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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 See the review in Zbl 0633.14023.
The translation incorporates corrections submitted by the author. finite Honda system; finite commutative group schemes Abrashkin, V., Honda systems of group schemes of period \(p\), Math. USSR Izvestiya, 30, 419-453, (1988) Formal groups, \(p\)-divisible groups, Group schemes Honda systems of group schemes of period p | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 convex polytopes \(P_1,\dots ,P_r \subset \mathbb R^n\) and finite subsets \(\mathcal{W}_I\) of the Minkowski sums \(P_I=\sum_{i \in I} P_i\), we consider the quantity \(N(\mathbf{W})=\sum_{I \subset[r]} {(-1)}^{r-| I|} | \mathcal{W}_I |\). If \(\mathcal{W}_I=\mathbb Z^n \cap P_I\) and \(P_1,\dots ,P_n\) are lattice polytopes in \(\mathbb R^n\), then \(N(\mathbf{W})\) is the classical \textit{mixed volume} of \(P_1,\dots ,P_n\) giving the number of complex solutions of a general complex polynomial system with Newton polytopes \(P_1,\dots ,P_n\). We develop a technique that we call \textit{irrational mixed decomposition} which allows us to estimate \(N(\mathbf{W})\) under some assumptions on the family \(\mathbf{W}=(\mathcal{W}_I)\). In particular, we are able to show the nonnegativity of \(N(\mathbf{W})\) in some important cases. A special attention is paid to the family \(\mathbf{W}=(\mathcal{W}_I)\) defined by \(\mathcal {W}_I=\sum _{i \in I} \mathcal {W}_i\), where \(\mathcal {W}_1,\dots ,\mathcal{W}_r\) are finite subsets of \(P_1,\dots ,P_r\). The associated quantity \(N(\mathbf{W})\) is called \textit{discrete mixed volume} of \(\mathcal{W}_1,\dots,\mathcal{W}_r\). Using our irrational mixed decomposition technique, we show that for \(r=n\) the discrete mixed volume is an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports \(\mathcal{W}_1,\dots ,\mathcal{W}_n \subset \mathbb R^n\). We also prove that the discrete mixed volume associated with \(\mathcal{W}_1,\dots ,\mathcal{W}_r\) is bounded from above by the Kouchnirenko number \(\prod_{i=1}^r (| \mathcal{W}_i| -1)\). For \(r=n\) this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports \(\mathcal{W}_1,\dots ,\mathcal{W}_n \subset \mathbb R^n\). This conjecture was disproved, but our result show that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking. tropical geometry; convex polytopes; Ehrhart theory Bihan, Frédéric, Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems, Discrete Comput. Geom., 55, 4, 907-933, (2016) , Mixed volumes and related topics in convex geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The subject of this paper concerns computations with rational and irrational maps between irreducible projective varieties, using the Macaulay2 package. \par The computational package Cremona.m2, which is included in Macaulay2 (since version 1.9), performs computations on rational and birational maps between absolutely irreducible projective varieties over a given field. The algorithms given in this paper are derived from the classical mathematical definitions and tools of algebraic-geometry. Cremona.m2 provides general methods to compute projective degrees of rational maps, and using this, one can interpret them as methods to compute the push-forward to projective space of Segre classes. \par All the methods (where this may make sense) are available both in a probabilistic version and in a deterministic version, and one can switch from one to the other easily. \par The paper is organized into two parts, which we briefly present: section 1 gives an exposition of the main methods provided by the package and the algorithms implemented, in a general setting along with some classical definitions of algebraic geometry. Section 2 shows how these methods work, in particular examples are given. rational maps; birational maps; projective degrees; Segre class Rational and birational maps, Computational aspects of higher-dimensional varieties A Macaulay2 package for computations with rational maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The number of nodal curves that one finds in a family, is a relevant invariant for the classification of algebraic varieties and for the knowledge of their geometry. In general this number is very hard to compute, even for natural families and rather initial cases.
The authors introduce in details a method for the computation of the number. It relies on a refinement and a generalization of the procedure, outlined in a previous paper, for the computation of the class of the cycle \(U(r)\subset B\) which parameterizes \(r\)-nodal elements of a given family \(Y\to B\) of curves. When \(r\) is small (typically \(r\leq 8\)) the computation can be performed explicitly.
In particular, the authors show how their method allows to compute the number of \(r\)-nodal curves in the following families:
\noindent - reduced plane curves of degree \(m\), passing through \((m-3)m/2 - r\) general points (\(r\leq 8\));
\noindent - plane sections of general threefolds of degree \(m\geq 4\) in \(\mathbb P^4\) (\(r=6\));
\noindent - curves in a given homology class \(\alpha\) of an abelian surface \(A\) with Picard number \(1\), passing through \(1-r+\alpha^2/2\) general points (\(r\leq 8)\).
\noindent These computations cover and extend some cases which have been studied, through a different approach, in the literature. nodal curve; Bell polynomial; Enriques diagram; Hilbert scheme; Göttsche's conjecture; quintic threefold; abelian surfac S.\ L. Kleiman and R. Piene, Node polynomials for families: Methods and applications, Math. Nachr. 271 (2004), 69-90. Enumerative problems (combinatorial problems) in algebraic geometry Node polynomials for families: Methods 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 Jet schemes of monomial schemes are known to be equidimensional but are not reduced in general. We give a formula for the multiplicity along every component of the jet schemes of a simple normal crossing divisor, inspired by recent work of \textit{R. A. Goward} and \textit{K. E. Smith} [Commun. Algebra 34, No. 5, 1591--1598 (2006; Zbl 1120.14055)]. DOI: 10.1080/00927870701512168 Computational aspects in algebraic geometry, Computational aspects and applications of commutative rings The multiplicity of jet schemes of a simple normal crossing divisor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main results of the paper estimate the volume and metric entropy of a measurable function \(f: [0,1]^n\to\mathbb{R}\) using its piecewise polynomial \(\varepsilon\)-approximaton \(S_f:= \{Q_i, p_j\}_{1\leq j\leq p}\), where cubes \(Q_j\) covers \([0,1]^n\) and polynomials \(p_j\) are such that \(\max_{Q_j}|f- p_j|\leq \varepsilon\). Given \(\delta\geq 0\) set
\[
\nu(\delta,S_f):= 4n \sum^p_{j=1} |Q_j|\Biggl({\delta\over \max_{Q_i} |p_j|}\Biggr)^{1/\deg p_j}.
\]
Theorem A. The volume of \(Z(f):= \{x\in [0,1]^n; f(x)= 0\}\) satisfies
\[
|Z(f)|\leq \underset{\varepsilon> 0}{}{\text{inf}}\,\underset{S_f}{}{\text{inf}}\nu_\varepsilon, S_f),
\]
where \(S_f\) runs over all \(\varepsilon\)-approximants of \(f\).
Now let \(M_\varepsilon(S)\) be the minimal numbers of \(\varepsilon\)-balls covering \(S\subset\mathbb{R}^n\). To estimate this for \(Z(f)\) one defines the ``\(i\)-variation'' of the \(\varepsilon\)-approximant \(S_f\) by setting
\[
\nu_i(S_f):= \sum^p_{k=1} c(n,i)|Q_k|^{{i\over n}}(\deg p_k)^{1-{i\over n}},
\]
where \(c(n,i)\) are certain constants
Theorem B. It is true that
\[
M_\varepsilon(Z(f))\leq \underset{S_f}{}{\text{inf}}\,\Biggl(\sum^{n-1}_{i=0} \nu_i(S_f)\Biggl({1\over\varepsilon}\Biggr)^i+ \nu(\varepsilon, S_f)\Biggl({1\over\varepsilon}\Biggr)^n\Biggr),
\]
where \(S_f\) runs over all \(\varepsilon\)-approximants of \(f\).
The basic tools of this study are Vitushkin's and the Bruduyi-Ganzburg inequalities estimating, respectively, \(\varepsilon\)-entropy of semialgebraic sets and growth of polynomials outside measurable subsets of \(\mathbb{R}^n\). Real algebraic sets Zero sets of functions and their piecewise-polynomial approximations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce the Brauer loop scheme
\[
E:=\{M\in M_ N({\mathbb C}):\;M\bullet M=0\},
\]
where \(\bullet\) is a certain degeneration of the ordinary matrix product. Its components of top dimension, \(\lfloor N^2/2\rfloor\), correspond to involutions \(\pi\in S_ N\) having one or no fixed points. In the case \(N\) even, this scheme contains the upper--upper scheme from [\textit{A. Knutson}, J. Algebr. Geom. 14, No. 2, 283--294 (2005; Zbl 1074.14044), see also \url{arXiv:math.AG/0306275}] as a union of \((N/2)!\) of its components. One of those is a degeneration of the commuting variety of pairs of commuting matrices.
The Brauer loop model is an integrable stochastic process studied in [\textit{J. de Gier} and \textit{B. Nienhuis}, J. Stat. Mech. Theory Exp. 2005, No. 1, Paper P01006, 10 p., electronic only (2005; Zbl 1072.82585), see also math.AG/0410392], based on earlier related work in [\textit{M. J. Martins, B. Nienhuis} and \textit{R. Rietman}, An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett. 81, No. 3, 504--507 (1998; Zbl 0944.82006), see also cond-mat/9709051], and some of the entries of its Perron--Frobenius eigenvector were observed (conjecturally) to equal the degrees of the components of the upper--upper scheme.
Our proof of this equality follows the program outlined in [\textit{P. Di Francesco} and \textit{P. Zinn-Justin}, Commun. Math. Phys. 262, No. 2, 459--487 (2006; Zbl 1113.82026), see also math-ph/0412031]. In that paper, the entries of the Perron--Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on \(E\). As a consequence, we obtain a formula for the degree of the commuting variety, previously calculated up to \(4\times 4\) matrices. commuting variety; Brauer loop model; Brauer algebra; integrable models; chord diagrams; multidegrees A. Knutson and P. Zinn-Justin, A scheme related to the Brauer loop model. Adv. Math. 214(2007), no. 1, 40--77.MR 2348022 Zbl 1193.14068 Type yC Brauer loop schemes and loop model with boundaries255 Special varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Exactly solvable models; Bethe ansatz A scheme related to the Brauer loop model | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a nice short expository article about jet schemes and arc spaces. Its purpose is to be introductory rather than comprehensive. The author also gives an introduction to the Nash problem which asks when there is a bijection between certain exceptional divisors in a log resolution and certain irreducible components of the space of arcs at singular points. arc space; jets scheme; Nash problem Ishii, S., Jet schemes, arc spaces and the Nash problem, \textit{C. R. Math. Rep. Acad. Canada}, 29, 1-21, (2007) Formal methods and deformations in algebraic geometry, Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Jet schemes, arc spaces and the Nash problem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix a totally real number field \(F/\mathbb{Q}\) of degree \(g\) and a prime number \(p\), remaining prime in \(F\). In this paper we study the reduction mod \(p\) of Hilbert-Blumenthal varieties of level \(\Gamma_0(p)\) where \(p\) denotes a fixed prime number. This is a special case of a Shimura variety \(\text{Sh}_C (G,X)\) for which the \(p\)-primary part \(C_p\subset G(\mathbb{Q}_p)\) of the subgroup \(C\subset G(\mathbb{A}_f)\) is of parahoric type. For these Shimura varieties, one may address the two problems:
(1) Determine the local structure of the natural model \(M_C/\mathbb{Z}_{(p)}\) of this Shimura variety.
(2) Determine the global structure of the reduction mod \(p\) of this model.
Treating the second of these questions, the so called ``supersingular locus'' on \(M_C\otimes\mathbb{F}\) (where \(\mathbb{F}\) is an algebraic closure of \(\mathbb{F}_p)\) is of particular interest. It is always a closed subset of \(M_C\). The most well-known example of these Shimura varieties, given by the data \(G=\text{GL}(2)\), \(C_p=\left\{ \left(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix} \right)\in\text{GL}(2,\mathbb{Z}_p) | c\equiv 0\pmod p\right\}\), is studied by \textit{P. Deligne} and \textit{M. Rapoport} [in: Modular Functions one variable. II, Proc. Int. Summer School, Antwerp 1972, Lect Notes Math. 349, 143-316 (1973; Zbl 0281.14010)] and serves as a prototype in this paper. The Shimura variety associated to these data -- the elliptic moduli curve of level \(\Gamma_0(p)\) -- possesses a model \(M_C\) over \(\mathbb{Z}_{(p)}\). Putting \(C_p'=\text{GL}(2,\mathbb{Z}_p)\) and \(C'=C^pC_p'\) and denoting by \(M_{C'}\) the model of \(\text{Sh}_{C'}(G,X)\) over \(\mathbb{Z}_{(p)}\) (classifying elliptic curves with level-\(C^p\) structure), the main results of the Deligne-Rapoport paper cited above with respect to the two questions above are: (1) The scheme \(M_C\) is regular, has the relative dimension one over \(\mathbb{Z}_{(p)}\), and possesses semi-stable reduction. (2) Over \(M_{C'} \otimes \mathbb{F}_p\) there is a section \(Fr\) to the canonical projections \(p_1:M_C \otimes\mathbb{F}_p\to M_{C'}\otimes\mathbb{F}_p\) (resp. a section \(Ver\) to \(p_2:M_C\otimes \mathbb{F}_p\to M_{C'}\otimes\mathbb{F}_p)\) being given by the Frobenius morphism (resp. the Verschiebung). The special fiber \(M_C\otimes\mathbb{F}_p\) is the union of \(Fr(M_{C'} \otimes\mathbb{F}_p)\) and \(Ver(M_{C'}\otimes\mathbb{F}_p)\), these two closed one-dimensional subschemes intersecting transversally in exactly the supersingular points.
The aim of this paper is to generalize these results to Hilbert-Blumenthal varieties. We succeed to give answers to the questions posed above in the two-dimensional case. An \(F\)-cyclic isogeny \(f: (A,i)\to (A',i')\) between abelian schemes with real multiplication in \(F\) is an isogeny \(f:(A,i)\to(A',i')\) of degree \(p^g\) satisfying \(\ker(f)\leq{}_pA\). It is shown that the moduli scheme \(M_{\Gamma_0(p)}/\mathbb{Z}_{(p)}\) classifying \(F\)-cyclic isogenies (+ additional data) has a model with semi-stable reduction.
For \(g=2\), the paper gives a global description of the supersingular locus on \(M_{\Gamma_0(p)}\) and on \(M_{\text{abs}}\) where \(M_{\text{abs}}/\mathbb{Z}_{(p)}\) denotes the moduli scheme classifying abelian schemes with real multiplication: All irreducible components \(C_i\) of \((M_{\text{abs}}\otimes\mathbb{F}_{p^2})^{ss}\) are isomorphic to \(\mathbb{P}^1\), two components meeting in at most one point and each point in \(C_i(\mathbb{F}_{p^2})\) being the intersection of \(C_i\) with another component. The irreducible components \(R_i\) of \((M_{\Gamma_0(p)} \otimes \mathbb{F}_{p^2})^{ss}\) are all smooth surfaces; the set of these components is in bijection with the components \(C_i\). Each one of the canonical projections \(p_j\), \(j=1,2\), gives \(R_i\) the structure of a rationally ruled surface over some component \(C_i\). Hilbert-Blumenthal varieties; moduli scheme; supersingular locus Stamm, H., On the reduction of the Hilbert-blumenthal-moduli scheme with \({\Gamma}\)_{0}(\textit{p})-level structure, Forum Math., 9, 4, 405-455, (1997) Algebraic moduli of abelian varieties, classification, Modular and Shimura varieties On the reduction of the Hilbert-Blumenthal-moduli scheme with \(\Gamma_0(p)\)-level structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a number of finiteness results for algebraic tori (and, more generally, for algebraic groups with toric connected component) over two classes of fields: finitely generated fields and function fields of algebraic varieties over fields of type (F), as defined by J.-P. Serre. Group schemes, Arithmetic varieties and schemes; Arakelov theory; heights, Galois cohomology of linear algebraic groups, Coverings in algebraic geometry, Arithmetic aspects of modular and Shimura varieties Finiteness theorems for algebraic tori over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The topic of this book is resolutions of singularities of Schubert varieties. Much of the material is already contained in earlier publications of the author: [Adv. Math. 71, No. 2, 186--231 (1988; Zbl 0688.14046)], and mainly the author's Thèse [Géométrie des groupes semi-simples, résolutions equivariantes, et Lieu singulier de leurs cellules de Schubert. Montpellier (1981)]. Howevere it is not easy to find access to the latter. As a technical clarification, let us mention that in the book under review, and in this review, the term resolution is understood in the weak sense that we do not require that a resolution is an isomorphism over all of the smooth locus.
To describe the key questions and results of the book in a specific case, we set up some notation and at the same time recall some basic definitions. Let \(n\geq 1\), let \(k\) be a field, and let \(G=\mathrm{GL}_{n, k}\) be the general linear group over \(k\). Let us consider the flag variety \(\text{Flag}\), i.e., the projective variety of flags
\[
{\mathcal F}_1 \subset \cdots \subset {\mathcal F}_{n-1} \subset {\mathcal F}_n = k^n
\]
with \(\dim_k {\mathcal F}_i = i\) for all \(i\). Denoting by \(e_1, \dots, e_n\) the standard basis of \(k^n\), we have the standard flag given by \({\mathcal S}^{(i)} = \langle e_1, \dots, e_i\rangle_k\), \(i=1, \dots, n\). The \textit{Schubert cells} in \(\text{Flag}\) are the orbits under the action by the group \(B\) of upper triangular matrices in \(\mathrm{GL}_{n,k}\), the so-called standard Borel subgroup. It is well known that the set of orbits is in bijection with the symmetric group \(\underline{S}_n\) on \(n\) letters (the Weyl group of \(G\)). Explicitly, given a permutation \(w\in\underline{S}_n\), the corresponding Schubert cell is
\[
C_w = \{ ({\mathcal F}_i)_i;\;\dim {\mathcal F}_i\cap {\mathcal S}_j = m_{ij} \},
\]
where \(M= (m_{ij})_{i,j}\) is the \textit{relative position} matrix attached to \(w\), defined by
\[
m_{ij} = \#\{ k \leq i;\;w(k) \leq j \}.
\]
It is known that as a variety \(C_w\) is isomorphic to affine space over \(k\), of dimension the number of inversions of the permutation \(w\).
A \textit{Schubert variety} in \(\text{Flag}\) is the closure of a Schubert cell (with the reduced scheme structure). It is easy to see that as a set the closure of \(C_w\) is
\[
S_w = \overline{C_w} = \{ ({\mathcal F}_i)_i;\;\dim {\mathcal F}_i\cap {\mathcal S}_j \geq m_{ij} \}.
\]
In general, \(S_w\) is a singular variety, and it is an interesting question to describe the singularities, the singular locus, and resolutions of singularities of \(S_w\). The best known type of resolution is the Demazure resolution [\textit{M. Demazure}, Ann. Sci. Éc. Norm. Supér. (4) 7, 53--88 (1974; Zbl 0312.14009)] also called the Bott-Samelson resolution [\textit{R. Bott} and \textit{H. Samelson}, Am. J. Math. 80, 964--1029 (1961; Zbl 0101.39702)] or Hansen resolution [\textit{H. C. Hansen}, Math. Scand. 33, 269--274 (1974; Zbl 0301.14019)]. In the book under review, the construction of resolutions is approached somewhat differently (the relation to the usual construction of the Demazure resolution is discussed in Chapter 15, see the comments below).
For \(\mathrm{GL}_n\), one can obtain a resolution not depending on any choices in the following way: Given the numbers \(m_{ij}\) attached to the permutation \(w\) as above, consider the following variety:
\[
\begin{aligned} \widehat{S}_w := \{ ({\mathcal H}_{ij})_{i,j};\;& {\mathcal H}_{ij} \subseteq k^n,\;\dim_k {\mathcal H}_{ij} = m_{ij},\\ & \forall i \leq i', j\leq j': {\mathcal H}_{ij}\subseteq {\mathcal H}_{i'j'},\\ & \forall j: {\mathcal H}_{nj} = {\mathcal S}_j \} \end{aligned}
\]
Clearly this is a closed subvariety of a product of Grassmann varieties, and mapping \(({\mathcal H}_{ij})_{i,j}\) to \(({\mathcal H}_{in})_i\) we obtain a flag in \(k^n\) which lies in \(S_w\) because \({\mathcal H}_{in}\cap {\mathcal S}_j\) contains \({\mathcal H}_{ij}\) and hence has dimension \(\geq \dim {\mathcal H}_{ij} = m_{ij}\). On the other hand it is easy to write \(\widehat{S}_w\) as a successive fibration of Grassmann varieties which shows that it is smooth. Furthermore, over \(C_w\) we have an obvious section to the map \(\widehat{S}_w\to S_w\) by mapping \(({\mathcal F}_i)_i\) to \(({\mathcal F}_i\cap {\mathcal S}_j)_{i,j}\). This resolution and its variants for varieties of partial flags are the theme of Chapter 3 (Def.~3.2), after some basic notions have been discussed in Chapters 1 and 2.
Based on this resolution, a combinatorial description of the singular locus of a Schubert variety for \(\mathrm{GL}_n\) is given (Thm.~4.38). Because of the differences in the approach, and in notation, it is not immediately apparent (to the reviewer, at least), how this description compares with other combinatorial characterizations of the singular locus (e.g., in terms of pattern avoidance of permutations, see the papers cited in the next paragraph).
To complement the bibliography in the book, we list in chronological order the following papers (among many others) about combinatorial descriptions of singular loci of Schubert varieties: [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Bull. Am. Math. Soc., New Ser. 11, 363--366 (1984; Zbl 0549.14016)], relying on standard monomial theory. The state of the art at around the year 2000 is given in the book [\textit{S. Billey} and \textit{V. Lakshmibai}, Singular loci of Schubert varieties. Boston, MA: Birkhäuser (2000; Zbl 0959.14032)]. Afterwards, the following authors have obtained more precise information about the singular locus of Schubert varieties: [\textit{L. Manivel}, Int. Math. Res. Not. 2001, No. 16, 849--871 (2001; Zbl 1023.14022)], [\textit{S. C. Billey} and \textit{G. S. Warrington}, Trans. Am. Math. Soc. 355, No. 10, 3915--3945 (2003; Zbl 1037.14020)], [\textit{C. Kassel} et al., J. Algebra 269, No. 1, 74--108 (2003; Zbl 1032.14012)], [\textit{S. Gaussent}, Commun. Algebra 31, No. 7, 3111--3133 (2003; Zbl 1065.14060)]. Some of these results extend to the cases of other reductive groups; see the book by Billey and Lakshmibai and the references given there.
While the above definition of resolution is easy to write down, it does not generalize well, in this form, to other reductive algebraic groups, because there is no direct analog of the relative position matrix. Instead, as the author proposes, one can use the theory of buildings and generalized galleries in order to arrive at a construction of resolutions of singularities of Schubert varieties which has the construction described above as a special case, but applies to general reductive groups.
This is carried out in the following chapters of the book. In Chapter 5 (The Flag Complex) the definition of building is recalled, and the example of the general linear group is discussed. Here we also find the definition of \textit{generalized} gallery (Def.~5.1) which is of key importance in the sequel. To describe it, recall the usual notion of (non-generalized) gallery: a sequence of chambers such that every chamber in the sequence shares a codimension \(1\) facet with the next one (including the possibility that it equals the next one). In a generalized gallery, the simplices need not have full dimension (and, for the connecting facets, codimension \(1\), resp.). Rather, a generalized gallery is defined as a sequence of facets \((F_1, \dots, F_n)\) (or \((F_0, \dots, F_n)\)), such that \(F_{2i} \supset F_{2i+1}\) for all \(i\), and \(F_{2i+1} \subset F_{2i+2}\) for all \(i\). Chapter 6 is about gallery varieties. Given a generalized gallery as above, for each facet \(F_i\) we have the partial flag variety \(\text{Flag}_i\) of the same type. The associated gallery variety is the closed subvariety of the product \(\prod_i \text{Flag}_i\) of partial flag varieties consisting of tuple \((\mathcal D_i)_i\), where the relative position between \(\mathcal D_i\) and \(\mathcal D_{i+1}\) is the one specified by \(F_i\) and \(F_{i+1}\) (here the notion of relative position is to be understood in terms of the Weyl group of the underlying group, or double cosets thereof). In Chapter 7 it is shown that the resolution defined in terms of the relative position matrix as explained above can be obtained as a gallery variety.
In the following three chapters, the combinatorial ingredients required to handle general reductive groups are discussed. Chapter 8 is about the Coxeter complex, Chapter 9 about \textit{minimal} generalized galleries in a Coxeter complex, and Chapter 10 about the notion of minimal generalized galleries in the building of a reductive group.
With these tools at hand, the author generalizes the theory which was previously set up for \(\mathrm{GL}_n\) over a field to the case of a reductive group scheme over an arbitrary base scheme. To start with, in Chapter 11 basic notions about parabolic subgroups and their relative position are established; for a large part, these topics are already discussed in [\textit{M. Demazure} (ed.) and \textit{A. Grothendieck} (ed.), Schémas en groupes. III: Structure des schémas en groupes réductifs. Exposés XIX à XXVI. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), dirigé par Michel Demazure et Alexander Grothendieck. Revised reprint. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0212.52810)], see also [\textit{M. Demazure} (ed.) and \textit{A. Grothendieck} (ed.), Séminaire de géométrie algébrique du Bois Marie 1962-64. Schémas en groupes (SGA 3). Tome III: Structure des schémas en groupes réductifs. New annotated edition of the 1970 original published bei Springer. Paris: Société Mathématique de France (2011; Zbl 1241.14003)]). In Chapters 12 and 13, Schubert schemes and gallery schemes are defined in this context. Using these, the author defines a smooth resolution of Schubert schemes in Chapter 14. The main result (Theorem 14.18) is phrased as follows: Consider the universal Schubert scheme \(\overline{\Sigma}\) over the scheme \(\text{Relpos}\) of relative positions (i.e., we consider all Schubert varieties simultaneously; in the case of a split group scheme, the scheme of relative positions is just a disjoint union of copies of the base scheme). As above, the Schubert scheme is defined using the operation of schematic closure. Then the scheme of minimal galleries \(\mathcal C\) is a smooth resolution of the fiber product \(\Gamma^m \times_{\text{Relpos}} \overline{\Sigma}\), where \(\Gamma^m\) denotes the scheme of types of minimal galleries. For Schubert varieties in the flag variety of a split group, we can read this as saying that for each Weyl group element \(w\) (i.e., a point in the relative position scheme), we get a resolution of singularities once we choose a reduced expression for this element (in other words, a minimal gallery connecting the base chamber with the chamber corresponding to \(w\)).
Next, the construction via generalized gallery schemes is compared to the more common point of view of Demazure resolutions written as contracted products. It is shown that both approaches are basically equivalent (Prop.~15.4). Finally, Chapter 16 discusses questions of functoriality and compatibility with base change.
Demazure resolutions have been studied from many points of view. Let us mention as examples the papers by \textit{S. Gaussent} [Indag. Math., New Ser. 12, No. 4, 453--468 (2001; Zbl 1065.14065)] and by \textit{S. Gaussent} and \textit{P. Littelmann} [Duke Math. J. 127, No. 1, 35--88 (2005; Zbl 1078.22007)]. In the latter one, several concepts treated in the book at hand are developed in the context of a Kac-Moody group, i.e., for affine Grassmannians and affine flag varieties, with the Bruhat-Tits building replacing the Tits building.
The heavy notation used in the book is not always easy to digest. There is an index (unfortunately it is more difficult than necessary to find the desired information there, because the index is not sorted alphabetically, but according to chapters). Schubert varieties; Demazure resolutions; generalized galleries; Tits building; reductive group schemes Research exposition (monographs, survey articles) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Group schemes, Groups with a \(BN\)-pair; buildings Buildings and Schubert 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 [For the entire collection see Zbl 0583.00007.]
The theorem of Artin referred to in the title says that when S is an étale extension of a Noetherian commutative ring R and T is an étale covering of \(S\otimes S\), there is an étale covering S' of S and a map \(\alpha:\quad T\to S'\otimes S'\) making \(\beta:\quad S\otimes S\to S'\otimes S'\) commute via T, that is: \(\beta =\alpha \circ \gamma\), with \(\gamma:\quad S\otimes S\to T\) coming from the covering. (This result permits, among other things, an explicit embedding of the Brauer group of R in a suitable étale cohomology group.)
The author extends Artin's theorem to the \({\mathbb{Z}}\)-graded case. graded ring; étale extension; étale covering; Brauer group Caenepeel, S.: A graded version of Artin's refinement theorem. Lecture notes in mathematics 1197, 31-44 (1986) Extension theory of commutative rings, Brauer groups of schemes, Graded rings and modules (associative rings and algebras) A graded version of Artin's refinement 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 It is a classical result of \textit{V. G. Sarkisov} [Math. USSR, Izv. 20, 355--390 (1983; Zbl 0593.14034); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 371--408 (1982)] saying that, over an algebraically closed field of characteristic zero, any conic bundle over a smooth base is birational to a stand form (see the paper for the precise definition). In the paper under review, the author generalizes this result to conic bundles over a general perfect base field, providing the existence of resolution of singularities. conic bundles; root stacks; destackification; resolution of singularities Families, moduli, classification: algebraic theory, Generalizations (algebraic spaces, stacks), Rational and birational maps, Brauer groups of schemes Conic bundles and iterated root stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0604.00004.]
From the introduction: `The aim of this short note is to describe a direct and natural construction of Morgan's mixed Hodge structure on the homotopy groups of a smooth complex algebraic variety [\textit{J. W. Morgan}, Publ. Math., Inst. Hautes Étud. Sci. 48, 137-204 (1978; Zbl 0401.14003)] using Chen's iterated integrals [\textit{K.-T. Chen}, Bull. Am. Math. Soc. 83, 831-879 (1977; Zbl 0389.58001)].' This is a simplified version of the author's construction in K-theory 1, 271-324 (1987; Zbl 0637.55006)]. bar construction; mixed Hodge structure; homotopy groups; iterated integrals Transcendental methods, Hodge theory (algebro-geometric aspects), Rational homotopy theory, Homotopy theory and fundamental groups in algebraic geometry, Bar and cobar constructions Iterated integrals and mixed Hodge structures on homotopy 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 Recurrence relations for the coefficients of the \(n\)-th division polynomial for elliptic curves are presented. These provide an algorithm for computing the general division polynomial without using polynomial multiplications; also a bound is given for the coefficients, and their general shape is revealed, with a means for computing the coefficients as explicit functions of \(n\). recurrence relations; coefficients of the \(n\)-th division polynomial for elliptic curves McKee, J, Computing division polynomials, Math. Comput., 63, 767-771, (1994) Computational aspects of field theory and polynomials, Computational aspects of algebraic curves, Elliptic curves over global fields, Number-theoretic algorithms; complexity Computing division 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 [For the entire collection see Zbl 0626.00011.]
This paper develops a theory of algebraic cycles and intersection numbers on regular arithmetic schemes and uses it to establish a formula describing the behaviour of the Euler characteristic in a degenerating family of curves in mixed characteristic case. To explain this formula, let \(f: X\to S=Spec(A)\) be a flat and proper morphism, with A a complete discrete valuation ring, X a regular scheme and the generic fibre \(X_ g\) smooth. If \(X_ s\) is the closed fibre of f, then one defines the Euler number of \(X_ z\) (with \(z=s\) or \(z=\) geometric generic point \(\bar g\)) \(\chi(X_ z)\) using the étale cohomology. The formula one is interested in asks to compute the difference \(\chi(X_ s)-\chi(X_{\bar g})\). In characteristic zero such a formula is well known. For example, if \(X_ s\) has only isolated singularities, then \((- 1)^{\dim(X)}(\chi(X_ s)-\chi(X_{\bar g}))\) is just the sum of the Milnor numbers of the singularities of X. If S is of pure characteristic \(p>0\) such a formula is in general wrong, but it can be ``corrected'' using the socalled Swan conductor. In the arithmetic case the author is able to define the intersection number \((\Delta_ X\cdot \Delta_ X)_ s=(-1)^{\dim (X)}c_{\dim (X)}(\Omega\) \(1_{X/S})\) as a 0-cycle of \(X_ s\) and then he conjectures that \((\Delta_ X\cdot \Delta_ X)=- sw(X/S)+\chi(X_ s)-\chi (X_{\bar g})\), where \(\Delta_ X\) is the diagonal of X and \(sw(X/S)\) is the Swan correction. Finally, he shows that this formula is true if the fibre dimension of X/S is one.
[See also the author's paper in Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 85-90 (1987; see review 14016)]. Chern classes; algebraic cycles; intersection numbers; arithmetic schemes; Euler characteristic; family of curves; Swan conductor; intersection number Bloch, S., Cycles on arithmetic schemes and Euler characteristics of curves, Proceedings of Symposia in Pure Math. AMS46 (1987), 421--450 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Families, moduli of curves (algebraic), Schemes and morphisms, Cycles and subschemes, Characteristic classes and numbers in differential topology, Topological properties in algebraic geometry Cycles on arithmetic schemes and Euler characteristics of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A mixed trigonometric polynomial system, which rather frequently occurs in applications, is a polynomial system where every monomial is a mixture of some variables and sine and cosine functions applied to the other variables. Polynomial systems transformed from the mixed trigonometric polynomial systems have a special structure. Based on this structure, a hybrid polynomial system solving method, which is more efficient than random product homotopy method and polyhedral homotopy method in solving this class of systems, has been presented. Furthermore, the transformed polynomial system has an inherent partially symmetric structure, which cannot be adequately exploited to reduce the computation by the existing methods for solving polynomial systems. In this paper, a symmetric homotopy is constructed and, combining homotopy methods, decomposition, and elimination techniques, an efficient symbolic-numerical method for solving this class of polynomial systems is presented. Preservation of the symmetric structure assures us that only part of the homotopy paths have to be traced, and more important, the computation work can be reduced due to the existence of the inconsistent subsystems, which need not to be solved at all. Exploiting the new hybrid method, some problems from the literature and a challenging practical problem, which cannot be solved by the existing methods, are resolved. Numerical results show that our method has an advantage over the polyhedral homotopy method, hybrid method and regeneration method, which are considered as the state-of-art numerical methods for solving highly deficient polynomial systems of high dimension. mixed trigonometric polynomial system; polynomial system; symmetry; homotopy method; hybrid algorithm; symbolic-numeric computation B. Dong, B. Yu, and Y. Yu, \textit{A symmetric homotopy and hybrid polynomial system solving method for mixed trigonometric polynomial systems}, Math. Comp., 83 (2014), pp. 1847--1868, http://dx.doi.org/10.1090/S0025-5718-2013-02763-9. Solving polynomial systems; resultants, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Numerical computation of solutions to systems of equations, Computational aspects in algebraic geometry, Symbolic computation and algebraic computation A symmetric homotopy and hybrid polynomial system solving method for mixed trigonometric polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list [\textit{H. Sakai}, Commun. Math. Phys. 220, No. 1, 165--229 (2001; Zbl 1010.34083)]. For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries, translation elements of which define discrete Painlevé equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlevé-type through projective reduction, the resulting equations have fewer than the maximal number of free parameters corresponding to their surface type. We show that equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlevé equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature [\textit{T. Hoffmann}, Bobenko, Alexander I. (ed.) et al., Discrete integrable geometry and physics. Based on the conference on condensed matter physics and discrete geometry, Vienna, Austria, February 1996. Oxford: Clarendon Press. Oxf. Lect. Ser. Math. Appl. 16, 83--96 (1999; Zbl 0944.53006)]. We obtain a full-parameter generalisation of this equation from the Cremona action of a non-translation element of the extended affine Weyl group \(\widetilde{W}(D_4^{(1)})\) on a family of generic \(D_4^{(1)}\)-surfaces. discrete Painlevé equation; affine Weyl group; Cremona transformation; projective reduction Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Relationships between algebraic curves and integrable systems, Birational automorphisms, Cremona group and generalizations, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Discrete version of topics in analysis, Additive difference equations, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems Full-parameter discrete Painlevé systems from non-translational Cremona isometries | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We propose a new collapsing mechanism for \(G_2\)-metrics, with the generic region admitting a circle bundle structure over a \(K3\) fibration over a Riemann surface. The adiabatic description involves a weighted version of the maximal submanifold equation. In a local smooth setting we prove the existence of formal power series solutions, and the problem of compactification is discussed at a heuristic level. Issues of holonomy in differential geometry, \(K3\) surfaces and Enriques surfaces, \(G\)-structures Iterated collapsing phenomenon on \(G_2\)-manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we first prove that the incremental algorithm for computing triangular decompositions proposed by the author et al. in [ISSAC 2011, 75--82 (2011; Zbl 1323.14029)] in its original form preserves chordality, which is an important property on sparsity of variables. On the other hand, we find that the current implementation in \textsf{Triangularize} command of the RegularChains library in Maple may not always respect chordality due to the use of some simplification operations. Experimentation show that modifying these operations, together with some other optimizations, brings significant speedups for some super sparse polynomial systems. triangular decomposition; chordal graph; incremental algorithm; regular chain Symbolic computation and algebraic computation, Semialgebraic sets and related spaces, Computational aspects of higher-dimensional varieties Chordality preserving incremental triangular decomposition and its implementation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be a commutative ring and \(A[x]\) the polynomial ring in one variable over \(A\). A ring homomorphism \(\sigma:A\rightarrow A[x]\) is called an exponential map on \(A\) if whenever \(a\in A\) and \(\sigma(a) = \sum_{i=0}^{m}a_{i}x^{i}\) (\(a_{i}\in A\) for \(i=0,\dots, m\)), one has \(a_{0} = a\) and \(\sum_{i=0}^{m}\sigma(a_{i})y^{i} = \sum_{i=0}^{m}a_{i}(x+y)^{i}\) in \(A[x, y]\). (Note that for every such an exponential map \(\sigma\), one can consider a collection \((\delta_{i})_{i=0}^{\infty}\) of endomorphisms of the additive group of \(A\) defined by \(\sigma(a) = \sum_{i\geq 0}\delta_{i}(a)x^{i}\) for every \(a\in A\). The family \((\delta_{i})_{i=0}^{\infty}\) is called a locally finite iterative higher derivation on \(A\); this concept is naturally equivalent to the concept of an exponential map.)
In [Osaka J. Math. 15, 655--662 (1978; Zbl 0393.13007)] \textit{Y. Nakai} proved the following structure theorem: Let \(k\) be an algebraically closed field, \(A\) a \(k\)-domain, and \(\sigma\) a nontrivial exponential map on \(A\) over \(k\). If \(A^{\sigma} = \{a\in A|\sigma(a) = a\}\) is a finitely generated PID over \(k\) and every prime element of \(A^{\sigma}\) is a prime element of \(A\), then \(A\) is the polynomial ring in one variable over \(A^{\sigma}\).
The main result of the paper under review generalizes the Nakai's theorem by removing the conditions that the ground field \(k\) is algebraically closed, that \(A\) is an integral domain, and that the \(k\)-algebra \(A^{\sigma}\) is finitely generated. As one of the consequences of this generalization of Nakai's theorem, the author obtains the following result that generalizes the cancellation theorem of \textit{A. J. Crachiola} [J. Pure Appl. Algebra 213, No. 9, 1735--1738 (2009; Zbl 1168.14041)]: Let \(k\) be a field, \(\overline{k}\) its algebraic closure, and \(A\) and \(A'\) finitely generated \(k\)-domains with \(A[x]\simeq_{k}A'[x]\). If \(A\) and \(\overline{k}\bigotimes_{k}A\) are UFDs and trans.\(\deg_{k}A = 2\), then \(A\simeq_{k}A'\). S. Kuroda, A generalization of Nakai's theorem on locally finite iterative higher derivations, Osaka J. Math. 54 (2017), 335--341. Derivations and commutative rings, Actions of groups on commutative rings; invariant theory, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) A generalization of Nakai's theorem on locally finite iterative higher derivations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(m^{th}\)-jet scheme \(X_m\) of an algebraic variety \(X\) over an algebraically closed field \(k\) is a \(k\)-scheme of finite type which parametrizes morphisms \(\mathrm{Spec}(k[t]/t^{m+1})\to X\). Jet schemes have attracted, from various viewpoints, the attention of many authors such as \textit{J. F. Nash, jun.} [Duke Math. J. 81, No.1, 31--38 (1995; Zbl 0880.14010)] and, more recently, \textit{M. Mustată} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)], with \textit{L. Ein} and \textit{R. Lazarsfeld} [Compos. Math. 140, No. 5, 1229--1244 (2004; Zbl 1060.14004)].
In the present paper the author considers a curve \(C\) in the complex plane, with a singularity at \(O\) at which it is analytically irreducible (i.e. the formal neighborhood \((C,O)\) of \(C\) at \(O\) is a branch). He determines the irreducible components of the \(m\)-Jet scheme of \((C,O)\) and shows that their number is not bounded as \(m\) grows. He also gives formulas for their number and for their codimension, in terms of \(m\) and of the generators of the semigroup of \((C,O)\).
As it is well known, the semigroup of the branch \((C,O)\) and the topological type of \(C\) near \(O\) are equivalent data and characterize the equisingularity class of \((C,O)\), as defined by Zariski. The author shows in particular that the structure of the Jet Scheme determines the topological type of \(C\) and conversely. jet schemes; singularities of plane curves Mourtada, H., Jet schemes of complex branches and equisingularity, \textit{Ann. Inst. Fourier}, 61, 6, 2313-2336, (2011) Arcs and motivic integration, Singularities in algebraic geometry Jet schemes of complex plane branches and equisingularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 curve \(y^2 = x^6 + 2kx^3 + 1\) \((k^2 \neq 1)\) has genus two and admits an automorphism of order three, which can be linearly extended to an automorphism \(\sigma\) of the Jacobian \(J\) of the curve. The quotient \(J/ \sigma\) is a ``generalized'' Kummer surface. It embeds in \(\mathbb{P}^4\) as a complete intersection of a quadric and a cubic hypersurface. Its singular points give origin to a configuration similar to the \(16_6\) configuration on a Kummer surface. Very elegant explicit equations are also obtained. generalized Kummer surfaces; configuration J. Bertin - P. Vanhaecke , The even master system and generalized Kummer surfaces , Math. Proc. Cambridge Philos. Soc. 116 ( 1994 ), 131 - 142 . MR 1274163 | Zbl 0828.14022 \(K3\) surfaces and Enriques surfaces, Complete intersections, Projective techniques in algebraic geometry The even master system and generalized Kummer 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 According to the author's abstract, this paper is intended to be a survey on tangent bundles \(T_{X/Y}\) of (relative) schemes \(f:X\to Y\) drawing on previous results of algebraic nature about symmetric algebras as applied to the case of the module of relative differentials \(\Omega_{X/Y}\). As it turns out, the paper serves an interesting additional purpose by conveying a neat introduction to (relative) vector fields of schemes. After these preliminaries, the author illustrates in the context of tangent bundles of schemes the formulas for the dimension of a symmetric algebra. Pushing further these connections, he deduces other formulas for the fibres of the structural projection of the bundle and for the fibres of \(f\). As might be expected, in this generality, the formulas regarding \(\Omega_{X/Y}\) turn out to be a bit more complicated, involving the inseparability degree and the regularity defect of a local morphism \(R\to S\) of local rings which is essentially of finite type.
A special section is devoted to the case in which \(Y=\text{Spec}(K)\), where \(K\) is a field of arbitrary characteristic. Here the author is at his best by developing a notion of admissible fields that is suited to safely take the place of perfect fields. He then derives results on equidimensionality and criteria of regularity known to hold when the base field is perfect.
The author also devotes a section to obtaining a condition, back in the general relative case (with sufficient conditions on the scheme \(X)\), for equidimensionality to trigger irreducibility for the relative tangent bundle. This question had been essentially distilled by other authors before in the case of an abstract module, but here the author brings into the picture a different inequality by unveiling the role of the inseparable degree and the (local) regularity defect. He applies this in the case in which \(X\) is a one-dimensional scheme over a field \(K\), showing that the torsionfreeness of the symmetric algebra of \(\Omega_{X/K_0}\) \((K_0\) standing for an admissible subfield of \(K)\) triggers the regularity of \(X\) -- this is still a few yards from proving the Berger conjecture, which requires that only \(\Omega_{X/K_0}\) itself be torsionfree.
The last part of the paper deals with local (relative) complete intersections \(X\to Y\) and a criterion for the tangent bundle \(T_{X/Y}\) to be a local complete intersection over \(Y\). module of relative differentials; vector fields of schemes; tangent bundles of schemes; Berger conjecture Modules of differentials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) On the tangent bundle of a scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct \(G\)-Hilbert schemes for finite group schemes \(G\). We find a construction of \(G\)-Hilbert schemes as relative \(G\)-Hilbert schemes over the quotient that does not need the Hilbert scheme of \(n\) points, works under more natural assumptions and gives additional information about the morphism from the \(G\)-Hilbert scheme to the quotient. \(G\)-Hilbert scheme; McKay correspondence Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Hopf algebras and their applications, Ordinary representations and characters Construction of \(G\)-Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present article develops and explores some beautiful connections between piecewise polynomial functions and convex polytopes. A new approach to McMullen's polytope algebra forms the basis of the approach, which gives a natural relation with the equivariant cohomology of toric varieties and spherical homogeneous space. As a consequence of the theory a new proof is given for the Fulton and Sturmfels identification of the polytope algebra with the direct limit of all Chow rings of smooth, complete torus embeddings [see \textit{W. Fulton} and \textit{B. Sturmfels}, ``Intersection theory on toric varieties'', Topology 36, No. 2, 335-353 (1997)]. The Jurkiewicz-Danilov presentation of the Chow ring of a smooth, complete toric variety is also recovered. An interesting version of Bezout's theorem is given for any spherical homogeneous space. The technique gives a generalization of Hadwiger's characterization of the volume of convex polytopes [see \textit{P. McMullen} in: First internat. Conf. stochastic geometry, Convex bodies, empirical measures, Palermo 1993, Auppl. Rend. Circ. Mat. Palermo, II. Ser. 35, 203-216 (1994; Zbl 0808.52010)], and of Bernstein and Kouchnirenko's result on the number of common points to \(d\) hypersurfaces in general position in a \(d\)-dimensional torus [\textit{A. G. Kushnirenko}, Invent. Math. 32, 1-31 (1976; Zbl 0328.32007)]. Interestingly the Bezout theorem can even be used to study smooth quadrics of rank \(d\) in \(\mathbb{P}^r\). piecewise polynomial functions; convex polytopes; enumerative geometry; Bezout theorem; Chow rings; toric varieties; spherical homogeneous space; polytope algebra M. Brion, ''Piecewise Polynomial Functions, Convex Polytopes and Enumerative Geometry,'' in Parameter Spaces, Ed. by P. Pragacz (Inst. Math., Pol. Acad. Sci., Warszawa, 1996), Banach Center Publ. 36, pp. 25--44. Toric varieties, Newton polyhedra, Okounkov bodies, Enumerative problems (combinatorial problems) in algebraic geometry, Homogeneous spaces and generalizations, Convex sets in \(n\) dimensions (including convex hypersurfaces) Piecewise polynomial functions, convex polytopes and enumerative 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 a smooth and connected projective curve and \((E,V)\) a spanned coherent system on \(X\) of type \((n,d,n+1)\) such that \(E\) has no trivial factor. Here we prove that the coherent system \((E,V)\) is \(\alpha\)-stable for all \(\alpha\gg 0\). Furthermore, \((E,V)\) is \(\alpha\)-stable for all \(\alpha\geq 0\) (resp. \(\alpha>0\)) if and only if \(E\) is stable (resp. semistable). Vector bundles on curves and their moduli Stable coherent systems of type \((n, d, n+1)\) on smooth curves and maps to projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth and connected projective curve, \(E\) a non-trivial rank \(n\) vector bundle on \(X\) and \(V\) an \(m\)-dimensional linear subspace of \(H^0(X,E)\) spanning \(E\). Assume that \(\varphi_{E,V}:X\to\text{Gr}(n,m)\) is unramified. Fix a general \(n\)-dimensional linear subspace of \(W\). Then:
(a) for a general \((n-1)\)-dimensional linear subspace \(B\) of \(W\) the evaluation map \(u_A:{\mathcal O}_X\otimes W\to E\) is injective and with locally free cokernel;
(b) for every \((n-1)\)-dimensional linear subspace \(B\) of \(W\) the evaluation map \(u_B:{\mathcal O}_X\otimes W\to E\) is injective as a map of sheaves;
(c) there is a non-empty family \(S(W)\) of the hyperplanes of \(W\) such that the evaluation map \(u_B:{\mathcal O}_X\otimes W\to E\) is injective as a map of sheaves, but it has a non-locally free cokernel if and only if \(B\in S(W)\); at each \(P\in X\) the fiber at \(P\) of \(u_B\) has rank at least \(n-2\); the saturation of \(u_B({\mathcal O}_X\otimes B)\) in \(E\) has degree one; \(S(W)\) has pure dimension \(n-2\). Vector bundles on curves and their moduli Vector subspaces of sections and stable coherent systems on 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 \(K\) be an infinite field, whose characteristics is neither \(2\) nor \(3\). The paper deals with smoothable Gorenstein \(K\)-points in a punctual Hilbert scheme, getting the following main results
\begin{itemize}
\item every \(K\)-point defined by local Gorenstein \(K\)-algebras with Hilbert function \((1,7,7,1)\) is smoothable;
\item the Hilbert scheme \(\mathrm{Hilb}_{16}^7\) has at least five irreducible components.
\end{itemize}
A new elementary component in \(\mathrm{Hilb}_{15}^7\) is found, starting from the study of \(\mathrm{Hilb}_{16}^7\), The problem is studied via properties double-generic initial ideals and of marked schemes. We remark that the considered problem is deeply related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more in general, of the irreducibility of a Hilbert scheme, a relevant and open question. Gorenstein algebra; Hilbert scheme; strongly stable ideal; marked basis Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Smoothable Gorenstein points via marked schemes and double-generic initial ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the article describes, using the notion of admissible affine structures introduced by the author, when two collections of rings and transition maps form isomorphic schemes. affine structure; pseudogroup of affine transformations; ringed space; scheme Schemes and morphisms, Elementary questions in algebraic geometry Affine structures on a ringed space and schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y \subset \mathbb P^n\) be a smooth hypersurface of degree \(d\). For a point \(y \not \in Y\) the projection \(\pi: Y \to \mathbb P^{n-1}\) gives a finite morphism of degree \(d\) and one can consider the relative Hilbert scheme \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) parametrizing zero-dimensional subschemes of length \(m\) contained in the fibers of the projection for each \(1 \leq m \leq d\). It is easy to see that \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) is isomorphic to \(\mathbb P^{n-1}\) for \(m=d\) and to \(Y\) for \(m=1\) or \(m=d\), so the author focuses on the intermediate values \(1 < m < d\). Here it is known from work of \textit{L. Gruson} and \textit{C. Peskine} that \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) is a smooth connected projective variety of dimension \(n-1\) for general \(y \not \in Y\) [Duke Math. J. 162, No. 3, 553--578 (2013; Zbl 1262.14058)]. In the paper under review the author gives an explicit embedding \(j:\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1}) \hookrightarrow \mathbb P\) into a weighted projective space for \(n > 1\) and gives gives equations defining the image \(X \subset \mathbb P\). The equations show that \(X \subset \mathbb P\) is a smooth weighted complete intersection, and hence results of \textit{I. Dolgachev} [Lect. Notes Math. 956, 34--71 (1982; Zbl 0516.14014)] yield the generating function for the dimensions of the graded pieces of \(\oplus_{k=0}^\infty H^0(X, {\mathcal O}_X (k))\), the dualizing sheaf \(\omega_X\), and the Picard group for \(n \geq 4\). relative Hilbert scheme; weighted projective space Parametrization (Chow and Hilbert schemes) The relative Hilbert scheme of projection morphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a very useful book for all who wants to know something about schemes but never dared to ask the right questions.
Indeed, it is a successor to the authors': ``Schemes: The language of modern algebra'' [\textit{D. Eisenbud} and \textit{J. Harris} (1992; Zbl 0745.14002)]. There, the influence of \textit{D. Mumford} [``The red book of varieties and schemes'', Lect. Notes Math. 1358 (1988; Zbl 0658.14001)] was not to be overseen.
The additions are intended to show schemes at work in a number of topics in classical geometry. For example the authors define blow-ups and study the blow-up of the plane at various non-reduced points. They define duals of plane curves, and study how the dual degenerates as the curve does.
The many examples and explaining pictures make this book recommendable for students how want to get the feeling for the abstract version of curves, surfaces, tangents, etc. to come to moduli spaces. A general method is the functor of points.
The author introduces Fano schemes, a concept not to be found in the book by \textit{R. Hartshorne} [``Algebraic geometry'', Grad. Texts Math. 52 (1977; Zbl 0367.14001)], to explain some very nice classical examples, as are the 27 lines on a smooth cubic surface (which is then given as an exercise to do in detail).
Also, quartic surfaces are considered. moduli spaces; functor of points; Fano schemes; quartic surfaces D. Eisenbud and J. Harris, \textit{The Geometry of Schemes}, Grad. Texts in Math. 197, Springer-Verlag, New York, 2000, . Schemes and morphisms, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The geometry 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 There are two main parts of this paper. The unifying theme is that the algebraic structure of the canonical module \(\omega_R\) of the coordinate ring \(R\) of a zero-dimensional subscheme \(X\) of projective space determines whether or not \(X\) lies on a rational normal curve (resp. a rational normal scroll).
In the first part, the author gives a nice extension of some results of \textit{D. Eisenbud} and \textit{J. Harris} [J. Algebr. Geom. 1, No. 1, 15-30 (1992; Zbl 0804.14002) and 31-59 (1992; Zbl 0798.14029)] on Castelnuovo's lemma. Given a zeroscheme \(X\) (not necessarily reduced) of degree \(\geq 2r + 2 + d\) in \(\mathbb{P}^r\) in uniform position, which imposes only \(2r + d\) conditions on quadrics, the author shows that there is a \(d\)-dimensional rational normal scroll containing \(X\). The case \(d = 1\) was done by Eisenbud and Harris, who used it to study ``nearly extremal'' curves with respect to Castelnuovo theory. They also showed that if deg \(X = r + 3\) then \(X\) lies on a unique rational normal curve.
In the second half of the paper, the author examines the range \(r + 4 \leq \deg X \leq 2r + 2\), which is the range not covered in the results of Eisenbud and Harris mentioned above. He looks at the matrix corresponding to the multiplication \((\omega_R)_{-1} \otimes S_1 \to (\omega_R )_0\) (where \(S\) is the polynomial ring), and shows that this matrix is equivalent to the Hankel matrix if and only if \(X\) lies on a unique rational normal curve. He also gives some results on the graded Betti numbers of \(X\) in this case. The main tools used are some results of \textit{M. Kreuzer} on canonical modules [Can. J. Math. 46, No. 2, 357-379 (1994; Zbl 0826.14030)], and work of \textit{D. Eisenbud} [Am. J. Math. 110, No. 3, 541-575 (1988; Zbl 0681.14028)] on 1-generic matrices. The author has recently given a very nice generalization of his work under review here [J. Pure Appl. Algebra 105, No. 1, 107-116 (1995)]. canonical module; zero-dimensional subscheme; rational normal curve; rational normal scroll; Castelnuovo's lemma; Hankel matrix Yanagawa, K.: Some generalizations of Castelnuovo's lemma on zero-dimensional schemes. J. algebra 170, 429-431 (1994) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Curves in algebraic geometry Some generalizations of Castelnuovo's lemma on zero-dimensional 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 For a family of local systems on the punctured Riemann sphere, with moving singularities, its first parabolic cohomology is a local system on the base space. The authors derive universal formulas for the monodromy of the resulting local system. In particular, they prove that the simple groups PSL\(_2(p^2)\) admit regular realizations over the field \({\mathbb Q}(t)\) for primes \(p\not\equiv 1,4,16 \pmod{21}\). Hurwitz braid group; middle convolution M. Dettweiler, S. Wewers, Variation of local systems and parabolic cohomology. Isr. J. Math. 156, 157-185 (2006) Structure of families (Picard-Lefschetz, monodromy, etc.), Inverse Galois theory, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Variation of local systems and parabolic cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study \(d\times d\) rational Lax pair systems and show that in some cases the system satisfies the ``topological type property''. This leads to some interesting corollaries, e.g., that the WKB expansion of determinantal correlators satisfies the topological recursion. Applications include minimal model reductions of the KP hierarchy and six Painlevé systems. Lax pair; integrable systems; topological type property; WKB expansion Belliard, R.; Eynard, B.; Marchal, O., Integrable differential systems of topological type and reconstruction by the topological recursion, Ann. Henri Poincaré, 18, 10, 3193-3248, (2017) KdV equations (Korteweg-de Vries equations), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), General topics in linear spectral theory for PDEs, Relationships between algebraic curves and integrable systems, Singular perturbations, turning point theory, WKB methods for ordinary differential equations Integrable differential systems of topological type and reconstruction by the topological recursion | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0539.00006.]
Let X be a normal projective scheme over an algebraically closed field k of characteristic 0. Let \({\mathcal D}\) denote the set of the rational coefficient Weil divisors D such that there exists a positive integer m with mD Cartier and ample. One can associate with every \(D\in {\mathcal D}\) the graded ring \(A=A(X,D):=\oplus_{n\geq 0}H^ 0(X,{\mathcal O}_ X(nD))T^ n.\) It is known that \(X\cong \Pr oj(A)\) [\textit{M. Demazure}, ''Anneaux gradués normaux'' Séminaire Demazure-Giraud-Teissier, Singularités des surfaces, École Polytechnique 1979]. This paper proves that A(X,D) is factorial (respectively almost factorial) for some \(D\in {\mathcal D}\) if and only if \(Cl(X)={\mathbb{Z}}\) (respectively rk Cl(X)\(=1)\) and that in this case, D is uniquely determined up to linear equivalence (respectively A(X,D) is almost factorial for all \(D\in {\mathcal D})\). In particular, if \(Cl(X)={\mathbb{Z}}\) generated by some [D] and \(W=\sum^{s}_{i=1}p_ iV_ i/q_ i\in {\mathcal D},\) where \(V_ i\) are distinct prime divisors, \((p_ i,q_ i)=1\) and \(q_ i>0\), then A(X,W) is factorial if and only if \(q_ i\) are pairwise coprime positive integers and \(\sum^{s}_{i=1}p_ id(V_ i)/q_ i=1/(q_ 1...q_ s)\) where \(d(V_ i)\) is defined by the relation \(V_ i\sim d(V_ i)D.\) These results are illustrated by various interesting examples. weighted projective space; almost factorial ring; rational coefficient Weil divisors L. Robbiano, Factorial and almost factorial schemes in weighted projective spaces , Lect. Notes Math. 1092 , Springer-Verlag, New York, %62-84, ( 1984.
Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Divisors, linear systems, invertible sheaves Factorial and almost factorial schemes in weighted projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:C\to X\) be a multiple covering of smooth projective curves. Here we discuss the non-existence of base point free linear systems on \(C\). multiple coverings of curves; double coverings; base point free pencils Special divisors on curves (gonality, Brill-Noether theory), Coverings of curves, fundamental group Linear systems on multiple coverings of a smooth 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 connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, \textit{P. Lisoněk} [IEEE Trans. Inf. Theory 57, No. 9, 6010--6014 (2011; Zbl 1365.94674)] exploited it to reformulate the Charpin-Gong characterization [\textit{P. Charpin} and \textit{G. Gong}, IEEE Trans. Inf. Theory 54, No. 9, 4230--4238 (2008; Zbl 1184.94233)] of a large class of hyperbent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin-Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by \textit{S. Mesnager} [WAIFI 2010, Lect. Notes Comput. Sci. 6087, 97--113 (2010; Zbl 1232.94016)]. Doing so, a polynomial time and space test for the hyperbentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong (loc. cit.). Boolean functions; Walsh-Hadamard transform; maximum nonlinearity; hyperbent functions; hyperelliptic curves; Dickson polynomials Flori J.P., Mesnager S.: An efficient characterization of a family of hyper-bent functions with multiple trace terms. J. Math. Cryptol. \textbf{7}(1), 43-68 (2013). , Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry An efficient characterization of a family of hyper-bent functions with multiple trace terms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove the equivalence of two conjectures on linear systems through fat points on a generic \(K3\) surface. The first conjecture is exactly as Segre conjecture on the projective plane. Whereas the second characterizes such linear system and can be compared to the Gimigliano-Harbourne-Hirschowitz conjecture. Cindy De Volder and Antonio Laface, Linear systems on generic \?3 surfaces, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 4, 481 -- 489. Divisors, linear systems, invertible sheaves, \(K3\) surfaces and Enriques surfaces Linear system on generic \(K3\) surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(0\in C\) be a germ of a smooth curve and \(f_U:Y_U\rightarrow U=C\setminus \{0\}\) a family of smooth surfaces of general type over \(U\). Then \(f_U\) can be completed in a unique way to a family \(f:Y\rightarrow C\) such that \(\omega_{Y/C}^{[k]}\) is invertible and ample for some \(k>0\) and the central fiber \(X=f^{-1}(0)\) is a stable surface [see \textit{V. Alexeev}, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 515--536 (2006; Zbl 1102.14023) and \textit{J. Kollár} and \textit{N. I. Shepherd-Barron}, Invent. Math. 91, No. 2, 299--338 (1988; Zbl 0642.14008)]. Thus the moduli space of surfaces of general type can be compactified by adding stable surfaces and it is important to know which stable surfaces are smoothable (this is related with the minimal model program). The purpose of this paper is to study the deformation theory of schemes with non-isolated singularities and to write some smoothability and nonsmoothability criteria. minimal model program; deformation theory; semi-log-canonical singularities; stable surface Tziolas N.: Smoothings of schemes with nonisolated singularities. Michigan Math. J. 59(1), 25--84 (2010) Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Deformations of singularities, Fibrations, degenerations in algebraic geometry Smoothings of schemes with nonisolated 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 \(S\) be a scheme, and consider \(S\)-schemes \(Z\) such that \(g : Z \to S\) is proper and equidimensional of relative dimension \(d\); under additional hypotheses one can identify the relative dualizing sheaf \(g^ ! {\mathcal O}_ S\) with the sheaf \(\omega_ Z\) of regular \(d\)-forms [cf. \textit{E. Kunz} and \textit{R. Waldi}, ``Regular differential forms'', Contemp. Math. 79 (1988; Zbl 0658.13019)]. Now let \(f:X \to Y\) be a suitable restricted morphism of such schemes; by a result of \textit{S. L. Kleiman} [Compos. Math. 41, 39-60 (1980; Zbl 0423.32006)] there is a canonical map \(\eta : f^* \omega_ Y \otimes \omega_ f \to \omega_ X = f^ ! \omega_ Y\) where \(\omega_ f\) is the canonical dualizing sheaf for \(f\). On the other hand, \textit{R. Hübl} [Manuscr. Math. 65, No. 2, 213-224 (1989; Zbl 0704.13004)] gives a rather explicit description by methods of commutative algebra of a morphism \(\varphi : f^* \omega_ Y \otimes \omega_ f \to \omega_ X\).
The bulk of the present paper consists of showing that \(\eta = \varphi\). To prove this result the author makes use of the residue formalism developed by \textit{R. Hübl} and \textit{E. Kunz} [J. Reine Angew. Math. 410, 53-83 (1990; Zbl 0712.14006) and ibid. 84-108 (1990; Zbl 0709.14014)]. This paper contains many results on differential forms, residues, generalizations of the residue theorem of \textit{R. Hübl} and \textit{P. Sastry} [Am. J. Math. 115, No. 4, 749-787 (1993; Zbl 0796.14012)] and related topics and should be a must for anybody interested in these questions. With respect to residues of differential forms, one may also consult the Habilitationsschrift of \textit{R. Hübl} [``Residues of differential forms, de Rham cohomology and Chern classes'' (Regensburg 1994)]. sheaf of regular \(d\)-forms; canonical dualizing sheaf; residue formalism; differential forms J. Lipman and P. Sastry,Regular differentials and equidimensional scheme-maps, Journal of Algebraic Geometry1 (1992), 101--130. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Modules of differentials Regular differentials and equidimensional scheme-maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of \(n-1\) homogeneous polynomials in \(n\geqslant 2\) variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring \(k\), typically a domain, if \(2\neq 0\) in \(k\). The case where \(2=0\) in \(k\) is also analyzed in detail. elimination theory; discriminant of homogeneous polynomials; resultant of homogeneous polynomials; inertia forms Busé, L; Jouanolou, JP, On the discriminant scheme of homogeneous polynomials, Math. Comput. Sci., 8, 175-234, (2014) Solving polynomial systems; resultants, Computational aspects in algebraic geometry, Determinantal varieties, Polynomial rings and ideals; rings of integer-valued polynomials On the discriminant scheme of 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 \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] proved a recursion to enumerate gluings of a \(2d\)-gon that result in an orientable genus \(g\) surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer-Zagier recursion, but our methodology also applies to the enumeration of dessins d'enfant, to Bousquet-Mélou-Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion. Harer-Zagier formula; 1-point functions; holonomic functions; Schur functions; Hurwitz numbers; ribbon graphs; dessins d'enfant Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Enumerative problems (combinatorial problems) in algebraic geometry, Planar graphs; geometric and topological aspects of graph theory, Topological properties in algebraic geometry Generalisations of the Harer-Zagier recursion for 1-point functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas. Charles, D., Lauter, K.: Computing modular polynomials. LMS J. Comput. Math. 8, 195--204 (2005) Arithmetic aspects of modular and Shimura varieties, Number-theoretic algorithms; complexity, Elliptic curves, Applications to coding theory and cryptography of arithmetic geometry Computing modular 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 \(G\) be a reductive group scheme over the algebraic curve \(X\) over the field \(k\). Let \(K\) be the function field of \(X\). We show how \(G\) induces an additional structure on the root system \(\Phi = \Phi(G_ K,T)\) for any split maximal torus \(T\) of \(G_ K\), the pullback of \(G\) to \(\text{Spec }K\). We call this structure a complementary polyhedron for \(\Phi\). The study of these polyhedra leads to a proof of the existence and uniqueness of a canonical parabolic subgroup \(P\) of \(G\). If \(G =\text{GL}(V)\) for a vector bundle \(V\) over \(X\), the parabolic \(P\) is the stabilizer of the Harder-Narasimhan filtration of \(V\). reductive group scheme; algebraic curve; function field; root system; split maximal torus; complementary polyhedron; parabolic subgroup; vector bundle; Harder-Narasimhan filtration Behrend K, Semi-stability of reductive group schemes over curves, Math. Ann. 301 (1995) 281--305 Linear algebraic groups over adèles and other rings and schemes, Group schemes, Simple, semisimple, reductive (super)algebras, Classical groups (algebro-geometric aspects) Semi-stability of reductive group schemes over curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that if (R,\({\mathfrak M})\) is a 2-dimensional local domain having finite local cohomology module \(H^ 1_{{\mathfrak M}}(R)\) (equivalently, R is a 2-dimensional local unmixed domain), and having algebraically closed residue field, such that \({\mathfrak M}^ 2\) is included in \(Ann(H^ 1_{{\mathfrak M}}(R))\), then any sequence of \(\ell +2\) successive parametric blowings-up of R necessarily produces a Cohen-Macaulay scheme, where \(\ell:=[\frac{\ell_ R(H^ 1_{{\mathfrak M}}(R))}{2}]\) (greatest integer). This gives a measure of the ``non-Cohen-Macaulayness'' of the singularity, partially generalizing results of Goto, Schenzel, et al. Some related ideal-theoretic results and some examples are included. non-Cohen-Macaulayness of singularity; 2-dimensional local domain; finite local cohomology module; blowings-up Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Local cohomology and commutative rings, Global theory and resolution of singularities (algebro-geometric aspects), Integral domains, Multiplicity theory and related topics Toward parametric Cohen-Macaulayfication of two-dimensional finite local cohomology domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Take an algebraically closed field k with \(ch(k)=p>2\). A smooth curve X of genus g is said to be ordinary (resp. very special) if the p-rank of its Jacobian is g (resp. 0). Most curves are ordinary and the first examples of such curves where given by \textit{L. Miller} [Math. Ann. 197, 123-127 (1972; Zbl 0235.14009)]. Fix a polynomial \(f_ t\), deg f\({}_ t=t\), with t different roots. Here the author gives many examples of ordinary and very special hyperelliptic curves. He proves the following theorem: Let \(a(z)=\sum^{t}_{m=0}c_ mz^{p^ m}, c_ 0\neq 0\), be an additive separable polynomial, \(X_ 1\) (resp. \(X_ 2)\) the curve with equation \(y^ 2=f_{2g+1}(x)\) (resp. \(y^ 2=f_{2g+2}(x))\) and X'\({}_ 1\) (resp. X'\({}_ 2)\) the curve whose equation is obtained from that of \(X_ 1\) (resp. \(X_ 2)\) by the substitution \(x=a(z)\); assume \(X_ 1\) very special and \(X_ 2\) ordinary; then X'\({}_ 1\) is very special and X'\({}_ 2\) ordinary. The proof uses elementary properties of the Cartier operator. ordinary curve; holomorphic differential; Hasse-Witt matrix; p-division points; very special curve; p-rank of Jacobian; characteristic p; hyperelliptic curves; Cartier operator Jacobians, Prym varieties, Finite ground fields in algebraic geometry, Special algebraic curves and curves of low genus, Algebraic theory of abelian varieties An iterative construction for ordinary and very special hyperelliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper determines the irreducible components of jet schemes of a toric surface. For the jet schemes of sufficiently high order, the number of its irreducible components in a certain class is the same as the number of the exceptional divisors on the minimal resolution. toric variety; jet scheme Mourtada, Hussein Jet schemes of toric surfaces \textit{C.~R.~Math. Acad. Sci. Paris}349 (2011) 563--566 Math Reviews MR2802925 Arcs and motivic integration, Toric varieties, Newton polyhedra, Okounkov bodies Jet schemes of toric 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 Counting the number of projective plane curves of a given degree and number of nodes through the appropriate number of generic points is a wonderfully classical problem receiving modern attention and remarkable -- and beautiful -- results through the application of state-of-the-art enumerative techniques. The landmark result in this subject, called the Göttsche conjecture (which is based on a conjecture by Di Francesco and Itzykson, and which is now a theorem by work of Fomin and Mikhalkin) is that this number depends polynomially on the degree, once the number of nodes is fixed, and once the degree is large enough. See the introduction and references in the current paper for exact citations. The Fomin-Mikhalkin technique is to apply a ``Correspondence Theorem'' of Mikhalkin, which says that instead of counting algebraic curves, one can count tropical curves. This translates the very hard algebro-geometric problem into a very hard combinatorial problem, but Fomin-Mikhalkin succeed in establishing polynomiality through a clever enumerative apparatus called a floor diagram, first introduced (in a slightly less refined way) by Mikhalkin and Brugallé.
Fomin-Mikhalkin essentially describe their count of floor diagrams (and hence tropical curves, and hence algebraic curves) in terms of a discrete integral of a more basic ``building block'' enumerative gadget. Block, in the present paper, notes that one can actually evaluate these sums symbolically by applying quite classical machinery (one summation result in particular is from 1631!). This leads rather directly to an explicit algorithm for computing these enumerative polynomials. However, there are certain bottlenecks in the algorithm which impede an efficient computation, and Block nicely addresses these and refines the algorithm to obtain a more feasible version. This is then literally fed into a computer to compute these polynomials explicitly for parameters significantly larger than were previously known, thereby verifying more cases of several conjectures about their shape/coefficients in the process (see Block's intro for details), and the polynomials are nicely listed explicitly in an appendix. Block also proves some remarkable results about the polynomiality threshold of these nodal curve counts: he extends the known cases of a conjecture of Göttsche on this polynomiality threshold, disproves a conjecture on it by Di Francesco and Itzykson, and prove a new general bound on the threshold (the latter, it appears, is the only explicit result in the paper valid for all choices of parameter and not dependent on the output of an algorithm that necessarily can only be applied for fixed values of the degree/node parameters -- and Block's proof of it beautifully uses the machinery developed in this paper).
This paper is thus entirely combinatorial and computational -- algebraic geometry is non-existent here as that has been factored out by the work of Mikhalkin, Brugallé-Mikhalkin, and Fomin-Mikhalkin, yet remarkably within this combinatorial/tropical approach Block has managed to prove results and extend our knowledge of plane algebraic curves beyond what has been known for hundreds of years. Severi degree; Göttsche conjecture; node polynomials; floor diagram; tropical curves Block, F., \textit{computing node polynomials for plane curves}, Math. Res. Lett., 18, 621-643, (2011) Enumerative problems (combinatorial problems) in algebraic geometry, , Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative combinatorics, Plane and space curves Computing node polynomials for plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems According to the Göttsche conjecture (now a theorem), the degree \(N^{d, \delta}\) of the Severi variety of plane curves of degree \(d\) with \(\delta \) nodes is given by a polynomial in \(d\), provided \(d\) is large enough. These ``node polynomials'' \(N_\delta (d)\) were determined by Vainsencher and Kleiman-Piene for \(\delta \leq 6\) and \(\delta \leq 8\), respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute \(N_\delta (d)\) for \(\delta \leq 14\). Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal threshold up to \(\delta \leq 14\). We also determine the first 9 coefficients of \(N_\delta (d)\), for general \(\delta \), settling and extending a 1994 conjecture of Di Francesco and Itzykson. Severi degree; curve enumeration; plane curve; node polynomial; labeled floor diagram Enumerative problems (combinatorial problems) in algebraic geometry Computing node polynomials for plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a complete, irreducible and nonsingular curve over an algebraically closed field \(k\) and \(p\in C\) a closed point. Let \(f,g \in {\Sigma}^{*}_{C}\) be the elements of the multiplicative group of the function field of \(C.\) The multiplicative local symbol is defined as
\[
(f,g)_{p}=(-1)^{v_{p}(f)\cdot v_{p}(g)}{\frac{f^{v_{p}(g)}}{g^{v_{p}(f)}}}(p) \in k^{*}
\]
\textit{J. W. Milnor} [in his book ``Introduction to algebraic \(K\)-theory''. Princeton, N. J.: Princeton University Press and University of Tokyo Press (1971; Zbl 0237.18005)] introduced the tame symbol \(d_{v}\) associated with a discrete valuation \(v\) on a field \(F\). Let \(A_{v}\) be the valuation ring, \(p_{v}\) the unique maximal ideal and \(k_{v}=A_{v}/p_{v}\) the residue field. Then \(d_{v}: F^{*}\times F^{*}\rightarrow k^{*}_{v}\) is defined by the formula:
\[
d_{v}(x,y)=(-1)^{v(x)\cdot v(y)}{\frac{x^{v(y)}}{y^{v(x)}}}\pmod {p_v}.
\]
If \(C\) is defined over a finite field that contains \(m\)-th roots of unity the Hilbert norm residue symbol is defined as:
\[
(f,g)_{p}= ( N_{k(p)/k} [(-1)^{v_{p}(f)\cdot v_{p}(g)}{\frac{f^{v_{p}(g)}}{g^{v_{p}(f)}}}(p) ])^{\frac{q-1}{m}} \in {\mu}_m\, .
\]
In the paper the author gives an algebraic construction, which is used to define local symbols as morphism of schemes. The only input is a closed point on an irreducible, nonsingular, complete curve. As a result the author obtains morphisms of group schemes that generalize the multiplicative local symbol and the Hilbert norm residue symbol. algebraic curve; local symbols; reciprocity laws Group schemes, Symbols, presentations and stability of \(K_2\), Algebraic functions and function fields in algebraic geometry An algebraic-geometric method for constructing generalized local symbols on 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 Author's abstract: ``Runge's method is a tool to figure out integral points on algebraic curves effectively in terms of height. This method has been generalized to varieties of any dimension, but unfortunately the conditions needed to apply it are often too restrictive. We provide a further generalization intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain an explicit finiteness result for integral points on the Siegel
modular variety \(A_2(2)\).''
The paper is closely related to [the author, Algebra Number Theory 14, No. 3, 785--807 (2020; Zbl 1446.11125)]. Runge's method; integral points on varieties; abelian varieties Varieties over global fields, Abelian varieties of dimension \(> 1\), Rational points, Modular and Shimura varieties A tubular variant of Runge's method in all dimensions, with applications to integral points on Siegel modular varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper shows a relation between the tangent spaces of two arcs corresponding under the morphism induced from a morphism of base schemes of the same dimension with the source scheme non-singular.
The difference of the dimension of the tangent spaces at those arcs is described by the vanishing order of the ramification subscheme at the source arc. In particular for a morphism of non-singular varieties, the induced morphism on the arc spaces gives locally analytically closed immersion with the codimension described by that vanishing order. space of arcs; tangent space; formal neighborhood Lawrence Ein & Mircea Mustaţă, ''Generically finite morphisms and formal neighborhoods of arcs'', Geom. Dedicata139 (2009), p. 331-335 Infinitesimal methods in algebraic geometry, Formal neighborhoods in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc. Generically finite morphisms and formal neighborhoods of arcs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X/S\) be a noetherian scheme with a coherent \(\mathcal O_X\)-module \(M\), and \(T_{X/S}\) be the relative tangent sheaf acting on \(M\). We give constructive proofs that sub-schemes \(Y\), with defining ideal \(I_Y\), of points \(x\in X\) where \(\mathcal O_x\) or \(M_x\) is ``bad'', are preserved by \(T_{X/S}\), making certain assumptions on \(X/S\). Here bad means one of the following: \(\mathcal O_x\) is not normal; \(\mathcal O_x\) has high regularity defect; \(\mathcal O_x\) does not satisfy Serre's condition (\(R_n\)); \(\mathcal O_x\) has high complete intersection defect; \(\mathcal O_x\) is not Gorenstein; \(\mathcal O_x\) does not satisfy (\(T_n\)); \(\mathcal O_x\) does not satisfy (\(G_n\)); \(\mathcal O_x\) is not \(n\)-Gorenstein; \(M_x\) is not free; \(M_x\) has high Cohen--Macaulay defect; \(M_x\) does not satisfy Serre's condition (\(S_n\)); \(M_x\) has high type. Kodaira--Spencer kernels for syzygies are described, and we give a general form of the assertion that \(M\) is locally free in certain cases if it can be acted upon by \(T_{X/S}\). Rolf Källström, Preservation of defect sub-schemes by the action of the tangent sheaf, J. Pure Appl. Algebra 203 (2005), no. 1-3, 166 -- 188. Relevant commutative algebra, Derivations and commutative rings Preservation of defect sub-schemes by the action of the tangent sheaf | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 standard determinantal and good determinantal schemes, that is, schemes \(X\subset \mathbb P^{n+c}\) of codimension \(c\) with a homogeneous saturated ideal that can be generated by the maximal minors of a homogeneous \(t\times(t+c-1)\)-matrix, respectively if it, in addition, is a generic complete intersection. More precisely, given integers \(a_0, a_1,\dots, a_{t+c-2}\) and \(b_1, \dots, b_t\), they study the locus \(W(\underline b;\underline a)\subset \text{Hilb}^p(\mathbb P^{n+c})\) and \(W_s(\underline b;\underline a)\) of standard, respectively good, schemes \(X\subseteq \mathbb P^{n+c}\) of codimension \(c\) that are defined by the maximal minors of a \(t\times (t+c-1)\)-matrix \((f_{ij})^{i=1,\dots, t}_{j=0,\dots, t+c-2}\), where \(f_{ij}\in k[x_0,x_1,\dots, x_{n+c}]\) is a homogeneous polynomial of degree \(a_j-b_i\). The authors give an upper bound of the dimension of \(W(\underline b:\underline a)\) and \(W_s(\underline b;\underline a)\) in terms of \(a_j\) and \(b_i\), and conjecture that this bound is sharp. The conjecture is proved for \(2\leq c\leq 5\), and for \(c\geq 6\) under some restrictions on \(a_0, \dots, a_{t+c-2}\) and \(b_1, \dots, b_t\). They also investigate whether the closure of \(W(\underline b;\underline a)\) is an irreducible component of \(\text{Hilb}^p(\mathbb P^{n+c})\), and when \(\text{Hilb}^p(\mathbb P^{n+c})\) is generically smooth along \(W(\underline b;\underline a)\). They give affirmative answers when \(2\leq c\leq 4\) and \(n\geq 2\), and for \(c\geq 5\) under certain numerical assumptions. The results generalize results by \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 423--431 (1975; Zbl 0325.14002)] in codimension \(2\) and of [\textit{J. O. Kleppe, J. C. Migliore, R. Miró-Roig, U. Nagel} and \textit{C. Peterson} [``Gorenstein liaison, complete intersection liaison invariants and unobstructedness''. Mem. Am. Math. Soc. 732 (2001; Zbl 1006.14018)] in codimension 3. The methods are basically those of Ellingsrud [loc. cit.]. standard determinantal scheme; good determinantal scheme; Hilbert scheme Kleppe, J.O.; Miró-Roig, R.M., Dimension of families of determinantal schemes, Trans. am. math. soc., 357, 2871-2907, (2005) Determinantal varieties, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, Projective techniques in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Dimension of families of determinantal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $C$ be an integral proper complex curve with compactified Jacobian $J$. Letting $C^{[n]}$ denote the Hilbert scheme of length $n$ subschemes of $C$, the Abel-Jacobi morphism $\varphi: C^{[n]} \to J$ sends a closed subscheme $Z$ to ${\mathcal I}_Z \otimes {\mathcal O}(x)^{\otimes n}$, where $x \in C$ is a nonsingular point. When $C$ has at worst planar singularities, both $C^{[n]}$ and $J$ are integral schemes with local complete intersection singularities according to \textit{A. B. Altman} et al. [in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 1--12 (1977; Zbl 0415.14014)] and \textit{J. Briancon} et al. [Ann. Sci. Éc. Norm. Supér. (4) 14, 1--25 (1981; Zbl 0463.14001)]. Furthermore $\varphi$ has the structure of a $\mathbb P^{n-g}$-bundle for $n \geq 2g-1$ by work of \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50--112 (1980; Zbl 0427.14015)] so that the rational homology group $H_* (C^{[n]})$ is determined by $H_* (J)$. Recent work of \textit{D. Maulik} and \textit{Z. Yun} [J. Reine Angew. Math. 694, 27--48 (2014; Zbl 1304.14036)] and \textit{L. Migliorini} and \textit{V. Shende} [J. Eur. Math. Soc. (JEMS) 15, No. 6, 2353--2367 (2013; Zbl 1303.14019)] endows $H^* (J)$ with a certain perverse filtration $P$ for which $H^* (C^{[n]})$ can be recovered from the $P$-graded space $\text{gr}_*^P H^* (J)$. \par Motivated by these results and a suggestion of Richard Thomas, the author shows how $H_* (C^{[n]})$ can be recovered from a filtration on $H_* (J)$ using a method not reliant on perverse sheaves. Taking an approach inspired by work of \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)], he defines two pairs of creation and annihilation operators acting on $V(C)=\bigoplus_{n \geq 0} H_* (C^{[n]})$. The first pair $\mu_{\pm} [\text{pt}]$ corresponds to adding or removing a nonsingular point in $C$. The second pair $\mu_{\pm}[C]$ come from the respective projections $p,q$ from the flag Hilbert scheme $C^{[n,n+1]}$ to $C^{[n]}$ and $C^{[n+1]}$, namely $q_* p^{!}$ and $p_* q^{!}$ for appropriate Gysin maps $p^!$ and $q^!$. The main theorem states that the subalgebra of $\text{End} (V(C))$ generated by $\mu_{\pm} [\text{pt}], \mu_{pm}[C]$ is isomorphic to the Weyl algebra $\mathbb Q [x_1, x_2, \partial_1, \partial_2]$ and that the natural map $W \otimes \mathbb Q [\mu_+ [\text{pt}], \mu_+ [C]] \to V(C)$ is an isomorphism, where $W$ is the intersection of the kernels of $\mu_- [\text{pt}]$ and $\mu_- [C]$; moreover the Abel-Jacobi pushforward map $\varphi_*: V(C) \to H_* (J)$ induces an isomorphism $W \cong H_* (J)$. Dual variations for cohomology groups recover and strengthen the results of Maulik-Yun [Zbl 1304.14036] and Migliorini-Shende [Zbl 1303.14019]. locally planar curves; Hilbert scheme; compactified Jacobian; Weyl algebra Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Singularities of curves, local rings Homology of Hilbert schemes of points on a locally planar 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 This thesis focuses on the theory -- still under construction -- of differential schemes. The aim of our work is to provide two new perspectives to this theory.
The first perspective is geometric and consists in considering schemes endowed with vector fields instead of differential rings. In this context, we define what is a leaf and what is the trajectory of a point. With the help of these tools, we reinvest and generalize some results of differential Galois theory. Similarly, we show that the Carrà Ferro sheaf is the natural sheaf of the space of leaves of a scheme with vector field. It is also this approach that lead us to prove that, in the reduced case, the Kovacic and Keigher sheaves are isomorphic and that they have the same constant as the Carrà Ferro sheaf.{
}The second perspective is functorial, and is based on the notion of scheme due to \textit{B. Toën} and \textit{M. Vaquié} [``Au-dessous de Spec \(\mathbb{Z}\)'', Preprint, \url{arXiv:math/0509684}]. We prove that the category of differential schemes in the sense similar to the one of these authors is equivalent to the category of schemes endowed with a vector field. Symbolic computation and algebraic computation, Differential algebra, Schemes and morphisms Differential schemes: geometric approach and functorial approach | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 construct a family of Teichmüller curves in moduli space of genus \(3\), whose closure is a Hurwitz space of dimension \(3\). Such a construction is provided by origamis, that is, finite index subgroups of a free group of rank \(2\) (seen as a group of hyperbolic isometries of hyperbolic plane and quotient being a punctured torus). The construction is very explicit and well written. In the first part of the paper the notion of Teichmüller curves and the corresponding Veech groups (and also origamis) is discussed. They recall the construction, done by the same authors in a previous paper [Math. Nachr. 281, No. 2, 219--237 (2008; Zbl 1159.14012), preprint \url{arXiv:math/0509195}], of a particular origami which provides a one-dimensional family of Riemann surfaces of genus \(3\) admitting a group of conformal automorphisms of order \(16\). This origami produces a Teichmüller curve in the moduli space of genus \(3\). Then, by looking at conformal involutions on it, they are able to construct another origamis (an infinitely countable many of them) with the property that each of these origamis produces a Teichmüller curve that intersects the Teichmüller curve produced by the original one. As they are able to describe the algebraic projective quartics representing the (classes of) Riemann surfaces of genus \(3\) of these new origamis, they obtains a concrete Hurwitz space of dimension \(3\), which turns out to be the closure in moduli space of the union of all these origamis. Teichmüller curves; Veech groups; Hurwitz spaces Herrlich F., Schmithüsen G.: A comb of origami curves in the moduli space M 3 with three dimensional closure. Geometriae dedicata 124, 69--94 (2007) Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), \(G\)-structures A comb of origami curves in the moduli space \(M_{3}\) with three dimensional closure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Usually, moduli spaces in geometry are singular varieties. In order to avoid the difficulties related to their singular nature, the recently invented ``Derived Deformation Theory Program (DDT)'' aims at developing appropriate versions of the (non-abelian) derived functor of the respective moduli functor. Rather than ordinary varieties or schemes, the resulting geometric objects are sought to be ``dg-schemes'', i.e., geometric objects whose algebras of functions are commutative differential graded algebras, and which are considered up to quasi-isomorphisms. While the DDT program appears to be well-established in the formal case, mainly in view of the recent fundamental work of M. Kontsevich, S. Barannikov, V. Hinich, M. Manetti, and others, the structure of global derived moduli spaces is much less understood.
In this vein, the aim of the paper under review is to provide a comprehensive DDT-type construction of the derived Hilbert scheme. In a foregoing paper [cf.: \textit{I. Ciocan-Fontanine} and \textit{M. Kapranov}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 34, 403--440 (2001; Zbl 1050.14042)], the authors have already constructed a derived version of a first global algebro-geometric moduli space, namely of Grothendieck's wellknown ``Quot scheme''. Using a somewhat similar but nevertheless different approach, the authors are now investigating another important global moduli space in the context of the DDT program. While in the usual algebraic geometry, the Hilbert scheme is a particular case of the Quot scheme, the two constructions turn out to diverge considerably when passing to the framework of derived categories.
More precisely, let \(k\) be a field of characteristic zero, \(X\) a smooth projective variety over \(k\), and \({\mathcal O}_X(1)\) a very ample line bundle on \(X\) defining a projective embedding. For a given polynomial \(h\), the authors construct a dg-manifold \(\text{RHilb}^{\text{LCI}}_h(X)\) as the derived version of the classical geometric Hilbert scheme \(\text{Hilb}_h(X)\) of closed subschemes of \(X\) with Hilbert polynomial \(h\) relative to the polarization \({\mathcal O}_X(1)\). However, when the polynomial \(h\) is identically 1, then the derived Hilbert scheme turns out to coincide with the variety \(X\) whereas the derived Quot scheme \(\text{RQuot}({\mathcal O}_X)\) is known to be different from \(X\).
As for applications of these DDT-type constructions, the earlier constructed dg-manifolds RQuot are suitable for describing the derived moduli spaces of vector bundles on a fixed variety \(X\). In contrast, the dg-schemes \(\text{RHilb\,}h(X)\) established here are expected to play a similar rôle with regard to the derived moduli spaces of projective varieties themselves, which the authors corroborate by two striking examples. Namely, they use the explicit structure of the dg-schemes \(\text{RHilb\,}h(X)\) to construct two types of geometric derived moduli spaces:
(1) the derived space of maps \(\text{RMap}(C,Y)\) from a fixed projective scheme \(C\) to a fixed smooth projective variety \(Y\) and
(2) the derived stack of stable degree-\(d\) maps \(R\overline M_{g,n}(Y,d)\) from \(n\)-pointed nodal curves of genus \(g\) to a given smooth projective variety \(Y\). The latter example completes some earlier work of \textit{M. Kontsevich} [in: The moduli spaces of curves, Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)] and others (Behrend-Manin, Fulton-Pandharipande), thereby contributing to the mathematical theory of Gromov-Witten invariants. All in all, this is a very comprehensive paper of fundamental importance in derived moduli theory. The conceptual ingredients and refined techniques for the construction of derived Hilbert schemes include cotangent complexes, Harrison homology, derived moduli of operad algebras, derived schemes of ideals in finite-dimensional commutative algebras, and the theory of algebraic stacks. In spite of its highly advanced character, the exposition is very detailed, systematic and clear. moduli spaces; derived categories; operad algebras; Gromov-Witten invariants Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail M., Derived Hilbert schemes, J. Amer. Math. Soc., 15, 4, 787-815, (2002) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Supervarieties, Derived categories, triangulated categories, Nonabelian homological algebra (category-theoretic aspects) Derived Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth algebraic curve of genus \(g\) which admits a \(k\)-sheeted covering onto a general curve \(C\) of genus \(h > 0\), where \(k\) is an odd prime. In the first section, certain integers \(\varepsilon (h,k)\) and \(\delta (h,k)\) are defined. The authors prove that for \(g \geq \varepsilon(h,k)\), the variety \(W^1_d (X)\) of pencils of degree \(d\) on \(X\) is generically reduced and irreducible with the expected dimension, for all \(d \geq \delta(h,k)\). This result is sharp, in the sense that the authors also show that \(W^1_{\delta (h,k)-1}\) is not irreducible. The authors also discuss the validity of the theorem for \(k=2\) and they discuss sharpness for \(k=3\).
In the second section the authors show that for suitable \(d\) and \(g\), and for a general choice of \(X\) as above, there exists an irreducible component of \(W^1_d (X)\) containing a base-point-free \(g^1_d\) which is not composed with the given \(k\)-sheeted covering. Furthermore, every such component is generically smooth of dimension \(\rho(d,g,1)\) (the Brill-Noether number, which is the expected dimension). algebraic curve; linear series; branched covering; genus; pencils; Brill-Noether number Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Pencils, nets, webs in algebraic geometry, Families, moduli of curves (algebraic) Variety of special linear systems on \(k\)-sheeted coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field, and let \(A\) be a finitely generated associative \(k\)-algebra with unit. For each \(d\geq1\) we define the module scheme mod\(_{A}^{d}\) by mod\(_{A}^{d}\left( R\right) =\text{Hom} _{k\text{-alg}}\left( A,\mathbb{M}_{d}\left( R\right) \right) ,\) where \(\mathbb{M}_{d}\left( R\right) \) is the set of \(d\times d\) matrices with entries in \(R\). In particular, mod\(_{A}^{d}\left( k\right) \) can be identified with \(A\)-module structures on \(k^{d};\) furthermore, mod\(_{A}^{d}\) is an affine scheme, say mod\(_{A}^{d}=\text{Spec}\left( k\left[ \text{mod}_{A}^{d}\right] \right)\). The group scheme \(\text{GL} _{d}\) acts on mod\(_{A}^{d}\) by conjugation on its points -- let \(\mathcal{O} _{M}\) denote the \(\text{GL}_{d}\left( k\right) \)-orbit of a fixed \(M\in\)mod\(_{A}^{d}\left( k\right) .\) One can view \(\mathcal{O}_{M}\) as the \(A\)-module structures on \(k^{d}\) isomorphic to \(M.\) \ Understanding the closure of \(\mathcal{O}_{M}\) \ has proved difficult in general.
For \(N\in\)mod\(_{A}^{d},\) given a \(p\times q\) matrix \(\underline{a}\) with coefficients in \(A\) one can naturally construct a \(pd\times qd\) matrix \(N\left( \underline{a}\right) \). Let \(\mathcal{I}_{M}\subset k\left[ \text{mod}_{A}^{d}\right] \) be the ideal generated by the minors of such matrices of size \(1+\)rk\ \(M\left( \underline{a}\right) ,\) and let \(\mathcal{C}_{M}=\text{Spec}\left( k\left[ \text{mod}_{A} ^{d}\right] /\mathcal{I}_{M}\right) \) -- this is a closed subscheme of mod\(_{A}^{d}\) containing \(\mathcal{\bar{O}}_{M}\) since these minors vanish.
In the work under review, the authors study the properties of this scheme \(\mathcal{C}_{M}.\) Comparisons are made with schemes which arise from quiver representations. As an example, if \(Q\) us an equioriented Dynkin quiver os type \(\mathbb{A}\) then \(\mathcal{C}_{M}=\mathcal{\bar{O}}_{M}\) for \(M\) a representation in rep\(_{Q}^{\mathbf{d}}\left( k\right) \): this is a reformulation of a result from Lakshmibai and Magyar .
Using \(\mathcal{C}_{M}\) instead of the orbit closure allows for a module-theoretic interpretation of the tangent space at some \(N\in\) \(\mathcal{\bar{O}}_{M}.\) This allows for a characterization of the singular locus of \(\mathcal{C}_{M}\) when \(A\) is representation-finite. This is useful when trying to describe the singular locus of \(\mathcal{O}_{M}.\) module schemes; orbit closures; representations of quivers Riedtmann, Christine; Zwara, Grzegorz, Orbit closures and rank schemes, Comment. Math. Helv., 88, 1, 55-84, (2013) Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Representations of quivers and partially ordered sets Orbit closures and rank 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 In [Proc. Lond. Math. Soc. (3) 51, 385--414 (1985; Zbl 0622.12011)], \textit{R. W. K. Odoni} showed that in characteristic 0 the Galois group of the \( n\)-th iterate of the generic polynomial with degree \( d\) is as large as possible. That is, he showed that this Galois group is the \( n\)-th wreath power of the symmetric group \( S_d\). We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R. W. K. Odoni in an unpublished paper. Galois group; density results; iterate of generic polynomials Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Varieties over global fields, Global ground fields in algebraic geometry, Density theorems, Separable extensions, Galois theory, Polynomials in general fields (irreducibility, etc.) Iterates of generic polynomials and generic rational functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Phi:A^m\to A^n\) be a homomorphism of free \(A\) modules, where \(A\) is a complete noetherian local ring with \(\mathbb{Q}\subset A\), and consider the ideal generated by the \((r\times r)\)-minors of \(\Phi\), denoted by \({\mathcal I}_r (\Phi)\). -- \(V_r(\Phi) =\text{Spec} (A/{\mathcal I}_r(\Phi))\) is called the \(r\)-th degeneracy locus of \(\Phi\). The author proves that if this locus is non-empty and \(\text{Spec} A\) is \(d\)-connected for some \(d>(m-r)(n-r)\), then \(V_{r+1}(\Phi)\) is \((d-(m-r) (n-r))\)-connected. With the same techniques a generalization of a similar result given by Fulton and Lazarsfeld in the projective case is also proved. determinantal schemes; \(d\)-connectedness; degeneracy locus Steffen, F, Connectedness theorems for determinantal schemes, J. Algebr. Geom., 8, 169-179, (1999) Determinantal varieties Connectedness theorems for determinantal 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 paper is devoted to widespreading of cubic spline interpolation on Euclidean space to spline interpolation on curved non-Euclidean spaces. This problem is solved by the use of mimic of the acceleration minimizing property which leads to Riemannian cubics as generalization. The method requires the solution of a coupled set of nonlinear boundary value problems that cannot be, in general, integrated explicitly. There is also used the De Casteljau's algorithm leading to generalized Bézier curves. There is provided an iterative algorithm for \(C^2\)-splines on Riemann symmetric spaces and it is proved convergence of linear order. There is demonstrated the algorithm for these geometries \(n\)-sphere, complex projective space and real Grassmannian. De Casteljau; cubic spline; Riemannian symmetric space; Bézier curve Spline approximation, Numerical computation using splines, Numerical interpolation, Differential geometry of symmetric spaces, Local Riemannian geometry, Homogeneous spaces and generalizations A numerical algorithm for \(C^2\)-splines on symmetric 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 We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos-4 sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos-4 sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions. convolution recurrence; generating function; Catalan number; Schröder numbers; Riordan array; Hankel transform; Somos sequence; elliptic curve; orthogonal polynomial Recurrences, Exact enumeration problems, generating functions, Elliptic curves, Matrices of integers, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal 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 [For the entire collection see Zbl 0614.00006.]
Let \(H(d,g)_ S\) be the open subscheme of the Hilbert scheme of curves of degree \(d\) and arithmetic genus g in \({\mathbb{P}}^ 3\) parametrizing smooth irreducible curves. The first author who pointed out the existence of irreducible non reduced components of \(H(d,g)_ S\), was \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)] who found a non reduced component of \(H(14,24)_ S\), the general curve of which lies on a smooth cubic surface in \({\mathbb{P}}^ 3\). Mumford's example has been widely generalized by the author of the present paper in his thesis (``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space'', Preprint no. 5-1981, Univ. Oslo). Among other things it turns out from his analysis that if \(W\subseteq H(d,g)_ S\) is a closed irreducible subset whose general point corresponds to a curve C lying on a smooth cubic surface, W is maximal under this condition and \(d>9\), then W irreducible, non reduced component of \(H(d,g)_ S\) yields \(g\geq 3d-18\) and \(H^ 1({\mathcal J}_ C(3))\neq 0\) (the latter inequality implying that \(g\leq (d^ 2-4)/8.\)
The author conjectures that these necessary conditions are also sufficient for W to be a non reduced component of \(H(d,g)_ S\), and he proves this conjecture in the ranges \(7+(d-2)^ 2/8<g\leq (d^ 2-4)/8\), \(d\geq 18\) and \(-1+(d^ 2-4)/8<g\leq (d^ 2-4)/8\), 17\(\geq d\geq 14\). The proof consists in an interesting analysis of the tangent and obstruction space to the so called Hilbert-flag scheme (parametrizing pairs (curve, surface), the first contained in the latter) in particular for curves lying on surfaces of degree \(s\leq 4.\) space curves; Hilbert scheme; degree; arithmetic genus; obstruction space; Hilbert-flag scheme J. O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves. In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math. (Springer, Berlin, 1987), pp. 181-207. Zbl0631.14022 MR908714 Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry Non-reduced components of the Hilbert scheme of smooth space 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 Aiming for a systematic feature extraction from time series, we introduce the iterated-sums signature over arbitrary commutative semirings. The case of the tropical semiring is a central, and our motivating, example. It leads to features of (real-valued) time series that are not easily available using existing signature-type objects. We demonstrate how the signature extracts chronological aspects of a time series and that its calculation is possible in linear time. We identify quasisymmetric expressions over semirings as the appropriate framework for iterated-sums signatures over semiring-valued time series. time series analysis; time warping; tropical quasisymmetric functions Time series, auto-correlation, regression, etc. in statistics (GARCH), Applications of tropical geometry, Semirings, Signatures and data streams Tropical time series, iterated-sums signatures, and quasisymmetric 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 Für \(z\in\mathbb{C}\), \(\text{Im }z>0\), \(q:=e^{2\pi iz}\), \(g,h\in\{0,1\}\) und \((g,h)\neq(0,0)\) sind die Theta-Nullwerte \(\vartheta_{g,h}\) wie folgt definiert:
\[
\vartheta_{g,h}(z):=q^{(g-1)^ 2/8}(1-(-1)^ h q^{(g/2)}) \prod_{m=1}^ \infty (1-(-1)^ h q^{m+(g/2)}) (1-(- 1)^ h q^{m-(g/2)}).
\]
Sind \(M={{\alpha\;\beta} \choose {\gamma\;\delta}}\in SL_ 2(\mathbb{Z})\) und \(g^*,h^*\in\{0,1\}\) definiert durch die Kongruenz \((g^*,h^*)\equiv(g,h)M^{-1} \pmod 2\), so gilt:
\[
\log \vartheta_{g^*,h^*}(Mz)-\log \vartheta_{g,h}(z) -(1/2)\log(\gamma z+\delta)=(\pi i/4)\sigma(M\mid g,h)
\]
mit \(\sigma(M\mid g,h)\) einer von \(z\) unabhängigen ganzen rationalen Zahl. (Der Zweig von \(\log\vartheta_{g,h}(z)\) ist so festgelegt, daß \(\log \vartheta_{g,h}(i)\) positiv reell ist und als Zweig von \(\log(\gamma z+\delta)\) ist der Hauptzweig zu nehmen. Der Autor gibt für reduzierte hyperbolische Matrizen \(M\in SL_ 2(\mathbb{Z})\) einen Kettenbruchalgorithmus an, mit dem \(\sigma(M\mid g,h)\) effektiv berechnet werden kann. Die Arbeit stützt sich hauptsächlich auf Arbeiten von F. Hirzebruch und D. Zagier, sie ist interessant und rechnerisch ziemlich kompliziert. elliptic modular group; continued fraction algorithm; Theta-Nullwerte Holomorphic modular forms of integral weight, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Theta series; Weil representation; theta correspondences, Theta functions and abelian varieties On the theta-multiplier system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a scheme. A group scheme \(G\) over \(S\) is said to have finite order \(r\) over \(S\) if \(G\) can be written as \(\text{Spec}(A)\), where \(A\) is a sheaf of \(\mathcal{O}_S\)-algebras which is locally free of finite rank \(r\). In [\textit{J. Tate} and \textit{F. Oort}, Ann. Sci. Ec. Norm. Super. (4), No. 3, 1--21 (1970; Zbl 0195.50801)], the authors presented the following result, due to Deligne: A commutative group scheme \(G\) over \(S\) of order \(r\) is annihilated by \(r\). In the paper under review, the author proves a similar result for group schemes over symmetric monoidal categories. Namely, let \((C, \otimes, 1)\) be an abelian, closed, \(\mathbb{C}\)-linear symmetric monoidal category and let \(G\) be an affine commutative group scheme over \(C\) free and of finite rank \(r\). Then \(G\) is annihilated by \(r\). Here, an affine group scheme over \(C\) is a covariant functor from the category of algebras in \(C\) to the category of groups, with the usual compatibility conditions.
It must be said here that the subject of relative algebraic geometry over a symmetric monoidal category is has been developed in the papers [\textit{P. Deligne}, in: The Grothendieck Festschrift, Collect. Artic. in honor of the 60th birthday of A. Grothendieck. Vol II, Prog. Math. No. 87, 111--195 (1990; Zbl 0727.14010); \textit{M. Hakim}, Topos anneles et schemas relatifs. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 64. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0246.14004); \textit{B. Toen} and \textit{M. Vaquie}, J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)]. The current paper contributes another result to this theory. group schemes; symmetric monoidal categories Group schemes, Monoidal categories (= multiplicative categories) [See also 19D23] Affine group schemes over symmetric monoidal 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 Given a torsor \(P\) under a principally polarised abelian scheme \(J\), projective over a Noetherian base, we use the Picard scheme of \(P\) to write down an explicit extension of \(\mathbb{Z}\) by \(J\) giving the class of \(P\). As an application, we give a version of Bhatt's period-index result valid over an arbitrary Noetherian base. abelian scheme; torsor; Picard scheme; polarisation Picard schemes, higher Jacobians, Brauer groups of schemes Torsors under abelian schemes via Picard 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 Viro method of constructing real algebraic varieties with prescribed topology uses convex subdivisions of Newton polyhedra. The authors show that in the case of arbitrary (not necessarily convex) subdivisions of polygons corresponding to \(\mathbb{C} \mathbb{P}^2\) and rational ruled surfaces \(\Sigma_a\), \(a\geq 0\), the Viro method produces pseudo-holomorphic curves. The version of the Viro method discussed in the paper also gives a possibility to construct singular pseudo-holomorphic curves by gluing singular algebraic curves whose collections of singularities do not permit these curves to be glued in the framework of the standard Viro method. As an application, the authors construct a series of singular real pseudo-holomorphic curves in \(\mathbb{C} \mathbb{P}^2\) whose collections of singular points do not occur on known algebraic curves of the same degree. gluing of charts; topology of curves; Viro method; real algebraic varieties; Newton polyhedra; pseudo-holomorphic curves Itenberg I., Shustin E. (2002). Combinatorial patchworking of real pseudo-holomorphic curves. Turkish J. Math. 26(1): 27--51 Topology of real algebraic varieties, Plane and space curves, Topological properties in algebraic geometry, Pseudoholomorphic curves Combinatorial patchworking of real pseudo-holomorphic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article is an overview of the original construction by Nori of the fundamental group scheme as the Galois group of some Tannaka category \(EF(X)\) (the category of essentially finite vector bundles) with a special stress on the correspondence between fiber functors and torsors. Basic definitions and duality theorem in Tannaka categories are stated. A paragraph is devoted to the characteristic 0 case and to a reformulation of Grothendieck's section conjecture in terms of fiber functors on \(EF(X)\). fundamental group; groupoid; fundamental group scheme; tannaka duality; gerbes; torsors Singularities of curves, local rings, Coverings of curves, fundamental group, Group schemes, Affine algebraic groups, hyperalgebra constructions, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Fundamental groupoid 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 The authors present an algorithmic method for computing a set of sampling points in \(\mathbb{K}^n\), where \(\mathbb{K}\) is a real closed field, intersecting nontrivially each connected component of a zero set in \(\mathbb{K}^n\), defined over a computable subring \(D\) of \(\mathbb{K}^n\) by means of a quadratic map \(Q:=\mathbb{K}^n\mapsto\mathbb{K}^k\) and a \(k\)-variate polynomial \(p\) over \(D\). The authors prove that the procedure works in \((dn)^{O(k)}\) arithmetic operations in \(D\), where \(d\) is the degree of \(p\). The process consists of exact symbolic computations in \(D\), involving the extension of \(\mathbb{K}\) by means of infinitesimals. symbolic computation; complexity; semialgebraic set; quadratic map; univariate representation; infinitesimal deformation D. Grigoriev and D. V. Pasechnik, \textit{Polynomial-time computing over quadratic maps I. Sampling in real algebraic sets}, Comput. Complexity, 14 (2005), pp. 20--52. Effectivity, complexity and computational aspects of algebraic geometry, Symbolic computation and algebraic computation, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Semialgebraic sets and related spaces Polynomial-time computing over quadratic maps i: sampling in real 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 It remains an open problem to classify the Hilbert functions of double points in \(\mathbb{P}^2\). Given a valid Hilbert function \(H\) of a zero-dimensional scheme in \(\mathbb{P}^2\), we show how to construct a set of fat points \(Z\subseteq\mathbb{P}^2\) of double and reduced points such that \(H_Z\), the Hilbert function of \(Z\), is the same as \(H\). In other words, we show that any valid Hilbert function \(H\) of a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. For some families of valid
Hilbert functions, we are also able to show that \(H\) is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of line. fat points; star configuration points; Hilbert functions Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Configurations and arrangements of linear subspaces Hilbert functions of schemes of double and reduced 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 A finite group scheme over a field is a group scheme whose coordinate algebra is finite dimensional. The author reviews the structure of finite group schemes with particular interest in simple finite group schemes. A simple finite group scheme over an algebraically closed field is either the étale group scheme associated to a finite simple group or the height one infinitesimal group scheme associated to a simple restricted Lie algebra (with the latter case occurring only if the field has prime characteristic). The author then reviews the classifications of finite simple groups and simple restricted Lie algebras. Finally, the author considers the infinitesimal deformations of a finite simple group scheme or equivalently the second cohomology of the group scheme with coefficients in the adjoint representation (its Lie algebra). The state of knowledge for the various cases is summarized. In particular, previous results of the author for simple Lie algebras of Cartan type are presented showing that the cohomology is non-zero (i.e., the Lie algebra is non-rigid) in contrast to the case of Lie algebras of classical type where in most cases it is zero (i.e., the Lie algebra is rigid). finite group scheme; simple group; infinitesimal group scheme; restricted Lie algebra; infinitesimal deformations; Lie algebra cohomology Viviani, F.: Deformations of simple finite group schemes Group schemes, Modular Lie (super)algebras, Cohomology of Lie (super)algebras, Graded Lie (super)algebras, Finite simple groups and their classification Simple finite group schemes and their infinitesimal deformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is concerned with a functorial presentation of Quillen's (long) homotopy fiber sequence construction in pointed model categories, which a priori is functorial on underlying homotopy categories only.
Section 1 consists of a brief historical overview of the relevant homotopical algebraic notions and a summary of the paper. In Section 2, the authors gather strict diagrammatic presentations of homotopy pullbacks in general model categories, and state the preservation of such along right Quillen functors.
In Section 3, they define a relative category \(h(\mathbb{M})\) of homotopy fiber sequences and a relative category \(l(\mathbb{M})\) of long homotopy fiber sequences in a given pointed model category \(\mathbb{M}\). They then construct an equivalence of the homotopy categories \(\mathrm{Ho}(l(\mathbb{M}))\) and \(\mathrm{Ho}(h(\mathbb{M}))\) to the homotopy category \(\mathrm{Ho}(\mathbb{M}^{\rightarrow})\) of arrows in \(\mathbb{M}\) (Theorem 3.5 and Corollary 3.15). This equivalence is given by restriction of a given (long) homotopy fiber sequence to its tail end on the one hand, and (iterative) homotopy pullback of a given map in \(\mathbb{M}\) along the zero object in suitably functorial manner via Section 2 on the other hand. Corollary 3.16 states that every suitably well-behaved functor between pointed model categories lifts to a relative functor between the respective relative categories of (long) homotopy fiber sequences.
Sections 4 and 5 consider the special case of (long) homotopy fiber sequences for maps between fibrant objects in \(\mathbb{M}\), and reconstruct Quillen's concrete construction of long homotopy fiber sequences in this case (up to isomorphism rather than up to homotopy equivalence only). Theorem 4.10 states that every homotopy fiber sequence in \(\mathbb{M}\) yields an exact sequence of pointed hom-sets. Section 6 is an application to the pointed model categories of unbounded and of positive chain complexes over a unital ring \(R\), and hence rederives the classic long exact sequence of homology groups. Section 7 is a brief outlook regarding potential applications in algebraic geometry. pointed model categories; homotopy fiber sequences; homotopy coherence Localization of categories, calculus of fractions, Homotopical algebra, Quillen model categories, derivators, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.) A new approach to model categorical homotopy fiber sequences | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix a smooth genus \(g\geq 2\) curve \(X\). Here we prove an existence theorem for coherent systems on \(X\) with certain parameters using a specialization argumentand a stronger existence theorem for general genus \(g\) curves recently proved by \textit{M. Teixidor-i-Bigas} [Int. J. Math. 19, No. 4, 449--454 (2008; Zbl 1161.14024), preprint \url{arXiv:math.AG/0606348}]. Vector bundles on curves and their moduli Stable and semistable coherent systems on smooth 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 A short exact sequence \(0\longrightarrow {\mathcal A} \overset {a}\longrightarrow {\mathcal B}\overset {b}\longrightarrow {\mathcal C} \longrightarrow 0\) is called Auslander-Reiten sequence if \(a\) is left almost split and \(b\) is right almost split. In the paper under review the author considered a smooth projective scheme \(X\) of dimension \(d>0\) over the base field together with an indecomposable coherent sheaf \({\mathcal C}\) on \(X\). It was shown that in the category of quasi-coherent sheaves on \(X\) there exists an Auslander-Reiten sequence ending in \({\mathcal C}\). Furthermore, \({\mathcal A}\) is isomorphic to the tensor product of the \((d-1)\)st syzygy of a minimal injective resolution of \({\mathcal C}\) and the dualizing sheaf of \(X\). In particular, the author recovered the well-known result by \textit{I. Reiten} and \textit{M. Van den Bergh} [J. Am. Math. Soc. 15, No. 2, 295--366 (2002; Zbl 0991.18009)] that the category of coherent sheaves on a curve has Auslander-Reiten sequences. Auslander-Reiten theory; sheaves Jørgensen, P, Auslander-Reiten sequences on schemes, Ark. Mat., 44, 97-103, (2006) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Auslander-Reiten sequences on 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 What is meant by ``solving a system of polynomial equations'' has to be made precise, and depending on the information sought for, and the representation of solutions chosen, the efficiency of algorithms can vary a lot. In the multivariate case, two notions of multiplicities can be considered: either the arithmetical notion of multiplicity, the multiplicity being a number, the dimension as a vector space of the local ring or the algebraic notion, where the structure of the local algebra is requested. The main tools existing for this algebraic approach are the closed subspace of differential conditions at the point, Gröbner basis or standard basis, a local analog of Gröbner basis. The main new contribution of the paper is to give precise complexity bounds on the conversion of one of these representations into another (section 4). solving a system of polynomial equations; efficiency of algorithms; multiplicities; complexity bounds M. G. Marinari, H. M. Möller, and T. Mora, On multiplicities in polynomial system solving, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3283 -- 3321. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Multiplicity theory and related topics, Infinitesimal methods in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Numerical computation of solutions to systems of equations On multiplicities in polynomial system solving | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we study some Hermite and (in one-variable) Birkhoff polynomial interpolation problems in which we insert ``weights''. In several variables, this is just Hermite interpolation for weighted polynomials, but in one variable we allow greater generality. Projective techniques in algebraic geometry, Interpolation in approximation theory Weighted Hermite and Birkhoff interpolation for polynomial rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a projective \(K3\) surface and \(S^{[k]}\) be the Hilbert scheme of \(k\) points on \(S\), where \(k \geq 2\). The author proves that any dominant rational map from \(S^{[k]}\) that is not generically finite has a rationally connected image. As an application the author gives an alternative proof of \textit{C. Voisin}'s theorem that any symplectic involution of a projective \(K3\) surface \(S\) acts as the identity on \(\mathrm{CH}_0(S)\) [Doc. Math., J. DMV 17, 851--860 (2012; Zbl 1276.14012)]. Using theorems of Graber-Harris-Starr [\textit{T. Graber} et al., J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)], the author reduces his main theorem to the equivalent formulation that the image of a dominant rational map from \(S^{[k]}\) that is not generically finite is either a point or a uniruled variety. punctual Hilbert schemes; \(K3\) surfaces; rational connectedness 9. H.-Y. Lin, Rational maps from punctual Hilbert schemes of K3 surfaces, preprint (2013); arXiv:1311.0743. Rational and birational maps, \(K3\) surfaces and Enriques surfaces, Cycles and subschemes Rational maps from punctual Hilbert schemes of \(K3\) surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we study \(K\)-theory of spectral schemes by using locally free sheaves. Let us regard the \(K\)-theory as a functor \(K\) on affine spectral schemes. Then, we prove that the group completion \(\Omega B^{\mathcal{G}} (B^{\mathcal{G}}\mathrm{GL})\) represents the sheafification of \(K\) with respect to Zariski (resp. Nisnevich) topology \(\mathcal{G}\), where \(B^{\mathcal{G}}\mathrm{GL}\) is a classifying space of a colimit of affine spectral schemes \(\mathrm{GL}_n\). infinity category; derived algebraic geometry; \(K\)-theory Algebraic \(K\)-theory of spaces, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Categorical algebra, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) A representation for algebraic \(K\)-theory of quasi-coherent modules over affine spectral schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field and let \(A\) be a local integral noetherian \(k\)-algebra of dimension one such that its normalization \(\bar A\) is a discrete valuation ring with residue field \(k\). The authors study the Hilbert scheme of zero-dimensional subschemes of Spec\((A)\). They prove that its connected components \({\mathcal M}_ \tau\) parametrize the ideals of \(A\) of colength \(\tau\). Furthermore \({\mathcal M}_ \tau\) are embedded in a linear subspace \({\mathcal M}\) of a certain Grassmannian. The authors study the partition of \({\mathcal M}\) by its intersection with the Schubert cells. They end the paper investigating the structure of \({\mathcal M}\) in the case of rings \(A\) with monomial semigroups. curve singularities; Hilbert scheme; Spec; Schubert cells Pfister, G., Steenbrink, J.H.M.: Reduced Hilbert schemes for irreducible curve singularities. J. Pure and Appl. Alg.77, 103--116 (1992) Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Grassmannians, Schubert varieties, flag manifolds Reduced Hilbert schemes for irreducible curve 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 \(C^\infty\) one-parameter families of area-preserving mappings of \(\mathbb{R}^2\), \(f : \mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2\). An elementary \(n\)-furcation occurs at a point \((x,\mu)\) such that \(x\) is a periodic point of least period \(p\) for \(f_\mu\), the eigenvalues of \(Df^p_\mu (x)\) are \(n\)-th roots of unity, and some genericity hypotheses hold [\textit{K. R. Meyer}, Trans. Am. Math. Soc. 149, 95-107 (1970; Zbl 0198.429)]. If we set \(q = np\), then \(Df^q_\mu = I\). The authors calculate the multiplicity \(M_n\) of an elementary \(n\)- furcation point as a solution of the system
\[
f^q_\mu(x) - x = 0,\quad \text{det} (Df^q_\mu (x) - I) = 0.\tag{1}
\]
The left hand side of (1) is thought of as a mapping from \(\mathbb{C}^3\) to \(\mathbb{C}^3\) (if \(f\) is not analytic, the Taylor series of \(f\) is truncated at an appropriate level), and multiplicity is in the sense of algebraic geometry. The authors find that
\[
M_1 = 1,\;M_2 = 3,\;M_3 = 8,\quad \text{and} \quad M_n = n^2 + 2 \quad \text{for} \quad n \geq 4.
\]
For the area-preserving Hénon family
\[
f_\mu \left ( \begin{smallmatrix} x \\ y \end{smallmatrix} \right) = \left( \begin{smallmatrix} \mu - y - x^2 \\ x \end{smallmatrix} \right), \tag{2}
\]
the authors show that for each \(q \in \mathbb{N}\), the total multiplicity in \(\mathbb{C}^3\) of all solutions of (1) is \(q2^{q - 1}\). They conclude that the known (real) bifurcation diagram of (2) exhibits all complex periodic points of least period 1 through 4, but this is not true for period 5. bifurcations; area-preserving mappings; Hénon family; multiplicity Local and nonlocal bifurcation theory for dynamical systems, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry The multiplicity of bifurcations for area-preserving maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper generalizes some results of \textit{P.-J. Cahen} [Trans. Am. Math. Soc. 184 (1973), 73-85 (1974; Zbl 0276.13014)] on commutative Noetherian rings to the non-Noetherian case. Let A be a commutative ring and U a quasicompact open set. It is shown that every quasicoherent \({\mathcal O}_ U\)-module F can be represented as \(\tilde M| U\) for some A-module M. Moreover the category of quasicoherent \({\mathcal O}_ U\)-modules is equivalent to the full subcategory of A-modules M such that the natural homomorphism \(M\to Q_ U(M)\) is an isomorphism. The proof uses torsion theoretic methods. quasicoherent modules; torsion theory Torsion modules and ideals in commutative rings, Torsion theory for commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Quasi-coherent modules on quasi-affine 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 paper is a study of a special class of polynomial endomorphisms of \(\mathbb{C}^N\) that are called ``Locally Finite'' (LF) and have nice dynamical properties.
It is proven that such an endomorphism \(F\) may be equivalently defined in three ways:
(1) there is a polynomial in one variable \(p(T)\) such that \(p(F)=0,\) where \(F^n\) is the \(n\)th iteration of \(F;\) (2) \(\sup_{n\geq 0} \deg F^n<\infty;\) (3) \(\dim \text{Span}_{n\geq 0} \;r\circ F^n<\infty\) for each \(r\in\mathbb{C}[x_1,\dots,x_N].\) The special type of the vanishing polynomial \(p(T)\) is found.
Then the unipotent and semisimple LF polynomial automorphisms are defined and it is shown that any LF polynomial automorphism is a composition of uniquely defined commuting LF polynomial automorphisms \(F_s\) and \(F_u,\) where \(F_s\) is semisimple and \(F_u\) is unipotent. The authors prove also that the exponential defines a bijective map from the set of locally nilpotent derivations on the ring \(\mathbb{C}[x_1,\dots,x_N]\) to the set of unipotent polynomial automorphisms of \(\mathbb{C}^N.\)
For \(N=2\) the vanishing polynomial for a LF polynomial automorphism \(F\) of degree \(d\) is found explicitly and the degree of minimal vanishing polynomial is shown to be less than \(d+1.\) polynomial endomorphism; derivations Furter J.-P., Maubach S.: Locally finite polynomial endomorphisms. J. Pure Appl. Algebra 211(2), 445--458 (2007) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Automorphisms, derivations, other operators for Lie algebras and super algebras Locally finite polynomial endomorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 author's paper in C. R. Acad. Sci., Paris, Sér. I 295, 607-610 (1982; Zbl 0518.13016.] - It is shown here that \(k[[X_ 1,...,X_ n]]/(M_ 1,...,M_ r)\), where the \(M_ i\) are monomials, is an image of a complete intersection local ring under a ''Golod homomorphism''. Poincaré series; Golod homomorphism Jörgen Backelin, Monomial ideal residue class rings and iterated Golod maps, Math. Scand. 53 (1983), no. 1, 16 -- 24. (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complete intersections, Formal power series rings Monomial ideal residue class rings and iterated Golod maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The object of this work is to present the status of art of an open problem: to provide an analogue for the cohomology of Shimura curves of the \textit{Y. Ihara}'s lemma [in: Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq. 1973, 161--202 (1975; Zbl 0343.14007)] which holds for modular curves. We will formulate our conjecture and locate it in the more general setting of the congruence subgroup problem. We will exploit the relationship between cohomology of Shimura curves and certain spaces of modular forms to establish some consequences of the conjecture about congruence modules of modular forms and about the problem of raising the level. cohomology of Shimura curves; congruence subgroups; Hecke algebras Arithmetic aspects of modular and Shimura varieties, Congruences for modular and \(p\)-adic modular forms, Modular and Shimura varieties About an analogue of Ihara's lemma for Shimura 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 [For the entire collection see Zbl 0584.00015.]
Let \(C\subset {\mathbb{P}}^ r\) be an integral curve of degree d and arithmetic genus \(p_ a\), whose dualizing sheaf \(\omega\) is invertible, and let Z be a finite closed subscheme of C of degree \(\delta\). The aim of this paper is to give some information on the postulation of Z, by means of the embedding \(Z\subset C\). - More precisely we are interested in finding an upper bound for the least integer \(\sigma\) such that Z imposes independent conditions to the hypersurfaces of degree \(n\geq \sigma -1\). This is equivalent to study the vanishing of \(H^ 1({\mathbb{P}}^ r,I(n))\), where \(I\subset {\mathcal O}_{{\mathbb{P}}^ r}\) is the ideal sheaf of Z as a subscheme of \({\mathbb{P}}^ r\), and is strictly related to the vanishing of \(H^ 1(C,J(n))\), where \(J\subset {\mathcal O}_ C\) is the ideal sheaf of Z as a subscheme of C.
In order to study \(H^ 1(C,J(n))\) we introduce, in {\S} 1, a certain integer \(\tau\), depending on the embedding \(Z\subset C\), with the property that \(0\leq \tau \leq \delta\), and \(\tau =0\) if and only if Z is a (Cartier) divisor on C; and our main result states that \(H^ 1(C,J(n))=0\) whenever \(nd>\delta -\tau +2p_ a-2\), see 1.5.
The remaining four sections are devoted to applications. For example in section \(2\) we give conditions for the vanishing of \(H^ 1({\mathbb{P}}^ r,I(n))\) \((theorem^ 2.1).\) As a consequence we have, that if C is a complete intersection of r-1 hypersurfaces of degrees \(a_ 1,...,a_{r- 1}\) and \(a=\sum a_ i,\quad then\sigma \leq (\delta -\tau)/d+a-r+1\). This improves and generalizes the bound \(\sigma \leq \delta /d+d-1\) well known for plane curves.
In section 3 we consider the special case when J is the conductor sheaf of \({\mathcal O}_ C\). The main result implies, in particular, that if C is a complete intersection as above, then \(\sigma\leq a-r\) (hence the classical result, when \(r=2)\). Moreover when C is a nondegenerate curve in \({\mathbb{P}}^ 3\) we get, that \(\sigma\leq d-1\). All these facts are strictly related with the theory of adjoint hypersurfaces for space curves.
In section 4 we use the results of section 1 to give conditions for Z to be a divisor on C, and to be a complete intersection of C with a hypersurface. For example if C is a complete intersection as the one considered above, Z is a complete intersection of C with a hypersurface of degree \(e>0\) if and only if \(\delta =de\) and \(H^ 1({\mathbb{P}}^ r,I(e+a-r-1))\neq 0.\)
In section 5 we consider the subschemes \(Z\subset {\mathbb{P}}^ 2\) corresponding to homogeneous ideals of the form \({\mathfrak A}={\mathfrak p}_ 1^{e_ 1}\cap...\cap {\mathfrak p}_ s^{e_ s}\), where \({\mathfrak p}_ i\) is the homogeneous ideal corresponding to the point \(P_ i\) of \({\mathbb{P}}^ 2\). We show that if \(s\geq 4\), \(e_ i\geq 2\) and no three of the \(P_ i's\) lie on a line, then \(\sigma\leq \sum e_ i-s+1\). finite closed subscheme of integral curve; postulation; embedding; complete intersection; divisor Projective techniques in algebraic geometry, Special algebraic curves and curves of low genus, Complete intersections, Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the postulation of finite subschemes of projective integral 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 Using an adapted coordinate system on a tangent bundle one gives an elementary and explicit proof of the coding of geodesic flows on a modular surface (it is a quotient of hyperbolic setting by action of \(\text{SL}(2,\mathbb{R})\)). coding; geodesic flows; modular surface Arnoux, P., Le codage du flot géodésique sur la surface modulaire, Enseign. Math. (2), 40, 29-48, (1994) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), Geodesic flows in symplectic geometry and contact geometry, Modular and Shimura varieties, Special surfaces The coding for geodesic flows on a modular 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 Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residue mappings and residues. Basic ideas of our approach are the use of Grothendieck local duality and a transformation law for local cohomology classes. A new tool is devised for efficiency to solve the extended ideal membership problems in local rings. The resulting algorithms are described with an example to illustrate them. An extension of the proposed method to parametric cases is also discussed as an application. Grothendieck point residue mappings; local cohomology Residues for several complex variables, Local cohomology of analytic spaces, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Local cohomology and algebraic geometry An effective method for computing Grothendieck point residue mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known since the work of \textit{C. Peskine} and \textit{L. Szpiro} [Inv. Math. 26, 271-302 (1974; Zbl 0298.14022)] that the intersection of two \(c\)-codimensional arithmetically Cohen-Macaulay linked schemes is a \((c+1)\)-codimensional arithmetically Gorenstein (aG) scheme. In this paper we study under which conditions a 3-dimensional aG scheme comes as such an intersection and we investigate relationships between the graded Betti numbers and Hilbert functions for these schemes. In particular, we show that this property holds in \(\mathbb{P}^3\) for aG schemes on smooth quadric surfaces and, for some Hilbert functions, for aG schemes lying on a smooth surface of minimal degree. arithmetically Gorenstein scheme; intersection of schemes; graded Betti numbers; Hilbert functions Ragusa, A.; Zappalà, G., Properties of 3-codimensional Gorenstein schemes, Comm. Algebra, 29, 1, 303-318, (2001) Low codimension problems in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Properties of 3-codimensional Gorenstein schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A conjecture of \textit{B. Harbourne} [in: Algebraic Geometry, Proc. Conf., Vancouver 1984, CMS Conf. Proc. 6, 95-111 (1986; Zbl 0611.14002)] and \textit{A. Hirschowitz} [J. Reine Angew. Math. 397, 208-213 (1989; Zbl 0686.14013)] implies that \(r\geq 9\) general points of multiplicity \(m\) impose independent conditions to the linear system of curves of degree \(d\) when \(d(d+3)\geq rm(m+1)-2\). In this paper we prove that the conditions are independent provided \(d+2\geq (m+1)(\sqrt {r+1.9}+ \pi/8)\). multiplicity; linear system of curves; degree J. Roé. Linear systems of plane curves with imposed multiple points, Illinois J. Math. 45 (2001), 895-906. Divisors, linear systems, invertible sheaves, Singularities of curves, local rings, Plane and space curves Linear systems of plane curves with imposed multiple points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an abelian variety over an algebraically closed field \(k\). The author considers the functor of germs of formal curves on \(X\); on \(S\)-points, this is \({\mathcal C}X(S) = \text{ Hom}(\text{Spf}k[\![t]\!] \times S, X)\). For any \(k\)-scheme \(Y\), let \(Y[n] = Y\times \text{ Spec}k[t]/t^n\). The main theorem states that \({\mathcal C}X\) is represented by \(\text{Pic}^0(\lim_{\rightarrow n}X[n])\). The proof relies on a characterization of \(\ker (\text{Pic}(Y[n]) \rightarrow \text{ Pic}(Y))\) for a proper scheme \(Y\) and on the identification of \(\text{Hom}(S[n], \text{Pic}(\hat X))\) with \(\text{Pic}(\hat X[n])(S)\). (In fact, the proof of Theorem 3.4 shows that \({\mathcal C}X\) is represented by \(\text{Pic}^0(\lim_{\rightarrow n}\hat X[n])\); there is a missing duality in the last paragraph on page 101.) abelian variety; Picard scheme; formal curve Picard schemes, higher Jacobians, Algebraic theory of abelian varieties The scheme of formal curves on an Abelian variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 extend the notions of generalised Poincarè series and divisorial Poincarè series (of motivic nature) introduced by \textit{A. Campillo, F. Delgado} and \textit{S. M. Gusein-Zade} [Monatsh. Math. 150, No. 3, 193--209 (2007; Zbl 1111.14020)] for complex curve singularities to curves defined over perfect fields, as well as to express then in terms of an embedded resolution of curves.
Let \(k\) be a perfect field, let \(\mathcal V_k\) be the category of reduced quasi-projective schemes of finite type over \(k\), and let \(K_0(\mathcal V_k)\) be the Grothendieck ring of \(\mathcal V_k\). Let \(\mathbb L\) be the class of \(\mathbb A^1_k\) in \(K_0(\mathcal V_k)\); \(\mathbb L\) is a regular element of \(K_0(\mathcal V_k)\). Set \(\mathcal M_k=K_0(\mathcal V_k)_{\mathbb L}\).
Let \(R\) be a two-dimensional regular local ring which contains \(k\), is of finite type over \(k\) and has \(k\) as residue field. Let \(f\in R\setminus\{0\}\) be a non-unit which is analytically reduced, and let \(C\) be the curve defined by \(f\). Let \(f=f_1\cdots f_r\) be the factorization of \(f\) as a product of analytically irreducible elements. The branches \(f_1,\ldots,f_r\) determine pseudo-valuations \(v_1,\ldots,v_r\) of \(R\). For every \(\underline n=(n_1,\ldots,n_r)\in\mathbb N_0^r\) set \(J(\underline n)=\{z\in R\mid v_i(z)\geq n_i\,\text{for }i\in\{1,\ldots,r\}\}\). The generalised Poincarè series given by the multi-index filtration \(J(\underline n)\) is
\[
P_g(t_1,\ldots,t_r;\mathbb L)=\int_{\mathbb PR} \underline t^{\underline v(h)}d\chi_g\in \mathcal M_k[\![\,t_1,\ldots,t_r\,]\!]\eqno(*)
\]
where \(\underline t^{\underline v(h)}=t_1^{v_1(h)}\cdots t_r^{v_r(h)}\) is considered as a function of the projectivization \(\mathbb PR\) of \(R\) with values in \(\mathbb Z[\![\,t_1,\ldots,t_r\,]\!]\) and the lower index \(g\) in \(P_g\) stands for ``generalized'' (we set \(\underline t^{\underline v(h)}=0\) as soon as at least one of the \(v_i(h)\) equals \(\infty\)). (For the definition of the integral in \((*)\) (cf. [\textit{S. M. Gusein-Zade, F. Delgado} and \textit{A. Campillo}, Russ. Math. Surv. 54, No. 3, 634--635 (1999); translation from Usp. Mat. Nauk 54, No. 3, 157--158 (1999; Zbl 0976.32017)]). The notion of a cylindric set \(X\) and the generalized Euler characteristic \(\chi_g(X)\) make also sense in the situation considered in this paper.)
The author considers an embedded resolution of the curve \(C\) and can give an explicit formula for \(P_g\) (cf.\ Th. 4.23). Also, he can give an explicit formula for the generalised divisorial Poincarè series \(P^D_g(t_1,\ldots,t_r;\mathbb L)\) (for the definition of this series cf.\ Def.\ 5.3, for the result cf.\ Th.\ 5.4. In order to get these formulae, the author introduces a semigroup \(Y\) [\,cf. \((\dag)\) on p.\ 208\,] and a surjective homomorphism of semigroups \(\text{Init}:\mathbb PR^*\to Y\).
Reviewer's remark: 1. p.\ 204, 2.7.: read \(p_j\in E_{ij}\) and \(p_{ij}= -1\) if \(p_j \succ p_i\). 2. p.\ 204, 3.1, add ``reduced''. 3. p.\ 205, second paragraph of 3.6: read \(X=\pi_p^{-1}(Y)\). 4. section 3.7: \( P_g(t_1,\ldots,t_r;\mathbb L)\in \mathcal M_{k_R}[[t_1,\ldots,t_r]].\) 5. p.\ 207, l.\ 7. from top: \(\underline n(g)\). 6. The notations \(w_i(\underline n)\) and \(v_j(\underline n)\), which are used in Cor.\ 4.12, are defined in~4.19. curve singularity; Poincaré series; divisorial valuations; motivic integration Moyano-Fernández, J. J.: Generalised Poincaré series and embedded resolution of curves, Monatshefte math. 164, No. 2, 201-224 (2011) Singularities of curves, local rings, Complex singularities Generalised Poincaré series and embedded resolution of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main purpose of this paper is to simulate the results of the curves \(K(p,k,r,n)\) with a JAVA program. It will be shown that these curves constitute fractals.
This work is devoted to a first simulation of the trajectories of the roots of the trinomial equation using a program which will give the appearance of these roots which are discussed in the theory by several researchers and mathematicians. fractal; trinomial curve; JAVA program; trinomial equation Questions of classical algebraic geometry, Software, source code, etc. for problems pertaining to geometry, Fractals, Special algebraic curves and curves of low genus Simulation of trinomial arcs \(K(p, k, r, n)\) | 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 nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class (NP intersection coNP) of problems with ``good characterizations''. This suggests a new algebraic combinatorics viewpoint on complexity theory. This paper focuses on the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the \(n \times n\) grid, together with a theorem of \textit{A. Fink} et al. [Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)], which proved a conjecture of \textit{C. Monical} et al. [Sel. Math., New Ser. 25, No. 5, Paper No. 66, 37 p. (2019; Zbl 1426.05175)]. Schubert polynomials; Newton polytopes; computational complexity Combinatorial aspects of representation theory, Symmetric functions and generalizations, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Computational complexity, Newton polytopes, and Schubert 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 We are going to observe special algebraic Turing machines designed for different assignments of cryptography such as classical symmetric encryption, public key algorithms, problems of secure key exchange, development of hash functions. The security level of related algorithms is based on the discrete logarithm problem (DLP) in Cremona group of free module over finite commutative ring. In the case of symbolic computations with ``sufficiently large number of variables'' the order of generator (base of DLP) is impossible to evaluate and we have ``hidden discrete logarithm problem''. In the case of subgroups of Cremona group DLP is closely connected with the following classical difficult mathematical problems:
(1) solving the system of nonlinear polynomial equations over finite fields and rings,
(2) problem of finding the inverse map of bijective polynomial multivariable map.
The complexity of discrete logarithm problem depends heavily from the choice of base. Generation of good ``pseudorandom'' base guarantees the high complexity of (1) and (2) and security of algorithms based on corresponding DLP. We will use methods of theory of special combinatorial time dependent dynamical systems for the construction of special Turing machines for the generation of the nonlinear DLP bases of large (or hidden) order and small degree. symbolic computations; algebraic transformations over commutative rings; Cremona groups; cryptography; discrete logarithm problem; dynamical systems of large girth; dynamical systems of large cycle indicator Ustimenko, V., Romańczuk, U.: On dynamical systems of large girth or cycle indicator and their applications to multivariate cryptography. In: Artificial Intelligence, Evolutionary Computing and Metaheuristics, In the Footsteps of Alan Turing Series: Studies in Computational Intelligence, vol. 427. Springer, Berlin, June (2012) Cryptography, , Applications to coding theory and cryptography of arithmetic geometry, Symbolic computation and algebraic computation On dynamical systems of large girth or cycle indicator and their applications to multivariate cryptography | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) denote a torsionfree coherent sheaf of rank \(n\), degree \(d\) and \(V \subset H^{0} (E)\) be a subspace of dimension \(k\) on a nodal curve \(X\). We show that for \(k \leq n\) the moduli space of coherent systems \((E,V)\) which are stable for sufficiently large values of a real parameter stabilizes. We study the nonemptiness and properties like irreducibility, smoothness, seminormality for this moduli space \(G_L\). Vector bundles on curves and their moduli Coherent systems on a nodal 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 Inspired by the influential work of \textit{A. Suslin} and \textit{V. Voevodsky} [Invent. Math. 123, No. 1, 61--94 (1996; Zbl 0896.55002)] and \textit{V. Voevodsky, A. Suslin} and \textit{E. M. Friedlander} [``Cycles, transfers, and motivic homology theories'', Ann. Math. Stud. 143 (2000; Zbl 1021.14006)], the article under review introduces a singular homology theory on schemes of finite type over a Dedekind domain and verifies several basic properties.
As an application to class field theory, the author constructs a reciprocity isomorphism from the zeroth integral singular homology and the abelianized modified tame fundamental group for an arithmetic scheme. algebraic cycles; class field theory; arithmetic schemes Schmidt A.: Singular homology of arithmetic schemes. Algebra Number Theory 1(2), 183--222 (2007) Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Class field theory, Motivic cohomology; motivic homotopy theory, Singular homology and cohomology theory Singular homology of arithmetic schemes | 0 |
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