where \(\mathbb{K}\) is an algebraically closed field of charcteristic \(0\). For a homogeneous ideal \(I\) we define the initial degree \(\alpha(I)\) to be the least integer \(t\) such that \(I_{t} \neq 0\). The Waldschmidt constant of \(I\) is defined as \[ \widehat{\alpha}(I) = \lim_{m \rightarrow \infty} \frac{\alpha(I^{(m)})}{m}, \] where \(I^{(m)}\) is the \(m\)-th symbolic power of \(I\). In the present paper the authors provide a complete classification of fat point schemes of the form \(I = m_{1}P_{1} +\dots+ m_{s}P_{s}\) for which the Waldschmidt constant \(\widehat{\alpha}(I)\) is strictly less than \(\frac{5}{2}\). Let us define the following fat subschemes: i) \(Z\) which consists of three simple points, plus one double point, such that the simple points lie on a line \(L_1\) and the double point lies on another line \(L_2\). Formally, we denote it as \(Z = 2q + p_1 + p_2 + p_3\). ii) \(W\) which consists of \(r+s\) simple points, plus one double point, where \(r,s \geq 1\), such that \(r\) simple points out of them lie on a line \(L_1\), and \(s\) simple points out of them lie on another line \(L_2\), and the single double point lies at the intersection of these two lines. We denote it as \(W = 2q +q_1 + q_2 +\cdots+ q_s + p_1 + p_2 +\cdots+ p_r\). iii) \(T\) which consists of \(r\) double points, plus \(s\) simple points, where \(r \geq 1\) and \(s \geq 0\), such that all of these points are contained in a single line. We denote it as \(T = 2p_1 +\cdots+ 2p_r + q_1 + q_2 +\cdots+ q_s\). Main Theorem. Let \(\mathcal{A}\) be a fat point subscheme of \(\mathbb{P}^{2}_{\mathbb{K}}\) with defining ideal \(I\), then \(\widehat{\alpha}(I) < \frac{5}{2}\) if and only if \(\mathcal{A}\) is one of the fat point subschemes \(Z\), \(W\), or \(T\). fat points; Waldschmidt constant; symbolic power; star configuration; configuration of points Configurations and arrangements of linear subspaces, Graded rings, Projective techniques in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials All fat point subschemes in \(\mathbb{P}^2\) with the Waldschmidt constant less than 5/2	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\phi : \mathbb P^1 \to \mathbb P^1\) be a rational map defined over a field \(K\), let \(N\) be a positive integer. A point \(P\in \mathbb P^1\) is called periodic of primitive period \(N\) if \(N\) is the smallest number such that \(\phi^N(P)=P\). The notion of a point of formal period \(N\) is a slight generalization of this notion. The points of formal period \(N\) are by definition the roots of the so called \(N\)th dynatomic polynomial for \(\phi\). Let \(M_d\) denote the moduli space parameterizing the rational maps of degree \(d\) up to coordinate change, i.~e., up to conjugacy by elements from \(\mathrm{PGL}_2(\overline K)\). The author constructs the moduli space \(M_d(N)\) parameterizing conjugacy classes of degree-\(d\) maps together with a point of formal period \(N\). One of the main results of the paper is Theorem~4.5 stating that \(M_{2}(N)\) is geometrically irreducible for \(N>1\). For a rational map \(\phi\) an element \(h \in \mathrm{PGL}_2(\overline K)\) is called an automorphism of \(\phi\) if \(h^{ - 1}\circ \phi \circ h=\phi\). Let \(p\) be a prime number, let \(\mathfrak C_p\) denote the cyclic group of order \(p\). Then the author considers the moduli space \(M_d(N, \mathfrak C_p)\) of rational maps of degree \(d\) with a point of formal period \(N\) having an automorphism of order \(p\). Another important result of the paper is Theorem~7.1, whose essence is expressed by Corollary~7.10 saying that for \(d\geq 2\) the moduli space \(M_d(pN, \mathfrak C_p)\) is reducible for all but finitely many prime integers \(N\). In the case of the field of zero characteristic it is reducible for all but finitely many positive integers \(N\). In particular the curves \(M_2(2N, \mathfrak C_2)\) are reducible for infinitely many \(N\). The paper consists of 9 sections. Section~1 is an introduction. An overview of the paper and its main results are given here. Some important notions and objects (\(N\)th dynatomic polynomial, points of formal period \(N\), moduli space \(M_d\), etc.) are defined and described in Section~2. In Section~3 a construction of moduli spaces with level structures is presented. An algebraic proof of the irreducibility of \(M_2(N)\) is given in Section~4 (Theorem~4.5). Section~5 deals with maps with automorphisms. In particular for a rational map \(\phi\) on \(\mathbb P^1\) and for its automorphism \(h\) the notion of \(h\)-periodic points as well as the notion of \(h\)-tuned dynatomic polynomials are introduced. In Section~6 basic properties of \(h\)-tuned dynatomic polynomials are studied. Section~7 contains another main result of the paper, i.~e., Theorem~7.1 and Corollary~7.10. It turns out that \(h\)-tuned dynatomic polynomials, which are defined as rational functions, are in fact polynomials, which justifies their name. In generic situation they turn out to be non-trivial divisors of dynatomic polynomials. Hence the latter are reducible and this leads to a constructive proof of the statements mentioned above. Section~8 provides complete factorizations of the dynatomic polynomials for the cases \(\phi(z)=z^d\) and \(\phi(z)=z^{-d}\). In these cases the dehomogenized dynatomic polynomials turn out to be products of certain cyclotomic polynomials. In Section~9 the author shows in Proposition~9.2 that \(M_2(N, \mathfrak{C}_2)\) is irreducible if \(2^N-1\) is a prime number. rational map; moduli space; level structure; dynatomic polynomial; irreducibility; reducibility Manes, M., Moduli spaces for families of rational maps on \(\mathbb{P}\)1, J. Number Theory, 129, 1623-1663, (2009) Algebraic moduli problems, moduli of vector bundles, Families and moduli spaces in arithmetic and non-Archimedean dynamical systems Moduli spaces for families of rational maps on \(\mathbb P^1\)	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 this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial \(\delta\)-uniform permutations is obtained, for any \(\delta\), as well as an upper bound for differentially two-valued functions. Concerning quadratic functions, significant relations between nonlinearity and differential uniformity are exhibited. In particular, we show that the quadratic differentially 4-uniform permutations should be differentially two-valued and possess the best known nonlinearity. vectorial Boolean function; quadratic function; nonlinearity; differential uniformity; differentially two-valued function Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry New links between nonlinearity and differential uniformity	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) be a hypersurface of degree \(m\) with an isolated singular point \(p\in RP^ n\). Let \(F_ t\), \(t\in(0,1]\), be a continuous family of hypersurfaces of degree \(m\), where \(F_ 0=F\) and \(F_ t\), \(t\in(0,1]\), are nonsingular and isotropic in a closed spherical regular neighbourhood \(D(p)\) of \(p\). The sets \(F_ t\cap D(p)\) are called smoothings of the singular point \(p\) of \(F\). The smoothings are called \(q\)-hyperbolic if for some continuous family \(q_ t\in RP^ n\), \(q_ 0=q\) for each real line \(L\supset\{q_ t\}\) we have \(L\cap F_ t\subset D(p)\). Author shows that all \(q\)-hyperbolic smoothings of a given singular point are rigidly isotopic, i.e. they are joined by continuous families of smoothings. He proved that the number of isotopic types of minimal smoothings of a solitary singular point of order \(k\) equals \(k/2+1\), and that the number of isotopic types of minimal smoothings of a nonsolitary singular point equals \((2r)!/(r!(r+1)!)\), where \(r\) is the number of real branches. It is also proved that by a small perturbation of the coefficients of a plane algebraic curve, one can realize the smoothings of all its singular points, isotopic to their preassigned minimal smoothings. isotopic hypersurfaces; smoothings of singular point of hypersurface; hyperbolic smoothings; family of hypersurfaces; plane algebraic curve Shustin, E.: Hyperbolic and minimal smoothings of singular points. Selecta math. Sov. 10, 19-25 (1991) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Real algebraic sets, Hypersurfaces and algebraic geometry Hyperbolic and minimal smoothings of singular 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 For a scheme \(X\) let \(GW_{0}(X)\) denote the Grothendieck-Witt group of symmetric bilinear spaces over \(X\). This is the abelian group generated by isometry classes \([\mathcal V,\phi ]\) of vector bundles \(\mathcal V\) over \(X\) with a nonsingular symmetric bilinear form \(\phi: \mathcal V\otimes \mathcal V\to O_X\) subject to the relations \([(\mathcal V,\phi) \perp (\mathcal V ',\phi ' ]= [{\mathcal V},{\phi}]+[\mathcal V ',\phi ']\) and \([\mathcal M,\phi]=[\mathcal H(\mathcal N)]\) for every metabolic space \((\mathcal M, \phi)\) with Lagrangian subbundle \(\mathcal N=\mathcal N^{\perp} \subset \mathcal M\) and associated hyperbolic space \(\mathcal H(\mathcal N)\). The higher Grothendieck-Witt groups \(GW_{i}(X), \, i\in {\mathbb N}\) were defined by the author [``Higher Grothendieck-Witt groups of exact categories'', J. K-theory (to appear)]. In the paper the author proves the Mayer-Vietoris sequence for open covers (Theorem 1). This main theorem of the paper (in fact Theorem 16 of the paper is more general and includes versions for skew symmetric forms and coefficients in line bundles different than \(O_{X}\) ) is derived from the localization (Theorem 2) and Zariski excision (Theorem 3) theorems. The author proves also additivity, fibration and approximation theorems for the hermitian \(K\)-theory of exact categories with weak equivalences and duality. As the author noticed, \textit{P. Balmer} [K-Theory 23, No.~1, 15--30 (2001; Zbl 0987.19002)] and \textit{J. Hornbostel} [Topology 44, No.~3, 661--687 (2005; Zbl 1078.19004)] proved similar results to theorems 1--3. However, their assumptions are stricter than these of the author. Grothendieck-Witt groups; hermitian K-theory; Mayer-Vietoris sequence; localization; excision Berrick, J., Karoubi, M., Østvær Paul, A., Schlichting, M.: The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-Theory, arxiv:1011.4977v2 ( 2011) Hermitian \(K\)-theory, relations with \(K\)-theory of rings, \(K\)-theory of schemes, Witt groups of rings, Algebraic theory of quadratic forms; Witt groups and rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus \(g\) over \(\mathbb{F}_q\) \((q \equiv 1\pmod 3)\) as the curve varies in an irreducible component of the moduli space. They show that for \(q\) fixed and \(g\) increasing, the limiting distribution of the trace of Frobenius equals the sum of \(q+1\) independent random variables taking the value \(0\) with probability \(2/(q+2)\) and \(1,\,e^{2\pi i/3},\,e^{4\pi i/3}\) each with probability \(q/(3(q+2))\). This extends the work of \textit{P. Kurlberg} and \textit{Z. Rudnick} [J. Number Theory 129, No. 3, 580--587 (2009; Zbl 1221.11141)] who considered the same limit for hyperelliptic curves (although not on the moduli space, which makes a slight difference explained in Theorem 1.1). They also show that when both \(g\) and \(q\) go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to \(p\)-fold covers of the projective line. Frobenius; traces of cyclic trigonal curves; Gaussian distribution Bucur, A.; David, C.; Feigon, B.; Lalın, M., Statistics for traces of cyclic trigonal curves over finite fields, Int. Math. Res. Not., 2010, 5, 932-967, (2009) Curves over finite and local fields, Finite ground fields in algebraic geometry Statistics for traces of cyclic trigonal curves over finite fields	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a log. smooth log. scheme \((X,M)\), defined over a discrete valuation ring \(V\), with residue field \(k\) of characteristic \(p\). Fix a uniformizer \(\pi\) of \(V\). The author studies the extensions of automorphisms \(\sigma\) of the special fiber \(X_s\) of \(X\) to the whole family \(X\). Assuming that the smooth locus is dense in \(X_s\), the author proves the existence of an integer \(m\) (actually computable) such that \(\sigma\) lifts to the log. structure \((X_s,M_s)\) over the special fiber if and only it lifts locally to the infinitesimal neighbourhood \(X/\pi^m\). This explains in greater detail the known fact that sheaves of nearby cycles to the special fiber depend only on the completion of \(X\). In the case of (relative) curves, the author shows a trace formula for the number of points collapsing to fixed points of \(\sigma\), in its lifting to \(X\). endomorphisms; valuation ring; logarithmic schemes Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry Endomorphisms of logarithmic schemes	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hilbert schemes \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) parametrizing closed subschemes \(X \subset \mathbb P^m\) with Hilbert polynomial \(p(t)\) have received much attention from algebraic geometers since their construction in the early 1960s. Although connected [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)], \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) exhibits many bad behaviors, for example it can have non-reduced components [\textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and every singularity type appears in some Hilbert scheme [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. In the paper under review, the authors classify all smooth Hilbert schemes. Recall that a Hilbert polynomial of a closed subscheme \(X \subset \mathbb P^m\) can be uniquely written in the form \(p(t) = \sum_{i=1}^r \binom{t+\lambda_i-i}{\lambda_i-1}\) where \(\lambda = (\lambda_1, \dots, \lambda_r)\) is a partition of integers satisfying \(\lambda_1 \geq \dots \geq \lambda_r \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. The authors prove that the partitions corresponding to a smooth Hilbert scheme are precisely those in the following list: (1) \(m = 2 \geq \lambda_1\). (2) \(m \geq \lambda_1\) and \(\lambda_r \geq 2\). (3) \(\lambda = (1)\) or \(\lambda = (m^{r-2}, \Lambda_{r-1},1)\), where \(r \geq 2\) and \(m \geq \lambda_{r-1} \geq 1\). (4) \(\lambda = (m^{r-s-3}, \lambda_{r-s-s}^{s+2},1)\), where \(r-3 \geq s \geq 0\) and \(m-1 \geq \lambda_{r-s-2} \geq 3\). (5) \(\lambda = (m^{r-s-5}, 2^{s+4},1)\), where \(r-5 \geq s \geq 0\). (6) \(\lambda = (m^{r-3},1^3)\), where \(r \geq 3\). (7) \(\lambda = (m+1)\) or \(r=0\). Moreover, the authors describe the schemes parametrized by each family listed. For example, the general member of family (3) is a union of a hypersurface of degree \(r-2\), a linear subspace of dimension \(\lambda_{r-1}\) and a point while the general member of family (5) is a union of a hypersurface of degree \(r-s-3\), a hypersurface of degree \(s+2\) of a linear subspace of dimension \(\lambda_{r-s-2}\) and a point. The families in (7) correspond to the one-point Hilbert scheme parametrizing \(\mathbb P^m\) and the empty scheme. All families in the theorem were previously known to be smooth: smoothness for families (1) and (6) follows from work of \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], families (2) and (3) were shown smooth by \textit{A. P. Staal} [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)] and families (4) and (5) were shown smooth by \textit{R. Ramkumar} [J. Algebra 617, 17--47 (2023; Zbl 1503.13007)] in his work on Hilbert schemes with at most two Borel-fixed ideals. The main contribution here is that this list is complete. As to the strategy of the proof, it is known from work of \textit{A. Reeves} and \textit{M. Stillman} that the lexicographical point is always smooth on the Hilbert scheme and determines a unique irreducible component of \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) of computable dimension [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)]. In theory one could attempt to show smoothness by computing the dimension of the Zariski tangent space at the other Borel-fixed points, but this is unwieldy. Instead, the authors construct families of subschemes corresponding to points on Hilbert schemes that necessarily singular. To describe these points, define a \textit{residual inclusion} \(X \subset Y \subset \mathbb P^m\) to be a closed immersion such that there is a linear subspace \(\Lambda \subset \mathbb P^m\) containing \(X\) and a hypersurface \(D \subset \Lambda\) with \(Y\) the residual scheme of \(D \subset X\) in \(\Lambda\). A \textit{residual flag} is a flag \(\emptyset \subset X_e \subset X_{e-1} \subset \dots \subset X_1\) of residual inclusions. For most Hilbert polynomials not in the list, the authors produce such an \(X_1\) near the lexicographic point which corresponds to a singular point on the Hilbert scheme. The others not on the list are handled with three other singular families. The proof is valid over \(\mathrm{Spec}\,\mathbb Z\). The authors credit \texttt{Macaulay2} [\url{http://www.math.uiuc.edu/Macaulay2/}] for many experimental computations that were indispensable in discovering their results. Hilbert schemes; Borel fixed ideals; partial flag varieties Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, Geometric invariant theory, Actions of groups on commutative rings; invariant theory Smooth Hilbert schemes: their classification and geometry	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an integral variety over a perfect field \(k\), \(\mathcal L_m(X)\) its jet scheme of level \(m\in \mathbb N\) and \(\mathcal L_{\infty}(X)\) its arc scheme. The general component \(\mathcal G_m(X)\) of \(\mathcal L_m(X)\) is the Zariski closure of \(\mathcal L_m(\mathrm{Reg}(X))\). If \(X\) is smooth on \(k\) then the geometry and topology of \(\mathcal L_m(X)\) are well understood. In this paper the authors consider the case \(X\) is not smooth and study some properties of the general component \(\mathcal G_m(X)\) by means of a smooth birational model of \(X\). Indeed, under the further hypothesis that \(X\) is affine embedded in \(\mathbb A^N_k\), the authors prove that a birational model of \(X\) provides a description of \(\mathcal G_m(X)\) that gives rice to an algorithm which computes a Groebner basis of the defining ideal of \(\mathcal G_m(X)\) in \(\mathbb A^N_k\) as a subscheme of \(\mathcal L_m(X)\) (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7. The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all \(\mathcal G_m(X)\)'' (Sections 8 and 9). computational aspects of algebraic geometry; derivation module; jet and arc scheme; singularities in algebraic geometry Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Computational aspects of algebraic curves, Computational aspects of higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry, Local complex singularities Two algorithms for computing the general component of jet scheme and 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 \(G\) be a semisimple algebraic group defined over \(\mathbb {Q}_p\), and let \(\Gamma \) be a compact open subgroup of \(G(\mathbb {Q}_p)\). We relate the asymptotic representation theory of \(\Gamma \) and the singularities of the moduli space of \(G\)-local systems on a smooth projective curve, proving new theorems about both: \begin{itemize} \item[(1)] We prove that there is a constant \(C\), independent of \(G\), such that the number of \(n\)-dimensional representations of \(\Gamma \) grows slower than \(n^{C}\), confirming a conjecture of \textit{M. Larsen} and \textit{A. Lubotzky} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 351--390 (2008; Zbl 1142.22006)]. In fact, we can take \(C=3\cdot {{\mathrm{dim}}}(E_8)+1=745\). We also prove the same bounds for groups over local fields of large enough characteristic. \item[(2)]We prove that the coarse moduli space of \(G\)-local systems on a smooth projective curve of genus at least \(\lceil C/2\rceil +1=374\) has rational singularities. \end{itemize} For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. 1 A. Aizenbud and N. Avni, 'Representation growth and rational singularities of the moduli space of local systems', \textit{Invent. Math.}204 (2016) 245-316. MR 3480557. Algebraic moduli problems, moduli of vector bundles, Singularities in algebraic geometry, Asymptotic properties of groups, Linear algebraic groups over local fields and their integers, Deformations of singularities, Symplectic structures of moduli spaces Representation growth and rational singularities of the moduli space of local systems	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book systematically develops a theory of ``continuously varying families'' of Berkovich spaces. In algebraic geometry, the idea of ``continuously varying families'' of schemes are formalized by \textit{flat} morphisms between schemes: in the words of \textit{D. Mumford} [The red book of varieties and schemes. Includes the Michigan lectures (1974) on ``Curves and their Jacobians''. 2nd, expanded ed. with contributions by Enrico Arbarello. Berlin: Springer (1999; Zbl 0945.14001)], ``the concept of flatness is a riddle that comes out of algebra, but which technically is the answer to many prayers.'' A standard result in algebraic geometry and commutative algebra, exemplifying the kind of ``continuous variation'' that is imposed by flatness, states that all fibers of a flat morphism between two equidimensional noetherian schemes must have the same dimension. Note also that, like all reasonable classes of morphisms of schemes, flat morphisms are stable under base change and ground field extension. In Berkovich geometry, one could consider the class morphisms of Berkovich spaces which are flat as morphisms of locally ringed spaces, just as in the case of schemes. Ducros calls these \textit{naively flat} morphisms. As the addition of the adverb ``naively'' suggests, this class is ill-behaved: it turns out to not be stable under base change and ground field extension, and Ducros describes a counterexample due to Temkin in sections 1.3 and 4.4. Ducros then remedies this roughly by defining a \textit{flat} morphism between Berkovich spaces to be one which is naively flat and remains naively flat after any base change and ground field extension. Ducros proceeds to show that this class of flat morphisms of Berkovich spaces has many of the same properties that flat morphisms of schemes do in algebraic geometry; for example, the result mentioned above about the constancy of dimension is discussed in section 4.5. More importantly, Ducros uses the machinery of flat morphisms between Berkovich spaces to prove the following. Given a morphism \(\phi : Y \to X\), we can consider the fibers \(Y_x\) over points \(x \in X\), which is a Berkovich space over the completed residue field \(\mathscr{H}(x)\). Suppose also that \(P\) assigns to each \(x \in X\) some property \(P(x)\) of Berkovich spaces over \(\mathscr{H}(x)\). Ducros proves that the set of points \(y \in Y\) for which \(\mathscr{O}_{Y,y}\) is flat over \(\mathscr{O}_{X,\phi(y)}\) and for which \(Y_{\phi(y)}\) satisfies \(P(\phi(y))\) is Zariski-open when \(P\) is either empty or any of the following: geometrically regular, Gorenstein, complete intersection, Cohen-Macaulay, geometrically reduced (if \(Y\) is relatively equidimensional), or geometrically normal (if \(Y\) is relatively equidimensional). Ducros also proves that the set of points \(y \in Y\) for which \(Y_{\phi(y)}\) satisfies \(P(\phi(y))\) is locally constructible when \(P\) is any of the following: geometrically regular, Gorenstein, complete intersection, Cohen-Macaulay, geometrically reduced, or geometrically normal. In rough outline, the book is structured as follows. Chapters 1, 2, and 3 review some background information on Berkovich spaces. Flatness is defined formally in chapter 4, where its basic properties are also developed. Chapters 5 through 9 form the technical heart of the book, though they are very likely to be of independent interest. Their contents are summarized in the subsequent paragraphs. The results mentioned above, about open-ness and local constructibility of various loci, are described in chapter 10. Chapter 11 discusses descent of various properties along flat morphisms. Parts of the results of this book generalize earlier work of \textit{R. Kiehl} [Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl., 2. Abhdl. 25--49 (1968; Zbl 0177.06101)] in the setting of affinoid rigid spaces. Chapter 5 introduces quasi-smooth morphisms. These are morphisms which satisfy a property akin to the Jacobian criterion of smoothness for schemes. The main result of the section is proving that quasi-smooth morphisms of Berkovich spaces are precisely the ones which are flat and fiberwise geometrically regular. Ducros's approach is similar to that of \textit{S. Bosch} et al. [Néron models. Berlin etc.: Springer-Verlag (1990; Zbl 0705.14001)], and expands on the quasi-étale morphisms of \textit{V. G. Berkovich} [Invent. Math. 115, No. 3, 539--571 (1994; Zbl 0791.14008)]. Chapter 6 discusses a technique of ''spreading out from the generic fiber.'' Suppose \(\phi : Y \to X\) is a morphism of Berkovich spaces and \(x\) is a point of \(X\) such that \(\mathscr{O}_{X,x}\) is a field (such a point should be thought of as the generic point of an irreducible component of \(X\)), and \(y \in Y_x\). There is a natural map of local rings \(\mathscr{O}_{Y,y} \to \mathscr{O}_{Y_x,y}\). Were \(\phi\) a morphism of schemes, this map would be an isomorphism. This is not true when \(\phi\) is a morphism of analytic spaces, but Ducros proves that this map is often a flat map; this ensures that many algebraic properties can be descended along this map. Chapters 7 and 9 study images of morphisms of analytic spaces. The main result, which appears in chapter 9, states that if \(\phi : Y \to X\) is a map of affinoid spaces and \(Y\) is the suppert of a flat coherent sheaf, then \(\phi(Y)\) is an analytic domain in \(X\). This is a generalization of an earlier result by \textit{S. Bosch} and \textit{W. Lütkebohmert} [Math. Ann. 296, No. 3, 403--429 (1993; Zbl 0808.14018), cor 5.11]. The proof relies on an analysis of the local situation, which is conducted in chapter 7. Chapter 8 develops a theory of dévissage as in the work of \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)]. The author defines dévissages and proves that they always exist. Berkovich analytic spaces; flatness; dévissage Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, moduli of curves (analytic), Rigid analytic geometry, Foundations of algebraic geometry, Topology of analytic spaces Families of Berkovich spaces	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a group of \(n \times n\) matrices over the field \(\mathbb{R}\) of real numbers acting on a finite dimensional vector space \(M\) over \(\mathbb{R}\). Assume there is a function \(\phi:G\times M\to M\) with the following properties: \(\phi(I,x)=x\) for all \(x\in M\), and \(\phi(A,\phi(B,x))=\phi(AB,x)\) for all \(A, B \in G\) and all \(x\in M\). An element \(x\in M\) has the global [local] Lipschitz property if there is a constant \(K > 0\) [are constants \(K > 0\), \(\varepsilon > 0\)] depending on \(x\) such that for every vector \(y \in O(x)=\{z \in M: \phi(A,x) =z\) for some \(A\in G\}\) [with \(\|y -z\|_2 < \varepsilon]\) there is an \(A\in G\) satisfying \(y=\phi(A,x)\) and \(\|I - A\|_1 \leq K \|y - x\|_2\) where \(\|\;\|_1\) is a norm on \(\mathbb{R}^{n \times n}\) and \(\|\;\|_2\) is a norm on \(M\). The authors study Lipschitz properties for matrix groups over the ground fields \(\mathbb{R}\), \(\mathbb{C}\), and \(\mathbb{H}\). Let \(G\) be a matrix group with elements \((G_1, G_2)\) where \(G_1 \in \mathbb{R}^{m \times m}\), \(G_2 \in \mathbb{R}^{n \times n}\) are invertible, and assume the action \(\phi\) of \(G\) on \(\mathbb{R}^{m \times n}\) is defined by \(\phi((G_1, G_2),x)=G^{-1}_1 x G_2\), \(x \in \mathbb{R}^{m \times n}\), \((G_1, G_2) \in G\). Then the action \(\phi\) has the local Lipschitz property. This result can be extended to complex or quaternionic matrices. Further, this theorem encloses some important particular cases, namely simultaneous similarity, restricted simultaneous similarity, equivalence, and simultaneous equivalence for \(m\)-tuples of matrices. From the local Lipschitz property the authors deduce the global Lipschitz property for (restricted) simultaneous similarity and for (restricted) simultaneous equivalence over \(\mathbb{R}\), \(\mathbb{C}\), and \(\mathbb{H}\). The authors also study simultaneous congruence. They consider a set of \(m\)-tuples \((A_1, \dots, A_m)\) of complex Hermitian \(n \times n\) matrices viewed as a real vector space. Let \(\phi\) be the action \(\phi(S, (A_1, \dots, A_m))=(S^* A_1 S, \dots, S^* A_m S)\). Assume that the \(n\times n\) complex Hermitian matrices \(A_1, \dots, A_m\) have no common isotropic vector. Then the \(m\)-tuple \((A_1, \dots, A_m)\) has the global Lipschitz property with respect to the action \(\phi\). The crucial point in the proof of this theorem is the proof of the local property which takes the most part of the paper. Moreover, a generalization for simultaneous congruence of not necessarily Hermitian matrices is stated. complex Hermitian matrices; matrix groups; actions; simultaneous similarity; simultaneous equivalence; local Lipschitz property; global Lipschitz property; simultaneous congruences DOI: 10.1016/0024-3795(94)90435-9 Linear algebraic groups over the reals, the complexes, the quaternions, Hermitian, skew-Hermitian, and related matrices, Group actions on varieties or schemes (quotients), Other matrix groups over fields, Matrices over special rings (quaternions, finite fields, etc.) Lipschitz properties of some actions of matrix groups	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author (along with G. Brumfiel) has defined abstract semialgebraic functions on the constructible subsets of the real spectrum of a ring A. The ring R of these functions on the real spectrum is called the real closure of A [the author, Mem. Am. Math. Soc. 397 (1989; Zbl 0697.14015)]. The abstract semialgebraic functions also form a sheaf for the ``inverse'' topology (for which the usual closed constructible subsets of the real spectrum form a basis of open sets), giving a ringed space \(Spec^R\). These ringed spaces are the building blocks of inverse real closed spaces [the author, Illinois J. Math. (to appear)], an abstraction of weakly semialgebraic spaces [\textit{M. Knebusch}, ``Weakly semialgebraic spaces'', Lect. Notes Math. 1367 (1989; Zbl 0681.14008)]. The purpose of the paper is to characterize \(Spec^R\) by a universal property. Let \({\mathcal D}\) be the category of ringed spaces (X,\({\mathcal O}_ X)\) whose stalks are real closed (in the above sense) integral domains, with morphisms of ringed spaces \(\phi: X\to Y\) such that \(\phi^({\mathcal O}_ Y)\to {\mathcal O}_ X\) is injective. Then \(Spec^R\) is in \({\mathcal D}\), and represents the functor \(X\mapsto Hom(A,H^ 0(X,{\mathcal O}_ X)).\) The first section of the paper contains a characterization of real closed integral domains which gives a first order axiomatisation of these rings. - The second section is devoted to the ``inverse affine scheme'' \(Spec^*A\) of any ring A. This is the same as the ``integral spectrum'' of \textit{P. T. Johnstone} [J. Algebra 49, 238-260 (1977; Zbl 0369.13019)], which was used to characterize those rings which are representable as rings of global sections of sheaves of integral domains [cf. \textit{J. F. Kennison}, Math. Z. 151, 35-56 (1976; Zbl 0346.18012)]. semialgebraic functions; real spectrum; inverse real closed spaces Schwartz, N.: Eine universelle eigenschaft reell abgeschlossener rǎume, Commun. algebra 18, No. 3, 755-774 (1990) Semialgebraic sets and related spaces, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Eine universelle Eigenschaft reell abgeschlossener Räume. (A universal property of real closed spaces)	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The first question raised in this paper, as suggested by its title, is whether one can define higher-dimensional formal loops in a scheme. Such a notion has been developed previously, generalizing earlier work of Contou-Carrère. Here, the author chooses instead to generalize an approach of Kapranov and Vasserot. Given a scheme \(X\), one can take the space of maps from the formal neighborhood of 0 in affine \(d\)-space and likewise the space of maps from a punctured formal neighborhood. The definition given here is that the \(d\)-dimensional formal loops are given by the completion of the latter space in the former. Making such a definition precise is subtle, but this loop space is shown to be representable by a derived ind-pro-scheme. The author uses derived algebraic geometry as the appropriate context in which to make these definitions, and indeed allows for generalization from schemes to stacks. The paper begins with an introduction to the higher categorical methods and derived algebraic geometry results that are needed. The second question is whether, if \(X\) has a symplectic structure, the higher loop space can also be endowed with a natural symplectic structure. While an initial answer might be that such a structure is impossible, since this loop space is typically infinite-dimensional, the author develops a notion of Tate stacks as the appropriate context in which to frame the question. If \(X\) is symplectic, then its higher dimensional formal loop space has the structure of a Tate stack. A variant, called the bubble space of \(X\), is also shown to have this structure. formal loops; derived algebraic geometry; shifted symplectic structures Categories in geometry and topology, Topological categories, foundations of homotopy theory, Abstract and axiomatic homotopy theory in algebraic topology, Schemes and morphisms, Generalizations (algebraic spaces, stacks) Higher dimensional formal loop 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 This paper is devoted to the Poisson geometry behind directed networks on surfaces. In previous works, the authors have already used Postnikov's construction in order to parametrize cells in Grassmannians and proved that the space of weights of networks in a disk can be naturally endowed with Poisson brackets. The purpose of this paper is to generalize the construction for directed weighted networks in an annulus. To this aim, the authors first associate with every network a matrix of boundary measurements. Then, they characterize all universal Poisson brackets on the space of edge weights. They show that the universal Poisson brackets induce a family of Poisson structures on rational matrix-valued functions and on the space of loops in the Grassmannian. Poisson geometry; network; annulus; weight; Grassmannian; R-matrix M. Gekhtman, M. Shapiro and A. Vainshtein, \textit{Poisson geometry of directed networks in an annulus}, arXiv:0901.0020. Poisson manifolds; Poisson groupoids and algebroids, Grassmannians, Schubert varieties, flag manifolds Poisson geometry of directed networks in an annulus	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 notion of regular separation for solutions of systems of ODEs \(y^\prime = F (x, y)\), where \(F\) is definable in a polynomially bounded o-minimal structure and \(y = (y_1, y_2)\). Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author. solutions of ODEs; non-oscillating trajectories of vector fields; o-minimality; Hardy field; transcendental formal solutions Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations, Model theory of ordered structures; o-minimality, Asymptotic properties of solutions to ordinary differential equations, Semi-analytic sets, subanalytic sets, and generalizations, Real-analytic and semi-analytic sets Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(M\) be a smooth manifold and \(\mathfrak Z\) a subspace of the space of complex vector fields on \(M\), which is closed under multiplication by complex valued smooth functions. Then one considers weak \(L^2_{loc}\) solutions of the system of equations \(Zu=0\) for all \(Z\in\mathfrak Z\). The main aim of the article is to find conditions on \(\mathfrak Z\) which imply that such solutions satisfy a maximum modulus principle and/or unique continuation. The basic conditions to be imposed are versions of hypoellipticity on one hand and conditions on the behavior of \(\mathfrak Z\) under the Lie bracket on the other hand. The latter conditions are either expressed as a Hörmander condition or via the Sussmann leaf associated to \(\mathfrak Z\). The authors first show that Lipschitz-hypoellipticity together with openness of the Sussmann leaf implies a local maximal modulus principle. If in addition weak unique continuation holds, one obtains a global maximal modulus principle. Second, the authors study weak unique continuation, deriving results in the real analytic case, and for embedded CR manifolds satsifying weakenings of pseudoconcavity. complex vector fields; maximum modulus principle; weak unique continuation; \(CR\) manifold CR structures, CR operators, and generalizations, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras, Homotopy groups of topological groups and homogeneous spaces Complex vector fields, unique continuation and the maximum modulus principle	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((X, 0)\) be the germ of a complex normal surface singularity, with link \(Sigma\). A smoothing of \((X, 0)\) is a morphism \(f : (X , 0) \to (\mathbb{C}, 0)\), with \((X , 0)\) a threedimensional isolated Cohen-Macaulay singularity, equipped with an isomorphism \((f ^{1} (0), 0)\simeq(X, 0)\). The Milnor fibre \(M\) is the general fibre \(f^{-1} (\delta)\). The second Betti number of \(M\) is called \(\mu\), the Milnor number of the smoothing. We say that \(f\) is a \(QHD\) smoothing if \(\mu = 0\). There are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a \(QHD\) smoothing. The fundamental group of the Milnor fibre has been known for all except three exceptional families. In this work the authors settle these cases, they present a new explicit construction for one of the exceptional families, showing the fundamental group is non-abelian. They show that the fundamental groups for the remaining two exceptional families are abelian. In all three cases, they show that it is possible to use the plane curves and their blow-ups to compute the fundamental group of the complement. surface singularity; rational homology sphere; Milnor fibre Singularities of curves, local rings, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Fundamental group, presentations, free differential calculus Fundamental group of rational homology disk smoothings of surface singularities	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the following classical interpolation problem: Given a finite set of points \(\{P_1,\dots, P_n\}\) in the projective plane and a set \(\{m_1,\dots, m_n\}\) of non negative integers, determine the dimension of the linear system \(\|L\|\) of plane curves of degree \(d\) passing through each \(P_i\) with multiplicity greater or equal than \(m_i\). Since every \((m_i)\)-uple point imposes \(m_i(m_i-1)/2\) conditions to curves, it is natural to expect that \(\|L\|\) has dimension either \((d(d-3)-\sum_1^n m_i(m_i-1))/2\) or is empty, when the previous number is negative. On the other hand it is easy to write down examples in which the true dimension is different from the expected one. Harbourne and Hirschowitz conjectured that the true and the expected dimensions differ exactly when the the proper transform of \(\|L\|\) in the blow up of \({\mathbb P}^2\) at the \(P_i\)'s has a fixed rational component of self intersection \(-1\). By now, only few particular cases of the conjecture are established. The authors examine here the \textit{homogeneous} case, in which all the multiplicities are the same (say equal to \(m\)). In this situation, they give an explicit description of all the cases in which the \((-1)\)-curve quoted above actually exists. Then they use an inductive procedure, which relies on the degeneration of the plane to a union of rational surfaces, to attack the conjecture. The authors are able to control the inductive steps and prove the conjecture for all \(m\leq 12\). plane curves; multiplicity; interpolation problem; linear system C. Ciliberto and R. Miranda, Linear systems of plane curves with base points of egual multiplicity , Trans. Amer. Math. Soc. 352 (2000), 4037-4050. JSTOR: Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves Linear systems of plane curves with 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 Let \(\kappa\) be a field and \(K\) its algebraic closure. For every positive integer \(m\in \mathbb{N}\), let \(A^m(\kappa)\) be the \(m\)-dimensional affine space over \(\kappa\). A subset \(C \subset A^m(K)\) is called constructible if it is equal to the image of an algebraic variety under the projection map from \(A^M(K)\) to \( A^m(K)\) with \(M \ge m\). In the literature, the degree of a constructible set \(C\) is defined to be the degree of its Zariski closure; however, as is shown in this paper, Bézout's inequality does not hold for this definition. In the first part of the paper, the authors present two different notions of degree for a constructible set and prove that they satisfy a Bézout inequality. Furthermore, the authors study several properties of the new notions. The second part of the paper is devoted to correct test sequences. Let \(X\) be a set, \(K\) a field and \(m\in \mathbb{N}\) a positive integer. Let \(\Omega\) be a set whose elements are sequences of length \(m\) of functions from \(X\) to \(K\). Let \(\Sigma \subset \Omega\). A correct test sequence of length \(L\) for \(\Omega\) with discriminant \(\Sigma\) is a finite set of \(L\) elements \(\{x_1, \ldots , x_L \} \subseteq X\) such that for each \(f\in \Omega\), if \(f(x_1)=\cdots=f(x_L)=(0,\ldots ,0)\) then \(f\in \Sigma\). Correct test sequences were introduced by Heintz and Schnorr in 1982 and they appeared in different places in mathematics. In this paper, by using the results of the first part, the authors prove that correct test sequences for lists of polynomials are densely distributed in any constructible set of accurate co-dimension and degree. Finally, they apply correct test sequences of asymptotically optimal length to describe a randomized algorithm to decide whether a list of polynomials forms a ``suite sécante'': a sequence of polynomials \(f_1,\ldots,f_m\) in terms of \(n\) variables is called suite sécante if the associated variety has dimension \(n-m\). degree of constructible and locally closed affine sets; Bézout's inequality; correct test sequences; probability; existence; equality test of lists of functions; bounded error probability; polynomial-time algorithms Symbolic computation and algebraic computation, Polynomials in general fields (irreducibility, etc.), Polynomial rings and ideals; rings of integer-valued polynomials, Solving polynomial systems; resultants, Effectivity, complexity and computational aspects of algebraic geometry, Analysis of algorithms and problem complexity, Randomized algorithms, Analysis of algorithms A promenade through correct test sequences. I: Degree of constructible sets, Bézout's inequality and density	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to prove a criterion for projectively Cohen- Macaulay-two-codimensional subschemes \(Y\) of \(\mathbb{P}_ k^ N\) (\(k\) algebraically closed of characteristic 0) to be smoothable. The considered schemes are determinantal, i.e. the homogeneous ideal \(I(Y)\), defining \(Y\), is generated by the \(t\times t\)-minors of a homogeneous \(m\times n\)-matrix \(M\) with \(\text{ht}(I(Y))= (m-t+1) (n-t+1)\). Determinantal schemes are equidimensional with a Cohen-Macaulay coordinate ring. It is easy to see that a closed subscheme \(Y\subset \mathbb{P}_ k^ N\) of equicodimension two with a Cohen-Macaulay coordinate ring is necessarily determinantal. The author determines first the codimension of the singular locus of \(Y\) (see propositions 1 and 2 in section 2.1) under certain conditions on the entries of \(M\). Then she proves the existence of a deformation \(f:X\to Z\) of \(Y\) (determinantal and of codimension 2) over an irreducible \(k\)-scheme \(Z\) (i.e. \(Y=f^{- 1}(z_ 0)\) over a closed point \(z_ 0\)); and \(Y\) is then smoothable iff \(2\leq N\leq 5\) (see theorem in section 2.2). This result generalizes a corresponding criterion of \textit{T. Sauer} [Math. Ann. 272, 83-90 (1985; Zbl 0546.14023)] for curves in \(\mathbb{P}_ k^ 3\). determinantal schemes; smoothability Frauke, S.:Generic determinantal schemes and the smoothability of determinantal schemes of codumension 2, Manuscripta Math.82 (1994) 417--431. Determinantal varieties, Linkage, complete intersections and determinantal ideals, Deformations and infinitesimal methods in commutative ring theory Generic determinantal schemes and the smoothability of determinantal schemes of codimension 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 provides a new proof of the result in [\textit{J.-P. Rolin} et al., J. Amer. Math. Soc. 16, 751--777 (2003; Zbl 1095.26018)] and a result similar to that in [\textit{J.-P. Rolin} et al., Proc. London Math. Soc. 95, 413--442 (2007; Zbl 1123.03031)] using the result of [\textit{J.-M. Lion}, J. Symbolic Logic 67, 1616--1622 (2002; Zbl 1042.03030)] on o-minimality of the structure generated by geometric, regular and \(0\)-regular family of functions. They provide examples of o-minimal structures without \(\mathcal C^\infty\)-cell decompositions. The first main result (Theorem 4.3) of this paper is as follows: Let \(\mathcal A = \{\mathcal A_m\}_{m \in \mathbb N}\) be a family of algebras of restricted smooth functions closed under taking partial derivatives. If the geometric family \(S(\mathcal A)\) generated by the germs of \(\mathcal A\) at every point in \([-1,1]^m\) is quasi-analytic, then \(\mathcal A\) is contained in a certain o-minimal structure. In his proof, the author introduces the notion of a geometric family of almost \(\mathcal C^{\infty}\)-germs and develops a normal crossings transformation theorem on their quasianalytic algebras, which is an adaptation of the classical reduction of analytic functions to normal crossings. Using this theorem together with Lion's result, the author gets another proof of the above result. Using Lion's result again, the author constructed an o-minimal structure generated by the sum of a function which is semialgebraic outside every neighborhood of zero and infinitely flat at zero and a strongly transcendental smooth function (Theorem 6.6). This result implies the existence of two functions such that each of their germs is contained in the Hardy field of an o-minimal structure, but both of them are not simultaneously contained in the Hardy field of any o-minimal structure (Proposition 7.6). o-minimality; theorem of Lion; Hardy field Model theory of ordered structures; o-minimality, Semialgebraic sets and related spaces Theorem of lion and o-minimal structures which do not admit \(\mathcal{C}^{\infty}\)-cell 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 purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings \(f:C \to\mathbb{P}^1\) of the projective line by smooth curves. Thus, from the very outset, one sees that there are essentially two ways to approach the Hurwitz scheme: (1) We start with \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\). (2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\). One finds that one can obtain the most information about the Hurwitz scheme and its compactification by exploiting interchangeably these two points of view. Our first main result is the following theorem. Let \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite. The number \(g\) is the genus of the ``curve \(C\) upstairs'' of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). -- Note that although in the statement of the theorem we spoke of ``the'' Hurwitz ``scheme,'' there are in fact several different Hurwitz schemes used in the literature, some of which are, in fact, not schemes, but stacks. The main idea of the proof is that by combinatorially analyzing the boundary of the compactification of the Hurwitz scheme, one realizes that there are essentially three kinds of divisors in the boundary, which we call excess divisors, which are ``more important'' than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally. Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings: Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\). finite Picard group; Hurwitz scheme; Hurwitz coverings; admissible coverings S. Mochizuki, ''The geometry of the compactification of the Hurwitz scheme,'' Publ. Res. Inst. Math. Sci., vol. 31, iss. 3, pp. 355-441, 1995. Families, moduli of curves (algebraic), Picard groups, Coverings in algebraic geometry The geometry of the compactification of the Hurwitz 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 Let \(S\) be a regular scheme and \({\mathcal S}m/S\) the category of smooth and separated schemes of finite type over \(S\). \textit{F. Morel} and \textit{V. Voevodsky} proved [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] representability of the algebraic \(K\)-theory of regular schemes in the \(\mathbb A^1\)-homotopy theory: Theorem 0.1 [Morel, Voevodsky]: Let \(S\) be a regular scheme. Then, for any \(n\in\mathbb N\) and \(X\in{\mathcal S}m/S\), there is a canonical isomorphism \[ \Hom_{{\mathcal H}_\bullet(S)}(S^n\wedge X_+,\mathbb Z\times\mathbb G\text{r})\cong K_n(X). \] \(\mathbb G\text{r}\) is a colimit of the system \((\mathbb G\text{r}_{d,r})_{(d,r)\in\mathbb N^2}\) in the category of presheaves over \({\mathcal S}m/S_{\text{Nis}}\) and \(\mathbb G\text{r}_{d,r}\) denotes the Grassmann scheme parametrizing subbundles of rank \(d\) in the trivial bundle of rank \(d+r\). From the isomorphism in the above theorem it is clear that the endomorphisms of \(\mathbb Z\times\mathbb G\text{r}\) in \({\mathcal H}_\bullet(S)\) act on algebraic \(K\)-groups of schemes in \({\mathcal S}m/S\). The author proves the following theorem: Theorem 0.2: Let \(S\) be a regular scheme. We let \(K_{0}(-)\) be the presheaf of sets on \({\mathcal S}m/S\) which maps \(X\) to \(K_{0}(X)\). Then the map induced by Theorem 0.1 is a bijection: \[ \text{End}_{{\mathcal H}(S)}(\mathbb Z\times\mathbb G\text{r}) \overset\sim \rightarrow \text{End}_{{\mathcal S}m/S^{\text{opp}}{\mathcal S}ets}(K_{0}(-)). \] The theorem shows that operations defined on the level of \(K_{0}\) [\textit{P. Berthelot, A. Grothendieck} and \textit{L. Illusie} (eds.), Séminaire de géométrie algébrique du Bois Marie 1966/67. Lect. Notes Math. 225 (1971; Zbl 0218.14001)] for example \({\lambda}^n\) or \({\Psi}^{n}\) lift uniquely to \({\mathcal H}(S)\) and therefore via Theorem 0.1 extend uniquely to operations on higher algebraic \(K\)-theory. This also works for operations which involve several operands, for example products. In section 2 the author studies the structures on \(\mathbb Z\times\mathbb G\text{r}\) that come from from algebraic structures on \(K_{0}\). In particular he shows that \(\mathbb Z\times\mathbb G\text{r}\) is equipped with a structure of a special \(\lambda\)-ring with duality. In section 3 the author compares the constructions of products, \(\lambda\)-operations and Adams' operations by various authors [cf. \textit{J.-L. Loday}, Ann. Sci. Éc. Norm. Supér. (4) 9, 309--377 (1976; Zbl 0362.18014); \textit{F. Waldhausen}, Lect. Notes Math. 1126, 318--419 (1985; Zbl 0579.18006); \textit{Ch. Kratzer}, Comment. Math. Helv. 55, 233--254 (1980; Zbl 0444.18008); \textit{C. Soulé}, Can. J. Math. 37, 488--550 (1985; Zbl 0575.14015); \textit{D. R. Grayson}, K-Theory 6, No. 2, 97--111 (1992; Zbl 0776.19001); \textit{F. Lecomte}, Algebraic \(K\)-theory and its applications. Proceedings of the workshop and symposium, ICTP, Trieste, Italy, September 1-19, 1997. Singapore: World Scientific. 437--449 (1999; Zbl 0969.55008) and \textit{M. Levine}, Fields Inst. Commun. 16, 131--184 (1997; Zbl 0883.19001)] to his constuction. In section 4 the author relates his constructions to virtual categories of \textit{P. Deligne} [Contemp. Math. 67, 93--117 (1987; Zbl 0629.14008)]. In section 5 the author examines the operations \({\tau}: K_{0}(-)\rightarrow K_{0}(-)\) such that \({\tau}(x+y)={\tau}(x)+{\tau}(y).\) These correspond to \(H\)-group endomorphisms of \(\mathbb Z\times\mathbb G\text{r}\). In section 6 a version of homotopical Riemann-Roch formulas is derived. operations in algebraic \(K\)-theory; Riemann-Roch theorems; Chern character Joël Riou, ``Algebraic \(K\)-theory, \(A^1\)-homotopy and Riemann-Roch theorems'', J. Topol.3 (2010) no. 2, p. 229-264 \(Q\)- and plus-constructions, Motivic cohomology; motivic homotopy theory, Homotopy theory Algebraic \(K\)-theory, \(A^1\)-homotopy and Riemann-Roch theorems	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The so-called Brenner-Hochster-Kollár problem about vector-valued continuous solutions of finite systems of linear equations was solved by \textit{C. Fefferman} and \textit{J. Kollár} [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)]. In the paper under review, the analytic method to settle this problem is extended to solve it for the space \(C^m(\mathbb{R}^n)\) of real-valued functions whose derivatives up to order \(m\) are continuous and bounded in \(\mathbb{R}^n\), and for the space \(C^{m,\omega}(\mathbb{R}^n)\) of real-valued functions whose \(m\)-th derivatives have modulus of continuity \(\omega\). These cases had been left open. For \(C^m(\mathbb{R}^n)\), the authors solve a more general problem that includes also a general Whitney extension problem for vector-valued functions and jets. The more general problem is formulated in terms of sections for bundles. A variant of this general problem for \(C^{m,\omega}(\mathbb{R}^n,\mathbb{R}^d)\) is also solved by means of a theorem which is a type of ``finiteness principle''. The dependence of a certain ``finiteness constant'' with respect to \(m\), \(n\) and \(d\) is also discussed. This paper continues recent, deep work by Bierstone-Milman-Pawlucki, Brudnyi-Shvartsman, Fefferman-Kollár, among others. generalized Whitney problems; extending vector-valued functions; continuous closures; differentiable closures; Brenner-Hochster-Kollár problem Fefferman, C.; Luli, G.K., The Brenner-hochster-Kollár and Whitney problems for vector-valued functions and jets, Revista Matematica Iberoamericana, 30, 875-892, (2014) Topological linear spaces of continuous, differentiable or analytic functions, Functions of several variables, Computational aspects in algebraic geometry The Brenner-Hochster-Kollár and Whitney problems for vector-valued functions and jets	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A real linear operator \({\mathcal M}\) acts on \(\mathcal{C}^n\) according to \(z\rightarrow {\mathcal M}z=Mz+M_{\#}\bar{z}\) for a pair of matrices \(M,M_{\#} \in \mathcal{C}^{n\times n}\) called the linear and antilinear parts of \({\mathcal M}.\) If \({\mathcal M}_{\#}=0\) (resp. \({\mathcal M}=0\)) then \({\mathcal M}\) is called \(\mathcal{C}\)-linear (resp. antilinear). The spectrum of \({\mathcal M}\) consists of those points \(\lambda \in \mathcal{C}\) for which \(\lambda I-{\mathcal M}\) is not invertible giving rise to a bounded, possibly empty, real algebraic plane curve of degree \(2n.\) The authors presents results concerning the location of the eigenvalues of \({\mathcal M}\) and classify components of the spectrum. Path continuation techniques are implemented for computing components and subsets of the spectrum once an eigenvalue is available. Three numerical examples illustrating the aspect studied have been presented. real linear operator; spectrum; characteristic bivariate polynomial; path following techniques; real algebraic plane curve; location of eigenvalues; numerical examples Huhtanen M., von Pfaler J.: The real linear eigenvalue problem in \$\$\{\(\backslash\)mathbb C\^n\}\$\$ . Linear Algebra Appl. 394, 169--199 (2005) Eigenvalues, singular values, and eigenvectors, Linear transformations, semilinear transformations, Computational aspects of algebraic curves, Numerical computation of eigenvalues and eigenvectors of matrices The real linear eigenvalue problem in \(\mathbb C^n\)	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 In this paper, we study length categories using iterated extensions. We fix a field \(k\), and for any family \(\mathsf{S}\) of orthogonal \(k\)-rational points in an Abelian \(k\)-category \(\mathcal{A} \), we consider the category \(\mathbf{Ext}(\mathsf{S})\) of iterated extensions of \(\mathsf{S}\) in \(\mathcal{A} \), equipped with the natural forgetful functor \(\mathbf{Ext}(\mathsf{S}) \to \mathcal{A}(\mathsf{S})\) into the length category \(\mathcal{A}(\mathsf{S})\). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel's criterion, we give a complete classification of the indecomposable objects in \(\mathcal{A}(\mathsf{S})\) when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family \(\mathsf{S}\) in \(\mathcal{A}\). As an application, we classify all graded holonomic \(D\)-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when \(D\) is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish. finite length categories; uniserial categories; iterated extensions; noncommutative deformations Abelian categories, Grothendieck categories, Representation theory of associative rings and algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Iterated extensions and uniserial length categories	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper extends previous work by \textit{R. M. Skjelnes} and \textit{D. Laksov} [Compos. Math. 126, 323-334 (2001; Zbl 1056.14500)] and gives a description of the Hilbert scheme of \(n\) points on the affine scheme \(C:=\text{Spec}(K[x]_U)\), where \(K\) is a field and \(K[x]_U\) is a fraction ring of the polynomial ring in one variable. The Hilbert functor of \(n\) points on \(C\), \(\text{Hilb}^n\), associates to a \(K\)-algebra \(A\) the set \(\text{Hilb}^n(A)\), formed by the ideals \(I\) of \(A\otimes{_KK}[x]_U\) such that the residue class ring \(A\otimes{_KK}[x]_U/I\) is locally free of finite rank \(n\) (as \(A\)-module). The main result is that \(\text{Hilb}^n\) is represented by the fraction ring \(H=K[s_1,...,s_n]_{U(n)}\), where \(s_1,...,s_n\) are the elementary symmetric functions in the variables \(t_1,...,t_n\) and \(U(n)=\{f(t_1)\dots f(t_n)\mid f\in U\}\). Moreover, the universal family of \(n\)-points on \(C\) is isomorphic to \(\text{Sym}_K^{n-1}(C)\times_K C\), as in the case where \(C\) is a smooth curve. The result relies on a characterization of the characteristic polynomial of the multiplication by the residue of a polynomial in \(A[x]/(F)\), where \(A\) is any commutative ring and \(F\) a monic polynomial in \(A[x]\). Hilbert scheme; resultants; points on the line; characteristic polynomial of the multiplication Skjelnes, R. M.: Resultants and the Hilbert scheme of the line. Ark. mat. 40, No. 1, 189-200 (2002) Parametrization (Chow and Hilbert schemes), Algebraic functions and function fields in algebraic geometry, Fine and coarse moduli spaces, Polynomial rings and ideals; rings of integer-valued polynomials Resultants and the Hilbert scheme of points on the line	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a basic compact semialgebraic set \(\mathbf{K}\subset\mathbb{R}^n\), we introduce a methodology that generates a sequence converging to the volume of \(\mathbf{K}\). This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on \(\mathbf{K}\) can be approximated as closely as desired, which permits the approximation of the integral on \(\mathbf{K}\) of any given polynomial; the extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed. computational geometry; volume; integration; \(\mathbf{K}\)-moment problem; semidefinite programming D. Henrion, J. B. Lasserre, and C. Savorgnan, \textit{Approximate volume and integration for basic semialgebraic sets}, SIAM Rev., 51 (2009), pp. 722--743, . Semialgebraic sets and related spaces, Sums of squares and representations by other particular quadratic forms, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Convex programming Approximate volume and integration for basic semialgebraic sets	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``\textit{Appell's hypergeometric functions} \(F_2\) and \(F_3\) are the analytic continuations of Appell's hypergeometric series'' defined by the relations \[ {F_2}\left( {\alpha ,\beta ,\beta ',\gamma ,\gamma ';x,y} \right) = \sum_{m,n \geqslant 0} {\frac{{{(\alpha)_{m + n}}{(\beta)_m}{(\beta')_n}}} {{{(\gamma)_m}{(\gamma')_n}m!n!}}{x^m}{y^n}} ,\quad \left\| x \right\| + \left\| y \right\| < 1 \tag{1} \] \[ {F_3}\left(\alpha,\alpha',\beta,\beta',\gamma ;x,y \right) = \sum_{m,n \geqslant 0} {\frac{{{(\alpha)_m}{(\alpha')_n}{(\beta)_m}{(\beta')_n}}} {{{(\gamma)_{m + n}}m!n!}}{x^m}{y^n}} ,\quad \left\| x \right\| < 1,\left\| y \right\| < 1 \tag{2} \] where, \({(a)_n}\) is the Pochhammer symbol (shifted factorial) defined by \[ {(a)_n} = a(a+1)(a+2)\cdots (a+n-1) \tag{3} \] The functions \(F_2\) and \(F_3\) are known to satisfy the following systems of partial differential equations denoted by \(E_2\) and \(E_3\) (each of rank 4) respectively: \[ \left({E_2}\right)\begin{cases} {\left[ {x\left( {1 - x} \right)\frac{{{\partial ^2}}} {{\partial {x^2}}} - xy\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma - \left( {\alpha + \beta + 1} \right)x} \right\}\frac{\partial } {{\partial x}} - \beta y\frac{\partial } {{\partial y}} - \alpha \beta } \right]F = 0,} \\ {\left[ {y\left( {1 - y} \right)\frac{{{\partial ^2}}} {{\partial {y^2}}} - xy\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma ' - \left( {\alpha + \beta ' + 1} \right)y} \right\}\frac{\partial } {{\partial y}} - \beta 'x\frac{\partial } {{\partial x}} - \alpha \beta '} \right]F = 0} \end{cases} \tag{4} \] and \[ \left( {{E_3}} \right)\begin{cases} {\left[ {x\left( {1 - x} \right)\frac{{{\partial ^2}}} {{\partial {x^2}}} + y\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma - \left( {\alpha + \beta + 1} \right)x} \right\}\frac{\partial } {{\partial x}} - \alpha \beta } \right]F = 0,} \\ {\left[ {y\left( {1 - y} \right)\frac{{{\partial ^2}}} {{\partial {y^2}}} + x\frac{{{\partial ^2}}} {{\partial x\partial y}} + \left\{ {\gamma - \left( {\alpha ' + \beta ' + 1} \right)y} \right\}\frac{\partial } {{\partial y}} - \alpha '\beta '} \right]F = 0} \end{cases} \tag{5} \] both on the space \[ {\mathbb{C}^2}\backslash \left\{ {\left\{ {x = 0} \right\} \cup \left\{ {x = 1} \right\} \cup \left\{ {y = 0} \right\} \cup \left\{ {y = 1} \right\} \cup \left\{ {x + y = 1} \right\}} \right\} \subset {\left( {{\mathbb{P}^1}} \right)^2}. \] E. Nakagiri had earlier in his Master's thesis [\textit{E. Nakagiri}, Monodromy representations of hypergeometric differential equations of two variables (Japanese). Master's Thesis. Kobe: Kobe University (1979)] investigated the monodromy representations associated with \(F_2\) and \(F_3\), where he gave ``explicit formulas about the matrix elements of the representations'' by imposing the following conditions \[ \begin{cases} {\text{Re} \left( {1 - \alpha } \right),\text{Re} \left( \beta \right),\text{Re} \left( {\beta '} \right),\text{Re} \left( {\gamma - \beta } \right),\text{Re} \left( {\gamma ' - \beta '} \right) > 0,} \\ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}{\text{ and }}\gamma ,\gamma ' \notin \mathbb{Z}}\end{cases} \tag{6} \] in the \(E_2\) case and \[ \begin{cases} {\text{Re} \left( {1 - \alpha } \right),\text{Re} \left( {1 - \alpha '} \right),\text{Re} \left( \beta \right),\text{Re} \left( {\beta '} \right),\text{Re} \left( {\gamma - \beta - \beta '} \right) > 0,} \\ {\gamma - \alpha - \alpha ',\gamma - \alpha - \beta ',\gamma - \alpha ' - \beta \notin {\mathbb{Z}_{ \leqslant 0}}{\text{ and }}\beta - \alpha ,\beta ' - \alpha ' \notin \mathbb{Z}} \end{cases}\tag{7} \] in the \(E_3\) case. The author of the present paper relaxes these two conditions (i.e. (6) and (7)) of Nakagiri [loc. cit.] to the following: \[ \begin{cases} { - \alpha ,\beta - 1,\beta ' - 1,\gamma - \beta - 1,\gamma ' - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}},} \\ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \end{cases} \tag{8} \] in the \(E_2\) case and to \[ \begin{cases} { - \alpha , - \alpha ',\beta - 1,\beta ' - 1,\gamma - \beta - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}},} \\ {\gamma - \alpha - \beta ',\gamma - \alpha ' - \beta ,\gamma - \alpha - \alpha ' - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \end{cases} \tag{9} \] in the \(E_3\) case and derives ``Nakagiri's matrix elements of the five generators of the monodromy representations associated with \(F_2\) and \(F_3\).'' He utilizes the integrals of multivalued functions and deforms the ``the chains on which the integrals are defined'' by showing ``each of the chains in the two dimensional complex space by the product of the paths in the complex plane''. He also gives an elegant derivation of the sufficient conditions for the systems of the partial differential equations \(E_2\) and \(E_3\) being irreducible. In the first section of the paper the author introduces his \textit{auxiliary integrals} as \[ \int {\int_C {t_1^{{\lambda _1}}{{\left( {{t_1} - {z_1}} \right)}^{{\lambda _2}}}t_2^{{\lambda _3}}{{\left( {{t_2} - {z_2}} \right)}^{{\lambda _4}}}{{\left( {{t_1} + {t_2} - 1} \right)}^{{\lambda _5}}}d{t_1}d{t_2}} } \tag{10} \] over a suitable chain \(C\). The following abbreviated symbols \[ e(A) = \exp \left( {2\pi \sqrt { - 1} A} \right),\quad {\lambda _{ijk \ldots l}} = {\lambda _i} + {\lambda _j} + {\lambda _k} + \cdots + {\lambda _l},\quad {e_{ijk \ldots l}} = e\left( {{\lambda _{ijk \ldots l}}} \right) \] \[ {\partial _x} = \frac{\partial } {{\partial x}},{\theta _x} = x{\partial _x},\quad t = \left( {{t_1},{t_2}} \right),\quad dt = d{t_1}d{t_2} \] the author uses to formulate his results throughout this paper. By letting \(z = \left( {{z_1},{z_2}} \right)\) to be a point of \[ Z = {\mathbb{C}^2}\backslash \left\{ {\left\{ {{z_1} = 0} \right\} \cup \left\{ {{z_1} = 1} \right\} \cup \left\{ {{z_2} = 0} \right\} \cup \left\{ {{z_2} = 1} \right\} \cup \left\{ {{z_1} + {z_2} = 1} \right\}} \right\} \subset {\left( {{\mathbb{P}^1}} \right)^2} \] and \(u(t) = u\left( {z;t} \right),\) a function defined by \[ u\left( {z;t} \right) = \prod_i {{f_i}{{(t)}^{{\lambda _i}}}} = t_1^{{\lambda _1}}{\left( {{t_1} - {z_1}} \right)^{{\lambda _2}}}t_2^{{\lambda _3}}{\left( {{t_2} - {z_2}} \right)^{{\lambda _4}}}{\left( {{t_1} + {t_2} - 1} \right)^{{\lambda _5}}} \tag{11} \] on \({T_z} = {\mathbb{C}^2}\backslash \bigcup_{i,{\lambda _i} \notin {Z_{ \geqslant 0}}} {\left\{ {{f_i}(t) = 0} \right\}} \) where, \[ {f_1}(t) = {t_1},{f_2}(t) = \left( {{t_1} - {z_1}} \right), \dots ,{f_5}(t) = \left( {{t_1} + {t_2} - 1} \right) \] and for a pair of complex variables \(\left( {{z_1},{z_2}} \right)\) which are real and sufficiently small positive numbers by allotting certain domains \(D_j\), \(1 \leqslant j \leqslant 4\), of the real manifold \(T_{\mathbb{R}}\) and letting \({u_{{D_j}}}\left( {z;t} \right) = \prod_i {{{\left( {{\varepsilon _i}{f_i}(t)} \right)}^{{\lambda _i}}}} \) ``where \({\varepsilon _i} = \pm \) is so determined that each \({{\varepsilon _i}{f_i}(t)}\) is positive and the argument of \({{\varepsilon _i}{f_i}(t)}\) is assigned to be zero on \(D_j\)'', the author, by following \textit{M. Kita} [Jpn. J. Math., New Ser. 18, No. 1, 25--74 (1992; Zbl 0767.33009)] ``for a way of constructing the cycle \(\text{reg} {D_j}\) when \({\lambda _i} \notin Z\) for all \(1 \leqslant i \leqslant 5\)'' and \textit{K. Mimachi} and \textit{T. Sasaki} [Kyushu J. Math. 66, No. 1, 35--60 (2012; Zbl 1259.33007)] ``for the regularization of the chain when some of \(\lambda_i\) are integers'' states the following notable results: Theorem 1.1. If \({\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4},{\lambda _5} \notin {\mathbb{Z}_{ < 0}}\) then each integral \[ {I_j}\left( z \right) = \int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {x,y;t} \right)dt} \] for \(1 \leqslant j \leqslant 4\) satisfies the system of differential equations \[ \begin{cases} {\left[ {{\theta _{{z_1}}}\left( {{\theta _{{z_1}}} - {\lambda _{12}} - 1} \right) - {z_1}\left( {{\theta _{{z_1}}} - {\lambda _2}} \right)\left( {{\theta _{{z_1}}} + {\theta _{{z_2}}} - 2 - {\lambda _{12345}}} \right)} \right]F = 0,} \\ {\left[ {{\theta _{{z_2}}}\left( {{\theta _{{z_2}}} - {\lambda _{34}} - 1} \right) - {z_2}\left( {{\theta _{{z_2}}} - {\lambda _4}} \right)\left( {{\theta _{{z_1}}} + {\theta _{{z_2}}} - 2 - {\lambda _{12345}}} \right)} \right]F = 0.} \end{cases} \tag{12} \] Theorem 1.2. If \({\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4},{\lambda _5} \notin {\mathbb{Z}_{ < 0}}\) and \({\lambda _{125}} + 2,{\lambda _{345}} + 2,{\lambda _{12345}} + 3 \notin {\mathbb{Z}_{ \leqslant 0}}\) then the integrals \({I_j}\left( z \right) = \int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {z;t} \right)dt} \) for \(1 \leqslant j \leqslant 4\) are linearly independent. A combination of these two theorems is the important consequence: Corollary 1.3. If \({\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4},{\lambda _5} \notin {\mathbb{Z}_{ < 0}}\) and \({\lambda _{125}} + 2,{\lambda _{345}} + 2,{\lambda _{12345}} + 3 \notin {\mathbb{Z}_{ \leqslant 0}}\) then the integrals \({I_j}\left( z \right) = \int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {z;t} \right)dt} \) for \(1 \leqslant j \leqslant 4\) constitute a fundamental set of solutions of (12). The lengthy proof of the Theorem 1.1 is given in Subsection 1.1 which is based on the statement and proof of a lemma, while the author derives a Wronskian formula for \(\int_{\text{reg} {D_j}} {{u_{{D_j}}}\left( {t} \right)dt} ,1 \leqslant j \leqslant 4\) at the beginning of the Subsection 1.2 to ultimately prove the Theorem 1.2 in this subsection by stating and proving two more lemmas. Remarking that from Section 2 of the paper and onwards ``the superscript (2) is used to indicate the case of Appell's \(F_2\) (similarly, the superscript (3) is used to indicate the case of Appell's \(F_3\))'', the author gives fundamental sets of solutions of \(E_2\) and \(E_3\) in Section 2 of the paper. For a point \(\left(x,y\right)\) of \({X^{(2)}} = {\mathbb{C}^2}\backslash \left\{ {\left\{ {x = 0} \right\} \cup \left\{ {x = 1} \right\} \cup \left\{ {y = 0} \right\} \cup \left\{ {y = 1} \right\} \cup \left\{ {x + y = 1} \right\}} \right\} \subset {\left( {{\mathbb{P}^1}} \right)^2}\) on \(T_{\left( {x,y} \right)}^{(2)} = {\mathbb{C}^2}\backslash \bigcup\nolimits_{i,{\lambda _i} \notin {Z_{ \geqslant 0}}} {\left\{ {f_i^{(2)}(t) = 0} \right\}} \) where, \(f_1^{(2)}(t) = {t_1},f_2^{(2)}(t) = {t_1} - 1, \ldots ,f_5^{(2)}(t) = 1 - x{t_1} - y{t_2}\) when \(\left(x,y\right)\) are sufficiently small positive numbers, by assigning certain domains \(D_j^{(2)}\) for \(1 \leqslant j \leqslant 4\) of the real manifold \(T_{\mathbb{R}}^{(2)}\) and by defining \(u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right) = \prod_i {{{\left( {{\varepsilon _i}f_i^{(2)}(t)} \right)}^{{\lambda _i}}}} \) in which \({\lambda _1} = \beta - 1,{\lambda _2} = \gamma - \beta - 1,{\lambda _3} = \beta ' - 1,{\lambda _4} = \gamma ' - \beta ' - 1,{\lambda _5} = - \alpha \) and \({\varepsilon _i} = \pm \) is so chosen ``that each \({{\varepsilon _i}f_i^{(2)}(t)}\) is positive and the argument of \({{\varepsilon _i}f_i^{(2)}(t)}\) is assigned to be zero on \(D_j^{(2)}\)'' the author proves: Theorem 2.1. If \(\beta - 1,\gamma - \beta - 1,\beta ' - 1,\gamma ' - \beta ' - 1, - \alpha \notin {\mathbb{Z}_{ < 0}}\) then each of the integrals \(I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)} \) for \(1 \leqslant j \leqslant 4\) satisfies the system of differential equations (4). Next he proves: Theorem 2.2. If \(\beta - 1,\gamma - \beta - 1,\beta ' - 1,\gamma ' - \beta ' - 1, - \alpha \notin {\mathbb{Z}_{ < 0}}\) and \[ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \] the integrals \(I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)}dt \) for \(1 \leqslant j \leqslant 4\) are linearly independent. A combination of these two theorems is the result: Corollary 2.3. If \(\beta - 1,\gamma - \beta - 1,\beta ' - 1,\gamma ' - \beta ' - 1, - \alpha \notin {\mathbb{Z}_{ < 0}}\) and \[ {\gamma - \alpha ,\gamma ' - \alpha ,\gamma + \gamma ' - \alpha - 1 \notin {\mathbb{Z}_{ \leqslant 0}}} \] the integrals \(I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)}dt \) for \(1 \leqslant j \leqslant 4\) constitute a fundamental set of solutions of \(E_2\). The author proves parallel results for the Appell's function \(F_3\) in the Theorems 2.4, 2.5 and Corollary 2.6 by showing ``that four integrals in the form \[ \int {\int_C {t_1^{\beta - 1}t_2^{\beta ' - 1}{{\left( {1 - {t_1} - {t_2}} \right)}^{\gamma - \beta - \beta ' - 1}}{{\left( {1 - x{t_1}} \right)}^{ - \alpha }}{{\left( {1 - y{t_2}} \right)}^{ - \alpha '}}d{t_1}d{t_2}} } \tag{13} \] satisfy (5) under the condition \( - \alpha , - \alpha ',\beta - 1,\beta ' - 1,\gamma - \beta - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}}\) and that the four integrals are linearly independent and constitute a fundamental set of solutions of \(E_3\)'' when the conditions stated in (9) hold. Returning to the \textit{auxiliary integrals}, introduced in the first section of the paper, the author presents monodromy representation associated with these integrals (10) in the third section of the paper where he computes the `` matrix elements of the five generators of the monodromy representation with respect to the four integrals by deforming the chains appropriately'' following the approach in [\textit{K. Mimachi}, Int. Math. Res. Not. 2005, No. 33, 2031--2057 (2005; Zbl 1129.32017); Funkc. Ekvacioj, Ser. Int. 51, No. 1, 107--133 (2008; Zbl 1157.33308)] and [\textit{K. Mimachi} and \textit{T. Sasaki}, Kyushu J. Math. 66, No. 1, 35--60 (2012; Zbl 1259.33007); ibid. 66, No. 1, 61--87 (2012; Zbl 1259.33025); ibid. 66, No. 1, 89--114 (2012; Zbl 1259.33028)]. The results of these investigations are reported in the Theorem 3.1 ``which gives rise to the matrix elements in the cases \(E_2\) and \(E_3\)''. By considering the monodromy representation \(\rho :{\pi _1}\left( {{X^{(2)}},\left( {{x^0},{y^0}} \right)} \right) \to GL\left( {4,\mathbb{C}} \right)\) around a base point \({\left( {{x^0},{y^0}} \right)}\) of real and sufficiently small positive numbers on the solution space \(V = \sum_{1 \leqslant j \leqslant 4} {\mathbb{C}I_j^{(2)}\left( {x,y} \right)} \) of \(E_2\) for \(I_j^{(2)}\left( {x,y} \right) = \int_{\text{reg} D_j^{(2)}} {u_{D_j^{(2)}}^{(2)}\left( {x,y;t} \right)dt}\), the author establishes ``Nakagiri's matrix elements in the \(E_2\) case'' in Theorem 4.1 under the conditions \({ - \alpha ,\beta - 1,\beta ' - 1,\gamma - \beta - 1,\gamma ' - \beta ' - 1 \notin {\mathbb{Z}_{ < 0}}}.\) Similarly, the author deduces Nakagiri's matrix elements for the \(E_3\) case in the Theorem 4.2 under the conditions \(- \alpha\), \(-\alpha'\), \(\beta - 1\), \(\beta' - 1\), \(\gamma - \beta - \beta ' - 1 \notin {\mathbb{Z}_{< 0}}.\) The reviewer feels that the Theorems 3.1, 4.1 and 4.2 are the very important results of this paper, only because their statements are quite lengthier, so, they are not quoted here for the reasons of brevity. In the fifth section of the paper the author utilizes the matrix elements developed in Theorem 3.1 to establish ``a sufficient condition for the monodromy representation associated with the auxiliary integrals being irreducible'' where he proves the notable result: Theorem 5.1. If none of \(\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_{125}, \lambda_{345}\) and \(\lambda_{12345}\) are integers, the monodromy representation \(\rho\) in Section 3 is irreducible. As an important corollary of this theorem the author deduces: Theorem 5.2. {\parindent=6mm\begin{itemize}\item[(1)] If none of \(\alpha, \beta, \beta',\gamma - \beta, \gamma'- \beta', \gamma - \alpha, \gamma' - \alpha, \gamma + \gamma' - \alpha\) is an integer, then the monodromy representation for \(E_2\) is irreducible. \item[(2)] If none of \(\alpha, \alpha',\beta, \beta',\gamma-\beta-\beta',\gamma-\alpha-\beta',\gamma-\alpha'-\beta,\gamma-\alpha-\alpha'\) is an integer, then the monodromy representation for \(E_3\) is irreducible. \end{itemize}} In an earlier work the author has proved (see [\textit{K. Mimachi} and \textit{T. Sasaki}, Kyushu J. Math. 69, No. 2, 429--435 (2015; Zbl 1329.33020)]) a complementary result of the Theorem 5.2 above which states that: Theorem 5.3. {\parindent=6mm\begin{itemize}\item[(1)] If at least one of \(\alpha, \beta, \beta',\gamma - \beta, \gamma'- \beta', \gamma - \alpha, \gamma' - \alpha, \gamma + \gamma' - \alpha\) is an integer, then the monodromy representation of the system \(E_2\) is reducible. \item[(2)] If at least one of \(\alpha, \alpha',\beta, \beta',\gamma-\beta-\beta',\gamma-\alpha-\beta',\gamma-\alpha'-\beta,\gamma-\alpha-\alpha'\) is an integer, then the monodromy representation of the system \(E_3\) is reducible. \end{itemize}} A combination of these two last theorems leads us to ``the necessary and sufficient conditions for the systems \(E_2\) and \(E_3\) being irreducible obtained by \textit{E. Bod} [J. Differ. Equations 252, No. 1, 541--566 (2012; Zbl 1236.33026)] and \textit{M. Kato} [Kyushu J. Math. 66, No. 2, 325--363 (2012; Zbl 1259.33027)]''. Finally in the sixth section (appendix) of the paper the author shows that if \(\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_{125}, \lambda_{345}, \lambda_{12345} \notin \mathbb{Z}\) then his monodromy representation \(\rho\) of Theorem 3.1 may also be considered as the irreducible monodromy representations on the twisted homology groups. The reviewer has noticed a few misprints in the paper which are not listed here. The reviewer opines that this well organized and nicely written paper is a valuable addition to our existing knowledge of the monodromy representations associated with the Appell's hypergeometric functions in general, and the Appell's functions \(F_2\) and \(F_3\) in particular. hypergeometric integrals; Nakagiri's monodromy representations; irreducibility; auxiliary integrals; chains; cycles; twisted homology; Appell's hypergeometric functions; Appell's function \(F_2\); Appell's function \(F_3\) Appell, Horn and Lauricella functions, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Monodromy on manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.) Nakagiri's monodromy representations associated with Appell's hypergeometric functions \(F_2\) and \(F_3\)	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)], \textit{B. Toën} and \textit{M. Vaquié} defined a scheme theory for a closed monoidal category \((\mathcal C, \otimes, 1)\). In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category \(s\) of simplicial objects in a monoidal category and all the categories \(sA\)-mod, \(sA\)-\(\mathrm{alg }(A \in \mathrm{sComm}(\mathcal C))\) with compatible model structures using the work of \textit{C. Rezk} [Topology Appl. 119, No. 1, 65--94 (2002; Zbl 0994.18008)]. We then give a general notion of smoothness in \(\mathrm{sComm}(\mathcal C)\). We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in \(\mathbb N\)-alg and Comm(Set). symmetric monoidal category; simplicial category; model category; relative geometry; smoothness; cohomology DOI: 10.1017/is013010008jkt242 Monoidal categories (= multiplicative categories) [See also 19D23], Étale and other Grothendieck topologies and (co)homologies, Simplicial sets, simplicial objects (in a category) [See also 55U10] Smoothness in relative 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 F be a surface in \({\mathbb{C}}^ 3\) with an isolated singularity at the origin 0. 0 is called a quasi-ordinary singularity if there is a finite map p: (F,0)\(\to ({\mathbb{C}}^ 2,0)\) such that the branched locus of p is \(xy=0\) or \(x=0\) in the degenerate case. It is known by \textit{J. Lipman}, ''Quasi-ordinary singularities of embedded surfaces''. Thesis, Harvard Univ. (1965) that there is a fractional power series \(\zeta =H(x^{1/n},y^{1/n})\) with \(H(0,0)=0\) so that \(f(x,y,z)=\prod (z- \zeta_ i)\) where f is the defining function of F and \(\zeta_ i\) has the form \(H(\mu^ j x^{1/n},\mu^ k y^{1/n})\), \(\mu =\exp (2\pi i/n)\). (u/n,v/n) is a distinguished pair if the monomial \(x^{u/n} y^{v/n}\) appears as the leading term of some \(\zeta_ i-\zeta_ j\). The normalized parametrization has the following property: The distinguished pairs are linearly ordered \((0,0)<(\lambda_ 1,\mu_ 1)<...<(\lambda_ s,\mu_ s)\). Monomial \(x^{\alpha}y^{\beta}\) with non-zero coefficient in \(\zeta\) satisfies \((\alpha,\beta)\geq (\lambda_ 1,\mu_ 1)\). \((\lambda_ 1,...,\lambda_ s)\geq (\mu_ 1,...,\mu_ s)\) (lexicographically) and \(\mu_ 1=0\) if \(\lambda_ 1\geq 1\). The main result in this paper is: Theorem (2.2.4). The topological type of a quasi-ordinary singularity determines and is determined by its distinguished pairs. isolated surface singularity; characteristic pairs; multiplicity; quasi- ordinary surface singularity; distinguished pair; normalized parametrization Gau, Topology of the quasi-ordinary singularities, Topology 25 (4) pp 495-- (1986) Complex singularities, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Topology of the quasi-ordinary surface singularities	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(p\) and \(q\) be multiplicatively independent positive integers and consider the so-called Mahler operators \(x\mapsto x^p\) and \(x\mapsto x^q\). A power series \(F(x)\in K[[x]]\) satisfies a \(p\)-Mahler fuctional equation if the vector space generated by \(F(x), F(x^p), F(x^{p^2}), \ldots\) has finite dimension. \textit{B. Adamczewski} and \textit{J. P. Bell} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 4, 1301--1355 (2017; Zbl 1432.11086)] proved that a power series \(F(x)\) satisfies both a \(p\)- and a \(q\)-Mahler functional equation if and only if it is a rational function. \textit{J.-P. Bézivin} and \textit{A. Boutabaa} [Collect. Math. 43, No. 2, 125--140 (1992; Zbl 0778.39009)] gave a similar result for dilations \(x\mapsto px\) and \(x\mapsto qx\). This paper considers similar objects called elliptic \((p,q)\)-difference modules for an elliptic curve \(\mathbb{C}/\Lambda\), where \(\Lambda\) is a lattice, and the field of \(\Lambda\)-elliptic functions. These are finite-dimensional vector spaces with commuting automorphisms given by \(\sigma f(z)=f(z/p), \tau f(z)=f(z/q)\) on the field of \(\Lambda\)-elliptic functions. The main theorem is a structure theorem for elliptic \((p,q)\)-difference modules in the case when \(p\) and \(q\) are relatively prime. As a consequence, it is shown that a power series which satisfies simultaneously a \(p\)-difference equation and a \(q\)-difference euqation with coefficients Laurent expansions of \(\Lambda\)-elliptic functions lies in the ring generated over the field of \(\Lambda'\)-elliptic functions by \(z^{\pm1}\) and \(\zeta(z,\Lambda')\) for some lattice \(\Lambda'\subset \Lambda\). Here, \(\zeta(z,\Lambda)\) is the Weierstrass zeta function for the elliptic curve. Conversely, every function from this ring satisfies a \(p\)-difference equation and a \(q\)-difference equation with elliptic functions as coefficients. The derivation explains the appearance of \(\zeta(z,\Lambda)\) in the elliptic case, unlike the cases for the simpler additive and multiplicative groups. In the 1930s, Mahler built a method for studying the transcendence of functions satisfying functional equations in the multiplicative case \(x\mapsto x^p\) and its higher-dimensional generalisations. These ideas found applications in computability because the generating functions of sets recognised by finite automata are related to Mahler functions. In the 1960s, Cobham proved that if a set is recognised by finite automata in base \(p\) and in base \(q\) then it is a union of a finite set and a finite number of arithmetic progressions. However, the precise link between finite automata and Mahler functions was elusive. Difference algebra, as further developed in this paper, has delivered the recent progress in this area. difference equations; simultaneous difference equations; elliptic functions Difference algebra, Elliptic curves, Additive difference equations Elliptic \((p,q)\)-difference modules	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring with quotient field of characteristic zero and algebraically closed residue field \(k\) of characteristic \(p>0\). Let \(\phi: \mathbb{R} \to \mathbb{R}\) be the unique lifting of the Frobenius of \(k\), and define \(\delta: \mathbb{R} \to \mathbb{R}\) by \(\delta x =\frac{\phi (x) - x^p}{p}\). Let \(G/R\) be a smooth group scheme of finite type. The author defines a \(\delta\)-formal function of order \(\leq n\) on \(G(R)\) to be an \(R\) valued function which can locally (in the Zariski topology of \(G\)) be expressed as a polynomial in affine coordinate functions and (\(n\) or fewer) iterates of \(\delta\). These form a ring \(O^n(G)\). For each \(n\), these rings have comultiplications, coinverses, and counits coming from the group scheme structure of \(G\). A coherent family \(\mathcal J\) of ideals \(J_n \subset O^n(G)\) respecting these cooperations determines a subgroup of \(G(R): \mathcal J^{\text{sol}} = \{x \in G(R) \mid f(x)=0\) for all \(f \in J_n\), \(n \geq 0\}\). The author calls such subgroups \(\delta\)-subgroups. The author calls a subgroup \(\Gamma \subset G(R)\) Zariski dense modulo \(p\) if the image of \(\Gamma\) in the closed fibre \(G_0(k)\) of \(G/R\) is Zariski dense. The main result of the paper is the following theorem: Let \(G/R\) be such that \(G_0/k\) is a simple algebraic group such that the adjoint representation of \(G_0\) on \(\text{Lie}(G_0)\) is irreducible. Let \(\Gamma\) be a \(\delta\)-subgroup which is Zariski dense modulo \(p\). Then \(\Gamma = G(R)\). The proof depends heavily on the author's previous work on \(p\) formal groups and arithmetic jet theory. complete discrete valuation ring; lifting of the Frobenius; characteristic \(p\); smooth group scheme; \(p\) formal groups; arithmetic jet theory Buium, A.: Differential subgroups of simple algebraic groups over p-adic fields. Amer. J. Math. 120, 1277-1287 (1998) Differential algebra, Linear algebraic groups over local fields and their integers, Derivations, actions of Lie algebras, Group schemes, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Differential subgroups of simple algebraic groups over \(p\)-adic 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 The purpose of this work is to construct a general semiregularity map for cycles on a complex analytic or algebraic manifold and to show that such semiregularity map can be obtained from the classical tool of the Atiyah-Chern character. The first part of the paper is fairly detailed, deducing the existence and explicit form of a generalized semiregularity map from known results and constructions. As a corollary we obtain in the second part as well a description of the infinitesimal Abel-Jacobi map for smooth cycles as the leading term of this generalized semiregularity map, indicate why for locally complete intersections the appropriate component of our semiregularity map coincides with the one constructed by \textit{S. Bloch} [Invent. Math. 17, 51--66 (1972; Zbl 0254.14011)], and give applications to embedded deformations and deformations of coherent modules. We restrict ourselves mainly to the case of a smooth ambient space, avoiding thus the formidable technical machinery of resolvents, powers of cotangent complexes, traces and so on. The underlying ``metaresult'' in its utmost generality is only stated at the end with details to be given in a later paper [\textit{R. O. Buchweitz} and \textit{H. Flenner}, Compos. Math. 137, 135--210 (2003; Zbl 1085.14503)]. semiregularity map; Atiyah-Chern character; Abel-Jacobi map; smooth cycles R.-O. Buchweitz, H. Flenner, The Atiyah-Chern character yields the semiregularity map as well as the infinitesimal Abel-Jacobi map, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proceeding Lecture Notes vol. 24, American Mathematical Society, Providence, RI, 2000, pp. 33-46. Algebraic cycles, Transcendental methods of algebraic geometry (complex-analytic aspects), Formal methods and deformations in algebraic geometry The Atiyah-Chern character yields the semiregularity map as well as the infinitesimal Abel-Jacobi 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 Consider a noetherian abelian hereditary Ext-finite category \(\mathcal{H}\) over a field \(k\), and assume that \(\mathcal{H}\) is equipped with an equivalence \(\tau\) which satisfies Serre duality. In the paper under review, two major invariants are attached to \(\mathcal{H}\), namely its function field \(k(\mathcal{H})\) and its Euler characteristic \(\chi_{\mathcal{H}}\), and they are proved to determine the shape of \(\mathcal{H}\) under many aspects. Several examples of these categories are outlined in the paper, the most relevant being that of coherent sheaves on a smooth projective curve \(C\) over \(k\), in which case the invariants of \(\mathcal{H}\) equal those of \(C\). Consider the full subcategory \(\mathcal{H}_{0}\) consisting of objects of finite length. This is the union over a set \(C(\mathcal{H})\) (called the set of points of \(\mathcal{H}\)) of categories whose Auslander-Reiten quiver can be of type \(\mathbb{Z\,A}/\tau^{p}\) (a finite point) or of type \(\mathbb{Z\,A}_{\infty}\) (an infinite point). The authors first prove that these alternatives cannot coexist, and that in the latter \(\mathcal{H}\) can have at most two points. On the full subcategory \(\mathcal{H}_{+}\) of objects (called bundles) with no indecomposable subobject, a convenient notion of rank takes positive value. Once chosen a line bundle on \(\mathcal{H}\) the authors provide a definition of \(\chi_{\mathcal{H}}\), a degree function, and a definition of semistable bundles in analogy with the case of smooth projective curves [see \textit{C. S. Seshadri}, Astérisque 96 (1982; Zbl 0517.14008)]. The case of \(\chi_{\mathcal{H}}\) being positive (\(\mathcal{H}\) is then called domestic) is studied in great detail. The condition \(\chi_{\mathcal{H}} > 0\) takes place, for instance, if and only if the rank function is bounded on some component of \(\mathcal{H}_{+}\), or if and only if each indecomposable is stable (or exceptional). The case of \(\mathcal{H}\) having an infinite point is proved to be domestic too. Moreover, in analogy with the case of \(\mathbb{P}^{1}\), it is proved that \(\mathcal{H}\) is domestic if and only if \(\mathcal{H}\) has a hereditary tilting class, which in turn amounts to \(\mathcal{H}\) being derived equivalent to \(\roman{mod}(\Lambda)\), where \(\Lambda\) is a hereditary locally bounded category. This allows to complete the classification of domestic categories over an algebraically closed field, which turn out to be associated to path algebras of extended simply laced Dynkin graphs, or of an infinite quiver of type \(\mathbb{A}_{\infty}^{\infty}\) or \(\mathbb{D}_{\infty}\). This complements the classification appearing by \textit{I. Reiten} and \textit{M. Van den Bergh} [J. Am. Math. Soc. 15, 295--366 (2002; Zbl 0991.18009)]. The tubular case (that is, when \(\chi_{\mathcal{H}}=0\)) is also studied, and the Auslander-Reiten quiver of \(\mathcal{H}\) must consist of infinitely many points of finite period. A number of open questions is listed at the end of the paper. hereditary noetherian categories; Euler characteristic; domestic categories; path algebras; infinite points; non-commutative curves H. Lenzing and I. Reiten, \emph{Hereditary {N}oetherian categories of positive {E}uler characteristic}, Math. Z. \textbf{254} (2006), no.~1, 133--171. \MR{2232010 (2007e:18008)} Categorical embedding theorems, Module categories in associative algebras, Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc., Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Families, moduli of curves (algebraic) Hereditary noetherian categories of positive Euler characteristic	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author gives a necessary and sufficient condition for the infinitesimal Torelli theorem to hold and identifies the \(\mu =const\). stratum in the discriminant of the miniversal deformation S. After examining the geometric reasons causing Torelli to fail for certain deformations, the author defines a local moduli space and proves a local Torelli theorem for singularities. Definition: The subset \({\mathcal K}\) of S is the union of all analytic arcs through 0 in S corresponding to the deformations \(f_ r=f+rg_ 1+r^ 2g_ 2+..\). with \[ g_ 1\in ''V^ 1{\mathcal O},\quad g_ 2=g^ 2_ 1\partial_ t,\quad g_ 3=2g_ 1g_ 2\partial_ t-g^ 3_ 1\partial^ 2_ t,\quad.... \] Theorem. The \(\mu =const\). stratum \(D_{\mu}={\mathcal K}.\) [For the notations and further explanations see the paper itself.] \(\mu \) \(=const\). deformation; mixed Hodge structure; Torelli theorem KARPISHPAN (Y.) . - Torelli theorems for singularities , Inv. Math., t. 100, 1990 , p. 97-141. MR 91b:32040 \| Zbl 0694.32013 Deformations of complex singularities; vanishing cycles, Local complex singularities, Deformations of singularities, Period matrices, variation of Hodge structure; degenerations Torelli theorems for singularities	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review proves existence of a parametrization of bounded definable subsets of real Euclidean space such that the number of parametrizing functions needed is explicitly bounded. More specifically, the sets under consideration are definable in some fixed o-minimal expansion of a real field and by a parametrization of such a set \(X \subset {\mathbb R}^n\) the authors mean a finite collection of definable maps from \((0,1)^m\) to \({\mathbb R}^n\) where \(m:=\dim(X)\), whose range covers \(X\). The fact that such a parametrization exists is a consequence of the cell decomposition theorem, and in an earlier paper the second and the third author constructed parametrizing functions with certain differentiability conditions and bounds on derivatives. Further, in the same paper it was shown that if \({\mathcal X}=\{X_t: t\in T\}\) is a definable family of \(m\)-dimensional subsets of \((0,1)^n\) in the sense that relation \( ``t \in T \text{ and } x\in X_t\)'' is definable in both \(x\) and \(t\) and the parametrizing functions have the \(r\)-th continuous derivatives, then there exists an integer \(N_r\) such that for each \(t\) at most \(N_r\) functions are required to parametrize \(X_t\) and each such function is definable. The earlier work however does not give an explicit bound on \(N_r\), and one of the results in the paper under review in the case of the ambient o-minimal structure being the restricted analytic field \({\mathbb R}_{\mathrm{an}}\) where the bounded definable sets are precisely the bounded subanalytic sets, shows that \(N_r\) can be taken to be polynomial in \(r\). The authors use the explicit bound on the number of the parametrizing functions to achieve in some settings a better estimate of the number of rational points contained in definable sets. rational points of bounded height; \(C^r\)-parameterizations; quasi-parameterizations; pre-parameterizations; subanalytic sets; restricted Pfaffian functions; weakly mild functions; a-b-m functions; Gevrey functions; power maps; entropy; dynamical system theory; Weierstrass preparation Model theory of ordered structures; o-minimality, Counting solutions of Diophantine equations, Heights, Model theory (number-theoretic aspects), Real-analytic and semi-analytic sets Uniform parameterization of subanalytic sets and Diophantine 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 \(R\) be a complete discrete valuation ring of mixed characeristic, let \(K\) be its fraction field and let \(k\) be its residue field. Let \(X\) be a proper strictly semistable scheme over \(R\). Then the generic fibre \(X_K\) is smooth over \(K\), whereas the special fibre \(X_s\) is a strictly normal crossing divisor in \(X\), in particular, it is the union of smooth proper \(k\)-schemes \(Y_1,\ldots, Y_n\) intersecting transversally. Let \({\mathcal X}\) denote the formal completion of \(X\); for any subscheme \(E\) of \(Y\), the preimage \(]E[_{\mathcal X}\) of \(E\) under the specialization map \({\mathcal X}_K\to Y\) is an admissible open subset in the rigid analytic \(K\)-space \({\mathcal X}_K\) associated with \({\mathcal X}\). For a non empty subset \(I\) of \(\{1,\ldots,n\}\) put \(Y_I=\cap_{i\in I}Y_i\) and \(U_I=Y_I-\cup_{I'\supsetneq I}Y_{I'}\). Let \(\Omega^{\bullet}_{c,I;{\mathcal X}}\) be the total complex of the bicomplex \[ \Omega^{\bullet}_{]Y_I[_{{\mathcal X}}/K}\longrightarrow (\text{ push forward of }\Omega^{\bullet}_{]Y_I-U_I[_{{\mathcal X}}/K}). \] Theorem. If \(\|I\|\geq2\) then there is a spectral sequence converging to \(H^*(]Y_I[_{{\mathcal X}},\Omega^{\bullet}_{c,I;{\mathcal X}})\) with \[ E_2^{pq}=H_{c,\text{rig}}^p(U_I/K)\otimes_K\bigwedge(V'_I) \] for a certain \(K\)-vector space \(V'_I\) of dimension \(\|I\|-1\). As an application one obtains two formulae comparing the Euler Poincaré characteristic \(\chi_{\text{rig}}(X_s)\) of the rigid cohomology of \(X_s\) with the Euler Poincaré characteristic \(\chi_{dR}(X_K)\) of the de Rham cohomology of \(X_K\): Proposition. \[ \chi_{\text{rig}}(X_s)-\chi_{dR}(X_K)=\sum_{\|I\|\geq2}\chi_{c}(U_I), \] \[ \chi_{\text{rig}}(X_s)-\chi_{dR}(X_K)=\sum_{\|I\|\geq2}(-1)^I(\|I\|-1)(\Delta Y_I.\Delta Y_I) \] where \(\chi_{c}(U_I)\) is the Euler Poincaré characteristic of the rigid cohomology with proper support of \(U_I\), and \(\Delta Y_I.\Delta Y_I\) is the self intersection number of \(Y_I\). One important ingredient in the proof is to elaborate further certain local cohomology calculations in a semistable reduction situation occuring in [\textit{E. Grosse-Klönne}, Duke Math. J. 113, No. 1, 57--91 (2002; Zbl 1057.14023)]. rigid cohomology; strictly semistable reduction; Euler Poincare characteristic de Rham cohomology and algebraic geometry, Schemes and morphisms A spectral sequence for de Rham 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 In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a 2-\((6, 3, 78)_5\) design by computer, which corresponds to a halving \(\operatorname{LS}_5 [2](2, 3, 6)\). The application of the new recursion method to this halving and an already known \(\operatorname{LS}_3 [2](2, 3, 6)\) yields two infinite two-parameter series of halvings \(\operatorname{LS}_3 [2](2, k, v)\) and \(\operatorname{LS}_5 [2](2, k, v)\) with integers \(v \geq 6\), \(v \equiv 2\pmod 4\) and \(3 \leq k \leq v - 3\), \(k \equiv 3 \pmod 4\). Thus in particular, two new infinite series of nontrivial subspace designs with \(t = 2\) are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with \(t = 2\). \(q\)-analog; combinatorial design; subspace design; large set; halving Braun, M; Kiermaier, M; Kohnert, A; Laue, R, Large sets of subspace designs, J. Comb. Theory Ser. A, 147, 155-185, (2014) Other designs, configurations, Combinatorial aspects of block designs, Grassmannians, Schubert varieties, flag manifolds Large sets of subspace designs	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 examines conditions under which the effective monoid \(M(X)\) of a rational surface \(X\) is finitely generated. The monoid \(M(X)\) of a surface \(X\) is the set of effective divisor classes on \(X\) modulo algebraic equivalence. The author considers the case of surfaces \(X\) obtained by blowing up the projective plane along a \(0\)-dimensional subscheme \(Z \subset {\mathbb P}^ 2\). \textit{B. Harbourne} [Duke Math. J. 52, 129--148 (1985; Zbl 0577.14025)] studied the case when \(Z\) is contained in a smooth plane cubic curve and gave a characterization of finite generation of \(M(X)\) in terms of the \(({-}2)\)-curves lying on \(X\). Here, the author assumes that \(Z\) is contained in a degenerate cubic that is the union of an integral conic and a line. The subscheme \(Z\) can contain infinitely near points and is not necessarily disjoint from the singular locus of the cubic curve. Whereas smooth projective rational surfaces with finitely generated effective monoids and small Picard number are well understood, little is known in the case of large Picard number. Note that \(X\) may have arbitrarily large Picard number. It is known that the effective monoid \(M(X)\) of a rational surface obtained by blowing up \({\mathbb P}^ 2\) along the points of a constellation with the origin a single point is not necessarily finitely generated. The article treats the case when the constellation has ``support'' on a degenerate cubic curve of the type above. The article determines a numerical criterion for the finite generation of \(M(X)\). As a byproduct of this criterion, the author shows that nef divisors on \(X\) are regular. finite generation of monoids; rational surfaces; constellations Lahyane, M.: On the finite generation of the effective monoid of rational surfaces. J. Pure Appl. Algebra. 214, 1217-1240 (2010) Rational and ruled surfaces, Divisors, linear systems, invertible sheaves, Riemann-Roch theorems, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps, Families, moduli, classification: algebraic theory, Special surfaces, Enumerative problems (combinatorial problems) in algebraic geometry On the finite generation of the effective monoid of rational 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 An infinite \((1+1)\)-dimensional chain \(u_{\alpha,t}=\phi_{\alpha,1}u_{1,x}+...+\phi_{\alpha,\alpha+1}u_{\alpha+1,x}\) is investigated. The authors introduce the notion of its hydrodynamical reduction in the following way. It is a chain of semi-Hamiltonian systems \(r_t^i=p_i(r^1,\dots,r^N)r^i_x\) and a function \(u(r^1,\dots,r^N)\) such that for every solution \(r^1,\dots,r^N\) of the semi-Hamiltonian system the function \(u(r^1,\dots,r^N)\) is a solution of the infinite chain. The notion of equivalence of chains is given. By integrality of the chain the authors mean the existence of an infinite set of hydrodynamical reductions (see details in Theor. Math. Phys. 161, No. 1, 1340--1352 (2009; Zbl 1180.37096)). The purpose of the paper is a classification of integrable \((1+1)\)-dimensional chains of hydrodynamical type. Some examples of \((2+1)\)-dimensional chains of hydrodynamical type are constructed. The problem of classification of chains is reduced to the classification of Gibbon-Tsarev systems (in the paper -- the system (1.5),(1.6)). The structure of the paper is the following: at first the main definitions are given. Only \(N=3\) case of hydrodynamical reduction is considered since the existence of a reduction with \(N>3\) gives nothing new. Then the statements of the previous results used in the paper are presented. In the next section the classification of admissible polynomial coefficients Gibbon-Tsarev systems is obtained. Then the constructions of integrable chains in general case and in some degenerate cases are given. Also some examples of \((2+1)\)-dimensional integrable chains in the sense of this paper of hydrodynamical type are given. Infinitesimal symmetries of Gibbon-Tsarev system are studied. integrable chains of hydrodynamical type; hydrodynamical reduction Odesskii, A.V.; Sokolov, V.V., Classification of integrable hydrodynamic chains, J. Phys. A: Math. Gen., 43, 434027, (2010) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Applications of Lie algebras and superalgebras to integrable systems, Applications of holomorphic fiber spaces to the sciences, Relationships between algebraic curves and integrable systems Classification of integrable hydrodynamic 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 In this short article, given a smooth diagonalizable group scheme \(G\) of finite type acting on a smooth quasi-compact separated scheme \(X\), we prove that (after inverting some elements of representation ring of \(G)\) all the information concerning the additive invariants of the quotient stack \([X/G]\) is ``concentrated'' in the subscheme of \(G\)-fixed points \(X^G\). Moreover, in the particular case where \(G\) is connected and the action has finite stabilizers, we compute the additive invariants of \([X/G]\) using solely the subgroups of roots of unity of \(G\). As an application, we establish a Lefschtez-Riemann-Roch formula for the computation of the additive invariants of proper push-forwards. Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Stacks and moduli problems Motivic concentration 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 The author studies a function \(k\mapsto N_ x(k)=\{y\in X:\) there is a sequence \((x_ 0,x_ 1,...,x_ k)\), \(x_ 0=x\), \(x_ k=y\); \((x_{i- 1},x_ i)\in \Gamma_{\phi}\), \(1\leq i\leq k\}\) where \(\Gamma_{\phi}\) is the graph of a multivalued map (correspondence) \(\phi\) from a set X into itself. The author defines the notions of a symplectic, Liouville integrable, algebraic and abelian correspondence and gives several results concerning the estimation of the number \(N_ x(k)\) for generic points x. The main result states that for an integrable m-valued correspondence there is a constant A such that \(N_ x(k)=Ak^{m-1}\) contrary to the natural expectation that \(N_ x(k)=m^ k\). The paper contains some examples and conjectures. No proofs are given. set-valued mapping; symplectic manifolds; Liouville integrability; correspondence Veselov, A. P.: On the growth of the numbers of images of the point under the iterations of multiple-valued mapping. Mat. Zametki,49 (2) (1991) (Russian) Set-valued and function-space-valued mappings on manifolds, Rational and birational maps Growth of the number of images of a point under iterates of a multivalued 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 The aim of this paper is to study the variation of mixed Hodge structure arising from a projective morphism \(f: X\to D^\) near the puncture of the disc \(D\) in \({\mathbb{C}}\). Namely to construct a limit mixed Hodge structure satisfying the condition given by \textit{P. Deligne} in his article in Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014). We use the results on filtered mixed Hodge complex giving rise to spectral sequences of mixed Hodge structures [previous note, C. R. Acad. Sci., Paris, Sér. I 295, 669-672 (1982; Zbl 0511.14004)]. First we construct such a complex in the case where \(X\) is a normal crossing divisor in some ambiant space smooth over \(D^\), and then we give the construction for a projective morphism. This paper contains the detailed proof with comments on previous work by W. Schmid, J. Steenbrink and H. Clemens in the case of smooth morphism f. [For a short announcement of this paper see the following title.] variation of mixed Hodge structure Fouad El Zein, Théorie de Hodge des cycles évanescents, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 107 -- 184 (French, with English summary). Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), de Rham cohomology and algebraic geometry Théorie de Hodge des cycles évanescents. (Hodge theory of vanishing cycles)	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article gives ``{evidence for a connection between the work of Kuo-Tsai Chen on iterated integrals and de Rham homotopy theory on the one hand, and the work of Wei-Liang Chow on algebraic cycles on the other.}'' It surveys the subject, presents new results, and prospects further developments. Sometimes certain constructions are presented in detail, whereas at others only a rough idea is given. The result is an overview for novices and experts about the role iterated integrals play in algebraic (and differential) geometry. In the opinion of the reviewer, this article merits being worked into a book on this subject. Below are short descriptions of the content of the 14 sections of this article. Section 1: The definition of differential forms on the path space \(PX\) of a smooth manifold \(X\) is given. Furthermore, there exists an exterior differential, making \(E^\bullet(PX)\) into a differential graded algebra (dga). Basic properties (pull back, integration along compact fibers, Stokes' theorem) are given. Section 2: Iterated integrals on \(PX\) which form the Chen complex \(Ch^\bullet(PX) \subset E^\bullet(PX)\) are introduced. For \(\omega_1,\ldots,\omega_r \in E^\bullet(X)\), the iterated integral \(\int \omega_1 \ldots \omega_r\) on \(PX\) is the push forward of the form \(p_1^\omega_1 \wedge \ldots \wedge p_r^\omega_r\) on \(\Delta^r \times PX\) to \(PX\). Here, \(\Delta^r\) is the \(r\) dimensional simplex and \(p_i\) maps \((t_1,t_2,\ldots,t_r, \alpha)\) to \(\alpha(t_i)\). The Chen complex is filtered by length and again a dga. These constructions can be carried through for natural subspaces of \(PX\), like the pointed loop space \(P_{x,x}X\) or \(P_{x,y}\). In the case of \(Ch^\bullet(P_{x,x}X)\), a Hopf algebra structure is obtained. Section 3: The cohomology of this algebra is related to the cohomology and homotopy of \(P_{x,x}X\) and \(X\). Section 4: It is shown how iterated integrals of holomorphic one forms give multi valued holomorphic functions. For example, the iterated integral \(\int \frac{dz}{1-z} \frac{dz}{z}\) on \({\mathbb P}^1 \setminus \{ 0, 1, \infty \}\) is Euler's dilogarithm function. Section 5: Bruno Harris' construction of the harmonic volume is presented. Iterated integrals are used to calculate the volume (mod \({\mathbb Z}\)) of a submanifold with given boundary. Section 6: The notation of iterated integrals is extended to currents. Here certain transversality conditions are needed to define this product. Section 7: Presentation of Chen's reduced bar construction and its connection to the Chen complex \(Ch(P_{x,y}X)\) and the (co)homology of \(P_{x,y}X\). Section 8: Exact sequences are deduced from the above construction which relate homotopy and cohomology of a simply connected manifold \(X\). Section 9: Hodge theory is taken into account. This way Hodge and weight filtrations on \(Ch^\bullet(P_{x,y}X)\) are obtained. Section 10: Application to algebraic cycles, to intermediate Jacobians, and the theorem of \textit{J. Carlson}, \textit{H. Clemens} and \textit{J. Morgan} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 14, 323--338 (1981; Zbl 0511.14005)] on the image of homological trivial cycles of codimension two in the intermediate Jacobian of a projective manifold \(X\). Section 11: The coordinate ring of the unipotent closure of \(\pi_1(X,x)\) is identified with the ring \(H^0(Ch^\bullet(P_{x,x}X))\). Section 12: The above construction is extended to the algebraic completion of \(\pi_1(X,x)\) relative to a Zariski dense homomorphism \(\rho:\pi_1(X,x) \to S\) to a reductive algebraic group. Section 13: Several approaches to extend iterated integrals to algebraic de Rham theory are presented. Section 14: \textit{J.F. Adams}' cobar construction [Proc. Natl. Acad. Sci. USA 42, 409--412 (1956; Zbl 0071.16404)] is explained. This gives a cosimplicial space which models the path space \(PX\). The iterated integrals are its de Rham realization. path spaces; mixed Hodge stucture; intermediate Jacobian Richard Hain, Iterated integrals and algebraic cycles: examples and prospects, Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000) Nankai Tracts Math., vol. 5, World Sci. Publ., River Edge, NJ, 2002, pp. 55 -- 118. Algebraic cycles, Algebraic topology on manifolds and differential topology, Homotopy theory and fundamental groups in algebraic geometry, Homotopy and topological questions for infinite-dimensional manifolds Iterated integrals and algebraic cycles: examples and prospects.	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 variety \({\mathcal N}_n({\mathbb F})\) of all nilpotent \(n\times n\) matrices over \({\mathbb F}\) are considered, where \({\mathbb F}\) is an algebraic closed field. A partition of \(n\) is a nonincreasing sequence of positive integers \(\underline{\lambda}=(\lambda_1,\lambda_2,\dots,\lambda_s)\) with \(\sum_{i=1}^s=n\). \({\mathcal P}(n)\) denotes the set of all partitions of \(n\). The nilpotent commutator of a matrix \(B\in{\mathcal N}_n({\mathbb F})\) is the set \({\mathcal N}_B\) of all matrices \(A\in{\mathcal N}_n({\mathbb F})\) such that \(AB=BA\). If the Jordan canonical form of \(B\) is given by a partition \(\underline{\lambda}\in{\mathcal P}(n)\), then the orbit of \(B\) under the conjugated action of \(GL_n({\mathbb F})\) on \({\mathcal N}_n({\mathbb F})\) is denoted \({\mathcal O}_B={\mathcal O}_{\underline{\lambda}}\); therefore \({\mathcal O}_{\underline{\lambda}}\) is the set of all nilpotent matrices with the Jordan canonical form given by \({\mathcal O}_{\underline{\lambda}}\). The set of all partitions that are Jordan canonical forms of matrices in \({\mathcal N}_B\) is denoted by \({\mathcal P}({\mathcal N}_B)\). The structure of the nilpotent commutator \({\mathcal N}_B\) is studied. It is shown that for \(n\geq 3\) \({\mathcal P}({\mathcal N}_B)={\mathcal P}(n)\), while for \(n\geq 4\) \({\mathcal P}({\mathcal N}_B)={\mathcal P}(n)\) if and only if \(B^2=0\). A characterization of the pairs of Jordan canonical forms of two commuting nilpotent matrices is provided in the case of matrices with two Jordan blocks. Some sufficient conditions and some necessary conditions are given in order for partitions to be Jordan canonical forms of matrices which have the dimension of the kernel equal to 2. The map \({\mathcal D}\) on \({\mathcal P}(n)\) is studied, where \({\mathcal D}(\underline{\lambda})=\underline{\mu}\) = the largest partition such that \({\mathcal O}_{\underline{\mu}}\cap{\mathcal N}_B\neq\emptyset\). The partitions \(\underline{\lambda}\) such that the parts of \({\mathcal D}(\underline{\lambda})\) differ by 2 are characterized and the preimage of \({\mathcal D}\) for certain partitions is described. nilpotent matrices; commuting matrices; nilpotent commutator; nilpotent orbit; Jordan canonical form; maximal partition Oblak, P., On the nilpotent commutator of a nilpotent matrix, Linear Multilinear Algebra, 60, 599-612, (2012) Commutativity of matrices, Canonical forms, reductions, classification, Hermitian, skew-Hermitian, and related matrices, Group actions on varieties or schemes (quotients) On the nilpotent commutator of a nilpotent matrix	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Branched covers of the complex projective line ramified over \(0,1,\) and \(\infty \) (Grothen\-dieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over \(\infty\) and given numbers of preimages of \(0\) and \(1\) are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers. dessins d'enfant; branched covers; Kadomtsev-Petviashvili hierarchy P. Zograf, \textit{Enumeration of Grothendieck's dessins and KP hierarchy}, arXiv:1312.2538 [INSPIRE]. Arithmetic aspects of dessins d'enfants, Belyĭ theory, Dessins d'enfants theory Enumeration of Grothendieck's dessins and KP hierarchy	0
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study some actions on the set \(M_{n,m}\) of matrices with coefficients in the ring of formal power series \(R:=K[[x_1,\ldots, x_n]]\), where \(K\) is a field. They study the action of \(G:=Gl_n(R)\) on \(M_{n,m}\) by multiplication from the left (or from the right, or from both sides). Then they give a condition to insure the finitely determinacy of a matrix \(A\). One says that a matrix \(A:=(a_{ij}(x))\) is finitely determined (resp. \(G\)-finitely determined) if, for every matrix \((b_{ij}(x))\) whose entries coincide with those of \(A\) up to some high power of the maximal ideal of \(R\), there exists an automorphism \(\varphi\) of \(R\) such that \((\phi(b_{ij}(x)))=(a_{ij}(x))\) (resp. such that \((\phi(b_{ij}(x)))\) belongs to the orbit of \(A\) under the action of \(G\)). The condition given in this paper is expressed in terms of the tangent image of the orbit map. In characteristic zero, they prove that this condition is not only sufficient but also necessery. These results generalize previous classical results concerning the finite determinacy of power series or of vectors of power series. matrices with power series coefficients; finite determinacy Singularities in algebraic geometry, Formal power series rings, Matrices over function rings in one or more variables On finite determinacy for matrices of power series	0