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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{N. Schwartz} [in: Ordered algebraic structures, The 1991 Conrad Conf., 169-202 (1993; Zbl 0827.13008)] gave the following definitions: Let \(A\) be a ring or \(\mathbb{Q}\)-algebra contained in its real closure \(R(A)\), the so-called ring of abstract semialgebraic functions. For \(\alpha\) in the real spectrum \(\text{Sper}(A)\), we denote by \(\rho(\alpha)\) the real closure of the quotient field \(qf(A/\text{supp}(\alpha))\) with respect to the total order specified by \(\alpha\). This makes \(\prod \rho(\alpha)\) a lattice ordered ring with respect to the pointwise lattice order containing \(R(A)\). An \(a \in R(A)\) is piecewise polynomial if the real spectrum \(\text{Sper}(A)\) of \(A\) can be covered by finitely many closed constructible sets \(C_1, \ldots, C_r\) such that for certain \(a_1, \ldots, a_r\in A\) there holds \(a |C_i = a_i\) for all \(i\). Denoting the collection of piecewise polynomial functions by \(PW(A)\) and the collection of elements generated in the sense below by the elements of \(A\), the so-called sup-inf definable functions, by \(L(A)\), one has a chain \(L(A)\subseteq PW(A) \subseteq R(A) \subseteq \prod \rho (\alpha)\) of lattice ordered (sub)rings. A ring \(A\) is Pierce-Birkhoff if \(L(A) = PW(A)\). Defining inductively \(L_0(A) = A\), \(L_{n+1}(A) = L_n(A) [|a |: a \in L_n(A)]\), one has \(L(A) = \bigcup_n L_n (A)\).
The unresolved conjecture these authors made in 1956 [cf. \textit{G. Birkhoff} and \textit{R. S. Pierce}, Anais Acad. Bras. Cic. 28, 41-69 (1956; Zbl 0070.26602)] is that this equality holds for the case \(A=\mathbb{Q}[X_1,\dots,X_n]\), in which indeed \(L(A)\) and \(PW(A)\) assume the meaning the terminology suggests [see \textit{J. J. Madden}, Arch. Math. 53, No. 6, 565-570 (1989; Zbl 0691.14012) or \textit{C. N. Delzell}, Rocky Mt. J. Math. 53, No. 3, 651-668 (1989; Zbl 0715.14047)]. In the paper under review a construction corresponding to the inductive algebraic construction of \(L(A)\) is provided for \(PW(A)\). Interesting corollaries for understanding the Pierce-Birkhoff conjecture are derived. Assume (real) intermediate rings \(A \subset B\subset C\subset PW(A)\). Let \((\text{Spec}B)_{\text{re}}=\{\text{real primes of }B\) Pierce-Birkhoff conjecture; lattice ordered ring; semialgebraic function; piecewise polynomial functions; sup-inf definable; constructible sets Semialgebraic sets and related spaces, Real algebra An algebraic construction of the ring of piecewise polynomial functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Sklyanin algebras are graded complex algebras \(A=\bigoplus_{n\in\mathbb{N}_0} A_n\) which fulfill the following properties: (1) \(A_0=\mathbb{C}\), (2) \(A\) is generated by \(A_1\), and (3) \(A\) has the same graded dimension as the polynomial algebra in \(n\) variables. Next to the polynomial algebra itself, the elliptic (noncommutative) algebras \(O_n({\mathcal E},\tau)\) are of fundamental importance. They depend on an elliptic curve \(\mathcal E\), on a point \(\tau\in{\mathcal E}\), and on \(n\in\mathbb{N}\). In this case \(A_1\) is given as the space of holomorphic sections of a certain theta line bundle of degree \(n\) on \(\mathcal E\). The spaces \(A_k\) are given by the \(k\)-th symmetric power of the space of sections. The twisted multiplication depends on \(\tau\).
The article deals with such kind of algebras. Starting from certain subspaces \(W\) of \(A_1\), graded representations \(L(W)\) of the Sklyanin algebras are considered. In particular the case \(\dim W=1\) is studied. Here \(W\) is fixed by the \(n\) zeros of the defining section for \(W\). Between such representations homomorphisms are studied. The corresponding intertwiners satisfy deformed Plücker relations. They give a deformation of the coordinate ring of the Grassmannian of 2-dimensional planes in \(n\)-dimensional space. quantum algebras; elliptic algebras; Sklyanin algebras; quantum deformations; graded representations Deformations of associative rings, Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Grassmannians, Schubert varieties, flag manifolds Coordinate ring of the quantum Grassmannian and intertwiners for the representations of Sklyanin algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the value at \(-1\) of the characteristic polynomial of the Frobenius \(H^d(\bar{X},\mathbb{Q}_l(d/2))\). They prove the following:
``Theorem 1.5. Let \(X\) be a smooth proper variety of even dimension \(d\) over a finite field \(\mathbb{F}_q\) of characteristic \(p\), and let \(\Phi \in \mathbb{Q}[T]\) be the characteristic polynomial of the Frobenius on \(H^d(X_{\bar{\mathbb{F}}_q},\mathbb{Q}_l(d/2))\). Then \((-2)^{\mathrm{deg}{\Phi}}\Phi(-1)\) is a square or \(p\) times a square in \(\mathbb{Q}\).''
They also prove that \((-2)^{\mathrm{deg}{\Phi}}q^{a(X)}\Phi(-1)\) is a square for a certain integer \(a(X)\) depending on the abstract Hodge numbers of \(X\) in degree \(d\). proper variety over finite field; middle cohomology; Frobenius; characteristic polynomial; Artin-Tate formula A.-S. Elsenhans, J. Jahnel, On the characteristic polynomial of the Frobenius on étale cohomology. Duke Math. J. 164, 2161-2184 (2015) Étale and other Grothendieck topologies and (co)homologies, Finite ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, \(4\)-folds On the characteristic polynomial of the Frobenius on étale cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Among the most interesting invariants one can associate with a link \(\mathcal{L}\subset S^3\) is its HOMFLY polynomial \(P(\mathcal{L},v,s)\in\mathbb{Z}[v^{\pm 1},(s-s^{-1})^{\pm 1}]\). \textit{A. Oblomkov} et al. [Geom. Topol. 22, No. 2, 645--691 (2018; Zbl 1388.14087)] conjectured that this polynomial can be expressed in algebraic geometric terms when \(\mathcal{L}\) is obtained as the intersection of a plane curve singularity \((C,p)\subset\mathbb{C}^2\) with a small sphere centered at \(p\): if \(f=0\) is the local equation of \(C\), its Hilbert scheme \(C_p^{[n]}\) is the algebraic variety whose points are the length \(n\) subschemes of \(C\) supported at \(p\), or, equivalently, the ideals \(I\subset\mathbb{C}[[x,y]]\) containing \(f\) and such that \(\dim C[[x,y]]/I=n\). If \(m:C_p^{[n]} \to\mathbb{Z}\) is the function associating with the ideal \(I\) the minimal number \(m(I)\) of its generators, they conjecture that the generating function \(Z(C,v,s)=\sum_n s^{2n}\int_{C_p^{[n]}}(1-v^2)^{m(I)}d\chi(I)\) coincides, up to a renormalization, with \(P({\mathcal L},v,s)\). In the formula the integral is done with respect to the Euler characteristic measure \(d\chi\). A more refined version of this surprising identity, involving a ``colored'' variant of \(P(\mathcal{L},v,s)\), was conjectured to hold by E. Diaconescu, Z. Hua and Y. Soibelman [\textit{D.-E. Diaconescu} et al., Commun. Number Theory Phys. 6, No. 3, 517--600 (2012; Zbl 1276.14065)].
The seminar will illustrate the techniques used by \textit{D. Maulik} [Invent. Math. 204, No. 3, 787--831 (2016; Zbl 1353.32032)] to prove this conjecture. plane curve singularities; Hilbert scheme; stable pairs. algebraic links; HOMFLY polynomials; skein algebra Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) HOMFLY polynomials from the Hilbert schemes of a planar curve [after D. Maulik, A. Oblomkov, V. Shende, \dots] | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The quantum cohomology algebra of a projective manifold \(X\) is the cohomology of \(X\) endowed with a different algebra structure, which takes into account the geometry of rational curves in \(X\). The aim of this note is to show that this algebra becomes remarkably simple for complete intersections when the dimension is high in comparison with the degree. The main result is:
Theorem. Let \(X\subset \mathbb{P}^{n+r}\) be a smooth complete intersection of degree \((d_1, \dots,d_r)\) and dimension \(n\geq 3\), with \(n\geq 2\sum (d_i-1)-1\). Let \(d=d_1 \dots d_r\) and \(\delta= \sum(d_i-1)\). The quantum cohomology algebra \(H^*(X, \mathbb{Q})\) is the algebra generated by the hyperplane class \(H\) and the primitive cohomology \(H^n (X,\mathbb{Q})_0\), with the relations:
\[
H^{n+1} =d_1^{d_1} \dots d_r^{d_r} H^\delta, \quad H\cdot \alpha=0,
\]
\[
\alpha\cdot \beta= (\alpha\mid \beta) {1\over d} (H^n-d_1^{d_1} \dots d_r^{d_r} H^{\delta-1}),
\]
for \(\alpha,\beta \in H^n (X,\mathbb{Q})_0\). quantum cohomology algebra; geometry of rational curves; complete intersections A. Beauville, Quantum cohomology of complete intersections. \textit{Mat. Fiz. Anal. Geom}. \textbf{2} (1995), 384-398. MR1484335 Zbl 0863.14029 Complete intersections, Quantization in field theory; cohomological methods, Rational and unirational varieties, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Quantum cohomology of complete intersections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a previous article [On subfields of the Hermitian function field, Compos. Math. (to appear)], the authors, together with \textit{C. P. Xing}, considered subfields of the Hermitian function field \(H\) over the finite field \({\mathbb F}_{q^2}\). It is shown that some of these subfields, which were obtained as fixed fields of tame subgroups of the group of automorphisms of \(H\), may be described by using Chebyshev polynomials. (The Chebyshev polynomial \(\Phi_n(T)\) here is the monic polynomial in \({\mathbb Z}[T]\) that expresses \(X^n+X^{-n}\) in terms of the variable \(T=X+X^{-1}\). This polynomial may be obtained from the classical Chebyshev polynomial of the first kind that expresses \(\cos n\theta\) in terms of \(\cos \theta\) by setting \(X=\exp(i\theta)\).) Conversely, the authors use known properties of these subfields to derive special properties of certain Chebyshev polynomials. Hermitian function field; Chebyshev polynomial; finite field Garcia A., Stichtenoth H.: On Chebyshev polynomials and maximal curves. Acta Arith. 90, 301--311 (1999) Curves over finite and local fields, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On Chebyshev polynomials and maximal curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Sigma:=S_N\) denote the symmetric group on \(N\) letters. Then \(\Sigma\) acts on the polynomial ring \(K[X_1,\dots, X_N]\) by permuting the variables. In case when the ideal \(I\) is invariant under the action of \(\Sigma\), then the quotient ring \(A=K[X_1,\dots, X_N]/I\) is \(\Sigma\)-invariant. In this article, we define for such quotient rings \(A\) a generalization of the Gröbner basis, the universal \(\Sigma\)-(Gröbner)-basis. The universal \(\Sigma\)-Gröbner-basis is the computational tool of choice when working with algebraic varieties that are \(\Sigma\)-invariant. In certain cases the universal \(\Sigma\)-Gröbner-basis coincides with the usual Gröbner basis with the total degree reverse lexicographic ordering. We illustrate such a case by explicit computations of Gröbner bases for the ideals defining the singular locus of a class of hypersurfaces \(A\) in \(\mathbb{A}_K^N\) with only isolated singularities. The number of generators of the torsion modules of differentials Torsion\((\Omega_{A/K}^{N-1})\) of these hypersurfaces is \(N!\). Ruth I. Michler, Gröbner bases of symmetric quotients and applications, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000) Springer, Berlin, 2004, pp. 627 -- 637. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities of surfaces or higher-dimensional varieties, Computational aspects of algebraic surfaces Gröbner bases of symmetric quotients and applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book is the second edition of \textit{Yu. I. Manin}'s celebrated and inspiring lectures at the Université de Montréal [Quantum groups and non-commutative geometry. Montréal: Université de Montréal, Centre de Recherches Mathématiques (CRM) (1988; Zbl 0724.17006)].
This new edition has an additional chapter by Raedschelders and Van den Bergh, based on the PhD Thesis of \textit{Th. Raedschelders} [Manin's universal Hopf algebras and highest weight categories. Brussels: Vrije Universiteit Brussel (PhD Thesis) (2017)] and the authors joint works [Adv. Math. 305, 601--660 (2017; Zbl 1405.16044); J. Noncommut. Geom. 11, No. 3, 845--885 (2017; Zbl 1383.16025)]. The new chapter aims to expand upon a discussion in Manin's lectures on the possibility of ``hidden-symmetry'' in algebraic geometry. More precisely, the chapter concerns several aspects of the representation theory of certain universal bialgebras \(\underline{\mathrm{end}}(A)\) and Hopf algebras \(\underline{\mathrm{gl}}(A)\) introduced in Manin's book, where \(A\) is an algebra subject to appropriate conditions. quantum group; quantum matrix space; quadratic Hopf algebra; compact matrix pseudogroup; Yang-Baxter equation; tensor category Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to group theory, Noncommutative algebraic geometry, Hopf algebras and their applications, Bialgebras, Ring-theoretic aspects of quantum groups, Yang-Baxter equations, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Noncommutative geometry in quantum theory Quantum groups and noncommutative geometry. With a contribution by Theo Raedschelders and Michel Van den Bergh | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, to define projective functors, and to construct ``noncommutative resolutions'' of Slodowy slices via translation functors. V. Ginzburg, ''Harish-Chandra bimodules for quantized Slodowy slices,'' Represent. Theory, vol. 13, pp. 236-271, 2009. Coadjoint orbits; nilpotent varieties, Noncommutative algebraic geometry, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Harish-Chandra bimodules for quantized Slodowy slices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ''... the main purpose of this paper is to give a unification of the following two basic theories for coherent sheaves on analytic varieties: (1) a type of cohomology theory in which what we call polynomial growth \((=p.g.\)) conditions on cochains and coverings are involved; and (2) completion theory along subvarieties of a given analytic variety. Our theory is given to affine varieties and their analytic analogues... The main body of this paper is devoted to certain explicit uniform estimations... Our main application is to give an analogue of Theorems A and B of H. Cartan in our unified theory... We will apply such a result to generalize, to varieties with singularities, the theorems of A. Grothendieck on algebraic and analytic de Rham theory... Cohomology theories with p.g. conditions were studied by Deligne-Maltsiniotis [\textit{G. Maltsiniotis}, Astérisque 17, 141-160 (1974; Zbl 0297.14006)] and \textit{M. Cornalba} and \textit{P. Griffiths} [Invent. Math. 28, 11-106 (1975; Zbl 0293.32026)] for locally free sheaves over smooth affine varieties by \({\bar \partial}\)-estimations. The situation in our theory is more general than theirs. Our method, depending on Cousin integrals, differs from theirs...''
Partly because of the complexity of notation and the author's style of exposition, the reviewer is unable to quote even a small sample of the results in the paper. theorem A; theorem B; coherent sheaves; polynomial growth; p.g.; completion theory; affine varieties; de Rham theory; Cousin integrals N. Sasakura , Cohomology with polynomial growth and completion theory . Publ. Res. Inst. Math. Sci. Kyoto Univ. 17 ( 1981 ), 371 - 552 . Article | MR 642649 | Zbl 0561.32005 Analytic sheaves and cohomology groups, de Rham cohomology and algebraic geometry, Stein spaces, Sheaves and cohomology of sections of holomorphic vector bundles, general results Cohomology with polynomial growth and completion theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The necessity to process data which live in nonlinear geometries (e.g. capture data, unit vectors, subspace, positive definite matrices) has led to some recent development in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and \(C^1\)- and \(C^2\)-smoothness of subdivision schemes which operate in the matrix groups or general Lie groups, and which are defined by the so-called log-exponential analogy. It is shown that a large class of such schemes has essentially the same smoothness as the linear schemes derived from them.
The bibliography contains 17 sources. Lie group; matrix group; subdivision scheme; smoothness properties; nonlinear multiscale representation; convergence; log-exponential analogy P. Grohs and J. Wallner, \textit{Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties}, in Approximation Theory XII, M. Neamtu and L. L. Schumaker, eds., Nashboro Press, Nashville, TN, 2008, pp. 181--190. Numerical aspects of computer graphics, image analysis, and computational geometry, Representations of Lie and linear algebraic groups over real fields: analytic methods, Group actions on varieties or schemes (quotients), Group actions on affine varieties, Associated Lie structures for groups, Analysis on real and complex Lie groups Log-exponential analogues of univariate subdivisional schemes in Lie groups and their smoothness properties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 convergence condition for adelic zeta functions associated to irreducible regular prehomogeneous vector spaces is given and applied to fundamental regular prehomogeneous vector spaces. Let \((G, X)\) be a regular irreducible prehomogeneous vector space defined on an algebraic number field \(k\) and let \(D\subset G\) be the derived subgroup of \(G\). We denote by \(f(x)\) the irreducible relatively invariant polynomial, i.e., \(f(g \cdot x)= \nu(g) f(x)\) for all \(g\in G\). We denote \(X_A\) the adelization of \(X\) and \(X_k\) the \(k\)-rational points in \(X\). It is a fundamental problem to see whether the adelic integral \(I' (\Phi)= \int_{D_A/ D_k} (\sum_{\xi\in Y_k} \Phi (g\cdot \xi)) \mu_D (g)\) is convergent or not. Here, \(D_A\) is the adelization of \(D\); \(D_k\) is the \(k\)-rational points in \(D\); \(Y_k= \{x\in X_k\); \(f(x)\neq 0\}\); \(\mu_D\) is the Haar measure on \(D\); \(\Phi\in {\mathcal S} (X_A)\) where \({\mathcal S} (X_A)\) is the Schwartz-Bruhat space of \(X_A\).
The author of this paper proves that if the generic isotropy subgroup \(H_0^1\) \((=\{ g\in D\); \(g\cdot \xi= \xi\}\) with \(\xi\in Y_k)\) is connected semi-simple, then \(I' (\Phi)\) is absolutely convergent for every \(\Phi\in {\mathcal S} (X_A)\), and this convergence condition is inherited under castling transform. convergence; adelic zeta functions; prehomogeneous vector spaces; Schwartz-Bruhat space DOI: 10.2307/2374924 Other Dirichlet series and zeta functions, Homogeneous spaces and generalizations, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On the convergence of the adelic zeta functions associated to irreducible regular prehomogeneous vector spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a finite Borel measure \(\mu\) on \(\mathbb{R}^n\) and basic semialgebraic sets \(\Omega_i \subset \mathbb{R}^n\), \(i = 1, \dots, p\), we provide a systematic numerical scheme to approximate as closely as desired \(\mu (\cup_i \Omega_i)\), when all moments of \(\mu\) are available (and finite). More precisely, we provide a hierarchy of semidefinite programs whose associated sequence of optimal values is monotone and converges to the desired value from above. The same methodology applied to the complement \(\mathbb{R}^n \smallsetminus (\cup_i \Omega_i)\) provides a monotone sequence that converges to the desired value from below. When \(\mu\) is the Lebesgue measure, we assume that \(\Omega := \cup_i \Omega_i\) is compact and contained in a known box \(\mathbf{B} := [ - a, a ]^n\), and in this case the complement is taken to be \(\mathbf{B} \smallsetminus \Omega\). In fact, not only \(\mu (\Omega)\) but also every finite vector of moments of \(\mu_{\Omega} \) (the restriction of \(\mu\) on \(\Omega)\) can be approximated as closely as desired and so permits to approximate the integral on \(\Omega\) of any given polynomial. Lebesgue and Gaussian measures; semialgebraic sets; moment problem and sums of squares; semidefinite relaxations; convex optimization Semidefinite programming, Nonconvex programming, global optimization, Semialgebraic sets and related spaces, Length, area, volume, other geometric measure theory Semidefinite relaxations for Lebesgue and Gaussian measures of unions of basic semialgebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a finite partition \(\Delta\) of \(\mathbb R^{n}\) into semialgebraic cells, the authors consider the ring \(S^{\mu}(\Delta)\) of the \(C^{\mu}\)-multivariate splines (where \(\mu\) is a positive integer), that is, piecewise (with respect to \(\Delta\)) polynomial functions, whose parts are glued in a \(C^{\mu}\) manner. A subset of \(\mathbb R^{n}\) is then called \(C^{\mu}\) piecewise semialgebraic, if it is a finite union of sets of the form \(\{x\in \mathbb R^{n}:P(x)=0,\) \(Q_{i}(x)>0,\) \(i=1,\dots,m\},\) where \(m\in N\) and \(P,Q_{i}\in S^{\mu}(\Delta)\). Obviously, every such set is in particular semialgebraic. It is shown that the class of \(C^{\mu}\) piecewise semialgebraic sets (notion depending on the partition, but not on the choice of coordinates) satisfies a Tarski-Seidenberg principle (stability under projection \(\pi:\mathbb R^{n+1}\rightarrow \mathbb R^{n}\)) if the partition \(\Delta\) satisfies some conditions (namely, it is hyper-rectangular and the set \(A\in \mathbb R^{n+1}\) is generated by a sequence of splines \(\{p_{1},\dots,p_{m}\}\) satisfying a degree-condition with respect to the variable \(x_{n+1}\)). Dimension for \(C^{\mu}\)-piecewise semialgebraic sets is also defined. semialgebraic set; multivariate spline; Tarski-Seidenberg principle Zhu, C. G., Wang, R. H.: Piecewise semialgebraic sets. J. Comput. Math., 23(5), 503--512 (2005) Semialgebraic sets and related spaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Numerical smoothing, curve fitting, Approximation by arbitrary nonlinear expressions; widths and entropy Piecewise semialgebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure \(\mu(\mathbf{\Omega})\) of (not necessarily compact) basic semi-algebraic sets \(\mathbf{\Omega} \subset \mathbb{R}^n\). We obtain two monotone (non-increasing and non-decreasing) sequences of upper and lower bounds \((\overline{\omega}_d)\), \((\underline{\omega}_d)\), \(d \in \mathbb{N}\), each converging to \(\mu(\mathbf{\Omega})\) as \(d \rightarrow \infty\). For each \(d\), computing \(\overline{\omega}_d\) or \(\underline{\omega}_d\) reduces to solving a semidefinite program whose size increases with \(d\). Some preliminary (small dimension) computational experiments are encouraging and illustrate the potential of the method. The method also works for any measure whose moments are known and which satisfies Carleman's condition. computational geometry; statistics; probability; Gaussian measure; moments; semi-algebraic sets; semidefinite programming; semidefinite relaxations Length, area, volume, other geometric measure theory, Semialgebraic sets and related spaces, Semidefinite programming, , Numerical mathematical programming methods Computing Gaussian \& exponential measures of semi-algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Subdivision-based algorithms recursively subdivide an input region until the smaller subregions can be processed. It is a challenge to analyze the complexity of such algorithms because the work they perform is not uniform across the input region. Continuous amortization was introduced in [\textit{M. A. Burr} et al., Continuous amortization: a non-probabilistic adaptive analysis technique. Techn. Rep. TR09-136, Electronic Colloquium on Computational Complexity (2009)] as a way to bound the complexity of subdivision-based algorithms. The main features of this new technique are that (1) the technique can be applied, uniformly, to a variety of subdivision-based algorithms, (2) the technique considers a function directly related to the subdivision-based algorithm under consideration, and (3) the output of the technique is often explicitly expressed in terms of the intrinsic complexity of the problem instance.
In this paper, the theory of continuous amortization is generalized and applied in several directions. The theory is generalized (1) to allow the domain to be higher dimensional or an abstract measure space, (2) to allow more general subdivisions than bisections, and (3) to bound the value of general functions on the regions of the final partition. The theory is applied to seven examples of subdivision-based algorithms. These applications include (1) bounding the number of subdivisions performed by algorithms for isolating real and complex roots of polynomials, (2) bounding the bit-complexity of subdivision-based algorithms for isolating the real roots of polynomials, and (3) bounding the expected run-time of an algorithm for approximating a biased coin. In each of these applications, by using continuous amortization, we achieve or improve the best-known complexity bounds. continuous amortization; subdivision algorithms; bisection algorithms; recursion tree; bit-complexity Burr, M.A., Continuous amortization and extensions: with applications to bisection-based root isolation, J. symb. comput., 77, 78-126, (2016) Symbolic computation and algebraic computation, Effectivity, complexity and computational aspects of algebraic geometry, Analysis of algorithms and problem complexity Continuous amortization and extensions: with applications to bisection-based root isolation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R=k[x_1,\dots,x_n]\) denote a polynomial ring and let \(h:\mathbb{N} \to\mathbb{N}\) be a numerical function. Consider the set of all graded Artin level quotients \(A=R/I\) having Hilbert function \(\underline h\). This set (if nonempty) is naturally in bijection with the closed points of a quasiprojective scheme \({\mathfrak L}^\circ (\underline h)\). The object of this note is to prove some specific geometric properties of these schemes, especially for \(n=2\). The case of Gorenstein Hilbert functions (i.e., where \(A\) has type 1) has been extensively studied, and several qualitative and quantitative results are known. Our results should be seen as generalizing some of them to the non-Gorenstein case. We derive an expression for the tangent space to a point of \({\mathfrak L}^\circ(\underline h)\). In the case \(n=2\), we give a geometric description of a point of \({\mathfrak L}^\circ (\underline h)\) in terms of secant planes to the rational normal curve, which generalizes the one just given for \(t=1\). We relate this description to Waring's problem for systems of algebraic forms and solve the problem for \(n=2\). In the last section we prove a projective normality theorem for a class of schemes \({\mathfrak L}(i,r)\) using spectral sequence techniques. The results are largely independent of each other, so they may be read separately. DOI: 10.1307/mmj/1049832900 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Classical problems, Schubert calculus On parameter spaces for Artin level algebras. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 concerns the openness, the Hölder metric (sub) regularity, and the Hölder continuity properties of set-valued maps between finite dimensional vector spaces. The authors offer a new approach that allows to derive relations between the openness, the Hölder metric regularity, and the Hölder continuity properties for the class of semialgebraic maps. In this paper the authors, assuming F is a semialgebraic set-valued map with a closed graph, show that F is Hölder metrically subregular and that the following conditions are equivalent: (i) the map F is open; (ii) the map F is Holder metrically regular; (iii) the inverse map of F is lower pseudo-Hölder continuous; (iv) the inverse map of F is lower pseudo-Hölder continuous. As an application of this result, the authors obtain the following result in the context of semialgebraic variational inequalities: if the solution map (as a map of the parameter vector) is lower semicontinuous, then the solution map is finite and pseudo-Hölder continuous. Finally, several remarks/examples are provided. openness; Hölder metric (sub)regularity; Hölder continuity; Łojasiewicz inequality; variational inequality; semialgebraicity Sensitivity, stability, parametric optimization, Variational inequalities, Nonsmooth analysis, Set-valued and variational analysis, Semialgebraic sets and related spaces Openness, Hölder metric regularity, and Hölder continuity properties of semialgebraic set-valued 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 main new results in the dissertation under review are geometric characterizations of Dirichlet regularity of boundary points for a large class of domains in \({\mathbb R}^N\), where \(N\geq2\). The domains under consideration are more general than the bounded semialgebraic domains but have similar geometric properties. (A semialgebraic set in \({\mathbb R}^N\) is a finite union of sets of the form \(\{x\colon\;f(x)=0, g_1(x)>0,\dots,g_k(x)>0\}\) where \(f,g_1,\dots,g_k\) are polynomials.) We shall call the domains that the author considers admissible -- their definition is too elaborate to explain in this review.
A boundary point \(x\) of an admissible domain \(G\) is called accessible if \(x\notin(\overline G)^\circ\). Let \(x\) be an accessible boundary point of an admissible domain \(G\) in \({\mathbb R}^N\). It is shown that \(x\) must be a regular boundary point of \(G\) in the cases \(N=2,3\) but \(x\) need not be regular in the case \(N\geq4\). (This reflects the fact that the classical Lebesgue spin in \({\mathbb R}^3\) has an exponentially sharp spike but its counterpart in higher dimensions need only be polynomially sharp.) In the case \(N\geq4\) a necessary and sufficient geometric condition is given for \(x\) to be a regular boundary point of \((\overline G)^\circ\); the condition is a modification of the classical (sufficient but not necessary) exterior cone condition. The proofs of these results rely upon the Wiener criterion; much calculation is required for the estimates of capacity.
The regularity results for accessible boundary points are extended to boundary points at which the local dimension of \({\mathbb R}^N\setminus G\) is at least \(N-1\). Boundary value and inverse problems for harmonic functions in higher dimensions, Semialgebraic sets and related spaces, Boundary values of solutions to elliptic equations and elliptic systems Dirichlet singularities on polynomially bounded \(o\)-minimal structures on \(\mathbb R\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Maps \(f:\mathbb{C}[[x_1,\dots, x_n]]^p\to\mathbb{C}[[x_1,\dots, x_n]]^q\) between products of complex power series rings, formal or convergent, appear naturally in singularity theory and local analytic geometry, and are the subject of the article under review. It splits into two parts. The first one develops the algebraic and analytic machinery necessary for working with some classes of such maps. The tactile maps, defined by substitution in a vector of formal power series \(g\in k[[y_1,\dots, y_p]]^q\), and the more general textile maps, each component of which being a polynomial in the coefficients of the power series in the argument, are the main examples (over a field \(k\) of characteristic \(0\)). To them are associated their zero sets, the felts, a typical example being the arc space of an affine variety. Then are obtained infinite-dimensional generalizations of the of the inverse function theorem and rank theorem for differentiable or analytic maps, in the case of textile maps. They are presented in a purely algebraic setting, extending the case of tactile maps established by \textit{H. Hauser} and \textit{G. Müller} [Publ. Math., Inst. Hautes Étud. Sci. 80, 95--115 (1995; Zbl 0831.58008)], and allowing textile maps as automorphisms. The Rank Theorem is extended furthermore in two directions -- for families (parametric rank theorem), and for spaces of power series with coefficients in a test ring.
The second part of the paper demonstrates the universality of the rank theorem. There are proved, as direct corollaries or special cases of the rank theorem, six important results from arc spaces, local analytic geometry, polynomial differential equations and Cauchy-Riemann manifolds. All proofs are based on the principle of linearization: the problem is formulated using maps between spaces of power series, and its solution is obtained by linearizing the map at a point. In this way the proof of the fibration theorem in arc spaces of \textit{J. Denef} and \textit{F. Loeser} [Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004)] gives explicitly the trivializing map. The local factorization theorem of \textit{M. Grinberg} and \textit{D. Kazhdan} [Geom. Funct. Anal. 10, No. 3, 543--555 (2000; Zbl 0966.14002)] and \textit{V. Drinfeld} [On the Grinberg-Kazhdan formal arc theorem, \url{arXiv:math.AG/0203263}], describing the formal neighborhood of a \(k\)-arc is proved in the case of hypersurfaces as a consequence of the rank theorem (this restriction is rather for keeping the notation simpler). The next result is on systems of polynomial ordinary differential equations, studied in the works of \textit{J. Denef} and \textit{L. Lipshitz} [Math. Ann. 267, 213--238 (1984; Zbl 0518.12015)] and \textit{R. Winkel} [On the coefficients of power series solutions for polynomial vector fields, preprint]. \textit{J. C. Tougeron's} implicit function theorem [Ann. Inst. Fourier 18, No. 1, 177--240 (1968; Zbl 0188.45102)] is proved as a special case of the rank theorem. \textit{J. J. Wavrik's} approximation theorem [Math. Ann. 216, 127--142 (1975; Zbl 0303.32018)], obtained by M. Artin for polynomials and by J. Wavrik for power series, is proved here in a particular case, but the proof requires an index much lower than the one in the Wavrik's paper. Finally, \textit{B. Lamel} and \textit{N. Mir's} inversion theorem [J. Am. Math. Soc. 20, No. 2, 519--572 (2007; Zbl 1112.32017)], in its formal version is a special instance of the rank theorem. In the convergent case the same reasoning could be applied, referring to the rank theorem from (Hauser-Müller).
The article is concluded with a set of unsolved problems and questions related to the applications of linearization principle. textile map; tactile map; rank theorem; scission; linearization principle Drnovšek, B.D., Forstnerič, F.: Holomorphic curves in complex spaces. Duke Math. J. \textbf{139}(2), 203-253 (2007). MR 2352132 (2008g:32019) Infinitesimal methods in algebraic geometry, Analytic algebras and generalizations, preparation theorems, Singularities in algebraic geometry Arcs, cords, and felts -- six instances of the linearization principle | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theory of linear systems on graphs and tropical curves, introduced by Baker, has rapidly become an important area of mathematics bridging combinatorics and algebraic geometry. Some striking applications include a recent essentially optimal bound on the number of rational points on algebraic curves, by Katz and Zureick-Brown. The key property of these graphical linear systems (in addition to the fact that they are combinatorial and hence often more easily computed than their classical geometric counterparts) is the Specialization Inequality, which states that rank can only increase when tropicalizing. One thinks of this as a form of upper-semicontinuity, where the classical curve is viewed as the generic fiber of a family and the tropicalization somehow the special fiber. One also has upper-semicontinuity purely on the tropical side, where one has a natural topology on the moduli space of tropical curves (i.e., metric graphs). Fundamental to both these results is that the rank of a tropical linear system, as defined by Baker and Norine, is not the dimension of the obvious \(\mathbb{T}\)-module of sections, but rather is a purely combinatorial definition that they introduce (which coincides with one of the equivalent definitions on the classical side)
Another step in this story is that rather than considering a single divisor/linear system at a time, one can consider an entire moduli space of them. Specifically, there is a graphical/tropical analogue of Brill-Noether loci, and these have been used by Payne and his collaborators to find new tropical proofs of the classical Brill-Noether and Gieseker-Petri theorems on the dimensions and smoothness of these loci. However, the dimensions of these loci do not vary upper-semicontinuously on the moduli space of tropical curves, as they do classically. There is an obvious obstruction given by separating edges, but even after contracting the corresponding loci in the moduli of tropical curves, the present papers finds counterexamples to upper-semicontinuity (and along the way, counterexamples to a conjecture of Luo on renk-determining sets as well). To rectify this, the authors introduce a new notion of dimension for the Brill-Noether loci. As with Baker-Norine rank, their notion is more of a direct combinatorial definition rather than the topological dimension of a \(\mathbb{T}\)-module. Remarkably, this fixes the previous issues as the authors prove a specialization and upper-semicontinuity result for their proffered definition. Surely the notion arrived at in the present paper is the correct one, and it will be interesting to see how it is used in future research, as the saga continues to unfold. tropical curves; metric graphs; linear systems; Brill-Noether; upper-semicontinuity Lim, C.M., Payne, S., Potashnik, N.: A note on Brill--Noether theory and rank determining sets for metric graphs. Int. Math. Res. Not. (2012). doi: 10.1093/imrn/rnr233 , Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Special divisors on curves (gonality, Brill-Noether theory) A note on Brill-Noether theory and rank-determining sets for metric graphs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an integral projective scheme defined over a number field \(F\) and \(v\) be a place of \(F\). Given a generic sequence \((x_n)_{n\geqslant 1}\) of algebraic points of \(X\), the equidistribution problem asks for the conditions under which the sequence of measures \((\eta_n)_{n\geqslant 1}\) defined by the average on the Galois orbit of \(x_n\) in the analytic space \(X_v^{\mathrm{an}}\) (in the sense of Berkovich if \(v\) is finite) converges weakly. In the Arakelov geometry approach of this problem, the appropriate condition is that the sequence \((x_n)_{n\geqslant 1}\) is small with respect to a suitable adelic line bundle \(\overline L\) on \(X\). That is, the sequence of heights \((h_{\overline L}(x_n))_{n\geqslant 1}\) converges to the normalized height of \(X\).
In the article under review, the authors consider a logarithmic variant of the equidistribution problem: given an adelic line bundle \(\overline M\) on \(X\) an a non-zero global section \(s\) of \(M\), does the sequence of integrals \((\int_{X_v^{\mathrm{an}}}\log\|s\|_v\,\mathrm{d}\eta_n)_{n\geqslant 1}\) converge? Note that this does not follow from the weak convergence of \((\eta_n)_{n\geqslant 1}\) since the function \(\log\|s\|_v\) many take the value \(-\infty\) on \(X_v^{\mathrm{an}}\). A conter-example is given in the article to show that the sequence of integrals need not converge in general. The authors prove that, if the normalized height of \(\mathrm{div}(s)\) coincides with that of \(X\), then the logarithmic equidistribution holds. They also establish a similar result for the logarithmic equidistribution of subvarieties of \(X\). Mahler measure; equidistribution; Arakelov geometry; points of small height; arithmetic dynamical system; Berkovich space [13] Antoine Chambert-Loir &aAmaury Thuillier, &Mesures de Mahler et équidistribution logarithmique
nn. Inst. Fourier (Grenoble)59 (2009) no. 3, p. 977-Cedram | &Zbl 1192. Arithmetic varieties and schemes; Arakelov theory; heights, Rigid analytic geometry, Plurisubharmonic functions and generalizations Mahler measures and logarithmic equidistribution | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 achievements of the article are short and elementary proofs of the following uniformization resp. rectilinearization theorem for subanalytic sets (originally due to Hironaka, using resolution of singularities): Let M be a real analytic manifold and \(X\subset M\) a subanalytic subset.
Uniformization theorem: If X is closed then there exists a real analytic manifold N of the same dimension as X and a proper real analytic mapping \(\phi\) : \(N\to M\) such that \(\phi (N)=X.\)
Rectilinearization theorem: If M is of pure dimension m and \(K\subset M\) a compact subset, then there are finitely many real analytic mappings \(\phi_ i: {\mathbb{R}}^ m\to M\) such that: (1) For each i there is a compact subset \(L_ i\subset {\mathbb{R}}^ m\) such that \(\cup \phi_ i(L_ i)\) is a neighbourhood of K in M. (2) For each i, \(\phi_ i^{- 1}(X)\) is a union of quadrants in \({\mathbb{R}}^ m.\)
As the authors say, from the point of view of analysis, these two theorems express the most important aspects of resolution of singularities. On the other hand, the theorems are still far from resolution of singularities since the morphisms are not required to be bimeromorphic. Moreover, they are actually local. Using the fact that each subanalytic set is locally the image of a closed analytic set under a projection, the proof of the uniformization theorem is reduced to the case where \(X\subset M\) is analytic. By considering the product of the defining equations for X one can further reduce to the case of a hypersurface.
The authors give an easy proof of the fact, that any analytic function on an analytic manifold (real or complex) can be transformed to normal crossings by blowings-up. The proof uses only the Weierstrass preparation theorem and elementary but clever induction on dim M. (The induction hypothesis guarantees that the coefficients of the Weierstrass polynomial in question can be transformed by blowings-up to monomials (up to a factor by units) such that the exponents are totally ordered with respect to the induced partial ordering from \({\mathbb{N}}^{m-1}\), \(m=\dim M.)\)
The rectilinearization theorem follows by applying the normal crossing theorem to appropriate distance functions.
In the last two sections the authors apply the uniformization theorem to give a proof of Łojasiewicz's inequality and of the fact (due to Tamm) that the set of smooth points of a subanalytic set is subanalytic. The first three sections of the article consists of a useful selfcontained treatment of semialgebraic resp. semianalytic resp. subanalytic sets.
In addition to the bibliography the referee likes to mention that the first complete treatment of semialgebraic sets (even for real ordered fields) including the triangulation theorem appeared in 1954 in the thesis of H. Brakhage. This thesis was never officially published but a copy is available from Prof. H. Brakhage (Univ. Kaiserslautern). semialgebraic sets; semianalytic sets; uniformization; rectilinearization; subanalytic sets; resolution of singularities Bierstone, E.; Milman, P. D., Semianalytic and subanalytic sets, Publ. Math. Inst. Hautes Études Sci., 67, 5-42, (1988) Semi-analytic sets, subanalytic sets, and generalizations, Analytic algebras and generalizations, preparation theorems, Real-analytic manifolds, real-analytic spaces, Modifications; resolution of singularities (complex-analytic aspects), Real algebraic and real-analytic geometry Semianalytic and subanalytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb F\) be an algebraically closed field of characteristic zero and let \({\mathcal M}_n \;(n\geq 2)\) be the set of all \((n\times n)\)-matrices whose entries are elements of \(\mathbb F\). For an element \(A\in {\mathcal M}_n\) and a positive integer \(j\leq n\), \({\text s}_j(A)\) denotes the sum of all principal minors of size \(j\) of the matrix \(A\). The regular map \(\chi: {\mathcal M}_n\to {\mathbb F}^n\) defined by \(A\mapsto ({\text s}_1(A), {\text s}_2(A), \ldots, {\text s}_n(A))\) is called the characteristic map. For a linear subspace \({\mathcal L}\subseteq {\mathcal M}_n\), we define a subset \({\mathcal S}({\mathcal L})\) as follows:
\[
{\mathcal S}({\mathcal L})=\{ A\in {\mathcal M}_n: \chi(A+{\mathcal L}) \;\text{ is not dense in} \;{\mathbb F}^n\},
\]
which is called a singular set related to the chracteristic map \(\chi\).
The author gives a complete characterization of the subspaces \({\mathcal L}\subseteq {\mathcal M}_2\) such that \(\emptyset\neq {\mathcal S}({\mathcal L})\neq {\mathcal M}_2\). Moreover, a complete characterization of the singular sets \({\mathcal S}({\mathcal L})\) in the case of \(n=2\) is given. Finally, a characterization of the \(n\)-dimensional subspaces \({\mathcal L}\subseteq {\mathcal M}_n\) such that \({\mathcal S}({\mathcal L})=\emptyset\) is obtained by their intersections with conjugacy classes. characteristic map; linear subspace; singular sets Vector spaces, linear dependence, rank, lineability, Eigenvalues, singular values, and eigenvectors, Varieties and morphisms, Classical groups (algebro-geometric aspects), Algebraic systems of matrices On linear subspaces of \(\mathcal {M}_n\) and their singular sets related to the characteristic map | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We investigate the smoothing problem of limit linear series of rank one on an enrichment of the notions of nodal curves and metrized complexes called saturated metrized complexes. We give a finitely verifiable full criterion for smoothability of a limit linear series of rank one on saturated metrized complexes, characterize the space of all such smoothings, and extend the criterion to metrized complexes. As applications, we prove that all limit linear series of rank one are smoothable on saturated metrized complexes corresponding to curves of compact-type, and prove an analogue for saturated metrized complexes of a theorem of Harris and Mumford on the characterization of nodal curves contained in a given gonality stratum. In addition, we give a full combinatorial criterion for smoothable limit linear series of rank one on saturated metrized complexes corresponding to nodal curves whose dual graphs are made of separate loops. limit linear series; metrized complex Smoothing of limit linear series of rank one on saturated metrized complexes of algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate \(A\)-hypergeometric systems \(H_A(\beta)\) with parameter vector \(\beta\in \mathbb{C}^d\), \(d\in \mathbb{N}\). In particular, the formal basic Nilsson solutions of \(H_A(\beta)\) in the direction of a weight vector \(w\in \mathbb{R}^n_{> 0}\) are studied. The \(\mathbb{C}\)-span of the basic Nilsson solutions of \(H_A(\beta)\) in the direction of \(w\) is called the space of formal Nilsson series solutions of \(H_A(\beta)\) in the direction of \(w\) and is denoted by \(\mathcal{N}_w (H_A(\beta))\). The relationship between \(\mathcal{N}_w (H_A(\beta))\) and the solution space of an associated regular holonomic hypergeometric system is explained. A linear map \(\rho\) between the aforementioned spaces is constructed and it is proved that \(\rho\) is injective and describes its image.
Moreover, a combinatorial formula for the dimension of \(\mathcal{N}_w (H_A(\beta))\) over \(\mathbb{C}\) is presented, and it is shown that the logarithmic-free basic Nilsson solutions of \(H_A(\beta)\) in the direction of \(w\) span the vector space of formal logarithmic-free \(A\)-hypergeometric systems in the direction \(w\).
Finally, the convergence of formal Nilsson solutions to \(A\)-hypergeometric systems is analyzed. In particular, an explicit construction for a basis of the space of multivalued holomorphic solutions of \(H_A(\beta)\) is given. \(A\)-hypergeometric functions; irregular holonomic \(D\)-modules; formal Nilsson series; Gröbner degenerations in the Weyl algebra Dickenstein, Alicia; Martínez, Federico N.; Matusevich, Laura Felicia, Nilsson solutions for irregular \(A\)-hypergeometric systems, Rev. Mat. Iberoam., 28, 3, 723-758, (2012) Other hypergeometric functions and integrals in several variables, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Toric varieties, Newton polyhedra, Okounkov bodies, Sheaves of differential operators and their modules, \(D\)-modules Nilsson solutions for irregular A-hypergeometric 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 authors prove two non-intersecting extensions of result due to \textit{A. S. Gorodetski} et al. [Funct. Anal. Appl. 39, No. 1, 21--30 (2005); translation from Funkts. Anal. Prilozh. 39, No. 1, 27--38 (2005; Zbl 1134.37347)]. The first result states the existence of a \(C^2\)-open set of iterated function systems (IFS) with fully supported ergodic measures and only zero Lyapunov exponents. Such measures are constructed as weak-star limits of sequences of measures supported on periodic orbits of the appropriate skew-products. The second result asserts the existence of a \(C^1\)-open set of IFS's having not fully supported ergodic measures with only zero Lyapunov exponents and positive entropy. The proofs rely on the construction of non-hyperbolic measures for the induced IFS's on the flag manifold. Also some related questions are listed. non-hyperbolic measure; fibered Lyapunov exponents; Furstenberg vector; bootstrapping procedure; flag manifold Bochi, J.; Bonatti, C.; Díaz, LJ, Robust vanishing of all Lyapunov exponents for iterated function systems, Math. Z., 276, 469-503, (2014) Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents, Smooth ergodic theory, invariant measures for smooth dynamical systems, Grassmannians, Schubert varieties, flag manifolds Robust vanishing of all Lyapunov exponents for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with a generalization of the construction presented by \textit{M. Fontana} and \textit{K. A. Loper} [Commun. Algebra 36, No. 8, 2917--2922 (2008; Zbl 1152.13003)]. In effect the motivation comes from the construction in [loc. cit.]. The author introduces ultrafilter topology in a more general setting. Starting from a set \(X\) and a collection of subsets \(\mathcal F\) of \(X\), the author defines a topology on \(X\) by utilizing ultrafilters. Then he demonstrates that this kind of topology is always a collection of clopen sets. The author offers several examples and results concerning the fundamental properties of \(\mathcal F\)-ultrafilter topology. Moreover, the author shows that the constructive topology and the ultrafilter topology are the same when \(X\) is the prime spectrum of a ring [loc. cit., Theorem 8]. In Section 3, the last section of the paper, the author gives an application of the general theory developed in Section 2. Among others, he shows the relation between the original topology \(\mathcal F\) and the \(\mathcal F\)-ultrafilter topology, and he also gives a new criterion for deciding when a topological space is spectral. constructible topology; ultrafilters; spectral space; Zariski topology Finocchiaro, Carmelo A., Spectral spaces and ultrafilters, Commun. Algebra, 42, 4, 1496-1508, (2014) Relevant commutative algebra, Special constructions of topological spaces (spaces of ultrafilters, etc.), General commutative ring theory Spectral spaces and ultrafilters | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 theorem of this paper is a uniformization theorem for the (rigid) strongly subanalytic sets introduced by \textit{H. Schoutens} [ibid., 269-295 (1994; see the following review)]. It says that after finitely many local blow-ups, a strongly subanalytic subset of an affinoid manifold becomes semianalytic (we are restricted to fields of characteristic zero here). The proof uses Hironaka's resolution of singularities together with the explicit representation (by quantifier elimiation techniques) of a strongly subanalytic set obtained by Schoutens in the above-cited paper. A strongly subanalytic subset of a reduced affinoid variety \(\text{Sp }A\) may be represented by a finite system of inequalities in absolute value among strongly \(D\)-functions over \(A\). A strongly \(D\)-function over \(A\) is obtained by iterating composition of overconvergent power series over \(A\) with functions \(D(f,g)\), where \(f\) and \(g\) are \(D\)-functions and \(D(a,b) =a/b\) if \(|a,b|\leq 1\), else \(D(a,b)=0\). At the first stage of complexity, a \(D\)-function is given by \(D(f,g)\), where \(f\) and \(g\) are elements of \(A\). After enough local blow-ups, one of \(f\), \(g\) divides the other in a Zariski-open set, thus permitting replacement of the function \(D(f,g)\) by an analytic function. To prove uniformization, one iterates this operation, successively eliminating \(D\)'s in favor of analytic functions, eventually obtaining a semianalytic set. This result is a generation of the uniformization theorem for subanalytic sets over \(\mathbb{Q}_p\) obtained by \textit{J. Denef} and \textit{L. P. D. van den Dries} [Ann. Math., II. Ser. 128, No. 1, 79-138 (1988; Zbl 0693.14012)].
In addition to this, Schoutens shows that a strongly subanalytic set in a (reduced) affinoid variety is the image of a semianalytic set of a rigid analytic variety under a proper analytic map. Thus, Schoutens' class of rigid subanalytic sets is a close analogue of the class of real subanalytic sets. The main idea here is roughly that a semianalytic set of a unit polydisk \({\mathbf B}^n\) defined using overconvergent power series is even semianalytic in the proper variety \({\mathbf P}^n\).
Finally, Schoutens shows that if \(S\) is a strongly subanalytic subset of the rigid analytic variety \(N\) and if \(\varphi:N \to M\) is a proper map into the quasi-compact rigid analytic variety \(M\), then there is some integer \(A\) such that for all \(x\in M\), \(\#(S\cap \varphi^{-1}(x))\in \{1,\dots,A\}\cup \{\infty\}\). uniformization; rigid subanalytic sets Schoutens, H.: Uniformization of rigid subanalytic sets. Compositio math. 94, 227-245 (1994) Non-Archimedean analysis, Semi-analytic sets, subanalytic sets, and generalizations, Local ground fields in algebraic geometry, Real-analytic and semi-analytic sets, Quantifier elimination, model completeness, and related topics Uniformization of rigid subanalytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors consider a problem on the construction of sufficient sets for a space of analytic functions of exponential growth on the algebraic Riemann surface \(R=\{z=(z_ 1,z_ 2)\in\mathbb{C}^ 2\): \(P(z_ 1,z_ 2)=0\}\) that can be defined by a polynomial \(P(z_ 1,z_ 2)\). Let \(H(R)\) be the space of functions analytic on \(R\), with the topology of uniform convergence on compact subsets; let \(V=\{v_ n\}\), \(n=1,2,\dots\), \(v_ 1\leq v_ 2\leq\dots\), be a sequence of weights on \(R\); let \(E=\{f(z)\in H(R)\): \(| f(z)|\leq cv_ m(z)\), \(z\in R\), \(c=c(f)\), \(m=m(f)\}\). If \(S\subset R\) is a set of uniqueness for \(E\), then put \(E(v_ n,S)= \{f\in E\): \(\sup_ S| f(z)|/v_ n(z)=\| f\|_{n,S}<\infty\}\), \(n=1,2,\dots\). The set \(E=\bigcup^ \infty_{n=1} E(v_ n,S)\) is equipped with the inductive limit topology \(\tau_ S\) generated by the normed spaces \(E(v_ n,S)\). Let \({\mathcal K}=\{k(z)\in C(R)\): \(k(z)\geq\varepsilon v_ n(z)\) for all \(n\), \(\varepsilon=\varepsilon(n,k(z))\}\). Then on \(E\) one can define the new topology \(\mu_ S\) by means of seminorms of the type \(\| f\|_{S,k}= \sup_ S(| f(z)|/k(z))\), \(k(z)\in{\mathcal K}\). The set \(S\) is said to be sufficient [weakly sufficient] for \(E\) if \(\mu_ R=\mu_ S\) \([\tau_ R=\tau_ S]\).
Theorem A: If \(S\) is weakly sufficient for the space \(E\), then \(\tau_ S=\mu_ S\) (i.e., the concepts of weakly sufficient and sufficient sets coincide). The authors give a method for local parametrization that allows one to construct discrete sufficient sets on an arbitrary algebraic Riemann surface.
Theorem B. On the Riemann surface \(R\) there always exists a discrete set \(S\) that is weakly sufficient for the space \(E\). Let \(P(z_ 1,z_ 2)\) be an irreducible polynomial. It is well known that the entire function \(u(\zeta_ 1,\zeta_ 2)\) is the solution of a differential equation of the form (1) \(P(\partial/\partial \zeta_ 1,\partial/\partial \zeta_ 2)u=0\) if and only if \(u(\zeta_ 1,\zeta_ 2)\) can be represented in the form of an integral over the characteristic manifold of \(R\): \(u(\zeta)=\int_ R \exp(i\zeta\cdot z)d\mu(z)/ k(z)\), \(\zeta=(\zeta_ 1,\zeta_ 2)\), \(z=(z_ 1,z_ 2)\), where \(\mu\) is a complex measure with bounded variation and \(k(z)\) belongs to the family \({\mathcal K}\).
Theorem C: Each integral solution of the equation (1) has an integral representation of the form \(u(\zeta)=\int_ S \exp(i\zeta\cdot z)d\mu(z)/k_ 1(z)\), where \(\mu\) is a complex measure of bounded variation, \(k_ 1(z)\in{\mathcal K}\). algebraic Riemann surface Compact Riemann surfaces and uniformization, Riemann surfaces; Weierstrass points; gap sequences, Differentials on Riemann surfaces, Ordinary differential equations in the complex domain Concerning sufficient sets on algebraic Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The origin of the problem solved in the present paper is the André-Oort conjecture. Particularly, this conjecture suggests that if we have a ``generic'' sequence of points on a Siegel variety over \(\mathbb Q\), then points of their Galois orbits are uniformly distributed (see below for the exact definition).
The authors prove some versions of this statement considering images under Hecke correspondences instead of Galois orbits. Proofs are given for 2 types of reductive groups \(G\): \(G=\text{GL}_n\) and \(G=\text{GS}p_{2g}\), and various types of Hecke correspondences \(T_{r,N}\), on quotient spaces \(\Gamma\backslash G(\mathbb R)\).
Roughly speaking, the main result is the following. Let \(x\in \Gamma\backslash G(\mathbb R)\) be fixed and \(N\to\infty\). Then points of the set \(T_{r,N}(x)\) tend to be uniformly distributed. This means that if we denote by \(\mu_N(x)\) the normalized measure concentrated on the set \(T_{r,N}(x)\), then \(\mu_N(x)\) tends to the normalized Haar measure \(\mu\) on \(\Gamma\backslash G(\mathbb R)\) while \(N\to\infty\).
Moreover, the velocity of convergence is estimated: for any function \(f\) on \(\Gamma\backslash G(\mathbb R)\) an upper bound of \((\mu_N(x)-\mu)(f)\) is given. For the case \(G=\text{GS}p_{2g}\) this upper bound depends on a Ramanujan constant \(\theta\). Clozel, Laurent; Ullmo, Emmanuel, Équidistribution des points de {H}ecke. Équidistribution des points de {H}eckeContributions to Automorphic Forms, Geometry, and Number Theory, 193-254, (2004) Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Uniform distribution of Hecke 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 We prove the continuity of Weierstrass-Hironaka division of finite order linear differential operators over a complex analytic manifold \(X\) with respect to the induced topology by a canonical one of Fréchet nuclear on the sheaf \({\mathcal D}^\infty_X\). As a consequence, admissible modules over \({\mathcal D}_X^\infty\) and coherent modules over \({\mathcal D}_X\) inherit a canonical locally convex structure and admit finite free resolutions with strict morphisms. This structure allows, as example, to give a topological characterisation of regularity and to prove that the existence of a regular Bernstein-Sato functional equation for a coherent \({\mathcal D}_X\)-module, \({\mathcal M}\), with respect to an arbitrary divisor \(Y \subset X\), implies the comparison theorem \({\mathcal D}_X^\infty \otimes_{{\mathcal D}_X} {\mathcal M} [*Y] \simeq j_* j^{-1} {\mathcal M}^\infty\). continuity; Weierstrass-Hironaka division; finite order linear differential operators; complex analytic manifold Mebkhout, Z.; Narváez-Macarro, L.: Le théorème de continuité de la division dans LES anneaux d'opérateurs différentiels. J. reine angew. Math. 503, 193-236 (1998) Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials The continuity theorem of the division in the ring of differential operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a Noetherian separated base scheme of finite Krull dimension, \(Sch_X\) the category of schemes of finite type over \(X\) and \(Sm_X\) the subcategory of smooth schemes over \(X\) regarded as a site with the Nisnevich topology. Let \({\mathcal M}\) be the category of pointed simplicial presheaves in \(Sm_X\) equipped with motivic Quillen model structure. By \(Spt(M)\) we will denote Jardine's category of symmetric \(T\)-spectra on \({\mathcal M}\), equipped with the motivic model structure, and by \(SH\) its homotopy category. \textit{V. Voevodsky} [in: Open problems in the motivic stable homotopy theory. I: Int. Press Lect. Ser. 3, 3--34, Int. Press, Somerville, Ma (2002; Zbl 1047.14012)] has defined a slice filtration on \(SH\) by means of a family of triangulated subcategories
\[
\cdots\subseteq \Sigma^{q+1}_T SH^{\text{eff}}\subseteq \Sigma^q_T SH^{\text{eff}}\subseteq\Sigma^{q-1}_T SH^{\text{eff}}\subseteq\cdots .
\]
Let \(E\in Spt({\mathcal M})\) be a symmetric \(T\)-spectrum. Then \(E\) is said to be \(n\)-orthogonal if, for all \(K\in\Sigma^n_T SH^{\text{eff}}\)
\[
\Hom_{SH}(K, E)= 0.
\]
This paper contains a proof of an equivalence of categories between the orthogonal components of the slice filtration and the birational motivic stable homotopy categories, which are constructed following the work of \textit{B. Kahn} and \textit{R. Sujatha} [in: Birational Motives I. K-theory Preprint Archives, October 2002, \url{http://www.math.uiuc.edu/K-theory/0596/}]. birational invariants; motivic homotopy theory; motivic spectral sequence; slice filtration Pelaez, P., Birational motivic homotopy theories and the slice filtration, Doc. math., 18, 51-70, (2013) Motivic cohomology; motivic homotopy theory, (Equivariant) Chow groups and rings; motives Birational motivic homotopy theories and the slice filtration | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \((G, V)\) be an irreducible prehomogeneous vector space defined over a number field \(k\), \(P\in k[V]\) a relative invariant polynomial, and \(\chi\) a rational character of \(G\) such that \(P(gx)= \chi(g) P(x)\). Let \(V^{ss}_k= \{x\in V_k\mid P(x)\neq 0\}\). For \(x\in V^{ss}_k\), let \(G_x\) be the stabilizer of \(x\), and \(G^0_x\) the connected component of 1 of \(G_x\). We define \(L_0\) to be the set of \(x\in V^{ss}_k\) such that \(G^0_x\) does not have a non-trivial rational character. Then we define the zeta function for \((G, V)\) by the following integral
\[
Z(\Phi, s)= \int_{G_\mathbb A/ G_k} |\chi(g)|^s \sum_{x\in L_0} \Phi(gx)\,dg,
\]
where \(\Phi\) is a Schwartz-Bruhat function, \(s\) is a complex variable, and \(dg\) is an invariant measure.
In this paper, we prove the convergence of \(Z(\Phi, s)\) for \(\text{Re}(s)\gg 0\) for prehomogeneous vector spaces of the form \((G/\widetilde T, V)\), where \(G\), \(V\) are as follows:
(1) \(G= \text{GL}(2)\times \text{GL}(2)\times \text{GL}(2)\), \(V= k^2\otimes k^2\otimes k^2\),
(2) \(G= \text{GL}(3)\times \text{GL}(3)\times \text{GL}(2)\), \(V= k^3\otimes k^3\otimes k^2\),
(3) \(G= \text{GL}(4)\times \text{GL}(2)\), \(V= \Lambda^2 k^4\otimes k^2\),
(4) \(G= \text{GL}(6)\times \text{GL}(2)\), \(V= \Lambda^2 k^6\otimes k^2\),
and \(\widetilde T= \text{Ker}(G\to \text{GL}(V))\) for all the cases.
Previously, the convergence of the zeta function was known for 23 types of irreducible reduced prehomogeneous vector spaces, most of which is due to the result of \textit{F. Sato} [Tôhoku Math. J., II. Ser. 35, 77--99 (1983; Zbl 0513.14011)], and a few to the result of the author [Lond. Math. Soc. Lect. Note Ser. 183 (Cambridge University Press, Cambridge 1993, Zbl 0801.11021)]. In his recent preprint [On the generalized global zeta functions associated to irreducible regular prehomogeneous vector spaces (Preprint, 1994)], \textit{K. Ying} proves the convergence of the zeta function for a few cases including the cases (1), (3), and he should get the credit for these cases even though our proof is totally different from his. prehomogeneous vector space; zeta function; Schwartz-Bruhat function; convergence Yukie, A.: On the convergence of the zeta function for certain prehomogeneous vector spaces. Nagoya Math. J. 140, 1-31 (1995) Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Prehomogeneous vector spaces, Homogeneous spaces and generalizations, Homogeneous complex manifolds On the convergence of the zeta function for certain prehomogeneous vector spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper the authors consider the problem of stable sampling of multivariate real polynomials defined on an affine real algebraic variety \(M\). Suppose \(H_k(M)\) is a given sequence of Hilbert spaces with reproducing kernels \(K_k(x,y)\). A sequence \(\Lambda_k\) of sets of points on \(M\) is said to be sampling for \(H_k(M)\) if
\[
\frac1{C} \|f\|^2 \leq \sum_{\lambda\in\Lambda_k} \frac{|f(\lambda)|^2}{K_k(\lambda,\lambda)} \leq C\|f\|^2
\]
for all \(f\in H_k(M)\). In the paper \(M\) is a variety cut out by a finite number of polynomials in \(\mathbb R^m\) and \(H_k(M)\) is the space of polynomials of total degree \(k\) restricted to \(M\) and equipped with the \(L^2\) norm
\[
\|p_k\|^2_{L^2(e^{-k\phi}\mu)} := \int_M |p_k|^2 e^{-k\phi} d\mu,
\]
where \(\mu\) is a compactly supported measure on \(M\) and \(\phi\) is a continuous function on \(M\). In the paper it is assumed that the pair \((\mu,\phi)\) (1) satisfies the Bernstein-Markov property, i.e. for any \(\epsilon>0\) there exists a positive constant \(C_\epsilon\) such that \(K_k(x,x) \leq C_\epsilon e^{\epsilon k}\) uniformly on the support of \(\mu\), and (2) has moderate growth, i.e., \(K_{k+1}(x,x) \leq C K_k(x,x)\) on the support of \(\mu\). In Theorem 1 the authors prove that a necessary condition for a sequence \(\Lambda_k\) of sets of points in \(M\) to be sampling for the space \(H_k(M) \) of polynomials of degree at most \(k\), with respect to the weight \(k\phi\) and measure \(\mu\) is that
\[
\liminf_{k\to\infty} \frac1{N_k}\sum_{\lambda\in\Lambda_k} \delta_\lambda \geq \mu_{eq}
\]
in the weak topology on the measures on \(M\), where \(\mu_{eq}\) denotes the normalized equilibrium measure of the weighted measure \((\mu,\phi)\) and \(N_k =\text{dim\,}(H_k(M))\). The paper contains more specific results for sampling on compact real algebraic varieties equipped with a volume form and sampling of multivariate real polynomials on convex domains. R. Berman and J. Ortega-Cerdà, Sampling of real multivariate polynomials and pluripotential theory.arXiv:1509.00956 Plurisubharmonic extremal functions, pluricomplex Green functions, General pluripotential theory, Complex Monge-Ampère operators, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Computational aspects of higher-dimensional varieties, Sampling theory in information and communication theory, Integral representations; canonical kernels (Szegő, Bergman, etc.) Sampling of real multivariate polynomials and pluripotential theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present multimatrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over \(n\) fixed points \(z_i\), \(i = {1},\dots, n\) (generalized Grothendieck's dessins d'enfants) of fixed genus, degree, and ramification profiles at two points \(z_{1}\) and \(z_n\). We sum over all possible ramifications at the other \(n-{2}\) points with a fixed length of the profile at \(z_{2}\) and with a fixed total length of profiles at the remaining \(n-{3}\) points. All these models belong to a class of hypergeometric Hurwitz models and are therefore tau functions of the Kadomtsev-Petviashvili hierarchy. In this case, we can represent the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type \({\mathrm{tr}} M_iM_{i+1}^{-1}\). We describe the technique for evaluating spectral curves of such models, which opens the way for obtaining \( 1/N^{2}\)-expansions of these models using the topological recursion method. These spectral curves turn out to be algebraic. Hurwitz number; random complex matrix; Kadomtsev-Petviashvili hierarchy; matrix chain; bipartite graph; spectral curve Ambjørn, J; Chekov, L, The matrix model for hypergeometric Hurwitz numbers, Theor. Math. Phys., 181, 1486-1498, (2014) Other special orthogonal polynomials and functions, Relationships between surfaces, higher-dimensional varieties, and physics, Hurwitz and Lerch zeta functions A matrix model for hypergeometric Hurwitz numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Basing on results by \textit{P. Tworzewski} and \textit{T. Winiarski} [Ann. Pol. Math. 42, 387--393 (1983; Zbl 0576.32013), Zesz. Nauk. Uniw. Jagielloń. 661, Pr. Mat. 24, 151--153 (1984; Zbl 0551.32008), Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 2, 227--271 (1990; Zbl 0717.32006)] concerning convergence of analytic and algebraic sets (in a local uniform convergence in the family of closed sets in \(\mathbb{C}^{n}\) -- called in the paper the Kuratowski convergence) the authors prove similar results for Nash sets (they are analytic sets which locally are described by algebraic equations) in open sets of \(\mathbb{ C}^{n}\). The main result is the following.
Let \(\Omega \subset \mathbb{C}^{n}\) be an open set and \( A\subset \Omega .\) If \(N_{\nu },\) \(\nu =1,2,\ldots ,\) are Nash subsets of \( \Omega \) such that \(N_{\nu }\rightarrow A\) and \(\deg \left( N_{\nu }^{ \text{ext}}\right) <d\) for \(\nu =1,2,\ldots ,\) then \(A\) is a Nash subset of \(A\) (\(\deg \left( N_{\nu }^{\text{ext}}\right) \) is the degree of a ``smallest in some sense'' algebraic extension of \(N_{\nu }\) to an algebraic set \(N_{\nu }^{\text{ext}}\) in the whole \(\mathbb{C}^{n}).\) Nash set; Kuratowski convergence; minimal structure Denkowski, M. P.; Pierzchała, R., On the Kuratowski convergence of analytic sets, Ann. Polon. Math., 93, 2, 101-112, (2008) Nash functions and manifolds, Real-analytic sets, complex Nash functions On the Kuratowski convergence of analytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to \(\varepsilon\), where \(\varepsilon\) is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator \(\mathcal{H}_\varepsilon\) with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in \(\epsilon \). We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of \(\mathcal{H}_\varepsilon\) are analytic in \(\varepsilon\). This allows us to represent the resolvent of \(\mathcal{H}_\varepsilon\) by a uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of \(\varepsilon\). In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in \(\varepsilon\)-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic \(\varepsilon\)-dependence of the coefficients in the differential expression and vertex conditions. graph; small edge; resolvent; analyticity; Taylor series; approximation Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), Spectrum, resolvent, Real-analytic and semi-analytic sets, Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) Analyticity of resolvents of elliptic operators on quantum graphs with small edges | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In recent years, the asymptotic properties of graded sequences of ideals \(\mathfrak{a}_{\bullet}= (\mathfrak{a}_{m})_{m\geq 1}\) on algebraic varieties have been studied systematically (see for example [\textit{L. Ein} et al., Pure Appl. Math. Q. 1, No. 2, 379--403 (2005; Zbl 1139.14008)] and [Ann. Inst. Fourier 56, No. 6, 1701--1734 (2006; Zbl 1127.14010)]). The main motivating example is the graded sequence \(\mathfrak{a}_{\bullet}^{L}\) associated to a line bundle \(L\), where \(\mathfrak{a}_{k}^{L}\) is the multiplier ideal sheaf associated to the linear system \(|L^{k}|\). Another more analytic example is the graded sequence \(\mathfrak{a}_{\bullet}\) associated to a quasi-psh function \(\varphi\), where \(\mathfrak{a}_{k}\) is the multiplier ideal sheaf associated to the quasi-psh function \(k\cdot\varphi\). It has become more and more evident that in order to study a single linear series, we should study it in an asymptotic way.
In this article, to answer a question raised by M.Păun (Păun's question was motivated by the infinite blow-up process in Siu's analytic approach to the problem of the finite generation of the canonical ring, cf. Section 6.3 of [``Techniques for the analytic proof of the finite generation of the canonical ring'', \url{arXiv:0811.1211}]), the authors study the log canonical threshold of graded sequence.
Recall first that for a single ideal sheaf \(\mathfrak{a}\), the log canonical threshold \(\mathrm{lct}(\mathfrak{a})\), which measures the singularity of the multiplier ideal sheaf, is defined as
\[
\mathrm{lct}(\mathfrak{a})=\inf\limits_{E}\frac{A( \mathrm{ord}_{E} ) }{ \mathrm{ord}_{E} ( \mathfrak{a} ) },
\]
where the infimum is taken over all divisors \(E\) over \(X\) (i.e., all prime divisors on normal varieties \(Y\) that have a proper birational morphism to \(X\)), and \(A( \mathrm{ord}_{E})\) is the log discrepancy of \(E\) over \(X\). For a graded sequence \(\mathfrak{a}_{\bullet}\), one can define the log canonical threshold as
\[
\mathrm{lct}(\mathfrak{a}_{\bullet})=\lim\limits_{m\rightarrow \infty}m\cdot \mathrm{lct}(\mathfrak{a}_{m}) .
\]
One of the most remarkable properties of the above asymptotic definition is that, if \(L\) is a big line bundle on a smooth projective variety, then \(\mathrm{lct} (\mathfrak{a}_{\bullet}^{L})=\infty\) if and only if \(L\) is numerically effective (see [``Valuations and asymptotic invariants for sequences of ideals'' \url{arXiv:1011.3699}] by the same authors).
The main result of the article can be now state as follows. Let \(\mathfrak{a}_{\bullet}\) be a graded sequence of ideals on a smooth variety \(X\). Let \(I\subseteq \mathbb{Z}_{> 0}\) be a subset such that for all \(m\in I\) we have a divisor \(E_{m}\) over \(X\) that computes \(\mathrm{lct}(\mathfrak{a}_{m})\). If \(\{A ( \mathrm{ord}_{E_{m}})\}_{m\in I}\) is bounded, then the set \(\{E_{m}\}_{I}\) is finite. As a direct consequence, the infinite blow-up process in Siu's paper can be ``stopped'' if \(\{A ( \mathrm{ord}_{E_{m}})\}\) is bounded.
This paper looks very nice and interesting. graded sequence of ideals; log canonical threshold Multiplier ideals, Singularities in algebraic geometry A finiteness property of graded sequences of ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The 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 In this article, both the Noether and the Cayley-Bacharach theorems are proved with arbitrary PD multiplicities. The theorems are concerned with a mathematical scheme that is essentially a generalisation of the Lagrange interpolation process; when Lagrange interpolation with polynomials is computed, the approximant is required to match the approximant pointwise at certain data points. The function evaluation at those data points can be generalised to linear operators of various kinds. These can include derivatives and evaluations (leading to Hermite interpolation) or integrals (averages) etc.
In this case, Lagrange interpolants are computed with conditions that are defined via differential equations. If such interpolation (linear) problems are uniquely solvable, they are called \(n\) correct (of certain orders \(n\)). Lagrange interpolants can also be expressed by fundamental functions (``Lagrange functions'') or fundamental polynomials. If a set of operators provides a fundamental function of order \(n\) for each of its elements, it is called \(n\) independent.
The Noether theorem which the authors prove in this context provides a decomposition of polynomials \(f\) of degree \(k\), say, as \(f=Ap+Bq\) where the degrees of \(A\), \(B\), \(p\) and \(q\) are \(k-m\), \(k-n\), \(m\) and \(n\), respectively. The polynomials \(p\) and \(q\) must not have an intersection at \(\infty\) and \(f\) must vanish at \(M_\lambda(p,q)\) for all \(\lambda\) in \(p\) and \(q\). The \(M_\lambda(p,q)\) is the intersection of the multiplicity spaces \(M_\lambda(p)\) and \(M_\lambda(q)\), where \(M_\lambda(p)\) and \(M_\lambda(q)\) are the
\[ M_\lambda(r)=\{h\mid D^\alpha h(D)r(\lambda)=0\forall\alpha\in\mathcal{Z}^2_+\} \]
with \(r=p,q\).
Additionally to the proof of the Noether theorem which is given as an example here, the authors prove the so-called
Cayley-Bacharach theorem in a similar context. polynomial interpolation; \(n\)-independent set; PD multiplicity space; arithmetical multiplicity Interpolation in approximation theory, Multidimensional problems, Plane and space curves On the Noether and the Cayley-Bacharach theorems with PD multiplicities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(p_0, p_1,\dots, p_n\) be \(n+1\) fixed points in \(\mathbb{P}^2\) and let \({\mathcal L}= {\mathcal L}(d, m_0, n, m)\) denote the linear system of plane curves of degree \(d\) with multiplicity \(m_0\) and at \(p_0\) and multiplicity \(m\) at \(p_1,\dots, p_n\). \({\mathcal L}\) is called quasi-homogeneous. Define the virtual dimension of \({\mathcal L}\) to be
\[
v(d,m_0,n,m)= d(d+3)/2- m_0(m_0+1)/2- nm(m_1)/2.
\]
The actual dimension of \({\mathcal L}\) cannot be less than \(-1\) (corresponding to an empty \({\mathcal L}\)) and the authors define the expected dimension of \({\mathcal L}\) to be
\[
e= e(d,m_0,n,m)= \max \{-1, v(d,m_0,n,m)\}.
\]
As the points \(p_i\) vary in \(\mathbb{P}^2\) the dimension of \({\mathcal L}\) is upper semicontinuous and therefore it assumes a minimum value that the authors call the dimension of \({\mathcal L}\) and denote by \(\ell= \ell(d,m_0,n,m)\). Of course \(\ell\geq e\) and equality implies that the conditions imposed by the points \(p_i\) are independent. If \(\ell=e\) the linear system \({\mathcal L}\) is called nonspecial and if \(\ell>e\) the linear system \({\mathcal L}\) is called special. Here the authors discuss the speciality of \({\mathcal L}\). The ``homogeneous'' cases \(m_0=0\) and \(m\leq 2\) have been considered by \textit{E. Arbarello} and \textit{M. Cornalba} [Math. Ann. 256, 341-362 (1981; Zbl 0454.14023)] and by \textit{A. Hirschowitz} [Manuscr. Math. 50, 337-388 (1985; Zbl 0571.14002)]. These kind of questions have a long history and one may consult the survey by \textit{A. Gimigliano} [in: Curves Semin. at Queen's, Vol. VI, Kingston 1989, Queen's Pap. Pure Appl. Math. 83, Exposé B (1989; Zbl 0743.14005)].
In the present paper the authors propose a new approach which is based on a degeneration of the plane and the corresponding linear system. The failure of \({\mathcal L}\) to be non-special is always due (in the example) to the presence of multiple \((-1)\)-curves in the base locus (i.e. of irreducible rational curves \(A\in{\mathcal L}\) such that \(A^2=-1\) on the blow-up of \(\mathbb{P}^2\)). This is formalized by the authors. They define those linear systems \((-1)\)-special and give a list of them for \(m\leq 3\). Finally the authors prove that all quasi-homogeneous special systems with \(m\leq 3\) are \((-1)\)-special. linear system of plane curves; degeneration C. CILIBERTO - R. MIRANDA, Degenerations of planar linear systems, J. reine angew. Math., 501 (1998), pp. 191-200. Zbl0943.14002 MR1637857 Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry Degenerations of planar linear systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known that the methods originated from noncommutative geometry work successfully in the geometric analysis of smooth manifolds (Connes and Connes-Moscovic). This is because the topological and geometric properties can be reconstructed by the corresponding algebra of regular smooth functions. The authors of the paper under the review are interested to apply a similar argument to the study of singular spaces. More concretely, they consider different homological theories for the algebra \(C^\infty(X)\) of Whitney functions over a subanalytic set \(X\) of \(\mathbb{R}^n\).
The first result in this paper is the proof of a Hochschild-Konstant-Rosenberg type theorem for \(C^\infty(X)\) when \(X\) is a regular subset of \(\mathbb{R}^n\) with regularly situated diagonals. The last condition for \(X\) includes the case of a subanalytic \(X\). The mentioned theorem is obtained with the help of some localization techniques used in the construction of the Hochschild homology of a fine commutative algebra.
It is to remark that the techniques introduced by the authors generalize the method of localization developed by Teleman and others. By the way, Peetre's theorem about the local operator acting on the smooth functions on \(\mathbb{R}^n\) is generalized for Whithey functions (see p. 16 of the paper).
The second result of the authors concerns the Hochschild homology of \(C^\infty(X)\) of Whitney functions on \(X\), assuming that \(X\) has regularly situated diagonals. The mentioned cohomology is computed. The cyclic and periodic cyclic homologies are computed with the help of Connes boundary map. A generalization of the Feigen-Tsygan result about the periodic cyclic theory is obtained. Very interesting is the theorem which says that for a subanalytic set \(X\) the de Rham cohomology of \(C^\infty(X)\) coincides with the singular cohomology. For the proof a new notion of the authors, namely, bimeromorphic subanalytic triangulation is used, and it is proved that such a triangulation exists for every bounded subanalytic set.
Final reviewer remarks: This paper is very rich in content. Hochschild homology is presented in a larger framework. A number of generalizations synthesize ideas from algebraic topology, geometric differential analysis and functional analysis (locally convex topological spaces, Frechet modules, currents, nuclear spaces, differential operators, etc.). The most impressive is the sketch of some general setting of the category of Whitney ringed spaces. More concretely, the algebra \(C^\infty(X)\) of Whitney functions on a stratified set \(X\) depends on the embedding \(X\subset\mathbb{R}^n\). Working in the spirit of the Grothendieck's algebraic de Rham homology, the authors show how to construct correctely a category \((X, C^\infty)\) of smooth structures on \(X\) with a structure sheaf \(C^\infty\) of Whitney functions. This new category allows us to interpret the main results in the paper as assertions about local homological properties of the structure sheaf \(C^\infty\) in the particular case \(X\) to be subanalytic. A question arises (at least for non-specialists like the reviewer): is it a new Grothendieck era which emerges in singular algebraic geometry (eventually noncommutative)? We have in mind that the de Rham cohomology of the formal completions coincides with the complex cohomology, and the authors' conjecture that the same holds for Whitney functions over appropriate singular spaces. regular smooth functions; Whitney functions; Hochschild homology; subanalytic sets; de Rham cohomology; formal completing; singular cohomology Epstein, C.L.: Subelliptic Spin\(\mathbb{C}\) Dirac operators III. Ann. Math. 167, 1--167 (2008) Semi-analytic sets, subanalytic sets, and generalizations, Noncommutative algebraic geometry, de Rham cohomology and algebraic geometry On the homology of algebras of Whitney functions over subanalytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this work is to present a general and simple strategy for the construction of compactly supported fundamental spline (piecewise-polynomial) functions for local interpolation, that are defined over quadrangulations of the real plane with extraordinary vertices. The proposed strategy -- which extends the univariate framework introduced in Antonelli et al. (Adv Comput Math 40:945-976, 2014) and Beccari et al. (J Comput Appl Math 240:5-19, 2013) -- consists in considering a suitable combination of bivariate polynomial interpolants with blending functions that are the natural generalization of odd-degree tensor-product B-splines. These blending functions are constructed as basic limit functions of the bivariate, primal subdivision schemes developed simultaneously in Stam (Comput Aided Geom Des 18:397-427, 2001) and Zorin et al. (Comput Aided Geom Des 18:483-502, 2001). As an application example of our constructive strategy we present the compactly supported \(C^2\) fundamental functions for local interpolation that arise by considering as blending functions the basic limit functions of the celebrated Catmull-Clark subdivision scheme proposed in Catmull and Clark (Comput Aided Des 46:103-124, 2016). local interpolation; piecewise polynomial; quadrilateral mesh; extraordinary vertex; \(C^2\) smoothness Spline approximation Fundamental functions for local interpolation of quadrilateral meshes with extraordinary vertices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Throughout this review `maps' and `functions' are supposed to be continuous. Given a semialgebraic set \(M\subseteq\mathbb{R}^m\) (endowed with euclidean topology) the family of semialgebraic real valued functions defines an \(\mathbb{R}\)-algebra \(\mathcal{S}(M)\) containing the family of bounded functions \(\mathcal{S}^*(M)\) as a subalgebra. The authors introduce the notations \(\text{Spec}_S(M)=\text{Spec}(\mathcal{S}(M))\) and \(\beta_SM=\text{Spec}_{\text{max}}(\mathcal{S}(M))\) for the prime and maximal spectra of \(\mathcal{S}(M),\) and similarly \(\text{Spec}_S^*(M), \beta_S^*M\) in the case of \(\mathcal{S}^*(M).\) They write \(\text{Spec}_S^\diamond (M), \beta_S^\diamond M\) in statements that apply to \(\mathcal{S}(M)\) as well as \(\mathcal{S}^*(M).\) \quad Note that a basic open set for the Zariski topology of \(\text{Spec}_S(M)\) is of the form \(\text{Spec}_S(M)\setminus \mathcal{Z}(f),\) where for \(f\in \mathcal{S}(M)\) define \(\mathcal{Z}(f)=\{\mathfrak{p}\in \text{Spec}_S(M): f\in \mathfrak{p} \}.\)
The article's goal is to prove the most important algebraic and topological properties of the Zariski and maximal spectra of the rings \(\mathcal{S}^\diamond(M)\) and the functorial properties of maps from \(\mathcal{S}^\diamond(M)\) to \(\mathcal{S}^\diamond(N)\) and from \(\text{Spec}_S^\diamond (M)\) to \(\text{Spec}_S^\diamond (N)\), naturally induced by semialgebraic maps \(N@>\varphi>> M\) (often the inclusion map \(N\hookrightarrow M\)). The authors say that their initial results could surely be obtained by Niels Schwartz's theory of real closed spaces [\textit{N. Schwartz}, Mem. Am. Math. Soc. 397, 122 p. (1989; Zbl 0697.14015)] but their approach, valid only for \(\mathbb{R}\) and not over arbitrary real closed fields, permits to use the theory of rings of continuous functions expounded in [\textit{L. Gillman} and \textit{M. Jerison}, Rings of continuous functions. The University Series in Higher Mathematics. Princeton-Toronto-London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0093.30001)] and admits a less involved presentation.
Sections 2 and 3 recall and survey in a readable manner basic facts on semialgebraic sets and functions and about spectra of rings of such functions; but for six out of seven cited papers by Fernando and Gamboa, the reader is relegated to Fernando's web page. This hopefully justifies this long review of what might be called a research survey guide to papers by the authors. The main results are below cited as Theorem 4.8 and Theorem 5.1.
In Section 2 properties of the closure \(\text{Cl}_{\mathbb{R}^n}(M)\) and the operators \(\rho_0,\rho_1\) defined by \(\rho_0(M)= \text{Cl}_{\mathbb{R}^n}(M)\setminus M\) and \(\rho_1=\rho_0\circ \rho_1\) are studied and a semialgebraic Tietze-Urysohn Lemma (see [\textit{J. Bochnak} et al., Real algebraic geometry. Transl. from the French. Rev. and updated ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 36. Berlin: Springer. (1998; Zbl 0912.14023); \textit{H. Delfs} and \textit{M. Knebusch}, Pac. J. Math. 114, 47--71 (1984; Zbl 0548.14008)] ) is given: For example, if \(N\subset M \subset \mathbb{R}^n\) are semialgebraic, then the restriction \(\mathcal{S}(M) \ni f \mapsto f|_N \in \mathcal{S}(N)\) defines a homomorphism \(\phi\) which is surjective iff \(N\) is closed. Conditions for surjectivity in the case where \(\mathcal{S}^*(.)\) substitutes \(\mathcal{S}(.)\) are also given. An ideal \(\mathfrak{a}\) in \(\mathcal{S}(M)\) is a \(z\)-ideal if \(f\in \mathfrak{a}\) and \(Z_M(f)\subset Z_M(g)\) implies \(g\in \mathfrak{a}\). In earlier papers [\textit{J. F. Fernando} and \textit{J. M. Gamboa}, ``On Łojasiewicz's inequality and the Nullstellensatz for rings of semialgebraic functions'', \url{http://http://eprints.ucm.es/24697/}] and [``On the Krull dimension of rings of semialgebraic functions'', \url{arxiv:1306.4109}] the authors have shown a Nullstellenatz: If M is locally compact then a nontrivial ideal in \(\mathcal{S}(M)\) is \(z\)-ideal iff it is radical; concerning the semialgebraic depth \(d_M(\mathfrak{p})=\min\{\dim Z_M(f):f\in \mathfrak{p}\}\) they got for prime \(z\)-ideals \(\mathfrak{p}\supset \mathfrak{q},\) that \(d_M(\mathfrak{p})<d_M(\mathfrak{q})\). In Section 3 the usual relation between what in [Zbl 0912.14023] is called prime cone of an algebra and the orderings of associated quotient fields is recalled in the context of the algebra \(\mathcal{S}^\diamond(M).\) Next the following observations are made: a radical ideal \(\mathfrak{a}\) of the ring \(\mathcal{S}^\diamond(M)\) enjoys a convexity property: \(0\leq f \leq g \in \mathfrak{a}\) implies \(f\in \mathfrak{a};\) the set of prime ideals in \(\mathcal{S}^\diamond (M)\) containing a fixed prime form a chain with respect to inclusion; the quotient \(\mathcal{S}^\diamond(M)/\mathfrak{a},\) \(\mathfrak{a}\) radical is an \(f\)-ring; the usual spectral topology of the real spectrum of the algebra \(\mathcal{S}^\diamond(M)\) coincides with the Zariski topology. Furthermore the map \(M\ni p \mapsto \mathfrak{m}_p=\{f\in \mathcal{S}(M):f(p)=0 \}\in \text{Spec}_S(M)\) embeds \(M\) into \(\text{Spec}_S(M)\) as a subspace; in fact \(\beta_S^\diamond M\) is a Hausdorff compactification of \(M\).
The main difference between \(\mathcal{S}(M)\) and \(\mathcal{S}^*(M)\) is that while a function \(f\in \mathcal{S}^*(M)\) with \(Z_M(f)=\emptyset\) is of course a unit in \(\mathcal{S}(M),\) it needs not be invertible in \(\mathcal{S}^*(M).\) Seeing why one finds with \(\mathcal{W}(M)=\{f\in \mathcal{S}^*(M): Z_M(f)=\emptyset\},\) that \(\mathcal{S}^*(M)\) is the localization of \(\mathcal{S}(M)\) at \(\mathcal{W}(M):\) \(\mathcal{S}(M)=\mathcal{S}^*(M)_{\mathcal{W}(M)}.\) This provides a correspondence between the primes (points) of the spectrum of \(\mathcal{S}(M)\) and those in the complement of \(\mathcal{W}(M)\) in \(\text{Spec}_S^*(M).\) The last part of Section 3 is dedicated to properties of \(\beta_S M.\) It is mentioned that the map \(\beta _S M \ni \mathfrak{m}\mapsto \mathfrak{m}^*:=\)unique maximal ideal containing \(\mathfrak{m}\cap \mathcal{S}^*(M)\) \quad provides a homeomorphism \(\beta_S M \cong \beta_S^* M. \) Maximal ideals defined by semialgebraic paths are also considered.
Section 4 has title `Functoriality of \(\text{Spec}_S\)'. If \(N\subset \mathbb{R}^n\) and \(M\subset \mathbb{R}^m\) are semialgebraic and \(\varphi: N\rightarrow M\) a semialgebraic map, we have a homomorphism \(\phi:\mathcal{S}^\diamond(M)\ni f \mapsto f\circ \varphi \in \mathcal{S}^\diamond(N)\) which in turn defines the map \(\text{Spec}^\diamond(\varphi):\text{Spec}_S^\diamond(N)\ni \mathfrak{q} \mapsto \phi^{-1}(\mathfrak{q}) \in \text{Spec}_S^\diamond(M).\) This map is the unique continuous map that extends \(\varphi;\) it maps prime \(z\)-ideals in \(\mathcal{S}(N)\) to \(z\)-ideals and, given \(\psi:M\rightarrow P\subset \mathbb{R}^p,\) there holds \(\text{Spec}_S^\diamond (\psi)\circ \text{Spec}_S^\diamond(\phi)= \text{Spec}_S^\diamond(\psi\circ \phi).\) Next natural questions concerning the closure operator in \(\text{Spec}_S^\diamond (M),\) are answered in corollaries 4.4, 4.5, 4.6. For example: given \(N\subset M,\) when is a primeideal in \(\text{Cl}_{\text{Spec}_S(M)}(N)\)?; what is \(\text{Cl}_{\text{Spec}_S(M))}(C_1\cap C_2) \)?; when do we have a homeomorphism \(\text{Cl}_{\text{Spec}_S^\diamond(M)}(N)\cong \text{Spec}_S^\diamond (N),\)?; etc. Then one of the main result of this paper is stated although its proof is deferred to still later when more lemmas are available - local compactness assumptions are made to ensure validity of the Lojasiewicz inequality.
Theorem 4.8: Let \(N\subset M \subset \mathbb{R}^n\) be semialgebraic sets such that \(N\) is open in \(M\) and locally compact and let \(Y=M\setminus N.\) Let \(\mathcal{L}(Y)\) be the set of all prime ideals that contain some \(f\in \mathcal{S}(M)\) whose zero set in \(M\) is \(Y.\) Let \(j:N\hookrightarrow M\) be the inclusion map. Then \(\text{Spec}_S(j)\) yields a homeomorphism \(\text{Spec}_S(N) \cong \text{Spec}_S(M)\setminus \mathcal{L}(Y).\) Furthermore the preimage of a maximal ideal of \(\mathcal{S}(M)\) under this homeomorphism is a maximal ideal of \(\mathcal{S}(N)\) while for the image of maximal ideals under \(\text{Spec}_S(j)\) needs not be maximal.
In the remainder of Section 4, lemmas 4.11, 4.13 and 4.15 characterize prime ideals in the spectral envelope \(\mathcal{L}(Y),\) explains the behaviour of the semialgebraic depth under extension and contraction of ideals, and shows that a \(\mathfrak{p}\in \mathcal{S}(M)\) so that every \(f\in \mathfrak{p}\) has a zero in the largest locally compact dense of \(M\) is a \(z\)-ideal.
Sections 5 and 6 deal with the functoriality of \(\text{Spec}_S^*(.)\) and \(\beta_S^\diamond(.),\) respectively. Many results are formulated for what the authors call a `suitably arranged tuple' \((M,N,Y,j,i)\) determined by a pair \(N\subset M \subset R^n\) of semialgebraic sets where \(N\) is locally compact and dense in \(M,\) \(Y=M\setminus N,\) and \(j:N\hookrightarrow M\) and \(i:Y=M\setminus N\hookrightarrow M\) are inclusion maps. \quad It is shown in lemmas 5.5, 5.6, 5.7 e.g. that if \(\mathfrak{p}\in \text{Spec}_S^*(M)\) is so that \(\mathfrak{p}\mathcal{S}^*(N)=\mathcal{S}^*(N),\) then \(\mathfrak{p}\in \text{Cl}_{\text{Spec}_S^*(M)}(Y)\); that a radical ideal in \(\mathcal{S}^*(M)\) containing a prime is itself radical; that if \(N\subset M\) is dense in \(M\) and \(\mathfrak{q}\) minimal in \(\mathcal{S}^*(N),\) then \(\mathfrak{q}\cap \mathcal{S}^*(M)\) is minimal in \(\mathcal{S}^*(M).\) \quad In dramatic difference to the case of the \(\text{Spec}_S(j)\) map (see T4.1), given a semialgebraic map \(\varphi: N\rightarrow M,\) Lemma 5.9 says that the map \(\text{Spec}_S^*(\phi)\) maps (maximal) ideals in \(\beta_s^*N\) into \(\beta_s^*M.\) Given a suitably arranged tuple \((M,N,Y,j,i)\) and a chain of prime ideals \(\mathfrak{p}_0 \subset \cdots \subset \mathfrak{p}_r\) in \(\mathcal{S}^*(M),\) by Corollary 5.11 there exists a chain \(\mathfrak{q}_0 \subset \cdots \subset \mathfrak{q}_r\) in \(\mathcal{S}^*(N),\) so that \(\mathfrak{q}_k\cap \mathcal{S}^*(M)=\mathfrak{p}_k,\) for \(k=1,\dots,r;\) furthermore the map \(\text{Spec}_S^*(j)\) is surjective.
Suppose now we have a semialgebraic situation \(\mathbb{R}^n \supset N @>\varphi>> M\subset \mathbb{R}^m\) and there is a \(Y\subset M\) so that \(M_1=M\setminus Y\) is locally compact and dense in \(M\) and the map \(\psi=\varphi_{|N_1}:N_1=N\setminus \varphi^{-1}(Y) \rightarrow M_1\) is a semialgebraic homeomorphism. Let \(Z=\text{Cl}_{\text{Spec}_S^*(M)}(Y).\) Then the second main result of the paper, Theorem 5.1, says that the map \(\text{Spec}_S^*(\varphi):\text{Spec}_S^*(N)\rightarrow \text{Spec}_S^*(M)\) is surjective and its restriction to \(\text{Spec}_S^*(N)\setminus \text{Spec}_S^*(\varphi)^{-1}(Z) \) is a homeomorphism onto \(\text{Spec}_S^*(M)\setminus Z.\) As Corollary 5.13 to lemmas leading up to the just cited main result one has e.g. a going up lemma for prime ideals for suitably arranged tuples \((M,N,Y,i,j)\): assume \(\mathfrak{p}_0 \subset \mathfrak{p}_1\) are prime ideals of \(\mathcal{S}^*(M)\) and \(\mathfrak{q}_0\) is a prime ideal of \(\mathcal{S}^*(N)\) so that \(\mathfrak{q}_0 \cap \mathcal{S}^*(M)=\mathfrak{p}_0.\) Then there exists a prime ideal \(\mathfrak{q}_1 \supset \mathfrak{q}_0 \) of \(\mathcal{S}^*(N)\) so that \(\mathfrak{q}_1 \cap \mathcal{S}^*(M)=\mathfrak{p}_1.\) \quad In the last part of Section 5, an analysis of the spectrum of \(\mathcal{S}^*(M)\) in case that \(M\) is not necessarily locally compact is given via stratifications of \(M\) into locally compact pieces constructed from the \(\rho_1\) operator defined above. It is also shown in Proposition 5.19 that given any semialgebraic set there exist semialgebraic sets \(A_1,\dots,A_r\) whose union is \(M\) and which are Nash diffeomorphic respectively to euclidean spaces \(\mathbb{R}^{d_i}\) with \(0\leq d_i \leq \dim M,\) \(i=1,\dots,r\) and so that \(\text{Spec}_S^*(M)\) is the disjoint union of open subsets that are homeomorphic to certain open subsets of \(\text{Spec}_S^*(\mathbb{R}^{d_i})\) constructed via the mentioned diffeomorphisms.
In Section 6 results of sections 4 and 5 are transferred to results for \(\beta_S^*M.\) This is done via the known fact that the map \(s_M:\text{Spec}_S(M) \rightarrow \beta_S(M)\) which maps each prime ideal to the unique maximal ideal of \(\mathcal{S}(M)\) containing it is a continuous map. The map \(\beta_S \varphi:= s_M\circ \text{Spec}_S(\varphi)|_{\beta_SN}: \beta_SN \rightarrow \beta_S M\) has properties that can be obtained as translations of properties of the map \(\beta_S^*\varphi=\text{Spec}_S^*(\varphi)|_{\beta_S^*N}. \) As an example, from connectedness results in Section 4, in Corollary 6.6 it is shown that if \(M_1,\dots,M_k\) are the connected components of semialgebraic \(M\subset \mathbb{R}^n,\) then their closures \(\text{Cl}_{\beta_S^*M}(M_i)\cong \beta_S^*M_i\) are connected components of \(\beta_S^*M.\) Finally, an easy corollary to the main result of this section follows after obvious notational changes (e.g. change \(\text{Spec}_{S^*}(M)\) to \(\beta_S^* M\) etc.) from Theorem 5.1. semialgebraic function; semialgebraic set; Zariski spectrum; maximal spectrum; functoriality; local compactness; semialgebraic depth; \(z\)-ideal; Nash diffeomorphism; radical ideal; Tietze Urysohn extension; Lojasiewicz inequality; rings of continuous functions; going up lemma; curve selection lemma Fernando, On the spectra of rings of semialgebraic functions, Collect. Math. 63 (3) pp 299-- (2012) Semialgebraic sets and related spaces, Extension of maps, Real-valued functions in general topology, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Chain conditions, finiteness conditions in commutative ring theory On the spectra of rings of semialgebraic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local \(k\)- algebra \(R\), not necessarily commutative, where \(k\) is a field. The class of topological modules we consider includes all those of finite rank over \(k\) and some of infinite rank as well, namely those with a Schauder basis in the sense of \S1. This generalizes the results of the second author [Ph. D. thesis, Brandeis 1987], where the result was obtained in a different way in case the ring \(R\) is the completion of the local ring of a plane curve singularity and the module is \(k^ n\).
Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called ``growth functions'' to handle explicit epsilonics involving the convergence of formal power series in non- commuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of \textit{M. Schlessinger} [Trans. Am. Math. Soc. 130, 208-222 (1968; Zbl 0167.495)] which were used in \textit{P. Shukla}'s thesis cited above and is more conceptual. growth functions; formal moduli space; complete topological modules; completion of the local ring of a plane curve singularity; convergence of formal power series in noncommuting variables Noncommutative algebraic geometry, Power series rings, Topological and ordered rings and modules, Valuations, completions, formal power series and related constructions (associative rings and algebras), Formal power series rings, Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) Formal moduli of modules over local \(k\)-algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 deals with developing classical scheme theory. Namely, the authors consider scheme-theoretical background subject -- relations between spectralization and topology.
The authors show that for a field \(K\) and \(n\geq 1\), the soberification \(\mathcal{S}(\mathbb{A}^n(K))\) of the affine \(n\)-space \(\mathbb{A}^n(K)\) over \(K\) is homeomorphic to its spectralization \(\mathcal{BS}(\mathbb{A}^n(K))\), and it can be embedded into the spectrum \(\mathrm{Spec}(K[X_1,\dots,X_n])\). Moreover, if the field \(K\) is algebraically closed, then there are homeomorphisms \(\mathcal{S}(\mathbb{A}^n(K)) \approx\mathcal{BS}(\mathbb{A}^n(K))\approx \mathrm{Spec}(K[X_1,\dots,X_n])\). They also show that for a space \(X\), the subspace \(z\mathrm{Spec}(C(X))\subseteq\mathrm{Spec}(C(X))\) of prime z-ideals of the ring \(C(X)\) of real-valued continuous functions on \(X\) is homeomorphic to the space \(z\mathcal{SR}(X)\) of prime \(z\)-filters with an appropriate topology and there is a homeomorphism \(\mathcal{BS}(X)\approx z\mathrm{Spec}(C(X))\) provided \(X\) is perfectly normal. (Notions of \textit{Sober space} and \textit{Soberification} deals with space improving. They are defined at pages 514 and 515 resp.)
The paper is self-contained elementary and easy for reader. affine space; perfectly normal space; reflective subcategory; ring of continuous functions; sober space; spectral space; spectrum of a ring; Zariski topology; \(z\)-ideal Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Rings and algebras of continuous, differentiable or analytic functions, Special constructions of topological spaces (spaces of ultrafilters, etc.) On the spectralization of affine and perfectly normal 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 Gelfand, Graev, Kapranov, and Zelevinsky defined certain linear systems of partial differential equations, now known as \(A\)-\textit{hypergeometric} or \textit{GKZ hypergeometric} systems \(H_A(\beta)\), whose solutions generalize the classical hypergeometric series. These holonomic systems are constructed from discrete input consisting of an integer \(d \times n\) matrix \(A\) along with continuous input consisting of a complex vector \(\beta\in\mathbb C^d\). Assume that the convex hull conv\((A)\) of the columns of \(A\) does not contain the origin. The matrix \(A\) determines a semigroup ring \({\mathbb C}[\mathbb NA]\), and the dimension rank\((H_A(\beta))\) of the space of analytic solutions of \(H_A(\beta)\) is independent of \(\beta\) whenever \({\mathbb C}[\mathbb NA]\) is Cohen-Macaulay. In this note, the authors use the combinatorics of \(\mathbb Z^d\)-graded local cohomology to characterize the set of parameters \(\beta\) for which the rank goes up, in the simplicial case. The premise is the standard fact that a semigroup ring \({\mathbb C}[\mathbb NA]\) fails to be Cohen-Macaulay iff a local cohomology module \(H_{\mathfrak m}^i({\mathbb C}[\mathbb NA])\) is nonzero for some cohomological index \(i\) strictly less than the dimension \(d\) of \({\mathbb C}[\mathbb NA]\). After gathering some facts about \(A\)-hypergeometric systems, the authors prove the simplicial case of the following: Assume that conv\((A)\) has dimension \(d-1\). The set of parameters \(\beta\in \mathbb C^d\) such that rank\((H_A(\beta))\) is greater than the generic rank equals the Zariski closure (in \(\mathbb C^d\)) of the set of \(\mathbb Z^d\)-graded degrees where the local cohomology \(\bigoplus_{i<d}H_{\mathfrak m}^i({\mathbb C}[\mathbb NA])\) is nonzero. Using a different approach, the authors prove the full conjecture in the paper \textit{L. F. Matusevich, E. Miller}, and \textit{U. Walther} [Homological methods for hypergeometric families, J. Am. Math. Soc. 18, No. 4, 919-941 (2005; Zbl 1095.13033)]. \(A\)-hypergeometric systems; simplicial; combinatorics of \(\mathbb Z^d\)-graded local cohomology; local cohomology for semigroup rings; \(A\)-hypergeometric module; local cohomology module; rank-jumping parameter Laura Felicia Matusevich and Ezra Miller, Combinatorics of rank jumps in simplicial hypergeometric systems, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1375 -- 1381. Other hypergeometric functions and integrals in several variables, Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings of differential operators and their modules, Local cohomology and commutative rings, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Cohen-Macaulay modules, Ordinary and skew polynomial rings and semigroup rings, Semigroup rings, multiplicative semigroups of rings Combinatorics of rank jumps in simplicial hypergeometric systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be an algebraic scheme (over a fixed field of characteristic zero) embedded in a regular \(scheme\quad W.\) A constructive resolution of the singularities of X in the sense of the author consists, roughly speaking, of the following:
(i) An upper semicontinuous function \(\phi\) defined on a fixed Samuel stratum of X such that \(Max(\phi):=\{x| \quad \phi (x)\quad is\quad \max imum\}\) is the center of a permissible transformation \(\pi:\quad X_ 1\to X.\)
(ii) If \(\pi\) : \(X_ 1\to X\) is not a resolution of X, then there is an upper semicontinuous function \(\phi_ 1\) defined at \(X_ 1\) such that \(Max(\phi_ 1)\) is permissible at \(X_ 1\), \(\phi (\pi (x))\geq \phi_ 1(x)\), for all x and \(\phi (\pi (x))=\phi_ 1(x)\) if \(\pi\) (x) if \(\pi\) (x)\(\not\in Max(\phi)\). - If there is no improvement of the Hilbert- Samuel function at \(X_ 1\) then there is an improvement of this function.
(iii) Repeating (i) and (ii) a finite number of times one can force an improvement of the Hilbert-Samuel function and finally a resolution of X.
The first half of the paper consists of notations and some results (without proofs) due to Hironaka concerning the local idealistic presentation of X. These are used in the second half where the upper semicontinuous functions are constructed. constructive resolution of the singularities; Hilbert-Samuel function Villamayor, Orlando, Constructiveness of Hironaka's resolution, Ann. Sci. École Norm. Sup. (4), 22, 1, 1-32, (1989) Global theory and resolution of singularities (algebro-geometric aspects) Constructiveness of Hironaka's resolution | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(E\subset \mathbb R^s\) be compact and let \(d_n^E\) denote the dimension of the space of polynomials of degree at most \(n\) in \(s\) variables restricted to \(E\). The authors introduce the notion of an asymptotic interpolation measure (AIM) \(\mu^E\) describing, \textit{if it exists}, the asymptotic behavior of any scheme \(\tau_n = \{{\mathbf X}_{k,n}\}_{k=1}^{d_n^E}\), \(n\in\mathbb N\), of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with \(n\). Some simple consequences of Auerbach's theorem that are essential for the proofs of the main results are given. It is proved that the union of finitely many subsets of algebraic curves having AIMs again has an AIM. Using the theory of logarithmic potential with external fields in \(\mathbb R^2\), it is shown that the union of finitely many compact subsets of positive capacity of algebraic curves of genus \(0\) in \(\mathbb R^2\) has an AIM and their \(\mu^E\) is determined. The essential feature of such curves is that they admit a rational parametrization. The obtained result applies in the case when \(E\) is a compact subset of the image of the unit circle under a rational mapping \(w =p(z)/q(z)\) in the complex variable \(z\) or a compact subset of a curve consisting of piecewise conies. The inverse problem of constructing good interpolation points when \(\mu^E\) is known, is considered. asymptotic interpolation measure; Lebesgue constants; Fekete points; equilibrium distributions; algebraic curves of genus \(0\); multivariate polynomial interpolation; Auerbach's theorem; piecewise conies; rational mapping; constructing good points for interpolation [GMS] Götz, M., Maymeskul, V. V. \& Saff, E.B., Asymptotic distribution of nodes for near-optimal polynomial interpolation on certain curves in \$\$ \{\(\backslash\)mathbb\{R\}\^2\} \$\$ . Constr. Approx., 18 (2002), 255--283. Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Proceedings, conferences, collections, etc. pertaining to functions of a complex variable, Varieties and morphisms, Multidimensional problems, Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions, Computational aspects of algebraic curves Asymptotic distribution of nodes for near-optimal polynomial interpolation on certain curves in \(\mathbb{R}^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In their article ``Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes'' [Commun. Algebra 35 (4), 1341--1367 (2007; Zbl 1124.14006)], the authors adapted the theory of smooth morphisms to the case of noetherian formal schemes based on properties of the completion of the module of Kähler differentials. In particular the property of formal smoothness in terms of the infinitesimal liftings of maps in the category of algebraic schemes was described. It was proved that smooth morphisms of noetherian formal schemes are flat and the associated module of differentials is locally free. In this paper a detailed study of the relationship between the infinitesial lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes is made. A characterization of completion morphisms as pseudo--closed immersions that are flat is given. The local structure of smooth and étale morphisms between locally noetherian formal schemes is described. formal scheme; smooth maps; étale morphisms Tarrío, L. Alonso; López, A. Jeremías; Rodríguez, M. Pérez: Local structure theorems for smooth maps of formal schemes. J. pure appl. Algebra 213, 1373-1398 (2009) Infinitesimal methods in algebraic geometry, Formal neighborhoods in algebraic geometry Local structure theorems for smooth maps of formal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F: \mathbb{C}^n \to \mathbb{C}^m\) be a polynomial mapping and let
\[
N(F)= \Bigl\{\delta \in\mathbb{R} \mid \exists C,\;R>0,\;\bigl(|x|>R\bigr) \Rightarrow \biggl(\bigl |F(x) \bigr|\geq C|x|^\delta \biggr) \Bigr\}.
\]
If \(N(F)\) is not empty then \(N(F)\) is bounded from above and the number \({\mathcal L}_\infty (F)= \sup N(F)\) is called the ``Łojasiewicz exponent of \(F\) at infinity''. If \({\mathcal L}_\infty (F)>0\) then \(F\) is proper, i.e. inverses of compact sets are compact. A general estimation of the Łojasiewicz exponent at infinity can be found in a paper by \textit{J. Kollár} [J. Am. Math. Soc. 1, No. 4, 963-975 (1988; Zbl 0682.14001)]. The most difficult (generally) is the calculation of \({\mathcal L}_\infty (F)\) for dominating mappings which are not proper.
The main goal of this paper is to present an algorithm for an estimation of \({\mathcal L}_\infty (F)\) for such dominating mappings \(F\) which are not proper. The calculation is supported by Gröbner bases algorithms. The Łojasiewicz exponent at infinity can also be defined, in the same way, in the real case. However, using our algorithm we will estimate \({\mathcal L}_\infty (F)\) for \(F\) treated as a mapping from \(\mathbb{C}^n\) into \(\mathbb{C}^m\). It is surprising that in many calculated examples we obtain \({\mathcal L}_\infty (F)\). Example 1 shows that in the general case we can only expect an estimation. Łojasiewicz exponent at infinity; polynomial mapping; dominating mappings Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Birational geometry, Local analytic geometry Estimation of growth of dominating mappings via Gröbner bases | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article is the first part of the interesting and rich survey of numerous remarkable results obtained recently concerning singularities in nonlinear infinite-dimensional problems of analysis, the theory of differential and integral equations, etc. This first part presents numerous results about mappings of fold type or, in other words, mappings \(F\), which in a suitable (nonlinear) system of ``coordinates'' \(\mathbb{R} \times E\) can be represented in the form \(F(t,v) =(t^2,v)\) or \(F(t,v) =(-t^2,v)\). The account of concrete results is based on the abstract global characterization of the fold-like mappings which was obtained by the authors in 1992. As a result, this survey presents numerous theorems about fold-like mappings from the unique point of view.
The contents of this part are: (1) Introduction; (2) Fréchet derivatives; (3) Fredholm maps; (4) Local structure of folds (Local characterization and Ambrosetti-Prodi local folds); (5) Abstract global characterization of the fold map (global structures, tools and examples); (6) Ambrosetti-Prodi and Berger-Podolak-Church fold maps; (7) McKean-Scovel fold maps (Riccati operator and a one-dimensional elliptic operator with \(u^2\) nonlinearity); (8) Giannoni-Micheletti fold maps; (9) Mandhyan fold map; (10) Oriented global fold maps; (11) A second Mandhyan fold map; (12) Jumping singularities.
This survey is both useful for specialists in the field as well as for all who want to study singularities of infinite-dimensional mappings.
[For Part II, see the following review, Zbl 0879.58008]. survey; singularities; fold-like mappings; infinite-dimensional mappings P.T. Church, J.G. Timourian, Global structure for nonlinear operators in differential and integral equations, I. Folds, Topological Nonlinear Analysis, II, 109 -- 160, Prog. Nonlinear Differ. Eq. Appl., vol. 27, Birkhaüser Boston, Boston, MA, 1997. Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Equations involving nonlinear operators (general), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Global structure for nonlinear operators in differential and integral equations. I: Folds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal H}_N\) be the vector space of Hermitian \(N\times N\) matrices with the unitary group \(U_N\) acting by conjugation. For a positive definite Hermitian matrix \(\Lambda\) one defines a \(U_N\)-invariant measure
\[
d\mu_{\Lambda}={\exp (-(1/2)\text{ Tr}(\Lambda {\mathbb X}^2))\over (2\pi)^{N^2/2}}d{\mathbb X},
\]
where \(d{\mathbb X}\) is the standard Lebesgue measure on \({\mathcal H}_N\). The techniques in [\textit{D. Bessis, C. Itzykson} and \textit{J.-B. Zuber}, Adv. Appl. Math. 1, 109-157 (1980; Zbl 0453.05035)] provide a method of computing integrals of the form
\[
\langle F({\mathbb X})\rangle ={\int_{{\mathcal H}_N}F({\mathbb X})d\mu_{\Lambda}\over\int_{{\mathcal H}_N}d\mu_{\Lambda}},
\]
where \(F({\mathbb X})\) is a \(U_N\)-equivariant function on \({\mathcal H}_N\). One can give an asymptotic expansion of \(\langle F({\mathbb X})\rangle\), \(N\to\infty\), through a collection of graphs known as ``ribbon graphs.'' A theorem of \textit{P. Di Francesco, C. Itzykson} and \textit{J.-B. Zuber} [Commun. Math. Phys. 151, No. 1, 193-219 (1993; Zbl 0831.14010)] relates the asymptotic expansion of such integrals for different choices of \(F({\mathbb X})\). As pointed out in \textit{E. Arbarello} and \textit{M. Cornalba} [J. Algebr. Geom. 5, 705-749 (1996; Zbl 0886.14007)], this theorem should yield new relations among cohomology classes of the moduli space of stable pointed curves. In the paper under review the author gives a combinatorial algorithm which can determine the coefficients appearing in these new relations. The algorithm involves graphs and two families of symmetric functions, Schur \(Q\)-functions associated to strict partitions and power sums for odd partitions. symmetric functions; cohomology classes; module space; pointed curves Symmetric functions and generalizations, Families, moduli of curves (algebraic), Combinatorial aspects of partitions of integers A combinatorial algorithm related to the geometry of the moduli space of pointed 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 authors study compactifications of a semi-algebraic set \(M \subseteq \mathbb{R}^n\). They show that there exists a universal compactification \(\hat{M}\), which they call the \textit{semi-algebraic Stone-Čech compactification}. The name is due to the analogy with the Stone-Čech compactification in topology. They give three different presentations of \(\hat{M}\), namely as the space of closed points of the prime spectrum of the ring of continuous semi-algebraic functions on \(M\), or as the space of closed points of the prime spectrum of the ring of bounded continuous semi-algebraic functions on \(M\), or as the projective limit of all semi-algebraic spaces that compactify \(M\). The space \(\hat{M}\) is rarely semi-algebraic. The authors study properties of the growth \(\hat{M} \setminus M\), in particular the number of connected components.
Many results in the paper are special cases of far more general properties of real closed rings. Every ring of continuous semi-algebraic functions is a real closed ring. The connections between the spectra of a real closed ring and a convex subring (e.g., the subring of bounded functions in a ring of continuous semi-algebraic functions) are well-known, see e.g. [\textit{N. Schwartz}, The basic theory of real closed spaces. Regensburger Math. Schr. 15, 257 p. (1987; Zbl 0634.14014), Chapter V.7; in: Proceedings of the Curaçao conference, Netherlands Antilles, June 26--30, 1995. Dordrecht: Kluwer Academic Publishers. 277--313 (1997; Zbl 0885.46024); Manuscr. Math. 102, No. 3, 347--381 (2000; Zbl 0966.13018); Math. Nachr. 283, No. 5, 758--774 (2010; Zbl 1196.13015)]. In particular the existence of the semi-algebraic Stone-Čech compactification has been known for a long time.
The authors do not explain the connections of their results with the existing literature, even though this would lead to significant simplifications and a deeper understanding. semi-algebraic set; semi-algebraic function; bounded function; compactification; connected component; local compactness Fernando, J.F.; Gamboa, J.M., On the semialgebraic stone-čech compactification of a semialgebraic set, Trans. am. math. soc., 364, 7, 3479-3511, (2012) Semialgebraic sets and related spaces, Real algebra, Real-valued functions in general topology, Extensions of spaces (compactifications, supercompactifications, completions, etc.) On the semialgebraic Stone-Čech compactification of a semialgebraic set | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb C\{x_1,\ldots,x_d\}=\mathbb C\{x\}\) be the ring of convergent power series. Let \(f\in\mathbb C\{x\}[z]\) be a quasi-ordinary (q.o) polynomial, i.e., \(f\) is a Weierstrass polynomial in \(z\) with discriminant \(\Delta_z(f)=x^\delta u\) where \(\delta\in \mathbb Z^d_{\geq0}\) and \(u\in \mathbb C\{x\}\) is a unit. The Jung-Abhyankar theorem guaranties that the roots of \(f\), called quasi-ordinary branches, are fractional powers series in \(\mathbb C\{x^{1/m}\}\) for some integer \(m\geq1\). The difference \(\zeta^{(s)}-\zeta^{(t)}\) of two different roots of \(f\) has the form \(x^{\lambda_{st}}H_{st}\) where \(H_{st}\in\mathbb C\{x\}\) is a unit; \(\Lambda(f):= \{\lambda_{st}\mid f(\zeta^{(s)})=f(\zeta^{(t)})=0,\zeta^{(s)}\neq\zeta^{(t)}\}\subset \mathbb Q^d_{\geq0}\) is the set of characteristic exponents of \(f\).
Let \(f\) be an irreducible q.o polynomial; then one may take \(m=\text{deg}(f)=:n\). In this case let \(\zeta\) be a root of \(f\); \(\text{frac}(\mathbb C\{x^{1/n}\})/\text{frac}(\mathbb C\{x\}[z]/(f))\) is a Galois extension, the roots of \(f\) are the conjugates of \(\zeta\), the fractional monomials \(x^{\lambda_{st}}\) (resp.\, the vector exponents \(\lambda_{st}\)) are called characteristic monomials (resp.\ exponents) of the q.o branch \(\zeta\), and we write \(M(\zeta)=M(f)\). We consider the preordering \(\alpha\leq \beta\) on \(\mathbb Q^d\) given by \(\alpha\leq\beta\) if \(\beta\in\alpha+\mathbb Q^d_{\geq0}\); the characteristic exponents can be ordered \(\lambda_1\leq\cdots\leq\lambda_g\). Let \(M_0=\mathbb Z^d\), \(M_j=M_{j-1}+\lambda_j\mathbb Z\), \(n_j=[M_{j-1}:M_j]\) for \(j\in\{1,\ldots,g\}\)---the integers \(n_j\) are called characteristic integers of \(f\). Set \(\gamma_1=\lambda_1\), \(\gamma_{j+1}=n_j\gamma_j+\lambda_{j+1}-\lambda_j\) for \(j\in\{1,\ldots,g-1\}\); the semigroup \(\Gamma=\mathbb Z^d_{\geq0}+ \gamma_1\mathbb Z_{\geq0}+\cdots+\gamma_g\mathbb Z_{\geq0}\) is the semigroup of \(f\). If \(f\) is not irreducible, then \(M(f)\) is, in general, not totally ordered. If \(\zeta\) is a q.o branch and \(f\) its minimal polynomial over \(\text{frac}(\mathbb C\{x\})\), then \(f\in\mathbb C\{x\}[z]\) is q.o, and we set \(M(\zeta)=M(f)\).
Let \(f(x,z)=\sum c_{\alpha,\beta}x^\alpha z^\beta \in\mathbb C\{x\}[z]\) be q.o, let \(\Delta(f)\subset \mathbb N_0^{d+1}\) be the support of \(f\), and let \(\mathcal N(f)\subset \mathbb R^{d+1}_{\geq0}\) be the Newton polyhedron of \(f\). For any compact face \(\gamma\) of \(\Delta(f)\) set \(f_\gamma(x,z)=\sum_{(\alpha,\beta)\in\gamma}c_{\alpha,\beta}x^\alpha z^\beta\). There exist systems of P-good coordinates for \(f\) such that \(\mathcal N(f)\) is a monotone polygonal path (see Def.\ 2.13 for a definition of P-good. For the name P-good coordinates (cf. \textit{E. Artal Bartolo} et al. [Mem. Am. Math. Soc. 841, 85 p. (2005; Zbl 1095.14005), p.\ 23]; see \textit{J. McDonald} [J. Pure Appl. Algebra 104, No. 2, 213--233 (1995; Zbl 0842.52009)]) for the notion of monotone polygonal path.).
Let \(N(f)\) be a monotone polygonal path. The one-dimensional (compact) faces \(\gamma_1,\ldots,\gamma_n\) of \(N(f)\) can be ordered with respect to their slopes. Let \(s_i=(q_1/p,\ldots,q_d/p)\) with \(q_1,\ldots,q_d\in \mathbb Z_{\geq0}\), \(p>0\) and \(\gcd(q_1,\ldots,q_d,p)=1\) be the slope of \(s_i\) and \(h_i>0\) be the height of \(s_i\); \(h=h_1+\cdots+h_d\) is the height of \(N(f)\). Let \(f\in \mathbb C\{x\}[z]\) be q.o, and choose P-good coordinates for \(f\). We assume that \(f\) is of order \(>1\) in \(z\). Let \(s=(q_1/p,\ldots,q_d/p)=(\overline q_1/\overline p_1,\ldots,\overline q_d/\overline p_d)\) with \(\gcd (\overline q_i,\overline p_i)=1 \) for \(i\in\{1,\ldots,d\}\) be the slope of a line segment \(\mathcal E\) of \(N(f)\); then we may write \(f_{\mathcal E}(x,z)=x^nz^{n_{d+1}}\prod_{j=1}^k(z^p-\mu_hx^q)\) with \(n\in\mathbb Q_{\geq0}^d\) and \(\mu_j\in\mathbb C^*\). Now choose \(j\in\{1,\ldots,k\}\) and \(\alpha_j\in\mathbb C^*\) with \(\alpha_j^p=\mu_j\). The Newton map corresponding to these data is the morphism \(\sigma_{s,\alpha_j}: \text{Spec}(\mathbb C\{y\}[z_1])\to \text{Spec}(\mathbb C\{x\}[z])\) defined by \(x_i\mapsto y_i^{\overline p_i}\), \(z\mapsto \prod_i(y_i^{\overline q_i}(\alpha_j+z_1)\). Then \(f\circ \sigma_{s,\alpha_j}(y,z_1)\in\mathbb C\{y\}[z_1]\) can be written as \(y^ef_1(y,z_1)\) where \(f_1\in\mathbb C\{y\}[z_1]\) is q.o. In general, \(f_1\) is not in P-good coordinates; also, if \(f\) is irreducible, then \(f_1\) is, in general, not irreducible---a phenomenon of false reducibility which was detected and analyzed by the author in his thesis [Función zeta motívica de singularidades quasiordinarias irreducibles. Tesis Doctoral, Universidad Complutense de Madrid (2010)].
Now we can repeat these steps for the q.o polynomial \(f_1(y,z_1)\). This procedure can be repeated until we reach some \(f_k\) of order \(1\) in \(z_k\). This situation always appears after a finite number of steps; such a sequence of iterations is called the Newton process for the q.o polynomial \(f\). From a result of \textit{P. D. González Pérez} [Can. J. Math. 52, No. 2, 348--368 (2000; Zbl 0970.14027)] it follows that the Newton process is independent of the choice of P-good coordinates.
If \(f\) is irreducible, the output of the Newton process is a q.o branch \(\zeta\) from which one may read the characteristic exponents of \(f\) (Proposition 3.8). A Newton process determines also if \(f\) is irreducible (Theorem 4.9). In Proposition 5.3 the author shows under what conditions all the iterations of \(f\) in a Newton process remain irreducible. Lastly, the author---in case \(d=2\)---gives a sufficient condition for the normalization of a q.o surface singularity defined by an irreducible q.o polynomial \(f\) to be smooth; here he uses results of \textit{P. D. Gonzalez Perez} [Ann. Inst. Fourier 53, No. 6, 1819--1881 (2003; Zbl 1052.32024)] and \textit{P. Popescu-Pampu} [C. R., Math., Acad. Sci. Paris 336, No. 8, 651--656 (2003; Zbl 1038.32024)]. In an appendix the author illustrates graphically a Newton process associated to a q.o polynomial \(f\) and compares this illustration with the Newton trees introduced in the papers of \textit{E. Artal Bartolo} et al. [Mem. Am. Math. Soc. 841, 85 p. (2005; Zbl 1095.14005); Contemp. Math. 538, 321--343 (2011; Zbl 1214.14004)]; [Mosc. Math. J. 13, No. 3, 365--398 (2013; Zbl 1396.14006)].
Reviewer's remark: The correct references for Lemma 2.14 are [17], Lemma 3.6 and [5], Lemma 4.6. quasi-ordinary power series; Newton process; semi-group; characteristic monomials González Villa, Manuel, Newton process and semigroups of irreducible quasi-ordinary power series, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 1578-7303, 108, 1, 259-279, (2014) Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings Newton process and semigroups of irreducible quasi-ordinary power series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The study of infinitesimal deformations of a variety embedded in projective space requires, at ground level, that of deformation of a collection of points, as specified by a zero-dimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes. An optimal Hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of R. Fröberg and A. Iarrobino. We discuss these conjectures from a geometric point of view. They give, for each such collection, a function (based on dimension, number of points, and order of each neighbourhood) which should serve as an upper bound to its Hilbert function (``weak conjecture''). The ``strong conjecture'' predicts when the upper bound is sharp, in the case of equal order throughout. In general we refer to the equality of the Hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the ``strong hypothesis''. We interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points. We develop techniques and insight toward the conjectures' verification and refinement. The main result gives an upper bound on the Hilbert function of a collection of infinitesimal neighbourhoods in \(\mathbb P^n\) based on Hilbert functions of certain such subschemes of \(\mathbb P^{n-1}\). Further, equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand. It follows that an infinitesimal neighbourhood scheme obeys the weak conjecture provided that the schemes identified in codimension one satisfy the strong hypothesis. This observation is then applied to show that the weak conjecture does hold valid in \(\mathbb P^n\) for \(n \leqslant 3\). The main feature here is that the result is obtained although the strong hypothesis is not known to hold generally in \(\mathbb P^2\) and, further, \(\mathbb P^2\) presents special exceptional cases. Consequences of the main result in higher dimension are then examined. We note, then, that the full weight of the strong conjecture (and validity of the strong hypothesis) are not necessary toward using the main theorem in the next dimension. We end with the observation of how our viewpoint on the strong hypothesis pertains to extra algebraic information: namely, on the structure of the minimal free resolution of an ideal generated by linear forms. Chandler, K, The geometric interpretation of fröberg-iarrobino conjectures on infinitesimal neighbourhoods of points in projective space, J. Algebra, 286, 421-455, (2005) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The geometric interpretation of Fröberg-Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal N} (M)\) be the ring of Nash functions on an affine Nash manifold \(M\), \(I\) an ideal of \({\mathcal N} (M)\) and \({\mathcal N}\) the sheaf of Nash functions on \(M\). We are dealing with the extension problem which asks whether the canonical morphism \(\varphi:{\mathcal N}(M) \to H^0(M, {\mathcal N}/I{\mathcal N})\) is surjective. We give an affirmative answer to this problem when \(H^0(M, {\mathcal N}/I {\mathcal N})\) is noetherian and \(\varphi\) regular, using the result of Spivakovsky-Popescu-André [cf. \textit{D. Popescu}, Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14001); see also: \textit{M. André}, ``Cinq exposés sur la désingularisation'' (preprint 1992), and \textit{M. Spirakovsky}, ``Smoothing of ring homomorphisms, approximation theorems and the Bass-Quillen conjecture'' (preprint 1992)] on approximation of regular morphisms. These hypotheses are consequences of the so-called ``idempotency of the real spectrum''; unfortunately there is a gap in its proofs given in the literature.
We show that the idempotency holds under an assumption of normality. idempotency of the real spectrum; Nash manifold Quarez, R.: The idempotency of the real spectrum implies the extension theorem for Nash functions. Math. Z. 227, 555-570 (1998) Nash functions and manifolds, Real-analytic and Nash manifolds The idempotency of the real spectrum implies the extension theorem for Nash functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed into very homogeneous semialgebraic pieces up to a small error (see e.g. [\textit{J. Pach} and \textit{J. Solymosi}, J. Comb. Theory, Ser. A 96, No. 2, 316--325 (2001; Zbl 0989.05031); \textit{N. Alon} et al., J. Comb. Theory, Ser. A 111, No. 2, 310--326 (2005; Zbl 1099.14048); \textit{J. Fox} et al., J. Reine Angew. Math. 671, 49--83 (2012; Zbl 1306.05171); \textit{J. Fox} et al., SIAM J. Comput. 45, No. 6, 2199--2223 (2016; Zbl 1353.05090)]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model-theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary o-minimal structures and in \(p\)-adics. NIP; VC-dimension; distal theories; o-minimality; \(p\)-adics; Erdős-Hajnal conjecture; regularity lemma Classification theory, stability, and related concepts in model theory, Models of other mathematical theories, Hypergraphs, Ramsey theory, Graphs and abstract algebra (groups, rings, fields, etc.), Semialgebraic sets and related spaces, Model theory of ordered structures; o-minimality Regularity lemma for distal structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be an open semi-algebraic subset of \(\mathbb{R}^n\) and let \(Z(P,A;s)= \sum_{m\in A\cap\mathbb{Z}^n} P(m)^{-s}\), where \(P\) is a polynomial in \(\mathbb{R} [x_1,\dots, x_n]\). The aim is to study the convergence and meromorphic continuation of Dirichlet series of this type and generality. Suppose \(P(x)\to \infty\) as \(\| x\|\to \infty\) in \(A\) and \(P(m)\neq 0\) for \(m\) in \(A\cap \mathbb{Z}^n\). The author shows that \(Z(P,A;s)\) is absolutely convergent in a half-plane \(\text{Re} s> \sigma_0\), say. If the complex zeros of \(P\) are separated from \(A\), then \(Z(P,A;s)\) has a meromorphic continuation to \(\mathbb{C}\) with poles of order at most \(n\) in the set \({\mathcal S}= \sigma_0- \frac 1M N\) and satisfies \(| Z(P,A;s)|\leq C(1+| \tau|^{D(\sigma_p- \sigma)+\varepsilon})\) uniformly in \(s= \sigma+ i\tau\) with \(\sigma> \sigma_0-N\) and \(d(s,{\mathcal S})\geq \varepsilon'\) for any \(\varepsilon,\varepsilon '>0\) and constants \(D= D(P,A)\) and \(C= C(\varepsilon, \varepsilon',P,A)\).
As an application, the theory yields the asymptotic formula
\[
\#\{m\in A\cap \mathbb{Z}^n: P(m)\leq t\}= t^{\sigma_0} Q_0(\log t)(1+ O(t^{-\theta})),
\]
as \(t\to\infty\), with \(\theta>0\) and \(Q_0\) a polynomial constructed explicitly from \(Z(P,A;s)\). A refinement which is probably optimal is obtained in the case \(n=2\). semi-algebraic subset; convergence; meromorphic continuation; Dirichlet series; asymptotic formula Other Dirichlet series and zeta functions, Semialgebraic sets and related spaces, Lattice points in specified regions, Meromorphic functions of several complex variables Meromorphic continuation of Dirichlet series with support in a semi-algebraic subset of \(\mathbb{R}^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main theme of this paper is a systematical study on characteristic variety of a holonomic system with regular singularity on a complex manifold X, especially extensions of index theorems on characteristic cycles. A holonomic system of \({\mathcal D}_ X\)-modules is originally introduced by Sato-Kawai-Kashiwara in 1972, which is a \({\mathcal D}_ X\)- module \({\mathcal M}\) whose dimension of its characteristic variety SS(\({\mathcal M})\) coincides with the dimension of X. For two holonomic systems \({\mathcal N}\) and \({\mathcal M}\) with regular singularity, the author of this paper proves a global index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})=I(SS({\mathcal M}),SS({\mathcal N}))\) where I(A,B) is the multiplicity of the intersection of the cycles A and B. Moreover the author proves the local index theorem: \(\Sigma_ k(- 1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})_ X=I_ X(SS({\mathcal M}),SS({\mathcal N})),\) and the microlocal index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal E}_ X^{{\mathbb{R}}}}({\mathcal N}^{{\mathbb{R}}},{\mathcal M}^{{\mathbb{R}}})_{\xi}=I_{\xi}(SS({\mathcal M}),SS({\mathcal N})).\)
The present work arise from the author's interest to characteristic varieties of the highest-weight modules over complex semi-simple Lie algebras. The detailed treatment of that subject appears in the next paper [Adv. Math. 61, 1-48 (1986)]. index theorems on characteristic cycles; holonomic system of \({\mathcal D}_ X\)-modules; characteristic varieties Ginzburg, V., \textit{characteristic varieties and vanishing cycles}, Invent. Math., 84, 327-402, (1986) Complex manifolds, Analytic subsets and submanifolds, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps Characteristic varieties and vanishing cycles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal A}\) be a spanning subset of \(\mathbb{Z}^{n+1}\) consisting of \(r\) elements, and let \(\alpha\in \mathbb{C}^{n+1}\). In the late eighties Gel'fand, Kapranov and Zelevinskij associated with \({\mathcal A}\) and \(\alpha\) a holonomic system of differential equations in \(\mathbb{C}^r\), called the \({\mathcal A}\)-hypergeometric system with exponent (or parameter) \(\alpha\). Its solutions are called the \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\) [see \textit{I. M. Gel'fand}, \textit{A. V. Zelevinskij} and \textit{M. M. Kapranov}, Funct. Anal. Appl. 23, No. 2, 94-106 (1989; Zbl 0721.33006); Adv. Math. 84, No. 2, 255-271 (1990; Zbl 0741.33011)]. In the literature \({\mathcal A}\)-hypergeometric systems are also called GKZ-systems. The paper under review studies the case of \({\mathcal A}\)-hypergeometric systems associated with monomial curves, which corresponds to the case \(n=1\). All rational \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\) are shown to be Laurent polynomials. This property is proven by counterexample not to be true in the general case \(n>1\). The rational \({\mathcal A}\)-hypergeometric functions with parameter \(\alpha\in \mathbb{Z}^2\) are shown to span a space of dimension at most 2. The value 2 is attained if and only if the monomial curve is not arithmetically Cohen-Macaulay. For all values of \(\alpha\), the holonomic rank \(r(\alpha)\) of the system is proven to satisfy the inequalities \(d\leq r(\alpha)\leq d+1\). Moreover \(r(\alpha)= d+1\) exactly for all \(\alpha\in \mathbb{Z}^2\) for which the space of rational solutions has dimension 2. The inequalities for the holonomic rank have also been obtained using different methods by \textit{M. Saito}, \textit{B. Sturmfels} and \textit{N. Takayama} [Gröbner deformations of hypergeometric differential equations. Springer-Verlag (2000; Zbl 0946.13021)]. GKZ-systems; \({\mathcal A}\)-hypergeometric function; \({\mathcal A}\)-hypergeometric system Eduardo Cattani, Carlos D'Andrea, and Alicia Dickenstein, The \?-hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179 -- 207. Other hypergeometric functions and integrals in several variables, Families, fibrations in algebraic geometry, Deformations of analytic structures, Basic hypergeometric functions in one variable, \({}_r\phi_s\) The \({\mathcal A}\)-hypergeometric system associated with a monomial 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 From the Introduction: ``The importance of approximation in the Nash setting arises from the fact that this category is highly rigid to work with; for instance, it does not admit partitions of unity, a usual and fruitful tool when available. On the other hand, there exist finite differentiable semialgebraic partitions of unity and therefore approximation becomes interesting as a bridge between the differentiable semialgebraic category and the Nash one. There are already relevant results concerning absolute approximation in the Nash setting (e.g. \textit{G. A. Efroymson}'s approximation theorem [Lect. Notes Math. 959, 343--357 (1982; Zbl 0516.14020)]). An even more powerful tool is relative approximation that allows approximation having a stronger control over certain subsets. In the 80s \textit{M. Shiota} developed a thorough study of Nash manifolds and Nash sets (see [Nash manifolds. Berlin etc.: Springer-Verlag (1987; Zbl 0629.58002)] for the full collected work); among other things, he devised approximation on an affine Nash manifold relative to a Nash submanifold. Our purpose is to generalize this type of results developing Nash approximation on an affine Nash manifold relative to a Nash subset.''
From the Abstract: ``This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold \(M\), and here we extend it to functions defined on Nash sets \(X \subset M\) whose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets \(X\). Namely, we prove that (i) \(X\) is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space, and (ii) every function on \(X\) which is Nash on every irreducible component of \(X\) extends to a Nash function on \(M\). Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that \(m\)-dimensional affine Nash manifolds with divisorial corners which are class \(k\) semialgebraically diffeomorphic, for \(k > m^2\), are also Nash diffeomorphic.'' semialgebraic; Nash; approximation; extension; monomial singularity; manifold with corners Baro, Elías; Fernando, José F.; Ruiz, Jesús M., Approximation on Nash sets with monomial singularities, Adv. Math., 262, 59-114, (2014) Nash functions and manifolds, Real-analytic and Nash manifolds, Real-analytic manifolds, real-analytic spaces Approximation on Nash sets with monomial singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is devoted to generic translation flows corresponding to abelian differentials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by the Avila-Forni theorem. Our main result gives first quantitative estimates on their spectrum, establishing the Hölder property for the spectral measures of Lipschitz functions. The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdos and Kahane. abelian differentials; Avila-Forni theorem; Birkhoff integrals Bufetov, A. I.; Solomyak, B., The Hölder property for the spectrum of translation flows in genus two, Israel J. Math., 223, 1, 205-259, (2018) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), Flows on surfaces, Analytic theory of abelian varieties; abelian integrals and differentials, Ergodic theorems, spectral theory, Markov operators, The Hölder property for the spectrum of translation flows in genus two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 considers multipliers of periodic orbits for complex polynomial dynamical systems in one variable, addressing in particular the behavior of such multipliers under perturbation of a polynomial map. The main results show that the multipliers of a set of distinct periodic orbits of a polynomial map on \(\mathbb{C}\) will in general vary independently if the coefficients of the polynomial are perturbed.
Given a monic polynomial \(g(z)\) of degree \(n\) and points \(\beta_1,\dots,\beta_l\) in \(\mathbb{C}\) with exact periods \(r_1,\dots,r_l\) under \(g(z)\), the behavior of the orbit of each \(\beta_i\) is understood via the map \(\text{Mult}_{\beta_i}: f(z) \mapsto (f^{\circ r_i})'(\beta_i(f))\) whose domain is a neighborhood (in the space of monic polynomials of degree \(n\)) of \(g(z)\) defined so that each \(\beta_i\) can be viewed as a holomorphic function on the neighborhood such that \(\beta_i(f)\) has exact period \(r_i\) under \(f(z)\). So the multipliers of the orbits of \(\beta_1,\dots,\beta_l\) vary independently at \(g(z)\) if the gradients of \(\text{Mult}_{\beta_1},\dots,\text{Mult}_{\beta_l}\) at \(g(z)\) are linearly independent in \(\mathbb{C}^n\). The paper concludes that independence is achieved for a generic choice of \(g(z)\) if \(n \geq 3\), \(r_1,\dots,r_l < n\) (with equality allowed if no \(\beta_i\) is a fixed point), and the orbits of \(\beta_1,\dots,\beta_l\) are pairwise disjoint. The conclusion follows from computations of the gradients of \(\text{Mult}_{\beta_1},\dots,\text{Mult}_{\beta_l}\) which show that maps of the form \(z\mapsto z^n\) achieve independence under the given assumptions, combined with an argument that the set of monic polynomial maps achieving linear independence must be Zariski open. complex polynomial dynamics in one variable; multipliers of periodic points DOI: 10.1070/IM2013v077n04ABEH002657 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Polynomials and rational functions of one complex variable, Elementary questions in algebraic geometry One-dimensional polynomial maps, periodic points and multipliers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author gives a unified, abstract treatment of the process of setting up a category \({\mathcal D}\) of geometric objects which -- like \(C^ \infty\)-manifolds, schemes or algebraic spaces -- are constructed by gluing together spaces from a (more) basic category \({\mathcal C}\) (the category of open subspaces of Euclidean space and \(C^ \infty\)-maps in the case of \(C^ \infty\)-manifolds, and the dual of the category of rings for the other two examples). The key ingredients qualifying the category \({\mathcal C}\) for such a local structure are:
(i) A family Sub of distinguished maps called formal subsets, closed under composition and stable (the pullback of a formal subset exists and is a formal subset); the collection of formal subsets of each \(C \in {\mathcal C}\) is essentially small. Thus, schemes and algebraic spaces differ due to different choices of formal subsets in the category of affine schemes.
(ii) For each \(C \in {\mathcal C}\) a collection \(\text{Cov} (C)\) of stable effective covers of \(C\) by formal subsets (epimorphic families of formal subsets which become pullback-stable colimiting cones for their canopies, the latter term referring to the diagram obtained from the domain-objects of a cover of \(C\) by filling in the projections of the obvious pairwise fibered products over \(C)\). The system Cov is required to satisfy certain axioms, amongst which are (essentially) those for a Grothendieck topology.
A morphism of local structures or continuous functor preserves formal subsets, their pullbacks and the specified covers.
A local structure \({\mathcal C}\) (equipped with Sub and Cov) becomes a global structure if each abstract canopy or ``cut-and-paste specification'' definable in it can be realized as that of a stable effective cover of some \(C \in {\mathcal C}\). The main result states that any local structure \({\mathcal C}\) can be universally completed to a global structure \({\mathcal D}\) via a fully faithful continuous functor \({\mathcal C} \to {\mathcal D}\) by iterating (twice) a certain ``plus-construction''. The classical examples (including rigid analytic spaces and Douady's espaces analytique banachique) fit in this mould, and can thus be usefully recognized as universal constructions. \(C^ \infty\)-manifolds; cut-and-paste specification; plus-construction; Banach analytic spaces; schemes; algebraic spaces; local structure; formal subsets; pullback-stable; canopies; Grothendieck topology; continuous functor; global structure; rigid analytic spaces; espaces analytique banachique; universal constructions Feit P., Axiomization of Passage from 'Local' Structure to 'Global' Object 91 pp 485-- (1993) Grothendieck topologies and Grothendieck topoi, Abstract manifolds and fiber bundles (category-theoretic aspects), Generalizations (algebraic spaces, stacks), Research exposition (monographs, survey articles) pertaining to category theory, Topoi, Banach analytic manifolds and spaces Axiomization of passage from ``local'' structure to ``global'' object | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring and let \(K\) be its fraction field. Let \(\mathfrak{X}\) be the formal spectrum of a quotient \(R\langle T_{1},\dots,T_{n}\rangle/I\). In [Bull. Soc. Math. Fr., Suppl., Mém. 39--40, 319--327 (1974; Zbl 0299.14003)], \textit{M. Raynaud} defined its generic fiber \(\mathfrak{X}^{\mathrm{rig}}\) as the maximal spectrum of \(R\langle T_{1},\dots,T_{n}\rangle/I \otimes_{R} K\). This construction extends to a functor from the category of formal \(R\)-schemes locally of topologically finite type to the category of rigid \(K\)-analytic spaces, which is well understood: adding finiteness conditions and localizing by admissible blowups at the source, we may turn it into an equivalence of categories.
Let us now consider formal \(R\)-schemes locally of formally finite type, i.e. locally formal spectra of quotients of \(R[[S_{1},\dots,S_{m}]]\langle T_{1},\dots,T_{n}\rangle\). Berthelot extended the functor rig to this setting at the cost of losing some of its nice properties: some morphisms between rigid spaces in the image fail to come from morphisms between models. In the text under review, Kappen explains how to recover the desired properties by considering uniformly rigid spaces (a notion that he defines) instead of rigid spaces.
The building blocks of uniformly rigid spaces are semiaffinoid spaces, i.e. spectra of quotients of \(R[[S_{1},\dots,S_{m}]]\langle T_{1},\dots,T_{n}\rangle \otimes_{R} K\). Loosely speaking, this means that uniformly rigid spaces (and their functions) are constructed using bounded functions on products of open and closed discs. Those spaces are endowed with a Grothendieck topology and Kappen proves the analogues of the classical results from rigid geometry in this setting: the presheaf of semiaffinoid functions is a sheaf and coherent ideals on semiaffinoid spaces are associated to their ideals of global sections.
We refer to the author's dissertation [Uniformly rigid spaces and Néron models of formally finite type. Univ. Münster (2009; Zbl 1219.14002)] for a more detailed review as well as applications to the notion of Néron models of formally finite type. rigid analytic spaces; uniformly rigid spaces; formal schemes; semiaffinoid; formally finite type C. KAPPEN. Uniformly rigid spaces. Algebra Number Theory, 6(2):341\pm 388, 2012. Rigid analytic geometry Uniformly rigid 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 \(f \in \mathbb{C}[x_1, \dots, x_d]\) with \(f(O) = 0\) be nondegenerate with respect to its Newton polyhedron \(\Gamma\). This means that for any compact face \(\gamma \subset \Gamma\) the corresponding function \(f_{\gamma} = \Sigma_{\alpha \in \gamma}c_{\alpha} x^{\alpha}\) is smooth in \((\mathbb{C}^*)^d\). For \(n \in \mathbb{N}\) the \(n\)-iterated contact locus of \(f\) at \(O\) is defined as \(\mathfrak{X}_{n, O}(f) = \{\alpha \in (t\mathbb{C}[t]/t^{n+1})^d : f(\alpha) = t^n \pmod {t^{n+1}}\} \). Let \(\textit{S}_{f, O}\) be the motivic Milnor fiber of \(f\) at \(O\), that is, \[\textit{S}_{f, O}:= -\lim_{T \rightarrow \infty}Z_{f, O} \in K_0(Var_{\mathbb{C}, \hat{\mu}})[\mathbb{L}^{-1}]\] of the motivic zeta function of \(f\) at \(O\), where \[\hat{\mu} = \lim_{\leftarrow}\mu_n\] is the limit of the groups of roots of 1.
A main result in the article under review claims that, given a sheaf \(\mathcal{F}\) on \(\mathfrak{X}_{n, O}(f)\) there is a spectral sequence of cohomology groups with compact support of \(\mathcal{F}\) which is degenerate at \(E_1\). This is used to calculate in the case of certain sheaves these cohomologies.
The problem of computing \(\textit{S}_{f, O}\) in terms of \(\Gamma\) was proposed by \textit{G. Guibert} [Comment. Math. Helv. 77, No. 4, 783--820 (2002; Zbl 1046.14008)]. Another question, asked by \textit{Lê Dũng Tráng} is about the relation between the monodromy of \((f, O))\), and the monodromy of its restriction to a generic hyperplane \(H\) [J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 409--427 (1975; Zbl 0355.32012)]. In the article that problem is explored in the case of nondegenerate \(f\) by using a refined decomposition of \(\textit{S}_{f, O}\), permitting to revisit \textit{G. Guibert}'s result [Comment. Math. Helv. 77, No. 4, 783--820 (2002; Zbl 1046.14008)] in a more general setting. A main result, considered as a motivic analogue of Lê's theorem in the case of nondegenerate \(f\), represents as a sum \[\textit{S}_{f, O} = \textit{S}_{\tilde{f}, \tilde{O}} + \textit{S}^{\Delta}_{f, x_d, O}\] the first term being the class of the motivic Milnor fiber at the origin of the restriction to the hyperplane \(x_d=0\) with \(\tilde{f} = f(x_1, \dots, x_{d-1}, 0)\), and the second term is the class of motivic Milnor fiber of the pair \((f, x_d)\) at \(O\).
A detailed version with complete proofs of this article could be found in [\textit{Q. T. Lê} and \textit{T. T. Nguyen}, ``Contact loci and motivic nearby cycles of nondegenerate polynomials'', Preprint, \url{arXiv:1903.07262}]. There as a corollary of the latter theorem is obtained also a proof of Kontsevich-Soibelman's integral identity conjecture in the case of nondegenerate \(f\). arc spaces; contact loci; motivic Milnor fiber; nondegenerate singularity Singularities in algebraic geometry, Arcs and motivic integration, Toric varieties, Newton polyhedra, Okounkov bodies, Local complex singularities, Milnor fibration; relations with knot theory Contact loci, motivic Milnor fibers of nondegenerate singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies some generalisations of the classical Verdier hypercovering theorem. This theorem approximates the morphisms \([X,Y]\) in the category of simplicial sheaves and presheaves by simplicial homotopy classes of maps. In its standard form this approximation is given by the comparison function \(\lim_{[p] : Z \rightarrow X} {\pi}(Z,Y) \rightarrow [X,Y]\) defined by mapping the element
\[
\begin{tikzcd} X & Z \lar["{[p]}"]\rar["f"] & Y \end{tikzcd}
\]
to the morphism \(f\cdot p^{-1}\) . Here, \({\pi}(Z,Y )\) denotes simplicial homotopy classes of maps corresponding to a hypercover \(p : Z\rightarrow X\). The theorem asserts that the comparison function is an isomorphism provided \(X\) an \(Y\) are locally fibrant. Based on results developed by him in [Algebraic topology. The Abel symposium 2007. Proceedings of the fourth Abel symposium, Oslo, Norway, August 5--10, 2007. Berlin: Springer. Abel Symposia 4, 185--218 (2009; Zbl 1182.55006)] the author gives a new easier proof of the theorem. This new proof yields a pointed generalisation which does not require the assumption that \(X\) is fibrant. simplicial presheaf; hypercover; cocycle Jardine, The Verdier hypercovering theorem, Canad. Math. Bull. 55 pp 319-- (2012) Homotopy theory and fundamental groups in algebraic geometry, Simplicial sets, simplicial objects (in a category) [See also 55U10], Abstract and axiomatic homotopy theory in algebraic topology, Coverings of curves, fundamental group, Simplicial sets and complexes in algebraic topology The Verdier hypercovering theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\subset \mathbb{R}^{n}\) be a closed semi-algebraic set and let \( g:X\rightarrow \mathbb{R}^{k},\) \(f:X\rightarrow \mathbb{R}^{m}\) be continuous semi-algebraic mappings. The Łojasiewicz exponent at infinity of \(g\) near a fiber \(f^{-1}(\lambda ),\) where \(\lambda \in \mathbb{R}^{m},\) is defined by
\[
\mathcal{L}_{\infty ,f\rightarrow \lambda }(g):=\sup \{\theta :\left| g(x)\right| \geq C\left| x\right| ^{\theta }\text{ \;for }x\in X \text{ \;and \;}f(x)\rightarrow \lambda \}.
\]
The authors prove the following properties of the function \(\mathcal{L} _{\infty ,f\rightarrow \lambda }(g):\)
(1) \(\mathcal{L}_{\infty ,f\rightarrow \lambda }(g)\in \mathbb{Q\cup } \{-\infty ,+\infty \}\);
(2) the function \(\mathbb{R}^{m}\ni \lambda \mapsto \mathcal{L}_{\infty ,f\rightarrow \lambda }(g)\) is upper semi-continuous;
(3) there exists a semi-algebraic stratification of \(\mathbb{R}^{m}=S_{1}\cup S_{2}\cup \dots \cup S_{j}\) such that the function defined in (2) is constant on each stratum.
They apply these results to describe the set of generalized critical values of \(f.\) Lojasiewicz exponent at infinity; generalized critical values; stratification [23]T. Rodak and S. Spodzieja, Łojasiewicz exponent near the fibre of a mapping, Proc. Amer. Math. Soc. 139 (2011), 1201--1213. Affine fibrations, Semialgebraic sets and related spaces, Asymptotic behavior of solutions to equations on manifolds Łojasiewicz exponent near the fibre of a mapping | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Suppose that \(f : \mathbb P^k \dashrightarrow \mathbb P^k\) is a birational map of the \(n\)-dimensional projective space over the complex numbers. Write \(d\) for the degree of \(f\) and \(I(f)\) for the indeterminacy locus. The map \(f\) is said to be generic regular birational if the condition
\[
\sum_{n=0}^\infty \frac{1}{d^n} \log \text{dist}(I(f),f^n(I(f^{-1}))) > -\infty
\]
is satisfied, as is the analogous condition for \(f^{-1}\). This notion was originated by \textit{E. Bedford} and \textit{J. Diller} [Duke Math. J. 128, No. 2, 331--368 (2005; Zbl 1076.37031)] in dimension \(2\), where the authors construct an invariant measure which is mixing; it was subsequently proved by \textit{R. Dujardin} that the measure is hyperbolic and of maximal entropy [Duke Math. J. 131, No. 2, 219--247 (2006; Zbl 1099.37037)]. Many of these results were extended to higher-dimensional settings by \textit{H. de Thélin} and the author [Mém. Soc. Math. Fr., Nouv. Sér. 122, 93 p. (2010; Zbl 1214.37004)], under the hypothesis that \(f\) satisfies \(\dim(I(f)) + \dim(I(f^{-1})) = k-2\).
The subject of this paper is the rate of mixing for these measures. The main result is as follows: for any \(0 < \alpha \leq 2\) there exists a constant \(C_\alpha\) such that for any two \(C^\alpha\) functions \(\phi\) and \(\psi\) on \(\mathbb P^k\), we have
\[
\left| \mu \left( ( \phi \circ f^N ) \psi \right) - \mu(\phi) \mu(\psi) \right| \leq C_\alpha \| \phi \|_{C^\alpha} \| \psi \|_{C^\alpha} d^{-\frac{\alpha s N}{4k}}.
\]
Cremona transformation; invariant measures; decay of correlations G. Vigny, Exponential decay of correlations for generic regular birational maps of {\mathbb{P}^{k}}, Math. Ann. 362 (2015), 1033-1054. Ergodicity, mixing, rates of mixing, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Birational automorphisms, Cremona group and generalizations Exponential decay of correlations for generic regular birational maps of \(\mathbb P^k\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(M\) be a \(d\)-dimensional non-singular real algebraic variety and \(Y \subset M\) a codimension 1 semialgebraic set. Let \(\sigma\) be a sign distribution on \(M \setminus Y\), in other words, a continuous assignment of \(+ 1\) or \(- 1\) to every point of \(M \setminus Y\). The question addressed by this paper is: When does \(\sigma\) correspond to the sign of a regular function \(f\) on \(M\) vanishing on \(Y\)?
For \(M\) compact, the authors give a proof that \(\sigma\) corresponds to the sign of a regular function if and only if the set \(\overline {\sigma^{-1} (-1)} \cap \overline {\sigma^{-1} (1)}\) where \(\sigma\) changes sign is the union of the \((d - 1)\)-dimensional parts of some irreducible components of \(Y\). As is pointed out in the paper, this follows from the work of L. Bröcker using fans, but the proof given here uses more traditional methods from algebraic geometry (general position, algebraic homology) and rests on the factoriality of the ring of regular functions on the complement of the set of points containing an open neighborhood in \(M\) on which \(\sigma\) is constant. The authors go on to consider a smooth function \(\varphi\) on \(M\) whose zero-set \(Y\) is that of a regular function. They show that such a function can be approximated in \(C^ 0 (M)\) by a regular function if the sign distribution of \(\varphi\) corresponds to that of a regular function. Furthermore, if \(Y\) is almost regular (the germs of a complexification are the complexifications of the germs), then the approximation can be done with smooth functions in the compact-open topology of \(C^ \infty (M)\). If we assume simply that the ring of regular functions on \(M\) is factorial, than all of the above results follow more simply, and a recipe for a general example is given which shows that this assumption is necessary if \(M\) is not compact.
In the analytic case, one considers a non-singular real-analytic manifold \(M\) and a codimension 1 subset \(Y\) which is the common zeros of a finite number of global analytic functions. The authors prove that the same results hold with the notion of a component of \(Y\) replaced by a minimal proper closed subspace which is itself the common zeros of a finite number of global analytic functions and under the additional hypothesis that, for every such component \(Z\), either the ideal of functions vanishing on \(Z\) is principal or \(Z\) is analytically irreducible. The topology used for approximation is the Whitney topology. -- As an application the authors prove the following theorem:
Let \(M\) be a \(d\)-dimensional real analytic manifold, let \(I\) be a coherent ideal sheaf in the sheaf of analytic functions on \(M\), and let \(Y\) be the support of the quotient sheaf. Suppose that \(I\) is locally principal and that \([Y] = 0\) in \(H^ \infty_{n - 1} (M, \mathbb{Z}_ 2)\). Then \(I\) is principal. approximation by regular function; \(R\)-space; real algebraic variety; semialgebraic set; sign; factoriality of the ring of regular functions; real-analytic manifold F. Acquistapace, F. Broglia:More about signatures and approximation, to appear in Geom. Dedicata. Semialgebraic sets and related spaces, Real-analytic and semi-analytic sets, Real-analytic manifolds, real-analytic spaces More about signatures and approximation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This work introduces a new framework for understanding uniform behavior of singularity measures such as Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity, and F-rational signature, for ideals varying in families of rings. Namely, the author calls the combination of a ring map $R \rightarrow A$ with an ideal $I$ of $A$ an \textit{affine $I$-family} if $A/I$ is module-finite over $R$, $I \cap R = 0$, and certain dimension formulas hold. (This is a restricted version of Lipman's notion of an $I$-family from [\textit{J. Lipman}, Lect. Notes Pure Appl. Math. 68, 111--147 (1982; Zbl 0508.13013)]). Then one analyzes the ideals $I(\mathfrak p)$, $I(\mathfrak p)^{[p^e]}$ (when char $k(\mathfrak p) = p>0$) and $I(\mathfrak p)^n$ for $\mathfrak p \in $Spec$(R)$ and various $e, n \in \mathbb N$ in the rings $R(\mathfrak p)$, where the notation $(\mathfrak p)$ means to tensor over $R$ with the residue field $k(\mathfrak p)$ of $\mathfrak p$.
This framework allows the author to recover Lipman's result [loc. cit.] on upper semicontinuity of Hilbert-Samuel multiplicity on the prime spectrum, by showing that the \textit{terms} defining Hilbert-Samuel multiplicity as a limit are also upper semicontinuous in the family.
He also recovers some of his own results (see [Compos. Math. 152, No. 3, 477--488 (2016; Zbl 1370.13006)]) on semicontinuity of Hilbert-Kunz multiplicity on the prime spectrum, again by analyzing the terms, where in this case one has an affine $I$-family $R \rightarrow S$ with reduced fibers of dimension $=$ height$(I)$, where $R$ is F-finite. He further obtains upper semicontinuity of Hilbert-Kunz multiplicity in an affine family where char $R=0$ and the characteristics of the fibers can vary but all residue fields of $R$ are F-finite when they are positive characteristic. In particular, when $R = \mathbb Z$, this answers a question of Claudia Miller from [\textit{H. Brenner} et al., J. Algebra 372, 488--504 (2012; Zbl 1435.13014)] in pursuance of obtaining a sensible notion of Hilbert-Kunz multiplicity in equal characteristic zero.
The author also parlays his methods to show that for local algebras essentially of finite type over a prime characteristic field, the infimum in Hochster and Yao's definition of F-rational signature is actually achieved. He thus recovers a special case of the result in [\textit{M. Hochster} and \textit{Y. Yao}, ``F-rational signature and drops in the Hilbert-Kunz multiplicity'', Preprint] that the F-rational signature of the ring is positive if and only if the ring is F-rational. multiplicity; Hilbert-Samuel polynomial; Hilbert-Kunz multiplicity; semicontinuity; families Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Singularities in algebraic geometry, Deformations of singularities, Fibrations, degenerations in algebraic geometry On semicontinuity of multiplicities in families | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The goal of this paper is to construct a category of constructible convergent \(F\)-\(\nabla\)-modules over a proper smooth curve \(X\) over a \(p\)-adic field with good reduction, which under the specialization map, is equivalent to the category of perverse holonomic \(F\)-\(\mathcal D^\dagger_{\hat X, \mathbb Q}\)-modules. One may view this as an ``overconvergent Deligne-Kashiwara correspondence''.
Let \(\mathcal V\) be a complete discrete valuation ring of mixed characteristic with perfect residue field \(k\) and fraction field \(K\), and let \(X\) denote a geometrically connected smooth proper curve over \(\mathcal V\). Let \(X_K^{\mathrm{an}}\) denote the Berkovich space associated to \(X\). The author defines a constructible overconvergent \(\nabla\)-module to be an \(\mathcal O_{X_K^{\mathrm{an}}}\)-module \(E\) with a convergent connection, such that there exists a finite covering of the special fiber \(X_k\) by locally closed subsets \(Y\) with the property that if \(i_Y:]Y[ \hookrightarrow X_K^{\mathrm{an}}\) denotes the inclusion map, then \(i_Y^{-1}E\) is a coherent \(i_Y^{-1}\mathcal O_{X_K^{\mathrm{an}}}\)-module. (Note that the pullbacks are taken for the Berkovich topology, so when \(Y\) is an open subspace of \(X_k\), \(i_Y^{-1}\mathcal O_{X_K^{\mathrm{an}}}\) is the limit of functions on strict neighborhoods of \(]Y[\).) The first main result of this paper is to describe the direct image under the specialization map \(\mathrm{Rsp}_* E_0\), where \(E_0\) is the corresponding module for the rigid topology. Explicitly, if \(D\) is a smooth relative divisor on \(X\) with affine open complement \(\mathrm{Spec}\; A\). Let \(A_K^\dagger\) denote the generic fiber of the weak completion of \(A\). For each \(a \in D_K\), write \(\mathcal R_a\) for the corresponding Robba ring at \(a\) over the residue field \(K(a)\). Then a constructible convergent \(\nabla\)-module is equivalent to the following data (for sufficiently large divisor \(D\)): (1) an overconvergent \(\nabla\)-module \(M\) over \(A_K^\dagger\), (2) for each \(a \in D_K\), a finite dimensional \(K(a)\)-vector space \(H_a\), and (3) a horizontal \(A_K^\dagger\)-linear map
\[
M \to \bigoplus_{a \in D_K}\mathcal R_a \otimes_{K(a)} H_a.
\]
The image of \(E\) under the specialization, namely \(\mathrm{Rsp}_*E_0\) is roughly just (the sheaf version of) the complex \(M \to \bigoplus_{a \in D_K}\delta_a \otimes_{K(a)} H_a\), where \(\delta_a = \mathcal R_a / \mathcal O_a^{\mathrm{an}}\) is the arithmetic analogue of the delta \(D\)-module. This is a perverse complex of \(\mathcal D_{\hat X, \mathbb Q}^\dagger\)-modules in the sense that it has \(\mathcal O_{\hat X, \mathbb Q}\)-cohomology in degree \(0\), finite support in degree \(1\), and no other cohomology.
Now, if one starts with a constructible overconvergent \(F\)-\(\nabla\)-module \(E\), by a theorem of \textit{D. Caro} [Bull. Soc. Math. Fr. 137, No. 4, 453--543 (2009; Zbl 1300.14021)], \(\mathrm{Rsp}_*E_0\) is a holonomic \(F\)-\(\mathcal D_{\hat X, \mathbb Q}^\dagger\)-module. The main result of this paper says that this construction in fact gives an equivalence of categories between the category of constructible overconvergent \(F\)-\(\nabla\)-modules and the category of perverse holonomic \(F\)-\(\mathcal D_{\hat X, \mathbb Q}^\dagger\)-modules.
The paper is elegantly written, mostly self-contained. arithmetic \(\mathcal D\)-modules; constructible sheaves; overconvergent differential modules; overconvergent Deligne-Kashiwara correspondence; Berkovich space Le Stum, Bernard, Constructible \(\nabla\)-modules on curves, Selecta Math. (N.S.), 20, 2, 627-674, (2014) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, \(p\)-adic cohomology, crystalline cohomology Constructible \(\nabla \)-modules 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 We prove that the radii of convergence of the solutions of a \(p\)-adic differential equation \(\mathcal{F}\) over an affinoid domain \(X\) of the Berkovich affine line are continuous functions on \(X\) that factorize through the retraction of \(X\to\Gamma\) of \(X\) onto a finite graph \(\Gamma\subseteq X\). We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on \(X\) is controlled by a \textit{finite} family of data. \(p\)-adic differential equations; Berkovich spaces; radius of convergence; Newton polygon; spectral radius; controlling graph; finiteness Andrea Pulita, The convergence Newton polygon of a \textit{p}-adic differential equation I: Affinoid domains of the Berkovich affine line, preprint, 2012, 44 pp., http://arxiv.org/abs/1208.5850. \(p\)-adic differential equations, Rigid analytic geometry The convergence Newton polygon of a \(p\)-adic differential equation. I: Affinoid domains of the Berkovich affine line | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a perfect field of characteristic \(p>0\) and let \(X\) be a smooth separated scheme over \(k\) of dimension \(d\). Let \(D\) be a divisor with simple normal crossings, \(U=X-D\) its complement and \(D_1,\dots, D_h\) its irreducible components. Let \(\Lambda\) be a local ring over \(\mathbb{Z}[1/p,\zeta_p]\) and let \(\mathcal F\) be a locally constant constructible sheaf of free \(\Lambda\)-modules of finite rank on \(U\). Assume that the ramification of \(\mathcal F\) is non-degenerate along \(D\), meaning that the ramification along \(D\) is uniformly controlled at the generic points of the irreducible components and it is satisfied after shrinking \(X\) to a neighborhood of each generic point. Then one can define the characteristic cycle \(\mathrm{Char}(\mathcal F)\) of \(\mathcal F\) as an effective \(d\)-cycle with rational coefficients on the cotangent bundle \(T^*X\).
Under these hypotheses, the author shows that the characteristic cycle has coefficients in \(\mathbb{Z}[1/p]\). At the generic point of each irreducible component of the divisor at the boundary, the characteristic cycle of a sheaf is defined by some differential forms. For a morphism \(f: X' \to X '\) between smooth schemes over \(k\), of dimension \(d\) and \(d'\), with \(U =X -D\) and \(U' =X' -D'\), such that \(f^{-1}(U) =U'\), one defines \(f\) to be non-characteristic with respect to \(\mathcal F\) if \(f\) is non-characteristic with respect to \(\mathrm{Char}(\mathcal F) \in Z_d(T^*(X)_{\mathbb{Q}}\). The construction of the characteristic cycles commutes with pull-backs by non-characteristic morphisms, a result that allows one to deduce a characterization of the support of the characteristic cycle in terms of the restrictions to curves transversally meeting the boundary.
The results of this paper are non-logarithmic variants of the logarithmic version considered by the same author in [J. Inst. Math. Jussieu 8, No. 4, 769--829 (2009; Zbl 1177.14044)]. Saito, T.: Wild ramification and the cotangent bundle. J. Algebr. Geom. \textbf{26}(2017), 399-473 (2016) Étale and other Grothendieck topologies and (co)homologies, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Wild ramification and the cotangent bundle | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a field of characteristic \(0\), \(n\in N\) and \(\mathbb{P}^n = \mathbb{P}^n(K)\). In this paper the authors study the postulation of general fat point schemes of \(\mathbb{P}^3\) with multiplicity up to \(5\). A fat point \(mP\) is a zero dimensional subscheme of \(\mathbb{P}^3\) supported on a point \(P.\) A general fat point scheme \(Y = m_1P_1+\dots+m_kP_k\), with \(m_1,\dots, m_k\) is a general zero-dimensional scheme such that its support \(Y_{\mathrm{red}}\) is a union of \(k\) points and for each \(i\) the connected component of \(Y\) supported on \(P_i\) is the fat point \(m_iP_i\). Studying the postulation of \(Y\) means to compute the dimension of the space of hypersurfaces of any degree containing the scheme \(Y.\) In other words this problem is equivalent to computing the dimension \(\delta\) of the space of homogeneous polynomials of any degree vanishing at each point \(P_i\) and with all their derivatives, up to multiplicity \(m_i- 1\), vanishing at \(P_i\). \(Y\) has good postulation if \(\delta\) is the expected dimension, that is, either the difference between the dimension of the polynomial space and the number of imposed conditions or just the dimension of the polynomial space (when \(\delta\) would exceed it).
In this paper, they focus on the case of general fat point schemes \(Y\subset \mathbb{P}^3\). In this case a general conjecture which characterizes all the general fat point schemes not having good postulation was proposed by Laface and Ugaglia. The good postulation of general fat point schemes of multiplicity \(4\) was proved for degrees \(d\geq 41\) by the first two authors. Then Dumnicki showed how to check the cases with degree \(9\leq d \leq 40\). The authors, here, prove that if \(Y\subset \mathbb{P}^3\) is a general union of \(w\) \(5\)-points, \(x\) \(4\)-points, \(y\) \(3\)-points and \(z\) \(2\)-points with fixed non- negative integers \(d,w, x, y, z\) such that \(d\geq 11\), then \(Y\) has good postulation with respect to degree-\(d\) forms. They also classify the exceptions in degree \(9\) and \(10\). Polinomial interpolation; fat points; zero-dimensional scheme; projective space E. Ballico, M. C. Brambilla, F. Caruso, and M. Sala, Postulation of general quintuple fat point schemes in \Bbb P³, J. Algebra 363 (2012), 113 -- 139. Projective techniques in algebraic geometry, Vector and tensor algebra, theory of invariants, Numerical interpolation Postulation of general quintuple fat point schemes in \(\mathbb P^3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use topological methods to study various semicontinuity properties of the local spectrum of singular points of algebraic plane curves and spectrum at infinity of polynomial maps in two variables. Using the Seifert form, the Tristram-Levine signatures of links, and the associated Murasugi-type inequalities, we reprove (in a slightly weaker form) a result obtained by \textit{J. H. M. Steenbrink} [Invent. Math. 79, 557--565 (1985; Zbl 0568.14021)] and \textit{A. N. Varchenko} [Sov. Math., Dokl. 27, 735--739 (1983; Zbl 0537.14003); translation from Dokl. Akad. Nauk SSSR 270, 1294--1297 (1983)] on semicontinuity of the spectrum of singular points under deformations and result of the second author and \textit{C. Sabbah} [Abh. Math. Semin. Univ. Hamb. 69, 25--35 (1999; Zbl 0973.32014)] on semicontinuity of the spectrum at infinity regarding families of polynomial maps. We also relate the spectrum at infinity of a polynomial map with the collection of the spectra of singular points of a chosen fiber. plane curves; local spectrum of singular points; Seifert form; semicontinuity of the spectrum Maciej Borodzik and András Némethi, Spectrum of plane curves via knot theory, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 87 -- 110. Milnor fibration; relations with knot theory, Deformations of singularities, Variation of Hodge structures (algebro-geometric aspects), Plane and space curves, Period matrices, variation of Hodge structure; degenerations Spectrum of plane curves via knot theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves the existence of a form of Puiseux-type parametrization of complex analytic singularities, under general conditions. A classical result in this direction is the Abhyankar-Jung theorem, for a germ of an \(m\)-dimensional analytic set such that, via a general linear projection on a \(m\)-dimensional linear space, the discriminant locus is contained in a normal crossings divisor. In this case a parametrization is achieved by means of holomorphic functions.
In the present paper, the author proves that given an \(m\)-dimensional analytic set \({\mathcal A}\subset\mathbb{C}^{m+ n}\), such that the projection \((z_1,\dots, z_{m+n})\to (z_1,\dots, z_m)\) induces a finite morphism \(\pi:{\mathcal A}\to \mathbb{C}^m\) near \(0\in{\mathcal A}\), then there is an integer \(k> 0\) and Laurent series \(s_1,\dots, s_n\), convergent near \(0\), such that for any function \(f\), vanishing on \({\mathcal A}\) (near \(0\)), we have
\[
f(z^k_1,\dots, z^k_m, s_1(z_1,\dots, z_m),\dots, s_n(z_1,\dots, z_m))= 0.
\]
Moreover, we have some ``control'' on the series \(s_1,\dots, s_n\). Namely, if \(\delta(z_1,\dots, z_m)\) is an analytic function vanishing on the discriminat locus of \(\pi\) and \(\sigma\subset\mathbb{R}^m\) is a cone of the Newton polyhedron of \(\delta\), then the series \(s_1,\dots, s_n\) can be chosen in such a way that, for any \(j\in \{1,\dots, n\}\), if \(a_{r_1\cdots r_m} t^{r_1}_1\cdots t^{r_m}_m\) is any term if the expansion of \(s_j\) with \(a_{r_1\cdots r_m}\neq 0\), then the point \((r_1,\dots, r_m)\) is in \(\sigma\). In this general situation a similar result, but using power rather than Laurent series cannot be achieved. The author uses her main result to obtain another proof of the theorem of Abhyankar-Jung.
Although the proofs are rather technical the paper is reasonably self-contained. There are several sections where the necessary background on polyhedra, cones, wedges, etc., is discussed and propositions essential in the proof of the main theorem are proved. Puiseux parametrization; Newton polygon; cone; wedge; quasi-ordinary singularity Aroca F., Proc. Amer. Math. Soc. 132 (10) pp 3035-- (2004) Local complex singularities, Germs of analytic sets, local parametrization, Toric varieties, Newton polyhedra, Okounkov bodies Puiseux parametric equations of analytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors study the Hilbert scheme parametrizing schemes and families of subschemes of a fixed projective space \(\mathbb{P}^n\) with a fixed Hilbert polynomial \(p(t)\). \textit{G. Gotzmann} [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)] gave an optimal upper bound of the Castelnuovo-Mumford regularity of schemes parametrized by a given Hilbert scheme. However, in many cases this bound turns out to be far higher than the regularity of the most meaningful schemes from a geometric point of view. Hence, it is natural and interesting to study loci of Hilbert schemes with bounded regularity.
The authors tackle the problem of determining equations for these loci, starting from the classical construction of the Hilbert scheme as subscheme of a suitable Grassmannian. By semicontinuity, imposing an upper bound of the regularity corresponds to an open condition, so these loci turn out to be locally closed subschemes of the Grassmannian. The authors prove that such subschemes can be defined by equations of degree at most \(\deg p(t)+2\) in an open subscheme of the Grassmannian defined by the non-vanishing of several linear forms. The tools and techniques used in this paper come from another paper of the third author with \textit{J. Brachat}, \textit{P. Lella} and \textit{B. Mourrain} [``Extensors and the Hilbert scheme'', to appear on Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2014); \url{doi:10.2422/2036-2145.201407\_003}]. Hilbert scheme; Castelnuovo-Mumford regularity; Borel-fixed ideal; marked basis Ceria, M.: JMBConst.lib. A library for Singular 4-0-2 which constructs J-Marked Schemes. Available at: http://www.singular.uni-kl.de/svn/trunk/Singular/LIB/JMSConst.lib (2012) Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The locus of points of the Hilbert scheme with bounded regularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the stack \(\mathcal{L}og_X\) parametrizing log schemes over a log scheme \(X\), and weak and strong properties of log morphisms via \(\mathcal{L}og_X\), as defined by \textit{M. C. Olsson} [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 747--791 (2003; Zbl 1069.14022)]. We give a concrete combinatorial presentation of \(\mathcal{L}og_X\), and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato's chart criterion of logarithmic smoothness. Logarithmic algebraic geometry, log schemes Logarithmically regular morphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\phi\) be the Hilbert function of a finite subscheme of \( \mathbb P^2\). It is known that the family of subschemes with Hilbert function \(\phi\) is irreducible, and the generic element, \(Z^\phi\), is reduced. Consider the linear system of curves of degree \(d\) passing through \(Z^\phi\), for any given \(d\). We say that \(\phi\) is \(d\)-uniform if this linear system has no fixed component. In this case we define the \(d\)-saturation, \(\overline \phi_d\), of \(\phi\) to be the Hilbert function of the base locus of this linear system (which contains \(Z^\phi\) but may contain more), i.e.\ the Hilbert function of the saturation of the ideal generated by the degree \(d\) component of \(I_{Z^\phi}\). Note that term by term, \(\overline \phi_d\) is greater than or equal to \(\phi\). The main result of this paper shows how to compute \(\overline \phi_d\). As a corollary, the authors characterize those Hilbert functions \(\phi\) and degrees \(d\) for which \(\overline \phi_d = \phi\), i.e.\ the ideal generated by the components of \(I_{Z^\phi}\) of degrees \(\leq d\) is already saturated and defines \(Z^\phi\) itself. The results are similar in flavor to the Cayley-Bacharach theorems, and indeed liaison is an important tool in the proofs. These results are presented very nicely via \(h\)-vectors, which are the first difference of the Hilbert functions, \(\phi\), and are referred to here as ``staircases''. The precise criterion for \(\overline \phi = \phi\) (omitted here) is called the ``one-step property.'' These results can be rephrased as follows. We denote by \({\mathcal L}_{Z^\phi}(d) := \pi^* {\mathcal O}(d)(-E)\) the line bundle on the blow-up of \(\mathbb P^2\) at \(Z^\phi\) whose sections correspond to curves of degree \(d\) passing through \(Z^\phi\). Then \({\mathcal L}_{Z^\phi} (d)\) is globally generated if and only if \(\phi\) is \(d\)-uniform and has the one-step property. The authors present, and answer, the next natural question: Given \(\phi\), what is the least degree \(d\) for which \({\mathcal L}_{Z^\phi}(d)\) is very ample? The answer is somewhat technical. liaison; very ample; global generation; saturation; Cayley-Bacharach Geertsen J. A., Math. Z. 245 pp 155-- (2003) Linkage, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Low codimension problems in algebraic geometry Saturation theory and very ample Hilbert 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 A scheme \(X\subseteq \mathbb{P}^n= \mathbb{P}^n_k\) of ``fat points'' is a 0-dimensional scheme defined by a homogeneous ideal
\[
I={\mathfrak p}_1^{m_1}\cap {\mathfrak p}_2^{m_2}\cap \dots\cap{\mathfrak p}_s^{m_s}\subseteq k[x_0,x_1,\dots, x_n],
\]
where \(k\) is an algebraically closed field with \(\text{char }k=0\) and each \({\mathfrak p}_i\) is the homogeneous ideal of a point \(P_i\). Every vector space \(I_t\) represents the linear system of hypersurfaces of degree \(t\) having multiplicity at least \(m_i\) at each \(P_i\), \(i=1,\dots, r\). Let the \(P_i\)'s be generic: it is an open problem (even for \(n=2\)), to determine the Hilbert function of \(X\). In this paper the case \(r\leq n+1\) is studied: under this hypothesis the ideal \(I\) is a monomial one and this allows to find the Hilbert function (here the ``genericity'' of the \(P_i\)'s is simply the fact that no \(s\) of them lie is an \((s-1)\)-linear space). Then the case \(r=2\) is considered, and a minimal system of generators and the graded Betti numbers of \(I\) are given by showing that in this case \(I\) is a splittable ideal, i.e. that there are two monomial ideals \(U,V\) such that \(I=U+V\), the minimal set of generators \(T(I)\) of \(I\) is the disjoint union of \(G(U)\), \(G(V)\) and there is a splitting function \(G(U\cap V)\to G(U)\times G(V)\) satisfying some technical hypotheses. The Betti numbers of \(I\) can be obtained from those of \(U, V\), which are computed. Eventually it is shown that \(I\) is splittable when \(m_1=\dots= m_r=2\). minimal resolution; fat points; Hilbert function; graded Betti numbers G. Fatabbi, Ideals of fat points and splittable ideals, inThe Curves Seminar at Queen's, Vol. 10, Queen's Papers in Pure and Applied Mathematics, Vol. 102, pp. 242--255. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Exposé II H: Ideals of fat points and splittable ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal M_1,\mathcal M_2\) be manifolds in a finite-dimensional real Hilbert space \(\mathcal K\), and \(\mathcal M=\mathcal M_1\cap\mathcal M_2\). Denote by \(\pi_1\), \(\pi_2\) and \(\pi\) the corresponding metric projections. Given a point \(B\in \mathcal K\), the sequence \((B_k)\) of alternating projections is defined by \(B_0=B,\, B_{k+1}=\pi_1(B_k)\) if \(k\) is even, and \(B_{k+1}=\pi_2(B_k)\) if \(k\) is odd. A classical result of von Neumann (1949) asserts that if \(\mathcal M_1\), \(\mathcal M_2\) are affine linear manifolds, then \((B_k)\) converges to \(\pi(B)\). The authors study the convergence of this sequence in the case of nonlinear manifolds. The key notion used in the paper is that of nontangential points. A point \(A\in \mathcal M_1\cap\mathcal M_2\) is called nontangential if \(T_{\mathcal M_1}(A)\cap T_{\mathcal M_2}(A)=T_{\mathcal M_1\cap\mathcal M_2}(A)\) (where by \(T_{\mathcal N}(A)\) one denotes the tangent space to a manifold \(\mathcal N\) at a point \(A\in \mathcal N),\) a condition that can be also expressed in terms of the positivity of the angle \(\sigma(A)\) between \(\mathcal M_1,\mathcal M_2\) at \(A\). A point \(A\) is called transversal if \(T_{\mathcal M_1}(A)+ T_{\mathcal M_2}(A)=\mathcal K\). Since any transversal point is nontangential, the results obtained in the present paper extend those obtained by \textit{A. S. Lewis} and \textit{J. Malick} [Math. Oper. Res. 33, No. 1, 216--234 (2008; Zbl 1163.65040)] for transversal points.
The main result of the paper (Theorem 5.1) asserts that if \(\mathcal M_1\), \(\mathcal M_2\) are \(C^2\)-manifolds and \(A\in \mathcal M_1\cap\mathcal M_2\) is a nontangential point, then for every \(\varepsilon >0\) and \( 1>c>\sigma(A),\) there exists \(r>0\) such that for any \(B\in\mathcal B(A,r)\) (the open ball) the corresponding sequence \((B_k)\) of alternating projections converges to a point \(B_\infty\in \mathcal M_1\cap\mathcal M_2 ,\; \|B_\infty-\pi(B)\|\leq\varepsilon \|B-\pi(B)\|,\) and \(\|B_\infty- B_k\|\leq\)const\(\cdot c^k \|B-\pi(B)\|\).
The second part of the paper is concerned with the case of real algebraic varieties.
Numerous examples, illustrative drawings, and a numerical example concerned with the correlation matrix of rank \(k\) that is closest (in the Hilbert-Schmidt norm) to an \(n\times n\) symmetric matrix \(B,\) are also included. The bibliography at the end of the paper counts 55 items. metric projection; alternating projections; algorithms; rate of convergence; smooth manifold; low-rank approximation; real algebraic variety Harshman, R.A.: Foundations of the PARAFAC procedure: models and conditions for an ''explanatory'' multi-modal factor analysis. UCLA Working Papers in Phonetics \textbf{16}, 1-84 (1970) Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc., Applications of operator theory in optimization, convex analysis, mathematical programming, economics, Numerical mathematical programming methods, Semialgebraic sets and related spaces Alternating projections on nontangential manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:=(f_1,\dots ,f_m):\mathbb {R}^n\rightarrow \mathbb {R}^m\) be a map. We say that \(f\) is polynomial if its components \(f_k\) are polynomials. The map \(f\) is regular if its components can be represented as quotients \(f_k=\frac {g_k}{h_k}\) of two polynomials \(g_k,h_k\) such that \(h_k\) never vanishes on \(\mathbb {R}^n\). More generally, the map \(f\) is Nash if each component \(f_k\) is a Nash function, that is, an analytic function whose graph is a \textsl{semialgebraic set}. Recall that a subset \(\mathcal {S}\subset \mathbb {R}^n\) is semialgebraic if it has a description as a finite boolean combination of polynomial equalities and inequalities. By Tarski-Seidenberg's principle the image of a map whose graph is a semialgebraic set is a semialgebraic set. Consequently, the images of polynomial, regular and Nash maps are semialgebraic sets. In 1990 {Oberwolfach reelle algebraische Geometrie} week, the second author proposed a kind of converse problem: To characterize the semialgebraic sets in \(\mathbb {R}^m\) that are either polynomial or regular images of some \(\mathbb {R}^n\). In the same period Shiota formulated a conjecture that characterizes Nash images of \(\mathbb {R}^n\), that has been recently proved by the first author. In this survey we collect our main contributions to these problems and present some new examples. We have approached our contributions along the last two decades in three directions:
(i) To construct explicitly polynomial and regular maps whose images are the members of large families of semialgebraic sets whose boundaries are piecewise linear.
(ii) To find obstructions to be polynomial/regular images of \(\mathbb {R}^n\).
(iii) To prove Shiota's conjecture and some relevant consequences. semialgebraic set; polynomial map; polynomial image; regular map; regular image; Nash map; Nash image; well-welded set Fernando, J. F.; Gamboa, J. M.; Ueno, C., Polynomial, regular and Nash images of Euclidean spaces, (Ordered Algebraic Structures and Related Topics, Contemp. Math., vol. 697, (2017), Amer. Math. Soc. Providence, RI), 145-167 Semialgebraic sets and related spaces, Real-valued functions in general topology, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) Polynomial, regular and Nash images of Euclidean 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 derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if \(\Omega\) is a skew-Hermitian matrix and \(t\) is a sufficiently small scalar, then there exists a polynomial of degree \(n\) in \(t\Omega\) (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of \(e^{t\Omega}\) with error \(O(t^{2n+1})\). We prove this fact and then leverage it to derive high-order approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices. matrix manifold; retraction; geodesic; Grassmannian; Stiefel manifold; Riemannian exponential; matrix exponential; polar decomposition; unitary group; geometric mean; Karcher mean E. S. Gawlik and M. Leok, \textit{High-Order Retractions on Matrix Manifolds Using Projected Polynomials}, preprint, , 2017. Other matrix algorithms, Local Riemannian geometry, Grassmannians, Schubert varieties, flag manifolds, Factorization of matrices High-order retractions on matrix manifolds using projected 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 The paper under review is devoted to exhibiting a simplified construction of a rational simplicial completion for any rational simplicial semifan in a finite-dimensional real vector space equipped with a rational structure. Unlike Sumihiro's proof of his famous Equivariant Compactification Theorem, the proof in the paper relies on the ideas of Ewald and Ishida and is constructive by means of polyhedral geometry. The paper consists of 5 sections.
In Sect. 1 the author fixes some notation and also reminds basic facts about rational structures \(W\) on \(\mathbb R\)-vector spaces \(V\). In order to make use of techniques from polyhedral geometry, all the necessary terminology and some basic topological and combinatorial properties of polycones are introduced. Then (Theorem 1.13) it is proved that every \(W\)-polycone has a \(W\)-decomposition, and a \(W\)-polycone has a unque \(W\)-decomposition if and only if it is sharp or a line. This is related to proving the well-known fact that every fan has a simplicial strict subdivision (1.16).
In Sect. 2 for a \(W\)-semifan \(\Sigma\) a combinatorial description of its support frontier \(|\Sigma|\) is given. Namely, let \(\Sigma_{\max}\) to be the set of maximal elements of \(\Sigma\) and \(\Sigma_k=\{\sigma\in\Sigma|\;\dim(\sigma)=k\}\). Consider \(D(\Sigma)=\Sigma_{\max}-\Sigma_n\) and \(F(\Sigma)\) to be the subset of \(\Sigma_{n-1}\) s.t. for each element \(\sigma\) there exists one and only one element \(\tau\in\Sigma_n\) with \(\sigma\leq\tau\). Then (Theorem 2.8): \(\mathrm{fr}(|\Sigma|)=(\cup D(\Sigma))\cup(\cup F(\Sigma))\).
In Sect. 3 for a \(W\)-fan \(\Sigma\) three notions concerning the properties of its \(W\)-extension \(\Sigma'\) are introduced, namely, relative simpliciality, separability and tight separability. To introduce the main notion of strong \(W\)-completion, construction of which is the aim of the paper, \(W\)-(quasi)packings of \(\Sigma\) as special types of extensions are defined, based on the three geometrical types of extensions above (3.6).
In Sect. 4 (subsections A,B,C) more techniques based on the previous considerations is introduced to construct a complete semifan from a given fan \(\Sigma\) (in general, not a completion of \(\Sigma\) yet). A strong \(W\)-completion of \(\Sigma\) is a pair of its \(W\)-extensions, consisting of its \(W\)-packing and a \(W\)-completion of the latter one that is relatively simplicial over \(\Sigma\). To construct a packing of \(\Sigma\) each of its 1-dimensional cones is ``packed'' recursively by induction on the dimension of \(V\).
In Sect. 5 the main result of the paper (Theorem 5.3) is proved: every \(W\)-fan has a strong \(W\)-completion. The crucial observation for this is Lemma 5.2, which is based on the results of Sect. 3,4: if \(\Sigma\) is a \(W\)-fan with \(\Sigma_1\neq\varnothing\) and has a \(W\)-packing then it has a strong \(W\)-completion. As a corollary of the main theorem (5.3) one gets Corollary 5.4: every simplicial \(W\)-semifan has a simplicial \(W\)-completion. fan; semifan; completion; polycone; extension; subdivision F. Rohrer, \textit{Completions of fans}, arXiv:1107.2483. Toric varieties, Newton polyhedra, Okounkov bodies, Completion of commutative rings Completions of fans | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 authors apply Fourier-Mukai transforms to study coherent systems on elliptic curves. Let \(X\) be an elliptic curve. A coherent system \((E,V)\) of type \((r,d,k)\) on \(X\) is a pair consisting of a vector bundle \(E\) of rank \(r\) and degree \(d\) and a \(k\)-dimensional subspace \(V\) of \(H^0(X,E)\). Semistability of such objects depends on a positive rational parameter \(\alpha\) and was defined in \textit{J. Le Potier} [Systèmes cohérents et structures de niveau, Astérisque 214, (1993; Zbl 0881.14008)]. In the same paper, moduli spaces of semsitable coherent systems were constructed. They are usually denoted by \(G ({\alpha}; r,d,k)\).
In order to study this moduli spaces, the authors consider a Fourier-Mukai transform \(\Phi: D(X) \to D(X)\) whose kernel is a sheaf on \(X \times X\). Since such a functor acts naturally on sheaf maps, it is enough to determine which transforms leave invariant the category of coherent systems. To this aim, the authors start by studying the behaviour under Fourier-Mukai transform of the category whose objects are maps \(\phi: V \otimes O_X \to F\), where \(V\) is a finite dimensional vector space and \(F\) a coherent sheaf. The category of coherent system is then a subcategory of the latter. Once the problem solved for the bigger category, one can use the spectral sequence \(F^{p,q}_2 = H^p (X, \Phi^q(F)) \to H^{p+q} (Y, \Phi(F))\) to understand how the Fourier-Mukai acts on coherent systems.
In general, the behaviour of (semistable) coherent systems under Fourier-Mukai is not easy to determine. However, there are two moduli spaces \(G_0(r,d,k)\) and \(G_L(r,d,k)\), with \(r < k\), for which the authors establish the preservation of stability. Finally, they are able to describe Fourier-Mukai transforms \(\Phi_a\), with \(a\) positive integer, such that any \(\Phi_a\) induces isomorphisms of moduli spaces
\[
\begin{aligned} \Phi_a^0: G_0 (r,d,k) &\to G_0(r+ad,d,k),\\ \Phi_a^L: G_L (r,d,k) &\to G_L(r+ad,d,k) \end{aligned}
\]
for \(k<0\) in the latter.
As an application, the authors study birational types of moduli spaces \(G(\alpha;r,d,k)\). In particular, for a given degree \(d\) of the vector bundle and a given dimension \(k\) of the vector subspace, there are at most \(d\) possible different birational types for the moduli space. Fourier-Mukai functors; coherent systems; moduli spaces; elliptic curves Ruipérez, D. Hernández; Prieto, C. Tejero: Fourier-Mukai transforms for coherent systems on elliptic curves, J. lond. Math. soc. 77, 15-32 (2008) Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Elliptic curves Fourier-Mukai transforms for coherent systems on elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies approximation of continuous or \(\mathcal C^{\infty}\) maps by real regular maps when linear real algebraic groups act the target spaces. Let \(X\) be a real algebraic variety (resp. nonsingular real algebraic variety) and \(Y\) be a homogeneous space for some linear real algebraic group. Then, for a continuous (resp. \(\mathcal C^{\infty}\)) map \(f: X \to Y\), the following are equivalent:
\begin{itemize}
\item \(f\) can be approximated by regular maps in the \(\mathcal C^0\) (resp. \(\mathcal C^{\infty}\)) topology.
\item \(f\) is homotopic to a regular map.
\end{itemize}
Applying this result to the unit \(n\)-spheres \(\mathbb S^n\), the authors solved problems on the approximations of maps between unit spheres, which had been open since the 1980's. Several other corollaries and three conjectures together with their partial solutions are also introduced.
The notion of malleable varieties and its generalization to submersions are used in the proof. By definition, a malleable variety admits an algebraic vector bundle \((E,p)\) over \(Y\) and a regular map \(s:E \to Y\) such that \(s(0_y)=y\) for all \(y \in Y\) and \(d_{0_y}(T_{0_y}E_y)=T_yY\). A homogeneous space for a linear real algebraic group is proven to be malleable. real algebraic variety; regular map; approximation; homotopy; homogeneous space Real algebraic sets, Topology of real algebraic varieties, Real algebraic and real-analytic geometry On approximation of maps into real algebraic homogeneous 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 investigate the connection between Osserman limit series (on curves of pseudocompact type) and limit linear series of \textit{O. Amini} and \textit{M. Baker} [Math. Ann. 362, No. 1--2, 55--106 (2015; Zbl 1355.14007)] (on metrized complexes with corresponding underlying curve) via a notion of pre-limit linear series on curves of the same type. Then, applying the smoothing theorems of Osserman limit linear series, we deduce that, fixing certain metrized complexes, or for certain types of Amini-Baker limit linear series, the smoothability is equivalent to a certain ``weak glueing condition''. Also for arbitrary metrized complexes of pseudocompact type the weak glueing condition (when it applies) is necessary for smoothability. As an application we confirm the lifting property of specific divisors on the metric graph associated with a certain regular smoothing family, and give a new proof of a result of Cartright, Jensen, and Payne for vertex-avoiding divisors, and generalize it for divisors of rank one in the sense that, for the metric graph, there could be at most three edges (instead of two) between any pair of adjacent vertices. curves; limit linear series; metrized complex Vector bundles on curves and their moduli, Geometric aspects of tropical varieties Smoothing of limit linear series on curves and metrized complexes of pseudocompact type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 iterated Darboux transformations of an ordinary differential operator are constructively parametrized by an infinite-dimensional Grassmannian of finitely supported distributions. In the case that the operator depends on time parameters so that it is a solution to the \(n\)-KdV hierarchy, it is shown that the transformation produces a solution of the KP hierarchy. The standard definitions of the theory of \(\tau\)-functions are applied to this Grassmannian and it is shown that these new \(\tau\)-functions are quotients of KP \(\tau\)-functions. The application of this procedure for the construction of `higher rank' KP solutions is discussed. Darboux transformations; KP hierarchy; tau functions; higher rank solutions Kasman, A.: Darboux transformations from n-KdV to KP. Acta Appl. Math. 49, 179--197 (1997) KdV equations (Korteweg-de Vries equations), Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds, Riemann surfaces; Weierstrass points; gap sequences, Grassmannians, Schubert varieties, flag manifolds Darboux transformations from \(n\)-KdV to KP | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a quasi-compact, quasiseparated scheme and \(R\) a Noetherian ring. Let \(\text{Const} (X)\) denotes the abelian category of constructible sheaves of abelian groups or \(R\)-modules. The author uses Waldhausen's \(K\)-theory machine [\textit{F. Waldhausen}, ``Algebraic \(K\)-theory of spaces'', Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)] to define the \(K\)-theory \(K_* (\text{Const} (X))\). The main conclusion is (theorem 5): Let \(X\) be a normal connected irreducible scheme of dimension one. Then there is an isomorphism:
\[
K_i \bigl( \text{Const} (X) \bigr) \cong \bigoplus_{x \in X_1} K_i \biggl( \text{Const} \bigl( k(x) \bigr) \oplus K_i \bigl( {\mathcal R} (X) \bigr) \biggr)
\]
where \({\mathcal R} (X)\) is a category related to the representations of the absolute Galois group of the generic point of \(X\). The method of proof is to use Waldhausen's fibration theorem to show that there exists a spectral sequence similar to Quillen's spectral sequence of the \(K\)-theory of coherent sheaves [Theorem 5.4, page 131 of \textit{D. Quillen}, ``Higher algebraic \(K\)-theory. I'', Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)]. The \(E_1\) term of this spectral sequence is calculated using Waldhausen's approximation theorem. \(K\)-theory of sheaves; higher \(K\)-theory; Waldhausen \(K\)-theory of spaces; constructible sheaves; spectral sequence; Waldhausen's approximation theorem \(K\)-theory of schemes, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic \(K\)-theory of spaces, Spectral sequences, hypercohomology A note on the \(K\)-theory of constructible sheaves over a 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 article is a global survey of approximation theorems; more precisely, it contains approximation theorems for sections of sheaves \({\mathcal F}\otimes{\mathcal E}_X\), where \({\mathcal F}\) is an analytic (or algebraic) coherent sheaf defined over a coherent real analytic space \(X\) (or a real affine variety \(X)\) and \({\mathcal E}_X\) is the sheaf of germs of \(C^\infty\) functions on \(X\).
The paper is divided into three parts: in the first one, the analytic case, we prove that the set of sections of \({\mathcal F}\) is dense in the strong (or Whitney) topology in the space of sections of \({\mathcal F}\otimes {\mathcal E}_X\).
The second part deals with the algebraic case. In this context we use the classical Weierstrass theorem and obtain approximation in the weak (or compact open) topology. In the real algebraic context, Theorems A and B are not true, so we study two subcategories of coherent sheaves: \(A\)-coherent and \(B\)-coherent sheaves. They are studied in \S 6 and \S 7; we prove approximation theorems for sections of \({\mathcal F}\otimes {\mathcal E}_X\) by means of sections of \({\mathcal F}\).
The third part of this article is devoted to fiber bundles. Approximation theorems were known also for sections of fiber bundles. In \S 8, via a duality theory, we find new approximation results for sections of fiber bundles of a more general class, which are not, in particular, locally trivial. Locally trivial algebraic vector bundles do not have the usual good behaviour. For example, the fiber is not in general generated by global sections. Bundles having the latter property are called strongly algebraic and are studied in \S 9. We prove that an algebraic vector bundle is strongly algebraic if, and only if, its total space is an affine variety. By applying the approximation results to the sheaf \(\Hom({\mathcal F}, {\mathcal G})\), we can extend to general vector bundles the results about the equivalence of the analytic and \(C^\infty\) classification. For strongly algebraic vector bundles, if \(X\) is compact, we have similar results about \(C^\infty\) and algebraic classification. survey; approximation theorems; fiber bundles Tognoli, A.: Approximation theorems in real analytic and algebraic geometry, Lectures in real geometry (Madrid, 1996), de Gruyter Exp. Math., 23, pp. 113-166. Berlin, (1996) Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs, Real-analytic manifolds, real-analytic spaces, Real-analytic and semi-analytic sets Approximation theorems in real analytic and algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review represents a milestone in the understanding of the local geometry of a semialgebraic set around a singular point. The approach of the authors follows the classical paper by \textit{Lê Dung Trang} and \textit{B. Teissier} [Comment. Math. Helv. 63, No. 4, 540--578 (1988; Zbl 0658.32010)] but, as they point out, ``the extension of the complex analytic results to the reals is not as straightforward as one may hope''. For the sake of simplicity the authors just consider algebraic surfaces, although most of their results seem to be true in a much more general setting. Let us just present one of the main results of this work. Let \(V\) be a real reduced algebraic surface embedded in \(\mathbb{R}^3\) and containing the origin \(\mathbf{0}\). Assume that \(V\) contains smooth points arbitrarily close to \(\mathbf{0}\). Let \(C^+(V,\mathbf{0})\) be the set of vectors \(v\in\mathbb{R}^3\) such that there exist sequences \(\{x_n\}\subset V-\{\mathbf{0}\}\) and \(\{t_n\}\subset\mathbb{R}\) such that \(t_n> 0\), \(\{x_n\}\) converges to \(\mathbf{0}\) and \(\{t_n, x_n\}\) converges to \(v\).
Let \({\mathcal N}(V,\mathbf{0})\) be the set of planes \(T\subset\mathbb{R}^3\) such that there exists a sequence of smooth points \(\{x_n\}\subset V\) converging to \(\mathbf{0}\) such that \(T\) is the limit of the sequence \(T_{x_n}V\) of tangent planes to \(V\) at \(x_n\). In such a case it is easily seen that, after substituting \(\{x_n\}\) by a subsequence if necessary, the sequence \(\{x_n\}\) approaches \(\mathbf{0}\) tangent to some ray \(l\), and it is said that \(T\in{\mathcal N}_l(V,\mathbf{0})\).
The authors prove that there exist finitely many rays tangent at \(\mathbf{0}\) to the singular locus \(S(V)\) of \(V\). Moreover, there exist finitely many rays \(l_1,\dots, l_r\) in \(C^+(V,\mathbf{0})- C^+(S(V),\text\textbf{0})\) such that each \({\mathcal N}_{l_i}(V,\mathbf{0})\) is connected, closed and one-dimensional, and for any ray \(l\in C^+(V,\mathbf{0})- C^+(S(V),\text\textbf{0})\) other than \(l_1,\dots, l_r\), the set \({\mathcal N}_l(V,\mathbf{0})={\mathcal N}_l(C^+(V,\text\textbf{0}),\text\textbf{0})\) consists of a single point.
This work, whose proofs and structure are very clear, contains some interesting examples and proposes a realistic plan to attack two intriguing questions, namely, how to compute the Nash fiber and the tangent semicone of a surface. real surfaces; limits of tangents spaces; Nash fiber D. B. O'Shea and L. C. Wilson, \textit{Limits of tangent spaces to real surfaces}, Amer. J. Math., 126 (2004), pp. 951--980. Topology of real algebraic varieties Limits of tangent spaces to real 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 the author investigate properties of a new notion of hypercovering introduced by himself in a previous paper [``Is Alexander property étale local?'' (preprint)]: A scheme valued ordered system is a contravariant functor \(F\) from the category of finite strictly ordered sets to the category of schemes.
Given a scheme valued ordered system \(F\), the author defines, for every collection \(\{A_1, \ldots A_n\}\) of finite subsets of \(\mathbb N\), a closed subscheme \(F[A_1,\ldots,A_n]\) of \(F(A_1) \times \ldots \times F(A_n)\). Set theoretically, a geometric point of \(F[A_1,\ldots ,A_n]\) is a point \((x_1,\ldots,x_n) \in F(A_1) \times \ldots \times F(A_n)\), such that any two components are compatible in \(F (A_i \cap A_j)\). -- If for all \(A_i=\{1,\ldots, n\}-\{i\}\), then \(F[A_1,\ldots , A_n]\) is said to be the cosquelton \(\cos k_n(F)\).
A scheme valued ordered system is said to be a hypercovering if the canonical morphism \(F(\{1,\ldots ,n\}) \rightarrow\cos k_n(F)\) belongs to a given Grothendieck topology. The main result of this paper is that, given a hypercovering \(F\) and finite subsets of \(\mathbb N\), \(A_1, \ldots, A_n, B_1, \ldots B_m\), such that for all \(i\) there is a \(j\) with \(B_i \subset A_j\), then there is a natural map \(F[A_1,\ldots, A_n]\rightarrow F[B_1,\ldots B_s]\), and it still belongs to the given Grothendieck topology.
As explained by the author in the last section of the paper, this research should help in investigating the following conjecture:
Let \(F\) be a hypercovering with all \(F(\{1,\ldots,n\})\) Alexander. Then for any bivariant sheaf \(\mathcal F\) on \(F(\emptyset)\), the Čech cohomology \(H^i({\mathcal F},F)\) vanishes for all \(i>0\). hypercovering; cosquelton; scheme valued ordered system; vanishing of Cech cohomology Coverings in algebraic geometry, Vanishing theorems in algebraic geometry On hypercoverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Every \(d\)-dimensional real multisequence \(\beta ^{(\infty )}=(\beta _{i})_{i\in \mathbb{N}^{d}}\) can be seen as a linear operator \(L_{\beta ^{(\infty )}}:\mathcal{P}\rightarrow \mathbb{R}\) acting on the space \(\mathcal{P=}\mathbb{R}[x_{1},\ldots ,x_{d}]\) of real polynomials in \(\mathbb{R}^{d}\) by setting \(L_{\beta ^{(\infty )}}(x^{i})=\beta _{i}\) (where \(x^{i}:=x_{1}^{i_{1}}\cdots x_{d}^{i_{d}}\)). Given a closed subset \(K\) of \(\mathbb{R}^{d}\) the Haviland generalization of the Riesz theorem asserts that \(L_{\beta ^{(\infty )}}\) can be represented by a measure \(\mu \) supported in \(K\) (that is, \(L_{\beta^{(\infty)}}(p)=\int p\,d\mu \) for all \(p\in \mathcal{P}\)) if, and only if, \(L_{\beta ^{(\infty )}}\) is \(K\)-positive (that is, \(L_{\beta ^{(\infty )}}(p)\geq 0\) whenever the polynomial \(p\) is positive on \(K\)). The truncated moment problem concerns the case where the data are restricted to \(\beta ^{(2n)}=(\beta _{i})_{i\in \mathbb{N}^{d},|i|\leq 2n}\) defining thus a linear operator \(L_{\beta ^{(2n)}}:\mathcal{P}_{2n}\rightarrow \mathbb{R}\), where \(\mathcal{P}_{2n}\) stands for the space of polynomials whose degree is less or equal to \(2n\). In this context, a similar representation result is true whenever \(K\) is compact: \(L_{\beta ^{(2n)}}\) admits a \(K\)-representing measure if and only if it is \(K\)-positive (that is, \(L_{\beta ^{(2n)}}(p)\geq 0\) for all polynomials of \(\mathcal{P}_{2n}\) that are positive on \(K\)). The authors present an example (Example~2.1) where the sufficiency fails if the compactness condition is removed. They subsequently establish the following criterium (Theorem~2.2):
\(L_{\beta ^{(2n)}}\) admits a \(K\)-representing measure if and only if it admits a \(K\)-positive linear extension \(\widetilde{L}:\mathcal{P}_{2n+2}\rightarrow \mathbb{R}\). They also discuss the particular case of a closed semialgebraic set \(S=\{x\in \mathbb{R }^{d}:q_{i}(x)\geq 0,\{q_{i}\}_{i\in \{1,\ldots ,m\}}\in \mathcal{P}\}\). In such a case they show that \(S\) solves the truncated moment problem in terms of natural degree-bounded positivity conditions if and only if every polynomial \(p\) that is strictly positive on \(S\) admits a degree-bounded weighted SOS-representation (in terms of the polynomails \(q_{i}\) defining \(S\)). truncated moment problem; measure representation; semialgebraic sets; positive functional; Riesz-Haviland theorem; \(K\)-moment problems; Riesz functional; moment matrix extension; flat extensions of positive matrices; localizing matrices Curto RE, Fialkow LA (2008) An analogue of the Riesz-Haviland theorem for the truncated moment problem. \textit{J. Funct. Anal.} 255(10):2709-2731. CrossRef Moment problems, Linear operator methods in interpolation, moment and extension problems, Semialgebraic sets and related spaces An analogue of the Riesz-Haviland theorem for the truncated moment 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 New methods for computing parametric local \(b\)-functions are introduced for \(\mu\)-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic \(\mathcal{D}\)-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of \(b\)-functions associated with \(\mu\)-constant deformations. In the case of inner modality two, local \(b\)-functions associated with \(\mu\)-constant deformations are obtained by the resulting method and given the list of parametric local \(b\)-functions. \(b\)-function; \(\mu\)-constant deformations; comprehensive Gröbner system; local cohomology; \(\mathcal{D}\)-modules Singularities of curves, local rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Local cohomology and commutative rings Methods for computing \(b\)-functions associated with \(\mu\)-constant deformations: case of inner modality two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 very interesting paper presents a breakthrough in an old problem motivated by the classical \textit{H. Whitney} extension papers [Trans. Am. Math. Soc. 36, 63--89 (1934; Zbl 0008.24902 and JFM 60.0217.01), 369--387 (1934; Zbl 0009.20803 and JFM 60.0217.02)]. In rough outline, the problem can be formulated as follows.
Let \(B\) be a Banach space of ``smooth'' functions on \(\mathbb{R}^n\) and \(X\subset \mathbb{R}^n\).
Problem. Characterize continuous functions on \(X\) admitting extensions to functions of the space \(B\).
``Smoothness'' means either the existence of continuous derivatives of fixed order or a prescribed behaviour of difference characteristics (moduli of continuity). Usually one considers ``mixed'' definitions where the derivatives of higher order have such a behaviour. The typical examples are the classical Sobolev and Besov spaces \(W^k_\infty\) and \(B_\infty^s\). The last two families are parts of a more general family consisting of Lipschitz spaces of higher order. A common member of this family is defined by the seminorm
\[
|f|_{\Lambda_\omega^k}:=\sup \left\{\frac{\bigl|\Delta_h^kf(x)\bigr|}{\omega \bigl(|h|\bigr)}\,\biggl|\,x, \lambda\in\mathbb{R}^n\right\}.
\]
Here \(\Delta_h^k:= (\Delta_h)^k\) is the \(k\)-th difference and \(\omega\) is a homeomorphism of \([0,+\infty)\) such that \(t\mapsto\omega(t^{1/k})\) is a concave function. Note that \(W^k_\infty=\Lambda^k_\omega\) with \(\omega(t)=t^k\) and \(B^\rho_\infty= \Lambda_\omega^k\) with \(\omega(t)=t^\rho\), \(0< \rho<K\).
Finiteness Conjecture (Yu. A. Brudnyi, 1983): Given the space \(\Lambda_\omega^k\) and subset \(X\subset \mathbb{R}^n\), there is an integer \(N=N(k,n)\) with the following property.
A continuous function \(f:X\to\mathbb{R}\) belongs to the trace space \(\Lambda^k_\omega(X)\) iff for every subset \(Y\subset X\) consisting of \(N\) points there is a function \(f_y\in\Lambda_\omega^k\) which agrees with \(f\) on \(Y\) and such that
\[
\sup\left\{|f_y|_{\Lambda_\omega^k} \mid Y\subset X,\text{card}\,Y=N \right\}<\infty.
\]
The optimal \(N\) is denoted by \(N(\Lambda^k_\omega)\).
The Whitney extension method presented in his second paper leads, in particular, to the sharp results for Sobolev and Besov spaces on \(\mathbb{R}\). Namely,
\[
N\bigl( W_\infty^k(\mathbb{R})\bigr)=k+1\text{ and }N\bigl(B_\infty^\rho(\mathbb{R}) \bigr)=\widetilde k
\]
where \(\widetilde k\) is the least integer bigger than \(\rho\).
The similar result for the space \(\Lambda_\omega^k(\mathbb{R})\) (with the optimal constant \(k+1)\) follows from the theorem of \textit{I. A. Shevchuk} [Anal. Math. 10, 249--273 (1984; Zbl 0596.41049)], see also his book ``Approximation by polynomials and traces of functions continuous on a segment of the real line'' (Russian) [Kiev, Naukova Dumka (1992)].
The multidimensional case is much more complicated. The reviewer proposed the following program to prove the conjecture. Using a local polynomial description of the space \(\Lambda_\omega^k(\mathbb{R}^n)\) [\textit{Yu. Brudnyi}, Math. USSR, 11, 157--170 (1970); translation from Mat. Sb., N. Ser. 82(124), 175--191 (1970; Zbl 0204.13501)] and the extension method preserving local polynomial approximation [\textit{Yu. A. Brudnyi}, Funct. Anal. Appl. 4, 252--253 (1970); translation from Funkts. Anal. Prilozh. 4, No. 3, 97--98 (1970; Zbl 0221.46008)], it is possible to characterize the trace space \(\Lambda_\omega^k(X)\) in terms of local polynomial approximation [\textit{Yu. A. Brudnyi} and \textit{P. Shvartsman}, Issled. Teor. Funkts. Mnogikh Veshchestv. Perem. 1982, 16--24 (1982; Zbl 0598.41033)]. The idea is to reformulate this characteristic in geometric terms which, in particular, leads to the following conjecture yet unproven.
Conjecture (Yu. A. Brudnyi, 1984): Let \(\varphi\) be a map from a metric space \(({\mathcal M},d)\) to compact subsets of \(\mathbb{R}^n\). Then \(\varphi\) admits a Lipschitz selection \(f:{\mathcal M}\to\mathbb{R}^n\) iff the restriction of \(\varphi\) to every \(2^n\) point subset \(S\subset{\mathcal M}\) has a Lipschitz selection \(f_s\) and the Lipschitz constants \(\text{Lip}(f_s)\) are uniformly bounded:
The first considerable success was achieved in the PhD thesis of P. Shvartsman. He proved a special case of the conjecture for \(\varphi\) taking values in the set of all affine subspaces of \(\mathbb{R}^n\), and derived based on that the following beautiful result
\[
N\left( \Lambda^2_\omega(\mathbb{R}^n)\right)=3.2^{n-1}.
\]
The abridged version of \textit{P. A. Shvartsman}'s proof appeared in [Sib. Math. J. 28, No. 5, 853--863 (1987); translation from Sib. Mat. Zh. 28, No. 5(165), 203--215 (1987; Zbl 0634.46025)]. The method is also applied to the space \(C^{1,\omega} (\mathbb{R}^n)\) where the finiteness constant is the same [\textit{Yu. Brudnyi} and \textit{P. Shvartsman}, Trans. Am. Math. Soc. 353, No. 6, 2487--2512 (2001; Zbl 0973.46025)].
The further advance towards higher than second smoothness, e.g., \(W_\infty^k (\mathbb{R}^k)\) with \(k>2\) or \(B^s_\infty(\mathbb{R}^k)\) with \(\rho\geq 2\) requires new ideas. The first result in this direction is Ch. Fefferman's Theorem A of the paper under review; it asserts that the Finiteness Conjecture is valid for the Sobolev space \(W^k_\infty(\mathbb{R}^n)\). The proof is a very impressive coup de force. It skillfully combines methods of modern real analysis, algebraic geometry (resolution of singularities) and convex geometry. One hopes that the clarification of the basic ideas of this proof may lead to the discovery of new important results in the now actively developing (but unnamed) field devoted to problems with incomplete information. Problems of this kind have appeared in a wide range of research areas (mathematical physics, numerical analysis, signal and image processing, the theory of computation, learning theory, etc.). \(C^{K-1,1}\)-function; trace to subsets of \(\mathbb{R}^n\); smoothness preserving extension; finiteness extension problem C. L. Fefferman, ''A sharp form of Whitney's extension theorem,'' Ann. of Math., vol. 161, iss. 1, pp. 509-577, 2005. Differentiable maps on manifolds, Global theory and resolution of singularities (algebro-geometric aspects), Special properties of functions of several variables, Hölder conditions, etc., Interpolation between normed linear spaces, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems A sharp form of Whitney's extension 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 Linear matrix inequalities (LMIs) \(I_d + \sum _{j=1}^g A_jx_j + \sum _{j=1}^g A_j^*x_j^* \succeq 0\) play a role in many areas of applications. The set of solutions of an LMI is a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. Free spectrahedra are exactly the free semialgebraic convex sets. This paper studies free analytic maps between free spectrahedra and, under certain (generically valid) irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in \(g\) variables sit in correspondence with \(g\)-dimensional algebras. If two bounded free spectrahedra \({\mathcal {D}}_A\) and \({\mathcal {D}}_B\) meeting our irreducibility assumptions are free bianalytic with map denoted \(p\), then \(p\) must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a \(g\)-dimensional algebra spanned by \((U-I)A_1,\dots ,(U-I)A_g\) for some unitary \(U\). Furthermore, \(B\) and \(UA\) are unitarily equivalent. The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from \({\mathcal {D}}_A\) to \({\mathcal {D}}_B\) (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps. Operator spaces (= matricially normed spaces), Semialgebraic sets and related spaces, Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Operator spaces and completely bounded maps, Convex sets without dimension restrictions (aspects of convex geometry) Bianalytic maps between free spectrahedra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a projective, connected, nodal algebraic curve of arithmetic genus greater than one, defined over an algebraically closed field \(k\) of characteristic zero, and let \(C_1, \dots, C_t\) be its components. Let \(S\) be the spectrum of a discrete valuation ring with residue field \(k\); a regular smoothing of \(C\) is a flat, projective map \(f: \mathcal{X} \rightarrow S\) whose generic fiber is smooth, whose special fiber is isomorphic to \(C\) and such that the total space \(\mathcal{X}\) is regular. Let \(K\) be the canonical sheaf on \(C\) and \(\beta\) a positive integer, then \(H^0(C, K^{\otimes \beta})\) defines the so-called \(\beta\)-canonical system; consider on the generic fiber of \(f\) the Weierstrass divisor of the \(\beta\)-canonical system and let \(W\) be the fundamental cycle on \(C\) associated to the subscheme that is the limit divisor.
In this paper the authors describe \(W\) as a Weil divisor, under certain hypothesis (among them, \(t > 1\), \(C_i\) has positive genus for all \(i =1, \dots, t\) and hypothesis on the intersection of the components of \(C\)); conversely, they also find conditions for a Weil divisor to be a fundamental cycle of a limit Weierstrass divisor. limit of ramification points; nodal curves Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences Limit Weierstrass points on nodal reducible curves | 0 |
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